Drilled Shafts in Rock Analysis and Design

DRILLED SHAFTS IN ROCK

Drilled Shafts in Rock Analysis and Design

LIANYANG ZHANG ICF Consulting, Lexington, MA, USA

A.A.BALKEMA PUBLISHERS LEIDEN/LONDON/NEW YORK/PHILADELPHIA/SINGAPORE

Library of Congress Cataloging-in-Publication Data A Catalogue record for the book is available from the Library of Congress Copyright © 2004 Taylor & Francis Group plc, London, UK All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publishers. Although all care is taken to ensure the integrity and quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: A.A.Balkema Publishers, a member of Taylor & Francis Group plc. http://www.balkema.nl/, www.tandf.co.uk/books This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to http://www.ebookstore.tandf.co.uk/.” ISBN 0-203-02442-7 Master e-book ISBN

ISBN 90 5809 650 5 (Print Edition)

Contents Preface

vi

1 Introduction

1

2 Intact rock and rock mass

9

3 Characterization of discontinuities in rock

36

4 Deformability and strength of rock

80

5 Site investigation and rock testing

173

6 Axial load capacity of drilled shafts in rock

214

7 Axial deformation of drilled shafts in rock

258

8 Lateral load capacity of drilled shafts in rock

285

9 Lateral deformation of drilled shafts in rock

299

10 Stability of drilled shaft foundations in rock

349

11 Drilled shafts in karstic formations

359

12 Loading test of drilled shafts in rock

372

References

404

Index

423

Preface Drilled shafts are widely used to transfer heavy structural loads (both axial and lateral) through the overburden soil to the underlying rock mass. The last three decades have seen sustained research and development efforts around the world to improve the technology of drilled shafts in rock. Although much has been learned on the analysis and design of drilled shafts in rock, all the major findings are reported in the form of reports and articles published in technical journals and conference proceedings. The main purpose of this book is to assist the reader in the analysis and design of drilled shafts in rock by summarizing and presenting the latest information in one volume. The primary difference between foundations in soil and those in rock is that rock masses contain discontinuities. Compared to intact rock, jointed rock masses have increased deformability and reduced strength. Also, the existence of discontinuities in a rock mass creates anisotropy in its response to loading and unloading. To analyze and design drilled shafts in rock, a geotechnical engineer has to know the properties of rock. Chapters 2–5 are devoted to the discussion of rocks: • Chapter 2 Intact rock and rock mass • Chapter 3 Characterization of discontinuities in rock • Chapter 4 Deformability and strength of rock • Chapter 5 Site investigation and rock testing The details of analysis and design procedures for drilled shafts in rock are presented in Chapters 6–11, specifically • Chapter 6 Axial load capacity of drilled shafts in rock • Chapter 7 Axial deformation of drilled shafts in rock • Chapter 8 Lateral load capacity of drilled shafts in rock • Chapter 9 Lateral deformation of drilled shafts in rock • Chapter 10 Stability of drilled shaft foundations in rock • Chapter 11 Drilled shafts in karstic formations These chapters contain worked examples illustrating the practical application of the analysis and design methods. Load tests are a key method in the study of drilled shafts in rock. Chapter 12 describes the various techniques used in testing drilled shafts in rock, with special treatment of the subject of interpretation. The anticipated audience for this book is the design professional of drilled shafts in rock. It may also be used as a reference text for courses of geological engineering, rock mechanics and foundation engineering. Portions of Chapters 3, 4, 6, 8 and 9 are based on the author’s doctoral research conducted at the Massachusetts Institute of Technology. The author acknowledges the support and advice given by Professor Herbert Einstein.

Mr. Ralph Grismala of ICF Consulting, on the author’s request, found time to look through the manuscript and made suggestions for improvement. His suggestions as an experienced geotechnical engineer have been particularly valuable and the author is grateful to him. Finally, the author wants to thank Dr. Francisco Silva of ICF Consulting for his support during the preparation of this book. Lianyang Zhang Lexington, MA, USA

1 Introduction 1.1 DEFINITION OF DRILLED SHAFTS A drilled shaft is a deep foundation that is constructed by placing concrete in an excavated hole. Reinforcing steel can be installed in the excavation, if desired, prior to placing the concrete. A schematic example of a typical drilled shaft socketed into rock is shown in Figure 1.1. The drilled shaft can carry both axial and lateral loads. Drilled shafts are sometimes called bored piles, piers, drilled piers, caissons, drilled caissons, or cast-inplace piles. To increase the bearing capacity, drilled shafts are commonly socketed into rock. The portion of the shaft drilled into rock is referred to as a rock socket. In many cases where there is no overburden soil, drilled shafts entirely embedded in rock are also used. Because of the high bearing capacity of rock sockets, the analysis and design of them are extremely important. This book discusses both rock sockets and drilled shafts entirely embedded in rock.

1.2 HISTORICAL DEVELOPMENT OF DRILLED SHAFTS The ancient “well foundation” can be considered the earliest version of drilled shafts. Such foundations were stone masonry pedestals built in hand-excavated holes, long before hydraulic cements came into common use. During the late nineteenth and early twentieth centuries when taller and heavier buildings began to appear, high-capacity foundations became necessary in large cities such as Chicago, Cleveland and Detroit. These cities are underlain by relatively thick deposits of medium to soft clays overlying deep glacial till or bedrock. Because traditional spread foot foundations settled excessively under the heavier building load, engineers began to use shafts such as the hand-dug “Chicago” and “Gow” caissons. The shafts were constructed by making the excavation and by placing sections of permanent liners (wooden lagging or steel rings) to retain the soil by hand. Hand excavation methods were slow and tedious, so machine-drilled shafts soon superseded the hand-dug caissons. A few examples of horse and engine-driven augers appeared between 1900 and 1930, but they had limited capabilities. By the late 1920s, manufacturers were building practical truck-mounted engine-driven augers, thus bringing drilled shaft construction into its maturity.

Drilled shafts in rock

2

Fig. 1.1 A drilled shaft socketed into rock. During the next three decades, manufacturers and contractors developed larger and more powerful drilling equipment, which allowed more economical and faster construction of drilled shafts. In the late 1940s and early 1950s, drilling contractors introduced techniques for drilling in rock. By introducing casing and drilling mud into boreholes, a process long established by the oil industry, boreholes could be drilled through difficult soils economically. By the 1960s, drilled shafts had become a strong competitor to driven piles. In the past decade, the use of drilled shafts has increased dramatically. In 1997, the value of drilled shaft construction in the United States reached more than one billion US dollars (O’Neill, 1998). Today, drilled shafts support different structures including one-

Introduction

3

story wood frame buildings to the largest skyscrapers, highway bridges, and retaining structures.

1.3 USE OF DRILLED SHAFTS Compared to other types of deep foundations, drilled shafts have the following major advantages: 1. The costs of mobilizing and demobilizing a drill rig are much less than those for a pile driver. 2. The construction process generates less vibration and noise, making drilled shafts appropriate for urban construction. 3. The quality of the bearing material can be inspected visually and tests can be run to determine its physical properties. For end-bearing designs, the soil/rock beneath the base can be probed for cavities or weak layers if desirable. 4. The diameter or length of the drilled shaft can be easily changed during construction to compensate for unanticipated soil/rock conditions. 5. The drilled shafts can penetrate through soils with cobbles or boulders. They can also be socketed into rock. 6. It is usually possible to support very large loads with one large drilled shaft instead of several piles, thus eliminating the need for a pile cap. 7. Large-diameter drilled shafts are particularly well-suited as foundations for structures that must resist extreme events that produce large lateral loads (e.g. earthquake and vessel impact loading) because of the very large moments of inertia. Drilled shafts also have the following major disadvantages: 1. The quality and performance of drilled shafts is very dependent on the contractor’s skills. Poor workmanship can produce weak foundations that may not be able to support the design load. 2. Since shaft construction removes soil/rock from the ground, it may decrease the competency of the bearing stratum. 3. The construction of drilled shafts through contaminated soils/rocks is problematic because of the expenses associated with disposing of the spoil. Because of the above advantages, drilled shafts have become an appropriate and economical foundation system for heavily loaded structures. When deep foundations are required, drilled shafts should always be considered as an option. An application example of drilled shafts in rock O’Neill and Reese (1999) presented an application example of drilled shafts in rock. It clearly shows the advantages of drilled shafts over pile-footings for the foundations of the interior bents of a river bridge in the United States. The bridge is a two-lane bridge with four spans. Siltstone near the surface at one end dips to a depth of about 6.1 m (20 ft) near the other end of the bridge. Mixed fine sediments exist above the siltstone. Two alternate foundation designs were considered by the design agency before the project bid. The first one called for the construction of one

Drilled shafts in rock

4

spread footing and two capped groups of steel H-piles for the three interior bents that were required to be placed in the river. Both the spread footing and driven piles (with pile caps) were to be constructed within cofferdams because of the need to construct footings/caps. The second one called for the replacement of the spread footing and driven pile groups by three large-diameter drilled shafts. The drilled shafts could be drilled during low water using a crane-mounted drill rig positioned on timber mats within the river and pouring the concrete for the shafts to an elevation above the water level, eliminating the need for cofferdams. Comparison of the pile-footing alternate with the drilled shaft alternate is shown in Table 1.1. The cost savings realized by using drilled shafts were $422,000 (50%).

Table 1.1 Comparison of the pile-footing alternate with the drilled shaft alternate—Queens River Bridge, Olympic Peninsula, Washington, USA (after O’Neill & Reese, 1999). Details

Pile-Footings

Drilled Shafts

25 capped H-piles driven into the soft siltstone for each of the two interior bents and a spread-footing at the other interior bent. All pile driving, cap construction and spread footing construction were within cofferdams. A single-bent column was formed on top of the spread footing or the pile cap prior to removal of the cofferdams. The construction of work trestle was required so that cofferdams could be constructed prior to installing the foundations. Because of the length of time required to construct the trestle and cofferdams, construction of pile groups, caps and footing could not proceed until the following working season, since operations in the river had to be suspended during the salmon runs.

Three 3.2 m (10.5 ft) diameter drilled shafts socketed about 10m (30 ft) into the siltstone, with casing extending from the top of the siltstone to high water level. The casing was used as a form, and the drilled shaft concrete was poured directly up to the top of the casing. The single columns for the bents were formed on top of the extended sections of drilled shafts, with no requirement to construct cofferdams.

Estimated $842,000 Cost

$420,000

1.4 CHARACTERISTICS OF DRILLED SHAFTS IN ROCK The characteristics of drilled shafts in rock are closely related to the special properties of rock masses. The following briefly describes some of the special rock mass properties that will affect the performance of drilled shafts.

Introduction

5

1.4.1 Effect of discontinuities The primary difference between drilled shafts in rock and those in soil is that rock masses contain discontinuities. The intact rock may have a high strength but the presence of discontinuities in the rock may result in very low strength of the rock mass. Wedges or blocks formed by sets of unfavorably orientated discontinuities may fail by sliding or toppling, causing excessive movement or failure of drilled shaft foundations. Figure 1.2 shows the drilled shaft foundations of a bridge across a river. The rock at this site consists of two sets of discontinuities with about the same dip angles; but set A is discontinuous and more widely spaced than set B. At the East side, the drilled shaft foundation would be stable because the discontinuities approximately parallel to the rock slope face are not continuous. In contrast, at the West side, the discontinuities approximately parallel to and dipping out of the slope face are continuous and movement of the entire foundation along these discontinuities is possible.

Fig. 1.2 Effect of discontinuities on the stability of drilled shafts. 1.4.2 Effect of groundwater Groundwater may affect the performance of drilled shafts in the following ways: 1. The most obvious is through the operation of the effective stress law. Water under pressure in the discontinuities defining rock blocks reduces the normal effective stress between the rock surfaces, and thus reduces the potential shear resistance which can be mobilized by friction. In porous rocks, such as sandstones, the effective stress law is obeyed as in granular soils. In both cases, the effect of fissure or pore water pressure

Drilled shafts in rock

6

is to reduce the ultimate strength of the rock mass, and thus decrease the bearing capacity of the drilled shaft foundation. 2. Groundwater affects rock mechanical properties due to the deleterious action of water on particular rocks and minerals. For example, clay seams may soften in the presence of groundwater, reducing the strength and increasing the deformability of the rock mass. Argillaceous rocks, such as shales and argillitic sandstones, also demonstrate marked reductions in material strength following infusion with water. According to Hoek and Brown (1997), strength losses of 30–100% may occur in many rocks as a result of chemical deterioration of the cement or clay binder. 3. Groundwater flow into the excavation of a drilled shaft can make cleaning and inspection of bearing surfaces difficult and result in decreased bearing capacity for the drilled shaft. 1.4.3 Effect of karstic formations A number of problems may arise when drilled shafts are built in karstic formations (Brown, 1990; Goodman, 1993; Sowers, 1996): 1. An existing cavity may underlie the base of the drilled shaft and collapse when the building is under construction or in service. The collapse of the cavity may be caused by excessive construction loading or erosion by acid groundwater [Fig. 1.3(a)]. 2. The tip of the shaft may slide along a steeply inclined rock instead of penetrating into bedrock, especially when part of the tip is located on top of a pinnacle with existing joints or cracks [Fig. 1.3(b)]. 3. The drilled shaft is placed on a cantilever rock over cavities or soft clay, so that excessive loads or continuing water erosion may cause rock collapse [Fig. 1.3(c)]. 4. A shifting rock slab or rock block floating in the residual soil may lead people to mistakenly believe that the bedrock has been reached and the bearing stratum has been located (d)]. Chapter 11 will discuss the performance of drilled shafts in karstic formations in more detail.

1.5 CONSIDERATIONS IN THE DESIGN OF DRILLED SHAFTS As for the design of any foundations, the design of drilled shafts must satisfy criteria related to strength, deformation and durability. For the strength, criteria are applied to both the structural strength of the shaft itself and the geotechnical strength, i.e., the load carrying capacity of the soil/rock. The structural and geotechnical strength criteria depend on the basis of the design method. The traditional working stress design method, sometimes referred to as the allowable stress design (ASD) method, relies on an overall safety factor against ultimate failure and the corresponding design criteria can be expressed as (1.1)

Introduction

7

where Qu is the ultimate load bearing capacity; FS is the global factor of safety; and Q is the allowable working load or the allowable design load. Equation (1.1) applies to both axial and lateral loadings. Typical factors of safety for the geotechnical strength of drilled shafts range between two and three, depending on the method of capacity calculation, the extent of the designer’s experience and knowledge of the site and the geotechnical conditions, and the likely consequences of failure. In cases where there is extensive experience of the site and field shaft load tests have been carried out, values of safety factor as low as 1.5 may be appropriate. On the other hand, where knowledge of the site is limited, and the consequences of failure may be extreme, safety factors of three or higher may be appropriate.

Fig. 1.3 Failure modes of drilled shafts in karstic formations (after Tang, 1995).

Drilled shafts in rock

8

In recent years, there has been an increasing tendency to use load and resistance factor design (LRFD) for drilled shafts and other structural components (AASHTO, 1994; FHWA, 1996a). With this method, various factors, with values of 1 or above, are applied to the individual components of load. Other factors, with values of 1 or less, are applied to the total resistance, or individual components of resistance, in such a way to assure a margin of safety consistent with historical practice using global factors of safety. The design criterion for the LRFD approach can be written as (1.2) where η is factor varying from 0.95 to 1.05 to reflect ductility, redundancy and operational importance of the structure; γi is the load factor for load type i; Qi is the nominal value of load type i; is the resistance factor for resistance component j; Quj is the estimated (nominal) value of ultimate resistance component j. Equation (1.2) applies to both axial and lateral loadings, and to structural and geotechnical strengths. The LRFD approach to foundation design has the advantages that (a) foundations are easier to design if the superstructure is designed using LRFD and (b) it offers a means to incorporate reliability into the design process in a rational manner. For the serviceability limit state, the design criteria for deformations may be stated generally as: Estimated deformation≤Allowable deformation (1.3) Estimated differential deformation≤Allowable differential deformation (1.4) Equations (1.3) and (1.4) apply to both axial and lateral deformations. The allowable deformations and differential deformations depend primarily on the nature of the structure. Grant et al. (1982) and Moulton et al. (1985) listed typical values of allowable deformations and differential deformations for different structures. For the durability, the usual design criterion is the drilled shafts shall have a design life that exceeds the design life of the structure to be supported; this is usually 50 years or more for permanent structures. In recent years, the influence of environmental factors on the design and construction of drilled shaft foundations has become more and more important. Requirements that impact the excavation, handling, and disposal of river bottom sediments are continually more restrictive. Consequently, design and construction techniques are being developed and modified to lessen the need for excavation.

2 Intact rock and rock mass 2.1 INTRODUCTION Rock differs from most other engineering materials in that it contains discontinuities of one type or another which render its structure discontinuous. Thus a clear distinction must be made between the intact rock or rock material on the one hand and the rock mass on the other. The intact rock may be considered as a continuum or polycrystalline solid between discontinuities consisting of an aggregate of minerals or grains. The rock mass is the in situ medium comprised of intact rock blocks separated by discontinuities such as joints, bedding planes, folds, sheared zones and faults. The properties of the intact rock are governed by the physical properties of the materials of which it is composed and the manner in which they are bonded to each other. The parameters which may be used in a description of intact rock include petrological name, color, texture, grain size, minor lithological characteristics, degree of weathering or alternation, density, porosity, strength, hardness and deformability. Rock masses are discontinuous and often have heterogeneous and anisotropic properties. Since the behavior of a rock mass is, to a large extent, determined by the type, spacing, orientation and characteristics of the discontinuities present, the parameters used to describe a rock mass include the nature and geometry of discontinuities as well as its overall strength and deformability. This chapter describes the types and important properties of intact rocks and different rock mass classification systems that will be useful in the analysis and design of drilled shafts in rock. Chapter 3 will discuss the characterization of discontinuities in rock masses.

2.2 INTACT ROCK Intact rocks may be classified from a geological or an engineering point of view. In the first case the mineral content of the rock is of prime importance, as is its texture and any change which has occurred since its formation. Although geological classifications of intact rocks usually have a genetic basis, they may provide little information relating to the engineering behavior of the rocks concerned since intact rocks of the same geological category may show a large scatter in strength and deformability, say of the order of 10 times. Therefore, engineering classifications of intact rocks are more related to the analysis and design of foundations in rock.

Drilled shafts in rock

10

2.2.1 Geological classification (a) Rock-forming minerals Rocks are composed of minerals, which are formed by the combination of naturally occurring elements. Although there are hundreds of recognized minerals, only a few are common. Table 2.1 summarizes the common rock-forming minerals and their properties. Moh’s scale, used in the table, is a standard of ten minerals by which the hardness of a mineral may be determined. Hardness is defined as the ability of a mineral to scratch another. The scale is one for the softest mineral (talc) and ten for the hardest (diamond).

Table 2.1 Common rock-forming minerals and their properties. Mineral

Hardness (Moh’s scale, 1–10)

Relative Density

Fracture

Structure

Orthoclase feldspar

6

2.6

Good cleavage at right angles

Monoclinic. Commonly occurs as crystals

Plagioclase feldspar

6

2.7

Cleavage nearly at right angles—very marked

Triclinic. Showing distinct cleavage lamellae

Quartz

7

2.65

No cleavage. Choncoidal fracture

hexagonal

Muscovite

2.5

2.8

Perfect single cleavage Monoclinic. Exhibiting into thin easily separated strong cleavage plates lamellae

Biotite

3

3

Perfect single cleavage Monoclinic. Exhibiting into thin easily separated strong cleavage plates lamellae

Hornblende

5–6

3.05

Good cleavage at 120°

Hexagonal—normally in elongated prisms

Augite

5–6

3.05

Cleavage nearly at right angles

Monoclinic

Olivine

6–7

3.5

No cleavage

No distinctive structure

Calcite

3

2.7

Three perfect cleavages. Hexagonal Rhomboids formed

Dolomite

4

2.8

Three perfect cleavages

Hexagonal

Kaolinite

1

2.6

No cleavage

No distinctive structure (altered feldspar)

Hematite

6

5

No cleavage

Hexagonal

Intact rock and rock mass

11

(b) Elementary rock classification Intact rocks are classified into three main groups according to the process by which they are formed: igneous, sedimentary and metamorphic. Igneous rocks are formed by crystallization of molten magma. The mode of crystallization of the magma, at depth in the Earth’s crust or by extrusion, and the rate of cooling affect the rock texture or crystal size. The igneous rocks are subdivided into plutonic, hypabyssal and extrusive (volcanic), according to their texture. They are further subdivided into acid, intermediate, basic and ultrabasic, according to their silica content. Table 2.2 shows a schematic classification of igneous rocks. Sedimentary rocks are formed from the consolidation of sediments. Sedimentary rocks cover three-quarters of the continental areas and most of the sea floor. In the process of erosion, rocks weather and are broken down into small particles or totally dissolved. These detritic particles may be carried away by water, wind or glaciers, and deposited far from their original position. When these sediments start to form thick deposits, they consolidate under their own weight and eventually turn into solid rock through chemical or biochemical precipitation or organic process. As a result of this process, sedimentary rocks almost invariably possess a distinct stratified, or bedded, structure. Table 2.3 shows the classification of sedimentary rocks. Metamorphic rocks are derived from pre-existing rocks by temperature, pressure and/or chemical changes. Table 2.4 shows a classification of the metamorphic rocks according to their physical structure, i.e., massive or foliated. (c) Weathering of rock Weathering is the disintegration and decomposition of the in situ rock, which is generally depth controlled, that is, the degree of weathering decreases with increasing depth below the surface. The engineering properties of a rock as discussed in next section can be, and often are, altered to varying degrees by weathering of the rock material. Intact rocks can be divided into 5 groups according to the degree of weathering (see Table 2.5).

Table 2.2 Geological classification of igneous rocks. Type Grain size

Acid >65% silica

Intermediate 55– 65% silica

Basic 45–55% silica

Ultrabasic <45% silica

Plutonic (coarse)

Granite Granodiorite

Diorite

Gabbro

Picrite Peridotite Serpentinite Dunite

Hypabyssal

Quartz Plagioclase Orthoclase porphyries porphyries

Dolerite

Basic dolerites

Extrusive (fine)

Rhyolite Dacite

Basalt

Basic olivine basalts

Pichstone Andesite

Drilled shafts in rock

Major mineral Quartz, orthoclase, constituents sodium-plagioclase, muscovite, biotite, hornblende

Quartz, orthoclase, plagioclase, biotite, hornblende, augite

12

Calciumplagioclase, augite, olivine, hornblende

Calciumplagioclase, olivine, augite

Table 2.3 Classification of sedimentary rocks. Method of formation

Classification

Rock

Description

Major mineral constituents

Mechanical

Rudaceous

Breccia Conglomerate

Large grains in clay matrix

Various

Arenaceous

Sandstone

Medium, round grains in calcite matrix

Quartz, calcite (sometimes feldspar, mica)

Quartzite

Medium, round Quartz grains in silica matrix

Gritstone

Medium, angular grains in matrix

Quartz, calcite, various

Breccia

Coarse, angular grains in matrix

Quartz, calcite, various

Claystone

Micro-fine-grained plastic texture

Kaolinite, quartz, mica

Shale Mudstone

Harder-laminated compacted clay

Kaolinite, quartz, mica

Calcareous

Limestone

Fossiliferous, coarse or fine grained

Calcite

Carbonaceous (siliceous, ferruginous, phosphatic)

Coal

Ferruginous

Ironstone

Impregnated limestone or claystone (or precipitated)

Calcite, iron oxide

Precipitated or replaced limestone, fine grained

Dolomite, calcite

Argillaceous

Organic

Chemical

Calcareous (siliceous, Dolomite saline) limestone

Table 2.4 Classification of metamorphic rocks. Classification Rock Massive

Description

Hornfels Micro-fine grained

Major mineral constituents Quartz

Intact rock and rock mass

Foliated

13

Quartzite Fined grained

Quartz

Marble

Fine—coarse grained

Calcite or dolomite

Slate

Micro-fine grained, laminated

Kaolinite, mica

Phyllite

Soft, laminated

Mica, kaolinite

Schist

Altered hypabyssal rocks, coarse grained

Feldspar, quartz, mica

Gneiss

Altered granite

Hornblende

Table 2.5 Degree of weathering of intact rocks. Degree of weathering

Description

Unweathered

No evidence of any chemical or mechanical alteration

Slightly weathered

Slight discoloration on surface, slight alteration along discontinuities, less than 10 percent of the rock volume altered

Moderately weathered

Discoloring evident, surface pitted and altered with alteration penetrating well below rock surfaces, weathering “halos” evident, 10 to 50 percent of the rock altered

Highly weathered Entire mass discolored, alteration pervading nearly all of the rock with some pockets of slightly weathered rock noticeable, some minerals leached away Decomposed

Rock reduced to a soil with relict rock texture, generally molded and crumbled by hand

Table 2.6 Engineering classification of rock by strength (from ISRM, 1978; CGS, 1985; Marinos & Hoek, 2001). Grade Classification Field identification

Unconfined compressive strength (MPa)

Point Load Index (MPa)

Examples

R0

Extremely weak

Indented by thumbnail

<1

–1)

Stiff fault gouge

R1

Very weak

Crumbles under firm blows of geological hammer; can be peeled with a pocket knife

1–5

–1)

Highly weathered or altered rock, shale

R2

Weak

Can be peeled with a pocket knife with difficulty; shallow indentations made by a firm blow with point of

5–25

–1)

Chalk, claystone, potash, marl, siltstone, shale, rock salt

Drilled shafts in rock

14

geological hammer R3

Medium strong Cannot be scraped or 25–50 peeled with a pocket knife; specimen can be fractured with a single firm blow of geological hammer

1–2

Concrete, phyllite, schist, siltstone

R4

Strong

Specimen requires more 50–100 than one blow of geological hammer to fracture

2–4

Limestone, marble, sandstone, schist

R5

Very strong

Specimen requires many blows of geological hammer to fracture

100–250

4–10

Amphibolite, sandstone, basalt, gabbro, gneiss, granodiorite, peridotite, rhyolite, tiff

R6

Extremely strong

Specimen can only be chipped with the geological hammer

>250

>10

Fresh basalt, chert, diabase, gneiss, granite, quartzite

1)

Point load tests on rocks with unconfined compressive strength below 25 MPa are likely to yield highly ambiguous results.

2.2.2 Engineering classification The engineering classification of intact rocks is based on strength and/or deformation properties of the rock. Table 2.6 shows the classification system of the International Society of Rock Mechanics (ISRM, 1978). The ISRM classification is also recommended in the Canadian Foundation Engineering Manual (CGS, 1985). In this classification, the rock may range from extremely weak to extremely strong depending on the unconfined compressive strength or the approximate field identification. Based on laboratory measurements of strength and deformation properties of rocks, Deere and Miller (1966) established a classification system based on the ultimate strength (unconfined compressive strength) and the tangent modulus Et of elasticity at 50% of the ultimate strength. Figure 2.1 summarizes the engineering classification of igneous, sedimentary and metamorphic rocks, respectively, as given in Deere and Miller (1966). The modulus ratio in these figures is that of the elastic modulus to the unconfined compressive strength. A rock may be classified as AM, BH, BL, etc. Voight (1968), however, argued that the elastic properties of intact rock could be omitted from practical classification since the elastic moduli as determined in the laboratory are seldom those required for engineering analysis.

Intact rock and rock mass

15

2.2.3 Typical values of intact rock properties This section lists the typical values of intact rock properties, including density (Table 2.7), unconfined compressive strength (Table 2.8), elastic modulus (Table 2.9) and Poisson’s ratio (Table 2.10). These values are listed only for reference and should not be used directly in design.

2.3 ROCK MASS Numerous rock mass classification systems have been developed, including Terzaghi’s Rock Load Height Classification (Terzaghi, 1946); Lauffer’s Classification (Lauffer, 1958); Deere’s Rock Quality Designation (RQD) (Deere, 1964); RSR Concept (Wickham et al., 1972); the Rock Mass Rating (RMR) system (Bieniawski, 1973, 1976, 1989); the Q-System (Barton et al., 1974), and the Geological Strength Index (GSI) system (Hoek & Brown, 1997). Most of the above systems were primarily developed for the design of underground excavations. However, four of the above classification systems have been used extensively in correlation with parameters applicable to the design of rock foundations. These four classification systems are the Rock Quality Designation (RQD), the Rock Mass Rating (RMR), the Q-System, and the Geological Strength Index (GSI). 2.3.1 Rock quality designation (RQD) Rock Quality Designation (RQD) was introduced by Deere (1964) as an index assessing rock quality quantitatively. The RQD is defined as the ratio (in percent) of the total length of sound core pieces 4 in. (10.16 cm) in length or longer to the length of the core run. RQD is perhaps the most commonly used method for characterizing the jointing in borehole cores, although this parameter may also implicitly include other rock mass features such as weathering and core loss. Deere (1964) proposed the relationship between the RQD index and the rock mass quality as shown in Table 2.11. (a) Direct method for determining RQD For RQD determination, the International Society for Rock Mechanics (ISRM) recommends a core size of at least NX (size 54.7 mm) drilled with double-tube core barrel using a diamond bit. Artificial fractures can be identified by close fitting of cores and unstained surfaces. All the artificial fractures should be ignored while counting the core length for RQD. A slow rate of drilling will also give better RQD. The correct procedure for measuring RQD is shown in Figure 2.2.

Drilled shafts in rock

16

Fig. 2.1 Engineering classification of intact rocks (Et is the tangent modulus at 50% ultimate strength) (after Deere & Miller, 1966).

Intact rock and rock mass

17

Table 2.7 Typical values of density of intact rocks (after Lama & Vutukuri, 1978). Range of density (kg/m3)

Mean density (kg/m3)

Granite

2516–2809

2667

Granodiorite

2668–2785

2716

Syenite

2630–2899

2757

Quartz diorite

2680–2960

2806

Diorite

2721–2960

2839

Norite

2720–3020

2984

Gabbro

2850–3120

2976

Diabase

2804–110

2965

Peridotite

3152–3276

3234

Dunite

3204–3314

3277

Pyroxenite

3100–3318

3231

Anorthosite

2640–2920

2734

Sandstone

2170–2700

–

Limestone

2370–750

–

Dolomite

2750–2800

–

Chalk

2230

–

Marble

2750

–

Shale

2060–2660

–

Sand

1920–1930

–

Gneiss

2590–3060

2703

Schist

2700–3030

2790

Slate

2720–840

2810

Amphibolite

2790–3140

2990

Granulite

2630–3100

2830

Eclogite

3338–3452

3392

Rock type Igneous rocks

Sedimentary rocks

Metamorphic rocks

Note: The values listed in the table are for the bulk density determined at natural water content.

Drilled shafts in rock

18

Table 2.8 Typical range of unconfined compressive strength of intact rocks (AASHTO, 1989). Rock category

General description

Rock

Unconfined compressive strength, σc(1) (MPa)

A

Carbonate rocks with welldeveloped crystal cleavage

Dolostone Limestone Carbonatite Marble Tactite-Skarn

33–310 24–290 38–69 38–241 131–338

B

Lithified argillaceous rock

Argillite Claystone Marlstone Phyllite Siltstone Shale(2) Slate

29–145 1–8 52–193 24–241 10–117 7–35 145–207

C

Arenaceous rocks with strong crystals and poor cleavage

Conglomerate Sandstone Quartzite

33–221 67–172 62–379

D

Fine-grained igneous crystalline rock

Andesite Diabase

97–179 21–572

E

Coarse-grained igneous and metamorphic crystalline rock

Amphibolite Gabbro Gneiss Granite Quartz diorite Quartz monozonite Schist Syenite

117–276 124–310 24–310 14–338 10–97 131–159 10–145 179–427

(1)

Range of unconfined compressive strength reported by various investigations. Not including oil shale.

(2)

Table 2.9 Typical values of elastic modulus of intact rocks (AASHTO, 1989). Rock type

No. of values

Elastic modulus (GPa)

No. of rock types

Maximum Minimum Mean

Standard Deviation

Granite

26

26

100

6.41

52.7

24.5

Diorite

3

3

112

17.1

51.4

42.7

Gabbro

3

3

84.1

67.6

75.8

6.69

Intact rock and rock mass

Diabase

19

7

7

104

69.0

88.3

12.3

12

12

84.1

29.0

56.1

17.9

7

7

88.3

36.5

66.1

16.0

Marble

14

13

73.8

4.00

42.6

17.2

Gneiss

13

13

82.1

28.5

61.1

15.9

Slate

11

2

26.1

2.41

9.58

6.62

Schist

13

12

69.0

5.93

34.3

21.9

3

3

17.3

8.62

11.8

3.93

27

19

39.2

0.62

14.7

8.21

5

5

32.8

2.62

16.5

11.4

Shale

30

14

38.6

0.007

9.79

10.0

Limestone

30

30

89.6

4.48

39.3

25.7

Dolostone

17

16

78.6

5.72

29.1

23.7

Basalt Quartzite

Phyllite Sandstone Siltstone

Table 2.10 Typical values of Poisson’s ratio of intact rocks (AASHTO, 1989). No. of values

Poisson’s ratio

No. of rock types

Rock type

Standard Deviation

Maximum Minimum Mean

Granite

22

22

0.39

0.09

0.20

0.08

Gabbro

3

3

0.20

0.16

0.18

0.02

Diabase

6

6

0.38

0.20

0.29

0.06

11

11

0.32

0.16

0.23

0.05

Quartzite

6

6

0.22

0.08

0.14

0.05

Marble

5

5

0.40

0.17

0.28

0.08

Gneiss

11

11

0.40

0.09

0.22

0.09

Schist

12

11

0.31

0.02

0.12

0.08

Sandstone

12

9

0.46

0.08

0.20

0.11

Siltstone

3

3

0.23

0.09

0.18

0.06

Shale

3

3

0.18

0.03

0.09

0.06

Basalt

Drilled shafts in rock

20

Table 2.11 Correlation between RQD and rock mass quality. RQD (%)

Rock Mass Quality

<25

Very poor

25–50

Poor

50–75

Fair

75–90

Good

90–100

Excellent

(b) Indirect methods for determining RQD The seismic survey method can be used to determine the RQD indirectly. By comparing the in situ compressional wave velocity with laboratory sonic velocity of intact drill core obtained from the same rock mass, the RQD can be estimated by RQD(%)=Velocity ratio=(VF/VL)2×100 (2.1) where VF is the in situ compressional wave velocity; and VL is the compressional wave velocity in intact rock core. When cores are not available, RQD may be estimated from the number of joints (discontinuities) per unit volume JV. A simple relationship which may be used to convert JV into RQD for clay-free rock masses is (Palmstrom, 1982) RQD(%)=115−3.3JV (2.2) where JV is the total number of joints per cubic meter or the volumetric joint count. The volumetric joint count JV has been described by Palmstrom (1982, 1985, 1986) and Sen and Eissa (1992). It is a measure for the number of joints within a unit volume of rock mass defined by (2.3)

Intact rock and rock mass

21

Fig. 2.2 Procedure for measurement and calculation of rock quality designation RQD (after Deere, 1989). where si is the average joint spacing in meters for the ith joint set and N is the total number of joint sets except the random joint set. Random joints may also be considered by assuming a “random spacing”. Experience indicates that this should be set to sr=5 m (Palmstrom, 1996). Thus, the volumetric joint count can be generally expressed as (2.4)

where Nr is the total number of random joint sets and can easily be estimated from joint observations. In cases where random or irregular jointing occurs, JV can be found by counting all the joints observed in an area of known size.

Drilled shafts in rock

22

Though the RQD is a simple and inexpensive index, when considered alone it is not sufficient to provide an adequate description of a rock mass because it disregards joint orientation, joint condition, type of joint filling and other features. 2.3.2 Rock mass rating (RMR) The Geomechanics Classification, or Rock Mass Rating (RMR) system, proposed by Bieniawski (1973), was initially developed for tunnels. In recent years, it has been applied to the preliminary design of rock slopes and foundations as well as for estimating the in-situ modulus of deformation and rock mass strength. The RMR uses six parameters that are readily determined in the field (see Table 2.12): • Unconfined compressive strength of the intact rock • Rock Quality Designation (RQD) • Spacing of discontinuities • Condition of discontinuities • Ground water conditions • Orientation of discontinuities All but the intact rock strength are normally determined in the standard geological investigations and are entered on an input data sheet. The guidelines for assessing the discontinuity condition is shown in Table 2.13. The unconfined compressive strength of rock is determined in accordance with standard laboratory procedures but can be readily estimated in situ from the point-load strength index (see Table 2.12). Rating adjustments for discontinuity orientation are summarized for underground excavations, foundations and slopes in Part B of Table 2.12. A more detailed explanation of these rating adjustments for dam foundations is given in Table 2.14, after ASCE (1996). The six separate ratings are summed to give an overall RMR, with a higher RMR indicating better quality rock. Based on the observed RMR value, the rock mass is classified into five classes named as very good, good, fair, poor and very poor, as shown in Part C of Table 2.12. Also shown in Part C of Table 2.12 is an interpretation of these five classes in terms of roof stand-up time, cohesion, internal friction angle and deformation modulus for the rock mass. 2.3.3 Rock mass quality (Q) The Q-system, proposed by Barton et al. (1974), was developed specifically for the design of tunnel support systems. As in the case of the Geomechanics System, the Qsystem has been expanded to provide preliminary estimates of rock mass properties. Likewise, the Q-system incorporates the following six parameters and the equation for obtaining rock mass quality Q:

Intact rock and rock mass

23

Table 2.12 Geomechanics classification of jointed rock masses (after Bieniawski, 1976, 1989). A. Classification parameters and their rating Parameter

Range of values

1 Strength Point-load of intact strength rock index (MPa)

>10

4–10

2–4

1–2

Unconfined compressive strength (MPa)

>250

100–250

50–100

25–50

5– 25

1– <1 5

15

12

7

4

2

1

90–100

75–90

50–75

25–50

<25

20

17

13

8

3

>2

0.6–2

0.2–0.6

0.06–0.2

<0.06

20

15

10

8

5

Rating 2 Drill core quality RQD (%) Rating 3 Spacing of discontinuities (m) Rating 4 Conditions of discontinuities

Rating 5 Ground water

Very rough Slightly surfaces, Not rough continuous, surfaces, No separation, separation Unweathered <1 mm, Slightly wall rock weathered walls

For this low range, unconfined compressive test is preferred

0

Slightly Slickensided Soft gouge rough surfaces or >5 mm thick surfaces, Gouge <5 or Separation separation mm thick or >5 mm <1 mm, Separation 1– Continuous Highly 5 mm weathered continuous walls

30

25

20

10

0

Inflow per 10m tunnel length (1/min)

None or

<10 or

10–25 or

25–125 or

>125 or

Ratio of joint water pressure to major principal stress

0 or

<0.1 or

0.1–0.2 or

0.2–0.5 or

>0.5 or

General conditions

Completely dry

Damp

Wet

Dripping

Flowing

Drilled shafts in rock

Rating

15

24

10

7

4

0

B. Rating adjustment for joint orientations Strike and dip orientations of discontinuities Ratings

Very favorable

Favorable Fair Unfavorable

Very Unfavorable

Tunnels and mines

0

−2

−5

−10

−12

Foundations

0

−2

−7

−15

−25

Slopes

0

−5

−25

−50

−60

C. Rock mass classes and corresponding design parameters and engineering properties Class No.

I

II

III

IV

V

100→81

80→61

60→41

40→21

<20

Description

Very Good

Good

Fair

Poor

Very poor

Average stand-up time

20 years for 15 m span

1 year for 10m span

1 week for 5m span

10 hours for 2.5 m span

30 minutes for 1 m span

Cohesion of rock mass (MPa)

>0.4

0.3–0.4

0.2–0.3

0.1–0.2

<0.1

Internal friction angle of rock mass (°)

>45

35–45

25–35

15–25

<15

Deformation modulus (GPa)a)

>56

56–18

18–5.6

5.6–1.8

<1.8

RMR

a)

Deformation modulus values are from Serafim and Pereira (1983).

Table 2.13 Guidelines for classifying discontinuity condition (after Bieniawski, 1989). Parameter

Range of values

Discontinuity length (persistence/ continuity)

Rating

6

4

2

1

0

Measurement (m)

<1

1–3

3–10

10–20

>20

Separation (aperture)

Rating

6

5

4

1

0

None

<0.1

0.1–1

1–5

>5

Roughness

Rating

6

5

3

1

0

Very rough

Rough

Slight

Smooth

Slickensided

6

4

2

2

0

Measurement (mm)

Description Infilling (gouge)

Rating

Intact rock and rock mass

Description and Measurement (mm) Degree of weathering

Rating Description

25

None

Hard filling <5

Hard filling >5

Soft filling <5

Soft filling >5

6

5

3

1

0

None

Slight

Moderate

High

Decomposed

Note: Some conditions are mutually exclusive. For example, if infilling is present, it is irrelevant what the roughness may be, since its effect will be overshadowed by the influence of the gouge. In such cases, use Table 2.12 directly.

Table 2.14 Ratings for discontinuity orientations for dam foundations (after ASCE, 1996). Dip 10°–30° Dip 0°–10° Very favorable

Dip direction Upstream

Downstream

Unfavorable

Fair

Dip 30°–60°

Dip 60°–90°

Favorable

Very unfavorable

• Rock Quality Designation (RQD) • Number of discontinuity sets • Roughness of the most unfavorable discontinuity • Degree of alteration or filling along the weakest discontinuity • Water inflow • Stress condition (2.5) where RQD=Rock Quality Designation; Jn=joint set number; Jr=joint roughness number; Ja=joint alteration number; Jw=joint water reduction number; and SRF= stress reduction number. The meaning of the parameters used to determine the value of Q in Equation (2.5) can be seen from the following comments by Barton et al. (1974): The first quotient (RQD/Jn), representing the structure of the rock mass, is a crude measure of the block or particle size, with the two extreme values (100/0.5 and 10/20) differing by a factor of 400. If the quotient is interpreted in units of centimetres, the extreme ‘particle sizes’ of 200 to 0.5 cm are seen to be crude but fairly realistic approximations. Probably the largest blocks should be several times this size and the smallest fragments less than half the size. (Clay particles are of course excluded). The second quotient (Jr/Ja) represents the roughness and frictional characteristics of the joint walls or filling materials. This quotient is weighted in favor of rough, unaltered joints in direct contact. It is to be expected that such surfaces will be close to peak strength, that they will dilate strongly when sheared, and they will therefore be especially favorable to tunnel stability.

Drilled shafts in rock

26

When rock joints have thin clay mineral coatings and fillings, the strength is reduced significantly. Nevertheless, rock wall contact after small shear displacements have occurred may be a very important factor for preserving the excavation from ultimate failure. Where no rock wall contact exists, the conditions are extremely unfavorable to tunnel stability. The ‘friction angles’ (given in Table 2.15) are a little below the residual strength values for most clays, and are possibly dowti-graded by the fact that these clay bands or fillings may tend to consolidate during shear, at least if normal consolidation or if softening and swelling has occurred. The swelling pressure of monttnorillonite may also be a factor here. The third quotient (Jw/SRF) consists of two stress parameters. SRF is a measure of: 1) loosening load in the case of an excavation through shear zones and clay bearing rock, 2) rock stress in competent rock, and 3) squeezing loads in plastic incompetent rocks. It can be regarded as a total stress parameter. The parameter Jw is a measure of water pressure, which has an adverse effect on the shear strength of joints due to a reduction in effective normal stress. Water may, in addition, cause softening and possible out-wash in the case of clay-filled joints. It has proved impossible to combine these two parameters in terms of inter-block effective stress, because paradoxically a high value of effective normal stress may sometimes signify less stable conditions than a low value, despite the higher shear strength. The quotient (Jw/SRF) is a complicated empirical factor describing the ‘active stress’. So the rock mass quality (Q) may be considered a function of three parameters which are approximate measures of: (i) Block size (RQD/Jn): It represents the overall structure of rock masses. (ii) Inter block shear strength (Jr/Ja): It represents the roughness and frictional characteristics of the joint walls or filling materials. (iii) Active stress (Jw/SRF): It is an empirical factor describing the active stress. Table 2.15 provides the necessary guidance for assigning values to the six parameters. Depending on the six assigned parameter values reflecting the rock mass quality, Q can vary between 0.001 to 1000. Rock quality is divided into nine classes ranging from exceptionally poor (Q ranging from 0.001 to 0.01) to exceptionally good (Q ranging from 400 to 1000) as shown in Table 2.16. Since the Q and RMR systems are based on much the same properties, they are highly correlated and can be predicted one from the other. Various authors give a relationship in the following form (Rutledge & Preston, 1978; Cameron-Clarke & Budavari, 1981; Abad et al., 1984; Beniawski, 1989; Goel et al., 1995):

Intact rock and rock mass

27

Table 2.15 The Q-system and associated parameters RQD, Jn, Jr, Ja, SRF and Jw (after Barton et al., 1974). Rock Quality Designation RQD (%) Very poor

0–25

Note:

Poor

25–0

Fair

50–5

(i) Where RQD is reported or measured to be <10 a nominal value of 10 is used to evaluate Q in Equation (2.5)

Good

75–90

Excellent

90–100

(ii) Take RQD to be nearest 5%

Joint Set Number Jn Massive, none or few joints

0.5–1.0

One joint set

2

One joint set plus random

3

Two joint sets

4

Two joint sets plus random

6

Three joint sets

9

Three joint sets plus random

12

Four or more joint sets, random, heavily jointed, ‘sugar cube’, etc

15

Crushed rock, earthlike

20

Note: (i) For intersections use (3.0×Jn) (ii) For portals use (2.0×Jn)

Joint Roughness Number Jr (a) Rock wall contact and

Notes:

(b) Rock wall contact

(i) Add 1.0 if the mean spacing of the relevant joint set is greater than 3 m

before 10 cm shear

Drilled shafts in rock

Discontinuous joint

4

Rough or irregular,

3

undulating Smooth, undulating

28

(ii) Jr=0.5 can be used for planar slickensided joints having lineations, provided the lineations are favorably orientated

2

Slickensided, undulating

1.5

Rough and irregular, planar

1.5

Smooth or irregular

1

Slickensided, planar

0.5

(c) No rock wall contact when sheared Zone containing clay

1

(nominal)

1

(nominal)

minerals thick enough to prevent rock wall contact Sandy, gravelly or crushed zone thick enough to prevent rock wall contact

Joint Alternation Number

Approximate residual angle of friction (deg)

Ja (a) Rock wall contact A. Tightly healed, hard, nonsoftening,

0.75

–

impermeable filling, i.e. quartz or epidote B. Unaltered joint walls, surface staining only

1

25– 35

C. Unlatered joint walls. Non-softening mineral coatings, sandy particles, clay-free disintegrated rock, etc.

2

25– 30

D. Silty or sandy clay coatings, small clay fraction (non-softening)

3

20– 25

E. Softening or low friction clay mineral coatings, i.e. kaolinite, mica. Also chlorite, talc, gypsum and graphite, etc, and small quantities of swelling clays (discontinuous coatings, 1–2 mm or less in thickness)

4

8– 16

4

25–

(b) Rock wall contact before 10 cm shear F. Sandy particles, clay free disintegrated rock, etc

Intact rock and rock mass

29

30 G. Strongly over-consolidated, non-softening clay mineral fillings (continuous, <5

6

16– 24

H. Medium or low over-consolidation, softening, clay mineral fillings (continuous, <5 mm in thickness)

8

12– 16

J. Swelling clay fillings, i.e. montmorillonite (continuous, <5 mm in thickness). Value of Ja depends on percentage of swelling clay-sized particles and access to water, etc

8–12

6– 12

6, 8 or 8– 12

6– 24

(c) No rock wall contact when sheared K. Zones or bands of disintegrated or crushed

rock and clay (see G, H. J for description of clay condition) L. Zones or bands of silty or sandy clay, small clay fraction (nonsoftening) M. Thick, continuous zones or bands of clay (see G, H. J for description of clay condition)

5

–

10, 13 or 13– 20

6–24

Joint Water Reduction Factor

Approximate water pressure (kPa)

Jw A. Dry excavations or minor inflow, i.e. <5 l/min locally

1

<100

B. Medium inflow or pressure occasional outwash of joint fillings

0.66

100–250

C. Large inflow or high pressure in competent rock with unfilled joints

0.5

250–1000

D. Large inflow or high pressure, considerable occasional outwash of joint fillings

0.33

250–1000

E. Exceptionally high inflow or water pressure at blasting, decaying with time

0.1

>1000

0.1−0.05

>1000

F.

Exceptionally high inflow or water pressure continuing without decay

Note: (i) Factors C-F are crude estimates. Increase Jw if drainage measures are installed (ii) Special problems caused by ice formation are not considered

Drilled shafts in rock

30

Stress Reduction Factor SRF (a) Weakness zones intersecting excavation, which may cause loosening of rock mass when tunnel is excavated A. Multiple occurrences of

10

weakness zones containing clay or chemically disintegrated rock, very loose surrounding rock (any depth) B. Single weakness zones containing clay or chemically disintegrated rock (depth of excavation <50 m)

5

C. Single weakness zones containing clay or chemically disintegrated rock (depth of excavation >50 m)

2.5

D. Multiple shear zones in competent rock (clay free), loose surrounding

7.5

E. Single shear zones in rock (any depth) competent rock (clay free, depth of excavation <50 m)

5

F. Single shear zones in competent rock (clay free, depth of excavation >50 m)

2.5

G. Loose open joints, heavily jointed, or “sugar cube” etc (any depth)

5

(b) Competent rock rock

Note: (i) Reduce these values by 25–50% if the relevant shear zones only influence but do not intersect the excavation

Intact rock and rock mass

31

stress problems Strength/stress ratios σc/σ1

σt/σ1

H. Low stress, near surface

>200

>13

2.5

J. Medium stress

200– 10

13– 0.66

1

K. High stress, very tight structure (usually favorable to stability, maybe unfavorable to wall stability)

10–5

0.66– 0.33

0.5– 2.0

L. Mild rock burst (massive rock)

5–2.5

0.33– 0.16

5– 10

M. Heavy rock burst (massive rock)

<2.5

<0.16

10– 20

(c) Swelling rock; chemical swelling activity depending on presence of water P. Mild swelling rock pressure

5–10

R. Heavy swelling rock pressure

10–15

(ii) If stress field is strongly anisotropic: when 5<σ1/σ3< 10, reduce σc and σt to 0.8σc and 0.8σt; when σ1/σ3>10, reduce σc and σt to 0.6σc and 0.6σt. Where σc=unconfined compressive strength, σt=tensile strength, σ1 and σ3=major and minor principal stresses.

(iii) Few case records available where depth of crown below surface is less than span width. Suggest SRF increase from 2.5 to 5 for such cases (see H)

Table 2.16 Classification of rock mass based on Qvalues (after Barton et al., 1974). Group

Q

Classification

1

1000−400

Exceptionally good

400–100

Extremely good

100−40

Very good

40–10

Good

10–4

Fair

4–1

Poor

1–0.1

Very poor

0.1–0.01

Extremely poor

0.01–0.001

Exceptionally poor

2

3

Drilled shafts in rock

32

RMR=A loge Q+B (2.6) where A is typically in the range 5–15, and B in the range 35–60. Based on data from hard rock tunneling projects in several countries, Barton (1991) proposed a correlation between Q and seismic P-wave velocity: (2.7) where Vp is P-wave velocity in m/s (see Chapter 5 about the measurement of Vp). 2.3.4 Geological strength index (GSI) Hoek and Brown (1997) introduced the Geological Strength Index (GSI), both for hard and weak rock masses. Experienced field engineers and geologists generally show a liking for a simple, fast, yet reliable classification which is based on visual inspection of geological conditions. Hoek and Brown (1997) proposed such a practical classification for estimating GSI based on visual inspection alone (see Tables 2.17). In this classification, there are four main qualitative classifications, adopted from Terzaghi’s classification (Terzaghi, 1946): (i) Blocky (ii) Very blocky (iii) Blocky/Disturbed (iv) Disintegrated Engineers and geologists have been familiar with it for over 50 years. Further, discontinuities are classified into 5 surface conditions which are similar to joint conditions in RMR as described earlier: (i)Very good (ii) Good (iii) Fair (iv) Poor (v) Very poor Now a block in the 4×5 matrix of Table 2.17 is picked up according to actual rock mass classification and discontinuity surface condition, and the corresponding GSI value can then be read from the table. According to Hoek and Brown (1997), a range of values of GSI (or RMR) should be estimated in preference to a single value. GSI can also be estimated from Bieniawski’s Rock Mass Rating (RMR) and Barton et al.’s (1974) modified Tunneling Quality Index (Q) (Hoek & Brown, 1997). In using Bieniawski’s 1989 Rock Mass Rating (see Part A of Table 2.12) to estimate the value of GSI, the rock mass should be assumed to be completely dry and a rating of 15 assigned to the groundwater value. Very favorable discontinuity orientations should be assumed and the Adjustment for Discontinuity Orientation value set to zero. The minimum value

Intact rock and rock mass

33

which can be obtained for the 1989 classifications is 23. The estimated RMR is used to estimate the value of GSI as follows: (i)

Table 2.17 Characterisation of rock masses on the basis of interlocking and joint alteration (after Hoek & Brown, 1997).

For RMR>23

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34

GSI=RMR−5 (2.8) For RMR<23, Bieniawski’s classification cannot be used to estimate GSI and the Q value of Barton et al. (1974) should be used instead: GSI=9logeQ+44 (2.9) where Q is calculated from Equation (2.5) by setting a value of 1 for both Jw (discontinuity water reduction factor) and SRF (stress reduction factor). It should be noted that, in the 1976 version of the Rock Mass Rating system, a rating of 10 is assigned to the groundwater value when the rock mass is assumed to be completely dry. The minimum value which can be obtained for the 1976 classifications is thus 18. Using the 1976 Rock Mass Rating system, GSI is equal to the RMR obtained by assuming completely dry rock mass condition and very favorable discontinuity orientations.

3 Characterization of discontinuities in rock 3.1 INTRODUCTION The primary difference between structures in soil and those in rock is that rock masses contain discontinuities. The analysis and design of any structure in rock require information on discontinuities in one form or another (Goodman, 1976; Hoek & Bray, 1981; Brady & Brown, 1985; Wyllie, 1999). Blocks formed by sets of unfavorably orientated discontinuities may fail by sliding or toppling, causing excessive movement or failure of foundations. The discontinuity properties that have the greatest influence at the design stage have been listed by Piteau (1970, 1973) as follows: 1. orientation 2. size 3. frequency 4. surface geometry 5. genetic type, and 6. infill material As the rock discontinuities cannot be directly examined in three dimensions (3D), engineers must infer discontinuity characteristics from data sampled at exposed rock faces (including both natural outcrops and excavation walls) and/or in boreholes (Priest, 1993). Taking measurements at exposed rock faces, either at or below the ground surface, enables one to obtain data on orientation, spacing, trace lengths and number of traces. Many statistical sampling and data processing methods have been adopted (Priest, 1993). The most widely used of these methods are the scanline and window sampling techniques. These techniques have been described and discussed by a number of authors including Baecher and Lanney (1978), ISRM (1978), Priest and Hudson (1981), Einstein and Baecher (1983), Kulatilake and Wu (1984a, b, c), Kulatilake (1993), Mauldon (1998) and Zhang and Einstein (1998b, 2000a). One disadvantage of this approach is that the exposed rock face is often remote from the zone of interest and may suffer from blasting damage or degradation by weathering and be obscured by vegetation cover. Borehole sampling, in most cases, provides the only viable exploratory tool that directly reveals geologic evidence of subsurface site conditions. In normal-size borehole sampling various techniques can be used for acquiring discontinuity data either from core samples or through inspection of the borehole walls. However, since the borehole diameter is small, such information is of limited use only. In some cases, e.g., when drilling holes for installing drilled shafts, the hole is large and direct measurements can be conducted to obtain more accurate and precise discontinuity information. For example, trace length data can be obtained from the side and bottom of the drilled shaft holes.

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37

The data sampled at exposed rock faces and/or in boreholes contain errors due to sampling biases. Therefore, it is important to correct sampling biases when inferring 3D discontinuity characteristics. In addition, principles of stereology need be used in order to infer 3D discontinuity characteristics from the data sampled at exposed rock faces and/or in boreholes.

3.2 TYPES OF DISCONTINUITIES Discontinuities and their origins are well described in several textbooks on general, structural and engineering geology. From an engineer’s point of view, the discussions by Price (1966), Hills (1972), Blyth and de Freitas (1974), Hobbs (1976) and Priest (1993) are particularly helpful. The following lists the major types of discontinuities and briefly describes their key engineering properties. (a) Faults Faults are discontinuities on which identifiable shear displacement has taken place. They may be recognized by the relative displacement of the rock on the opposite sides of the fault plane. The sense of this displacement is often used to classify faults. Faults may be pervasive features which traverse a large area or they may be of relatively limited local extent on the scale of meters; they often occur in echelon or in groups. Fault thickness may vary from meters in the case of major, regional structures to millimeters in the case of local faults. This fault thickness may contain weak materials such as fault gouge (clay), fault breccia (recemented), rock flour or angular fragments. The wall rock is frequently slickensided and may be coated with minerals such as graphite and chlorite which have low frictional strengths. The ground adjacent to the fault may be disturbed and weakened by associated discontinuities such as drag folds or secondary faults. These factors result in faults being zones of low shear strength on which slip may readily occur. (b) Bedding planes Bedding planes divide sedimentary rocks into beds or strata. They represent interruptions in the course of deposition of the rock mass. Bedding planes are generally highly persistent features, although sediments laid down rapidly from heavily laden wind or water currents may contain cross or discordant bedding. Bedding planes may contain parting material of different grain size from the sediments forming the rock mass, or may have been partially healed by low-order metamorphism. In either of these two cases, there would be some ‘cohesion’ between the beds; otherwise, shear resistance on bedding planes would be purely frictional. Arising from the depositional process, there may be a preferred orientation of particles in the rock, giving rise to planes of weakness parallel to the bedding planes.

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(c) Joints Joints are the most common and generally the most geotechnically significant discontinuities in rocks. Joints are breaks of geological origin along which there has been no visible relative displacement. A group of parallel or sub-parallel joints is called a joint set, and joint sets intersect to form a joint system. Joints may be open, filled or healed. Discontinuities frequently form parallel to bedding planes, foliations or slaty cleavage, and they may be termed bedding joints, foliation joints or cleavage joints. Sedimentary rocks often contain two sets of joints approximately orthogonal to each other and to the bedding planes. These joints sometimes end at bedding planes, but others, called master joints, may cross several bedding planes. (d) Cleavage There are two broad types of rock cleavage: fracture cleavage and flow cleavage. Fracture cleavage (also known as false cleavage and strain slip cleavage) is a term describing incipient, cemented or welded parallel discontinuities that are independent of any parallel alignment of minerals. Spencer (1969) lists six possible mechanisms for the formation of fracture cleavage. In each mechanism, lithology and stress conditions are assumed to have produced shearing, extension or compression, giving rise to numerous closely-spaced discontinuities separated by thin slivers of intact rock. Fracture cleavage is generally associated with other structural features such as faults, folds and kink bands. Flow cleavage, which can occur as slaty cleavage or schistosity, is dependent upon the recrystallization and parallel allignment of platy minerals such as mica, producing interleaving or foliation structure. It is generally accepted that flow cleavage is produced by high temperatures and/or pressures associated with metamorphism in fine-grained rocks. Although cleavage is usually clearly visible in slates, phyllites and schists, most cleavage planes possess significant tensile strength and do not, therefore, contribute to the discontinuity network. Cleavage can, however, create significant anisotropy in the deformabilty and strength of such rocks. Geological processes, such as folding and faulting, subsequent to the formation of the cleavage can exploit these planes of weakness and generate discontinuities along a proportion of the better developed cleavage planes. The decision as to whether a particular cleavage plane is a discontinuity presents one of the most challenging problems to those undertaking discontinuity surveys in cleaved rocks.

3.3 IMPORTANT PROPERTIES OF DISCONTINUITIES This section lists and discusses the most important properties of discontinuities that influence the engineering behavior. A fuller discussion of these properties can be found in the ISRM publication Suggested methods for the quantitative description of discontinuities in rock masses (ISRM, 1978).

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Fig. 3.1 Definition of dip orientation (α) and dip (β). (a) Orientation Orientation, or the attitude of a discontinuity in space, is described by the dip of the line of maximum declination on the discontinuity surface measured from the horizontal, and the dip direction or azimuth of this line, measured clockwise from true north (see Fig. 3.1). Some geologists record the strike of the discontinuity rather than the dip direction. For rock mechanics purposes, it is usual to quote orientation data in the form of dip direction (three digits)/dip (two digits) such as 035°/75° and 290°/30°. The orientation of discontinuities relative to an engineering structure largely controls the possibility of unstable conditions or excessive deformations developing. The importance of orientation increases when other conditions for deformation are present, such as low shear strength and a sufficient number of discontinuities for slip to occur. The mutual orientation of discontinuities will determine the shape of the individual blocks, beds or mosaics comprising the rock mass. The procedures for presenting and analyzing orientation measurements using the stereonet are discussed in detail in Section 3.4. (b) Spacing and frequency Spacing is the perpendicular distance between adjacent discontinuities, and is usually expressed as the mean spacing of a particular set of discontinuities. Frequency (i.e. the number per unit distance) is the reciprocal of spacing (i.e. the mean spacing). The spacing of discontinuities determines the sizes of the blocks making up the rock mass. The mechanism of deformation and failure of rock masses can vary with the discontinuity spacing. As in the case of orientation, the importance of spacing increases when other conditions for deformation are present, such as low shear strength and a sufficient number of discontinuities for slip to occur.

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Like all other characteristics of a given rock mass, discontinuity spacings will not have uniquely defined values but, rather, will take a range of values, possibly according to some form of statistical distribution. The probabilistic analysis of discontinuity spacings will be discussed in detail in Section 3.5. Discontinuity spacing is a factor used in many rock mass classification schemes. Table 3.1 gives the terminology used by ISRM (1978).

Table 3.1 Classification of discontinuity spacing. Description

Spacing (mm)

Extremely close spacing

<20

Very close spacing

20–60

Close spacing

60–200

Moderate spacing

200–600

Wide spacing

600–2000

Very wide spacing

2000–6000

Extremely wide spacing

>6000

(c) Persistence and size Persistence is a term used to describe the areal extent or size of a discontinuity within a plane. It can be crudely quantified by observing discontinuity trace lengths on exposed rock faces. It is one of the most important rock mass parameters, but one of the most difficult to determine. The discontinuities of one particular set are often more continuous than those of the other sets. The minor sets tend to terminate against the primary features, or they may terminate in solid rock. The sets of discontinuities can be distinguished by terms of persistent, sub-persistent and non-persistent respectively. Figure 3.2 shows a set of simple plane sketches and block diagrams used to help indicate the persistence of various sets of discontinuities in a rock mass. Clearly, the persistence of discontinuities has a major influence on the shear strength developed in the plane of the discontinuity. ISRM (1978) uses the most common or modal trace lengths of each set of discontinuities measured on exposed rock faces to classify persistence according to Table 3.2.

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Fig. 3.2 Simple sketches and block diagrams indicating the persistence of various sets of discontinuities (after ISRM, 1978). Table 3.2 Classification of discontinuity persistence. Description

Modal trace length (m)

Very low persistence

<1

Low persistence

1–3

Medium persistence

3–10

High persistence

10–20

Very high persistence

>20

Persistence ratio PR is often used to describe the persistence of discontinuities. In the literature, discontinuity persistence ratio PR is usually defined as (3.1)

in which S is a region on the discontinuity plane with area As and is the area of the ith discontinuity in S (see Fig. 3.3). The summation in Equation (3.1) is over all discontinuities in S. Equivalently, discontinuity persistence ratio PR can be expressed as a limit length ratio along a given line on a discontinuity plane. In this case,

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42

(3.2)

is the length of the ith in which LS is the length of a straight line segment S and discontinuity segment in S (see Fig. 3.4). For a finite sampling length LS, PR can be simply estimated by (Fig. 3.5)

Fig. 3.3 Definition of PR as area ratio. (3.3) where ΣDL is the sum of the length of all discontinuities; and ΣRBL is the sum of the length of all rock bridges.

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Fig. 3.4 Definition of PR as length ratio.

Fig. 3.5 Estimation of PR for a finite sampling length.

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Fig. 3.6 Failure of “low-angle” transitions through intact rock: (a) Tensile fracture; and (b) Secondary shear fracture (after Zhang, 1999). The above definition of discontinuity persistence ratio PR considers only the discontinuities in the same plane. However, according to Einstein et al. (1983), when two discontinuities are at a low-angle transition (β<θt, see Fig. 3.6), the rock bridge may fail by the same mechanism as the in-plane rock bridge (see Fig. 3.7), where θt is the angle of the tension cracks which can be obtained from Mohr’s circle [see Fig. 3.7(a)]. For both the in-plane (Fig. 3.7) and the low-angle out-of-plane (Fig. 3.6) transitions, the intactrock resistance R can be calculated by R=τad (3.4)

Characterization of discontinuities in rock

45

where d is the “in-plane length” of the rock bridge; and τa is the peak shear stress mobilized in the direction of discontinuities which can be obtained by (3.5) where σt is the tensile strength of the intact rock; and σa is the effective normal stress on the discontinuity plane.

Fig. 3.7 In-plane failure of intact rock: (a) Tensile fracture and corresponding Mohr’s circle; and (b) Secondary shear fracture (after Zhang, 1999). Zhang (1999) proposed a definition of discontinuity persistence ratio PR that considers both in-plane and low-angle-transition discontinuities: (3.6)

in which LS is the total sampling length along the direction of the discontinuity traces, DLi is the length of the ith in-plane discontinuities and DLl is the length of the lth low-

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angle-transition discontinuities (see Fig. 3.8). For a finite sampling length, PR can be simply approximated by (3.7)

Fig. 3.8 Definition of PR considering both in-plane and low-angle-transition discontinuities (after Zhang, 1999). where m and n are the numbers respectively of the in-plane and low-angle-transition (β <θt) discontinuities within the sampling length LS (see Fig. 3.8). (d) Shape The planar shape of discontinuities has a profound effect on the connectivity of discontinuities and on rock mass properties (Dershowitz et al., 1993; Petit et al., 1994). However, since a rock mass is usually inaccessible in three dimensions, the real discontinuity shape is rarely known. Information on discontinuity shape is limited and often open to more than one interpretation (Warburton, 1980a; Wathugala, 1991). Discontinuities can be classified into two categories: unrestricted and restricted. Unrestricted discontinuities are blind and effectively isolated discontinuities whose growth has not been perturbed by adjacent geological structures such as faults and free

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47

surfaces. In general, the edge of unrestricted discontinuities is a closed convex curve. In many cases, the growth of discontinuities is limited by adjacent preexisting discontinuities and free surfaces. Such discontinuities are called restricted discontinuities. One way to represent restricted discontinuities is to use polygons, some of the polygon sides being those formed by intersections with the adjacent preexisting discontinuities and free surfaces. Due to the mathematical convenience, many investigators assume that discontinuities are thin circular discs randomly located in space (Baecher et al., 1977; Warburton, 1980a; Chan, 1986; Villaescusa & Brown, 1990; Kulatilake, 1993). With circular discontinuities, the trace patterns in differently oriented sampling planes will be the same. In practice, however, the trace patterns may vary with the orientation of sampling planes (Warburton, 1980b). Therefore, Warburton (1980b) assumed that discontinuities in a set are parallelograms of various sizes. Dershowitz et al. (1993) used polygons to represent discontinuities in the FracMan discrete fracture code. The polygons are formed by inscribing a polygon in an ellipse. Ivanova (1998) and Meyer (1999) also used polygons to represent discontinuities in their discrete fracture code GeoFrac. It is noted that polygons can be used to effectively represent elliptical discontinuities when the number of polygon sides is large (say >10) (Dershowitz et al., 1993). Zhang et al. (2002) assumed that discontinuities are elliptical and derived a general stereological relationship between trace length distributions and discontinuity size (expressed by the major axis length of the ellipse) distributions. Many researchers infer the discontinuity shape from the study of trace lengths in both the strike and dip directions. Based on the fact that the average strike length of a discontinuity set is approximately equal to its average dip length, Robertson (1970) and Barton (1977) assumed that discontinuities are equidimensional (circular). However, the average strike length of a discontinuity set being the same as its average dip length does not necessarily mean that the discontinuities of such a set are equidimensional; instead, there exist the following three possibilities (Zhang et al., 2002): 1. The discontinuities are indeed equidimensional [see Fig. 3.9(a)]. 2. The discontinuities are non-equidimensional such as elliptical or rectangular with long axes in a single (or deterministic) orientation. However, the discontinuities are oriented such that the strike length is approximately equal to the dip length [see Fig. 3.9(b)]. 3. The discontinuities are non-equidimensional such as elliptical or rectangular with long axes randomly oriented. The random discontinuity orientation distribution makes the average strike length approximately equal to the average dip length [see Fig. 3.9(c)]. Therefore, the conclusion that discontinuities are equidimensional (circular) drawn from the fact that the average strike length of a discontinuity set is about equal to its average dip length is questionable. Investigators assume circular discontinuity shape possibly because of mathematical convenience. Einstein et al. (1979) measured trace lengths of two sets of discontinuities on both the horizontal and vertical surfaces of excavations and found that discontinuities are nonequidimensional. Petit et al. (1994) presented results of a field study to determine the shape of discontinuities in sedimentary rocks. Pelites with isolated sandstone layers in the red Permian sandstones of the Lodeve Basin were studied. The exposed discontinuities

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(i.e., one of the discontinuity walls had been removed by erosion) appear as rough ellipses with a shape ratio L/H of about 2.0, where L and H are respectively the largest horizontal and vertical dimensions. For non-exposed discontinuities, the distributions of the dimensions of the horizontal and vertical traces were measured. The ratio of the mean L to the mean H of such traces is 1.9, which is very close to the L/H ratio of the observed individual discontinuity planes.

Fig. 3.9 Three possible cases for which the average strike length is about equal to the average dip length: (a) Discontinuities are equidimensional (circular); (b) Discontinuities are nonequidimensional (elliptical), with long axes in a single orientation. The

Characterization of discontinuities in rock

49

discontinuities are oriented so that the average strike length is about equal to the average dip length; and (c) Discontinuities are nonequidimensional (elliptical), with long axes randomly orientated so that the strike length is about equal to the dip length (after Zhang et al., 2002). (e) Roughness Roughness is a measure of the inherent surface unevenness and waveness of the discontinuity relative to its mean plane. The wall roughness of a discontinuity has an important influence on its shear strength, especially in the case of undisplaced and interlocked features such as unfilled joints. The importance of roughness declines with increasing aperture, filling thickness or previous shear displacement. The important influence of roughness on discontinuity shear strength is discussed in detail in Section 4.2. When the properties of discontinuities are being recorded from observations made on either borings cores or exposed faces, it is usual to distinguish between small-scale surface irregularity or unevenness and large-scale undulations or waveness of the surface (see Fig. 3.10). Each of these types of roughness may be quantified on an arbitrary scale of, say, one to five. Descriptive terms may also be used particularly in the preliminary stages of mapping. For example, ISRM (1978) suggests that the terms listed in Table 3.3 and illustrated in Figure 3.11 may be used to describe roughness on two scales—the small scale (several centimeters) and the intermediate scale (several meters). Large-scale waveness may be superimposed on such small- and intermediate-scale roughness.

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Fig. 3.10 Different scales of discontinuity roughness are sampled by different scales of test. Waveness can be characterized by the angle i (after ISRM, 1978). Table 3.3 Classification of discontinuity roughness. Class

Description

I

Rough or irregular, stepped

II

Smooth, stepped

III

Slickensided, stepped

IV

Rough or irregular, undulating

V

Smooth, undulating

VI

Slickensided, undulating

VII

Rough or irregular, planar

VIII

Smooth, planar

IX

Slickensided, planar

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Fig. 3.11 Typical roughness profiles and suggested nomenclature. The length of each profile is in the range of 1 to 10 meters. The vertical and horizontal scales are equal (after ISRM, 1978).

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(f) Aperture Aperture is the perpendicular distance separating the adjacent rock walls of an open discontinuity in which the intervening space is filled with air or water. Aperture is thereby distinguished from the width of a filled discontinuity (see Fig. 3.12). Large apertures can result from shear displacement of discontinuities having appreciable roughness, from outwash of filling materials (e.g. clay), from tensile opening, and/or from solution. In most subsurface rock masses, apertures are small, probably less than half a millimeter. Table 3.4 lists terms describing aperture dimensions suggested by ISRM (1978). Clearly, aperture and its areal variation will have an influence on the shear strength of discontinuities. (g) Filling Filling is a term used to describe material separating the adjacent rock walls of discontinuities, such as calcite, chlorite, clay, silt, fault gourge, breccia, quartz and pyrite. The perpendicular distance between the adjacent rock walls is termed the width of the filled discontinuity, as opposed to the aperture of a gapped or open discontinuity. Filling materials have a major influence on the shear strength of discontinuities. With the exception of discontinuities filled with strong vein materials (calcite, quartz, pyrite), filled discontinuities generally have lower shear strengths than comparable clean, closed discontinuities. The behavior of filled discontinuities depends on many factors of which the following are probably the most important:

Table 3.4 Classification of discontinuity aperture Description

Aperture (mm)

“Closed” features

Very tight Tight Partly open

<0.1 0.1–0.25 0.25–0.5

“Gapped” features

Open Moderately wide Wide

0.5–2.5 2.5–10 >10

“Open” features

Very wide Extremely wide Cavernous

10–100 100–1000 >1000

(1) Mineralogy of filling material (2) Grading or particle size (3) Over-consolidation ratio (4) Water content and permeability (5) Previous shear displacement (6) Wall roughness (7) Width (8) Fracturing, crushing or chemical alteration of wall rock

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Fig. 3.12 Suggested definitions of the aperture of open discontinuities and the width of filled discontinuities (after ISRM, 1978).

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If the filling materials are likely to influence the performance of foundations, samples of the filling materials (undisturbed if possible) should be collected, or an in situ test may be carried out.

3.4 STEREOGRAPHIC PROJECTION Stereographic projection is a procedure for mapping data located on the surface of a sphere on to a horizontal plane, and can be used for analysis of the orientation of planes, lines and forces (Donn & Shimmer, 1958; Phillips, 1971; Goodman, 1976; Hoek & Bray, 1981). There are several types of stereographic projections, but the one most suitable for geological applications is the equal area net, or Lambert projection, which is also used by geographers to represent the spherical shape of the Earth on a flat surface. In structural geology, a point or a line on the sphere representing the dip and dip direction of a discontinuity can be projected on to a horizontal surface. In this way an analysis of threedimensional data can be carried out in two dimensions. 3.4.1 Hemispherical projection of a plane Consider a reference sphere which is oriented in space, usually with respect to true north. When a plane (discontinuity) passes through the center of the reference sphere, the intersection between the plane and the sphere surface is a circle which is called the great circle. A line normal to the plane and passing through the center of the sphere intersects the sphere at two diametrically opposite points called the poles of the plane. Because the great circle and the pole representing the plane appear on both the upper and lower halves of the sphere, only one hemisphere need be used to plot and manipulate structural data. In rock mechanics and rock engineering, the lower hemisphere projection is almost always used. Figure 3.13 shows the lower hemisphere projection of a great circle and its pole onto the horizontal plane passing through the center of the sphere. This is known as the equalangle or Wulff projection. In this projection, any circle on the reference hemisphere projects as a circle on the plane of the projection. This is not the case for an alternative projection known as equal-area or Lambert projection. The latter is better suited than the equal-angle projection for use in the analysis of discontinuity orientation. The equal-area projection for any point on the surface of the reference sphere is accomplished by drawing an arc about the lower end of the vertical axis of the sphere from the point to the horizontal base plane (see Fig. 3.14).

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Fig. 3.13 (a) Stereographic projection of a great circle and its pole from the

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lower reference hemisphere onto the horizontal plane passing through the center of the reference sphere; and (b) Plan view of the equal-angle projection of the great circle and pole of a plane 230/50. Stereographic projections of the great circle and pole of a discontinuity can be prepared by hand by plotting the data on standard sheets (stereonets) with lines representing dip and dip direction values. For the detailed steps, the reader can refer to books on structural geology and rock mechanics such as those by Phillips (1971), Goodman (1976) and Hoek and Bray (1981). Alternatively, there are computer programs such as the one by Mahtab et al. (1972) available that will not only plot great circles and poles, but will also obtain orientation frequency contour plots. It should be noted that the projection technique only examines the orientation of discontinuities and there is no information on their position in space. That is, it is assumed that all the planes pass through the center of the reference sphere. If the stereographic projection identifies a plane on which the drilled shaft foundation could slide, its location on the geological map will have to be examined to determine if it intersects the drilled shaft foundation. 3.4.2 Plotting and analysis of discontinuity orientation data An elementary use of the stereographic projection is the plotting and analysis of field measurements of discontinuity orientation data. In plotting field measurements of dip and dip direction, it is convenient to work with poles rather than great circles since the poles can be plotted directly on a polar stereonet such as that shown in Figure 3.15.

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Fig. 3.14 (a) Vertical section through reference sphere showing lower hemisphere, equal-area projection of a great circle and its pole; and (b) Plan view of the equal-area projection of the great circle and pole of a plane 230/50.

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Fig. 3.15 Polar stereographic net for plotting poles of discontinuity planes. Suppose that a plane has dip direction and dip values 030/40, the pole is located on the stereonet by using the dip direction value of 30 given in bold and then measuring the dip value of 40 from the center of the net along the radial line. Figure 3.16 shows such a plot of the poles of 351 individual discontinuities whose orientations were measured at a particular field site (Hoek & Brown, 1980). Different symbols have been used for three different types of discontinuities—bedding planes, joints and a fault. The fault has a dip direction of 307° and a dip of 56°. Contours of pole concentrations may be drawn for the bedding planes and joints to give an indication of the preferred orientations of the various discontinuity sets present. The essential tool required for pole contouring is a counting net which divides the surface of the reference hemisphere into a number of equal areas. Figure 3.17 shows a counting net containing 100 equal areas for use with the polar stereonet shown in Figure 3.15. The most convenient way of using the counting net is to prepare a transparent overlay of it and to center this overlay on the pole plot by means of a pin through the center of the net. A

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piece of tracing paper is mounted on top of the overlay, pierced by the center pin but fixed by a piece of adhesive tape so that it cannot rotate with respect to the pole plot. Keeping the counting net in a fixed position, the number of poles falling within each counting cell can be counted and noted in pencil on the tracing paper at the center of each cell. The counting net is then rotated to center the densest pole

Fig. 3.16 Plot of poles of 351 discontinuities (after Hoek & Brown, 1980).

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Fig. 3.17 Counting net used in conjunction with the polar stereographic net shown in Figure 3.15.

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Fig. 3.18 Contours of pole concentrations for the data plotted in Figure 3.15 (after Hoek & Brown, 1980). concentrations in counting cells and the maximum percentage pole concentrations are determined. By further small rotations of the counting net, the contours of decreasing percentage which surround the maximum pole concentrations can be established. Figure 3.18 shows the contours of pole concentrations determined in this way from the data shown in Figure 3.16. The central orientations of the two major two joint sets are 347/22 and 352/83, and that of the bedding planes is 232/81. Computer programs such as the one by Mahtab et al. (1972) are also available for plotting and contouring discontinuity orientation data. For a large number of discontinuities, it will be more convenient to use computer programs to plot and contour orientation data. The assignment of poles into discontinuity sets is usually achieved by a combination of contouring, visual examination of the stereonet and a knowledge of geological conditions at the site. However, in many cases visual clustering is very difficult due to the overlap of clusters. A number of algorithms which are based on statistical or fuzzy-set approaches are available for numerically clustering orientation data (Einstein et al., 1979; Miller, 1983; Mahtab & Yegulalp, 1984; Harrison, 1992; Kulatilake, 1993).

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3.5 STATISTICAL ANALYSIS OF DISCONTINUITY PROPERTIES Since discontinuity characteristics are inherently statistical, statistical techniques are widely used in the data reduction, presentation and analysis of discontinuity properties. 3.5.1 Discontinuity orientation As seen in Figure 3.16, there is scatter of the poles of discontinuities when they are plotted on the stereonet. The mean orientation of a number of discontinuities can be calculated from the direction cosines as follows. The sampling bias on orientation can also be considered. The pole of a discontinuity in three-dimensional space can be represented by a unit vector (ux, uy, uz) associated with the direction cosines as shown in Figure 3.19: ux=cos αn cos βn, uy=sin αn cos βn, uz=sin βn (3.8) where αn and βn are respectively the trend and plunge of the pole, which can be obtained by (3.9a)

(3.9b)

The parameter Q is an angle, in degrees that ensures that αn lies in the correct quadrant and in the range of 0 to 360° (see Table 3.5).

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Fig. 3.19 Pole of a discontinuity represented by a unit vector. Tabte 3.5 The quadrant parameter Q in Equation (3.9a). ux

uy

Q

≥0

≥0

0

<0

≥0

180˚

<0

<0

180°

≥0

<0

360˚

The dip direction and dip angle α/β of a discontinuity are related to the trend and plunge αn/βn of its normal by the following expressions: αn=α+180° (for α≤180°) αn=α−180° (for α≥180°) (3.10a) βn=90°−β (3.10b) The mean orientation of a set of discontinuities intersecting a sampling line of trend/plunge αs/βs can be obtained using the procedure outlined below. This procedure corrects for orientation sampling bias through the introduction of weighted direction cosines.

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1. For discontinuity i, calculate the angle δi between its normal and the sampling line:

cos δi=|uxiuxs+uyiuys+uziuzs|

(3.11)

where (uxi, uyi, uzi) and (uxs, uys, uzs) are the direction cosines respectively of the normal to discontinuity i and the sampling line. 2. For discontinuity i, calculate the weighting factor wi based on the angle δi obtained in step 1:

(3.12) 3. After the weighting factor for each discontinuity is obtained, calculate the total weighted sample size Nw for a sample of size N by

(3.13) 4. Calculate the normalized weighting factor wni for each discontinuity by

(3.14) 5. Calculate the corrected direction cosines (nxi, nyi, nzi) for the normal of each discontinuity by

(nxi, nyi, nzi)=wni(uxi, uyi, uzi)

(3.15)

6. Calculate the resultant vector (rx, ry, rz) of the corrected normal vectors (nxi, nyi, nzi), i=1 to N:

(3.16) 7. The mean orientation of the N discontinuities is the orientation of the resultant vector whose trend and plunge can be found by replacing ux, uy and uz by rx, ry and rz in Equation (3.9). Several probability distributions have been suggested in the literature to represent the discontinuity orientations (Shanley & Mahtab, 1976; Zanbak, 1977; Einstein et al., 1979; Kulatilake, 1985a, 1986). The distributions are the hemispherical uniform, hemispherical normal or Fisher, bivariate Fisher, Bingham, bivariate normal and bivariate lognormal. The best means to check if a certain distribution is applicable to represent the orientation of a discontinuity set is to perform goodness-of-fit tests. Shanley and Mahtab (1976) and

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Kulatilake (1985a, 1986) have presented χ2 goodness-of-fit tests respectively for Bingham, hemispherical normal and bivariate normal distributions. Einstein et al. (1979) have tried all the aforementioned distributions to represent the statistical distributions for 22 data sets. They have reported that they could not find a probability distribution which satisfied χ2 test at 5 percent significance level for 18 of these data sets. This shows clearly the inadequacy of the currently available analytical distributions in representing the discontinuity orientations. In the case that no analytical distribution can represent the discontinuity orientation data, empirical distributions can be used. 3.5.2 Discontinuity spacing and frequency Discontinuity spacings usually take a range of values, possibly according to some form of statistical distribution. Priest and Hudson (1976) made measurements on a number of sedimentary rock masses in the United Kingdom and found that, in each case, the discontinuity spacing histogram gave a probability density distribution that could be approximated by the negative exponential distribution. The same conclusion has been reached by others, notably Wallis and King (1980) working on a Precambrian porphyritic granite, and Baecher (1983) working on a variety of igneous, sedimentary and metamorphic rocks. Thus the frequency f(s) of a given discontinuity spacing value s is given by the function f(s)=λe−λs (3.17) where is the mean discontinuity frequency of a large discontinuity population and is the mean spacing. Figure 3.20 shows the discontinuity spacing histogram and the corresponding negative exponential distribution calculated from Equation (3.17) for the Lower Chalk, Chinnor, Oxfordshire, UK (Priest & Hudson, 1976). The use of frequency distributions such as that given by Equation (3.17) permits statistical calculations to be made of such factors as possible block sizes and the likelihood that certain types of intersection will occur. Priest and Hudson (1976) found that an estimate of RQD (see Chapter 2 about the definition of RQD) could be obtained from discontinuity spacing measurements made

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Fig. 3.20 Discontinuity spacing histogram for all scanlines in the first 85 m of tunnel, Chinnor UK (after Priest & Hudson, 1976). on core or exposed rock face using the equation RQD=100e−0.1λ(0.1λ+1) (3.18) For values of λ in the range 6 to 16 m−1, a good approximation to measured RQD values was found to be given by the linear relation RQD=−3.68λ+110.4 (3.19) Figure 3.21 shows the relations obtained by Priest and Hudson (1976) between measured values of RQD and λ, and the values calculated using Equations (3.18) and (3.19). Please note the similarity of Equation (3.19) and Equation (2.2). 3.5.3 Discontinuity trace length In sampling for trace lengths, errors can occur due to the following biases (Baecher & Lanney, 1978; Einstein et al., 1979; Priest & Hudson, 1981; Kulatilake & Wu, 1984c; Mauldon, 1998; Zhang & Einstein, 1998b, 2000a): (1) Orientation bias: the probability of a discontinuity appearing at an exposed rock face depends on the relative orientation between the rock face and the discontinuity. (2) Size bias: large discontinuities are more likely to be sampled than small

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Fig. 3.21 Relationship between RQD and mean discontinuity frequency (after Priest & Hudson, 1976). discontinuities. This bias affects the results in two ways: (a) a larger discontinuity is more likely to appear at an exposed rock face than a smaller one; and (b) a longer trace is more likely to appear in a sampling area than a shorter one. (3) Truncation bias: Very small trace lengths are difficult or sometimes impossible to measure. Therefore, trace lengths below some known cutoff length are not recorded. (4) Censoring bias: Long discontinuity traces may extend beyond the visible exposure so that one end or both ends of the discontinuity traces can not be seen. In inferring the trace length distribution on an infinite surface from the measured trace lengths on a finite size area on this surface, biases (2b), (3) and (4) should be considered. Biases (1) and (2a) are of interest only in three-dimensional simulations of discontinuities, i.e., when inferring discontinuity size distributions as discussed in Section 3.5.4. (a) Probability distribution of measured trace lengths Many investigators have looked into the distribution of trace lengths (Table 3.6). Apart from Baecher et al. (1977), Cruden (1977), Einstein et al. (1979) and Kulatilake (1993) others have based their argument on inspection rather than on goodness-of-fit tests. It seems that only Baecher et al. (1977), Einstein et al. (1979) and Kulatilake (1993) have tried more than one distribution to find the best distribution to represent trace length data. To find the suitable distribution for the measured trace lengths of each discontinuity set, the distribution forms in Table 3.6 can be checked by using χ2 and KolmogorovSmirnov goodness-of-fit tests.

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(b) Corrected mean trace length In inferring the corrected mean trace length (i.e., the mean trace length on an infinite surface), from the measured trace lengths on a finite exposure, biases (2b), (3) and (4) should be considered. Truncation bias (3) can be corrected using the method of

Table 3.6 Distribution forms of trace length. Investigator

Distribution

Robertson (1970)

Exponential

McMahon (1974)

Lognormal

Bridges (1975)

Lognormal

Call et al. (1976)

Exponential

Barton (1977)

Lognormal

Cruden (1977)

Exponential

Baecher et al (1977)

Lognormal

Einstein et al. (1979)

Lognormal

Priest and Hudson (1981)

Exponential

Kulatilake (1993)

Exponential and Gamma (Gamma better)

Warburton (1980a). Decreasing the truncation level in discontinuity surveys can reduce effects of truncation bias on trace length estimates. It is practically feasible to observe and measure trace lengths as low as 10 mm both in the field and from photographs (Priest & Hudson, 1981). Truncation at this level will have only a small effect on the data, particularly if the mean trace length is in the order of meters (Priest & Hudson, 1981; Einstein & Baecher, 1983). Therefore, the effect of truncation bias on trace length estimates can be ignored. However, biases (2b) and (4) are important (Kulatilake & Wu, 1984c) and need be considered. Pahl (1981) suggested a technique to estimate the mean trace length on an infinite surface produced by a discontinuity set whose orientation has a single value, i.e., all discontinuities in the set have the same orientation. His technique is based on the categorization of randomly located discontinuities that intersect a vertical, rectangular planar rock face window of height h and width w, and whose traces make an angle with the vertical, as shown in Figure 3.22. Discontinuities intersecting the sampling window can be divided into three classes: (1) discontinuities with both ends censored, (2) discontinuities with one end censored and one end observable, and (3) discontinuities with both ends observable. If the numbers of traces in each of the above three types are N0, N1 and N2 respectively, the total number of traces, N, will be N=N0+N1+N2 (3.20) Pahl (1981) has derived the following expression for mean trace length µ

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(3.21) Although the approach in Equation (3.21) is both rigorous and easy to implement, it

Fig. 3.22 Discontinuities intersecting a vertical rock face (after Pahl, 1981). does rely on the discontinuities being grouped into a parallel or nearly parallel set. Kulatilake and Wu (1984c) extended Pahl’s technique to discontinuities whose orientation is described by a probabilistic distribution. A major difficulty in applying the extended technique is to determine the probabilistic distribution function of the orientation of discontinuities. Using different methods, Mauldon (1998) and Zhang and Einstein (1998b) independently derived the following expression for estimating the mean trace length on an infinite surface, from the observed trace data in a finite circular window (see Fig. 3.23): (3.22) where c is the radius of the circular sampling window. The major advantage of the method of Mauldon (1998) and Zhang and Einstein (1998b) over the methods of Pahl (1981) and Kulatilake and Wu (1984c) is that it does not need sampling data about the orientation of discontinuities, i.e., the method of Mauldon (1998) and Zhang and Einstein (1998b) is applicable to traces with arbitrary orientation distributions. Therefore, the method of Mauldon (1998) and Zhang and Einstein (1998b) can be used to estimate the mean trace length of more than one set of discontinuities. The orientation distribution-

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free nature of this method comes from the symmetric properties of the circular sampling windows. Trace length measurements are not needed when using Equations (3.21) and (3.22). In the derivation of Equations (3.21) and (3.22), discontinuity trace length l can be anywhere between zero and infinity. Hence, µ obtained by Equations (3.21) and (3.22) does not contain errors due to biases (2b) and (4) as described before. µ in Equations (3.21) and (3.22) is the population (thus correct or true) mean trace length, with N, N0 and N2 being respectively the expected total number of traces intersecting the window, the expected number of traces with both ends censored and the expected number of traces with both ends observable. In practice, the exact values of N, N0 and N2 are not known and thus µ has to be estimated using sampled data. From sampling in one rectangular or circular window, what we get is only one sample of N, N0 and N2 and from this sample only a point estimate of µ can be obtained. For example, for a sample of traces intersecting the sampling window, if and are respectively the numbers of discontinuities that appear on the window with both ends censored and both ends observable, the mean trace length of the sample,

, can be obtained by (3.23) (3.24)

Fig. 3.23 Discontinuities intersecting a circular sampling window (after Zhang & Einstein, 1998b).

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In other words, the of several samples can be used to evaluate µ. When applying Equation (3.23) or (3.24), the following two special cases may occur: , then . In this case, all the discontinuities intersecting the (1) If sampling window have both ends censored. This implies that the area of the window used for the discontinuity survey may be too small. (2) If , then . In this case, all the discontinuities intersecting the sampling window have both ends observable. According to Pahl (1981), this results is due to violation of the assumption that the midpoints of traces are uniformly distributed in the two dimensional space. These two special cases can be addressed by increasing the sampling window size and/or changing the sampling window position (Zhang, 1999). Another method to address these two special cases is to use multiple windows of the same size but at different locations and then use the total numbers from these windows to estimate 1998b).

(Zhang & Einstein,

(c) Trace length distribution on an infinite surface Two probability density functions (pdf) can be defined for trace lengths as follows: (1) f(l)=pdf of trace lengths on an infinite surface. (2) g(l)=pdf of measured trace lengths on a finite exposure subjected to sampling biases. It is necessary to obtain f(l) from g(l), because 3D size distribution of discontinuities is inferred from f(l). Zhang and Einstein (2000a) proposed the following procedure for obtaining f(l): (1) Use the corrected mean trace length µ as the mean value of f(l) (2) Use the coefficient of variation (COV) value of g(l) as the COV of f(l) (3) Find the distribution of g(l) as discussed before and assume that f(l) and g(l) have the same distribution form 3.5.4 Discontinuity size Zhang and Einstein (2000a) presented a method for inferring the discontinuity size distribution from the corrected trace length distribution obtained from circular window sampling as described in Section 3.5.3, based on the stereological relationship between trace lengths and discontinuity diameter distributions for area sampling of discontinuities (Warburton, 1980a): (3.25)

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where D is the diameter of discontinuities; l is the trace length of discontinuities; g(D) is the probability density function of the diameter of discontinuities; f(l) is the probability density function of the trace length of discontinuities; and µD is the mean of the diameter of discontinuities. Villaescusa and Brown (1992) presented a similar method for inferring the discontinuity size distribution from the corrected trace length distribution obtained from straight scanline sampling. They used the following stereological relationship between trace length and discontinuity diameter distributions for straight scanline sampling of discontinuities (Warburton, 1980a): (3.26)

where E(D2) is the mean of D2. Zhang et al. (2002) derived a general stereological relationship between trace length distributions and discontinuity size (expressed by the major axis length a of the ellipse) distributions for area (or window) sampling, following the methodology of Warburton (1980a, b): (3.27)

where (3.28)

in which k is the aspect ratio of the discontinuity, i.e., the length of the discontinuity minor axis is a/k (see Fig. 3.24); β is the angle between the discontinuity major axis and the trace line (note that β is measured in the discontinuity plane). Obviously, β will change for different sampling planes. For a specific sampling plane, however, there will be only one β value for a discontinuity set with a deterministic orientation. When k=1 (i.e., the discontinuities are circular), M=1 and Equation (3.27) reduces to Equation (3.25). Based on Equation (3.27), Zhang et al. (2002) extended the method of Zhang and Einstein (2000a) to elliptical discontinuities. Table 3.7 summarizes the expressions for determining µa and σa from µl and σl, respectively for the lognormal, negative exponential and Gamma distribution of discontinuity size a. Conversely, with known µa and σa, and the distribution form of g(a) the mean µl and standard deviation σl of trace lengths can also be obtained (see Table 3.8). Consider a discontinuity set having a lognormal size distribution with µa=8.0 m and σa=4.0 m (For other distribution forms, similar conclusions can be obtained). Figure 3.25 shows the variation of the mean trace length and the standard deviation of trace lengths with β. Since β is the angle between the trace line and the discontinuity major axis, it is

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related to the sampling plane orientation relative to the discontinuity. It can be seen that, despite the considerable difference between the maximum and the minimum,

Fig. 3.24 Parameters used in the definition of an elliptical discontinuity (after Zhang et al., 2002). respectively, of the mean trace length and the standard deviation of trace lengths, there are extensive ranges of sampling plane orientations, reflected by β, over which both the mean trace length and the standard deviation of trace lengths show little variation, especially for large k values. The results in Figure 3.25 could well explain why Bridges (1976), Einstein et al. (1979) and McMahon (Mostyn & Li, 1993) found different mean trace lengths on differently oriented sampling planes, whereas Robertson (1970) and Barton (1977) observed them to be approximately equal. In each of these papers or reports, the number of differently oriented sampling planes was very limited and, depending on the relative orientations of the sampling planes, the authors could observe either approximately equal mean trace lengths or significantly different mean trace lengths. For example, in Bridges (1976), Einstein et al. (1979) and McMahon (Mostyn & Li, 1993), the strike and dip sampling planes might be respectively in the β=0°–20° (or 160°–180°) range and the β=40°–140° range, or vice versa. From Figure 3.25, this would result in very different mean trace lengths. On the other hand, in Robertson (1970) and Barton (1977), the strike and dip sampling planes might be both in the β=40°–140° range (i.e., in the “flat” trace length part of Fig. 3.25) or respectively in some β ranges approximately symmetrical about β=90°. It should be noted that the comments above are assumptions because no information about the β values can be found in the original papers or reports. The implications of Figure 3.25 about field sampling are as follows: If different sampling planes are used to collect trace (length) data, the sampling planes should be oriented such that significantly different mean

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trace lengths can be obtained from different planes. For example, if two sampling planes are used, one should be oriented in the β=0°–20° (or 160° –180°) range and the other in the β=60°–120° range.

Table 3.7 Expressions for deteimining µa and σa from µl and σl. Distribution form of g(a)

µa

(σa)2

Lognormal

Negative exponential

Gamma

Table 3.8 Expressions for determining µl and σl from µa and σa. Distribution form of g(a) Lognormal

Negative exponential Gamma

µl

(σl)2

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Fig. 3.25 Variation of mean trace length and standard deviation (s.d.) of trace length with β. It is noted that, with the same µl and σl, one can have different µa and σa if the assumed distribution form of g(a) is different. This means that the estimation of discontinuity size distributions from the equations in Table 3.7 may not be robust. To overcome the problem of uniqueness, a relationship between the ratio of the 4th and 1st moments of the discontinuity size distribution and the 3rd moment of the trace length distribution is used to check the suitability of the assumed discontinuity size distribution form: (3.29)

For the three distribution forms of g(a) discussed above, Equation (3.29) can be rewritten as: (a) If g(a) is lognormally distributed with mean µa and standard deviation σa,

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(3.30) (b) If g(a) has a negative exponential distribution with mean µa,

(3.31) (c) If g(a) has a Gamma distribution with mean µa and standard deviation σa,

(3.32) The procedure for inferring the major axis orientation, aspect ratio k and size distribution g(a) (probability density function of the major axis length) of elliptical discontinuities from trace length sampling on different sampling windows is summarized as follows (the reader can refer to Zhang et al., 2002 for details): 1. Sampling (a) Trace length: Use two or more sampling windows at different orientations to conduct trace (length) sampling. The sampling windows (planes) should be oriented such that significantly different mean trace lengths can be obtained from different windows. (b) Orientation: Use exposed rock surface or borehole sampling so that the normal orientation of each discontinuity set can be obtained. 2. Conduct trace length analysis to estimate the true trace length distribution f(l) on different sampling windows: µl, σl and form of f(l). 3. Infer the major axis orientation, aspect ratio k and size distribution g(a) of discontinuities from trace length sampling on different sampling windows: (a) Assume a major axis orientation and compute the β (the angle between discontinuity major axis and trace line) value for each sampling window. (b) For the assumed major axis orientation, compute µa and σa from µl and σl of each sampling window, by assuming aspect ratios k=1, 2, 4, 6, 8 and lognormal, negative exponential and Gamma distribution forms of g(a). The results are then used to draw the curves relating µa (and σa) to k, respectively, for the lognormal, negative exponential and Gamma distribution forms of g(a). (c) Repeat steps (a) and (b) until the curves relating µa (and σa) to k for different sampling windows intersect in one point. The major axis orientation for this case is the inferred actual major axis orientation. The k, µa and σa values at the intersection points are the corresponding possible characteristics of the discontinuities. (d) Find the best distribution form of g(a) by checking the equality of Equation (3.29). The k, µa and σa values found in Step (c) and corresponding to the best distribution form of g(a) are the inferred characteristics of the discontinuity size.

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3.6 FRACTURE TENSOR FOR DESCRIBING DISCONTINUITY GEOMETRY Tensors have been used by several researchers to describe discontinuity geometry including intensity and orientation. Kachanov (1980) introduced a tensor αij to quantify the geometry of microcracks in rocks (3.33)

where V is the volume of the rock mass considered; S(k) is the area of the kth discontinuity; m(V) is the number of discontinuities in volume V; ui(k) and uj(k) (i, j=x, y, z) are components of the unit normal vector of the kth discontinuity with respect to orthogonal reference axes i and j (i, j=x, y, z) respectively (see Fig. 3.19 about the definition of the normal direction of a discontinuity). Oda (1982) also proposed a tensor Fij (called the crack tensor) for describing discontinuity geometry (3.34)

where r(k) is the radius of the kth discontinuity. Kawamoto et al. (1988) regarded discontinuities as damages, and defined a tensor Ωij called the damage tensor (3.35)

where is a characteristic length for a given discontinuity system. The tensors described above are non-dimensional due to some arbitrary operation included in their definitions: in Equation (3.33) the area S(k) of a discontinuity is multiplied by the square root of S(k); in Equation (3.34) the area S(k) of a discontinuity is multiplied by its radius r(k); and in Equation (3.35) a characteristic length for a given discontinuity system is included. Because of the arbitrary operation, the physical meaning of the discontinuity intensity expressed by those definitions is not clear and thus a little confusing (e.g., what is the physical meaning of [S(k)]3/2?). Dershowitz and Herda (1992) and Mauldon (1994) have shown that P32, defined as the mean area of discontinuities per unit volume of the rock mass, is the most useful measure of discontinuity intensity. However, P32 does not include the effect of discontinuity orientations. Zhang (1999) introduced the fracture tensor Fij, which is a combined measure of discontinuity intensity and orientation, defined as follows:

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(3.36)

Fij can be determined with the data obtained in the previous sections. Fracture tensor Fij can also be written in matrix form as follows (3.37)

Fij has three principal values F1, F2 and F3, which can be obtained by finding the eigenvalues of Fij. The principal orientation of Fij can be obtained by finding the eigenvectors corresponding to F1, F2 and F3. The first invariant of Fij is just P32, i.e., (3.38) In contrast to the tensors proposed by Kachanov (1980), Oda (1982) and Kawamoto et al. (1988), the fracture tensor defined in Equation (3.36) has a clear physical meaning. It represents the ratio of the total area of discontinuities and the volume of the rock mass considered. The fracture tensor defined in Equation (3.36) keeps the advantage of P32, i.e., P32 does not depend on the size of the sampled region as long as it is representative of the discontinuity network.

4 Deformability and strength of rock 4.1 INTRODUCTION Discontinuities have a profound effect on the deformability and strength of rock masses. Characterization of a rock mass depends not only on the nature of the rock material, but also on the discontinuities which are pervasive throughout almost all natural rock. The presence of discontinuities has long been recognized as an important factor influencing the mechanical behavior of rock masses. The existence of one or several discontinuity sets in a rock mass creates anisotropy in its response to loading and unloading. Also, compared to intact rock, jointed rock shows reduced shear strength along planes of discontinuity, increased deformability parallel to those planes, and increased deformability and negligible tensile strength in directions normal to those planes. One of the most important concerns in designing foundations in rock is the determination of deformation and strength properties of rock masses. For predicting the ultimate load capacity of a foundation in rock, a strength model of the rock mass is required. Alternatively, if predictions of the foundation movements caused by the applied loading are required, a constitutive (or deformation) model must be selected. With few exceptions, it is incorrect to ignore the presence of discontinuities when modeling rock mass response to loading and unloading. Therefore, it is important to account for the effect of discontinuities on the deformation and strength properties of rock masses. Currently, there are two ways to account for the effect of discontinuities on the deformation and strength properties of jointed rock masses: direct and indirect methods (Amadei & Savage, 1993; Kulatilake, et al. 1992, 1993; Wang & Kulatilake, 1993). Direct Methods Direct methods include laboratory and in situ tests, To obtain realistic results of rock mass deformability and strength, rock of different volumes having a number of different known discontinuity configurations should be tested at relevant stress levels under different stress paths. Such an experimental program is almost impossible to carry out in the laboratory. With in situ tests, such an experimental program would be very difficult, time-consuming and expensive. At the laboratory level, some researchers have performed experiments on model material to study mechanical behavior of jointed rock. Results of laboratory model studies (Brown, 1970a, b; John, 1970; Ladanyi & Archambault, 1970; Einstein & Hirschfeld, 1973; Chappel, 1974; Singh et al., 2002) show that many different failure modes are possible in jointed rock and that the internal distribution of stresses within a jointed rock mass can be highly complex. Since laboratory tests on small scale samples are often inadequate to predict the deformability and strength of rock masses, in situ tests are necessary. There are many types of in situ tests, including uniaxial compression, plate bearing, flat jack, pressure chamber, borehole jacking and dilatometer

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tests. Deformability properties may be estimated from such in situ test data, usually assuming that some idealized model describes the rock behavior in the test configuration. A few in situ tests have been carried out to study the effect of size on rock mass compressive strength (Bieniawski, 1968; Pratt et al., 1972; Bieniawski & Van Heerden, 1975) and on rock mass modulus (Bieniawski, 1978). Heuze (1980) has reviewed the previous work on scale effects on mass deformability and strength. The results of these investigations clearly show the reduction of mass strength and modulus with size up to a certain size at which changes become insignificant. It is important to note that these relations are highly site-dependent, since the scale effect is primarily governed by the discontinuity networks. Chapter 5 will discuss the in situ tests in more detail. Indirect Methods The indirect methods can be divided into the following three approaches: 1. The first indirect approach consists of empirically deducing the deformability and strength properties of rock masses from those measured on intact rock samples in the laboratory. Rock mass modulus and strength can be estimated in different ways. Deere et al. (1967), Coon and Merritt (1970), Gardner (1987) and Zhang and Einstein (2000b) presented correlations between rock quality designation (RQD) and modulus ratio Em/Er, where Em and Er are respectively the rock mass deformation modulus and the intact rock deformation modulus. Bieniawski (1978) and Serafim and Pereira (1983) proposed relationships between the deformation modulus of rock masses and the RMR ratings using the geomechanics classification system (Bieniawski, 1974). Based on practical observations and back analysis of excavation behavior in poor quality rock masses, Hoek and Brown (1997) modified the relation of Serafim and Pereira (1983). Rowe and Armitage (1984) correlated the rock mass modulus deduced from a large number of field tests of drilled shafts under axial loading with the average unconfined compressive strength of weak rock deposits in which the drilled shafts are founded. According to an extensive literature review, Heuze (1980) concluded that the deformation modulus of rock masses ranges between 20 and 60% of the modulus measured on intact rock specimens in the laboratory. Hoek and Brown (1980) proposed an empirical failure criterion for rock masses using two parameters, m and s, which are related to the degree of rock mass fracturing. Empirical expressions have also been proposed between those parameters and RQD (Deere et al., 1967), RMR (Bieniawski, 1974) and Q ratings (Barton et al., 1974). 2. The second indirect approach consists of treating jointed rock mass as an equivalent anisotropic continuum with deformability and strength properties that are directional and reflect the properties of intact rock and those of the discontinuity sets, i.e., orientation, spacing and normal and shear stiffnesses. The discontinuities are characterized without reference to their specific locations. Singh (1973), Kulhawy (1978), Gerrard (1982a, b, 1991), Amadei (1983), Oda et al. (1984), Fossum (1985), Yoshinaka and Yambe (1986), Oda (1988), Chen (1989) and Amadei and Savage (1993) have derived the deformation moduli (or compliances) for rock masses with continuous persistent discontinuities by considering the load-deformation relation for each component (intact rock and discontinuities) and assuming that the behavior of the jointed rock mass is the summation of each component response. Kulatilake et al. (1992, 1993) and Wang (1992) derived relationships between the deformation

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parameters of rock masses with non-persistent discontinuities and fracture tensor parameters based on DEM (discrete element method) analysis results of simulated rock mass blocks. Jaeger (1960) and Jaeger and Cook (1979) presented an equilibrium continuum strength model for jointed rock masses under axisymmetric loading condition. In their model, the strength of both the intact rock and the discontinuities are described by the Coulomb criterion. Since the effect of the intermediate principal stress is not considered in the model of Jaeger (1960) and Jaeger and Cook (1979), Amadei (1988) and Amadei and Savage (1989, 1993) derived solutions for the strength of a jointed rock mass under a variety of multiaxial states of stress. As in the model of Jaeger (1960) and Jaeger and Cook (1979), the modeled rock mass is cut by a single discontinuity set. In the formulations of Amadei (1988) and Amadei and Savage (1989, 1993), however, the intact rock strength is described by the HoekBrown strength criterion and the discontinuity strength is modeled using a Coulomb criterion with a zero tensile strength cut-off. 3. The third indirect approach is to treat discontinuities as discrete features. This is usually done in numerical methods, such as the finite element (Goodman et al., 1968; Ghaboussi et al., 1973; Desai et al., 1984), boundary element (Crouch & Starfield, 1983) and discrete element (Cundall, 1971; Lemos et al., 1985; Lorig et al., 1986; Cundall, 1988; Hart et al., 1988) methods, in which the complex response of discontinuities to normal and shear stresses can be introduced in an explicit manner. The main drawbacks of this approach is that so far, due to computer limitations, only rock masses with a limited number of discontinuities can be analyzed.

4.2 DEFORMABILITY AND STRENGTH OF ROCK DISCONTINUITIES The behavior of jointed rock masses is dominated by the behavior of discontinuities in the rock mass. To consider the effects of discontinuities on the deformability and strength of rock masses, the deformability and strength of rock discontinuities should be known first. 4.2.1 Deformation behavior of rock discontinuities The deformation properties of individual rock discontinuities are described by normal stiffness kn and shear stiffness ks. These refer to the rate of change of normal stress σn and shear stress τ with respect to normal displacement un and shear displacement us. Details about the definition of kn and ks are presented in the following. If a compressive normal stress σn is applied on a rock discontinuity, it would cause its closure by a certain amount, say un. Figure 4.1(a) shows a typical relationship between σn and un. The slope of the curve in Figure 4.1(a) gives the tangential normal stiffness kn of the discontinuity and, at any stress level, is defined as (4.1)

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where ∆ denotes an increment. It is noted that kn is small when σn is small but rapidly builds up as the discontinuity closes. There is actually a limit of discontinuity closure and σn→∞ as this limit (unc) is reached. Goodman et al. (1968) proposed a hyperbolic relationship given by (4.2)

Fig. 4.1 Typical stress-relative displacement relationship: (a) σn versus un; and (b) τ versus us.

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where α and β are constants defining the shape of the hyperbolic curve between σn and un. Differentiating Equation (4.2), we obtain the expression for kn as (4.3)

which can be rewritten as (4.4) It is noted that Equation (4.4) is valid for compressive normal stresses only. It is usual to assume that discontinuities do not offer any resistance to tensile normal stresses implying kn =0 if σn is tensile. If a shear stress τ is applied on the discontinuity, there will be a relative shear displacement us on the discontinuity. Figure 4.1(b) shows a typical relationship between τ and us. It is now possible to define a tangential shear stiffness ks exactly in the same way as was done for the normal stiffness. Thus (4.5) ks is roughly constant till a peak value of the shear stress is reached. Nonlinear values can, however, be adopted if justified by experimental results. It is noted that for discontinuities (especially rough discontinuities), an increment of a shear stress can produce an increment of relative displacement in the normal direction and vice versa an increment of a normal stress can produce an increment of relative displacement in the shear direction. This behavior is called dilation of discontinuities. If the relative shear displacement is broken into two components (along two perpendicular coordinate axes s and t on the discontinuity plane—see Fig. 4.2), the general constitutive relation for a discontinuity including the dilation behavior can be expressed as (4.6)

where the subscripts ‘s’ and ‘t’ represent two orthogonal directions in the discontinuity plane; the subscript ‘n’ represents the direction normal to the discontinuity plane; us and ut are the shear displacements respectively in directions s and t; un is the closure displacement; τs and τt are the shear stresses respectively in directions s and t; σn is the normal stress; and [Cij] (i, j=s, t, n) is the compliance matrix of the discontinuity. Elements of the compliance matrix can be found experimentally by holding two of the stresses constant (for example at zero) and then monitoring the three relative displacement components associated with changes in the third stress component (Priest, 1993).

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For simplicity, the following assumptions are often made for the behavior of a single discontinuity: (1) Deformation behavior is the same in all directions in the discontinuity plane. Thus Css=Ctt, Cst=Cts, Csn=Cta and Cns=Cnt. (2) The dilation (coupling) effect is neglected, i.e., Cij (i≠j) in Equation (4.6) are zero. With the above two assumptions, Equation (4.6) can be simplified to (4.7)

where ks and kn are respectively the discontinuity shear and normal stiffness as described above. 4.2.2 Shear strength of rock discontinuities Discontinuities usually have negligible tensile strength and a shear strength that is, under most circumstances, significantly smaller than that of the surrounding intact rock material. The following describes several shear strength models for rock discontinuities.

Fig. 4.2 A local coordinate system s, t, n.

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(a) Coulomb model The simplest shear strength model of discontinuities is the Coulomb failure criterion. This model can be expressed by the well-known equation (4.8) where τ is the shear strength of the discontinuities; cj and are respectively the cohesion and internal friction angle of the discontinuities; and σ′n is the effective normal stress on the discontinuity plane. It need be noted that the” primes” for cj and for brevity although they are for the effective stress conditions.

have been omitted

(b) Bilinear shear strength model Usually the shear stress-normal stress relation of the discontinuities is non-linear. Patton (1966) addressed this problem by formulating the bilinear model as shown in Figure 4.3. At normal stresses less than or equal to σ′0 the shear strength is given by (4.9) where is the basic friction angle for an apparently smooth surface of the rock material; and i is the effective roughness angle. Barton and Choubey (1977) have listed values of determined experimentally by a number of authors. Some representative values are listed in Table 4.1. At normal stresses greater than or equal to σ′0 the shear strength is given by (4.10) where ca is the apparent cohesion derived from the asperities; and is the residual friction angle of the rock material forming the asperities. Jaeger (1971) proposed the following shear strength model to provide a curved transition between the straight lines of the Patton model (4.11)

Table 4.1 Basic friction angles for different rock materials (after Barton & Choubey, 1977). Rock Types

dry (degrees)

wet (degrees)

Sandstone

26–35

25–34

Siltstone

31–33

27–31

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Limestone

31–37

27–35

Basalt

35–38

31–36

Fine granite

31–35

29–31

Coarse granite

31–35

31–33

Gneiss

26–29

23–26

Slate

25–30

21

Fig. 4.3 Bilinear shear strength models (Equations 4.9 and 4.10) with empirical transitions curve (Equation 4.11). where d is an experimentally determined empirical parameter which controls the shape of the transition curve.

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(c) Barton model A direct, practical approach to predicting the shear strength of discontinuities on the basis of relatively simple measurements was developed by Barton and his coworkers (Barton, 1976; Barton & Choubey, 1977; Barton & Bandis, 1990). According to the Barton model, the shear strength τ of a discontinuity subjected to a normal stress σ′n in a rock material with the basic friction angle

is given by (4.12)

where JRC is the discontinuity roughness coefficient; and JCS is the discontinuity wall compressive strength. The discontinuity roughness coefficient JRC provides an angular measure of the geometrical roughness of the discontinuity surface in the approximate range 0 (smooth) to 20 (very rough). The JRC can be estimated in a number of ways. Barton and Choubey (1977) present a selection of scaled typical roughness profiles (Fig. 4.4), which facilitate the estimation of JRC for real discontinuities by visual matching. Barton (1987) published a table relating Jr (discontinuity roughness number in the Q classification system) to JRC (see Fig. 4.5). Barton and Bandis (1990) suggest that JRC can also be estimated from a simple tilt shear test in which a pair of matching discontinuity surfaces are tilted until one slides over the other. The JRC can be back-figured from the tilt angle α (Fig. 4.6) using the following equation: (4.13)

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Fig. 4.4 Typical discontinuity roughness profiles and associated JRC values (after Barton & Choubey, 1977).

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Fig. 4.5 Relationship between Jr in Qsystem and JRC for 200 mm and 1 m samples (after Barton, 1987). The nail brush is one of the simple methods to record surface profiles. Tse and Cruden (1979) present a method for estimating JRC based on a digitization of the discontinuity surface into a total of M data points spaced at a constant small distance ∆x along the profile. If yi is the amplitude of the ith data point measured above (yi+) and below (yi−) the center line, the root mean square Z2 of the first derivative of the roughness profile is given by

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Fig. 4.6 Tilt test to measure the tilt angle α (after Barton & Bandis, 1990). (4.14)

By digitizing the ten typical roughness profiles presented in Figure 4.4 and then conducting a series of regression analyses, Tse and Cruden (1979) found that there is a strong correlation between JRC and Z2. On this basis, they proposed the following expression for estimating JRC: JRC≈32.2+32.47 log10Z2 (4.15) The increasing availability of image analysis hardware and low-cost digitizing pads makes the method of Tse and Cruden (1979) a valuable objective alternative for the assessment of JRC. This approach should be used with caution, however, since Bandis et al. (1981) have shown that both JRC and JCS reduce with increasing scale. The idea of applying statistical and probabilistic analysis of surface profiles to the calculation of JRC has recently been examined and extended by several authors, notably McWilliams et al. (1990), Roberds et al. (1990), Yu and Vayssade (1990), and Zongqi and Xu (1990). These last authors, noting that the value of JRC is dependent upon the sampling interval along the profile, propose the following extension to Equation (4.15) JRC≈AZ2−B (4.16) where the constants A and B depend on the sampling interval ∆x, taking values of 60.32 and 4.51, respectively, for an interval of 0.25 mm, 61.79 and 3.47 for an interval of 0.5 mm, and 64.22 and 2.31 for an interval of 1.0 mm. Lee et al. (1990), applying the concept of fractals to discontinuity surface profiles, obtained an empirical relation linking the fractal dimension D to the JRC value, as follows:

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(4.17)

Unfortunately Lee et al. (1990) do not explain adequately how the fractal dimension D should be determined in practice. Odling (1994) proposed a method for determining the fractal dimension D. In Odling’s method, the roughness of a fracture surface is represented by the structure function S. For a fracture surface profile, S is defined as (4.18)

where M is the number of data points at a sampling interval ∆x, and yi is the amplitude of the ith data point measured above (yi+) and below (yi−) the center line. The structure function is thus simply the mean square height difference of points on the profile at horizontal separations of ∆x. The structure function is related to the Hurst exponent H (Voss, 1988; Poon et al., 1992): S(∆x)=A(∆x)2H (4.19) Thus, if a log-log plot of S(∆x) versus ∆x gives an acceptably straight line, the slope of this line gives 2H. A is an amplitude parameter and is equivalent to the mean square height difference at a sampling interval of l unit, and is therefore dependent on the units of measurement. From H, the fractal dimension can be determined from the following equation (Voss, 1988): D=E−H (4.20) where E is the Euclidean dimension of embedding medium. E=2 for surface profiles. If the discontinuity is unweathered, JCS is equal to the unconfined compressive strength of the rock material σc, determined by point load index tests or compression tests on cylindrical specimens. If there has been softening or other forms of weathering along the discontinuity, then JCS will be less than σc and must be estimated in some way. Suggested methods for estimating JCS are published by ISRM (1978). Barton and Choubey (1977) explain how the Schmidt hammer index test can be used to estimate JCS from the following empirical expression log10JCS≈0.88γR+1.01 (4.21) where γ is the unit weight of the rock material (MN/m3), R is the rebound number for the L-hammer and JCS has the units MPa in the range 20 to approximately 300 MPa. Although the Schmidt hammer is notoriously unreliable, particularly for heterogeneous materials, it is one of the few methods available for estimating the strength of a surface coating of material.

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Equation (4.12) suggests that there are three factors which control the shear strength of rock discontinuities: the basic friction angle , a geometrical component JRC, and an asperity failure component controlled by the ratio JCS/σ′n. Research results show that both JRC (geometrical component) and JCS (asperity failure component) decrease with increasing scale (Bandis, 1990; Barton & Bandis, 1982) (see Fig. 4.7). Based on extensive testing of discontinuities, discontinuity replicas, and a review of literature, Barton and Bandis (1982) proposed the scale corrections for JRC and JCS: (4.22a)

(4.22b)

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Fig. 4.7 Influence of scale on the three components of the shear strength of a rough discontinuity (after Bandis, 1990; Barton & Bandis, 1990). where JRC0, JCS0 and L0 (length) refer to 100 mm laboratory scale samples and JRCn, JCSn and Ln refer to in situ block sizes.

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It is worth noting two important limitations on the use of Barton model for estimating the shear strength of discontinuities. Barton and Choubey (1977) suggest that the curves should be truncated such that the maximum allowable shear strength for design purposes is given by arctan(τ/σ′n)=70°. For example, curve 1 in Figure 4.8 has a linear “cut-off’ representing the maximum suggested design value of 70° for the total frictional angle. Barton (1976) cautioned that when the effective normal stress exceeds the unconfined compressive strength of the rock material, the measured shear strength is always appreciably higher than that predicted by Equation (4.12). Noting that this discrepancy was probably due to the effect of confining stresses increasing the strength of asperities, Barton

Fig. 4.8 Range of peak shear strength for 136 joints representing eight different rock types. Curves 1, 2 and 3 are evaluated using Equation (4.12) (after Barton & Choubey, 1977).

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proposed that a high stress version of Equation (4.12) could be obtained by replacing JCS by (σ′1−σ′3), i.e., (4.23)

where σ′1 is the effective axial stress required to yield the rock material under an effective confining stress σ′3. The failure stress σ′1 can either be determined experimentally or can be estimated from an appropriate yield criterion such as the Hoek-Brown criterion. (d) Comments The following comments should be noted when using the shear strength criteria described in the previous sections: 1. The Coulomb model is applicable to discontinuities with planar surfaces and the bilinear model to discontinuities with rough surfaces. Since the discontinuity roughness coefficient JRC is incorporated in the strength criterion, the Barton model is applicable to discontinuities with either planar or rough surfaces. 2. The shear strength criteria described in the previous sections are applicable to discontinuities in which rock wall contact occurs over the entire length of the surface under consideration. The shear strength can be reduced drastically when part or all of the surface is not in intimate contact, but covered by soft filling material such as clay gouge.

4.3 DEFORMABILITY OF ROCK MASS 4.3.1 Definition of modulus The modulus relates the change in applied stress to the change in the resulting strain. Mathematically, it is expressed as the slope of a given stress-strain response. Since a rock mass seldom behaves as an ideal linear elastic material, the modulus value is dependent upon the proportion of the stress-strain response considered. Figure 4.9 shows a stressstrain curve typical of an in-situ rock mass containing discontinuities with the various moduli that can be obtained. Although the curve, as shown, is representative of a jointed mass, the curve is also typical of intact rock except that the upper part of the curve tends to be concaved downward at stress levels approaching failure. As can be seen in Figure 4.9 there are at least four portions of the stress-strain curve that can be used for determining in-situ rock mass moduli: the initial tangent modulus, the elastic modulus, the tangent recovery modulus, and the deformation modulus: a. Initial tangent modulus. The initial tangent modulus is determined from the slope of a line constructed tangent to the initial concave upward section of the stress-strain curve (i.e. line 1 in Fig. 4.9). The initial curved section reflects the effects of discontinuity closure in in-situ tests and micro-crack closure in tests on small laboratory specimens.

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b. Elastic modulus. Upon closure of discontinuities/micro-cracks, the stress-strain curve becomes essentially linear. The elastic modulus, frequently referred to as the modulus of elasticity, is derived from the slope of this linear (or near linear) portion of the curve (i.e. line 2 in Fig. 4.9). In some cases, the elastic modulus is derived from the slope of a line constructed tangent to the stress-strain curve at some specified stress level. The stress level is usually specified as 50 percent of the maximum or peak stress. c. Recovery modulus. The recovery modulus is obtained from the slope of a line constructed tangent to the initial segment of the unloading stress-strain curve (i.e. line 3 in Fig. 4.9). As such, the recovery modulus is primarily derived from in-situ tests where test specimens are seldom stressed to failure. d. Deformation modulus. Each of the above moduli is confined to specific regions of the stress-strain curve. The deformation modulus is determined from the slope of the secant line established between zero and some specified stress level (i.e. line 4 in Fig. 4.9). The stress level is usually specified as half of the maximum or peak stress. Since the actual jointed rock masses do not behave elastically, deformation modulus is usually used in practice.

Fig. 4.9 Stress-strain curve typical of in-situ rock mass with various moduli that can be obtained.

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4.3.2 Empirical methods for estimating rock mass deformation modulus A number of empirical methods have been developed that correlate various rock quality indices or classification systems to in-situ deformation modulus of rock masses. The commonly used include correlations between the deformation modulus and RQD, RMR, GSI and Q. The definition of RQD, RMR, GSI and Q and the methods for determining them have been discussed in Chapter 2. (a) Methods relating deformation modulus with RQD Based on field studies at Dworshak Dam, Deere et al. (1967) suggested that RQD be used for determining the rock mass deformation modulus. By adding further data from other sites, Coon and Merritt (1970) developed a relation between RQD and the modulus ratio Em/Er, where Em and Er are the deformation moduli respectively of the rock mass and the intact rock (see Fig. 4.10). Gardner (1987) proposed the following relation for estimating the rock mass deformation modulus Em from the intact rock modulus Er by using a reduction factor αE which accounts for frequency of discontinuities by RQD:

Fig. 4.10 Variation of Em/Er with RQD (after Coon & Merritt, 1970).

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Em=αEEr (4.24a) αE=0.0231(RQD)−1.32≥0.15 (4.24b) This method is adopted by the American Association of State Highway and Transportation Officials in the Standard Specification for HighwayBridges (AASHTO, 1989). For RQD> 57%, Equation (4.24) is the same as the relation of Coon and Merritt (1970). For RQD< 57%, Equation (4.24) gives Em/Er=0.15. It is noted that the RQD−Em/Er relations by Coon and Merritt (1970) and Gardner (1987) have the following limitations (Zhang & Einstein, 2000b): (1) The range of RQD<60% is not covered and only an arbitrary value of Em/Er can be selected in this range. (2) For RQD=100%, Em is assumed to be equal to Er. This is obviously unsafe in design practice because RQD=100% does not mean that the rock is intact. There may be discontinuities in rock masses with RQD=100% and thus Em may be smaller than Er even when RQD=100%. Zhang and Einstein (2000b) added further data collected from the published literature to cover the entire range 0≤RQD≤100% (see Fig. 4.11). It can be seen that the data in Figure 4.11 shows a large scatter, which may be caused by the following three factors (Zhang & Einstein, 2000b): (1) Testing Methods The data in Figure 4.11 were obtained with different testing methods. For example, Deere et al. (1967) used plate load tests while Ebisu et al. (1992) used borehole jacking tests. Different testing methods may give different values of deformation modulus even for the same rock mass. According to Bieniawski (1978), even a single testing method, such as the flat jack test, can lead to a wide scatter in the results even where the rock mass is very uniform. (2) Discontinuity Conditions RQD does not consider the discontinuity conditions, such as the aperture and fillers. However, the discontinuity conditions have a great effect on the rock mass deformation modulus. Figure 4.12 shows the variation of Em/Er with the average discontinuity spacing s for different values of kn/Er using the Kulhawy (1978)model(see Section 4.3.3) It can be seen that kn/Er which represents the discontinuity conditions has a great effect on the rock mass deformation modulus.

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Fig. 4.11 Em/Er−RQD data and proposed Em/Er−RQD relations (after Zhang & Einstein, 2000b). (3) Insensitivity of RQD to Discontinuity Frequency RQD used in Figure 4.11 is defined in terms of the percentage of intact pieces of rock (or discontinuity spacings) greater than a threshold value t of 0.1 m. According to Harrison (1999), the adoption of a threshold value t of 0.1 m leads to the insensitivity of RQD to the change of discontinuity frequency λ or mean discontinuity spacing s. As discussed in Chapter 3, for a negative exponential distribution of discontinuity spacings, the theoretical RQD can be related to the discontinuity frequency λ by Equation (3.18). Figure 4.13 shows the variation of RQD with λ. It can be seen that, for a threshold value t of 0.1 m, when discontinuity frequency λ increases from 1 m−1 to 8 m−1 (i.e., the mean discontinuity spacing s decreases from 1 m to 0.125 m) RQD only decreases from 99.5% to 80.9%, which is a range of only 23%. However, when the mean discontinuity spacing s decreases from 1 m to 0.125 m, the rock mass deformation modulus will vary over a large range. As shown in Figure 4.12, with kn/Er=1, Em/Er changes from 0.5 to 0.11 when s decreases from 1 m to 0.125 m. Harrison

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(1999) showed that the sensitivity of RQD to the mean discontinuity spacing s is closely related to the adopted threshold value t. For example, if a threshold value t of 0.5 m is used, the corresponding RQD will change from 91.0% to 9.2% when A, increases from 1 m−1 to 8 m−1 (see.Fig. 4.13). Considering the data shown in Figure 4.11, Zhang and Einstein (2000b) proposed the following relations between the rock mass deformation modulus and RQD: Lower bound: Em/Em=0.2×100.0186RQD−1.91 (4.25a) Upper bound: Em/Er=1.8×100.0186RQD−1.91 (4.25b)

Fig. 4.12 Variation of Em/Er with average discontinuity spacing s for different values of kn/Er using Kulhawy (1978) model (after Zhang & Einstein, 2000b).

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Mean: Em/Er=100.0186RQD−1.91 (4.25c) The mean relation between Em/Er and RQD was obtained by regression of the data in Figure 4.11. The coefficient of regression, r2, is 0.76. The upper bound could be put somewhat higher but it was selected to be conservative. RQD is a directionally dependent parameter and its value may change significantly, depending on the borehole orientation. Therefore, it is important to know the borehole orientation when estimating the rock mass deformation modulus Em using the Em/Er−RQD relationship. To reduce the directional dependence of RQD, the relationship suggested by Palmstrom (1982) [Equation (2.2) in Chapter 2] can be used to estimate RQD. (b) Methods relating deformation modulus with RMR or GSI The empirical relationship between the RMR rating value and the in situ rock mass modulus is shown in Figure 4.14. Bieniawski (1978) studied seven projects and suggested the following equation to predict rock mass modulus from RMR: Em=2RMR−100 (GPa) (4.26)

Fig. 4.13 RQD—discontinuity frequency relations for threshold values of 0.1 and 0.5 m (after Harrison, 1999 but with different threshold values).

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The obvious deficiency of this equation is that it does not give modulus values for RMR values less than 50. Additional studies carried out on rock masses with qualities ranging from poor to very good indicated that the rock mass modulus could be related to RMR by (Serafim & Pereira, 1983): E=10(RMR−10)/40 (GPa) (4.27) Equation (4.27) has been found to work well for good quality rocks. However, for many of the poor quality rocks it appears to predict deformation modulus values which are too high (Hoek & Brown, 1997). Based on practical observations and back analysis of excavation behavior in poor quality rock masses, Hoek and Brown (1997) modified Equation (4.27) for unconfined compressive strength of intact rock σc<100 MPa as follows: (4.28)

where σc is in the unit of MPa. Note that GSI (Geological Strength Index) has been substituted for RMR in Equation (4.28).

Fig. 4.14 Correlation between in situ deformation modulus and RMR (after Serafim & Pereira, 1983). Johnston et al. (1980) also found that Equation (4.27) overestimates the rock mass modulus for poor quality rocks. They reported that the results of various in situ load tests in moderately weathered Melbourne mudstone of σc in the range 2 to 3 MPa yielded a rock mass modulus of about 0.5 GPa for estimated RMRs of about 70. If we use Equation

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(4.28) with σc=2.5 MPa and GSI=RMR−5=65, we can obtain an Em of 3.7 GPa which is much closer to the measured value of about 0.5 GPa than the value of 31.6 GPa calculated using Equation (4.27). (c) Methods relating deformation modulus with Q Barton et al. (1980) suggested the following relationships between in situ deformation modulus values and Q values: Lower bound: Em=10loge Q (GPa) (4.29) Upper bound: Em=40loge Q (GPa) (4.29) Mean: Em=25loge Q (GPa) (4.29) where Q is the rock quality index as discussed in Chapter 2 (d) Methods relating deformation modulus with σc Rowe and Armitage (1984) correlated the rock mass modulus deduced from a large number of field tests of drilled shafts under axial loading with the average unconfined compressive strength σc of weak rock deposits in which the drilled shafts was founded as follows: (4.30) Radhakrishnan and Leung (1989) found good agreement between the rock mass moduli obtained from back analysis of load-settlement relationship of large diameter drilled shafts in weathered sedimentary rocks and those computed from Equation (4.30). It is interesting to note that Equation (4.30) is equivalent to Equation (4.28) for GSI=23 which corresponds to rock masses of very poor quality. (e) Comments Although the methods of this category are most widely used in practice, there are some limitations: 1. The anisotropy of the rock mass caused by discontinuities is not considered. 2. Different empirical relations often give very different deformation moduli of rock masses at the same site.

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4.3.3 Equivalent continuum approach for estimating rock mass deformation modulus Equivalent continuum approach treats jointed rock mass as an equivalent anisotropic continuum with deformability that reflects the deformation properties of the intact rock and those of the discontinuity sets. (a) Rock mass with persistent discontinuities For rock masses with persistent discontinuities, analytical expressions for their deformation properties have been derived by a number of authors, including Duncan and Goodman (1971), Singh (1973), Kulhawy (1978), Gerrard (1982a, b, 1991), Amadei (1983), Oda et al. (1984), Fossum (1985), Yoshinaka and Yambe (1986), Oda (1988) and Amadei and Savage (1993). The basic idea used by different authors to derive the expressions for deformation properties is essentially the same, i.e., the average stresses are assumed to distribute throughout the rock mass and the overall average strains of the rock mass are contributed by both the intact rock and the discontinuities. The only difference is the method for determining the additional deformation due to the discontinuities. Some of the typical results are presented in the following. The three-dimensional equivalent continuum model presented by Kulhawy (1978) for a rock mass containing three orthogonal discontinuity sets is shown in Figure 4.15. The intact rock material is defined by the Young’s modulus Er and Poisson’s ratio νr, while the discontinuities are described by normal stiffness kn, shear stiffness ks, and mean discontinuity spacing s. For this model, the deformation properties of the equivalent orthotropic elastic mass are given as (4.31)

Fig. 4.15 Rock mass model of Kulhawy (1978).

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(4.32)

(4.33) for i=x, y, z with j=y, z, x and k=z, x, y. These equations describe the rock mass elastic properties completely. The single discontinuity model is a special case of the foregoing in which sx=sy=∞. Singh (1973), Amadei (1983), Chen (1989) and Amadei and Savage (1993) obtained the same expressions as above for deformation properties of rock masses containing three orthogonal discontinuity sets. For engineering convenience, it is useful to define a modulus reduction factor, αE, which represents the ratio of the rock mass to rock material modulus. This factor can be obtained by re-writing Equation (4.31) as (4.34)

The relationship is plotted in Figure 4.16. This figure shows smaller values of αE in rock masses with softer discontinuities (larger Er/kn values). Unfortunately, the mean discontinuity spacing is not easy to obtain directly and, in normal practice, RQD values are determined instead. Using a physical model, the RQD can be correlated with the number of discontinuities per 1.5 meters (5 ft) core run, a common measure in practice. This relationship is shown in Figure 4.17. Combining Figures 4.16 and 4.17 yields Figure 4.18, which relate αE and RQD with Er/kn as an additional parameter.

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Fig. 4.16 Modulus reduction factor versus discontinuity spacing (after Kulhawy, 1978).

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Fig. 4.17 RQD versus number of discontinuities per 1.5m run (after Kulhawy, 1978).

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Fig. 4.18 Modulus reduction factor versus RQD (after Kulhawy, 1978). Consider the jointed rock under uniaxial loading (as shown in Fig. 4.19). The constitutive relation in the n, s, t coordinate system can be defined from the single discontinuity model of Kulhawy (1978). In the global coordinate system x, y, z, the constitutive relation can be determined using second tensor coordinate transformation rules. In matrix form this gives (Amadei & Savage, 1993). (ε)xyz=(A)xyz(σ)xyz (4.35) where and . The components aij=aji (i, j=1−6) of the compliance (A)xyz depend on the dip angle θ as follows: (4.36a)

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(4.36b)

(4.36c)

Fig. 4.19 Jointed rock under uniaxial loading (after Amadei & Savage, 1993). (4.36d)

(4.36e) (4.36f)

(4.36g)

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(4.36h)

(4.36i)

(4.36j) (4.36k)

All other coefficients aij vanish. Note that for the orientation of the discontinuities considered here, the jointed rock has a plane of elastic symmetry normal to the z-axis. If the discontinuity set is inclined with respect to x and z axes or if the rock sample under consideration has two or three orthogonal discontinuity sets, then new expressions must be derived. Gerrard (1982a, b, 1991) presented an approximate method for determining the equivalent elastic properties for a rock mass containing several sets of discontinuities. His analysis is based on the assumption that the strain energy stored in the equivalent continuum is the same as that stored in the discontinuous system. The first step is to rank the various discontinuity sets according to their mechanical significance. Taking the least significant set first, a compliance matrix for the equivalent continuum is determined. This equivalent continuum is then regarded as the anisotropic ‘rock material’ for the next discontinuity set, and so on until all discontinuity sets have been incorporated. A rotation matrix must be applied to transform the equivalent continuum compliance matrix from local coordinate axes, associated with one discontinuity orientation, to axes associated with the next. The models for one, two and three sets of discontinuities are briefly described in the following: 1) A single set of discontinuities can be modeled by considering a system of alternating layers of approximately equal spacing. The interfacing planes are perpendicular to the z axis of the coordinate set x, y, z. Material ‘a’ represents the rock material with thickness Ta, material ‘b’ the discontinuity material with thickness Tb, and material ‘c’ is the homogeneous material equivalent to the system of alternating layers of ‘a’ and ‘b’ (see Fig. 4.20). The properties of material ‘c’ can be determined by using a series of equations which are not listed here because they are too cumbersome (Gerrard, 1982a). 2) A second set of planar parallel discontinuities can be incorporated, in this case the discontinuities being perpendicular to the x-axis. Alternating layers of the equivalent material ‘c’, with thickness Tc, and the discontinuity material ‘d’, with thickness Td, taken together can be represented by the equivalent homogeneous material (see Fig. 4.21). In order that material ‘c’ behaves in an effectively homogeneous fashion when it is incorporated into material ‘e’ it is necessary that Tc»Ta+Tb.

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3) The third set of planar parallel discontinuities are perpendicular to the y-axis. In this case the homogeneous equivalent material ‘g’ can represent the alternating layers of equivalent material ‘e’ with thickness Tc and the discontinuity material ‘f with thickness Tf (see Fig. 4.22). In this case, to ensure that material ‘e’ behaves in an effectively homogeneous fashion when it is incorporated into material ‘g’ it is necessary that Te»Tc+Td. Fossum (1985) derived a constitutive model for a rock mass that contains randomly oriented discontinuities of constant normal stiffness kn and shear stiffness ks. He assumed that if the discontinuities are randomly oriented, the mean discontinuity spacing would be the same in all directions taken through a representative sample of the mass. Arguing that the mechanical properties of the discontinuous mass would be isotropic, Fossum derived the following expressions for the bulk modulus Km and shear modulus Gm of the equivalent elastic continuum: (4.37)

Fig. 4.20 One dimensional system of discontinuities (after Gerrard, 1982a).

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Fig. 4.21 Two dimensional system of discontinuities (after Gerrard, 1982a). (4.38) The equivalent Young’s modulus and Poisson’s ratio can be obtained from (4.39) (4.40)

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Fig. 4.22 Three dimensional system of discontinuities: (a) Representation in the x-y plane; (b) Oblique view of discontinuities (after Gerrard, 1982a). At large values of mean discontinuity spacing s the equivalent modulus Em and Poisson’s ratio νm approach the values Er and νr for the intact rock material. At very small values of mean discontinuity spacing the equivalent modulus Em and Poisson’s ratio νm are given by the following expressions (4.41)

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(4.42) Considering the fact that the available methods do not consider the statistical nature of jointed rock masses, Dershowitz et al. (1979) present a statistically based analytical model to examine rock mass deformability. The statistical model is shown in Figure 4.23. The rock is taken as a three dimensional circular cylinder. Deformation is assumed to accrue both from the elasticity of intact rock and from displacement along discontinuities. Displacements along intersecting discontinuities are assumed to be independent. In this model compatibility of lateral displacements across jointed blocks is approximated by constraining springs. Inputs to the model include stififness and deformation moduli, stress state, and discontinuity geometry. Intact rock deformability is expressed by Young’s modulus Er, set at 200,000 kg/cm2, a typical value. Discontinuity stiffnesses are represented by normal stiffness kn set at 1,000,000 kg/cm3, and shear stiffness ks set at 200,000 kg/cm3. The stress state is described by vertical major principal stress σ1, horizontal “confining” stress σ3. “Confining” stress σ3 is determined from initial stress σ30 and a spring constant kg as follows σ3=σ30+kgδy (4.43) where δy is the calculated horizontal displacement; σ30 is set to 50 kg/cm2; and kg is set at 2500 kg/cm3, a value chosen to maximize the increase of stress with lateral strain without causing rotation of principal planes. Discontinuity geometry is described by three parameters: the mean spacing sm, the mean orientation θm and the dispersion according to the Fisher model κ. Spacing is assumed to follow an exponential distribution and orientation a Fisher distribution (Table 4.2).

Table 4.2 Distribution assumptions for deformation model (after Dershowitz et al., 1979). Discontinuity property

Distribution form

Spacing

Exponential: λe−λs, λ=(mean spacing)−1

Size (Persistence)

Completely persistent

Orientation

Normal stiffness

Deterministic

Shear stiffness

Deterministic

Some of the results are shown in Figures 4.24 to 4.27. The results show that the proposed model is consistent with the results of Deere et al. (1967) and Coon and Merritt (1970)

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(see Fig. 4.10), to the extent that the relationships between deformation and RQD are of similar form. The model proposed by Dershowitz et al. (1979) has the following limitations: 1) The analysis applies only to “hard” rock. Shears and weathering can only be accommodated through changes in discontinuity stiffnesses, which is inadequate. 2) The analysis is for infinitesimal strains. Finite strains would violate the assumption of independence among discontinuity displacements. 3) The analysis is for a homogeneous deterministic stress field specified extraneous to the discontinuity pattern. Real rock masses may have complex stress distributions strongly influenced by the actual jointing pattern. 4) Boundary conditions are highly idealized.

Fig. 4.23 Statistical model for jointed rock (after Dershowitz et al., 1979).

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Fig. 4.24 Relationship between Em/Er and RQD, parallel discontinuities (after Dershowitz et al., 1979). (b) Rock mass with non-persistent discontinuities For rock masses with non-persistent discontinuities, relationships between the deformation properties and the fracture tensor parameters in two and three dimensions have been derived by Kulatilake et al. (1992, 1993) and Wang (1992) from the discrete element method (DEM) analysis results of generated rock mass blocks. The procedure used to evaluate the effect of discontinuities and the obtained relationships between the deformation properties and the fracture tensor parameters in three dimensions are outlined in the following.

Fig. 4.25 Relationship between the mean of Em/Er, E[Em/Er], and the mean of RQD, E[RQD], subparallel discontinuities distributed according to Fisher (after Dershowitz et al., 1979).

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Fig. 4.26 Relationship between the standard deviation of Em/Er, SD[Em/Er], and the mean of RQD, E[RQD], subparallel discontinuities distributed according to Fisher (after Dershowitz et al., 1979). The procedure for evaluating the effect of discontinuities on the deformability of rock masses is shown in Figure 4.28. The first step is the generation of non-persistent discontinuities in 2 m cubical rock blocks. The discontinuities were generated in a systematic fashion as follows:

Fig. 4.27 Effect of stiffness values on modulus ratio Em/En parallel discontinuities (after Dershowitz et al., 1979).

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1) In each rock block, a certain number of discontinuities having a selected orientation and a selected discontinuity size were placed to represent a discontinuity set. 2) Discontinuities were considered as 2D circular discs. 3) Discontinuity center locations were generated according to a uniform distribution. 4) Either a single discontinuity set or two discontinuity sets were included in each rock block. The generated discontinuity networks in the rock blocks are given in Table 4.3.

Table 4.3. Generated discontinuity networks of actual discontinuities in the rock block for 3D DEM analysis (after Kulatilake et al., 1992, 1993). Number of Orientation discontinuity sets α/β

Discontinuity size/ block size

Number of discontinuities

Discontinuity location

One set

60°/45°

0.1–0.9 with step 0.1

5, 10, 20, 30

Uniform

94.42°/37.89°

0.3, 0.5, 0.6, 0.7, 0.9

5, 10, 20, 30

distribution

30°/45°

0.3, 0.5, 0.6, 0.7, 0.8, 0.9

5, 10, 20,

90°/45°

0.3, 0.5, 0.6, 0.7, 0.8, 0.9

5, 10, 20

68.2°/72.2°

0.3, 0.6, 0.7, 0.8

5, 10, 20, 30

248.9°/79.8°

0.3, 0.6, 0.7,0.8

5, 10, 20, 30

60°/45°

0.1, 0.2, 0.3, 0.5, 0.6, 0.7

10

Two sets

240°/60°

10

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Fig. 4.28 Procedure for evaluating the effect of discontinuity geometry parameters on the deformability properties of jointed rock (after Kulatilake et al., 1993). The second step is the generation of fictitious discontinuities according to the actual nonpersistent discontinuity network generated in the rock block. In order to use the DEM for 3D analyses of a generated rock block, the block should be discretized into polyhedra. Since a typical non-persistent discontinuity network in 3D may not discretize the block into polyhedra, it is necessary to create some type of fictitious discontinuities so that when they are combined with actual discontinuities, the block was discretized into polyhedra. Before the generation of fictitious discontinuities, the actual disc-shaped discontinuities are converted into square-shaped ones having the same area. In order for the fictitious discontinuities to simulate the intact rock behavior, an appropriate constitutive model and associated parameter values for the fictitious discontinuities have to be found. From the investigation performed on 2D rock blocks, Kulatilake et al. (1992) found that by choosing the mechanical properties of the fictitious discontinuities in the

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way given below, it is possible to make the fictitious discontinuities behave as the intact rock:

Table 4.4. Values for the mechanical parameters of intact rock, actual and fictitious discontinuities used by Kulatilake et al. (1992, 1993) and Wang (1992). Intact rock or Discontinuities

Parameter

Assigned value

Intact rock

Young’s modulus Er

60 GPa

Poisson’s ratio νr

0.25

Cohesion cr

50 MPa

Tensile strength tr

10 MPa

Friction coefficient

0.839

Normal stiffness kn

5000 GPa/m

Shear stiffness ks

2000 GPa/m

Cohesion cj

50MPa

Dilation coefficient dj

0

Tensile strength tj

10 MPa

Fictitious discontinuities

Friction coefficient Actual discontinuities

0.839

Normal stiffness kn

67.2 GPa/m

Shear stiffness ks

2.7 GPa/m

Cohesion cj

0.4 MPa

Tensile strength tj

0

Friction coefficient

0.654

(a) The strength parameters of the fictitious discontinuities are the same as those of the intact rock. (b) Gr/ks=0.008–0.012. (c) kn/ks=2–3, with the most appropriate value being Er/Gr. For the intact rock (granitic gneiss) studied by Kulatilake et al. (1992, 1993) and Wang (1992), the approximate parameters of the fictitious discontinuities are shown in Table 4.4. The mechanical parameters of the actual discontinuities used by them are also shown in this table. The constitutive models used for the intact rock and discontinuities (both actual and fictitious) are shown in Figures 4.29 and 4.30, respectively. The third step is the DEM analysis of the rock block (using the 3D distinct element code 3DEC) under different stress paths and the evaluation of the effect of discontinuities on the deformation parameters of the rock mass. In order to estimate different property

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values of the jointed rock block, Kulatilake et al. (1993) and Wang (1992) used the following stress paths: 1) The rock block was first subjected to an isotropic compressive stress of 5 MPa in three perpendicular directions (x, y, z); then, for each of the three directions, e.g. the zdirection, the compressive stress σz was increased, while keeping the confining stresses in the other two directions (σx and σy) the same, until the failure of the rock occurred (see Fig. 4.31). From these analysis results, it is possible to estimate the deformation modulus of the rock block in each of the three directions and the related Poisson’s ratios.

Fig. 4.29 Constitutive model assumed for intact rock: (a) stress versus strain; (b) Coulomb failure criterion with a tension cut-off (after Kulatilake et al., 1993).

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2) The rock block was first subjected to an isotropic compressive stress of 5 MPa in three perpendicular directions (x, y, z); then, on each of the three perpendicular planes, e.g. the x-y plane, the rock was subjected to an increasing shear stress as shown in Figure 4.32. These analysis results can be used to estimate the shear modulus of the rock block on each of the three perpendicular planes. In the DEM analysis, during the loading process, displacements were recorded simultaneously on each block face in the direction(s) needed to calculate the required block strains. On each block face, five points were selected to record the displacement. The average value of these five displacements was considered as the mean displacement of this face for block strain calculations. To make it possible to estimate the deformation properties of the rock block from the DEM analysis results, Kulatilake et al. (1993) and Wang (1992) assumed that the rock block was orthotropic in the x, y, z directions, regardless of the actual orientations of the discontinuities, i.e., With the above constitutive model, the deformation moduli Ex, Ey, Ez and Poisson’s ratios νxy, νxz, vyx, νyz, νzx, vzy can be estimated from the DEM analysis results of rock blocks under stress path 1 (Fig. 4.31). The shear moduli Gxy, Gxz and Gyz can be estimated from the DEM analysis results of rock blocks under stress path 2 (Fig. 4.32). To reflect the effect of discontinuity geometry parameters on the deformation properties, Kulatilake, et al. (1993) and Wang (1992) used the fracture tensor defined by Oda (1982) as an overall measure of the discontinuity parameters—discontinuity density, orientation, size and the number of discontinuity sets. For thin circular discontinuities, the general form of the fracture tensor at the 3D level for the kth discontinuity set can be expressed as (see also Chapter 3 about the discussion of fracture tensors) (4.45) where ρ is the average number of discontinuities per unit volume (discontinuity density), r is the radius of the circular discontinuity (discontinuity size), n is the unit vector normal to the discontinuity plane, f(n, r) is the discontinuity probability density function of n and r, Ω/2 is a solid angle corresponding to the surface of a unit hemisphere, and ni and nj (i, j=x, y, z) are the components of vector n in the rectangular coordinate system considered (see Fig. 4.33). The solid angle dΩ is also shown in Figure 4.33. If the distributions of the size and the orientation of the discontinuities are independent of each other, Equation

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Fig. 4.30 Constitutive model assumed for joints: (a) normal stress versus normal displacement; (b) shear stress versus shear displacement; and (c) Coulomb failure criterion with a tension cut-off (after Kulatilakeetal., 1993).

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Fig. 4.31 Stress paths of first type used to perform DEM analysis of generated rock blocks (after Kulatilakeetal., 1993).

Fig. 4.32 Stress paths of second type used to perform DEM analysis of generated rock blocks (after Kulatilakeetal, 1993).

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(4.44)

(4.45) can be rewritten as follows (4.46) where f(n) and f(r) are the probability density functions of the unit normal vector n and size r, respectively. If there are more than one discontinuity set in the rock mass, the fracture tensor for the rock mass can be obtained by (4.47) wher N is the number of discontinuity sets in the rock mass. Fracture tensor Fij can also be written in matrix form as follows (4.48)

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Fig. 4.33 Unit sphere used to define the solid angle dΩ (after Oda, 1982). Since the diagonal components of the fracture tensor Fxx, Fyy and Fzz express the combined effect of discontinuity density and discontinuity size in the x, y and z directions, respectively, Kulatilake et al. (1993) and Wang (1992) showed the obtained deformation properties as in Figures 4.34 and 4.35. Putting the data in Figures 4.34(a)–(c) and Figures 4.35(a)–(c) respectively together, Figures 4.36 and 4.37 were obtained, which show that the deformation properties of jointed rock masses are related to the corresponding components of the fracture tensor. As for the Poisson’s ratios of the generated rock blocks, Kulatilake et al. (1993) and Wang (1992) found that they were between 50 and 190% of the intact rock Poisson’s ratio. (c) Comments In the equivalent continuum approach, the elastic properties of the equivalent material are essentially derived by examining the behavior of two rock blocks having the same volume and by using an averaging process. One volume is a representative sample of the rock

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Fig. 4.34 Relations between rock block deformation moduli and fracture tensor components for different discontinuity networks: (a) Ez/Er vs Fzz; (b) Ey/Er vs Fyy; and (c) Ex/Er vs Fxx (after Kulatilake et al., 1993).

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Fig. 4.35 Relations between rock block shear moduli and summation of corresponding fracture tensor components for different discontinuity networks: (a) Gxy/Gr vs (Fxx+Fyy); (b) Gxz/Gr vs (Fxx+Fzz); and (c) Gyz/Gr vs (Fyy+Fzz) (after Kulatilake et al., 1993).

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Fig. 4.36 Relations between rock block deformation modulus in any direction Em and the fracture tensor components in the same direction (after Kulatilake et al., 1993).

Fig. 4.37 Relations between rock block shear modulus on any plane Gm and the summation of fracture tensor components on that plane (after Kulatilake et al., 1993).

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mass whereas the second volume is cut from the equivalent continuum and is subject to homogeneous (average) stresses and strains. Therefore, the equivalent continuum approach requires that the representative sample of the rock mass be large enough to contain a large number of discontinuities. On the other hand, the corresponding equivalent continuum volume must also be sufficiently small to make negligible stress and strain variations across it. This leads to a dilemma which is typical in modeling continuous or discontinuous composite media. Numerous authors have used the equivalent continuum approach and derived the expressions for the equivalent continuum deformation properties. Most of these expressions are based on the assumption that the discontinuities are persistent. This is a conservative assumption since, in reality, most of the discontinuities are non-persistent with finite size. For a rock mass containing non-persistent discontinuities, Kulatilake et al. (1992, 1993) and Wang (1992) derived relationships between the deformation properties and the fracture tensor parameters from the DEM analysis results of generated rock mass blocks. However, there exist limitations for the method they used and thus for the relationships they derived as follows: 1. The generated rock mass block is assumed to be orthotropic in the x, y, z directions, regardless of the actual orientations of the discontinuities. The appropriateness of this assumption is questionable. For example, the two blocks shown in Figure 4.38 have the same fracture tensor Fij, block 1 containing three orthogonal discontinuity sets while block 2 containing 1 discontinuity set. It is appropriate to assume that block 1 is orthotropic in the x, y, z directions. However, it is obviously inappropriate to assume that block 2 is orthotropic in the x, y, z directions. 2. To do DEM analysis on the generated rock mass block, fictitious discontinuities are introduced so that when they are combined with actual discontinuities, the block is discretized into polyhedra. To make the fictitious discontinuities behave as the intact rock, appropriate mechanical properties have to be assigned to the fictitious discontinuities. From the investigation performed on 2D rock blocks, Kulatilake et al. (1992) found a relationship between the mechanical properties of the fictitious discontinuities and those of the intact rock. However, even if the mechanical properties of the fictitious discontinuities are chosen from this relationship, the fictitious discontinuities can only approximately behave as the intact rock. So the introduction of fictitious discontinuities brings further errors to the final analysis results. 3. Discontinuity persistence ratio PR (defined as the ratio of the actual area of a discontinuity to the cross-section area of the discontinuity plane with the rock block) should have a great effect on the deformability of rock masses. However, the relationships derived by Kulatilake et al. (1992, 1993) and Wang (1992) does not show any effect of PR on the deformability of jointed rock masses. 4. The conclusion that Ei/Er (i=x,y,z) is related only to Fii (i=x,y,z) is questionable. This can be clearly seen from the two rock blocks shown in Figure 4.39. The two blocks have the same fracture tensor component Fzz. From Figure 4.36, the two blocks will have the same deformation modulus in the z-direction. However, block 2 is obviously more deformable than block 1 in the z-direction.

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Fig. 4.38 Two rock blocks having the same fracture tensor but different joint sets: (a) Rock block with three orthogonal joint sets; and (b) Rock block with one joint set.

Fig. 4.39 Two rock blocks having the same fracture tensor component in zdirection but different joint orientations: (a) Rock block with joint normal parallel to z-axis; and (b) Rock block with joint normal inclined from z-axis.

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4.3.4 Direct consideration of discontinuities in numerical analysis The direct consideration of discontinuities as discrete features is usually done in numerical methods, such as the finite element, boundary element and discrete element methods. Considering the fact that the finite element method (FEM) is the most widely used numerical method in foundation analysis and design, only the methods for representing discontinuities in finite element modeling are described in the following. The presence of rock discontinuities is considered in finite element analyses by employing special joint elements which describe the localized response of the discontinuities. The various joint elements can be grouped into three general classes (Curran & Ofoegbu, 1993): (1) joint elements which use the nodal displacements as the independent degrees of freedom (Goodman et al., 1968); (2) joint elements which use the relative nodal displacements as the independent degrees of freedom (Ghaboussi et al., 1973); (3) thin-layer continuum elements assigned the behavior of discontinuities (Zienkiewicz et al., 1970; Desai et al., 1984). These classes of joint elements are briefly discussed below. (a) Joint elements using nodal displacements as independent degrees-offreedom This approach to modeling discontinuities was originally proposed by Goodman et al. (1968) and is still commonly used today. A good summary of the development of the element equations is given in Pande et al. (1990). The basic geometry of the element, for 2D problems, is illustrated in Figure 4.40(a). It has a length L (along the s-axis, i.e. the discontinuity plane) and zero thickness (in the n-axis direction, i.e. normal to the discontinuity plane). It is a four-node element, nodes 1 and 2 lying on the bottom surface (subscript B), while nodes 3 and 4 lie on the top surface (subscript T). The relative displacements ws (shear) and wn (normal) are given by (4.49)

where the absolute displacements in s and n directions are denoted as u and ν, respectively. Assuming that displacements vary linearly along each boundary and with nodes 1 and 4 at s=−L/2 and nodes 2 and 3 at s=L/2 [Fig. 4.40(a)], the displacements at the bottom and top boundaries are respectively given by (4.50)

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(4.51)

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Fig. 4.40 Joint elements based on: (a) nodal displacements; (b) relative nodal displacements; and (c) thin-layer solid. Substituting Equations (4.50) and (4.51) into Equation (4.49), the following can be obtained

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(4.52)

where α=1−2s/L, β=1+2s/L and the vector on the right-hand side is the node displacement vector. For the Goodman model, the vector of the nodal force F is related to the relative displacements w through the equation (4.53)

where Ks and Kn are the shear and normal stiffness, respectively. Using the minimum energy principle, the equilibrium equation for the element can be obtained in the form Ku=F (4.54) where u is the vector of the nodal displacements in Equation (4.52), and (4.55)

The element’s contribution Kg to the global stiffness matrix is given by (4.56)

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where θ is the angle measured anti-clockwise from the discontinuity local s-axis to the global x-axis. The following remarks can be made about Goodman’s joint element: 1) In the derivation above, the properties of discontinuities are assumed to be represented by stiffness of discontinuities Ks and Kn, The stiffness matrix of the discontinuity

has no off-diagonal terms. This implies that there is no dilatancy of discontinuities and the normal and shear behavior are uncoupled. 2) It is possible to formulate higher-order joint elements on the basis of Goodman’s joint element. A procedure of numerical integration will have to be adopted as direct integration is quite cumbersome. 3) Mehtab and Goodman (1970) have extended the formulation of the joint element suitable for three-dimensional analysis. The joint element is a two-dimensional eightnode quadrilateral with the nodes in the thickness direction being coincident. (b) Joint elements using relative nodal displacements as independent degrees-of-freedom In this model, introduced by Ghaboussi et al. (1973), the joint element has a finite thickness t and its degrees of freedom are the relative displacements ws and wn, which vary linearly from s=−L/2 to s=L/2 [L is the length of the joint element and the local coordinate system in Fig. 4.40(b) is in effect]. The joint strains εs (shear) and εn (normal) are given by (4.57) Hence, the strain can be related to the relative displacements of the element as follows (4.58)

where subscripts 1 and 2 stand for the two ends of the element, end 1 being at s=−L/2 and end 2 at s=L/2. The stress-strain relation for the discontinuity is given by (4.59)

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where σs and σn are the shear and normal stresses respectively, and the 2×2 matrix D represents the discontinuity stiffness. The stiffness matrix K for the joint element in the local coordinate system is given by K=∫BTDBdv (4.60) where B is the 2×4 matrix in Equation (4.58). The element’s contribution Kg to the global stiffness matrix can be obtained using Equation (4.56). Wilson (1977) further developed the technique of using relative displacements for the joint element, including the expansion from two dimensions to three dimensions. (c) Thin-layer elements The two classes of joint elements described above differ from solid elements in some fundamental ways, such as structural stiffness matrix, nature of the stress and strain vectors and the strain-displacement relations. Because of these differences, their incorporation into a regular finite element program (which is usually designed for solid elements) requires significant modifications in the code. Desai et al. (1984) proposed the thin-layer element as a means of reducing this problem. The thin-layer element is basically a solid element, but its properties are assigned in such way that its behavior closely approximate that of a discontinuity. A typical thinlayer element is shown in Figure 4.40(c). This example is a six-node element but a fournode element is acceptable. The stress-strain relations are derived in exactly the same way as for other solid elements. The main issue for using thin-layer elements in FEM analysis is choosing appropriate material properties and thickness of the element. Desai et al. (1984) originally proposed the thin-layer element mainly for applications in soil-structure interaction problems. Since the interface is surrounded by the geological (soil) and structural materials, Desai et al. (1984) proposed that the normal stiffness (i.e., deformation modulus in the direction normal to the element plane) of the thin-layer element be chosen according to the properties of the interface zone and the structural and geological materials, i.e., En=λ1(En)i+λ2(En)g+λ3(En)st (4.61) where (En)i, (En)g and (En)st are respectively the deformation modulus of the interface zone and the geological and structural materials; and λ1, λ2 and λ3 are the participation factors varying from 0 to 1. In a series of soil-structure interaction examples, Desai and his coworkers chose λ2=λ3=0 and λ1=1 and obtained satisfactory results by assigning the interface zone the same properties as the geological material. Desai et al. (1984) proposed using shear testing devices [Fig. 4.41(a)] to obtain the shear modulus of the thin-layer element. The expression used for obtaining a tangent shear modulus is given by (4.62)

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where t is the thickness of the element [Fig. 4.41(b)] and us is the relative displacement.

Fig. 4.41 Behavior at interface: (a) schematic of direct shear test; and (b) deformation at interface. Fishman et al. (1991) used thin-layer elements for modeling rock discontinuities. Arguing that the discontinuity interface is smooth and thus the normal deformation will be small, they chose the Young’s modulus for the thin layer elements to be 10 times higher than that of the surrounding solid rock elements. As for the shear modulus, they used the method suggested by Desai et al. (1984) as described above. The thickness of the thin-layer element has a great effect on the quality of simulation of the interface (discontinuity) behavior. If the thickness is too large in comparison with the dimension of the surrounding elements, the thin-layer element will behave essentially as a solid element. If it is too small, computational difficulties such as numerical illconditioning may arise. Investigations by Desai et al. (1984) suggest that values of t/B (B is the smaller of the other two dimensions of the element) in the range 0.01≤t/B≤0.1 are likely to give good results. In applying thin-layer elements for numerical modeling jointed rock masses, Fishman et al. (1991) used thin-layer elements with t/B=0.018−0.054 and got satisfactory results.

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(d) Comments Special joint elements have been used widely in the area of soil/rock and structure interaction. The first two classes of joint elements as discussed in (a) and (b) differ from solid elements in some fundamental ways, such as structural stiffness matrix, nature of the stress and strain vectors and the strain-displacement relations. However, thin-layer element is basically a solid element with a small thickness and a particular constitutive relationship. Investigations by Ng et al. (1997) revealed that all these joint elements have limitations, such as the problems of numerical ill-conditioning: if the joint elements have a large aspect ratio (ratio of length to thickness), small values of the coefficients in the diagonal of the stiffness matrices can create problems in the solution routine with a loss in accuracy. It happens very often that there is filling in rock discontinuities. Since the filling itself is physically a solid, it is obviously more appropriate to use thin-layer elements than to use the first two classes of joint elements to represent them in the FEM analysis. To use thin-layer elements to represent discontinuities in the FEM analysis, appropriate mechanical properties should be assigned to them. For 2D thin-layer elements, Desai et al. (1984) proposed a procedure for determining the shear modulus and gave a general idea (no detailed procedure) of evaluating the normal deformation modulus. Since, to date, thin-layer elements have been used basically in 2D problems, no detailed suggestions about selecting the properties of 3D thin-layer element are available.

4.4 STRENGTH OF ROCK MASS 4.4.1 Empirical strength criteria for rock mass Several empirical strength criteria for rock masses have been formulated based on largescale testing and/or application experience and analysis. In the following, four typical empirical strength criteria are described and discussed. Since the Hoek-Brown criterion is the mostly widely used one, it is described and discussed in more details than the others. (a) Hoek-Brown criterion The Hoek-Brown criterion was originally published in 1980 (Hoek & Brown, 1980) and has evolved to being used under conditions which were not visualized when it was originally developed. For intact rock, the Hoek-Brown criterion may be expressed in the following form (4.63)

where σc is the uniaxial compressive strength of the intact rock material; σ′1 and σ′3 are respectively the major and minor effective principal stresses; and mi is a material constant for the intact rock. mi depends only upon the rock type (texture and mineralogy) as tabulated in Table 4.5.

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For jointed rock masses, the most general form of the Hoek-Brown criterion, which incorporates both the original and the modified form, is given by (4.64)

Table 4.5 Values of parameter mi for a range of rock types (after Hoek & Brown, 1997). Rock type

Class

Group

Clastic

Texture Coarse

Medium

Fine

Very fine

Conglomerate (22)

Sandstone 19

Siltstone 9

Claystone 4

Greywacke (18) Chalk 7

Organic NonClastic

Carbonate

Coal (8–21) Breccia (20)

Sparitic Limestone (10)

Micritic Limestone 8

Gypstone 16

Anhydrite 13

Marble 9

Hornfels (19)

Quartzite 24

Migmatite (30)

Amphibolite 25–31

Mylonites (6)

Gneiss 33

Schists 4–8

Phyllites (10)

Slate 9

Rhyolite (16) Dacite (17) Andesite 19

Obsidian (19)

Chemical Non-foliated Slightly foliated Foliated*

Light

Granite 33 Granodiorite (30) Diorite (28)

Dark

Gabbro 27 Norite 22

Dolerite (19)

Basalt (17)

Agglomerate (20)

Breccia (18)

Tuff (15)

Extrusive pyroclastic type

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*

These values are for intact rock specimen tests normal to bedding or foliation. The value of mi will be significantly different if failure occurs along a weakness plane.

where mb is the material constant for the rock mass; and s and a are constants that depend on the characteristics of the rock mass. The original criterion has been found to work well for most rocks of good to reasonable quality in which the rock mass strength is controlled by tightly interlocking angular rock pieces. The failure of such rock masses can be defined by setting a=0.5 in Equation (4.64), giving (4.65)

For poor quality rock masses in which the tight interlocking has been partially destroyed by shearing or weathering, the rock mass has no tensile strength or ‘cohesion’ and specimens will fall apart without confinement. For such rock masses the following modified criterion is more appropriate and it is obtained by putting s=0 in Equation (4.64) which gives (4.66)

Equations (4.64) to (4.66) are of no practical value unless the values of the material constants mb, s and a can be estimated in some way. Hoek and Brown (1988) proposed a set of relations between the parameters mb, s and a and the 1976 version of Bieniawski’s Rock Mass Rating (RMR), assuming completely dry conditions and a very favorable (according to RMR rating system) discontinuity orientation: (i) disturbed rock masses

(4.67a) (4.67b) a=0.5

(4.67c)

(ii) undisturbed or interlocking rock masses

(4.68a)

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(4.68b) a=0.5

(4.68c)

Equations (4.67) and (4.68) are acceptable for rock masses with RMR values of more than about 25, but they do not work for very poor rock masses since the minimum value which RMR can assume is 18 for the 1976 RMR system and 23 for the 1989 RMR system (see Chapter 2 for details). In order to overcome this limitation, Hoek (1994) and Hoek et al. (1995) introduced the Geological Strength Index (GSI). The relationships between mb, s and a and the Geological Strength Index (GSI) are as follows: (i) For GSI>25, i.e. rock masses of good to reasonable quality

(4.69a) (4.69b) a=0.5

(4.69c)

(ii) For GSI<25, i.e. rock masses of very poor quality

(4.70a) s=0

(4.70b) (4.70c)

It is noted that the distinction between disturbed and undisturbed rock masses is dropped in evaluating the parameters mb, s and a from GSI. This is based on the fact that disturbance is generally induced by engineering activities and should be allowed by downgrading the values of GSI. The methods for determining RMR and GSI have been discussed in Chapter 2. Since many of the numerical models and limit equilibrium analyses used in rock mechanics are expressed in terms of the Coulomb failure criterion, it is necessary to estimate an equivalent set of cohesion and friction parameters for given Hoek-Brown values. This can be done using a solution published by Balmer (1952) in which the normal and shear stresses are expressed in terms of the corresponding principal stresses as follows:

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(4.71)

(4.72)

For the GSI>25, when a=0.5: (4.73) For the GSI<25, when s=0: (4.74)

Once a set of (σ′n, τ) values have been calculated from Equations (4.71) and (4.72), average cohesion c and friction angle values can be calculated by linear regression analysis, in which the best fitting straight line is calculated for the range of (σ′n, τ) pairs. The uniaxial compressive strength of a rock mass defined by a cohesive strength c and a friction angle

is given by (4.75)

Water has a great effect on the strength of rock masses. Many rocks show a significant strength decrease with increasing moisture content. Typically, strength losses of 30– 100% occur in many rocks as a result of chemical deterioration of the cement or clay binder. Therefore, it is important to conduct laboratory tests at moisture contents which are as close as possible to those which occur in the field. A more important effect of water is the strength reduction which occurs as a result of water pressures in the pore spaces in the rock. This is why the effective not the total stresses are used in the HoekBrown strength criterion. The Hoek-Brown strength criterion was originally developed for intact rock and then extended to rock masses. The process used by Hoek and Brown in deriving their strength criterion for intact rock (Equation 4.63) was one of pure trial and error (Hoek et al., 1995). Apart from the conceptual starting point provided by the Griffith theory, there is no fundamental relationship between the empirical constants included in the criterion and any physical characteristics of the rock. The justification for choosing this particular criterion (Equation 4.63) over the numerous alternatives lies in the adequacy of its predictions of the observed rock fracture behavior, and the convenience of its application

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to a range of typical engineering problems (Hoek, 1983). The material constant mi is derived based upon analyses of published triaxial test results on intact rock (Hoek, 1983; Doruk, 1991; Hoek et al., 1992). The strength criterion for rock masses is just an empirical extension of the criterion for intact rock. Since it is practically impossible to determine the material constants mb and s using triaxial tests on rock masses, empirical relations are suggested to estimate these constants from RMR or GSI. The RMR and the GSI rating systems are also empirical. For these reasons the Hoek-Brown empirical rock mass strength criterion must be used with extreme care. In discussing the limitations in the use of their strength criterion, Hoek and Brown (1988) emphasize that it is not applicable to anisotropic rocks nor to elements of rock masses that behave anisotropically by virtue of containing only a few discontinuities. Alternative empirical approaches and further developments of the Hoek-Brown criterion which seek to account for some of its limitations are given by Amadei (1988), Pan and Hudson (1988), Ramamurthy and Arora (1991), Amadei and Savage (1993), and Ramamurthy (1993). (b) Bieniawski-Yudhbir criterion Bieniawski (1974) proposed a strength criterion for intact rock as follows (4.76)

This was changed by Yudhbir et al. (1983), based on tests on jointed gypsum-celite specimens, to the form (4.77)

to fit rock masses. Yudhbir et al. (1983) recommended that the parameters α and a be determined from α=0.65 (4.78a) (4.78b)

where Q is the classification index of Barton et al. (1974) and RMR is Bieniawski’s 1976 Rock Mass Rating (Bieniawski, 1976). Parameter b is determined from Table 4.6. Kalamaras and Bieniawski (1993) suggested that both a and b should be varied with RMR for better results. They proposed the criterion of Table 4.7 specifically for coal seams.

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(c) Johnston criterion Based on experimental data of a wide range of geotechnical material, from lightly overconsolidated clays through hard rocks, Johnston (1985) proposes the following strength criterion (4.79)

where σ′1n and σ′3n are the normalized effective principal stresses at failure, obtained by dividing the effective principal stresses, σ′1 and σ′3, by the relevant uniaxial compressive strength, σc; B and M are intact material constants; and s is a constant to account for the strength of discontinuous soil and rock masses in a manner similar to that proposed by Hoek and Brown (1980). However, in the development of the criterion, Johnston (1985) considers only intact materials.

Table 4.6 Parameter b in the Bieniawski-Yudhbir criterion (Yudhbir et al., 1983). Rock Type

b

Tuff, Shale, Limestone

2

Siltstone, mudstone

3

Quartzite, Sandstone, Dolerite

4

Norite, Granite, Quartz diorite, Chert

5

Table 4.7 Rock mass criterion for coal seams by Kalamaras and Bieniawski (1993). Equation

Parameters

For intact material, s=1, the criterion becomes (4.80)

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By placing σ′3n=0, the uniaxial compressive strength is correctly modeled with the righthand side of Equation (4.80) becoming unity. By putting B=1, the criterion simplifies to (4.81) which for (4.82) is identical to the normalized Coulomb criterion. The parameter B, which describes the nonlinearity of a failure envelope, is essentially independent of the material type, and is a function of uniaxial compressive strength: B=1−0.0172(logσc)2 (4.83) The parameter M, which describes the slope of a failure envelope at σ′3n=0, is found to be a function of both the uniaxial compressive strength and the material type. For the material types shown in Table 4.8, M can be estimated by (no result is obtained for type D material because of lack of data): Type A, M=2.065+0.170(logσc)2 (4.84a) 2 Type B, M=2.065+0.231(logσc) (4.84b) Type C, M=2.065+0.270(logσc)2 (4.84c) 2 Type E, M=2.065+0.659(logσc) (4.84d)

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Table 4.8 A range of rock types (after Hoek & Brown, 1980). Type General Rock Type

Examples

A

Carbonate rocks with well developed crystal cleavage

Dolomite, limestone, marble

B

Lithified argillaceous rocks

Mudstone, siltstone, shale, slate

C

Arenaceous rocks with strong crystals and poorlydeveloped crystal cleavage

Sandstone, quartzite

D

Fine grained polyminerallic igneous crystalline rocks

Andesite, dolerite, diabase, rhyolite

E

Coarse grained polyminerallic igneous and metamorphic crystalline rocks

Amphibolite, gabbro, gneiss, granite, norite, quartz diorite

(d) Ramamurthy criterion Ramamurthy and his coworkers (Ramamurthy et al., 1985; Ramamurthy, 1986; Ramamurthy, 1993) modified the Coulomb theory to represent the nonlinear shear strength behavior of rocks. For intact rock, the strength criterion is in the following form (4.85)

where σ′1 and σ′3 are the major and minor principal effective stresses; σc is the uniaxial compressive strength; αr is the slope of the curve between (σ′1−σ′3)/σ′3 and σc/σ′3, for most intact rocks the mean value of αr is 0.8; and Br is a material constant of intact rock, equal to (σ′1−σ′3)/σ′3 when σc/σ′3=1. The values of Br vary from 1.8 to 3.0 depending on the type of rock (Table 4.9). The values of αr and Br can be estimated by conducting a minimum of two triaxial tests at confining pressures greater than 5% of σc for the rock. The above expression is applicable in the ductile region and in most of the brittle region. It underestimates the strength when σ′3 is less than 5% of σc and also ignores the tensile strength of the rock. To account for the tensile strength, the following expression gives a better prediction for intact rock (4.86)

where σt is the tensile strength of rock preferably obtained from Brazilian tests; α=0.67 for most rocks; and B is a material constant. The values of α and B in Equation (4.86) can be obtained by two triaxial tests conducted at convenient confining pressures greater than

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5% of σc for the rock. In the absence of these tests, the value of B is estimated as 1.3(σc/σt)1/3. For rock masses, the strength criterion has the same form as for intact rock, i.e. (4.87)

Table 4.9 Mean values of parameter Br for different rocks (after Ramamurthy, 1993). Rock type

Metamorphic and sedimentary rocks Argillaceous Siltstone Shales

Arenaceous

Chemical

Sandstone Quartzite Limestone Marble

Igneous rocks Andesite Granite

Clays

Slates

Anhydrite Dolomite Diorite

Tuffs

Mudstone

Rock salt

Loess

Claystone

Charnockite

Norite Liparite Basalt

Br Mean value

1.8

2.2 2.0

2.2

2.6

2.4

2.4

2.8 2.6

2.6

3.0 2.8

where σcm is the rock mass strength in unconfined compression; Bm is a material constant for rock mass; and αm is the slope of the plot between (σ′1−σ′3)/σ′3 and σcm/σ′3, which can be assumed to be 0.8 for rock masses as well. σcm and Bm can be obtained by (4.88) (4.89) in which σc is the unconfined compressive strength of intact rock strength; and Br is a material constant for intact rock, as in Equation (4.83). (e) Comments In addition to the four empirical strength criteria for rock masses described above, there are many other criteria. All these criteria are purely empirical and thus it is impossible to say which one is correct or which one is not. However, the Hoek-Brown strength criterion is the most representative one of the empirical strength criteria for rock masses, because it is the mostly widely referred and used. Since its advent in 1980, considerable application experience has been gained by its authors as well as by others. As a result,

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this criterion has been modified several times to meet the needs of users who have applied it to conditions which were not visualized when it was originally developed. It is noted that all the empirical strength criteria for rock masses have the following limitations: 1. The influence of the intermediate principal stress, which in some cases is important, is not considered. 2. The criteria are not applicable to anisotropic rock masses. So they can be used only when the rock masses are approximately isotropic, i.e. when the discontinuity orientation does not have a dominant effect on failure. 4.4.2 Equivalent continuum approach for estimating rock mass strength (a) Model of Jaeger (1960) and Jaeger and Cook (1979) Figure 4.42(a) shows a cylindrical rock mass specimen subjected to an axial major principal stress σ′1 and a lateral minor principal stress σ′3. The rock mass is cut by welldefined parallel discontinuities inclined at an angle β to the major principal stress σ′1. The strength of both the intact rock and the discontinuities are described by the Coulomb criterion, i.e. (4.90) (4.91) where τr and τj are respectively the shear strength of the intact rock and the discontinuities; cr and

are respectively the cohesion and internal friction angle of the

intact rock; cj and are respectively the cohesion and internal friction angle of the discontinuities; and σ′n is the effective normal stress on the shear plane. For the applied stresses on the rock mass cylinder, the effective normal stress σ′n and the shear stress τ on a plane which makes an angle β′ to the σ′1 axis are respectively given by (4.92) (4.93) If shear failure occurs on the discontinuity plane, the effective normal stress σ′n and the shear stress τ on the discontinuity plane can be obtained by replacing β′ in Equations (4.92) and (4.93) by β. Adopting the obtained stresses on the discontinuity plane to substitute for σ′n and τj in Equation (4.91) and then rearranging, we can obtain the effective major principal stress required to cause shear failure along the discontinuity as follows

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(4.94)

If shear failure occurs in the intact rock, the minimum effective major principal stress can be obtained by (4.95) The model of Jaeger (1960) and Jaeger and Cook (1979) assumes that failure during compressive loading of a rock mass cylinder subject to a lateral stress σ′3 [see Fig. 4.42(a)] will occur when σ′1 exceeds the smaller of the σ′1f values given by Equations (4.94) and (4.95). Figure 4.42(b) shows the variation of σ′1f with β, from which we can clearly see the anisotropy of the rock mass strength caused by the discontinuities. (b) Model of Amadei (1988) and Amadei and Savage (1989, 1993) As seen above, the model of Jaeger (1960) and Jaeger and Cook (1979) assumes that the jointed rock mass is under axisymmetric loading, so the effect of the intermediate principal stress is not involved in their formulations. To address the limitation of the model of Jaeger (1960) and Jaeger and Cook (1979), Amadei (1988) and Amadei and Savage (1989, 1993) derived solutions for the strength of a jointed rock mass under a variety of multiaxial states of stress. As in the model of Jaeger (1960) and Jaeger and Cook (1979), the modeled rock mass is cut by a single discontinuity set. In the formulations of Amadei (1988) and Amadei and Savage (1989, 1993), however, the intact rock strength is described by the HoekBrown strength criterion and the discontinuity strength is modeled using a Coulomb criterion with a zero tensile strength cut-off. The principle used by Amadei (1988) and Amadei and Savage (1989, 1993) to derive the expressions of the jointed rock mass strength is the same as that used by Jaeger (1960) and Jaeger and Cook (1979). However, since the effect of the intermediate principal stress is included and since the nonlinear Hoek-Brown strength criterion is used, the derivation process and the final results are much more complicated. For reasons of space, only some of the typical results of Amadei and Savage (1989, 1993) are shown here. Consider a jointed rock mass cube under a triaxial state of stress σ′x, σ′y and σ′z. The orientation of the discontinuity plane is defined by two angles β and Ψ with respect to the xyz coordinate system (see Fig. 4.43). Let nst be another coordinate system attached to the discontinuity plane such that the n-axis is along the discontinuity upward normal and the s-and t-axes are in the discontinuity plane. The t-axis is in the xz plane. The upward unit vector n has direction cosines

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Fig. 4.42 Variation of compressive strength with angle β of the discontinuity plane (after Jaeger & Cook, 1979).

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(4.96) Defining m=σ′y/σ′x and n=σ′z/σ′x and introducing two functions (4.97) where σ′n and τ are respectively the normal and shear stresses acting across the discontinuity; and is the friction angle of the discontinuity, the limiting equilibrium (incipient slip) condition of the discontinuity can be derived as (4.98)

The nonnegative normal stress condition of the discontinuity is (4.99) So for a discontinuity with orientation angles β and ψ the condition Ff=0 corresponds to impending slip. No slip takes place when Ff is negative. Figure 4.44 shows a typical set of failure surfaces Ff(m,n)=Q for ψ equal to 40° or 80° and β ranging between 0° and 90°. In this figure the ranges Ff(m,n)>0 are shaded and Fn=0 is represented as a dashed straight line. The positive normal stress condition (Fn>0) is shown as the region on either side of the line Fn=0 depending on the sign of σx.

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Fig. 4.43 Discontinuity plane in a triaxial stress field (after Amadei & Savage, 1993). Depending on the ordering of σ′x, σ′y and σ′z, the Hoek-Brown strength criterion for intact rock (Equation 4.63) assume six possible forms as shown in Table 4.10. Using mi=

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Table 4.10 Forms of Equation (4.63) for different orderings of σ′x, σ′y and σ′z. Principal stress ordering

Major stress σ′1

Minor stress σ′3

σ′x>σ′y>σ′z

σ′x

σ′z

σ′x>σ′z>σ′y

σ′x

σ′y

σ′y>σ′x>σ′z

σ′y

σ′z

σ′y>σ′z>σ′x

σ′y

σ′x

σ′z>σ′x>σ′y

σ′z

σ′y

σ′z>σ′y>σ′x

σ′z

σ′x

Forms of Equation (4.63)

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Fig. 4.44 Shape of the failure surface Ff(m,n)=0 in the m=σ′y/σ′x, n=σ′z/σ′x space for (a) β= 38.935°, ψ=40°; (b) β=30°, ψ=40°; (c) β=20°, ψ=80°; and (d) β=70°, ψ=40°. The region Fn >0 is above the dashed line Fn=0 when σ′x is compressive and below that line when σ′x is tensile. Friction (after Amadei & Savage, 1993).

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7 and σc=42 MPa, the intact rock failure surfaces for different values of σ′x/σc can be obtained as shown in Figure 4.45. The failure surfaces of the jointed rock masses can be obtained by superposition of the discontinuity failure surfaces and the intact rock failure surfaces. Figure 4.46 is obtained by superposition of the failure surfaces in Figures 4.44 and 4.45. The following remarks can be made about the diagrams shown in Figure 4.46: 1. In general, for a given value of σ′x/σc, the size of the stable domain enclosed by the intact rock failure surface is reduced because of the discontinuities. The symmetry of the intact rock failure surface with respect to the m=n axis in the m, n space (Fig. 4. 46) is lost. The strength of the jointed rock mass is clearly anisotropic. 2. The strength reduction associated with the discontinuities is more pronounced for discontinuities with orientation angles β and ψ for which the discontinuity failure surface in the m, n space is ellipse than when it is an hyperbola or a parabola. 3. Despite the zero discontinuity tensile strength and the strength reduction associated with the discontinuities, jointed rock masses can be stable under a wide variety of states of stress σ′x, σ′y=mσ′x, σ′z=nσ′x. These states of stress depend on the values of discontinuity orientation angles β and ψ and the stress ratio σ′x/σc. (c) Comments In Section (a), a rock mass with one discontinuity set is considered. If we apply the model of Jaeger (1960) and Jaeger and Cook (1979) to a rock mass with several discontinuity sets, the strength of the rock mass can be obtained by considering the effect of each discontinuity set. For example, consider a simple case of two discontinuity sets A and B [see Fig. 4.47(a)], the angle between them being α. The corresponding variation of the compressive strength σ′1β, if the two discontinuity sets are present singly, is shown in Figure 4.47(b). As the angle βa of discontinuity set A is changed from 0 to 90°, the angle βb of discontinuity set B with the major stress direction will be βb=|α−βa| for α≤90° (4.100) When βa is varied from 0 to 90°, the resultant strength variation for α=60 and 90° will be as in Figure 4.47(c), choosing the minimum of the two values σ′1βa and the corresponding σ′1βb from the curves in Figure 4.47(b). Hoek and Brown (1980) have shown that with three or more discontinuity sets, all sets having identical strength characteristics, the rock mass will exhibit an almost flat strength variation (see Fig. 4.48), concluding that in highly jointed rock masses, it is possible to adopt one of the rock mass failure criteria presented in Section 4.4.1. It should be noted that, in the models of the equivalent continuum approach, discontinuities are assumed to be persistent and all discontinuities in one set have the same orientation. In reality, however, discontinuities are usually non-persistent and the discontinuities in one set have orientation distributions.

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Fig. 4.45 Geometrical representation of the Hoek-Brown failure surface for intact rock in the m= σ′y/σ′x, n=σ′z/σ′x space for different values of σ′x/σc with mi=7 and σc=42 MPa. (a) σ′x is compressive; and (b) σ′x is tensile (after Amadei & Savage, 1993).

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Fig. 4.46 Superposition of the joint failure surface with and the intact rock failure surface with mi=7 and σc=42 MPa in the m=σ′y/σ′x, n=σ′z/σ′x space for (a) β=38.935°, ψ=40°; (b) β= 30°, ψ=40°; (c) β=20°, ψ=80°; and (d) β=70°, ψ=40° (after Amadei & Savage, 1993).

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4.4.3 Direct consideration of discontinuities in numerical analysis The effect of discontinuities on the rock mass strength can be directly accounted for in numerical analyses. Special joint elements used to represent the discontinuities in FEM analysis have been discussed in Section 4.3.3. Here only the constitutive models of discontinuities will be described and discussed. To account for the effect of discontinuities on the rock mass strength in numerical analysis, the behavior of discontinuities is usually assumed to be elasto-plastic. The elastic behavior is represented by the initial elastic tangential normal and shear stiffnesses kne and kse (see Fig. 4.1). The peak strength and dilatancy of rock discontinuities is represented by a failure criterion and a flow rule, respectively. (a) Coulomb Model The Coulomb model is perhaps the crudest of rock discontinuity models but has been extensively used in engineering analysis and design of rock structures. The Coulomb strength criterion has been given in Equation (4.8) and is rewritten here in the following form (4.101) where |σs| is the absolute value of shear stress on the discontinuity plane; cj and are respectively the cohesion and friction angle of the discontinuities; and σ′n is the effective normal stress on the discontinuity plane. If an associated flow rule is adopted, rates of plastic normal strain

and shear strain

are given by (4.102)

where

is a positive proportionality constant. Equation (4.102) implies (4.103)

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Fig. 4.47 (a) Rock mass with two discontinuity sets A and B; (b) Strength variation with β if the discontinuity sets are present singly; and (c) Strength variation when both discontinuity sets are present.

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Fig. 4.48 Strength variation with angle β1 of discontinuity plane 1 in the presence of 4 discontinuity sets, the angle between two adjoining planes being 45° (after Hoek & Brown, 1980).

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The discontinuities are, therefore, dilatant, i.e. an increment of shear displacement ∆us along the discontinuity is accompanied by an increment in the normal displacement ∆un given by (4.104) The rate of dilation is constant and goes on unabated. This behavior is quite unrealistic. Roberds and Einstein (1978) presented a very comprehensive model for rock discontinuities. From various studies it has been established that the flow rule for rock discontinuities should be non-associated. By introducing a variable dilation angle Ψj, a plastic potential function can be written as (4.105) where Ψj can be identified from the experimental results on rock discontinuities. It is clear that when the average normal displacement on the rock discontinuity is equal to the average height of the asperities, dilation must cease, i.e. Ψj→0. The Coulomb model has another drawback. cj and in Equation (4.101) are not truly constants. They depend on σ′n. The values of σ′n on rock discontinuities can vary by several orders of magnitudes within the structure to be analyzed. Choosing a single appropriate value of cj and impossible.

for a discontinuity set, therefore, becomes difficult, if not

(b) Barton Model The Barton model has been described in Section 4.2.2 [Equation (4.12)] and is rewritten here in the following form (4.106)

where JRC is the discontinuity roughness coefficient; JCS is the discontinuity wall is the basic friction angle of the rock material. compressive strength; and If an associated flow rule is assumed, the dilation angle at peak strength can be readily computed by differentiating Equation (4.106). However, the computed dilation angles Ψj based on an associated flow rule do not match the experimentally observed values. This again shows that the flow rule for rock discontinuities should be non-associated. Pande and Xiong (1982) proposed the following plastic potential function to match the experimental results of Barton and Chaubey (1977): (4.107)

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where (4.108)

Table 4.11 shows the comparison of experimental values with those computed using Equation (4.107) as the plastic potential function. A close agreement can be seen. (c) Comments In addition to the two elasto-plastic models for rock discontinuities described above, there are many other models. Roberds and Einstein (1978) presented a very comprehensive model and critically examined Patton (1966) model, Ladanyi and Archambault (1970) model, Agbabian model (Ghaboussi et al., 1973), Goodman (1966, 1974) model and Barton (1976) model by comparing them with the comprehensive model. Since the comprehensive rock discontinuity model of Roberds and Einstein (1978) can treat the entire behavioral history from the creation of the discontinuity to its behavior before, during and after sliding, it provides a good basis for comparison of various models. With the comprehensive rock discontinuity model, it is possible to show where and to what extent the existing models are limited or simplified as compared to the comprehensive model and this makes it possible to appropriately modify the existing models, if so desired.

Table 4.11 Comparison of measured angle of dilation with that predicted by Equation (4.107). Rock Type

No. of Samples

Measured angle of dilation Computed angle of dilation (°) (°)

Alpite

36

25.5

23.0

Granite

38

20.9

20.2

Hornfels

17

26.5

26.2

Calcareous shale

11

14.8

19.1

Slate

7

6.8

–

Gneiss

17

17.3

15.5

Soapstone model

5

16.2

18.6

Fractures

130

13.2

–

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165

4.5 SCALE EFFECT Research results (see, e.g., Heuze, 1980; Hoek & Brown, 1980; Medhurst & Brown, 1996) indicate that rock masses show strong scale dependent mechanical properties. In the following, the scale effect on the strength and deformation properties of rock masses is briefly discussed. 4.5.1 Scale effect on strength of rock mass Experimental results show that rock strength decreases significantly with increasing sample size. Based upon an analysis of published data, Hoek and Brown (1980) suggested that the unconfined compressive strength σcd of a rock specimen with a diameter of d mm is related to the unconfined compressive strength σc50 of a 50 mm diameter specimen by (4.109)

This relationship, together with the data upon which it was based, is illustrated in Figure 4.49. Hoek and Brown (1997) suggested that the reduction in strength is due to the greater opportunity for failure through and around grains, the “building blocks” of intact rock, as more and more of these grains are included in the test sample. Eventually, when a sufficiently large number of grains are included in the sample, the strength reaches a constant value. Medhurst and Brown (1996) reported the results of laboratory triaxial tests on 61, 101, 146 and 300 mm diameter samples of coal from the Moura mine in Australia. The results of these tests are as summarized in Table 4.12 and Figure 4.50. It can be seen that the strength decreases significantly with increasing specimen size. This is attributed to the effects of cleat spacing. For this coal, the persistent cleats are spaced at 0.3–1.0 m while non-persistent cleats within vitrain bands and individual lithotypes define blocks of 1 cm or less. This cleating results in a “critical” sample size of about 1m above which the strength remains constant. Heuze (1980) conducted an extensive literature search and found results of 77 plate tests as shown in Figure 4.51. The test volume shown in this figure is calculated in the following way: 1. For a circular plate, the test volume is taken as that of a sphere having a diameter of 4 times the diameter of the plate. 2. For a rectangular or square plate of given area, the diameter of a circle of equal area is first calculated, and the test volume is then determined using the equivalent diameter.

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Fig. 4.49 Influence of specimen size on the strength of intact rock (after Hoek & Brown, 1980). Table 4.12 Peak strength of Moura coal in terms of the parameters in Equation (4.64), based upon a value of σc=32.7 Mpa. Diameter (mm)

mb

s

a

61

19.4

1.0

0.5

101

13.3

0.555

0.5

146

10.0

0.236

0.5

300

5.7

0.184

0.6

Mass

2.6

0.052

0.65

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Fig. 4.50 Peak strength for Australian Moura coal (after Medhurst & Brown, 1996). The number shown next to the open triangles in the figure indicates the number of tests performed; the mean value of these test results is plotted as the triangle. The test results [except those of Coates and Gyenge (1966) and Rhodes (1973)] show that the strength decreases with increasing test volume. Figure 4.52 (Hoek et al., 1995) shows a simplified representation of the influence of the relation between the discontinuity spacing and the size of the problem domain on the

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selection of a rock mass behavior model (Hoek-Brown strength criterion). As the problem domain enlarges, the corresponding rock behavior changes from that of the isotropic intact rock, through that of a highly anisotropic rock mass in which failure is controlled by one or two discontinuities, to that an isotropic heavily jointed rock mass. In determining the allowable bearing capacity of shallow foundations on rock using the Hoek-Brown strength criterion, Serrano and Olalla (1996), following the technique of Hoek (1983), divide the rock masses into three main structural groups depending on the conditions of rock masses and the foundation dimensions (Fig. 4.53):

Fig. 4.51 Effect of test volume on measured bearing strength of rock masses. The number next to the triangle indicates the number of tests performed (after Heuze, 1980).

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Fig. 4.52 Simplified representation of the influence of scale on the type of rock mass behavior (after Hoeketal, 1995). Group I: where the rock can be considered as intact. If the microstructure of the rock is isotropic, the rock mass can be considered isotropic and the Hoek-Brown criterion can be applied. Group II: where the rock mass is affected by only a few sets of discontinuities. The behavior of the rock mass is basically anisotropic and the Hoek-Brown criterion cannot generally be applied to the rock mass.

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Fig. 4.53 Simplified representation of scale effect on the type of rock mass model which should be used in designing shallow foundations on rock slope (after Serrano & Olalla, 1996). Group III: where the rock mass is affected by a number of sets of discontinuities giving rise to “small spacing” between discontinuities. This group of rock masses can be regarded as isotropically broken media and the Hoek-Brown criterion can be applied. “Small spacing” is a relative concept, in the sense that it depends on the foundation dimensions. Serrano and Olalla (1996) propose a parameter, the “spacing ratio of a foundation” (SR), for its quantification. SR is defined as (4.110)

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where B is the foundation width (in meters); smi is the discontinuity spacing of set i (in meters); λi is the frequency of discontinuity set i (m−1); and n is the number of discontinuity sets. As an initial and conservative proposal, a “relatively small spacing” is suggested when SR is greater than 60. A value of 60 means that, if there are four sets of discontinuities, each of them appears at least 15 times within the foundation width. When SR>60, the mass can be regarded as an isotropically broken medium and the Hoek-Brown criterion can be applied. For values of SR≤(0.8−4), in the case of four sets of discontinuities, the rock mass can be considered as an intact rock mass (Group I). 4.5.2 Scale effect on deformability of rock mass The scale effect on the deformability of rock masses can be seen from the difference of rock mass modulus measured in the field and intact rock mass modulus measured in the laboratory. Heuze (1980) concluded that the rock mass modulus measured in the field ranges between 20 and 60% of the intact rock mass modulus measured in the laboratory. One simple and apparent explanation to the reduction of rock mass modulus is that the effect of discontinuities is included in the rock masses.

4.6 DISCUSSION The structure of jointed rock masses is highly variable; the methods used to consider the effect of discontinuities on the mechanical behavior of jointed rock masses are also variable. The selection of the methods should be based on careful studies of the in situ situation of jointed rock masses. Laboratory and in situ tests (i.e., direct methods) can directly provide results about the mechanical properties of tested specimens. However, care need be exercised about the extent to which the measured behavior of the rock specimen reflects the actual behavior of rock masses. The extrapolation of the behavior induced by the experimental system to different circumstances can be very misleading. In addition, in situ tests are time consuming, expensive and difficult to conduct; it is extremely difficult to investigate the effects of discontinuity system on the mechanical properties of jointed rock masses through in situ tests. Indirect methods consist of the empirical methods, the equivalent continuum approach and numerical analysis methods. It is important to note that all the indirect methods need to use some of the mechanical properties of intact rock or discontinuities obtained through laboratory or in situ tests. Since they are simple and easy to use, and most importantly, since they originate from practical experience, the empirical methods are most widely used in design practice. However, it is important to note their limitations as described in Sections 4.3.1 and 4.4.1. The equivalent continuum approach usually assumes that all discontinuities are persistent and the discontinuities in one set have the same orientation. In reality, however, discontinuities are usually non-persistent and the discontinuities in one set are not in the same orientation. Kulatilake et al. (1992, 1993) and Wang (1992) considered rock masses

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containing non-persistent discontinuities and derived relationships between the deformation properties and the fracture tensor parameters from the DEM analysis results of generated rock mass blocks. However, this method is also limited as described in Section 4.3.2. The numerical methods have great potential for the complex mechanical analyses of jointed rock masses. The key problems associated with numerical methods are the representation of discontinuities and the determination of discontinuity constitutive models. The main drawback of this approach is that, due to computer limitations and difficulty in creating meshes for a heavily jointed rock mass, only rock masses with a limited number of discontinuities can be analyzed. In summary, the limitations for each method are as follows: 1. For laboratory tests, only small specimens can be used. Since rock masses show strong scale dependent mechanical properties, the measured behavior of small rock specimens may not reflect the actual behavior of rock masses in the field. 2. In situ tests are time consuming, expensive and difficult to conduct. 3. Empirical methods do not consider the anisotropy of rock masses caused by discontinuities and different empirical relations often give very different values. 4. The equivalent continuum approach assumes that the discontinuities are persistent and the discontinuities in one set have the same orientation. In reality, however, most of the discontinuities are non-persistent with finite size and the discontinuities in one set are not in the same orientation. 5. Numerical methods can only be used for rock masses with a limited number of discontinuities. Because each method has its own advantages and disadvantages, it is important to select the appropriate method(s) for different purposes. Following are the principles that can be used when selecting the methods according to the nature of the problems: 1. The empirical methods can always be used in the first stage of design. For heavily jointed rock masses, the empirical methods can be used in all design stages. 2. For rock masses with persistent discontinuities which are regularly distributed, the equivalent continuum approach can be used. 3. For rock masses with a limited number of discontinuities, the numerical methods and the limit equilibrium method can be used.

5 Site investigation and rock testing 5.1 INTRODUCTION As required for any geotechnical projects, site investigations need be conducted to obtain the information required for the design of drilled shafts in rock. The nature and extent of the information to be obtained from a site investigation will vary according to the project involved and the expected ground conditions. A site investigation is a process of progressive discovery, and, although there must be a plan and program of work at the beginning, the information emerging at any stage will influence the requirements of subsequent stages. Typically, a site investigation consists of the following three main stages: 1. Preliminary investigation including desk study and site reconnaissance 2. Detailed investigation including boring, drilling, in situ testing and lab testing 3. Review during construction and monitoring A distinguishing feature of site investigations for foundations in rock is that it is particularly important to focus on the details of the structural geology. The rock mass at a site may contain very strong intact rock, but the discontinuities in the rock mass may lead to excessive deformation or even failure of the drilled shaft foundations in the rock mass. 5.1.1 Preliminary investigation Prior to implementing a detailed site investigation program, certain types of preliminary information need be developed. The type and extent of information depends on the cost and complexity of the project. The information is developed from a thorough survey of existing information and field reconnaissance. Information on topography, geology and potential geologic hazards, surface and groundwater hydrology, seismology, and rock mass characteristics are reviewed to determine the following (ASCE, 1996): • Adequacy of available data • Type and extent of additional data that will be needed • The need for initiating critical long-term studies, such as ground water and seismicity studies, that require advance planning and early action • Possible locations and type of geologic features that might control the design and construction of drilled shaft foundations. Various types of published maps can provide an excellent source of geologic information to develop the regional geology and geological models of potential or final sites. Other geotechnical information and data pertinent to the project can frequently be obtained from a careful search of federal, state, or local governments as well as private industry in

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the vicinity. Consultation with private geotechnical engineering firms, mining companies, well drilling and development companies and state and private university staff can sometimes provide a wealth of information. After a complete review of available geotechnical data, a site reconnaissance should be made to gather information through visual examination of the site and an inspection of ground exposures in the vicinity. In some cases adjacent sites will also be examined. The primary objective of this field reconnaissance is to, insofar as possible, confirm, correct or expand geologic and hydrologic information collected from preliminary office studies. If rock outcrops are present, the field reconnaissance offers an opportunity to collect preliminary information on rock mass conditions that might influence the design and construction of drilled shafts. Notation should be made of the strike and dip of major discontinuity sets, discontinuity spacing, discontinuity conditions (i.e. weathering, wall roughness, tightness, fillings, and shear zones), and discontinuity persistence. The reconnaissance will assist in planning the detailed investigation program. Where the geology is relatively straightforward and the engineering problems are not complex, sufficient geological information may be provided by the desk study, subject to confirmation by the exploration which follows. In other cases detailed investigations may be carried out. Geophysical survey is becoming quicker and more robust to provide information on the depth of weathering, the bedrock profile, the location of major faults and solution cavities, and the degree of fracturing of the rock. So some geophysical work is often conducted in the stage of preliminary investigation rather than leaving it all to the stage of detailed investigation. In some cases it may be appropriate to put down pits or use relatively light and simple boring equipment during the preliminary investigation. However, the objectives of a boring program at this stage should be limited. The main boring program should be deferred until the stage of detailed investigation. 5.1.2 Detailed investigation Information from the desk study and site reconnaissance provides a preliminary conception of the ground conditions and the engineering problems that may be involved. Detailed investigation then proceeds with flexible planning so that the work can be varied as necessary as fresh information emerges. The extent of the detailed investigation will be governed by the type of project and the nature and variation of the ground. Other factors to be taken into account are access, time available and cost. However, technical requirements rather than cost should govern. Various methods are available for detailed investigation of rock masses. This chapter will describe discontinuity sampling on exposed rock faces (Section 5.2), boring (Section 5.3), geophysical exploration (Section 5.4), lab testing (Section 5.5) and in situ testing (Section 5.6). 5.7.3 Review during construction and monitoring It is essential to examine all excavations during construction to see whether the expectations of the preceding investigations have been realized. The examinations can be carried out after the excavation has been cleaned up and just prior to the placement of

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concrete. The identification of exceptions may lead to an early diagnosis and anticipation of problems. During construction and in the post-commissioning stage, monitoring will involve regular reading of instruments installed to check performance against design criteria. This should serve as an “early warning” system, which will initiate a contingency program, thus minimizing the delays that would occur as a result of an adverse situation.

5.2 DISCONTINUITY SAMPLING ON EXPOSED ROCK FACES From sampling on exposed rock faces, either above or below ground, information about the orientation, spacing, roughness and curvature of discontinuities can usually be satisfactorily obtained. It should be noted, however, that there exists sampling bias on discontinuity orientation and spacing (Terzaghi, 1965; Priest, 1993; Mauldon & Mauldon, 1997). This sampling bias should be corrected before inferring statistical distributions of orientation data. Although the locations of traces give some information about the locations of discontinuities, it is still impossible to determine the exact locations of discontinuities. From sampling on exposed rock faces, almost no information about the shape of discontinuities can be obtained. Measured trace lengths give some information about the size of discontinuities. However, because of the sampling biases and the unknown shape of discontinuities, the size of discontinuities can only be inferred based on assumptions (see Chapter 3). 5.2.1 Scanline sampling Intersections between discontinuities and the rock face produce linear traces which provide an essentially two-dimensional sample of the discontinuity network. In current geologic and rock engineering practice, straight scanlines are commonly used for discontinuity sampling on exposed rock faces (see Fig. 5.1). The straight scanlines can be simply measuring tapes pinned with masonry nails and wire to the rock face or chalk lines drawn on the rock face. The length and orientation of each discontinuity crossing the straight scanline are measured. Spacing between adjacent discontinuities intersecting the straight scanline can also be measured along the scanline. From measurements taken this way, three inferences can be made on the population of discontinuities: intensity (or spacing), size (length or persistence), and orientation. Straight scanlines provide a fast method for discontinuity sampling, but yield a sample biased by the relative orientation and size between the discontinuities and the rock face. The relative orientation introduces a bias to both the discontinuity spacing and the number of discontinuities that are measured. The bias arises because discontinuities subparallel to the rock face have less chance to intersect the rock face than discontinuities perpendicular to the rock face (Terzaghi, 1965). The bias in spacing can be corrected as follows (Terzahgi, 1965): s=sa sinθ (5.1)

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where s is the true spacing between discontinuities of the same set; sa is the measured (apparent) spacing between discontinuities of the same set on the rock face; and θ is the angle between the scanline and the discontinuity traces (see Fig. 5.1). The number of discontinuities in a set can be adjusted to account for the orientation bias as follows: (5.2) where N is the adjusted number of discontinuities; Na is the measured (apparent) number of discontinuities. The statistical analysis of discontinuity spacing and frequency has been discussed in Chapter 3.

Fig. 5.1 Straight scanline sampling. There is also sampling bias for trace lengths intersected by a straight scanline because the scanline tends to intersect preferentially the longer traces (Priest & Hudson, 1981). For detailed analysis of sampling biases on trace lengths, the reader can refer to Priest (1993). Circular scanlines (see Fig. 5.2) are also used for discontinuity sampling at exposed rock faces (Einstein et al., 1979; Titley et al., 1986; Davis & Reynolds, 1996; Mauldon et al., 2000). Circular scanline sampling measures only the traces intersecting the line of the circle. One advantage of circular scanlines over straight scanlines is the elimination of directional bias. Mauldon et al. (2000) derived a simple expression for estimating discontinuity intensity from circular scanline sampling: (5.3)

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where λ′ is the discontinuity intensity defined as the mean length of traces per unit area; N is the number of traces intersecting the circular scanline; and c is the radius of the scanline circle.

Fig. 5.2 Circular scanline sampling. 5.2.2 Window sampling Window or area sampling measures discontinuity traces within a finite size area (usually rectangular or circular in shape as discussed in Chapter 3) at an exposed rock face (see Figs. 3.22 and 3.23). It is important to note that in window sampling as it is defined here only the portions of the discontinuity traces within the window are measured, while the portions of traces intersecting such a window but lying outside are not considered. Window sampling reduces the sampling biases for orientation and size created by scanline sampling, but problems of discontinuity curtailment remain where the rock face is of limited extent. The estimation of mean trace length from sampling on rectangular or circular windows has been described in Chapter 3. The probability of intersecting a discontinuity with a sampling domain depends on the relative orientation of the discontinuity with respect to the sampling domain, the shape and size of the discontinuity, and the shape and size of the sampling domain. Therefore, observed frequencies of discontinuities do not represent the true frequencies in three dimensions. This is called the sampling bias on discontinuity orientation. This sampling bias should be corrected when inferring statistical distributions of orientation data. Kulatilake and Wu (1984a) proposed a method to find the probability that a finite size discontinuity intersects a finite size sampling plane, and presented a procedure for correcting sampling bias for circular discontinuities intersecting rectangular, vertical

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sampling planes. Kulatilake et al. (1990) extended the previous formulation to cover other discontinuity shapes such as parallelograms, rectangles, rhombuses and triangles. However, the procedures for correcting sampling bias cannot be applied directly for nonvertical, finite sampling areas. Wathugala et al. (1990) presented, using a vector approach, a more general procedure for correcting sampling bias on orientation, applicable for sampling planes of any orientation. 5.2.3 Photographic mapping Field sampling of discontinuities requires sufficient exposed and accessible rock faces, time and considerable cost. Photographic techniques provide a way of overcoming the above difficulties (Hagan, 1980; Thomas et al., 1987; Franklin et al., 1988; BlinLacroix et al., 1990; Tsoutrelis et al., 1990; Ord & Cheung, 1991; Crosta, 1997). Application of photographic techniques provides a partly automated method for estimating discontinuity characteristics including orientation, size, spacing and surface geometry. Photographs of the rock face have to get through the digitizing and processing phase before performing any correction and analysis. Many image processing software programs exist but, according to Crosta (1997), the best way is still the digitization of fracture lineaments (2D) by means of a digitizing board. This approach allows for a more correct representation of the fractures with respect to other automatic procedures. In fact, the operator, who preferably should be the same person who accomplished the field survey, is allowed to select or discard lineaments that can be ascribed to or not to the discontinuity network (vegetation, shadows, facets). The digitized photographs can then be interrogated by computerized sampling. The sampling techniques can vary from a single scanline or window to multiple scanlines or windows. Discontinuity characteristics obtained from photographic mapping have been found to agree well with values obtained using standard manual procedures (Tsoutrelis et al., 1990; Crosta, 1997).

5.3 BORINGS Borings, in most cases, provide the only viable exploratory tool that directly reveals geologic evidence of the subsurface site conditions. In addition to exploring geologic stratigraphy and structure, borings are necessary to obtain samples for laboratory engineering property tests. Borings are also frequently made for other purposes, such as collection of groundwater data, performing in situ tests, installing instruments, and exploring the condition of existing structures. Of the various boring methods, rock core borings are the most useful in rock foundation investigations. 5.3.7 Rock core boring Rock core boring is the process in which diamond or other types of core drill bits are used to drill exploratory holes and retrieve rock cores. Good rock core retrieval with a minimum of disturbance requires the expertise of an experienced drill crew. Core bits that produce 2.0 inch (nominal) diameter cores (i.e., NW or NQ bit sizes) are satisfactory for most exploration work in good rock as well as provide sufficient size samples for most

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rock index tests such as unconfined compression, density, and petrographic analysis. However, the use of larger diameter core bits ranging from 4.0 to 6.0 inches (nominal) in diameter are frequently required to produce good cores in soft, weak and/or fractured strata. The larger diameter cores are also more desirable for samples from which rock strength test specimens are prepared; particularly strengths of natural discontinuities. The number of borings and the depths to which bore holes should be advanced are dependent upon the subsurface geological conditions, the project site areas, types of projects and structural features. Where rock mass conditions are known to be massive and of excellent quality, the number and depth of borings can be minimal. Where the foundation rock is suspected to be highly variable and weak, such as karstic limestone or sedimentary rock containing weak and compressible seams, one or more borings for each major load bearing foundation element may be required. In cases where structural loads may cause excessive deformation, at least one of the boreholes should be extended to a depth equivalent to an elevation where the structure imposed stress acting within the foundation material is no more than 10 percent of the maximum stress applied by the foundation. While the majorities of rock core borings are drilled vertically, inclined borings and in some cases oriented cores are required to adequately define stratification and jointing. In near vertical bedding, inclined borings can be used to reduce the total number of borings needed to obtain core samples of all strata. Where precise geological structure is required from core samples, techniques involving oriented cores are sometimes employed. In these procedures, the core is scribed or engraved with a special drilling tool so that its orientation is preserved. In this manner, both the dip and dip orientation of any joint, bedding plane, or other planar surface can be ascertained. To ensure the maximum amount of data recovered from rock core borings it is necessary to correctly orient boreholes with respect to discontinuities present in the rock mass. If there is an outcrop present the main discontinuity sets should be established and the borehole(s) drilled to intersect these sets at as large an angle as possible. If no outcrop is present, the discontinuity pattern is unknown, and to ensure representative results, a minimum of three holes should be drilled as nearly orthogonal to each other as possible (ISRM, 1978; McMillan et al., 1996). (a) Core logging From core logging, one can obtain the total core recovery, discontinuity frequency, rock quality designation (RQD) and other discontinuity information such as orientation, spacing and aperture (ISRM, 1978). Before making detailed observations the core as a whole should be examined to determine the structural boundaries (domains) and geological features to be measured. The markers indicating depths of geological horizons and the start and end of each run should be carefolly checked for errors. Total core recovery is defined as the summed length of all pieces of recovered core expressed as a percentage of the length drilled and should be measured and recorded to the nearest 2% if possible. When the core is highly fragmented the length of such portions is estimated by assembling the fragments and estimating the length of core that the fragments appear to represent. Core recovery is normally used to describe individual core runs or whole boreholes, and not specific structurally defined rock units. The results obtained in a rock mass of poor quality will be strongly dependent on the drilling

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equipment and the skill of the drilling crew. Core grinding may result in excessive lost core. Core that is damaged in this way should always be recorded. The depth drilled at start and end of zones of core loss should be carefully recorded. The relevant lengths lost can be replaced by wooden blocks with markings on both ends. Frequency is defined as the number of natural discontinuities intersecting a unit length of recovered core and should be counted for each meter of core. Artificial fractures resulting from rough handling or from drilling process should be discounted only when they can be clearly distinguished from natural discontinuities. It should be noted that orientation bias need be corrected in order to obtain the true discontinuity frequency. This can be done by treating the core axis as a scanline and using Equation (5.2). Discontinuity spacing may also be estimated by matching the individual core pieces and measuring the length along the core axis between adjacent natural discontinuities of one set. Again the orientation bias need be corrected in order to obtain the true discontinuity spacing (Equation 5.1). Terzaghi’s (1965) method (Equations 5.1 and 5.2) for correcting the orientation bias assumes discontinuities of infinite size and does not consider the effect of borehole size. For discontinuities of finite size intersecting a borehole, the size of both the discontinuity and borehole will influence the probability of intersection. Mauldon and Mauldon (1997) developed a procedure for correcting orientation bias when sampling discontinuities using a borehole. In their approach, discontinuities are assumed to be discs of finite size and the borehole is assumed to be an infinitely long cylinder of circular cross section. Rock quality designation (RQD) is a modified core recovery percentage in which all the pieces of sound core over 4 in. (10.16 cm) long are counted as recovery, and are expressed as a percentage of the length drilled. The small pieces resulting from closer jointing, faulting or weathering are discounted. The detailed procedure for estimating RQD has been described in Chapter 2. (b) Core orientation To determine the dip and dip orientation of discontinuities from core samples, the core orientation need be known. The following briefly describes some of the techniques for determining the core orientation. Craelius Method (Goodman, 1976; ISRM, 1978) Orientation of the core is based on orienting the first piece of core in each core run. The orienting device is a cylinder, of about the same diameter as the core, with six locking extension feet. At the beginning of each core run, it is inserted on the front of the core barrel with the feet fully extended. It is lowered with the core barrel and when it hits the bottom, the feet are depressed differentially until they lock into position. The orienting device rides into the barrel as coring progresses. When the core barrel is emptied, the top of the core is laid in an alignment cradle against the Craelius device and rotated to find the proper fit of the feet against the rough top-of-core, and the orientation of the first piece of core can thus be determined. The orientation of other pieces is determined by laying them on a “V” trough in proper sequence and rotating them as necessary to fit all pieces together. This method works well if adjacent pieces of core can be matched. Zones of core loss and perpendicularly intersected discontinuities reduce the effectiveness of the method locally.

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Christensen-Huegel Method (Goodman, 1976; ISRM, 1978) The ChristensenHuegel barrel contains three knives on a shoe mounted on the end of the inner barrel, so that as the core enters the core holder, three grooves are cut longitudinally. The barrel also has a compass photo device to give the bearing of the hole and the orientation of a marker oriented relative to one of the scribing lines. Integral Sampling Method (Goodman, 1976; ISRM, 1978) The cores are first reinforced with a grouted bar whose azimuth is known from positioning rods. The reinforcing bar is co-axially overcored with a large diameter coring crown (see Fig. 5.3). Clay Core Barrel Method (Call et al., 1981) A modified inner core barrel is used with conventional diamond drilling equipment (see Fig. 5.4). The barrel is eccentrically weighted with lead and lowered into an inclined, fluid-filled borehole so that its orientation with respect to the vertical is known. Modeling clay protrudes from the downhole end of the inner barrel such that it also extends through the drill bit when the inner and outer tubes are engaged. The barrel assembly is pressed against the hole bottom which causes the clay to take an impression of the core stub left from the previous core run. The inner barrel is then retrieved with the wire-line and a conventional barrel is lowered to continue coring. At the completion of the run, the recovered core is fitted together and the core is oriented by matching the piece of core from the upper end of the core run with the oriented clay imprint. The clay barrel method can only be used in inclined holes within the dip range of 45° to 70° where the weighted barrel will orient itself as it is lowered down the hole. Like the Craelius method, this method works well only if adjacent pieces of core can be matched. Pendulum Orientation Method (Webber & Gowans, 1996) Orientation of the core is based on orienting the last piece of core in each core run. The pendulum orientation system incorporates a pendulum which moves under gravitational force while drilling to indicate the lowest position at an inclined borehole (see Fig. 5.5). The system depends on maintaining a fixed rotational relationship between the inner tube of the corebarrel and the orientation device containing the pendulum. This is achieved by rigidly fixing the orientation device to a modified spindle in the corebarrel head. Once the core run is complete an overshot trigger is lowered to activate the core orientator. The overshot device latches onto the core barrel assembly, triggering the pendulum by pushing it downwards against the action of a spring. The lowest position of the inner tube in the inclined borehole is then indicated by the point of the pendulum which emerges through one of the 72 small holes on the indicator plate. The pendulum and the inner tube are then fixed. The core barrel can then be removed. At the completion of the run, the last piece of core can be oriented by marking the lowest point from the point of the pendulum and the rest pieces of core can be oriented by matching the pieces of core from the lower end of the core run. The system is designed to operate in boreholes with a minimum inclination of 5° from the vertical. Like the Craelius method and clay barrel method, this method works well only if adjacent pieces of core can be matched.

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Fig. 5.3 A sketch showing stages in removing an integral sample: (1) positioning rod; (2) connecting sleeve; (3) cementing material; and (4) integral sample before drilling it free (after Goodman, 1976).

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Fig. 5.4 Clay core barrel used to orient diamond core in an inclined drill hole (after Call et al., 1981). 5.3.2 Inspecting borehole walls The sidewalls of the borehole from which the core has been extracted offer a unique picture of the subsurface where all structural features of the rock formation are still in their original position. This view of the rock can be important when portions of rock core have been lost during the drilling operation, particularly weak seam fillers, and when the true dip and dip direction of the structural features are required. The spacing of discontinuities can be determined by televiewing or photographing the borehole walls. Experience in examining a variety of rock types drilled using different methods suggests that, for most rock types, the drilling process does not create significant fractures in the borehole wall (Gunning, 1992). As a result discontinuity spacing collected from borehole walls may be inherently more accurate and precise than that derived from borehole cores which can be badly affected by drilling and handling process.

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Fig. 5.5 A pendulum orientation device attached to a conventional corebarrel (after Webber & Gowans, 1996). Borehole viewing and photography equipment include borescopes, photographic cameras, TV cameras, sonic imagery loggers, caliper loggers, and alignment survey devices. Borehole Periscopes can be used in small holes, but due to distortion of the optical path the depth is usually limited to about 30 m (Goodman, 1976; ISRM, 1978). Borehole cameras can be used to take photos of the borehole wall and the orientation and spacing data can be obtained by interpreting the photos (Goodman, 1976; ISRM, 1978). Closed Circuit Television (CCTV) provides a means of directly inspecting a borehole wall and if the direction of view of the camera can be orientated, it is possible to determine discontinuity orientation and spacing. In addition aperture, infilling and water seepage may also be assessed. Successful CCTV surveys can be conducted in either dry or waterfilled holes. For best results in either case the borehole wall should be clean and stable. If the borehole is full of water then measures should be taken to ensure that the water is clear enough to give a side view image of the borehole wall (Gunning, 1992; McMillan et al., 1996). Acoustic Televiewer and Dipmeter offer great potential for analyzing discontinuities. Advances in technology and digital instrumentation mean that such methods can provide effective data acquisition systems. Acoustic televiewers only operate in fluid filled boreholes (Gunning, 1992).

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5.3.3 Large-diameter borings Large-diameter borings, 2 feet or more in diameter (e.g., the borings of large drilled shafts), permit direct examination of the sidewalls and bottoms of the boring and provides access for obtaining high-quality undisturbed samples. Direct inspection of the sidewalls and bottoms may reveal details, such as thin weak layers or old shear planes, that may not be detected by continuous undisturbed sampling. However, direct measurements may not always be possible because of water in the borehole and concerns for the safety of personnel. It is very often that no outcrop can be used at a site to obtain trace lengths of discontinuities. In this case, trace length data can be obtained from the sidewalls and bottoms of large-diameter borings such as the borings of large drilled shafts. If only several traces or even no trace is present at a bottom, one can use bottoms at different depths during the drilling process to collect the trace length data and then use the entire data set (see Fig. 5.6).

5.4 EXPLORATORY EXCAVATIONS Test pits, test trenches, and exploratory tunnels provide access for larger-scaled observations of rock mass conditions, for determining top of rock profile in highly weathered rock/soil interfaces, and for some in situ tests which cannot be executed in a smaller borehole. 5.4.1 Test pits and trenches In weak or highly fractured rock, test pits and trenches can be constructed quickly and economically by surface-type excavation equipment. Final excavation to grade where samples are to be obtained or in situ tests performed must be done carefully. Test pits and trenches are generally used only above the groundwater level. Exploratory trench excavations are often used in fault evaluation studies. An extension of a bedrock fault into much younger overburden materials exposed by trenching is usually considered proof of recent fault activity.

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Fig. 5.6 Sampling at the bottom of the hole, during the drilling process at different depths, to obtain trace length data (L is the depth of the drilled shaft). 5.4.2 Exploratory tunnels Exploratory tunnels/adits permit detailed examination of the composition and geometry of rock structures such as joints, fractures, faults, shear zones, and solution channels. They are commonly used to explore conditions at the locations of large underground excavations and the foundations and abutments of large dam projects. They are particularly appropriate in defining the extent of marginal strength rock or adverse rock structure suspected from surface mapping and boring information. For major projects where high-intensity loads will be transmitted to foundations or abutments, tunnels/adits afford the only practical means for testing in-place rock at locations and in directions corresponding to the structure loading. The detailed geology of exploratory tunnels, regardless of their purpose, should be mapped carefully. The cost of obtaining an accurate and reliable geologic map of a tunnel is usually insignificant compared with the

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cost of the tunnel. The geologic information gained from such mapping provides a very useful additional dimension to interpretations of rock structure deduced from other sources. A complete picture of the site geology can be achieved only when the geologic data and interpretations from surface mapping, borings, and pilot tunnels are combined and well correlated. When exploratory tunnels are strategically located, they can often be incorporated into the permanent structure. Exploratory tunnels can be used for drainage and post-construction observations to determine seepage quantities and to confirm certain design assumptions. On some projects, exploratory tunnels may be used for permanent access or for utility conduits.

5.5 GEOPHYSICAL EXPLORATIONS 5.5.1 General description Geophysical techniques consist of making indirect measurements on the ground surface, or in boreholes, to obtain generalized subsurface information. Geologic information is obtained through analysis or interpretation of these measurements. Boreholes or other subsurface explorations are needed for reference and control when geophysical methods are used. Geophysical explorations are of greatest value when performed early in the field exploration program in combination with limited subsurface explorations. Geophysical explorations are appropriate for a rapid, though approximate, location and correlation of geologic features such as stratigraphy, lithology, discontinuities, ground water, and for the in situ measurement of dynamic elastic moduli and densities. The cost of geophysical explorations is generally low compared with the cost of core borings or test pits, and considerable savings may be realized by judicious use of these methods. Geophysical methods can be classified as active or passive techniques. Active techniques impart some energy or effect into the earth and measure the earth materials’ response. Passive measurements record the strengths of various natural fields which are continuous in existence. Active techniques generally produce more accurate results or more detailed solutions due to the ability to control the size and location of the active source. 5.5.2 Seismic methods Seismic methods are the most commonly conducted geophysical surveys for engineering investigations. Seismic surveys measure the relative arrival times, and thus the velocity of seismic waves traveling between an energy source and a number of geophones. The energy source may be a hammer blow, an explosion of a propaneoxygen mixture in a heavy chamber (gas-gun), or a light explosive charge. There are two major classes of seismic waves: body waves which pass through the volume of a material and surface waves that exist only near a boundary. The body waves consist of the compressional or pressure or primary wave (P-wave) and the secondary or transverse or shear wave (Swave). P-waves travel through all media that support seismic waves. P-waves in fluids, e.g. water and air, are commonly referred to as acoustic waves. S-waves travel slower than P-waves and can only transit material that has shear strength. S-waves do not exist in

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liquids and gases, as these media have no shear strength. The velocities of the P- and Swaves are related to the elastic properties and density of a medium by the following equations: (5.4)

(5.5)

(5.6) E=2G(1+ν) (5.7) where ν is the Poisson’s ratio; Vp is the velocity of the P-wave; Vs is the velocity of the Swave; G is the shear modulus; ρ is the density; and E is the Young’s or elastic modulus. It should be noted that these are not independent equations. Knowing two velocities uniquely determines only two unknowns of ρ, ν and E. Shear modulus is dependent on two other values. Usually the possible range of ρ is approximated and ν is estimated. The typical density values of intact rocks have been presented in Table 2.7. Table 5.1 provides some typical values of Vp and ν. The velocity of the S-wave in most rocks is about half the velocity of the P-wave. Surface waves are produced by surface impacts, explosions and wave form changes at boundaries. One of the surface waves is the Rayleigh wave which travels about 10% slower than the S-wave. The Rayleigh wave exhibits vertical and horizontal displacement in the vertical plane of the ray path. A point in the path of a Rayleigh wave moves back, down, forward, and up repetitively in an ellipse like ocean waves. The equipment used for seismic surveys includes the following components: 1) Seismic sources. The seismic source may be a hammer repetitively striking an aluminum plate or weighted plank, drop weights of varying sizes, a rifle shot, a harmonic oscillator, waterborne mechanisms, or explosives. The energy disturbance for seismic work is most often called the “shot,” an archaic term

Table 5.1 Typical/representative field values of Vp and ν (after ASCE, 1998). Material

Vp (m/s)

ν

Air

330

Damp loam

300–750

Dry sand

450–900

0.3–0.35

Clay

900–1,800

~0.5

Fresh, shallow water

1,430–1,490

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Saturated, loose sand

1,500

Basal/lodgement till

1,700–2,300

Rock

0.15–0.25

Weathered igneous & metamorphic rock

450–3,700

Weathered sedimentary rock

600–3,000

Shale

800–3,700

Sandstone

2,200–4,000

Metamorphic rock

2,400–6,600

Unweathered bsalt

2,600–4,300

Dolostone and limestone

4,300–6,700

Unweathered granite

4,800–6,700

Steel

6,000

from petroleum seismic exploration. Reference to the “shot” does not necessarily mean an explosive or rifle source was used. The type of survey dictates some source parameters. Smaller mass, higher frequency sources are preferable. Higher frequencies give shorter wavelengths and more precision in choosing arrivals and estimating depths. Yet sufficient energy needs to be entered to obtain a strong return at the end of the survey line. 2) Geophones. The geophones receiving seismic energy are either accelerometers or velocity transducers, and convert ground shaking into a voltage response. Most geophones are vertical, single-axis sensors to receive the incoming wave form from beneath the surface. Some geophones have horizontal-axis response for S-wave or surface wave assessments. Triaxial phones, capable of measuring absolute response, are used in specialized surveys. Geophones are chosen for their frequency band response. 3) Seismographs. The equipment that records input geophone voltages in a timed sequence is the seismograph. Current practice uses seismographs that store the channels’ signals as digital data in discrete time units. Earlier seismographs would record directly to paper or photographic film. Stacking, inputting, and processing the vast volumes of data and archiving the information for the client virtually require digital seismographs. In a homogeneous medium a bundle of seismic energy travels in a straight line. Upon striking a boundary between different material properties, wave energy is refracted, reflected, and converted. The properties of the two media and the angle at which the incident ray path strikes will determine the amount of energy reflected off the surface, refracted into the adjoining material, lost as heat, and changed to other wave types. Figure 5.7 shows the refraction and reflection of a seismic ray incident at an angle θ1 on the boundary between media with velocities V1 and V2. Refraction, as with any wave, obeys Snell’s Law relating the angle between the ray path and the normal to the boundary to the velocity V(VP or Vs appropriate). Thus in Figure 5.7(a), we have

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(5.8) If refraction continues through a series of such interfaces parallel to each other, we have (5.9) If the lower material has a higher velocity (V2>V1 in Fig. 5.7), a particular down-going ray making an angle (5.10)

with the normal will critically refract along the boundary and return to the surface at the same angle [see Fig. 5.7(b)].

Fig. 5.7 (a) Reflection and refraction of a seismic ray incident at an angle θ1 on the boundary between media with velocities V1 and V2; and (b) Critically refracted wave travels below the boundary and returns to the surface.

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(a) Seismic refraction method The method of seismic refraction is schematically shown in Figure 5.8. Waves of different types traveling by various paths to points on the surface at various horizontal distances, X, from the shot are detected by geophones. For geophones near the shot, the first arrivals will be directly from the shot. If the lower material has a higher velocity [V2>V1 in Fig. 5.8(a)], rays traveling along the boundary will be the first to arrive at geopehone away from the shot. If the time of first arrivals is plotted against distance X, a plot with two straight branches as shown in Figure 5.8(b) will be obtained. From an examination of Figure 5.8(b) one can obtain the following information: 1) The slopes of the two straight lines are equal to 1/V1 and 1/V2, respectively. 2) The depth to the interface, D, can be obtained by

Fig. 5.8 Simplified representation of seismic refraction method: (a) Shot, geophones and direct and refiracted wave paths; and (b) Time versus distance plot for seismic refraction survey as shown in (a).

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(5.11)

where XC is the crossover distance; and D is the depth to the horizontal refracting interface (Fig. 5.8). When the interface is dipping downwards from the shot towards the geophones, the velocity of the lower medium obtained as described above will be smaller than the true velocity. In the opposite situation, with the interface rising from the shot towards the geophones, the obtained velocity will be higher than the true one. By reversing shots and measuring the velocities in both directions (up- and down- dip) the dip of the interface can be estimated (ASCE, 1998). The method described above for finding the seismic wave velocities and the depths to the refracting interfaces can readily be extended to systems with three or more layers with boundaries that need not be planar and velocities that may show lateral changes. For details, the reader can refer to Griffiths and King (1981) and ASCE (1998). In simple cases, such as the two layer system described above, the seismic refraction method can predict depths to geological surfaces with an accuracy of ±10%. In complex formation, the accuracy drops considerably, and is much more dependent on the skill of the operators. The two most difficult geologic conditions for accurate refraction work are the existence of a thin water-saturated zone just above the bedrock and the existence of a weathered zone at the top of bedrock. The method fails completely, however, when a high velocity layer covers a low velocity one, since there is no refraction at this case. (b) Seismic reflection method A portion of the seismic energy striking an interface between two differing materials will be reflected from the interface (Fig. 5.7). The ratio of the reflected energy to incident energy is called the reflection coefficient. The reflection coefficient is defined in terms of the densities and seismic velocities of the two materials as: (5.12) where R is the reflection coefficient; ρ1 and ρ2 are densities respectively of the first and second layers; V1 and V2 are seismic velocities respectively of the first and second layers. Modern reflection methods can ordinarily detect isolated interfaces whose reflection coefficients are as small as 0.02. The physical process of reflection is illustrated in Figure 5.9, where the ray paths from the successive layers are shown. As in Figure 5.9, there are commonly several layers beneath the ground surface which contribute reflections to a single seismogram. Thus, seismic reflection data are more complex than refraction data because it is these later arrivals that yield information about the deeper layers. At later times in the record, more noise is present thus making the reflections difficult to extract from the unprocessed record. Figure 5.10 indicates the paths of arrivals that would be recorded on a multichannel seismograph. Another important feature of modern reflection data acquisition is

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illustrated by Figure 5.11. If multiple shots, S1 and S2, are recorded by multiple geophones, G1 and G2, and the geometry is as shown in the figure, the reflector point for both rays is the same. However, the ray paths are not the same length, thus the reflection will occur at different times on the two traces. This time delay, whose magnitude is indicative of the subsurface velocities, is called normalmoveout. With an appropriate time shift, called the normal-moveout correction, the two traces (S1 to G2 and S2 to G1) can be summed, greatly enhancing the reflected energy and canceling spurious noise.

Fig. 5.9 Schematic of seismic reflection method.

Fig. 5.10 Multi-channel recording of seismic reflections.

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Fig. 5.11 Common depth point recording. (c) Rayleigh wave method Of the surface waves, the Rayleigh wave is important in engineering studies because of its simplicity and because of the close relationship of its velocity to the shear wave velocity for earth materials. Approximation of Rayleigh wave velocities as shear-wave velocities causes less than a 10-percent error. Rayleigh wave studies for engineering purposes have most often been made in the past by direct observation of the Rayleigh wave velocities. One method consists of excitation of a monochromatic wave train and the direct observation of the travel time of this wave train between two points. As the frequency is known, the wavelength is determined by dividing the velocity by the frequency. (d) Cross hole method Cross hole testing takes advantage of generating and recording seismic body waves, both the P- and S-waves, at selected depth intervals where the source and receiver(s) are maintained at equal elevations for each measurement. Figure 5.12 illustrates a general field setup for the cross hole seismic test method. Knowing the distance between the source borehole and the geophone borehole and measuring the time of travel of the induced wave from the source to the geophone, the velocity of the rock mass can be simply obtained by (5.13) where D (D1 or D2 in Fig. 5.12) is the distance between the source borehole and the geophone borehole; and t is the time of travel of the induced wave from the source to the geophone. Particle motions generated with different seismic source types used during cross hole testing are three-directional. Therefore, three-component geophones with orthogonal orientations yield optimal results when acquiring cross hole P- and/or S-wave seismic

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signals. The requirement for multiple drill holes in cross hole testing means that care must be taken when completing each borehole with casing and grout. ASTM procedures call for PVC casing and a grout mix that closely matches the formation density. Another critical element of cross hole testing, which is often ignored,

Fig. 5.12 Schematic of cross hole method. is the requirement for borehole directional surveys. Borehole verticality and direction (azimuth) measurements should be performed at every depth interval that seismic data are acquired. Since seismic wave travel times should be measured to the nearest tenth of a millisecond, relative borehole positions should be known to within a tenth of a foot. Assuming that the boreholes are vertical and plumb leads to computational inaccuracies and ultimately to data which cannot be quality assured. Unlike surface seismic techniques previously described, cross hole testing requires a more careful interpretation of the wave forms acquired at each depth. For example, in cross hole testing, the first arrival is not always the direct ray path. As illustrated schematically in Figure 5.13, when the source and receivers are located within a layer that has a lower velocity than either the layer above or below it (this is termed a hidden layer in refraction testing), refracted waves can be the first arrivals. Both the source/receiver distance above or below the high velocity layer and the velocity contrast across the seismic interface determine if the refracted wave will arrive before the direct wave.

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(e) Down hole method and up hole method For the down hole method, the seismic waves are generated by a source located at the ground surface and the geophones are located inside a borehole drilled adjacent to the energy source (see Fig. 5.14). Each test involves the determination of the wave velocity based on the distance between the ground surface and the level at which the geophone is located, and the respective traveling time. The equipment is basically the same as for the cross hole method. The only significant difference is the energy source. The impulses are generated by hammering a plate after assuring a good contact with the ground. For generating P-waves the blow is normal to the surface of the plate (and to the ground surface) and for generating a S-waves the blow is horizontal (parallel to the ground surface). When measuring S-wave velocities, the test can be repeated with the geophone at the same level, by reversing the direction of the impact, which allows a second recording. In the opposite, for the up hole method, the seismic waves are generated in the borehole and the geophones are located at the ground surface. The main advantage of the down hole and up hole methods over the cross hole method lies in the fact that only one regular borehole is required to perform the test.

Fig. 5.13 Refracted ray paths in cross hole seismic test where V1>V2
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Fig. 5.14 Schematic of down hole method. 5.5.3 Electrical resistivity method Electrical resistivity surveying is based on the principle that the distribution of electrical potential in the ground around a current-carrying electrode depends on the electrical resistivities and distribution of the surrounding soils and rocks. The usual practice in the field is to apply an electrical direct current (DC) between two electrodes implanted in the ground and to measure the difference of potential between two additional electrodes that do not carry current (see Fig. 5.15). Usually, the potential electrodes are in line between the current electrodes, but in principle, they can be located anywhere. The current used is direct current, commutated direct current (i.e., a square-wave alternating current), or AC of low frequency (typically about 20 Hz). All analysis and interpretation are done on the basis of direct currents. The distribution of potential can be related theoretically to ground resistivities and their distribution for some simple cases, notably, the case of a horizontally stratified ground and the case of homogeneous masses separated by vertical planes (e.g., a vertical fault with a large throw or a vertical dike). For other kinds of resistivity distributions, interpretation is usually done by qualitative comparison of

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observed response with that of idealized hypothetical models or on the basis of empirical methods. Mineral grains composing soils and rocks are essentially nonconductive, except in some exotic materials such as metallic ores, so the resistivity of soils and rocks is governed primarily by the amount of pore water, its resistivity, and the arrangement of the pores. Since the resistivity of a soil or rock is controlled primarily by the pore water conditions, there are wide ranges in resistivity for any particular soil or rock type (see Table 5.2), and resistivity values cannot be directly interpreted in terms of soil type or lithology. Commonly, however, zones of distinctive resistivity can be associated with specific soil or rock units on the basis of local field or drill hole information, and resistivity surveys can be used profitably to extend field investigations into areas with very limited or nonexistent data. Also, resistivity surveys may be used as a reconnaissance method, to detect anomalies that can be further investigated by complementary geophysical methods and/or drill holes.

Fig. 5.15 Schematic of electrical resistivity method. Table 5.2 Typical resistivity values of soils and rocks (after Griffiths & King, 1981). Soil/Rock

Resistivity (ohm-m)

Soft shale, clay

1–10

Hard shale

10–500

Sand

50–1,000

Sandstone

50–5,000

Porous limestone

100–5,000

Dense limestone

>5,000

Igneous rock

100–1,000,000

Metamorphic rock

50–1,000,000

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5.5.4 Ground penetration radar Ground Penetrating Radar (GPR), also known as ground probing radar, ground radar or georadar, has been widely used in high-resolution mapping of soil and rock stratigraphy (Deng, 1996; Sharma, 1997). The GPR method uses high-frequency (80 to 1,000 MHz) electromagnetic (EM) waves transmitted from a radar antenna to probe the earth. The transmitted EM waves are reflected from various interfaces within the ground and are detected by the radar receiver. Reflecting interfaces may be soil horizons, the groundwater surface, soil/rock interfaces, man-made objects, or any other interfaces possessing a contrast in dielectric properties. The GPR method is analogous to seismic reflection except for the energy source (Sharma, 1997; ASCE, 1998). Contrasts in dielectric properties across an interface cause EM waves to be reflected. Fracture fillings with dielectric properties different from their adjacent rock materials can cause radar reflections and thus can be detected. One limitation of the GPR method is that the penetration depth of radar is limited usually less than 20 meters (Cummings, 1990; Kearey & Brooks, 1991; Sharma, 1997; ASCE, 1998). At the Gypsy Outcrop Site in Northeastern Oklahoma, the maximum depth with noticeable radar response is about 10 meters (Deng, 1996). Therefore, the GPR method can only be used for shallow depth survey. Similar to the seismic wave method, the processing and interpretation of recorded GPR data is critically important. Due to the kinematic similarities between radar and seismic wave propagation, seismic processing techniques are widely used to process the GPR data (Deng, 1996; Sharma, 1997).

5.6 LABORATORY TESTING Laboratory tests are usually performed to determine index values for identification and correlation, further refining the geologic model of the site, and provide values for engineering properties of the rock used in the analysis and design of foundations. The selection of samples and the number and type of tests are influenced by local subsurface conditions and the size and type of structure. Prior to any laboratory testing, rock cores should have been visually classified and logged. Selection of samples and the type and number of tests can best be accomplished after development of the geologic model using results of field observations and examination of rock cores, together with other geotechnical data obtained from earlier preliminary investigations. The geologic model, in the form of profiles and sections, will change as the level of testing and the number of tests progresses. Testing requirements are also likely to change as more data become available and are reviewed for project needs. Table 5.3 summarizes laboratory tests according to purpose and type. The tests listed are the types more commonly performed for input to rock foundation analysis and design. Details and procedures for individual test types can be found in books on rock mechanics and rock engineering. For rock specimens with the same geometrical shape, the strength decreases with increasing size, reaching a limit value asymptotically (see Section 4.5 for details on scale effect). This size, beyond which no further decrease in strength is observed, depends on the type of rock material. A simplified explanation for this phenomenon is that rock is not

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a continuous solid material, but may contain various types of discontinuities or flaws. The strength of any rock specimen is, therefore, a statistical value depending on how many and what type of discontinuities are present. In smaller specimens the probability of the presence of such discontinuities is smaller and thus the strength is higher. In addition to the size effect, the strength of a rock specimen is affected by its shape, i.e. the length-to-diameter ratio of the test specimen or the width-to-height ratio of the specimen with a square cross-section. Figure 5.16 shows the effect of length-to-

Table 5.3 Summary of purpose and type of laboratory tests for rock (after ASCE, 1996). Purpose of test

Type of test

Strength

Unconfined compression Direct shear Triaxial compression Direct tension Brazilian split Point load1

Deformability

Unconfined compression Triaxial compression Swell Creep

Permeability

Gas permeability

Characterization

Water content Porosity Density (unit weight) Specific gravity Absorption Rebound Sonic velocities Abrasion resistance

1. Point load tests are also frequently performed in the field.

diameter ratio on the unconfined compressive strength of cylindrical sandstone specimens (John, 1972). This effect may be explained by the variation in the stress distribution in the test specimen as a result of the end constraint. The influence of loading platens on the specimen ends in a compression test diminishes with increasing length of specimen. To minimize the effects of size and shape on the test results, minimum dimensions and minimum height-to-diameter ratios for test specimens have been recommended by ASTM (1971) and ISRM (1979a).

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5.7 IN SITU TESTING 5.7.1 General description In situ tests are often the best means for determining the engineering properties of subsurface materials and, in some cases, may be the only way to obtain meaningful results. Table 5.4 lists some of the most widely used in situ tests and their purposes. In situ rock tests are performed to determine in situ stresses and deformation properties of the jointed rock mass, shear strength of jointed rock mass or critically weak seams within the rock mass, residual stresses within the rock mass, and rock mass permeability. Largescale in situ tests tend to average out the effect of complex interactions. In situ tests in rock are frequently expensive and should be reserved for projects with large, concentrated loads. Well-conducted tests may be useful in reducing overly conservative assumptions. Such tests should be located in the same general area as the proposed structure and test loading should be applied in the same direction as the proposed structural loading. Some of the strength and deformability tests are described in more detail in the following.

Fig. 5.16 Influence of length-todiameter ratio on the unconfined compressive strength of dry sandstone (after John, 1972).

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Table 5.4 Summary of purpose and type of in situ tests for rock (modified from ASCE, 1996). Purpose of test

Type of test

Strength

Field vane shear1 Direct shear Pressuremeter2 Unconfined compression2 Borehole jacking2

Deformability

Seismic3 Pressuremeter or dilatometer Plate bearing Radial (tunnel) jacking2 Borehole jacking2 Chamber (gallery) pressure2

Bearing capacity

Plate bearing1 Standard penetration1

Stress conditions

Hydraulic fracturing Pressuremeter Overcoring Flat jack Uniaxial (tunnel) jacking2 Chamber (gallery) pressure2

Permeability

Constant head Rising or falling head Well slug pumping Pressure injection

Notes: 1. Primarily for clay shales, badly decomposed, or moderately soft rocks, and rock with soft seams. 2. Less frequently used. 3. Dynamic deformability.

5.7.2 Strength tests The most common in situ test for determining the strength of rock masses is the direct shear test. Triaxial tests have been conducted in particular situations, but due to difficulties in execution, they are of very restricted use. The scope of the direct shear test is to measure the peak and residual shear strength as a function of the normal stress on the sheared plane. At least three or four specimens should be tested at different normal stresses on the same test horizon. Figure 5.17 shows a typical in situ direct shear test arrangement. In the case of tests conducted in adits, the reaction for the normal load is obtained from the opposite wall of the adit. Tests can be conducted on a rock surface using cables anchored into the rock adjacent to the test site to supply the normal reaction. The procedure for conducting an in situ direct shear test can be referred to Oliveira and Charrua Graca (1987). By conducting

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direct shear test on a set (three or four) of blocks, the shear strength curves as shown in Figure 5.18 can be obtained. The peak and residual strength

Fig. 5.17 Schematic of in situ shear test (after Oliveira & Charrua Graca, 1987).

Fig. 5.18 Shear strength curves from in situ shear tests (after Oliveira & Charrua Graca, 1987).

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parameters (cohesion and friction angle) can then be determined from the shear strength curves. 5.7.3 Deformability tests As shown in Table 5.4, there are a number of in situ test methods available to estimate rock modulus (ASCE, 1996). The details about the seismic methods have been discussed in Section 5.5. In this section, the borehole dilatometer test, the boehole jack test, the plate bearing test, the flat jack test and the radial jacking test will be discussed. (a) Borehole dilatometer test The borehole dilatometer test expands a fluid filled flexible membrane in a borehole causing the surrounding wall of rock to deform. The fluid pressure and the volume of fluid equivalent to the volume of displaced rock are recorded. From the theory of elasticity, pressure and volume changes are related to the modulus. The primary advantage of the borehole dilatometer test is its low cost. The test is, however, restricted to relatively soft rock. Furthermore, the test influences only a relatively small volume of rock. Hence, modulus values derived from the borehole dilatometer tests are not considered to be representative of rock mass conditions. The deformation modulus Em of the rock mass can be calculated from the dilatometer test by (Goodman, 1980) (5.14) where νm is the Poisson’s ratio of the rock mass; d is the diameter of the borehole test section; ∆p is the change in pressure applied uniformly over the borehole surface; and ∆d is the measured radial deformation. For a Colorado School of Mines (CSM) dilatometer, a calibration test in a material of known modulus need be conducted to determine the stiffness of the membrane system. Figure 5.19 shows typical pressure-dilation curves for a calibration test and a test carried out in rock. A complete test usually consists of three loading and unloading cycles, with dilation and pressure readings being taken on both the loading and unloading cycles. The shear modulus Gm and the deformation modulus Em of the rock mass in the borehole test section are given by (ISRM, 1987) (5.15)

and Em=2(1+νm)Gm (5.16) where L is the length of the test section (cell membrane); d is the diameter of the borehole test section; νm is the Poisson’s ratio of the rock mass; ρ is the pump constant (the fluid

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volume displaced per turn of pump wheel); and km is the stiffness of the rock mass, which can be obtained by

Fig. 5.19 Typical pressure-dilation curves for a Colorado School of Mines (CSM) dilatometer (IRSM, 1987). (5.17) where ks is the stiffness of the hydraulic system [Equation (5.19)]; and kT is the stiffness of the overall system plus the rock mass (ratio D/C in Fig. 5.19). The rock mass stiffness km is calculated from calibration of the hydraulic system and the results of a pressuredilation test carried out in a calibration cylinder of known modulus. The steps for calculating the rock mass stiffness are as follows. If the shear modulus and Poisson’s ratio of the calibration cylinder are respectively Gc and νc, the stiffness of the calibration cylinder kc is (5.18)

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where ri and ro are respectively the inside and outside radii of the calibration cylinder. The stiffness of the hydraulic system ks is calculated from the stiffness of the calibration cylinder and the slope of the calibration pressure-dilation curve km (ratio B/A in Fig. 5.19) as follows (5.19) It is also necessary to make a correction for pressure losses due to the rigidity of the membrane. This is determined by inflating the dilatometer in the air without confinement to show the pressure required to inflate the membrane and the hydraulic system. pi,corr=pi−nmp (MPa) (5.20) where pi,corr is the corrected pressure; pi is the indicated pressure; n is the number of turns to attain pi; and mp is the slope of pressure-dilation curve for dilation in air (MPa/turn). Another correction is required to account for loss of volume in the hydraulic system that takes place in inflating and seating the membrane. For the test measurements shown in Figure 5.19, the net corrected number of turns ∆ncorr is calculated from (5.21)

(b) Borehole jack test Instead of applying a uniform pressure to the full cross-section of a borehole as in the borehole dilatometer tests, the borehole jack presses plates against the borehole walls using hydraulic pistons, wedges, or flat jacks. This technique allows the application of significantly higher pressures required to deform hard rock. The NX-borehole (76 mm in diameter) jack (also known as the hard-rock jack or Goodman Jack) is the best known device for this test (Fig. 5.20). The deformation modulus of the rock mass can be calculated from a NX-borehole jack test by (Heuze, 1984; Heuze & Amadei, 1985) Ecalc=0.86(0.93)d(∆Qh/∆d)T* (5.22) where 0.86 is the factor for the three-dimensional effect; 0.93 is the hydraulic efficiency; d is the diameter of the borehole; ∆d is the change of borehole diameter; ∆Qh is the increment of hydraulic-line pressure; and T* is a coefficient depending on the Poisson’s ratio νm of the rock mass (Table 5.5). In rock with a deformation modulus greater than about 7 GPa, there will be a longitudinal outward bending of the jack platens and the calculated modulus need be corrected to obtain the true modulus Em (Fig. 5.21). This correction is necessary because the bending gives larger displacements at the ends than at the center of the loading platens and the displacement gauges are located near the ends of the platens (Heuze, 1984).

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The advantage of the borehole jack test over the borehole dilatometer test is that the unidirectional pressure can be imposed in a given orientation. The limitation of the borehole jack test is that only a point modulus (for a small volume of rock mass) can be obtained.

Fig. 5.20 Schematic of loading of NXborehole jack (Heuze, 1984). Table 5.5 T* for full platen and rock contact (after Heuze, 1984). νm

0.1

0.2

0.25

0.3

0.33

0.4

0.5

T*

1.519

1.474

1.438

1.397

1.366

1.289

1.151

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Fig. 5.21 Curve of Ecalc versus Em (after Heuze, 1984). (c) Plate bearing test In a plate bearing test a plane circular area of the rock surface is loaded and the induced deformation of the rock is measured. Usually the test is performed in small tunnels or adits and two opposite surfaces are loaded, thus providing the reaction support for the forces employed (Fig. 5.22). When it is necessary to conduct a test at the ground surface, special structures such as cables anchored at some depth below the surface must be employed to support the reaction (Fig. 5.23). The site selected for a test should be large enough and carefully prepared. The areas to be loaded and their vicinities, from 0.5 to 1 diameters of the loading plate, must be cleaned of all disturbed rock. These areas should then be hand-prepared to become plane and parallel. Usually a tunnel diameter gauge connecting the centers of the loaded surface measures the relative deformation. If more sophisticated information on deformation is needed, multiple-position borehole extensonmeters (MPBX) installed in holes drilled into the

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rock along the load axis can be used. The depth of the extensometer holes must be such that the deepest anchor is beyond the zone of deformation, a distance of about six diameters of the loading plate [Fig. 5.22(b)]. Assuming that the loaded surface behaves like a homogeneous infinite half space and that the rock mass behaves like an isotropic elastic linear medium, the deformation modulus of the rock mass can be calculated from the deformation measurements. For a test condition in which the bearing plate is circular and has a circular hole in the center through which the deformation measurements are made, the deformation modulus Em at any depth z is given by the following expression (5.23)

where δz is the measured displacement at depth z below the lower surface of the loading plate; p is the applied pressure on the loading plate; νm is the Poisson’s ratio of the rock mass; R is the outer radius of the loading plate; r is the radius of the hole in the center of the loading plate; and C is a constant. For a perfectly rigid loading plate, the theoretical solution gives C as π/2. Since the actual loading plate has some flexibility, the measured deformation is somewhat greater than the theoretical deformation. This results in the calculated deformation modulus being smaller than the true modulus and for this reason the constant C is usually given the value of 2. For a loading plate with no center hole, the deformation modulus is given by (5.24) For measurements at the surface of the rock where z=0, this expression reduces to

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210

Fig. 5.22 (a) Simple plate bearing test in a gallery; and (b) Plate bearing test in a gallery using multiple-position borehole extensonmeters (MPBX) (after IRSM, 1979b).

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211

Fig. 5.23 (a) Plate bearing test using a single anchorage cable; and (b) Plate bearing test using multiple anchorage cables.

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212

(5.25)

(d) Flat jack test The flat jack test is a simple test in which slots cut in the rock mass are uniformly loaded by flat jacks inserted into them (Fig. 5.24). Deformation of the rock mass caused by pressurizing the flat jack is measured by the volumetric change in the jack fluid. The deformation modulus of the rock mass is derived from relationships between jack pressure and deformation. Using loading, unloading and reloading cycles permits calculation of the deformation modulus of the rock mass by (Jaeger & Cook, 1979) (5.26)

where p is the applied pressure; 2c is the length of the jack; 2∆y is the variation of pin separation; νm is the Poisson’s ratio of the rock mass; and y is the distance from the jack center to each of a pair of measuring pins. The primary advantages of the flat jack test lie in its ability to load a large volume of rock and its relatively low cost.

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Fig. 5.24 Flat jack test: (a) Front view; and (b) Section after flat jack installation (after Jaeger & Cook, 1979). (e) Radial jacking test In this test the load is applied uniformly to the complete surface of a test chamber with a circular cross section and the radial deformations are measured along a number of axes. The results of this test provide the deformation modulus of a larger volume of the rock mass than is possible with the tests described earlier and, in the case of anisotropic rock, can show the variation of the modulus with orientation. Although more information on rock conditions is provided by the radial jacking test than the tests described earlier, the high cost and time required to conduct the test means that very few will be carried out and thus the results may not be representative of the overall site. The best known and oldest variant of the radial jacking test is the pressure chamber test in which the pressure is applied by means of water filling the chamber. With this method, special care must be taken not to allow the water to flow from the chamber.

6 Axial load capacity of drilled shafts in rock 6.1 INTRODUCTION The design of axially loaded drilled shafts in rock usually involves computation of ultimate load capacity and prediction of settlement under working load. This chapter addresses the determination of the ultimate load capacity while the prediction of settlement at the working load will be discussed in Chapter 7. Axially loaded drilled shafts in rock are designed to transfer structural loads to rock in one of the following three ways (CGS, 1985): 1. Through side shear only; 2. Through end bearing only; 3. Through the combination of side shear and end bearing. Situations where support is provided solely by side shear resistance are those where the base of the drilled hole cannot be cleaned so that it is uncertain if any end bearing resistance will be developed. Alternatively, where sound bedrock underlies low strength overburden material, it may be possible to achieve the required support in end bearing only, and assume that no side shear support is developed in the overburden. However, where the shaft is drilled some depth into sound rock, a combination of side shear resistance and end bearing resistance can be assumed (Kulhawy & Goodman, 1980). The load bearing capacity of a drilled shaft in rock is determined by the smaller of the two values: the structural strength of the shaft itself, and the ability of the rock to support the loads transferred by the shaft.

6.2 CAPACITY OF DRILLED SHAFTS RELATED TO REENFORCED CONCRETE Axially loaded drilled shafts may fail in compression or by buckling. Buckling is possible in the long and slender part that extends above the ground surface. Scour of the soil/rock around the shaft will expose portions of the shaft, thus extending the unbraced length and making the shaft more prone to buckling. The capacity of a shaft as a reinforced concrete element is a function of the shaft diameter, the strength of the concrete and the amount and type of reinforcement. The shaft should be designed such that the working stresses are limited to the allowable concrete stresses as shown in Table 6.1. For the reinforcing steel, the allowable design stress should not exceed 40% of its specified minimum yield strength, nor 206.8 MPa (30,000 psi) (ASCE, 1997).

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In LRFD, the ultimate (factored) axial capacity of a drilled shaft can be calculated using the expression for reinforced concrete columns: (6.1) where is the capacity reduction (resistance) factor=0.75 for spiral columns and 0.70 for horizontally tied columns (ACI, 1995); Qu is the nominal (computed) structural capacity; β is the eccentricity factor=0.85 for spiral columns and 0.80 for tied columns; is the specified minimum concrete strength; Ac is the cross-sectional area of the concrete; fy is the yield strength of the longitudinal reinforcing steel; and As is the crosssectional area of the longitudinal reinforcing steel. The Standard Specifications for Highway Bridges adopted by the American Association of State Highway and Transportation Officials (AASHTO, 1989) stipulates a minimum shaft diameter of 18 inches, with shaft sizing in 6-inch increments. Where the potential for lateral loading is not significant, drilled shafts need to be reinforced for axial loads only. The design of longitudinal and spiral reinforcement should conform to the requirements of reinforced compression members.

Table 6.1 Allowable concrete stresses for drilled shafts (after ASCE, 1993). Uniform axial compression Confined

0.33f′c

Unconfined

0.27f′c

Uniform axial tension

0

Bending (extreme fiber) Compression

0.40f′c

Tension

0

Note: f′c is the specified minimum concrete strength.

6.3 CAPACITY OF DRILLED SHAFTS RELATED TO ROCK Assuming that the shaft itself is strong enough, its load capacity depends on the capacity of the rock to accept without distress the loads transmitted from the shaft. The required area of shaft-rock interface (i.e., the size of drilled shaft) depends on this factor. The ultimate axial load of a drilled shaft related to rock, Qu, consists of the ultimate side shear load, Qus, and the ultimate end bearing load, Qub (see Fig. 6.1): Qu=Qus+Qub (6.2)

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216

The ultimate side shear load and the ultimate end bearing load are respectively calculated as the average side shear resistance multiplied by the shaft side surface area and as the end bearing resistance multiplied by the shaft bottom area, i.e.

Fig. 6.1 Axially loaded drilled shaft. Qus=πBLτmax (6.3) (6.4) where L and B are respectively the length and diameter of the shaft; and τmax and qmax are respectively the average side shear resistance and the end bearing resistance. The ultimate side shear resistance and the end bearing resistance are usually determined based on local experience and building codes, empirical relations, or field load tests. Methods based on local experience and building codes and empirical relations are discussed in this chapter. The methods for conducting field load tests and interpretation of test results will be discussed in Chapter 12. 6.3.1 Side shear resistance The shear resistance mobilized at the shaft-rock interface is affected by many factors. These include the shaft roughness, strength and deformation properties of the concrete

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and the rock mass, geometry of the shaft, and initial stresses in the ground. The effect of shaft roughness is emphasized by most investigators and considered in a number of empirical relations for estimating the side shear resistance. (a) Correlation with SPT N value Standard Penetration Tests (SPT) are often carried out in weak or weathered rock. Table 6.2 shows the measured side shear resistances of drilled shafts and their corresponding SPT N values in weathered sedimentary rocks. It can be seen that the τmax/N ratio is generally smaller than 2.0 except the case reported by Toh et al. (1989). We can also see that the τmax/N ratio tends to decrease as N increases.

Table 6.2 Side shear resistance and SPT N values in weathered sedimentary rock. Rock

SPT N values (blows/0.3 m)

τmax (kPa)

τmax/N (kPa)

Reference

Highly weathered siltstone

230

>195– 226

>0.87– 1.0

Buttling (1986)

Highly weathered siltstone, silty sandstone and shale

100–180

100– 320

1.0–1.8

Chang and Wong (1987)

Very dense clayey/sandy silt to highly weathered siltstone

110–127

80–125 0.63– 1.14

Highly to moderately weathered siltstone

200–375

340

0.9–1.7

Completely to partly weathered interbedded sandstone, siltstone and shale/mudstone

100–150 150–200

– –

1.2–3.7 0.6–2.3

Toh et al. (1989)

Highly to moderately fragmented siltstone/shale

400–1000

300– 800

0.5–0.8

Radhakrishnan and Leung (1989)

Highly weathered sandy shale

150–200

120– 140

0.8–0.7

Moh et al. (1993)

Slightly weathered sandy shale and sandstone

375–430

240– 280

ave. 0.65

Buttling and Lam (1988)

(b) Empirical relations between side shear resistance and unconfined compressive strength of intact rock Empirical relations between the side shear resistance and the unconfined compressive strength of rock have been proposed by many researchers. The form of these empirical relations can be generalized as τmax= ασcβ (6.5)

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where τmax is the side shear resistance; σc is the unconfined compressive strength of the intact rock (if the intact rock is stronger than the shaft concrete, σc of the concrete is used); and α and β are empirical factors. The empirical factors proposed by a number of researchers have been summarized by O’Neill et al. (1996) and are shown in Table 6.3. Most of these empirical relations were developed for specific and limited data sets, which may have correlated well with the proposed equations. However, O’Neill et al. (1996) compared the first nine empirical relations listed in Table 6.3 with an international database of 137 pile load tests in intermediate-strength rock and concluded that none of the methods could be considered a satisfactory predictor for the database. Kulhawy and Phoon (1993) developed a relatively extensive load test database for drilled shafts in soil and rock and presented their data both for individual shaft load tests and as site-averaged data. The results are shown in Figures 6.2 and 6.3, in terms of adhesion factor, σc, versus normalized shear strength, cu/pa or σc/2pa (assuming cu≈ σc/2), where pa is atmospheric pressure (≈0.1 MPa). It should be noted that Kulhawy and

Table 6.3 Empirical factors a and β for side shear resistance (modified from O’Neill et al., 1996). Design method

α

β

Horvath and Kenney (1979)

0.21

0.50

Carter and Kulhawy (1988)

0.20

0.50

Williams et al. (1980)

0.44

0.36

Rowe and Armitage (1984)

0.40

0.57

Rosenberg and Journeaux (1976)

0.34

0.51

Reynolds and Kaderbek (1980)

0.30

1.00

Gupton and Logan (1984)

0.20

1.00

Reese and O’Neill (1987)

0.15

1.00

Toh et al. (1989)

0.25

1.00

Meigh and Wolshi (1979)

0.22

0.60

Horvath (1982)

0.20–0.30

0.50

Phoon (1993) defined αc as the ratio of the side shear resistance τmax to the undrained shear strength cu. Understandably, the results of individual load tests show considerably greater scatter than the site-averaged data. On the basis of the site-averaged data, Kulhawy and Phoon (1993) proposed the following relations for drilled shafts in rock: (6.6a) (6.6b)

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219

(6.6c) Equation (6.6) can be rewritten in a general form as (6.7) This leads to a general expression for the side shear resistance τmax=Ψ[paσc/2pa]−0.5 (6.8) It is very important to note that the empirical relations given in Equations (6.6b) and (6.6c) are bounds to site-averaged data, and do not necessarily represent bounds to individual shaft behavior. The coefficient of determination (r2) is approximately 0.71 for the site-averaged data, but is only 0.46 for the individual data, reflecting the much greater variability of the individual test results (Seidel & Haberfield, 1995).

Fig. 6.2 Adhesion factor αc(=τmax/0.5σc) versus normalized shear strength for site-averaged data (after Kulhawy & Phoon, 1993).

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220

Fig. 6.3 Adhesion factor αc(=τmax/0.5σc) versus normalized shear strength for individual test data (after Kulhawy & Phoon, 1993). (c) Empirical relations considering roughness of shaft wall The roughness of the shaft wall is an important factor controlling the development of side shear resistance. Depending on the type of drilling technique and the hardness of the rock, a drilled shaft will have a certain degree of roughness. Research has shown that the benefits gained from increasing the roughness of a shaft wall can be quite significant, both in terms of peak and residual shear resistance. Studies by Williams et al. (1980) and others showed that smooth-sided shafts exhibit a brittle type of failure, while shafts having an adequate roughness exhibit ductile failure. Williams and Pells (1981) suggested that rough shafts generate a locked-in normal stress such that there is practically no distinguishing difference between peak and residual side shear resistance. Classifications have been developed so that roughness can be quantified. One such classification proposed by Pells et al. (1980) is based on the size and frequency of grooves in the shaft wall (see Table 6.4). Based on this classification, Rowe and Armitage (1987b) proposed the following relation for shafts with different roughness: τmax=0.45(σc)0.5 for shafts with roughness R1, R2 or R3 (6.9a) τmax=0.60(σc)0.5 for shafts with roughness R4 (6.9b) where both τmax and σc are in MPa.

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221

Horvath et al. (1980) also developed a relation from model shaft behavior using various roughness profiles. They found that as shaft profiles go from smooth to rough, the roughness factor increases significantly, as does the peak side shear resistance. These findings were confirmed in a later study by Horvath et al. (1983), and the following equation was proposed for the roughness factor (RF): (6.10) where hm is the average roughness (asperity) height of the shaft; Lt is the total travel length along the shaft wall profile; R is the nominal radius of the shaft; and L is the nominal length of the shaft (see Fig. 6.4). Using Equation (6.10), the following relation was developed between the side shear resistance and RF: τmax= 0.8σc(RF)0.45 (6.11) Kodikara et al. (1992) developed a rational model for predicting the relationship of τmax to σc based on a specific definition of interface roughness, initial normal stress on the interface and the stiffness of the rock during interface dilation. The parameters needed to define interface roughness in the model are also shown in Figure 6.4. The model accounts for variability in asperity height and angularity, assuming clean, triangular interface discontinuities. Figure 6.5 shows the predicted adhesion factor, α(=τmax/σc), for Melbourne Mudstone with the range of parameters and roughnesses as given in Table 6.5. The adhesion factor is presented as a function of Em/σc, σc/σn and the degree of roughness, where Em is the elastic modulus of the rock mass and σn is the initial normal stress on the shaft-rock interface. It can be seen that the adhesion factor is affected not only by the interface roughness, but also by Em/σc and σc/σn.

Table 6.4 Roughness classes after Pells et al. (1980). Roughness Class

Description

R1

Straight, smooth-sided shaft, grooves or indentation less than 1.00 mm deep

R2

Grooves of depth 1–4 mm, width greater than 2 mm, at spacing 50 to 200 mm.

R3

Grooves of depth 4–10 mm, width greater than 5 mm, at spacing 50 to 200 mm.

R4

Grooves or undulations of depth greater than 10 mm, width greater than 10 mm, at spacing 50 to 200 mm.

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Fig. 6.4 Parameters for defining shaft wall roughness (after Horvath et al., 1980 and Kodikara et al., 1992).

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223

Fig. 6.5 Simplified design charts for adhesion factor α(=τmax/σc) for Melbourne Mudstone (after Kodikara et al., 1992).

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224

Table 6.5 Definition of borehole roughness and range of parameters for Melbourne Mudstone (after Kodikara et al., 1992). Range of values for shafts in Melbourne Mudstone Parameter

Smooth

im(degrees)

10–12

isd(degrees) hm(mm)

Medium

Rough 12–17

17–30

2–4

4–6

6–8

1–4

4–20

20–80

hsd/hm

0.35

B(m)

0.5–2.0

σc(MPa)

0.5–10.0

σn(MPa)

50–500

Em(MPa)

50–500

Notes: 1) Refer to Figure 6.4 for the definitions of im, isd, hm and hsi 2) B=diameter of the shaft. 3) σc=unconfined compressive strength of the intact rock. 4) σn=initial normal stress on the shaft-rock interface. 5) Em=deformation modulus of the rock mass.

Seidel and Collingwood (2001) introduced a nondimensional factor called Shaft Resistance Coefficient (SRC) to reflect the influence of shaft roughness and other factors on the shaft side shear resistance. The SRC is defined as follows: (6.12) where hm is the mean roughness height (either assessed directly by estimation or measurement, or computed as the product of asperity length, la, and the sine of the mean asperity angle); B is the shaft diameter; ηc is the construction method reduction factor as shown in Table 6.6; n is the ratio of rock mass modulus to the unconfined compressive strength of the rock (Em/σc), known as the modulus ratio; and ν is the Poisson’s ratio of the rock. Using SRC, Seidel and Collingwood (2001) have created shaft resistance charts as shown in Figures 6.6 and 6.7. These charts are based on results of a parametric study using a computer program called ROCKET. To develop these charts, the intact rock strength parameters were related to the unconfined compressive strength using the HoekBrown strength criteria described in Chapter 4. Mohr-Coulomb strength parameters adopted in the analyses were determined after the method of Hoek (1990) using the unconfined compressive strength of the rock and appropriate values of parameters s and m.

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(d) Estimation of roughness height of shaft wall Application of the empirical relations considering shaft wall roughness in design requires estimation of likely shaft wall roughness height. A small number of studies have produced actual roughness profiles which enable quantitative analysis. Detailed studies have been carried out into shafts in Melbourne Mudstone (Williams, 1980; Holden, 1984; Kodikara et al., 1992; Baycan, 1996). The results show that shaft wall roughness in this low- to medium-strength argillaceous rock can vary considerably and appears to be influenced by rock discontinuities, drilling techniques, and rate of advance. Shaft wall roughness profiles in medium-strength shale were also recorded by Horvath et al. (1983), but most of their shafts were artificially roughened by grooving. O’Neill & Hassan (1994) and O’Neill et al., (1996) recorded measurements of roughness profiles of shafts in clay shale, argillite and sandstone.

Table 6.6 Indicative construction method reduction factor ηc (after Seidel & Collingwood, 2001). Construction method

ηc

Construction without drilling fluid Best construction practice and high level of construction control

1.0

(e.g., shaft sidewalls free of smear and remoulded rock) Poor construction practice or low-quality construction control (e.g.,

0.3–0.9

smear or remoulded rock present on shaft sidewalls) Construction under bentonite slurry Best construction practice and high level of construction control

0.7–0.9

Poor construction practice or low-quality construction control

0.3–0.6

Construction under polymer slurry Best construction practice and high level of construction control

0.9–1.0

Poor construction practice or low-quality construction control

0.8

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226

Fig. 6.6 Adhesion factor α(=τmax/σc) versus σc (after Seidel & Collingwood, 2001).

Fig. 6.7 Adhesion factor α(=τmax/σc) versus SRC (after Seidel & Collingwood, 2001).

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227

Based on roughness heights back-calculated from load tests on shafts in rock, Seidel and Collingwood (2001) developed the effective roughness height versus the unconfined compressive strength plot as shown in Figure 6.8. The back-calculations were conducted using Equation (6.12) and assuming ηc=1.0. In the case of a shaft for which the concreterock interface is clean and unbounded, the roughness height back-calculated assuming ηc=1.0 should provide a reasonable estimate of the roughness height magnitude. However, if the shaft resistance is adversely influenced by construction procedures, the roughness height would be underestimated if ηc is assumed to be 1. Example 6.1 A drilled shaft of diameter 1.0 m is to be socketed 3.0 meters in rock. The rock properties are as follows: Unconfined compressive strength of intact rock, σc=15.0 MPa Deformation modulus of intact rock, Er=10.6 GPa RQD=76

Determine the side shear resistance.

Fig. 6.8 Effective roughness height versus σc (after Seidel & Collingwood, 2001).

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228

Solution: Method of Kulhawy and Phoon (1993)—Equations (6.6) to (6.8) Lower bound τmax=1.0[paσc/2]0.5=1.0[0.1×15.0/2]0.5=0.87 MPa Upper bound τmax=3.0[paσc/2]0.5=3.0[0.1×15.0/2]0.5=2.60 MPa Method of Seidel and Collingwood (2001) From Figure 6.8, the mean roughness height hm=1.64 mm (lower bound) and 6.19 mm (upper bound). Using Equation (4.24), the rock mass modulus: αE=0.0231(RQD)−1.32=0.297 Em=αEEr=0.297×10.6=3.15 GPa The modulus ratio n=Em/σc=210. The Poisson’s ratio of the rock is simply assumed to be ν=0.25. Using ηc=1.0, SRC can be obtained from Equation (6.12) as:

From Figure 6.6, the adhesion factor a can be obtained as α=0.102 (lower bound) α=0.225 (upper bound) So the side shear resistance can be obtained as τmax=ασc=0.102×15.0=1.53 MPa (lower bound) τmax=ασc=0.225×15.0=3.37 MPa (upper bound) The results show that the shaft wall roughness (reflected by the roughness height) has a great effect on the side shear resistance. (e) Factors affecting side shear resistance As stated above, the shaft wall roughness, which is an important factor controlling the development of side shear resistance, has been studied extensively. Other factors such as the discontinuities in the rock mass and the shaft geometry have also been studied by some researchers. Williams et al. (1980) suggested that the existence of discontinuities in

Axial load capacity of drilled shafts in rock

229

the rock mass reduces the side shear resistance by reducing the normal stiffness of the rock mass. They developed the following empirical relation that considers the effect of discontinuities on the side shear resistance: τmax=αwβwσc (6.13) where αw is a reduction factor reflecting the strength of the rock, as shown in Figure 6.9; and βw is the ratio of side shear resistance of jointed rock mass to side shear resistance of intact rock. βw is a function of modulus reduction factor, j, as shown in Figure 6.10, in which βw=f(j), j=Em/Er (6.14) where Em is the elastic modulus of the rock mass; and Er is the elastic modulus of the intact rock. When the rock mass is such that the discontinuities are tightly closed and seatns are infrequent, βw is essentially equal to 1.0. Comparing Equation (6.13) with Equation (6.5), it can be seen that αwβw is just the adhesion factor, a, for β=1. Since αw is derived from field test data, the effect of discontinuities is already included in αw. If αw is multiplied by βw which is obtained from laboratory tests (Williams et al., 1980), the effect of discontinuities will be considered twice. So Equation (6.13) may be too conservative. Pabon and Nelson (1993) studied the effect of soft horizontal seams on the behavior of laboratory model shafts. The study included four instrumented model shafts in manufactured rock, three of which have soft seams. They concluded that a soft seam significantly reduces the normal interface stresses generated in the rock layer overlying it. Consequently the side shear resistance of shafts in rock with soft seams is much lower than that of shafts in intact rock. The effect of shaft geometry on side shear resistance was studied by Williams and Pells (1981). They tested 15 shafts in Melbourne Mudstone, with diameters ranging from 335 mm to 1580 mm, and 27 shafts in Hawkesbury Sandstone, with diameters ranging from 64 mm to 710 mm. The results of these tests indicated that the shaft length, L, does not have a discernible effect on the side shear resistance. They argued that the interface dilation creates a locked-in normal stress with the result that the shear displacement behavior exhibits virtually no peak or residual behavior. They also reported that the shaft diameter has a negligible effect on the side shear resistance. On the other hand, tests by Horvath et al. (1983) indicated that the side shear resistance decreases as the shaft diameter increases. Williams and Pells (1981) explained this phenomenon by referring to the theory of expansion of an infinite cylindrical cavity, which suggests that cylinders with smaller diameters develop higher normal stresses for a given absolute value of dilation. However, they offered no physical explanation why the shaft diameter does not affect their own test results.

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230

Fig. 6.9 Side shear resistance reduction factor αw [Equation (6.13)] (after Williams & Pells, 1981). 6.3.2 End bearing resistance (a) End bearing behavior of drilled shafts The typical bearing capacity failure modes for rock masses depend on discontinuity spacing with respect to foundation width (or diameter), discontinuity orientation, discontinuity condition (open or closed), and rock type. Table 6.7 illustrates typical failure modes according to rock mass conditions (ASCE, 1996). Prototype failure modes may actually consist of a combination of modes. The failure modes shown in Table 6.7 are for foundations with the base at or close to the ground surface. The depth of shaft embedment may change the end bearing failure modes of drilled shafts. As shown in Figure 6.11, when the base of the shaft is at or close to the ground surface, a wedge type of failure is developed and the shaft undergoes both vertical settlement and rotation. When the depth of embedment is greater than twice the diameter of the shaft, a punching type of failure occurs and a truncated conical plug of fractured rock is formed below the base (Williams et al., 1980).

Axial load capacity of drilled shafts in rock

231

Fig. 6.10 Side shear resistance reduction factor βw [Equation (6.13)] (after Williams & Pells, 1981). In a study by Johnston and Choi (1985), stereo photogrammetric techniques were used to study the process of failure of a model pile socketed into simulated rock. As shown in Figure 6.12, the study suggests that failure progresses from initial radial cracking to a fan shaped wedge. These observations were compared to typical load displacement curve where four points are identified as: 1) at the end of elastic deformation; 2) a little before major yielding; 3) a little after major yielding; and 4) failure. (b) End bearing resistance based on local experience and codes Peck et al. (1974) suggested a correlation between the allowable bearing pressure and RQD for footings supported on level surfaces in competent rock (Fig. 6.13). This correlation can be used as a first crude step in determination of the end bearing resistance of drilled shafts in rock. It need be noted that this correlation is intended only for unweathered jointed rock where the discontinuities are generally tight. If the value of allowable pressure exceeds the unconfined compressive strength of intact rock, the allowable pressure is taken as the unconfined compressive strength. In Hong Kong design practice, for large diameter drilled shafts in granitic and volcanic rocks, the allowable end bearing resistance may be used as specified in Table 6.8. The presumptive end bearing resistance values range from 3.0 to 7.5 MPa, depending

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232

on the rock category which is defined in terms of the rock decomposition grade, strength and total core recovery.

Table 6.7 Typical bearing capacity failure modes associated with various rock mass conditions (after ASCE, 1996). Rock mass conditions Joint dip

Joint spacing

Failure Illustration

Mode Brittle rock: Local shear failure caused by localized brittle fracture

N/A

s»B

Ductile rock: General shear failure along well defined failure surfaces

Open joints: Compressive failure of individual rock columns. Near vertical joint set(s)

70°<β <90°

s
Axial load capacity of drilled shafts in rock

233

s>B

Open or closed joints: Failure initiated by splitting leading to general shear failure. Near vertical joint set(s)

20°<β sB <70° if failure wedge can develop along joints

General shear failure with potential for failure along joints. Moderately dipping joint set(s)

Rock mass conditions Joint Joint dip spacing

Failure Illustration

Mode Thick rigid upper layer: Failure is initiated by tensile failure caused by flexure of the thick rigid upper layer

Limiting 0°<β value of H <20° with respect to B

N/A

s«B

Thin rigid upper layer: Failure is initiated by punching tensile failure of the thin rigid upper layer

General shear failure with irregular failure surface through rock mass. Two or more closely spaced joint sets

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234

The Standard Specifications for Highway Bridges adopted by the American Association of State Highway and Transportation Officials (AASHTO, 1989) also provide presumptive allowable bearing pressures for spread footing foundations in rock (see Table 6.9). These presumptive values can be used as a first crude step in determination of the end bearing resistance of drilled shafts in rock. (c) End bearing resistance from pressuremeter test results The Canadian Foundation Engineering Manual (CGS, 1985) proposed a method for determining the end bearing resistance of drilled shafts based on in situ pressuremeter test results: qmax=Kb(Pl−Po)+σo (6.15)

Axial load capacity of drilled shafts in rock

235

Fig. 6.11 Typical failure mechanism for end bearing shafts: (a) Base of shaft bearing at ground surface; and (b) Shaft with length/diameter>2 (after Williams et al., 1980).

Drilled shafts in rock

236

Fig. 6.12 Observed progressive failure modes: (a) Typical load-displacement curve; and (b) Failure modes corresponding to the points in (a) (after Johnston & Choi, 1985).

Axial load capacity of drilled shafts in rock

237

Fig. 6.13 Allowable bearing pressure of jointed rock (after Peck et al., 1974). Table 6.8 Presumed safe vertical bearing stress for foundations on horizontal ground in Hong Kong [simplified from PNAP 141 (BOO, 1990)]. Category Granitic and volcanic rock

Presumed bearing stress (MPa)

1(a)

Fresh to slightly decomposed strong rock of material weathering grade II or better, with total core recovery>95 and minimum uniaxial compressive strength of rock material σc not less than 50 MPa (equivalent point load index strength PLI50a not less than 2 MPa)

7.5

1(b)

Slightly to moderately decomposed moderately strong rock of material weathering grade II or III or better, with total core recovery>85% and minimum unconfined compressive strength of rock material σc not less than 25 MPa (equivalent point load index strength PLI50a not less than 1 MPa)

5.0

1(c)

Moderately decomposed moderately strong to moderately weak rock of material weathering grade III or IV or better, with total core recovery>50%

3.0

a

Point load index strength PLI50 of rock quoted is equivalent value for 50-mm-diameter cores (ISRM, 1979a).

where pl is the limit pressure as determined from pressuremeter tests in the zone extending two shaft diameters above and below the shaft base; po is the at rest horizontal

Drilled shafts in rock

238

stress in the rock at the elevation of the shaft base; σo is the total overburden stress at elevation of the shaft base; and Kb is an empirical non-dimensional coefficient, which depends on the depth and shaft diameter ratio as shown in Table 6.10. (d) Empirical and Semi-Empirical Relations Unlike the side shear resistance, numerous theories have been proposed for estimating the end bearing resistance. According to Pells and Turner (1980), the theoretical approaches fall into three categories: 1. Methods which assume rock failure to be plastic. 2. Methods which idealize the zone of failure beneath the base in a form which allows either the brittleness strength ratio or the brittleness modulus ratio to be taken into account. 3. Methods based on limiting the maximum stress beneath the loaded area to a value less than required to initiate fracture. These methods assume essentially that once the maximum strength is exceeded at any point in a brittle material, total collapse will occur. There is a significant variation in the end bearing resistance predicted from different theories. For example, the predicted end bearing capacity of rock with an internal friction ranges from 4.9σc using the incipient failure theory (Category 3) based angle on the modified Griffith theory to 56σc using the classical plasticity theory (Category 1), where σc is the unconfined compressive strength of intact rock (Poulos

Table 6.9 Presumptive allowable bearing pressures for spread footing foundations, modified after Navy (1982) (simplified from AASHTO, 1989). Range of Allowable bearing pressure σc (MPa) (MFa) Type of bearing material

Consistency in place

Massive crystalline igneous and Very hard, metamorphic rock: granite, sound rock diorite, basalt, gneiss, thoroughly cemented conglomerate (sound condition allows minor cracks)

>250

Ordinary range

Recommended value for use

6–10

8

Foliated metamorphic rock: slate, schist (sound condition allows minor cracks)

Hard sound rock 100–250

3–4

3.5

Sedimentary rock: hard cemented shales, siltstone, sandstone, limestone without cavaties

Hard sound rock 50–100

1.5–2.5

2

Axial load capacity of drilled shafts in rock

239

Weathered or broken bedrock of any kind except highly argillaceous rock (shale)

Medium hard rock

25–50

0.8–1.2

1

Compaction shale or other highly argillaceous rock in sound condition

Medium hard rock

25–50

0.8–1.2

1

Notes: 1. Variations of allowable bearing pressure for size, depth, and arrangement of footings must be determined by analysis. 2. Presumptive values for allowable bearing pressures obtained from building codes and charts developed by various agencies based on local experience with satisfactory and unsatisfactory performance; usually the pressure that will limit total and differential settlements to 1 inch. Presumptive values are not based on thorough engineering analysis. 3. Allowable bearing pressure for rock is controlled by rock mass discontinuities, and should not exceed the unconfined compressive strength.

Table 6.10 Kb as fimction of depth and shaft diameter ratio (CGS, 1985). Depth/Diameter

0

1

2

3

5

7

Kb

0.8

2.8

3.6

4.2

4.9

5.2

& Davis, 1980). Because of the wide variation of theoretical results, empirical and semiempirical relations have been developed. Since they are more commonly used than the theoretical methods, only the empirical and semi-empirical relations are discussed in the following. Analogous to the side shear resistance, many attempts have been made to correlate the end bearing capacity, qmax, to the unconfined compressive strength, σc, of intact rock. Some of the suggested relations are: Coates (1967): qmax=3.0σc (6.16) Rowe and Armitage (1987b): qmax=2.7σc (6.17) ARGEMA (1992): qmax=4.5σc ≤10 MPa (6.18) Findlay et al. (1997): qmax=(1−4.5)σc (6.19) The bearing capacity of foundations on rock is largely dependent on the strength of the rock mass. Discontinuities can have a significant influence on the strength of the rock mass depending on their orientation and the nature of material within discontinuities (Pells & Turner, 1980). As a result, relations have been developed to account for the

Drilled shafts in rock

240

influence of discontinuities in the rock mass. The Standard Specifications for Highway Bridges adopted by the American Association of State Highway and Transportation Officials (AASHTO, 1989) suggests that the end bearing capacity be estimated using the following relationship: qmax=Nmsσc (6.20) where Nms is a coefficient relating qmax to σc. The value of Nms is a function of rock mass quality and rock type (Table 6.11), where rock mass quality, in essence, expresses the degree of jointing and weathering. Rock mass quality has a much stronger effect on Nms than rock type. For a given rock type, Nms for excellent rock mass quality is more than 250 times higher than Nms for poor quality. For a given rock mass quality, however, Nms changes little with rock type. For example, for a rock mass of very good quality, the values of Nms are 1.4, 1.6, 1.9, 2.0 and 2.3 respectively for rock types A, B, C, D and E (see Table 6.11). It should be noted however that rock type is implicitly related to the unconfined compressive strength. Equation (6.20) may thus represent a non-linear relation between qmax and σc. Although it is not explicitly mentioned in AASHTO (1989), Equation (6.20) and coefficient Nms can be simply derived from the lower bound solution suggested by Carter and Kulhawy (1988) (see Fig. 6.14): qmax=[s0.5+(mbs0.5+s)0.5]σc (6.21) in which the expression in the brackets is simply the coefficient Nms in Equation (6.20); and mb and s are the strength parameters for the Hoek-Brown strength criterion as discussed in Chapter 4. Values of mb and s for the rock categories in Table 6.11 are shown in Table 6.12. The values of Nms in Table 6.11 can be simply obtained by inserting the corresponding values of mb and s from Table 6.12 in the expression in the brackets of Equation (6.21). Equation (6.21) does not consider the influence of the overburden soil and rock (i.e., overburden stress qs=0 is assumed). Zhang and Einstein (1998a) derived an expression for the end bearing capacity that considers the influence of the overburden

Table 6.11 Values of Nms for estimating the end bearing capacity of drilled shafts in broken or jointed rock (after AASHTO, 1989). Rock Mask Quality

General Description

RMR(1) Q(2) RQD(3) Rating Rating Rating

Excellent

Intact rock with joints spaced > 10 feet apart

100

Very Good

Tightly interlocking, 85 undisturbed rock

Nms(4) A

B

C

D

E

500

95–100

3.8

4.3

5.0

5.2

6.1

100

90–95

1.4

1.6

1.9

2.0

2.3

Axial load capacity of drilled shafts in rock

241

with rough unweathered discontinuities spaced 3 to 10 feet apart Good

Fresh to slightly weathered rock, slightly disturbed with discontinuities spaced 3 to 10 feet apart

65

10

75–90

0.28

Fair

Rock with several sets of moderately weathered discontinuities spaced 1 to 3 feet apart

44

1

50–75

0.049 0.056 0.066 0.069 0.081

Poor

Rock with numerous 23 weathered discontinuities spaced 1 to 20 inches apart with some gouge

0.1

25–50

0.015 0.016 0.019 0.020 0.024

0.01

<25

Use qult for an equivalent soil

Very Poor Rock with numerous 3 highly weathered discontinuities spaced<2 inches apart

0.32

0.38

0.40

0.46

(1)

Geomechanics rock mass rating (RMR) system (Bieniawski, 1988)—See Chapter 2

(2)

Rock mass quality (Q) system (Barton et al., 1974)—See Chapter 2

(3)

Range of RQD values provided for general guidance only; actual determination of rock mass quality should be based on RMR or Q rating systems

(4)

Value of Nms as function of rock type; refer to Table 2.8 for typical range of values of σc for different rocks in each category

Table 6.12 Values of mb and s based on rock mass classification (modified from Carter & Kulhawy, 1988). Rock Mass Quality

General Description

Excellent Intact rock with joints spaced >10 feet apart

RMR(1) Q(2) RQD(3) Rating Rating Rating 100

500

95–100

s

1

mb A

B

C

D

E

7

10

15

17

25

Drilled shafts in rock

242

Very Good

Tightly interlocking, undisturbed rock with rough unweath-ered discontinuities spaced 3 to 10 feet apart

85

100

90–95

0.1

3.5

5

7.5

8.5

12.5

Good

Freshtoslightly weathered rock, slightly disturbed with discontinuities spaced 3 to 10 feet apart

65

10

75–90

0.004

0.7

1

1.5

1.7

2.5

Fair

Rock with several sets of moderately weathered discontinuities spaced 1 to 3 feet apart

44

1

50–75

10−4

0.14

0.2

0.3

0.34

0.5

Poor

Rock with numerous weathered discontnuities spaced 1 to 20 inches apart with some gouge

23

0.1

25–50

10−5

0.04 0.05 0.08

0.09

0.13

Very Poor

Rock with numerous highly weathered discontinuities spaced <2 inches apart

3

0.01

<25

0

0.007 0.01 0.015 0.017 0.025

stress (see Fig. 6.15): (6.22) where (6.23)

Axial load capacity of drilled shafts in rock

243

Fig. 6.14 Lower-bound solution for bearing capacity (after Kulhawy & Carter, 1992).

Fig. 6.15 Assumed failure mode of rock mass below the shaft base (after Zhang & Einstein, 1998a).

Drilled shafts in rock

244

Kulhawy and Goodman (1980) presented the following relationship originally proposed by Bishnoi (1968): qmax=JcNcr (6.24) where J is a correction factor depending on normalized spacing of horizontal discontinuities (spacing of horizontal discontinuities/shaft diameter) (see Fig. 6.16); c is the cohesion of the rock mass; and Ncr is a modified bearing capacity factor, which is a function of the friction angle of the rock mass and normalized spacing of vertical discontinuities (spacing of vertical discontinuities/shaft diameter) (see Fig. 6.17).

Fig. 6.16 Correction factor for discontinuity spacing (after Kulhawy & Goodman, 1980).

Axial load capacity of drilled shafts in rock

245

Fig. 6.17 Bearing capacity factor for open discontinuities (after Kulhawy & Goodman, 1980). Table 6.13 Suggested design values of strength parameters c and (from Kulhawy Goodman, 1987). Rock mass properties RQD (%)

Unconfined Compressive strength

Cohesion c

0–70

0.33σc

0.1σc

30°

70–100

0.33σc–0.8σc

0.1σc

30°–60°

Angle of friction

σc=unconfined compressive strength of intact rock

As indicated in the preceding text, the strength parameters c and are rock mass properties. Kulhawy and Goodman (1987) provided a table relating the rock mass properties c and to intact rock properties and RQD (Table 6.13). The correction factor J considers the effect of horizontal discontinuities and the variation of Jwith the discontinuity spacing is shown in Figure 6.16, where H is the spacing of horizontal discontinuities. For the value of Ncr the authors considered the discontinuities being either open or closed. According to Goodman (1980), the presence of open discontinuities would allow failure to occur by

Drilled shafts in rock

246

splitting (because the discontinuities are open, there is no confining pressure and failure is likely to occur by uniaxial compression of the rock columns), and this mode of failure needs to be included in the calculation of the end bearing capacity. Several charts are given by Kulhawy and Goodman (1980), following the method of Bishnoi (1968), to determine Ncr for both open and closed discontinuities. Figure 6.17 shows Ncr for open discontinuities. The Canadian Foundation Engineering Manual (CGS, 1985) proposed that the end bearing pressure be calculated using the following equation: qmax=3σcKspD (6.25) where Ksp=[3+s/B]/[10(1+300g/s)0.5] is an empirical factor; s is the spacing of the discontinuities; B is the shaft width or diameter; g is the aperture of the discontinuities; D=1+0.4(L/B)≤3.4 is the depth factor; L is the shaft length. In general the method will apply only if s/B ratios lie between 0.05 to 2.0 and the values of g/s between 0 and 0.02 (CGS, 1985). Zhang and Einstein (1998a) developed a database of 39 shaft load tests about the ultimate end bearing capacity (see Table 6.14). This database represents rocks of relatively low strength. Table 6.14 lists, in addition to shaft dimensions, the unconfined compressive strength of intact rock σc, the end bearing capacity qmax, and the end bearing capacity factor Nc(=qmax/σc). The ratio of the shaft base displacement sb at qmax to the shaft diameter B is also included in Table 6.14. A number of issues that need be considered when studying the relationship between the end bearing capacity and the unconfined compressive strength of intact rock are as follows: 1. Different interpretations of the load test data will give different capacities. For the test shafts in Table 6.14, different interpretation methods were used. For example, Goeke and Hustad (1979) took the load at plunging failure as the ultimate capacity of the shaft (plunging failure is defined as the point at which additional load cannot be applied to the shaft without experiencing continuous movement), while Jubenville and Hepworth (1981) defined the ultimate capacity of the shaft as the load at which the shaft head displacement reached 10% of the shaft diameter. Therefore, some uncertainties and variabilities are likely to be included in the database. However, the general trend reflected by the database will be useful. 2. The unconfined compressive strength is a property of the intact rock, not of the rock mass. Clearly rock mass discontinuities must affect the end bearing

Table 6.14 Summary of database of shaft load tests (Zhang and Einstein, 1998a). No. Rock description

1 Mudstone, weak, clayey

Diameter Depth σc qmax Nc=qmax/σc Sb/Ba Reference B (mm) to (MPa) (MPa) (%) base L (m) 670

6

1.09

6.88

6.31

7.0

Wilson (1976)

Axial load capacity of drilled shafts in rock

247

cretaceous 2 Clayshale, with occational thin limestone seams

762

8.8

0.81

4.69

5.79

6.2

Goeke and Hustad(1979)

3 Shale, thinly bedded with thin sandstone layers

457

13.7

3.82

10.8

2.83

>10.0 Hummert and Cooling (1988)

4 Shale, unweathered

305

2.4

1.08

3.66

3.39

10.0 Jubenville and Hepworth(1981)

5 Gypsumb

1064

4.20

2.1

6.51

3.1

15– 20

Leung and Ko (1993)

6 Gypsumb

1064

4.20

4.2

10.9

2.6

15– 20

Leung and Ko (1993)

7 Gypsumb

1064

4.20

5.4

15.7

2.9

15– 20

Leung and Ko (1993)

8 Gypsumb

1064

4.20

6.7

16.1

2.4

15– 20

Leung and Ko (1993)

9 Gypsumb

1064

4.20

8.5

23

2.7

15– 20

Leung and Ko (1993)

10 Gypsumb

1064

4.20

11.3

27.7

2.5

15– 20

Leung and Ko (1993)

11 Tillc

762

**

0.7

4

5.71

~1.3 Orpwoodetal. (1989)

12 Tillc

762

**

0.81

4.15

5.12

~4.6 Orpwood et al. (1989)

13 Tillc

762

**

1

5.5

5.5

~1.4 Orpwood et al. (1989)

14 Diabase, highly weathered

615

12.2

0.52

2.65

5.1

>4.0 Webb (1976)

15 Hardpan (hard bearing till)c

1281

18.3

1.38

5.84

4.23

~4.0 Baker (1985)

16 Tillc

1920

20.7

0.57

2.29

4.04

~1.9 Baker (1985)

17 Hardpan (hard bearing till)c

762

18.3

1.11

4.79

4.33

~7.3 Baker (1985)

18 Sandstone, horizontally bedded,

610

15.6

8.36

10.1

1.21

>1.7 Glos and Briggs (1983)

Drilled shafts in rock

248

shaley, RQD=74% 19 Sandstone, horizontally bedded, shaley, with some coal stringers, RQD=88%

610

16.9

9.26

13.1

1.41

>1.7 Glos and Briggs (1983)

20 Mudstone, highly weathered

300

2.01

0.65

6.4

9.8

6.4

Williams (1980)

21 Mudstone, highly weathered

300

1

0.67

7

10.5

5.7

Williams (1980)

22 Mudstone, moderately weathered

1000

15.5

2.68

5.9

2.2

1.1

Williams (1980)

23 Mudstone, moderately weathered

1000

15.5

2.45

6.6

2.7

0.7

Williams (1980)

24 Mudstone, moderately weathered

1000

15.5

2.45

7

2.9

0.6

Williams (1980)

25 Mudstone, moderately weathered

1000

15.5

2.68

6.7

2.5

0.7

Williams (1980)

26 Mudstone, moderately weathered

600

1.8

1.93

9.2

4.8

14.1 Williams (1980)

27 Mudstone, moderately weathered

1000

3

1.4

7.1

5

10.9 Williams (1980)

No. Rock description

Diameter Depth σc qmax Nc= Sb/Ba Reference B (mm) to (MPa) (MPa) qmax/σc (%) base L (m)

28 Shale

**

**

34

28

0.82

**

Thorne (1980)d

29 Sandstone

**

**

12.5

14

1.12

**

Thorne (1980)d

30 Sandstone, fresh, defect free

**

**

27.5

50

1.82

**

Thorne (1980)d

31 Shale, occational

**

**

55

27.8

0.51

**

Thorne (1980)d

Axial load capacity of drilled shafts in rock

249

recemented moisture fractures and thin mud seams, intact core lengths 75 to 250 mm 32 Clayshale

740

7.24

1.42

5.68

4

~8.8 Aurora and Reese (1977)

33 Clayshale

790

7.29

1.42

5.11

3.6

~8.9 Aurora and Reese (1977)

34 Clayshale

750

7.31

1.42

6.11

4.3

~6.0 Aurora and Reese (1977)

35 Clayshale

890

7.63

0.62

2.64

4.25

~6.6 Aurora and Reese (1977)

36 Siltstone, medium hard, fragmented

705

7.3

9

13.1

1.46

~12.0 Radhakrishnan and Leung (1989)

37 Marl, intact, RQD=100%

1200

18.5

0.9

5.3

5.89

**

Carrubba (1997)

38 Diabase Breccia, highly fractured, RQD=10%

1200

19

15.0

8.9

0.59

**

Carrubba (1997)

39 Limestone, intact, RQD=100%

1200

13.5

2.5

8.9

3.56

**

Carrubba (1997)

a

Sb is the shaft base displacement at qmax. Gypsum mixed with cement is used as pseudo-rock in centrifuge tests. The and depths are the equivalent prototype dimensions corresponding to 40 g in the centrifuge tests. The equivalent prototype depths to the shaft base range 4.04 m to 4.35 m with an average of 4.20 m. c Till is not a rock. It is used here because its σc is comparable to that of some rocks. d These tests were not conducted by Thorne (1980). He only reported the data other references b

capacity. Unfortunately, relevant information on this factor is unavailable for most of the cases in Table 6.14. 3. The conditions below the base of the shaft also influence the end bearing capacity. If the base of the drilled hole cannot be cleaned, little or no end bearing support will be developed. For all the test shafts in Table 6.14, the base of the drilled hole was cleaned. 4. Different methods are used to separate the side shear resistance from the end bearing capacity in load tests. 5. Clearly it would be interesting to have a relatively narrowly defined shaft base displacement which one can associate with the end bearing capacity. However, the values of sb/B in Table 6.14 indicate that the base displacement at qmax ranges from 0.6 to 20% of the shaft diameter, i.e., 6 to 210 mm. It is thus difficult to say at this

Drilled shafts in rock

250

point what typical base displacements at qmax are. [For comparison, the displacement at ultimate side shear resistance is smaller; examination of more than 50 loaddisplacement curves for large-diameter drilled shafts showed that an average displacement of only 5 mm was necessary to reach initial failure of side shear resistance (Horvath et al., 1983)].

Fig. 6.18 qmax versus σc (after Zhang & Einstein, 1998a). All the load test data in Table 6.14 are plotted in Figure 6.18. A log-log plot is used. It can be seen that there is a strong relation between qmax and σc. Using linear regression, the relationship between qmax and σc is as follows: qmax=4.83(σc)0.51 (6.26) The coefficient of determination, r2, is 0.81. Example 6.2 A drilled shaft of diameter 1.0 m is to be socketed 3.0 meters in siltstone. The rock properties are as follows: Unconfined compressive strength of intact rock, σc=15.0 MPa

Axial load capacity of drilled shafts in rock

251

The rock mass is heavily jointed and the average discontinuity spacing near the base of the shaft is 0.5 m The discontinuities are moderately weathered and filled with debris with thickness of 3 mm Deformation modulus of intact rock, Er=10.6 GPa RQD=45

Determine the end bearing resistance. Solution: Method of AASHTO (1989)—Equation (6.20) From Table 2.8, the rock is classified as Type B. From Table 6.11, the rock quality is classified as Fair and the value of Nms is 0.056. Using Equation (6.20), the end bearing resistance can be obtained as qmax=Nmsσc=0.056×15.0=0.84 MPa Method of Zhang & Einstein (1998a)—Equations (6.22) & (6.23) From Table 2.8, the rock is classified as Type B. From Table 6.12, the rock quality is classified as Fair and the values of s and mb are respectively 10−4 and 0.2. Assuming that the effective unit weight of the rock mass is 13.0 kN/m3 and ignoring the weight of the soil above the rock, the end bearing resistance can be obtained from Equations (6.22) and (6.23) as

Method of CGS (1985)—Equation (6.25) Empirical factor Ksp=[3+s/B]/[10(1+300g/s)0.5]=(3+0.5/1.0)/[10(1+300×0.003/0.5)0.5] =0.21 Depth factor D=1+0.4(L/B)=1+0.4(3.0/1.0)=2.2.

Drilled shafts in rock

252

The end bearing resistance can be calculated from Equation (6.25) as

Method of Zhang and Einstein (1998)—Equation (6.26) The end bearing resistance can be simply calculated from Equation (6.26) as

The results clearly show the wide range of the estimated end bearing capacity from different methods. It is therefore important not to rely on a single method when estimating the end bearing capacity.

6.4 CAPACITY OF DMLLED SHAFT GROUPS In many cases, drilled shaft foundations will consist not of a single drilled shaft, but of a group of drilled shafts. The drilled shafts in a group and the soil/rock between them interact in a very complex fashion, and the axial capacity of the group may not be equal to the axial capacity of a single isolated drilled shaft multiplied by the number of shafts. One way to account for the interaction is to use the group efficiency factor η, which is expressed as: (6.27) where QuG is the ultimate axial load of a drilled shaft group; N is the number of drilled shafts in the group; and Qu is the ultimate axial load of a single isolated drilled shaft, which can be determined using the methods described in Section 6.3. The group efficiency for axial load capacity depends on many factors, including the following: •The number, length, diameter, arrangement and spacing of the drilled shafts. •The load transfer mode (side shear versus end bearing). •The elapsed time since the drilled shafts were installed. •The rock type. Katzenbach et al. (1998) studied the group efficiency of a large drilled shaft group in rock. For the 300 m high Commerzbank tower in Frankfurt am Main, 111 drilled shafts are used to transfer the building load through the relatively weak Frankfurt Clay to the stiffer underlying Frankfurt Limestone. Of the 111 drilled shafts, 30 were instrumented

Axial load capacity of drilled shafts in rock

253

and monitored during the 2-year construction period. The measurements give a detailed view into the interaction between the drilled shafts in the group. Figure 6.19 shows the variation of the group efficiency factor with the shaft head settlement. At service loads of the building the value of the group efficiency factor is about 60%. When drilled shafts are closely spaced, the shafts in a group may tend to form a “group block” that behaves like a giant, short shaft (see Fig. 6.20). In this case, the bearing capacity of the drilled shaft group can be obtained in a similar fashion to that for a single isolated drilled shaft, by means of Equation (6.2), but now taking the shaft base area as the block base area and the shaft side surface area as the block surface area. It should be noted that the deformation required to mobilize the base capacity of the block will be larger than that required for a single isolated shaft.

6.5 UPLIFT CAPACITY In many cases, drilled shafts in rock may be required to resist uplift forces. Examples are drilled shaft foundations for structures subjected to large overturning moments such as tall chimneys, transmission lines, and highway sign posts. Drilled shafts through expansive soils and socketed into rock may also subject to uplift forces due to the swelling of the soil. Drilled shafts can be designed to resist uplift forces either by enlarging or belling the base, or by developing sufficient side shear resistance. Belling the base of a shaft is common in soils, but this can be an expensive and difficult operation in rock. Moreover, since large side shear resistance can be developed in drilled shafts socketed into rock, it is usually more economical to deepen the socket than to construct a shorter, belled socket. For drilled shafts subject to uplift forces, it is important to check the structural capacity of the shaft. This can be done using the methods presented in Section 6.1. The ultimate uplift resistance of a straight-sided drilled shaft related to rock can be determined by Quu=πBLτmax+Ws (6.28) where Quu is the ultimate uplift resistance; L and B are respectively the length and diameter of the shaft; τmax is the average side shear resistance along the shaft; and Ws is the weight of the shaft.

Drilled shafts in rock

254

Fig. 6.19 Variation of group efficiency factor with shaft head settlement (after Katzenbach et al., 1998).

Fig. 6.20 Treating the drilled shaft group as a group block. Uplift loading does not produce the same stress conditions in the shaft or rock mass as those produced by compression loading. Compression loading compresses the shaft,

Axial load capacity of drilled shafts in rock

255

causing outward radial straining in the concrete (positive Poisson effect), which results in higher frictional stresses at the interface with the rock mass; simultaneously it adds total vertical stress to the rock mass around the shaft through the process of load transfer, which consequently adds strength to rock masses that drain during loading. Uplift loading, however, produces radial contraction of the concrete (negative Poisson effect) and reduces the total vertical stresses in the rock mass around the shaft. Because of the different stress conditions, the average side shear resistance for uplift loading should usually be lower than that for compression loading.

Fig. 6.21 Measured side shear resistance from compression tests and pull-out tests. Figure 6.21 shows the variation of measured side shear resistance with the unconfined compressive strength of intact rock respectively from the compression load tests and the pull-out load tests. The data are collected from the published literature. We can see that the measured side shear resistances from the pull-out load tests are about the same as or even higher than those from the compression load tests. One of the reasons for this might be that the pull-out test shafts have rougher wall surfaces than the compression test shafts. However, we are not sure about this at this point since no information on the wall roughness is available for most of the test shafts shown in Figure 6.21.

Drilled shafts in rock

256

For preliminary design, the side shear resistance for uplift loading can be simply taken to be the same as that for compression loading and estimated using the methods presented in Section 6.3.1. Where vertical drilled shafts are arranged in closely-spaced groups the uplift resistance of the complete group may not be equal to the sum of the resistance of the individual shafts. This is because, at ultimate-load conditions, the block of rock enclosed by the shafts may be lifted. The uplift resistance of the block of rock may be determined by (see Fig. 6.20) (6.29) where QuuG is the total ultimate uplift resistance of the shaft group; B1 and B2 are respectively the overall length and width of the group (see Fig. 6.20); and WB is the combined weight of the block of rock enclosed by the shaft group plus the weight of the shafts.

7 Axial deformation of drilled shafts in rock 7.1 INTRODUCTION Predicting the axial load-displacement response of drilled shafts is in some cases as important as, or possibly more critical than, predicting the ultimate bearing capacity. Many methods are available for predicting the axial displacement of drilled shafts in rock. While the most reliable means for predicting the axial displacement of drilled shafts is probably to carry out an axial loading test of the prototype shaft (which will be discussed in Chapter 12), theoretical analyses may also be usefully employed. The main three theoretical methods used to predict the axial load-displacement response of drilled shafts in rock are the load-transfer (t-z) method, the continuum approach and the finite element method. The general load-displacement curve for a drilled shaft under axial loading can be simply illustrated in Figure 7.1. The whole curve can be described in three stages: 1. As load is first applied to the head of the shaft, a small amount of displacement occurs which induces the mobilization of side shear resistance from head to base. During this initial period, the shaft behaves essentially in a linear manner, and the displacement can be computed using the theory of elasticity. This linear behavior is illustrated in Figure 7.1 as the line OA. The side shear stress along the shaft is smaller than the ultimate side shear resistance (Fig. 7.2a). 2. As load is increased to point A in Figure 7.1, the shear stress at some point along the interface will reach the ultimate side shear resistance (Fig. 7.2b), and the shaft-rock ‘bound’ will begin to rupture and relative displacement (slip) will occur between the shaft and the surrounding rock. As the loading is increased further (beyond point A), this process will continue along the shaft, more of the shaft will slip, and a greater proportion of the applied load will be transferred to the end of the shaft (Fig. 7.2c). If loading is continued, eventually the side shear stress everywhere will reach the ultimate side shear resistance and the entire shaft will slip (point B in Fig. 7.1). 3. Beyond point B, a greater proportion of the total axial load will be transmitted directly to the end of the shaft. When both side shear resistance and end bearing resistance are fully mobilized (point C), any increase of load may produce significant displacement. This indicates that the ultimate bearing capacity of the drilled shaft has been reached.

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Fig. 7.1 Generalized load-displacement curves for drilled shafts under compressive loading. 7.2 LOAD-TRANSFER (t-z CURVE) METHOD The load-transfer method models the reaction of soil/rock surrounding the shaft using localized springs: a series of springs along the shaft (the t-z or τ-w curves) and a spring at the tip or bottom of the shaft (the q-w curve). τ is the local load transfer or side shear resistance developed at displacement w, q is the base resistance developed at displacement w, and w is the displacement of the shaft at the location of a spring. The physical drilled shaft is also represented by a number of blocks connected by springs to indicate that there will be compression of the drilled shaft due to the applied compressive load. The mechanical model is shown in Figure 7.3. The displacement of the shaft at any depth z can be expressed by the following differential equation: (7.1)

where Ep is the composite Young’s modulus of the shaft (considering the contribution of both concrete and reinforcing steel); A and B are respectively the cross-sectional area and diameter of the shaft; w is the displacement of the shaft at depth z; and τ is the side shear resistance developed at displacement w at depth z. Equation (7.1) can be solved analytically or numerically depending on the τ-w and q-w curves (linear or nonlinear), which is discussed in the sections below.

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260

7.2.1 Linear analysis For linear analysis, the relationship between τ and w at any depth z and that between q and w are assumed to be linear, i.e.,

Fig. 7.2 Shear stress at different values of applied load (QA is the applied load corresponding to point A in Fig. 7.1). (7.2a)

Axial deformation of drilled shafts in rock

261

(7.2b) where ks and kb are spring constants respectively of the side springs and the base spring. Substitution Equation (7.2a) into Equation (7.1) gives (7.3)

where

Fig. 7.3 Load-transfer (t-z curve) model of axially loaded drilled shaft. (7.4)

The general solution to Equation (7.3) is

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(7.5) where C1 and C2 are integration constants. The axial force at any depth is proportional to the first derivative of the displacement with respect to depth: (7.6) If a load Qt is applied at the top of the shaft (z=0) and the force transferred to the base of the shaft (z=L) is Qb, we have, from Equation (7.6), (7.7a) (7.7b) From Equations (7.2b) and (7.5), We have (7.8) Solving Equations (7.7) and (7.8), constants C1 and C2 can be obtained as (7.9a)

(7.9a)

The displacement at the top of the shaft (z=0) is then obtained from Equations (7.5) and (7.9) as

7.2.2 Nonlinear analysis In general, the τ-w and q-w curves are nonlinear. In this case, a convenient way to solve differential Equation (7.1) is to use the finite difference method (Desai & Christian, 1977). Computer programs can be easily written to do the computations. The main issue for the nonlinear analysis is the determination of the τ-w and q-w curves. There are several techniques for determining the load transfer curves in soils (Vijayvergiya, 1977; Kraft et al., 1981; Castelli et al., 1992) and rock masses (Baguelin et

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al., 1982; O’Neill & Hassan, 1994). However, research has not advanced to the point that the load transfer curves (τ-w and q-w curves) can be determined for all conditions with confidence (O’Neill & Reese, 1999). Construction practices and the particular response of a given formation to drilling and concreting will affect the load transfer curves. For major projects, therefore, it is advisable to measure the load transfer curves using fullscale loading tests of instrumented shafts. Chapter 12 will show how to obtain the experimental load transfer curves from the results of an axial loading test of an instrumented shaft. Based on measured load displacement curves, Carrubba (1997) conducted numerical analyses to evaluate the side shear resistance and the end bearing capacity and obtained the load transfer curves for five rock-socketed shafts. The model is based on a hyperbolic transfer function approach and solves the equilibrium of the shaft by means of finite element discretization. The interaction at the shaft-soil and shaft-rock interfaces is described by the following function (7.11) where f(z) is the mobilized resistance along a shaft portion (τ) or at the shaft base (q); and w(z) is the corresponding displacement (see Fig. 7.3). In the transfer function, parameters a and b represent the reciprocals of initial slope and limit strength, respectively: (7.12a) (7.12b) where flim is the end bearing capacity (qmax) in rock or the side shear resistance in soil or rock (τmax). Numerical analyses are carried out by selecting three transfer functions for each shaft: one representative of overall friction in soil, one for overall friction in rock, and the last one for end bearing resistance in rock. The friction transfer functions in soils, once selected, are maintained constant throughout the analyses. Transfer function parameters for rock, both along the shaft and at the base, are first estimated and then modified with an iterative process until the actual load displacement curve is reproduced. Figure 7.4 shows the comparison between the test results and the numerical simulations for the shaft in marl. Since the side and base strengths are not mobilized at the same time and the numerical model used cannot simulate this event, two different ideal shaft behaviors are examined. The first neglects the base reaction; the second takes into account the contemporary mobilization of side and base resistances from the beginning of the test. The rock properties and the transfer function parameters obtained for the five rocksocketed shafts are shown in Table 7.1. O’Neill and Hassan (1994) proposed an interim criterion for a hyperbolic τ-w curve in most types of rock until better solutions become accepted:

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(7.13)

where B is the diameter of the shaft; and Em is the deformation modulus of the rock mass. This model is based on the fact that the interface asperity pattern is regular and the asperities are rigid, even though in most cases the interface asperity pattern is not regular, some degree of smear exists, and asperities are deformable, which results in ductile, progressive failure among asperities. Equation (7.13) is a special form of Equation (7.11) with a=2.5B/Em.

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Fig. 7.4 Comparison between test results and numerical simulations for the drilled shaft in Marl. Curve a neglects base reaction; curve b takes into account cotemporary mobilization of side and base resistances (after Carrubba, 1997). Table 7.1 Rock properties and transfer function parameters (Carrubba, 1997). Rock type Marl

σc (MPa) 0.90

RQD (%)

Em (MPa)

Shaft side in rock

Base in rock

1/b (MPa)

1/b (MPa)

1/a (MN/m3)

1/a (MN/m3)

100

200a

0.14

100

5.30

220

b

0.49

70

8.90

300

Diabasic Breccia

15.00

10

200

Gypsum

6.00

60

2,000a

0.47

200

–

–

50

a

1.20

500

–

–

b

0.40

500

8.90

3,000

Diabase Limestone a b

40.00 2.50

100

10,000 500

From compression tests on specimens From plate bearing tests

The q-w curve is usually assumed to have an initial elastic response given by

where Eb and νb are respectively the deformation modulus and Poisson’s ratio of the rock below the shaft base. Nonlinear response is usually assumed to initiate between 1/3 and 1/2 of qmax. This response can be simply modeled using an equation similar to Equation (7.13).

7.3 CONTINUUM APPROACH The continuum approach assumes the soil/rock to be a continuum. Mattes and Poulos (1969) are among the first to investigate the load-displacement behavior of rock-socketed shafts by integration of Mindlin’s equations. Carter and Kulhawy (1988) provide a set of approximate analytical solutions to predict the load-displacement response of drilled shafts in rock by modifying the solutions of Randolph and Wroth (1978) for piles in soil.

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The majority of the theoretical continuum solutions for predicting the displacement of drilled shafts in rock, however, have been developed using finite element analyses (e.g., Osterberg & Gill, 1973; Pells & Turner, 1979; Donald et al., 1980; Rowe & Armitage, 1987a). Most of the techniques proposed for calculating the vertical displacements of drilled shafts in rock are based on the theory of elasticity. It has been usual to assume that the drilled shaft is essentially an elastic inclusion within the surrounding rock mass and that no slip occurs at the interface between the shaft and the rock mass, although the solutions of Rowe and Armitage (1987a) and Carter and Kulhawy (1988) can consider the possibility of slip. 7.3.1 Linear continuum approach (a) Solutions based on finite element results As stated in Chapter 6, axially loaded drilled shafts in rock are designed to transfer structural loads in one of the following three ways (CGS, 1985): 1. Through side shear only; 2. Through end bearing only; 3. Through the combination of side shear and end bearing. The following presents the elastic solutions based on the finite element results for estimating the axial deformation of the above three types of shafts. Side shear only shaft Based on finite element analysis, Pells and Turner (1979) presented the following general equation for calculating the axial deformation of side shear only shafts in a single elastic half space: (7.15) where wt is the axial deformation of the shaft at the rock surface; Qt is the applied load at the top of the shaft; Em is the deformation modulus of the rock mass; B is the diameter of the shaft; and I is the axial deformation influence factor given in Figure 7.5. The values of I given in Figure 7.5 have been calculated for a Poisson’s ratio of 0.25. It has been found that variations in the Poisson’s ratio in the range 0.1–0.3 for the rock mass and 0.15–0.3 for the concrete have little effect on the influence factors. The values of the influence factor shown in Figure 7.5 are for drilled shafts that are fully bonded from the rock surface. In many cases, the drilled shaft is recessed by casing the upper part of the drilled hole or for conditions where the shaft passes through a layer of soil or weathered rock where little or no side shear resistance will be developed. Recessment of the shaft will result in a decrease in axial deformation of the shaft at the head of the socket. This reduction can be expressed in terms of a reduction factor RF such that the axial deformation of the shaft at the ground surface is given by

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267

(7.16)

Fig. 7.5 Axial deformation influence factors for side shear only drilled shafts (after Pells & Turner, 1979). where Qt is the applied load at the top of the shaft; D and Bl are respectively the length and diameter of the recessed shaft; Ep is the composite Young’s modulus of the shaft (considering contributions of both concrete and reinforcing steel); RF is a reduction factor for the effect of recessment; B is the diameter of the socketed shaft; Em is the deformation modulus of the rock mass; and I is the influence factor for shaft with no recessment (see Fig. 7.5). The first portion of Equation (7.16) simply represents the elastic compression of the shaft over the length D. The second portion of Equation (7.16) gives the axial deformation of the socketed portion of the shaft. The reduction factor RF is given in Figure 7.6 for a range of situations.

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Fig. 7.6 Reduction factors for calculation of axial deformation of recessed drilled shafts (after Pells & Turner, 1979). End bearing only shaft

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An end bearing only shaft can be considered a shaft that is wholly recessed (See Fig. 7.7). The axial deformation of an end bearing only shaft at the ground surface consists of the elastic compression of the shaft and the axial deformation of the shaft base: (7.17)

where Qt is the applied load at the top of the shaft; D and Bl are respectively the length and diameter of the shaft; Ep is the composite Young’s modulus of the shaft (considering contributions of both concrete and reinforcing steel); Em and νm are respectively the deformation modulus and Poisson’s ratio of the rock mass; Cd is the shape and rigidity factor equal to 0.85 for a flexible footing and 0.79 for a rigid footing; and RF′ is a reduction factor for an end bearing only shaft as shown in Figure 7.7. The axial deformation of the shaft base is calculated in a similar manner to that of a footing on the surface. However, because the rock mass below the base of the shaft is more confined than surface rock mass, the axial deformation of the shaft base will be smaller than that of a footing at the surface. The effect of this confinement if accounted for by applying the reduction factor RF′ to the deformation equation as shown in Equation (7.17). The value of the reduction factor depends on the ratio of the shaft length D to the shaft diameter B1, and the relative stiffness of the shaft and the rock mass. Figure 7.7 shows the values of the reduction factor RF′ obtained by Pells and Turner (1979). Side shear and end bearing shaft For side shear and ending bearing shafts, the axial deformation at the rock surface can be calculated using Equation (7.15). Considering the interaction between the side shear and end bearing, the influence factors given in Figure 7.8 should be used. These factors have been developed for elastic behavior without slip along the side walls by Rowe and Armitage (1987a).

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Fig. 7.7 Reduction factors for calculation of axial deformation of end bearing only drilled shafts (after Pells & Turner, 1979). Comparison of Figure 7.8(a) (for Eb/Em=1) with Figure 7.5 shows that the influence factor for a side shear and end bearing shaft is smaller than that for a side shear only shaft, which demonstrates that a shaft with both side shear and end bearing will settle less than a shaft with side shear only. Figure 7.9 shows the percentage of the load carried in the end bearing. (b) Analytical solutions of Carter and Kulhawy (1988) Carter and Kulhawy (1988) provide a set of approximate analytical solutions to predict the load-displacement response of drilled shafts in rock. Two layers of rock mass as shown in Figure 7.10 are considered in the solutions. The solutions are for a shaft without slip or with full slip. The following presents the solution for a shaft without slip while the solution for a shaft with full slip will be presented in Section 7.3.2.

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Under an applied axial load, the displacements in the rock mass are predominantly vertical, and the load is transferred from the shaft to the rock mass by vertical shear stresses acting on the cylindrical interface, with little change in vertical normal stress in the rock mass (except near the base of the shaft). The pattern of deformation around the shaft may be visualized as an infinite number of concentric cylinders sliding inside each other (Randolph & Wroth, 1978). Randolph and Wroth (1978) have shown that, for this type of behavior, the displacement of the shaft w may be described adequately in terms of hyperbolic sine and cosine functions of depth z below the surface, as given below: w=A1 sinh(µz)+A2 cosh(µz) (7.18) in which, A1 and A2 are constants which can be determined from the boundary conditions of the problem. The constant µ is given by (7.19) where ζ=ln[2.5(1−νm)L/R]; R=B/2 is the radius of the shaft; λ=Ep/Gm; Ep is the Young’s modulus of the shaft; Gm=Em/[2(1+νm)] is the shear modulus of the rock mass surrounding the shaft; and Em and νm are respectively the deformation modulus and Poisson’s ratio of the rock mass surrounding the shaft. For side shear and end bearing shafts as shown in Figure 7.10(a), the shaft base can be approximated as a punch acting on the surface of an elastic half-space with Young’s modulus Eb and Poisson’s ratio νb. Using the standard solutions for the displacement of a rigid punch resting on an elastic half-space as the boundary condition at the base of the shaft, the elastic displacement at the head of the shaft can be obtained by (Randolph & Wroth, 1978): (7.20)

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Fig. 7.8 Axial deformation influence factors for side shear and end bearing drilled shafts (after Rowe & Armitage, 1987a).

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Fig. 7.9 Load distribution curves for side shear and end bearing drilled shafts (after Rowe & Armitage, 1987a).

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where ξ=Gb/Gm; Gb=Eb/[2(1+νb)] is the shear modulus of the rock mass below the shaft base; and Eb and νb are respectively the deformation modulus and Poisson’s ratio of the rock mass below the shaft base. The proportion of the applied load transmitted to the shaft base is (7.21)

For side shear only shafts as shown in Figure 7.10(b), the boundary condition at the shaft base is one of zero axial stress. For this case, the elastic displacement at the head of the shaft can be obtained by

Fig. 7.10 Axially loaded drilled shafts in rock (after Carter & Kulhawy, 1988).

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275

Fig. 7.11 Comparison of analytical solution with finite element solution for predicting axial elastic displacement (after Carter & Kulhawy, 1988). (7.22)

The solution given by Equations (7.20) and (7.22) are in general agreement with the finite element solutions by Pells and Turner (1979) and Rowe and Armitage (1987a) as presented in last sections (Fig. 7.11). Example 7.1 A drilled shaft of 3.0 meters long and 1.0 meter in diameter is to be installed in siltstone. The rock properties are as follows: Unconfined compressive strength of intact rock, σc=15.0 MPa Deformation modulus of intact rock, Er=10.6 GPa RQD=70

Determine the settlement of the drilled shaft at a work load of 10.0 MN.

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276

Solution: For simplicity, the Young’s modulus of the drilled shaft is simply assumed to be Ep=30 GPa. The Poisson’s ratio of 0.25 is selected for both the drilled shaft and the rock. Using Equation (4.24), the rock mass modulus: αE=0.0231×70−1.32=0.297 Em=0.297×10.6=3.15Gpa Using solutions based on finite element method L/B=3.0/1.0=3.0 Ep/Em=30/3.15=9.52 If the drilled shaft is side shear resistance only (i.e., the shaft base cannot be cleaned), from Figure 7.5, the axial deformation influence factor is I=0.462. Using Equation (7.15), the settlement of the drilled shaft at the rock surface is

If the drilled shaft has both side shear and end bearing resistance, from Figure 7.8, the axial deformation influence factor is I=0.417 for Eb/Em=1.0. Using Equation (7.15), the settlement of the drilled shaft at the rock surface is

From Figure 7.9, it can be seen that about 15% of the load is transmitted to the shaft base. Using analytical solutions of Carter and Kulhawy (1988)

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277

ξ=Gb/Gm=1.0 for Eb/Em=1.0 If the drilled shaft is side shear resistance only (i.e., the shaft base cannot be cleaned), the settlement of the drilled shaft at the rock surface can be calculated from Equation (7.22) as

If the drilled shaft has both side shear and end bearing resistance, the settlement of the drilled shaft at the rock surface can be calculated from Equation (7.20) as

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278

The percentage of the load transmitted to the shaft base can be calculated from Equation (7.21) as

The results from the solutions based on the finite element method are in good agreement with those from the analytical solutions of Carter & Kulhawy (1988). 7.3.2 Nonlinear continuum approach (a) Solutions based on finite element results Rowe and Armitage (1987a) performed an elastic-plastic finite element analysis that accounts for slip along the interface based on the technique developed by Rowe and Pells (1980). Two layers of rock are considered in the analyses. The interface behavior is established in terms of the Coulomb failure criterion. The roughness of the interface is modeled implicitly through the use of an angle of interface dilatancy that produces additional normal stress on the interface as the shaft deflects vertically due to the applied load. The contribution of the interface dilatancy commences once slip occurs at the interface. The results of this study are presented in three sets of design charts respectively for Eb/Em=0.5, 1.0 and 2.0. Although the analysis is carried out considering the behavior of a cohesive-frictional-dilative interface, the design charts are developed only for nondilative-cohesive interfaces. The procedure for using the design charts is described in Rowe and Armitage (1987b). (b) Analytical solutions of Carter and Kulhawy (1988) The case of slip along the entire length of the shaft has also been considered in detail by Carter and Kulhawy (1988). For this case, the shear strength of the interface is given by the Coulomb criterion: (7.23)

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279

where c is the interface cohesion; is the interface friction angle; and σr is the radial stress acting on the interface. As relative displacement (slip) occurs, the interface may dilate, and it is assumed that the displacement components follow the dilation law: (7.24) where ∆u and ∆w are the relative shear and normal displacements of the shaft-rock interface; and ψ is the angle of dilation defined by Davis (1968). To determine the radial displacements at the interface, the procedure suggested by Goodman (1980) and Kulhawy and Goodman (1987) is followed, in which conditions of plane strain are assumed, as an approximation, independently in the rock mass and in the slipping shaft. The rock mass is considered to be linear elastic, even after full slip has taken place, and the shaft is considered to be an elastic column. These assumptions, together with the dilatancy law, allow one to derive an expression for the variation of vertical stress in the compressible shaft. The distribution of the shear stress acting on the shaft can then be calculated from equilibrium conditions, and the vertical displacement can be determined as function of depth z by treating the shaft as a simple elastic column. The ‘full slip’ solution for the displacement of the shaft head is derived as (7.25)

in which F3=a1(λ1BC3−λ2BC4)−4a3 (7.26) (7.27) C3,4=D3,4/(D4−D3) (7.28) (7.29)

(7.30)

(7.31)

Drilled shafts in rock

280

(7.32) a1=(1+νm)ς+a2 (7.33) (7.34)

(7.35)

All other parameters in Equations (7.25) to (7.35) are as defined before. The adequacy of the closed-form expressions is demonstrated by comparing them with the finite element solution of Rowe and Armitage (1987a, b). The overall agreement between the closedform solutions and the finite element results is good (Fig. 7.12). It must be noted that the closed-form solutions of Carter and Kulhawy (1988) just consider “no slip” (presented in Section 7.3.1) and “full slip” conditions. They cannot predict the load-displacement response between the occurrence of first slip and full slip of the shaft. However, the finite element results indicate that the progression of slip along the shaft takes place over a relatively small interval of displacement. Therefore it seems reasonable, at least for most practical cases, to ignore the small region of the curves corresponding to the progressive slip and to assume that the load-displacement relationship is bilinear, with the slope of the initial portion given by Equation (7.20) and the slip portion by Equation (7.25) (Carter & Kulhawy, 1988).

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Fig. 7.12 Axial displacement of a drilled shaft in rock considering full slip (after Carter & Kulhawy, 1988). 7.4 FINITE ELEMENT METHOD (FEM) The finite element method is probably the most powerfiil and the most widely used numerical method currently available to engineers. Suitable elements can be used to simulate not only linearly elastic materials, but also nonlinear materials with different failure criteria, including rock discontinuities and shaft-rock interfaces (see Sections 4.3.4 and 4.4.3 for discussion of joint elements). However, the finite element method is time consuming and needs sophisticated soil or rock constitutive relations whose parameters are often difficult if not impossible in design practice to obtain. Therefore, the finite element method is, in general, used for analysis of important structures and for generation of parametric solutions for the load-displacement relations of axially loaded drilled shafts, such as the charts presented in Sections 7.3.1 and 7.3.2. Typical of many geotechnical problems, the analysis of drilled shafts in rock involves an unbounded domain. It is a common practice in finite element modeling of these problems to truncate the finite element mesh at a distance deemed far enough so as not to influence the near field solutions. These truncations are usually determined by trail and

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282

error until an acceptable solution is obtained. Such a method places a heavy demand on computer resources, both memory and time, as solutions for the far field which are of no interest are generated as well. In the last decade or so, several methods have been developed to model unbounded domains. Of these methods, the use of infinite elements with finite elements appears to be the most popular. Leong and Randolph (1994) successfully used finite elements and infinite elements in the modeling of axially loaded shafts in rock.

7.5 DRILLED SHAFT GROUPS Numerous methods exist for analyzing axially loaded pile groups in soil (Poulos, 2001), some of which can be applied to drilled shaft groups in rock and are briefly described in the following. 7.5.1 Settlement ratio method In the settlement ratio method, the group settlement is related to the single-shaft settlement as follows: (7.36) where wtG is the settlement of the shaft group; wtav is the settlement of a single shaft at the average load of a shaft in the group; and Rw is the settlement ratio. wtav can be estimated using the methods presented in the previous sections or from the results of load test on a prototype drilled shaft. Theoretical values of Rw for various pile groups in soil have been presented by Poulos and Davis (1980) and Butterfield and Douglas (1981). A particularly useful approximation for the settlement ratio has been derived by Fleming et al. (1992): (7.37) where n is the number of piles in the group; and e is an exponent depending on pile spacing, pile proportions, relative pile stiffness and the variation of soil modulus with depth. For typical pile proportions and pile spacings, Poulos (1989) suggested the following approximate values: e≈0.5 for piles in clay, and e≈0.33 for piles in sand. For drilled shafts in rock, the e values suggested by Poulos (1989) for soils may be used for the very preliminary design. For the final design of major projects, it is desirable, when feasible, to conduct axial load tests on groups of two or more drilled shafts in rock in order to confirm the e values of Poulos (1989) or to derive new, site-specific values. 7.5.2 Equivalent pier method The equivalent pier method, frequently used for pile groups in soils, treats the pile group as an equivalent pier consisting of the piles and the soil between them (Poulos & Davis,

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283

1980; Randolph, 1994). For closely spaced drilled shafts in rock, the shaft group may also be analyzed using the equivalent pier method. Consider the drilled shaft group as an equivalent pier (Fig. 7.13), the diameter of the equivalent pier Beq can be taken as (Randolph, 1994).

Fig. 7.13 Equivalent pier method treating drilled shaft group as a group block. (7.38) where Ag is the plan area of the drilled shaft group as a block. Deformation modulus of the equivalent pier Eeq is then calculated as Eeq=Em+(Ep−Em)Apt/Ag (7.39) where Ep is the Young’s modulus of the drilled shafts; Em is the deformation modulus of the rock mass; and Apt is the total cross-sectional area of the drilled shafts in the group. The load-settlement response of the equivalent pier can be calculated using the solutions as described in the previous sections for the response of a single drilled shaft. Based on the equivalent pier method and the load-transfer (t-z curve) approach, Castelli and Maugeri (2002) presented a simplified nonlinear analysis for settlement prediction of pile groups in soil. To take into account the group action due to pile-soilpile interaction, load-transfer functions are modified to relate the behavior of a single pile to that of a pile group. The bearing capacity of the equivalent pier can be evaluated using the procedure in Section 6.4. The initial stiffness of the equivalent pier is estimated by

Drilled shafts in rock

284

(7.40)

where Kgi is the initial stiffness of the equivalent pier and β is an empirical parameter. To take into account the increase of pile group head settlements with respect to the case of a single pile, the following expression is used (7.41)

where wg is the average settlement of the equivalent pier and ε is an empirical parameter. The empirical parameters β and ε can be derived on the basis of numerical analysis of field tests. Castelli and Maugeri (2002) derived values of 0.30 and 0.15 respectively for β and ε based on analysis of field test piles and pile groups in soils. For drilled shafts in rock, similar values of β and ε can be obtained from field tests of shafts and shaft groups. 7.5.3 Finite element method (FEM) The finite element method has been used to analyze axially loaded pile groups in soil by simplifying the group to an equivalent plane strain or axisymmetric system. If necessary, it can also be used to analyze drilled shaft groups in rock.

8 Lateral load capacity of drilled shafts in rock 8.1 INTRODUCTION In the design of drilled shafts subjected to lateral forces, two criteria must be satisfied: first, an adequate factor of safety against ultimate failure, second, an acceptable deflection at working loads. This chapter discusses the prediction of ultimate load of drilled shafts and drilled shaft groups. The calculation of lateral deflection will be discussed in Chapter 9. As the axial load capacity, the lateral load capacity of a drilled shaft in rock is determined by the smaller of the two values: the structural strength of the shaft itself, and the ability of the rock to support the loads transferred by the shaft.

8.2 CAPACITY OF DRILLED SHAFTS RELATED TO REINFORCED CONCRETE The structural capacity of a drilled shaft under lateral loading is controlled by the bending capacity and the shear capacity. The bending capacity is usually checked by considering the interaction between axial load and bending moment. Figure 8.1 shows the normalized axial load-moment intersection diagrams for fy=10f′c and fy=15f′c, where fy is the yield strength of the longitudinal reinforcing steel and f′c is the specified minimum concrete strength. The factored axial load ΣγiQi is normalized by dividing by the factored nominal axial capacity

, where γi is the load factor for axial load i, Qi is the nominal value

is the resistance factor for the nominal (computed) structural axial of axial load i, and load capacity Qu. The factored moment ΣγmMm is similarly normalized by dividing by the factored nominal moment capacity

, where γm is the load factor for moment m, Mm

is the nominal value of moment m, and is the resistance factor for the nominal (computed) structural moment capacity Mu. The factored axial capacity is estimated from Equation (6.1). Normalized axial load-moment interaction diagrams may be developed for any fy/f′c ratios and cage diameters other than 0.6B. With the axial load-moment interaction diagrams available, the structural capacity can be checked as follows: 1. Estimate the combined axial load ΣγiQi. 2. Compute the factored nominal axial capacity

from Equation (6.1).

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Fig. 8.1 Normalized axial loadmoment interaction diagrams for drilled shafts for (a)fy=10f′c and (b)fy=15f′c.

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Table 8.1 Nominal structural moment capacity Mu of drilled shaft. Mu/f′cBAg As/Ag

fy−10f′c

fy=15f′c

0.01

0.037

0.050

0.02

0.067

0.092

0.03

0.088

0.119

0.04

0.107

0.147

0.05

0.126

0.172

0.06

0.144

0.197

0.07

0.161

0.208

0.08

0.176

0.244

As is the cross-sectional area of the longitudinal reinforcing steel Ag is the gross cross-sectional area of the shaft fy is the yield strength of the longitudinal reinforcing steel f′c is the specified minimum concrete strength B is the diameter of the shaft

3. Estimate the factored (required) moment ΣγmMm. 4. Estimate the nominal structural moment capacity Mu of the drilled shaft. This may be based on complete analysis, or it may be obtained from design aids such as Table 8.1. 5. Compute the factored nominal moment capacity reinforced concrete.

where

for

6. Determine the ratios , and and with these values locate an appropriate point on the axial load-moment interaction diagram. If the point falls inside the area defined by the interaction curve, the shaft capacity is adequate. If this is not the case, the shaft size should be increased and the analysis repeated until the shaft capacity is adequate. The factored nominal shear capacity of a drilled shaft without special shear reinforcement can be calculated by (O’Neill & Reese, 1999): (8.1) where is the capacity reduction (resistance) factor for shear=0.85; Vu is the nominal (computed) shear resistance; νc is the limiting concrete shear stress; and Av is the area of the shaft cross section that is effective in resisting shear, which can be taken as B(0.5B+0.5756rls) for a circular drilled shaft, where r1s is the radius of the ring formed by the centroids of the longitudinal reinforcing steels. The limiting concrete shear stress νc can be evaluated from:

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(8.2a) (8.2b)

where both f′c and νc are in kPa. If the factored shear load is greater than the factored nominal shear resistance determined above, two options are available. The first and simplest solution is to increase the shaft diameter to increase the shear capacity. The second alternative is to provide properly designed shear reinforcement (O’Neill & Reese, 1999). O’Neill and Reese (1999) provide detailed discussion and examples on checking the structural load capacity of drilled shafts.

8.3 CAPACITY OF DRILLED SHAFTS RELATED TO ROCK 8.3.1 Method of Carter and Kulhawy (1992) Carter and Kulhawy (1992) presented a method to determine the lateral load capacity of drilled shafts related to rock. When a lateral load is applied at the rock surface, the rock mass immediately in front of the shaft will be subject to zero vertical stress, while horizontal stress is applied by the leading face of the shaft. Ultimately, the horizontal stress may reach the uniaxial compressive strength of the rock mass and, with further increase in the lateral load, the horizontal stress may decrease as the rock mass softens during postpeak deformation. Large lateral deformations may be required for the rock mass at depth to exert a maximum reaction stress on the leading face of the shaft. Therefore, Carter and Kulhawy (1992) assumed that the reaction stress at the rock mass surface, in the limiting case of loading of the shaft, is zero or very nearly zero as a result of the postpeak softening. Along the sides of the shaft, some shearing resistance may be mobilized. The shearing resistance varies along the perimeter and the average can be chosen as τmax/2, where τmax is likely to be approximately the same as the maximum unit side resistance under axial compression. Therefore, at the rock surface, the ultimate force per unit length resisting the lateral loading is Bτmax. At greater depth, Carter and Kulhawy (1992) assume that the stress in front of the shaft increases from the initial in situ horizontal stress to the limit stress, pL, reached during the expansion of a long cylindrical cavity, i.e., a plane strain condition will apply. Behind the shaft, the horizontal stress will decrease, and after tensile rupture of the bond between the concrete and the rock mass, the horizontal stress will reduce to zero. At the sides of the shaft, some shearing resistance may also be mobilized. Therefore, at depth, the ultimate force per unit length resisting the lateral loading is B(pL+τmax). To determine the depth at which the limit stress is mobilized, the result of Randolph and Houslby (1984) in a cohesive material is adopted, i.e., the depth is about three times shaft diameter. Therefore, the distribution of ultimate force per unit length resisting the shaft is as shown in Figure 8.2.

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The ultimate lateral force that may be applied can be obtained from the horizontal equilibrium as: For L<3B

Fig. 8.2 Distribution of ultimate lateral force per unit length (after Carter & Kulhawy, 1992) (8.3a) For L>3B (8.3b) where τmax is the shearing resistance along the sides of the shaft, which is assumed to be the same as the maximum side resistance under axial loading; pL is the limit stress reached during the expansion of a long cylindrical cavity. Closed-form solutions have been found for the limit stresses developed during the expansion of a long cylindrical cavity in an elasto-plastic, cohesive-frictional, dilatant material (Carter et al., 1986). This limit stress pL can be determined from the following parametric equation in the nondimensional quantity ρ (Carter et al., 1986): (8.4)

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with (8.5) in which (8.6)

(8.7)

(8.8)

(8.9) (8.10) (8.11) (8.12) (8.13) (814) and σhi is the initial in situ horizontal stress; Gm is the elastic shear modulus; νm is the Poisson’s ratio; cm is the cohesion intercept; φm is the friction angle; and ψm is the dilation angle, all of the rock mass. The rock mass is assumed to obey the Coulomb failure criterion, and dilatancy accompanies yielding according to the following flow rule (8.15)

in which dε1p and dε3p are the major and minor principal plastic strain increments, respectively. For convenience, solutions for the limit pressures pL have been plotted in Figure 8.3 for selected values of νm, φm, ψm. The central vertical axis on each plot

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291

indicates the ratio of the plastic radius at the limit condition R to the cavity radius a. These charts may be used by entering with a value of Gm/(σhi+cmcotφm) and working clockwise around the figure, determining in turn values of R/a, then ρL=(pL+cmcotφm)/(σR+ cmcotφm), and thus, determining the limit pressure pL. 8.3.2 Method of Zhang et al. (2000) Zhang et al. (2000) presented an approximate method for calculating the ultimate lateral resistance of drilled shafts in rock. As shown in Figure 8.4(a), the total reaction of the rock mass consists of two parts: the side shear resistance and the front normal resistance. So the ultimate resistance pult can be estimated by (Briaud & Smith, 1983; Carter & Kulhawy, 1992): (8.16)

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Fig. 8.3 Limit solution for expansion of cylindrical cavity (after Carter & Kulhawy, 1992)

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Fig. 8.4 (a) Components of rock mass resistance; and (b) Calculation of normal limit stress pL (after Zhang et al., 2000).

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Fig. 8.5 Distribution of ultimate lateral resistance with depth (after Zhang et al., 2000). where B is the diameter of the shaft; τmax is the maximum shearing resistance along the sides of the shaft; and pL is the normal limit resistance. For simplicity, τmax is assumed to be the same as the maximum side resistance under axial loading and can be determined using the methods presented in Chapter 6. To determine the normal limit stress pL, the strength criterion for rock masses developed by Hoek and Brown (1980, 1988) is used. Assuming that the minor principal effective stress σ′3 is the effective overburden pressure γ′z and the limit normal stress pL is the major principal effective stress σ′1 [see Fig. 8.4(b)], we have, from Equation (4.64), the following (8.17)

where γ′ is the effective unit weight of the rock mass; z is the depth from the rock mass surface; and mb, s and a are rock mass parameters as described in Chapter 4. With the distribution of pult along the depth determined, the ultimate lateral load that may be applied can be approximated by (see Fig. 8.5) (8.18)

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Fig. 8.6 Comparison of estimated pult and that from field shaft tests (after Cho, 2002). Cho (2002) used the method of Zhang et al. (2000) to estimate pult of field test shafts in rock. The estimated values agree well with the field test data (see Fig. 8.6). Example 8.1 A drilled shaft of diameter 1.0 m is to be installed 3.0 meters in siltstone. The rock properties are as follows: Unconfined compresive strength of intact rock, σc=15.0 Mpa RMR=55

Determine the ultimate lateral load capacity of the shaft. Solution: From Table 4.5, mi=9 for siltstone.

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Using Equation (4.68),

Using Equation (6.8) and choosing Ψ=1.0 (lower bound), the side shear resistance can be obtained as

Assuming that the effective unit weight of the rock mass is 13.0 kN/m3, the limit normal stress pL can be obtained from Equation (8.17) as follows

Using Equation (8.18), the ultimate lateral load capacity can be obtained as

It need be noted that the ultimate lateral load capacity obtained above does not consider the moment equilibrium of the shaft. The structural strength of the shaft should also be checked when using the ultimate lateral load capacity obtained above in design.

8.4 CAPACITY OF DRILLED SHAFT GROUPS The ultimate lateral load capacity of a drilled shaft group can be calculated in a similar way to calculating the axial load capacity of a drilled shaft group, i.e. (8.19) where HultG is the ultimate lateral load of a drilled shaft group; N is the number of drilled shafts in the group; Hult is the ultimate lateral load of a single isolated drilled shaft; and α is the group efficiency factor. Table 8.2 lists the values of α recommended by the American Association of State Highway and Transportation Officials (AASHTO, 1989).

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Equation (8.19) applies when the head boundary conditions of the single shaft and the shaft group are the same. If the head boundary conditions of the single shaft and the shaft group are different (e.g., the single shaft has a free-head boundary condition while the shaft group has a flxed-head boundary condition because of the cap), a modification factor should be added to Equation (8.19) to account for the difference in head boundary conditions (Frechette et al., 2002): HultG=αNHultR (8.20) where R is the modification factor to account for the difference in head boundary conditions. For the case of a single shaft at a free-head boundary condition and a shaft group at a fixed-head boundary condition, Frechette et al. (2001) recommended a R value of 2.2 based on five case studies while Matlock and Foo (1976) recommended a R value of 2.0 based on a single case study (Frechette et al., 2002). If the drilled shafts are closely spaced, the drilled shaft group can be represented by a group block and its ultimate load can be calculated using the methods described in Section 8.3 by treating the group block as a big single shaft.

8.5 DISCONTINUUM METHOD In Sections 8.3 and 8.4, the rock mass is treated as a continuum. Since most rocks contain discontinuities, drilled shafts may fail due to the sliding of the rock blocks or wedges along discontinuities (see Fig. 8.7). In such cases, the lateral resistance is only provided by the shear resistance along the discontinuities and the weight of wedge bounded by the shaft and the discontinuities. Obviously, the rock mass need be treated as a discontinuous medium in order to obtain the lateral resistance provided by the wedges.

Table 8.2 Group efficiency factor a recommended by AASHTO (1989). Center-to-Center Shaft Spacing in Direction of Loading

Group Efficiency Factor α

3B

0.25

4B

0.40

6B

0.70

8B

1.00

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Figure 8.7 Sliding of rock blocks due to laterally loaded shaft. To (1999) developed a discontinuum method for determining the lateral load capacity of drilled shafts in a jointed rock mass containing two or three discontinuity sets. The method consists of two parts: a kinematic analysis and a kinetic analysis. In the kinematic analysis, Goodman’s block theory (Goodman & Shi, 1985) is extended to analyze the movability of a combination of blocks laterally loaded by a drilled shaft. Based on the extended theory, a 2-dimensional (2D) graphical method was developed to select the possible combinations of movable blocks. This 2D graphical method can be easily implemented with CAD programs such as AutoCAD or with spreadsheet programs such as Excel. In the kinetic analysis, the stability of each kinematically selected movable combination of blocks or wedges is analyzed with the limit equilibrium approach. This analysis, similar to slope stability analysis, considers the axial and lateral forces exerted by the drilled shaft in addition to the weight of the wedge and the shearing resistance along the discontinuities. From the stability analysis, simple analytical relations were developed to solve for the lateral load capacity of the drilled shaft. The lateral load capacity can also be obtained by analyzing the load-displacement response of a drilled shaft using the discrete element method (DEM) as described in Section 9.5.

9 Lateral deformation of drilled shafts in rock 9.1 INTRODUCTION For drilled shafts in rock to resist lateral loads, the design criterion in the majority of cases is not the ultimate lateral capacity of the shafts, but the maximum deflection of the shafts. Predicting the deformation of laterally loaded drilled shafts is, therefore, the most important aspect in designing drilled shafts to withstand lateral loads. To date, it has been customary practice to adopt the techniques developed for laterally loaded piles in soil (Poulos, 1971a, b, 1972; Banerjee & Davies, 1978; Randolph, 1981) to solve the problem of drilled shafts in rock under lateral loading (Amir, 1986; Gabr, 1993; Wyllie, 1999). However, the solutions for laterally loaded piles in soil do not cover all cases for laterally loaded drilled shafts in rock in practice (Carter & Kulhawy, 1992). Carter & Kulhawy (1992), therefore, developed a method for predicting the deformation of laterally loaded drilled shafts in rock. This method treats the rock mass as an elastic continuum and has been found to give reasonable results of predicted deflections only at low load levels (20–30% capacity). At higher load levels, the predicted displacements are too small (DiGioia & Rojas-Gonzalez, 1993). Reese (1997) developed a p-y curve method for analyzing drilled shafts in rock under lateral loading. The major advantage of the p-y curve approach lies in its ability to simulate the nonlinearity and nonhomogeneity of the rock mass surrounding the drilled shaft. However, since it represents the rock mass as a series of springs acting along the length of the shaft, the p-y curve approach ignores the interaction between different parts of the rock mass. Also, the p-y curve approach uses empirically derived spring constants that are not measurable material properties. Advances in computer technology have made it possible to analyze laterally loaded piles using three-dimensional (3D) finite element (FE) models. p-y curves (Hoit et al., 1997) or sophisticated constitutive relations (Wakai et al., 1999) are usually used to represent the soil or rock behavior in the 3D FE analyses. However, p-y curves have the limitations as described above. As for sophisticated soil or rock constitutive relations, it is often difficult if not impossible in design practice to obtain the parameters in the constitutive relations. Zhang et al. (2000) developed a nonlinear continuum method for analyzing laterally loaded drilled shafts in rock. The method can consider drilled shafts in a continuum consisting of a soil layer overlying a rock mass layer. The deformation modulus of the soil is assumed to vary linearly with depth while the deformation modulus of the rock mass is assumed to vary linearly with depth and then stay constant below the shaft tip. The effect of soil and/or rock mass yielding on the behavior of shafts is considered by assuming that the soil and/or rock mass behaves linearly elastically at small strain levels and yields when the soil and/or rock mass reaction force exceeds the ultimate resistance.

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9.2 SUBGRADE-REACTION (p-y CURVE) APPROACH Treating the rock as a series of springs along the length of the shaft (see Fig. 9.1), the behavior of the shaft under lateral load can be obtained by solving the following differential equation (Reese, 1997)

where Q is the axial load on the shaft; y is the lateral deflection of the shaft at a point z along the length of the shaft; p is the lateral reaction of the rock; EpIp is the flexural rigidity of the shaft; and W is the distributed horizontal load along the length of the shaft. Equation (9.1) is the standard beam-column equation where the values of EpIp may change along the length of the shaft and may also be a function of the bending moment. The equation (a) allows a distributed load to be placed along the upper portion of a shaft; (b) can be used to investigate the axial load at which a shaft will buckle; and (c) can deal with a layered profile of soil or rock (Reese, 1997). Computer programs, such as COM624P and LPILE, are available to solve equation (9.1) efficiently. COM624P (version 2.0 and higher) and LPILEPLUS can also consider the variation of EpIp with the bending moment (see O’Neill & Reese, 1999 for the detailed procedure). To solve Equation (9.1), boundary conditions at the top and bottom of the shaft also need be considered. For example, the applied shear and moment at the shaft head can be specified, and the shear and moment at the base of the shaft can be taken to be zero if the shaft is long. For short shafts, a base boundary condition can be specified that allows for the imposition of a shear reaction on the base as a function of lateral base deflection. Full or partial head restraint can also be specified. Other formula that are used in the analysis are (9.2)

(9.3)

(9.4) where V, M and S are respectively the transverse shear, bending moment and deflection slope of the drilled shaft. The major difference between various methods lies in the determination of the variation of p with y or the p-y curve, which are described below.

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Fig. 9.1 Subgrade-reaction (p-y curve) model of laterally loaded drilled shafts. 9.2.1 Linear analysis For linear analysis, the relationship between rock reaction p and shaft deflection y at any point along the shaft is assumed to be linear, i.e., (9.5) where kh is the coefficient of subgrade reaction, in the unit of force/length3; and B is the width or diameter of the shaft. Substituting Equation (9.5) into Equation (9.1) and neglecting the influence of Q and W, the governing equation for the deflection of a laterally loaded shaft with constant EpIp can be simplified as (9.6)

Solutions to the above equation may be obtained analytically as well as numerically with a computer program. The analysis of the load-displacement behavior of a drilled shaft also requires knowledge of the variation of kh along the shaft. A number of distributions of kh along the

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depth have been employed by different investigators, which can be described by the following general expression proposed by Bowles (1996): kh=Ah+Bhzn (9.7) where Ah, Bh and n are empirical constants which can be determined for a particular site by working backward from the results of lateral shaft load tests. If the rock is considered homogeneous with a constant kh down the length of the drilled shaft, the deflection u (both y and u are used to denote lateral deflection in this book) and rotation θ at the ground level due to applied load H and moment M can be calculated by (9.8a)

(9.8b)

where Lc is the critical length given by (9.9)

It should be noted that Equation (9.8) is applicable only to flexible shafts, i.e., shafts longer than their critical length defined by Equation (9.9). For non-flexible shafts, solutions in closed-form expressions or in the form of charts are also available (Tomlinson, 1977; Reese & Van Impe, 2001). 9.2.2 Nonlinear analysis In general, the relationship between rock reaction p and shaft deflection y at any point along a shaft is nonlinear. Kubo (1965) used the following nonlinear relationship for soil between reaction p, deflection y, and depth z: p=kzmyn (9.10) where k, m, and n are experimentally determined coefficients. Equation (9.10) can also be used for rock if the corresponding coefficients k, m, and n can be determined. Since Matlock (1970) developed a method for deriving the variation of p with y, or the p-y curves, for soft clay, based on field test results, a number of methods for deriving p-y curves for different soils have been developed. Some of them are listed below (for details, the reader can refer to the listed references): 1.API RP2A (1982) or Reese et al. (1974) method for sand.

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2. Bogard and Matlock (1980) method for sand. 3. API RP2A (1991) or O’Neill and Murchison (1983) method for sand. 4. API RP2A (1982) or Matlock (1970) method for soft clay. 5. API RP2A (1982) or Reese et al. (1975) method for stiff clay. 6. Integrated method for clay by Gazioglu and O’Neill (1984). 7. Pressuremeter methods for all soils (Robertson et al., 1982, 1986; Briaud & Smith, 1983; Briaud, 1986). The method, developed by Reese (1997), specifically for calculating the p-y curves for rock is described in the following section. 9.2.5 p-y curves for rock Reese (1997) presented a p-y curve method for analyzing laterally loaded drilled shafts in rock. The concepts and procedures for constructing the p-y curves for rock are as follows (Reese, 1997): (1) The secondary structure of rock, related to joints, cracks, inclusions, fractures, and any other zones of weakness, can strongly influence the behavior of the rock and thus need be taken into account when applying the method described in this section. (2) The p-y curves for rock and the bending stiffness E0Ip for the shaft must both reflect nonlinear behavior in order to predict loadings at failure. (3) The initial slope Kmi of the p-y curves must be predicted because small lateral deflections of shafts in rock can result in resistances of large magnitudes. For a given value of compressive strength, Kmi is assumed to increase with depth below the ground surface. (4) The modulus of the rock Em, for correlation with Kmi, may be taken from the initial slope of a pressuremeter curve. Alternatively, the correlations presented in Chapter 4 can be used to determine Em. (5) The ultimate resistance pult for the p-y curves will rarely, if ever, be developed in practice, but the prediction of pult is necessary in order to reflect nonlinear behavior. (6) The component of the strength of rock from unit weight is considered to be small in comparison to that from compressive strength, and therefore the weight of rock is ignored. (7) The compressive strength σc of the intact rock for computing pult may be obtained from tests of intact specimens. (8) The assumption is made that fracturing will occur at the surface of the rock under small deflections; therefore, the compressive strength of intact rock specimens is reduced by multiplication by αm to account for fracturing. The value of αm is assumed to be 1/3 for RQD of 100 and to increase linearly to unity at RQD of zero. If RQD is zero, the compressive strength may be obtained directly from a pressuremeter curve. (a) Calculation of ultimate resistance pult of rock The following expressions are used for calculating the ultimate resistance pult of rock

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(9.11a)

(9.11b) where B is the diameter of the shaft; zm is the depth below the rock surface; σc is the unconfined compressive strength of the intact rock; and αm is the strength reduction factor considering that fracturing will occur at the surface of the rock under small deflections and thus reducing the resistance of the rock. (b) Calculation of the slope of initial portion of p-y curves The slope of the initial portion of p-y curves, kmi, is estimated by Kmi=kmiEm (9.12) where Em is the modulus of the rock (mass); and kmi is a dimensionless constant which can be determined by (9.13a) kmi=500 zm≥3B (9.13b) Equation (9.13) is developed from experimental data and reflect the assumption that the presence of the rock surface has a similar effect on kmi, as was shown for the ultimate resistance pult. (c) Calculation of p-y curves Referring to Figure 9.2, the p-y curve consists of three portions. The initial and the third portions are straight-lines and the second portion is a curve. The three portions can be expressed by First Portion: p=Kmiy; y≤yA (9.14a) (9.14b) Third Portion p=pult (9.14c)

Lateral deformation of drilled shafts in rock

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in which ym=kmB, where km is a constant, ranging from 0.0005 to 0.00005, that serves to establish overall stiffness of curves. The value of yA is found by solving the intersection of Equations (9.14a) and (9.14b), and is shown by (9.15)

Fig. 9.2 Sketch of p-y curve for rock (after Reese, 1997). (d) Comments The equations described above for constructing the p-y curves for rock are based on limited data and should be used with caution. An adequate factor of safety should be employed in all cases; preferably, field tests should be undertaken on full-sized shafts with appropriate instrumentation. If the rock contains joints that are filled with weak soil, the selection of strength and stiffness must be site-specific and will require a comprehensive geotechnical investigation. In those cases, the application of the method presented in this section should proceed with even more caution than normal (Reese, 1997). Cho et al. (2001) conducted lateral load tests on two drilled shafts embedded in weathered Piedmont rock. These shafts were instrumented with inclinometers and strain gauges. The field data obtained from the instrumented shafts were used to backcalculate the p-y curves. A comparison of the back-calculated p-y curves with the p-y curves predicted using the method of Reese (1997) shows that the method of Reese (1997) significantly overestimates the resistance of the weathered rock.

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9.3 CONTINUUM APPROACH The continuum approach assumes the soil and rock to be a continuum. Numerical solutions were developed by assuming that the soil and rock are ideally elastic, first with the boundary element method (Poulos, 1971a, b, 1972; Banerjee & Davies, 1978) and second with the finite element method (Randolph, 1981). Most of these elastic solutions were presented in the form of charts. Randolph (1981) published approximate but convenient closed-form expressions for the response of flexible piles to lateral loading. Considering the fact that the closed-form expressions of Randolph (1981) for the lateral response of flexible piles in soils may not cover the ranges of material and geometric parameters encountered in drilled shafts in rock, Carter and Kulhawy (1992) expanded the solutions by Randolph (1981). The solutions of Carter and Kulhawy (1992) give a reasonable agreement between measured and predicted displacements for drilled shafts in rock at low load levels (20–30% capacity). At higher load levels, however, the predicted displacements are too small (DiGioia & Rojas-Gonzalez, 1993). Zhang et al. (2000) developed a nonlinear continuum approach for the analysis of laterally loaded drilled shafts in rock. The approach can consider the effect of soil and/or rock mass yielding on the behavior of shafts. 9.3.1 Linear continuum approach (a) Approach of Poulos (1971a, b, 1972) and Poulos and Davis (1980) By modeling the soil as an elastic continuum and idealizing the pile as an infinitely thin strip of the same width and bending rigidity as the prototype pile, Poulos (1971a, b, 1972) and Poulos and Davis (1980) obtained the solutions for laterally loaded piles using the boundary element method. The solutions are presented in the form of charts and can be used to predict the deflection of drilled shafts in rock. For a free head drilled shaft, the lateral deflection u and rotation θ under lateral force H and overturning moment M at ground surface are given by (9.16a)

(9.16b)

where L is the length of the shaft; EmL is the deformation modulus of the rock mass at the level of shaft tip; and IuH, IuM, IθH and IθM (note that IuM=IθH) are deflection and rotation influence factors which are a function of the drilled shaft flexibility factor KR and the rock mass non-homogeneity η: (9.17)

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(9.18) where Em0 is the deformation modulus of the rock mass at the ground surface. A homogeneous rock mass is represented by η=1, whereas η=0 represents a rock mass with zero modulus at the surface. The deflection and rotation influence factors are plotted in Figures 9.3 to 9.5 for values of η of 0 and 1. If the shaft is partially embedded, the deflection of the free-standing portion due to shaft rotation and bending can be added to the groundline deflection to obtain the deflection at the shaft head. If the drilled shaft is fixed-headed, the horizontal deflection can be obtained by putting θ =0 in Equation (9.16b) and substituting for the obtained moment in Equation (9.16a), as (9.19)

Fig. 9.3 Deflection influence factor IuH (after Poulos & Davis, 1980)

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Fig. 9.4 Deflection and rotation influence factors IuM and IθH (after Poulos & Davis, 1980)

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Fig. 9.5 Rotation influence factor IθM (after Poulos & Davis, 1980). For a single raking shaft, Poulos and Madhav (1971) have shown that the force acting on the shaft head may be resolved into axial and normal components and the shaft then treated as a vertical shaft subjected to these forces and the applied moment. (b) Approach of Randolph (1981) and Carter and Kulhawy (1992) Randolph (1981) conducted a parametric study of the response of laterally loaded piles embedded in an elastic soil continuum. The study was conducted using the finite element method and the results were fitted with closed-form expressions from which the lateral response of piles may be readily calculated. Considering the fact that the closedform expressions for the lateral response of flexible piles in soils may not cover the ranges of material and geometric parameters encountered in drilled shafts in rock, Carter and Kulhawy (1992) expanded the solutions by Randolph (1981). The expressions were derived from the results of finite element studies of the behavior of laterally loaded drilled shafts in rock. For a drilled shaft wholly embedded in rock [Fig. 9.6(a)], the shaft response can be calculated in the following way (Carter & Kulhawy, 1992): (1) The shaft is considered flexible when (9.20)

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where Ee is the effective Young’s modulus of the shaft (9.21)

in which B and EpIp are respectively the diameter and flexural rigidity of the shaft; and G* is the equivalent shear modulus of the rock mass (9.22) in which Gm and νm are respectively the shear modulus and Poisson’s ratio of the rock mass. The shaft response can then be obtained by the closed-form expressions suggested by Randolph (1981), i.e., (9.23a)

(9.23b)

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Fig. 9.6 (a) Drilled shaft wholly embedded in rock; and (b) Drilled shaft embedded in soil and rock. (2) The shaft is considered rigid when (9.24)

The shaft response can then be obtained by the following closed-form expressions (9.25a)

(9.25b)

(3) The shaft can be described as having intermediate stiffness whenever the slenderness ratio is bounded approximately as follows

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312

(9.26)

The finite element results show that the displacements for an intermediate case exceed the maximum of the predictions for corresponding rigid and flexible shafts by no more than about 25%, and often by much less. For simplicity, it is suggested that the shaft displacement in the intermediate case be taken as 1.25 times the maximum of either: (a) The predicted response of a rigid shaft with the same slenderness ratio L/B as the actual shaft; or (b) the predicted response of a flexible shaft with the same modulus ratio (Ee/G*) as the actual shaft. Values calculated in this way should, in most cases, be slightly larger than those given by the more rigorous finite element analysis for a shaft of intennediate stiffness. If there exists a layer of soil overlying rock as shown in Figure 9.6(b), Carter and Kulhawy (1992) assume that the complete distribution of soil reaction on the shaft is known and that the socket provides the majority of resistance to the lateral load or moment. The groundline horizontal displacement u and rotation θ can then be determined after structural decomposition of the shaft and its loading, as shown in Figure 9.7. To determine the distribution of the soil reaction, they simply assume that the limiting condition is reached at all points along the shaft, from the ground surface to the interface with the underlying rock mass, and then use the reaction distribution suggested by Broms (1964a, b). For shafts through cohesive soils (Fig. 9.8), the lateral displacement uAO and rotation θAO of point A relative to point O are given by (9.27a)

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Fig. 9.7 Consideration of soil reaction: (a) Loading and displaced shape; and (b) Decomposition of loading (after Carter & Kulhawy, 1992). (9.27b) where Ls is the thickness of the soil layer; and su is the undrained shear strength of the soil. The shear force Ho and bending moment Mo at point O are determined by HO=H−9su(Ls−1.5B)B (9.28a) 2 MO=M−4.5su(Ls−1.5B) B+HLs (9.28b) The contribution to the groundline displacement from the loading transmitted to the rock mass can then be computed by analyzing a fully rock-socketed shaft of embedded length L, subject to horizontal force HO and moment MO applied at the level of the rock mass. For shafts through cohesionless soils (Fig. 9.9), the lateral displacement uAO and rotation θAO of point A relative to point O are given by (9.29a) (9.29b)

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314

where γ′ is the effective unit weight of the soil; and Kp is the Rankine passive earth pressure coefficient. The shear force HO and bending moment MO at point O are determined by (9.30a)

Fig. 9.8 Idealized loading of socketed shaft through cohesive soil (after Carter & Kulhawy, 1992).

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Fig. 9.9 Idealized loading of socketed shaft through cohesionless soil (after Carter & Kulhawy, 1992). (9.30b) Example 9.1 A drilled shaft of diameter 1.0 m is to be installed 3.0 meters in siltstone. The rock properties are as follows: Unconfined compressive strength of intact rock, σc=15.0 Mpa Deformation modulus of intact rock Er=10.6 GPa RQD=70

Determine the lateral displacement and rotation of the drilled shaft at the groundline by a horizontal force of 2.6 MN at 2.5 m above the groundline. Solution:

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316

For simplicity, the Young’s modulus of the drilled shaft is simply assumed to be Ep=30 GPa. A Poisson’s ratio of 0.25 is selected for both the drilled shaft and the rock. The flexural rigidity of the shaft is

Using Equation (4.24), the deformation modulus of the rock mass is αE=0.0231×70−1.32=0.297 Em=0.297×10.6=3.15 Gpa and the shear modulus of the rock mass is Gm=3.15/(1+0.25)=1.26 Gpa Using Equation (9.22), the equivalent shear modulus of the rock mass is G*=1.26×(1+3×0.25/4)=1.50 Gpa Since

the shaft is considered flexible and the lateral displacement and rotation of the drilled shaft at the groundline can be obtained from Equation (9.23) as

9.3.2 Nonlinear continuum approach Poulos and Davis (1980) presented an approximate nonlinear approach for calculating the deflection of laterally loaded piles in soil. This approach uses the elastic solutions presented in the last section, but introduces yield factors. The yield factors are a function of relative flexibility and load level and allow for the increased deflection and rotation of a pile due to the onset of local yielding of the soil adjacent to the pile. This approach can also be used to calculate the nonlinear deflection of drilled shafts in rock.

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For a drilled shaft subjected to a lateral load H at an eccentricity of e above the groundline, the groundline deflection u and rotation θ can be expressed as follows: (1) Uniform modulus with depth, i.e., η=1.0

(9.31a) (9.31b) (2) Linearly increasing modulus with depth, i.e., η=0

(9.32a) (9.32b) where uelastic and θelastic are respectively deflection and rotation from elastic solutions as described in the previous section; and Fu, Fθ, F′u, and Fθ are yield deflection and rotation factors which can be found from Poulos and Davis (1980). The yield factors are functions of a dimensionless load level H/Hu, where Hu is the ultimate lateral load capacity of the equivalent rigid shaft and can be estimated using the methods presented in Chapter 8. Zhang et al. (2000) developed a nonlinear continuum approach for the analysis of laterally loaded drilled shafts in rock. This approach adopts and extends the basic idea of Sun’s (1994) work on laterally loaded piles in soil. Sun’s model treats soil as a homogeneous elastic continuum with a constant Young’s modulus, which may apply to stiff clay, and it does not consider yielding of the soil. In the nonlinear approach developed by Zhang et al. (2000), drilled shafts in a soil and rock mass continuum (see Fig. 9.10) are considered, and the effect of soil and/or rock mass yielding on the behavior of shafts is included. For simplicity, the shaft is assumed to be elastic, while the soil/rock mass can be either elastic or elasto-plastic. It is, nevertheless, possible to also check whether the shaft concrete will yield or not using standard concrete design methods, as will be briefly mentioned later. (a) Method of analysis—elastic behavior Governing equations of shaft and soil/rock mass system Consider a drilled shaft of length L, radius R and flexural rigidity EpIp, embedded within a

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Fig. 9.10 (a) Shaft and soil/rock mass system; (b) Coordinate system and displacement components; and (c) Shear force V(z) and moment M(z) acting on shaft at z (after Zhang et al., 2000). soil/rock mass system (Fig. 9.10). The deformation modulus of the soil varies linearly from Es1 at the ground surface to Es2 at the soil/rock mass interface. The deformation modulus of the rock mass varies linearly from Em1 at the soil/rock mass interface to Em2 at the shaft tip and stays constant below the shaft tip. For convenience of presentation, nonuniformity indices defined by (9.33) (9.34) are introduced. The increase of the deformation moduli of the soil and the rock mass with depth, z, can then be expressed, respectively, by (9.35a)

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(9.35b) Em=Em2 (z>Ls+L) (9.35c) By adopting the basic idea of Sun (1994), the displacements usm, νsm and usm of the soil and/or rock mass can be approximated by separable functions of the cylindrical coordinates r, θ and z as (9.36a) (9.36b) wsm(r,θ,z)=0 (9.36c) is a where u(z) is the displacement of the shaft as a function of depth; and dimensionless function representing the variation of displacements of the soil and/or rock mass in the r-direction. For the displacements of Equation (9.36), the governing equations for the shaft can be obtained as (9.37a)

(9.37b)

with boundary conditions (9.38a)

(9.38b)

(9.38c) us−um=0 (z=Ls) (9.38d)

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(9.38e) (9.38f)

(9.38g)

(9.38h) (9.38i) where us and um are the displacement components u of the shaft in the soil and in the rock mass, respectively; and ts, ks and tm, km are parameters that can be expressed as (9.39a)

(9.39b)

(9.39c)

(9.39d) where m1 and m2 are parameters describing the behavior of the elastic foundations, which can be obtained by (9.40a) (9.40b)

Function

can be obtained by solving the following equation

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(9.41) where γ is a nondimensional parameter that can be expressed as (9.42)

and

The solution to Equation (9.41) that satisfies the unit condition at the finite condition at and m2 can then be expressed as (Sun, 1994)

can be obtained and the parameters m1 (9.43a)

(9.43b) where K0( ) is the modified Bessel function of the second kind of zero order; and K1( ) is the modified Bessel function of the second kind of first-order. The shear force V(z) acting on the shaft (see Fig. 9.10) can be obtained by (9.44a)

(9.44b) and the bending moment M(z) acting on the shaft (see Fig. 9.10) can be obtained by (9.45a)

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(9.45b)

The governing differential equations and the shear force V(z) and bending moment M(z) are solved using the classical finite difference method as described below. At this point it is also possible to check if the shaft concrete yields (recall that the basic assumption is non-yielding concrete). This can be done using the calculated shear force and moment together with the axial force on the shaft and using standard concrete design methods. Finite difference model The classical finite difference method (Desai & Christian, 1977) is employed to solve the governing differential Equation (9.37). By dividing the shaft in the soil into Ns equal segments (see Fig. 9.11) and using the central difference operator, for an interior node i (i= 0, 1, 2,…, Ns), the following equation is obtained:

Fig. 9.11 Dividing shaft into segments for finite-difference analysis, and estimating reaction force p of soil and rock from shear force V (after Zhang et al., 2000).

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(9.46)

where (9.47)

in which hs=Ls/Ns. Similarly, by dividing the shaft in the rock mass into Nm equal segments (see Fig. 9.11), the following equation is obtained for an interior nodey j(j=0, 1, 2,…, Nm): (9.48)

where (9.49)

in which hm=L/Nm. Equations (9.46) and (9.48) can be written recursively for each point i=0, 1, 2,…, Ns and j=0, 1, 2,…, Nm(see Fig. 9.11), resulting in a set of simultaneous equations in u. To solve the set of equations the boundary conditions must be introduced. By incorporating the boundaiy conditions expressed by Equation (9.38), the following finite difference equations can be obtained: at z=0 (9.50a)

(9.50b)

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−us(−1)−us(1)=0 (fixed-head) (9.50c) at z=Ls (9.50d) (9.50e) (9.50f)

(9.50g)

at z=Ls+L (9.50h) (9.50i)

The set of equations [Equations (9.46) and (9.48)] is modified by introducing the boundary conditions given in Equation (9.50). The resulting equations are solved simultaneously for u by using the Gaussian elimination procedure. After the shaft displacement u is obtained, the shear force V acting on the shaft can be obtained from Equation (9.44) as: (9.51a)

(9.51b)

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The bending moment M acting on the shaft can be obtained from Equation (9.45) as: (9.52a)

(9.52b) With the shear force V(z) obtained from Equation (9.51), the lateral reaction force p(z) (F/L) of the soil and rock mass acting on the shaft can be estimated by (see Fig. 9.11) (9.53a)

(9.53b)

Iteration procedure To solve for u, parameters ts, ks and tm, km should be known [see Equations (9.46) to (9.50)]. As can be seen from Equations (9.39) and (9.43), the parameter γ is needed to get ts, ks and tm, km. Note that γ defined by Equation (9.42) depends on u. Since we do not know the value of γ a priori, an iterative procedure is required to obtain it (Sun, 1994). The procedure consists of the following steps: 1. Assume γ=1.0 2. Calculate m1 and m2 from Equation (9.43) 3. Calculate ts, ks and tm, km from Equation (9.39) 4.Calculate the pile displacement u(z) along the shaft by solving Equations (9.46), (9.48) and (9.50) 5. Calculate the new value of γ using Equation (9.42) 6. Use the new value of γ and repeat steps 2–5. The iteration is continued until the ith and (i+1)th γ meet following criterion:

(9.54) where ε is a prescribed convergence tolerance, say, 0.0001. After γ is determined, the displacement of the shaft can be obtained. 7. Calculate the shear force and bending moment distribution along the shaft from Equations (9.51) and (9.52). 8. Calculate the lateral reaction force p(z) of the soil and rock mass acting on the shaft from Equation (9.53).

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(b) Method of analysis—including yielding of soil and rock mass Consideration of yielding of soil and rock mass For a laterally loaded shaft, the soil or rock mass near the top of the shaft may yield if the loads are large enough and, consequently, increased displacements will occur. Hence it is important to consider the effect of yielding of the soil or rock mass on the shaft behavior. A simple method is proposed to consider local yielding of the soil and rock mass by assuming that the soil and rock mass are elastic-perfectly plastic. The method consists of the following steps (Fig. 9.12): 1. For the applied lateral load H and moment M the shaft is analyzed by assuming the soil and rock mass are elastic and the lateral reaction force p of the soil and rock mass along the shaft is determined as described in the elastic analysis. 2. Compare the computed lateral reaction force p with the ultimate resistance pult (which will be discussed in detail in next section) and, if p>pult, determine the yield depth, zy, in the soil and/or rock mass.

Fig. 9.12 Consideration of yielding of soil and/or rock by decomposition of loading (after Zhang et al., 2000). 3. Consider the portion of the shaft in the unyielded ground (soil and/or rock mass) (zy ≤z≤Ls+L) as a new shaft and analyze it by ignoring the effect of the soil and/or rock mass above the level z=zy. The lateral load and moment at the new shaft head are

(9.55a)

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(9.55b) 4. Repeat steps 2–3. The iteration is continued until no further yielding of soil or rock mass occurs. 5. Obtain the final results by considering the two parts of the shaft separately. The part in the yielded soil and/or rock mass is analyzed as a beam with the distributed load pult acting on it. The part in the unyielded soil and/or rock mass is analyzed as a shaft with the soil and/or rock mass behaving elastically. A computer program has been written to execute the above iteration procedure including the process of elastic analysis. Determination of ultimate resistance pult of soil and rock mass To consider the yielding of soil and rock mass, the ultimate resistance pult of the soil and rock mass need be determined. For clays, it is usual to adopt a total stress approach and consider the ultimate soil resistance under undrained loading conditions. The simplest approach is to express pult as follows: pult=NpsuB (9.56) where B is the diameter of the shaft; su is the undrained shear strength of soil; and Np is the bearing capacity factor. A number of expressions for estimating Np are available in the literature (Hansen, 1961; Broms, 1964a; Matlock, 1970; Reese & Welch, 1975; Stevens & Audibert, 1979; Randolph & Houlsby, 1984). Zhang et al. (2000) recommended the following expression for Np, which was proposed by Matlock (1970) and Reese and Welch (1975) and are most widely employed in engineering practice: (9.57) where γ′ is the average effective unit weight of soil above depth z; and J is a coefficient ranging from 0.25 to 0.5. For sand, several methods are available in the literature for estimating pult (Broms, 1964b; Reese et al., 1974; Borgard & Matlock, 1980; Fleming et al., 1992). These methods often produce significantly different values of pult (Zhang et al., 2002). By analyzing the lateral soil resistance distribution along the width of piles and based on the test results of model rigid piles in sand collected from the published literature, Zhang et al. (2002) developed the following expression for calculating pult (9.58) where B is the diameter of the shaft; pmax is the maximum normal resistance against the shaft; τmax is the maximum shear resistance against the shaft; and η and β are the shape factors to account for the non-uniform distribution of the normal resistance and the shear

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resistance along the width of piles. According to Briaud et al. (1983), η and β can be respectively taken as 0.8 and 1.0 for round piles. pmax and τmax are calculated by (Zhang et al., 2002) (9.59) (9.60) where

γ′

is

the

average

effective

unit

weight

of

soil

above

depth

is the Rankine passive earth pressure coefficient, in which is the effective internal friction angle; K is the coefficient of lateral earth pressure (ratio of horizontal to vertical normal effective stress); and δ is the friction angle between the shaft and the soil. For shafts in rock, the method presented in Section 8.3 can be used to calculate the ultimate resistance pult. (c) Validation of method Zhang et al. (2000) verified the proposed method of analysis by comparing the results of the proposed method with those obtained by other methods available in the literature and with field test results including yielding. Comparison with available elastic solution The first verification concerns the elastic behavior of a shaft in a homogeneous half-space with a constant modulus of elasticity E and Poisson’s ratio ν. In Figure 9.13, the results obtained for a shaft having a length of 25 times its diameter (L/B=25) are shown as a function of parameter Ep/G*, where Ep is the elastic modulus of the shaft and G* is the modified shear modulus defined by G*=G(1+3ν/4) (9.61) The displacement of the shaft head is expressed by the nondimensional parameter uRG*/H, where R is the radius of the shaft. The general agreement between the results of Verruijt and Kooijman (1989), those of Poulos (1971a) and those obtained by the proposed

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Fig. 9.13 Dimensionless displacement of shaft head for soil or rock mass with constant E (after Zhang et al., 2000). method is good, although the displacements obtained by Verruijt and Kooijman (1989) are greater than those predicted by other continuum methods at small Ep/G* values. The dashed line in Figure 9.13 represents the results obtained by considering the soil or rock mass as springs having a subgrade modulus equal to the modulus of elasticity in the continuum model. The results for stiff shafts (large Ep/G* values) are remarkably good. For flexible shafts (small Ep/G* values), however, this method results in an overestimation of the displacements. As a second verification, the elastic behavior of a shaft in an elastic medium with a linearly increasing modulus of elasticity is considered, assuming zero stiffness at the ground surface, i.e., the shear modulus G of the soil or rock mass is expressed by G=mz (9.62) The nondimensional displacement uR2m*/H of the shaft head is shown in Figure 9.14, as a function of parameter Ep/m*R, where

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m*=m(1+3ν/4) (9.63)

Fig. 9.14 Dimensionless displacement of shaft head for soil or rock mass with G(E) increasing linearly with depth (after Zhang et al., 2000). As can be seen, the general agreement between the results of Verruijt and Kooijman (1989), of Randolph (1981), of Banerjee and Davies (1978), and those obtained with the proposed method is good, although the displacements of Randolph (1981) at small values of Ep/m*R are slightly smaller than those predicted with the other continuum methods. As in the case of a soil or rock mass with a constant modulus of elasticity, the spring (subgrade) model overestimates the displacements for flexible shafts (small Ep/m*R values), while the results for stiff shafts (large Ep/m*R values) correspond well to the other predictions. Comparison with field test results including yielding The next two verifications compare the results obtained with the proposed method with field test results including yielding at two sites by Frantzen and Stratten (1987). At each site, two drilled shafts 0.22 m (8.6 in.) in diameter and 4.57 m (15 ft) long, were

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constructed next to each other and were subjected to identical lateral loads. At the sandy shale test site, the average unconfined compressive strength σc of the sandy shale is 3.26 MPa (34 TSF) and the average RQD is 55% (Frantzen & Stratten, 1987). To predict the load-deflection response of the shaft, the deformation modulus Em and the ultimate resistance pult of the rock mass have to be determined first. Since Em is not given in the original report, the average deflection of the two shafts at the first recorded load 26.7 kN [see Fig. 9.15(a)] is used to back-calculate the value of Em, assuming that the shafts behave elastically at and below this load level. The back-calculated value of Em is 123 MPa. Next the ultimate resistance pult need be calculated. For sandy shale, the material constant mi can be obtained from Table 4.5 as mi=12. Since GSI is not given in the original report, it is approximately evaluated using the available information. Using Bieniawski’s 1989 Rock Mass Rating (RMR) system, GSI can be evaluated by GSI=RMR−5 (2.8) To evaluate RMR, we have to know the unconfined compressive strength, RQD, the spacing of discontinuities, the condition of discontinuities, the ground water conditions, and the discontinuity orientations (see Table 9.1). When evaluating RMR in Equation (2.8), a value of 15 is assigned to the groundwater rating and the adjustment for the discontinuity orientation value is set to zero. Since we lack the information about the spacing and condition of discontinuities for evaluating RMR, we assume the “average” condition for both the spacing and condition of discontinuities. With this assumption, RMR and thus GSI can be evaluated as shown in Table 9.1. Using the obtained GSI= 54>25, the material constants mb, s and a can be obtained from Equation (4.69), respectively, as mb=2.321, s=0.00603 and a=0.5. Assuming that the effective unit weight of the rock mass is 23 kN/m3, the ultimate resistance pult can be obtained, using the method in Section 8.3.2, as shown in Figure 9.16(a). Using the deformation modulus and ultimate resistance of the rock mass estimated above, the shaft head deflection at different load levels can then be predicted. The comparison of the shaft head deflection obtained from the field experiment and from the proposed method is shown in Figure 9.15(a). It can be seen that the predicted deflections are in a reasonable agreement with those measured. From the relative magnitudes of the predicted and measured deflections at high load levels, we can clearly see that the shaft surface condition is between smooth and rough.

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Fig. 9.15 Comparison of test (Frantzen & Stratten, 1987) and computed values of shaft head deflection at sandy shale and sandstone test sites (after Zhang et al., 2000).

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Fig. 9.16 Calculated ultimate lateral resistance at sandy shale and sandstone test sites (after Zhang et al., 2000).

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Table 9.1 Values of estimated RMR89 and GSI (after Zhang et al., 2000). Rock Mass Parameter A1

Sandy Shale

Sandstone

3.26 MPa

5.75 MPa

Rating

1

2

RQD

55%

45%

Ratinga)

13

8

Unconfined compressive strength a)

A2

A3

Spacing of discontinuities

Assume: 200–600 mm

a)

A4

A5

Rating

10

Condition of discontinuities

Assume: Slightly rough surfaces, Separation < 1 mm, Highly weathered walls

Ratinga)

20

Ground water

Hoek et al. (1995): Completely dry

a)

B

Rating

15

Rating adjustment for discontinuity orientations

Hoek et al. (1995): very favorable

Ratinga)

0

RMR

59

55

GSI=RMR−5

54

50

a)

Rating value is assigned according to Table 2.12(a).

At the sandstone test site, the average unconfined compressive strength of the sandstone is 5.75 MPa (60 TSF) and it has an average RQD of 45% (Frantzen & Stratten, 1987). Using the average deflection of the two shafts at load 26.7 kN [see Fig. 9.15(b)], the value of Em can be back-calculated as Em=170 MPa. GSI is evaluated as shown in Table 9.1. For sandstone, the material constant mi can be obtained from Table 4.5 as mi=19. Using GSI=50>25, the material constants mb, s and a can then be obtained as mb=3.186, s=0.00387 and a=0.5. Assuming again that the effective unit weight of the rock mass is 23 kN/m3, the ultimate resistance pult can be obtained as shown in Figure 9.16(b). The shaft head deflection at different load levels is predicted as shown in Figure 9.15(b). It can be seen that the predicted deflections are in a reasonable agreement with those measured, the predicted results for rough socket conditions being closer to the measured values than those for smooth socket conditions. The predicted results (Fig. 9.15) show that the socket condition at the sandy shale test site is somewhere in the middle between the smooth and rough conditions while the socket condition at the sandstone test site is closer to the rough condition. This is as

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expected because, for the same construction method, the socket in sandy shale should be smoother than that in sandstone. Example 9.2 In this example, the method of Zhang et al. (2000) is used to calculate the lateral displacement of drilled shaft foundations of a planned cable stayed bridge. The geologic profile is shown in Figure 9.17. The rock is a light brownish-gray to chocolate weathered and unweathered, fine grained, plagioclase-quartz-biotite granofels and phyllite and it includes thin beds of quartzite and fine grained schist. Since the geological conditions at the east pier site are worse than those at the west pier site (see Fig. 9.17), only the east shaft will be considered. Due to the magnitude of the expected loads, the drilled shafts are proposed to be socketed into the unweathered rock. Considering the influence of scour, the overburden soil layer is ignored in the design and the shaft is assumed to be in a two-layer (weathered and unweathered) rock mass system. The design parameters are summarized as follows: 1) diameter B and socket length L of the shaft: According to the construction methods and the axial load design, B=3.0 m and L=4.0 m were selected 2) deformation properties of the shaft: Ep=30 GPa and νp=0.25 3) applied lateral load (i.e., the working load) H and M: The designers used many different load combinations. For the most critical longitudinal loads acting in the EastWest direction, H=1.38 MN and M=51.1 MNm 4) Properties of the rock mass: RQD, σc, and Er of the weathered rock and the unweathered rock are respectively as follows: Weathered rock:

RQD=0 to 27 with an average=7, σc=6.9 MPa and Er=3.1 GPa. Unweathered rock:

RQD=40 to 93 with an average=76, σc=67.6 MPa and Er=20.7 GPa. The effective unit weight γ′ of both the weathered and unweathered rock masses is assumed to be 13 kN/m3. Solution: Using the Em/Er—RQD relationship presented in Chapter 4, Em of the weathered and unweathered rock masses can be obtained respectively as follows: Weathered rock mass: αE=0.0231(RQD)−1.32 (needs to be≥0.15) =−1.158→0.15 Em=αEEr=0.15×3.1=0.46 GPa Unweathered rock mass:

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αE=0.0231(RQD)−1.32 (needs to be≥0.15) =0.4356 Em=αEEr=0.4356×20.7=9.02 GPa

Fig. 9.17 Design example: Geological profiles of East and West pier sites (after Zhang et al., 2000). For granofels and phyllite, mi=10. GSI is estimated to be 35 and 65 respectively for the weathered rock mass and the unweathered rock mass. Using Equation (4.69), mb, s and a can be obtained as follows: Weathered rock mass (GSI=35>25): mb=0.9813, s=0.00073 and a=0.5 Unweathered rock mass (GSI=65>25): mb=2.865, s=0.0205 and a=0.5 The ultimate resistance pult is obtained, using the method in Section 8.3.2, as shown in Figure 9.18. The load-displacement and load-rotation relations of the shaft head, for both smooth and rough shaft surface conditions, can be obtained as shown respectively in Figures 9.19 and 9.20. It is noted that at the working load H=1.38 MN and M=51.1 MNm, the shaftrock mass system yields slightly for the smooth shaft surface condition and acts elastically for the rough shaft surface condition. The displacement and rotation of the shaft head at the working load H=1.38 MN and M= 51.1 MNm are as follows: For the smooth shaft surface condition: u=3.703×10−3m, θ=1.713×10−3 rad For the rough shaft surface condition: u=3.568×10−3 m, θ=1.658×10−3 rad

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The obtained displacement and rotation can then be checked against the allowable design values.

Fig. 9.18 Calculated ultimate lateral resistance of rock mass (after Zhang et al., 2000).

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Fig. 9.19 Predicted shaft head loaddisplacement relations for the design example with M/H= 37.0 m (after Zhang et al., 2000).

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Fig. 9.20 Predicted shaft head loadrotation relations for the design example with M/H=37.0 m (after Zhang et al., 2000). 9.4 FINITE ELEMENT METHOD (FEM) Randolph (1981) and Carter and Kulhawy (1992) used the finite element method (FEM) to generate the parametric solutions for the load-displacement relations of laterally loaded piles/shafts and, based on these solutions, they developed the closed-form expressions as described in Section 9.3.1. The finite element method can also be used for analysis of important structures and for study of the effect of important factors on the performance of drilled shafts. Zhang (1999) used the finite element code ABAQUS (1998) to study the effect of anisotropy of jointed rock mass on the deformation behavior of laterally loaded drilled shafts in rock.

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9.5 DISCRETE ELEMENT METHOD (DEM) The discrete element method (DEM) is widely used in studying problems related to fractured rock masses. Alfonsi et al. (1998) used the UDEC (Universal Distinct Element Code) software to analyze drilled shafts in fractured rock masses. UDEC is a 2D discrete element program specially designed to solve the discontinuous problem in which the mechanical behavior of discontinuities can be directly simulated (Cundall, 1980). Figure 9.21 shows the drilled shaft in a horizontal rock mass studied by Alfonsi et al. (1998). There are two sets of discontinuities in the rock mass. The first set is vertical and the second set has a dip angle of α (α=0° in Fig. 9.21). The intact rock elements are assumed elastic with deformation modulus Er=10 GPa and Poisson’s ratio νr=0.25. The discontinuities are assumed elasto-plastic with cj=0 kPa and . Assuming a constant axial load of Q=2.5 MN, Alfonsi et al. (1998) obtained the lateral loaddisplacement curves for three different values of α as shown in Figure 9.22. It can be seen that α=0 (the second set of discontinuities are horizontal) provides much higher lateral failure load than a =10° or 20° (the second set of discontinuities are inclined). This is because the rock mass cannot fail by sliding along the discontinuities when α=0 (see Fig. 9.23). Alfonsi et al. (1998) also analyzed drilled shafts in fractured rock slopes containing two sets of discontinuities: the first set of discontinuities are persistent and have a dip angle a, and the second set of discontinuities are non-persistent and perpendicular to the first set. Keeping a constant axial load of Q=55 MN and increasing the lateral load H, Alfonsi et al. (1998) obtained the failure modes for three different discontinuity orientations (expressed by α) as shown in Figure 9.24. When the persistent discontinuity set dips down with α= 60° [see Fig. 9.24(a)], no clear rupture line is formed and the maximum lateral load obtained is 4.5 MN. When the persistent discontinuity set is horizontal (α=0°) [see Fig. 9.24(b)], sliding occurs along the persistent discontinuities and the maximum lateral load obtained is only 3.5 MN. When the persistent discontinuity set dips up with α=30° [see Fig. 9.24(c)], a stair-shape rupture line is formed and the maximum lateral load obtained is very high, i.e., 18.5 MN.

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Fig. 9.21 Drilled shaft in rock analyzed using UDEC (after Alfonsi et al., 1998).

Fig. 9.22 Lateral load-displacement curves obtained using UDEC (after Alfonsi et al., 1998).

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Fig. 9.23 Influence of discontinuity orientation on the failure pattern of rock mass (after Alfonsi et al., 1998).

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Fig. 9.24 Failure modes for three different discontinuity orientations (after Alfonsi et al., 1998).

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It should be noted that the results shown in Figures 9.22 to 9.24 are from the 2D analyses. The actual 3D performance of drilled shafts in fractured rock masses will be affected by the third discontinuity set (Alfonsi et al., 1998, 1999). As shown in Figure 9.25, the rock mass volume moved depends on the pattern of the third discontinuity set.

9.6 DRILLED SHAFT GROUPS Numerous methods exist for analyzing laterally loaded pile groups in soil, some of which can be applied to drilled shaft groups in rock and are briefly described in the following. 9.6.1 Deflection ratio approach The deflection ratio approach for calculating the deflection and rotation of a pile group in soil has been described by Poulos and Davis (1980). The approach involves superposition of lateral interaction factors, and is similar in principle to the analysis described in Chapter 7 for axially loaded pile groups. For pinned-head piles, the group deflection can be expressed as follows uG=uavRu (9.64) where uG is the horizontal deflection of the pile group; uav is the horizontal deflection of a single pile at the average load level of a pile in the pile group; and Ru is the group deflection ratio for a pinned-head pile group. For fixed-head piles, the group deflection is uG=uavRF (9.65) where uav is as above; and RF is the group deflection ratio for a fixed-head pile group. For piles which are rigidly attached to the pile cap, but the pile cap can rotate, the response of the pile group is dependent on both the lateral and axial characteristics of the piles. However, for such groups, the lateral group deflection is found to be only slightly greater than that for a fixed-head group, so that, for practical purposes, Equation (9.65) may be used. The values of Ru and RF for different soil profiles can be found in Poulos (1979) for a variety of group configurations, pile spacings and relative stiffnesses. These values may be used with the single shaft deflection computed in Section 9.3 to estimate the deflection of a drilled shaft group in rock. However, since the values of Ru and RF are obtained specifically for piles in soil, they can only be used in the very preliminary design. For the final design of major projects, it is desirable, when feasible, to conduct lateral load tests on groups of two or more drilled shafts in rock in order to confirm the Ru and RF values of Poulos (1979) or to derive new, site-specific values.

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Fig. 9.25 Influence of discontinuity pattern on the rock mass volume moved (after Alfonsi et al., 1998). 9.6.2 p-y curve approach The p-y curve approach can be used to estimate group action by introducing a “pmultiplier” suggested by Brown et al. (1988) to modify the p-y curve for a single drilled shaft. That is, all of the values of rock resistance p are multiplied by a factor that is less than 1, the p-multiplier, depending upon the location of the shaft within the group and the spacing of the shafts within the group. That is, all along the p-y curve (Fig. 9.26): pgroup=ρPsingle (9.64)

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where ρ is the p-multiplier. This factor reflects a dominant physical situation that develops within a laterally loaded group of drilled shafts: The shafts in the leading row push into the rock in front of the group. The rock reacting against any drilled shaft in this “front row” is relatively unaffected by the presence of other drilled shafts in the group and only a minor adjustment needs to be made to the p-y curves. However, the shafts in the rows that “trail” the front row obtain resistance from rock that is pushed by the shafts into the voids left by the forward movement of the shafts in front of them. This phenomenon causes the value of rock resistance p on a p-y curve to be reduced at any given value of lateral deflection y

Fig. 9.26 Modification of p-y curves using the p-multiplier ρ (after Brown et al., 1988). relative to the values that would exist if the drilled shafts in the forward row were not there. In addition, the presence of all the shafts in the group produces a mass movement of the rock surrounding the shafts in the group, which reduces the p-value for a given displacement y to varying degrees for all drilled shafts in the group (O’Neill & Reese, 1999). Table 9.2 lists the most commonly used p-multiplier recommendations at 3 diameter center-to-center (3B) spacing. Rollins et al. (1998) recommend that the p-multipliers increase to unity at a center-to-center spacing of 6B. FHWA (1996b) recommend using the p-multipliers presented in Table 9.2 for all center-to-center spacings. It should be noted that the p-multipliers that have been developed are primarily for driven piles in soil. To apply the p-multipliers to drilled shafts in rock, the following two issues need be considered: 1. The difference between soil and rock in their performance 2. The stress relief around existing drilled shafts when new drilled shafts are installed adjacent to the existing drilled shafts.

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Therefore, it is desirable, when feasible, to conduct lateral load tests on groups of two or more drilled shafts in rock for major projects in order to confirm the p-multipliers in Table 9.2 or to derive new, site-specific values.

Table 9.2 Common p-multiplier recommendations at 3 diameter center-to-center spacing. Recommendation

FHWA (1996b)

Rollins et al. (1998)

Lead row

0.8

0.6

2nd row

0.4

0.4

3rd row

0.3

0.4

Software that uses p-y curves to analyze laterally loaded drilled shafts allow the user to input values of the p-multiplier, based on the recommendations in Table 9.2 or based on other information, such as site-specific load tests. When using a single-shaft computer code to analyze a group of identical, vertical, laterally loaded drilled shafts subjected to a shear load at the elevation of the shaft heads and a concentric axial load, it is advisable to use the lateral displacement rather than the lateral load (applied shear at the shaft head) as a head boundary condition (O’Neill & Reese, 1999). The head restraint condition (free, fixed or intermediate restraint) is used as the other head boundary condition, depending upon how the shaft is connected to the cap. A typical front row shaft is analyzed using the p-multiplier for the front row. This analysis gives the head shear, moment and rotation, as well as the deflected shape of the shaft and the shear and moment distribution along the shaft for the front-row shafts. This analysis is repeated for a typical drilled shaft on a trailing (back) row applying the same value of head deflection and using the value of pmultiplier for shafts on back rows to modify the p-y curves. Similar output is obtained. The shear load that must have been applied to the group to produce the assumed lateral head deflection is then equal to the head shear on a front row shaft times the number of shafts on the front row plus the head shear on a back row shaft times the number of shafts in the group that are not on the front row. If this shear is not equal to the applied shear, a different head displacement is selected and the process is repeated until the computed head shears of all of the shafts in the group sum to the applied group shear. The moment and shear distributions for the shafts in the front row will be different from those for the shafts not on the front row; therefore, different steel schedules will often be appropriate among the shafts in the various rows within the group (O’Neill & Reese, 1999). 9.6.3 Finite element method (FEM) Advances in computer technology have made it possible to analyze laterally loaded drilled shaft groups using 3D finite element (FE) models. p-y curves or sophisticated constitutive relations can be used to represent the rock behavior in the 3D FE analyses. It should be noted that, however, it is often difficult if not impossible in design practice to obtain the parameters in the sophisticated constitutive relations.

10 Stability of drilled shaft foundations in rock 10.1 INTRODUCTION Drilled shaft are frequently installed in rock slopes, for example, the foundations for power poles and bridges. In many cases, drilled shafts are also used to stabilize rock slopes. Because rock masses often contain discontinuities, it is important to check the stability of rock blocks or wedges formed by the discontinuities. Blocks formed by discontinuities may fail in different modes (Hoek & Bray, 1981). Figures 10.1(a) to (c) show respectively the planar sliding failure on a single discontinuity, the wedge sliding failure on two intersecting discontinuities, and the toppling failure of toppling blocks. In weathered or highly fractured rock masses, the failure surface is less controlled by single through-going discontinuities. In this case, an approximately circular failure surface may develop in a similar manner to failures in soil [Fig. 10.1(d)]. To be general, the drilled shafts in Figure 10.1 are subject to not only axial load but also lateral load. The failure surface may cut through the shaft or pass through below the shaft base.

10.2 PLANAR SLIDING FAILURE When discontinuities are approximately parallel to and dip out of the slope face, a planar sliding failure may be formed along the discontinuities as shown in Figure 10.2. The stability of the block is defined by the relative magnitude of two forces acting parallel to the potential sliding surface: the driving force F acting down the surface, and the resisting force R acting up the surface. The factor of safety (FS) of the block is simply defined by (10.1) For potential sliding along a persistent discontinuity as shown in Figure 10.2(a), the total driving force F acting down the sliding surface can be calculated by: F=Wsinα+Qsinα+Hcosα (10.2)

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Fig. 10.1 Failure modes of rock slopes: (a) Planar sliding failure on a single discontinuity; (b) Wedge sliding failure on two intersecting discontinuities; (c) Toppling failure of steeply dipping slabs; and (d) Approximately circular failure in weathered or highly fractured rock masses. where α is the angle between the discontinuity and the horizontal plane; Q and H are respectively the axial and lateral loads acting on the shaft head; and W is the total weight of the block including the drilled shaft. The resisting force R acting up the potential sliding surface is

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(10.3) are respectively the cohesion and internal friction angle of the where cj and discontinuity; and l and b are respectively the length and width of the sliding surface. The

Fig. 10.2 Potential sliding along: (a) A persistent discontinuity; and (b) Nonpersistent discontinuities. selection of b is critically important in analyzing the stability of the block and will significantly affect the FS calculated. If there is a row of drilled shafts in the longitudinal direction of the slope at spacing s, b can be simply selected equal to s. If there is no lateral load acting on the drilled shaft head and the cohesion of the discontinuity is zero, the FS is simply given by (10.4) that is, the limiting condition occurs when the dip of the sliding surface equals the friction angle of the discontinuity. For non-persistent discontinuities as shown in Figure 10.2(b), the failure surface will also pass through rock bridges. For this case, the method of Einstein et al. (1983) can be extended to calculate the stability of the block. The principle of this method is illustrated in a simplified form in Figure 10.3: the slope overlying the failure path is partitioned into a series of vertical slices, bounded at their bottom end by discontinuities or intact rock.

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The total driving force F can be calculated by summing slice contributions and the forces acting on the shaft head: (10.5) where α is the angle between the discontinuity and the horizontal plane; Q and H are respectively the axial and lateral loads acting on the shaft head; and Wi is the weight of the ith slice. For the slices containing the drilled shaft, it is important to include the weight of the shaft.

Fig. 10.3 Dividing the potential sliding wedge into slices to calculate total driving force F and total resisting force R (Modified from Einstein et al., 1983). The total resisting force R is calculated by summing slice contributions: (10.6)

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where Ri is the shear resistance mobilized by the portion of path underlying that slice. The ith portion of the path may be jointed or consist of intact rock. The calculation of Ri for those two cases is described in the following. For the ith portion of the path along a discontinuity, Ri can be simply calculated by (10.7) are respectively the cohesion and internal friction angle of the where cj and discontinuity; li and b are respectively the length and width of the ith portion of the path along the discontinuity; and σai is the effective normal stress on the discontinuity plane, which can be simply calculated by (10.8a) (10.8b) For the ith portion of the path through the intact rock, Ri need be calculated in two ways based on the failure modes of the rock bridges. For in-plane or low-angle out-ofplane transitions (βi<θt, see Fig. 10.4 for βi and Fig. 3.7 for θt), the intact-rock resistance Ri can be calculated by Ri=τaidib (10.9) where di is the “in-plane length” of the rock bridge (Fig. 10.4); and τai is the peak shear stress mobilized in the direction of discontinuities which can be obtained by (10.10) where σt is the tensile strength of the intact rock; and σai is the effective normal stress on the discontinuity plane, which can be simply calculated using Equation (10.8). For high angle transitions (βi>θt, see Fig. 10.4 for βi and Fig. 3.7 for θt), the intactrock resistance Rican be calculated by Ri=σtXib (10.11) where Xi is the distance between discontinuity planes that define the bridge (Fig. 10.4). In Figure 10.2, the drilled shaft is above the potential sliding surface. If the drilled shaft penetrates through the sliding surface as shown in Figure 10.5, the contribution of the drilled shaft to the stability of the block need be considered in the stability analysis. One simple way is to include the shear resistance of the drilled shaft in the resisting force R. It is important to note that the bending and shear capacity of the drilled shaft should be checked so that no structural failure will occur for the drilled shaft itself. The lateral load

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capacity provided by the portion of the shaft below the sliding surface should also be checked.

Fig. 10.4 Definition of βi, Xi and di (Modified from Einstein et al., 1983).

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Fig. 10.5 Drilled shaft penetrating through the sliding surface. Example 10.1 A row of drilled shafts of diameter 0.61 m is to be installed in a rock slope (Fig. 10.6). The properties of the drilled shafts and the rock mass are as follows: Spacing of the drilled shafts in the longitudinal direction, b=3.0 m Shear strength properties of the discontinuity, cj=0 and Unit weight of the rock mass, γ=23 kN/m3 Loads at the shaft head, Q=2,500 kN, H=100 kN Length of the shaft above the rock surface, e=4.0 m Length of the sliding surface along the discontinuity, l=14.4 m Other properties are shown in the figure

Evaluate the stability of the rock slope along the discontinuity. Solution: The gross cross-sectional area of the drilled shaft is Ag=0.257π×0.612=0.292 m2. Assuming that the drilled shaft has the same unit weight as the rock mass the weight

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of the rock block including the drilled shaft can be estimated as follows: W=[0.5×14.4×14.4×0.5×tan(30°)×3.0+0.292×4.0]×23=2,092 kN Using equation (10.2), the total driving force acting down the discontinuity surface is F=2,092×sin(30°)+2,500×sin(30°)+100×cos(30°)=2,383 kN Using equation (10.3), the total resisting force acting up the discontinuity surface is R=[2,092×cos(30°)+2,500×cos(30°)−100×sin(30°)]×tan(31°)=2,359 kN

Fig. 10.6 Stability analysis of a rock slope containing drilled shafts. Using Equation (10.1), the factor of safety of the rock slope along the discontinuity is FS=2,359/2,383=0.99 which is below 1, meaning the rock slope will slide along the discontinuity. It is noted that the factor of safety of the rock slope without the drilled shafts is

which is above 1, meaning the rock slope is stable. So the installation of the drilled shafts without passing through the discontinuity plane decreases the factor of safety of the rock slope because of the lateral load at the shaft head. If the drilled shafts pass through the discontinuity plane, the shear resistance of the drilled shafts will increase the stability of the rock slope. Assuming a concrete strength of f′c=28,000 kPa, the limiting concrete shear stress can be estimated from Equation (8.2) as

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νc=2.63×(1+25,000/13,780/0.292)×(28,000)0.5=714 kPa The area of the drilled shaft resisting shear along the discontinuity plane is estimated as Av=0.95Ag/cos(α)=0.95×0.292/cos(30°)−0.32 m2 So the shear resistance of the drilled shaft along the discontinuity plane is approximately Vu=νcAv=714×0.32=229 kN The factor of safety of the rock slope with drilled shafts passing through the discontinuity is FS=(2,359+229)/2,383=1.09 which is above 1. It is important that the drilled shafts be extended long enough below the sliding plane to provide sufficient lateral load capacity. The structural resistance of the drilled shafts should also be checked.

10.3 WEDGE SLIDING FAILURE A wedge failure is formed by two intersecting discontinuities dipping out of but aligned at an oblique angle to the slope face, the slope face and the upper slope surface [Fig. 10.1(b)]. The general failure mode is by sliding parallel to the line of intersection of the two discontinuities. The method of stability analysis of wedge blocks follows the same principles as that of the planar blocks, except that it is necessary to resolve forces on both of the sliding planes. For the detailed procedure for calculating the factor of safety of three-dimensional wedge blocks, the reader can refer to Hoek and Bray (1981) and Wyllie (1999). In the analysis, it is important to include the influence of the drilled shaft, including the size and location of the shaft and the applied axial and lateral loads at the shaft head. The drilled shaft may cause a stable wedge block to slide due to the loads at the shaft head or stabilize an unstable wedge block by extending beyond the potential sliding planes.

10.4 TOPPLING FAILURE Toppling failure may occur where discontinuities dip into the face and form either a single block, or series of slabs, such that the center of gravity of the block falls outside the base (Wyllie, 1999). The analysis of toppling failure of drilled shaft foundations can be conducted by examining the stability conditions of each block in turn starting at the top of the slope, following the procedure of Goodman and Bray (1976) and Wyllie (1999). In the analysis, it is important to include the influence of the drilled shaft, including the size and location of the shaft and the applied axial and lateral loads at the shaft head. The drilled shaft may cause a stable block to topple or stabilize an unstable block, depending on the size and location of the shaft and the applied axial and lateral loads at the shaft head.

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10.5 CIRCULAR FAILURE An approximately circular failure may occur in weathered or highly fractured rock masses, where the failure surface is less controlled by single through-going discontinuities but passes partially through intact rock and partially along existing discontinuities. The analysis of circular failure surfaces in rock may follow the same methodology for analyzing circular failure surfaces in soil. Slope stability analysis programs such as UTEXAS3 (Wright, 1991) and XSTABL (Sharma, 1991) can be readily used in the analysis of circular failure surfaces in rock. Again, the influence of the drilled shaft, including the size and location of the shaft and the applied axial and lateral loads at the shaft head, need be considered in the stability analysis.

11 Drilled shafts in karstic formations 11.1 INTRODUCTION Drilled shafts are frequently used in karstic formations. The challenges to using drilled shafts in karstic formations involve the highly irregular nature of the rock-overburden interface and the cavities in the bearing rock. The erratic nature of the bearing rock surface may require drilled shafts of different length be used. Two “depth of bedrock” borings cannot be simply connected by a straight line when inferring the rock surface from borings. An existing cavity underlying a drilled shaft may collapse after the building is in service. In blanketed active karst, new sinkholes may form and lead to collapse of drilled shafts. Drilled shafts in karstic formations may fail in different modes as shown in Figure 1.3. Since large structural load is supported by each drilled shaflt the failure of any one shaft may cause critical damage to the entire structure. Therefore, special care must be taken for drilled shafts in karstic formations.

11.2 CHARACTERISTICS OF KARSTIC FORMATIONS Approximately one-fourth of the earth’s land surface is underlain by rocks which are susceptible to solutioning activity (Cooper & Ballard, 1988). These rocks include limestone, dolomite, gypsum, anhydrite, and salt (halite) formations. Karst terrain develops through continuous erosion of soluble rock minerals over a long period of time. When rainwater falls onto the ground surface and percolates downward into cracks and fissures, it gradually dissolves the rock and leaves insoluble materials such as chert and clay behind. Since the weathering resistance of rocks is variable, areas of least weathering develop high rock pinnacles and areas of severe weathering develop deep slots and cavities. This results in an extremely irregular rock and overburden interface such as that shown in Figure 11.1. Rock solution can also result in enlargement of interparticle porosity, decreasing the rock strength and increasing the compressibility. Continuing solution enlargement of the interparticle porosity can result in coalescence of voids to form cavities.

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Fig. 11.1 Bedrock surface in a thinly mantled karst terrane, West Central Florida (after Wilson & Beck, 1988). When a rock cavity enlarges due to rock solution, the shear and tensile stresses in the cavity roof and the compressive stresses in the cavity walls increase, with the maximum shear stresses between. Continuing enlargement of a rock cavity may result in the collapse of the cavity roof.

11.3 INVESTIGATION OF KARSTIC FORMATIONS Because of the peculiar nature of karstic formations, it is extremely important to conduct site investigations to identity the degree of dissolution and the pattern and extent of specific hazards, such as cavities and sinkholes, erosion domes and the potential for their further development. Site investigations in karstic formations also include the three main stages as described in Chapter 5: (1) preliminary investigations, (2) detailed investigations, and (3) review during construction and monitoring. The following briefly describes the main points related to site investigations in karstic formations. Preliminary investigations begin with an intensive review of existing information. The fundamental data include geological maps and reports of the area to identify the underlying rock and, particularly, any carbonate rocks and their approximate geographic boundaries. In areas of significant cave development, local, regional and national cave exploration groups have prepared and compiled descriptions, detailed maps, and reports on specific caves. Table 11.1 is a summary of major karst areas in the United States. The second tool of preliminary investigations is remote sensing using air photographs, infrared imagery, and side-looking radar. For most of the United Sates, air photographs are available from the U.S. Geological Survey, the U.S. Forest Service, and the U.S. Department of Agriculture. Air photographs can be used to detect karstic terrain which is shown by such topographic features as basin-studded plains, narrow U-shaped valleys with vertical sides, rolling topography, and scalloped effect around river

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Table 11.1 Summary of major karst areas in the United States (after ASCE, 1996). Karst Area

Location

Characteristics

Southeastern coastal plain

South Carolina, Georgia

Rolling, dissected plain, shallow dolines, few caves; tertiary limestone generally covered by thin deposits of sand and silt.

Florida

Florida, southern Georgia

Level to rolling plain; tertiary, flat-lying limestone; numerous dolines, commonly with ponds; large springs; moderate sized caves, many water filled.

Appalachian

New York, Vermont, south to northern Alabama

Valleys, ridges, and plateau fronts formed south of Palaeozoic limestones, strongly folded in eastern part; numerous large caves, dolines, karst valleys, and deep shafts; extensive areas of karren.

Highland Rim

Central Kentucky, Tennessee, northern Georgia

Highly dissected plateau with Carboniferous, flat-lying limestone; numerous large caves, karren, large dolines and uvala.

Lexington Nashville

North-central Kentucky, central Tennessee, south eastern Indiana

Rolling plain, gently arched; lower Palaeozoic limestone; a few caves, numerous rounded shallow dolines.

Mammoth Cave Pennyroyal Plain

West-central, Rolling plain and low plateau; flat-lying Carboniferous southwestern Kentucky, rocks; numerous dolines, uvala and collapse sinks; very southern Indiana large caves, karren developed locally, complex subterranean drainage, numerous large “disappearing” streams.

Ozarks

Southern Missouri, northern Arkansas

Dissected low plateau and plain; broadly arched Lower Palaeozoic limestones and dolomites; numerous moderatesized caves, dolines, very large springs; similar but less extensive karst in Wisconsin, lowa, and northern Illinois.

Canadian River

Western Oklahoma, northern Texas

Dissected plain, small caves and dolines in Carboniferous gypsum.

Pecos Valley

Western Texas, southeastern New Mexico

Moderately dissected low plateau and plains; flat-lying to tilted Upper Palaeozoic limestones with large caves, dolines, and fissures; sparse vegetation; some gypsum karst with dolines.

Edwards Plateau

Southwestern Texas

High plateau, flat-lying Cretaceous limestone; deep shafts, moderate-sized caves, dolines; sparse vegetation.

Black Hills

Western South Dakota

Highly dissected ridges; folded (domed) Palaeozoic limestone; moderate-sized caves, some karren and dolines.

Kaibab

Northern Arizona

Partially dissected plateau, flat-lying Carboniferous limestones; shallow dolines, some with ponds; few

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moderate-sized caves. Western mountains

Wyoming, north western Utah, Nevada, western Montana, Idaho, Washington, Oregon, California

Isolated small areas, primarily on tops and flanks of ridges, and some area in valleys; primarily in folded and tilted Palaeozoic and Mesozoic limestone; large caves, some with great vertical extent, in Wyoming, Utah, Montana, and Nevada; small to moderate-sized caves elsewhere; dolines and shafts present; karren developed locally.

systems, with streams entrenched in bedrock on rectangular patterns. It is often useful to examine photographs over an extended time period which may show, for example, the progressive development of solutioning, or that sinkholes may have been obscured by human activities. Site reconnaissance is another important part of preliminary investigations and includes examining the area for verification of changes in previously observed or photographed features. Site reconnaissance should also be made to examine suspicious terrain details that are difficult to see from the air photographs because of tree and vegetation cover as well as overhangs and other obstructions. Geophysical methods, such as ground penetration radar (GPR), seismic survey and electrical resistivity, are widely used in site investigations in karstic terrains. As geophysical survey becomes quicker and cheaper, geophysical work is conducted in the stage of preliminary investigation as well as in the stage of detailed investigation. The selection of the most appropriate technique(s) for a site will depend on the particular site conditions. For example, GPR has been successfully used in Florida where the cavities are overlain by sand (Benson, 1984), but it is less successful in Pennsylvania where overlain is clayey soil with a high moisture content (Wyllie, 1999). The reliability of geophysics to detect cavities and predict their shape and size is limited because cavities may have irregular shapes and be filled with different materials such as air, water, clay and boulders. For this reason drilling is often carried out at the stage of detailed investigation. Because of the irregular rock and overburden interface and the cavities in rock, the number of borings required per unit of site area in karstic formations is usually much larger than for site investigations in other formations. In cases such as that the rock masses contain steep or vertical discontinuities, a few inclined borings may be required. Table 11.2 lists the boring and sampling techniques that may be used for foundation exploration in karstic formations.

11.4 CONSIDERATIONS IN THE DESIGN AND CONSTRUCTION OF DRILLED SHAFTS IN KARSTIC FORMATIONS Design of drilled shafts in karstic formations is generally based on end bearing resistance in the hard rock (Brown, 1990). Because it is difficult to form a clean socket in massive rock, the side friction of the rock socket is usually ignored or assigned an extremely conservative value. In areas where fault zones or other geological features have produced

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deep slots that have virtually no sound rock, drilled shafts may be designed for side friction only (Brown, 1990). In karstic formations, it is essential that the foundation rock below the bottom of each drilled shaft be explored for defects. The defects include cavities that could allow the rock below the drilled shaft to crush or break under the future foundation load or clayfilled seams that would allow the foundation to subside as the clay consolidates or extrudes outward under the concentrated foundation load. Inspection of the bearing rock below the bottom of a drilled shaft is performed both by inspecting the bearing surface and by drilling one or more probe holes to a depth of at least two shaft diameters below the bearing surface (Fig. 11.2). The walls of the probe holes can be probed to find any small open seams or cavities that would compress or allow the rock to fracture under load transferred by the shaft. This is done with a steel tube or rod fitted with a small, horizontal wedge-shaped tip that is pressed against the hole wall as it is lowered into, or pulled out of, the probe hole. The tip can find open seams less than 1 mm thick and clay seams as thin as 1 to 2 mm.

Table 11.2 Boring and sampling techniques for foundation exploration in karstic formations (after Sowers, 1996). • Percussion drilled holes to identify soil-rock interface. The observed drilling rate is an indicator of rock hardness and rock discontinuities such as fissures and voids. This requires recording the rate of penetration for short intervals of drilling: minutes per foot or per meter, as well as visual examination of the drill cuttings. However, it is difficult to differentiate between a large boulder, a pinnacle, and the upper surface of continuous rock. • Test borings with intact split-spoon samples and Standard Penetration Tests (SPT) in soillike materials, particularly in the soft zone immediately above rock and in the soil in cavities within the rock after drilling into the rock. The boreholes are made by augers or rotary cutters using air or drilling fluid to remove the cuttings. Any loss of drilling fluid is measured as an indication of the size and continuity of rock fissures and cavities. Laboratory tests of the samples provide data for accurate classification of the soils and for estimating some of the engineering characteristics such as hydraulic conductivity and response to loading. • Undisturbed sampling of the stiff overburden at representative intervals and, if possible, of the soft soil overlying the rock and filling slots and cavities. The size of the sample tube is determined by the boring diameter. Laboratory tests of the samples provide quantitative data for engineering analyses of the soil hydraulic conductivity and response to loading. • Cone penetration tests in soil, particularly in the soft soil zone and in soil in the rock (after core drilling or percussion drilling exposes the soil seams). The usual cone point and sleeve resistance can be supplemented by pore water pressure sensors using a piezocone. • Core boring preferably with triple tube diamond bits in rock. When the rock is so weak or closely fractured that the core recovery is less than approximately 90%, larger diameter cores, 4 to 6 in (100 to 150 mm), are preferred. • Oriented core drilling to determine the dip and dip azimuth of the strata and of fissures. • Large diameter drill holes that permit human access to examine the exposed materials directly, particularly at the soil-rock interface. The minimum diameter is approximately 30 in (760 mm); holes of 36 in (900 mm) or larger are preferred. Direct access requires casing in the hole for

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safety, which means alternating drilling, setting casing, and making the observations. This is often impractical below the groundwater level. • Test pits or test trenches to check the rock surface, exposing both slots and pinnacles. This requires either flat slopes or bracing to prevent cave-ins and to provide safety. It is often impractical below the groundwater level despite heroic pumping to dewater the bottom. Moreover, pumping could trigger sinkhole activity. Pits and trenches make it possible to view the stratification, the orientation of fissures, as well as the geometry of the soil-rock interface in three dimensions. • Borehole photography or video imaging of the borehole walls.

Fig. 11.2 Finding defects in the rock below the bottom of a drilled shaft with the aid of small diameter probe holes and a hand probe: (a) Probe holes drilled in the rock below the bottom of a drilled shaft; and (b) Probe rod for finding rock defects on the walls of probe holes.

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Fig. 11.3 Drilled shaft bearing over vertical soil filled seam (after Brown, 1990). If a vertical discontinuity or seam exists as shown in Figure 11.3, the rock that is present over the base of the shaft need be probed. If sound rock exists except for a small area (rock coverage over 75% of the shaft base), the hole may be accepted as adequate if the resulting bearing pressures are not excessive (Brown, 1990). If possible, the soil in the seam can be excavated and the seam backfilled with concrete so that higher end bearing capacity can be achieved. For deep shafts drilled below the groundwater table, this practice is not recommended because of the possibility of large and uncontrolled seepage into the shaft through such an excavated seam. If the soil filled seam is located at one side of the drilled shaft and the rock below the bottom of the shaft contains non-vertical discontinuities as shown in Figure 11.4, rock anchors may be used to provide continuity and load transfer across the discontinuities (Brown, 1990; Goodman, 1993; Sowers, 1996). If the rock is of insufficient quality to provide the required end bearing capacity, the shaft can be extended deeper so that the foundation load is partially transferred to the rock by side shear. Figure 11.5 shows that extension of the shaft reduces the vertical stress in the underlying rock by transferring much of the load from the shaft to the rock by side shear (Sowers, 1996). If a deep, near-vertical soil filled seam is directly under the bottom of the shaft, it is also possible to increase the load capacity of the shaft by extending the shaft deeper, which is true especially when the rock on both sides of the seam is intact and sound (Fig. 11.6).

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If there exist caves below the shaft, different remedial measures can be taken based on the size and location of the caves and the condition of the rock. If the roof of the cave is thin, the drilled shaft can be extended through the roof and cave and into the sound rock in the cave floor (Fig. 11.7). Casing is required for the shaft through the cave. If the roof of the cave is thick and the rock is sound, the drilled shaft can be used as a side shear only shaft [Fig. 11.8(a)]. In some cases, drilling may break through into the cave at the depth where drilling would normally terminate [Fig. 11.8(b)]. To use the hole for a shaft without extending it through the cave and into the rock in the cave floor, a wood plug can be inserted into the bottom of the hole before casting concrete. Obviously, the shaft with the wood plug will only provide side shear resistance. If the cave below the shaft is small, drilling can be extended through into the cave so that the cave can be filled with concrete. This will minimize subsidence and prevent catastrophic collapse.

Fig. 11.4 Rock anchors for drilled shaft bearing over rock containing discontinuities (after Brown, 1990).

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Fig. 11.5 Extending drilled shaft to reduce the vertical stress in the underlying rock by transferring load from shaft to rock by side shear: (a) Underlying rock cannot provide required end bearing capacity; and (b) Shaft extension and assumed average vertical rock stress distribution (modified from Sowers, 1996).

Fig. 11.6 Remedial measures for drilled shaft with a deep, near-vertical soil filled seam directly under its bottom: (a) Drilled shaft with a deep, near-vertical soil filled seam directly

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under its bottom; and (b) Shaft extension and added, deeper probe holes (modified from Sowers, 1996).

Fig. 11.7 Extending drilled shaft through the roof and cave to the sound rock in the cave floor (modified from Sowers, 1996).

Fig. 11.8 (a) Side shear only shaft with a cave below it; and (b) Using wood plug at the bottom of the hole in the case that drilling breaks through into the cave.

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11.5 AN EXAMPLE OF DRILLED SHAFT FOUNDATIONS IN KARSTIC FORMATIONS This example is from Erwin and Brown (1988). It shows the problems and the corresponding solutions to them for drilled shaft foundations in karstic formations. Since 1962, active limestone sinkholes had been causing problems and affecting the operation of the access railroad to the Military Ocean Terminals, Sunny Point, North Carolina (MOTSU). Many springs and sinkholes developed in areas close to the MOTSU access railroad (Fig. 11.9). Two sinkholes that developed in October 1984 following extensive rainfall caused all traffic to be stopped. To reactivate railroad traffic, a 450 ft (137.2 m) bypass alignment was selected based on a ground penetration radar (GPR) survey. An 8,500 lb (3,855 kg) concrete weight was dropped from a 30 ft (9.14 m) height every 15 ft (4.57 m) along the alignment. No sinkholes were activated by dropping the weight. Two potential sinkholes identified by the GPR survey were grouted. The two sinkholes were encircled with 6,500 ft3 (184.0 m3) of sanded grout, excavated 8 to 10 ft (2.44 to 3.05 m) deep, compacted by dropping concrete weight, and backfilled with compacted sand. The stratigraphy along the MOTSU access railroad consists of the following materials in descending order: 1. Sand and silt of Pleistocene age that are loose and noncemented: 10 to 40 ft (3.05 to 12.2 m) thick. 2. Silt, clayey sand, shell hash, and shell layers of the early Pleistocene/Pliocene age Waccamaw formation up to 30 ft (9.14 m) thick. 3. Limestone of the Castle Hayne formation of upper to middle Eocene age: 5 to 33 ft (1.52 to 10.1 m) thick. 4. Limestone, sandstone, sand, and silt of the Cretaceous age Pee Dee formation: 5 to 10 ft (1.52 to 3.05 m) thick. Extensive subsurface investigations indicated that there is a set of southeast-northwest trending joints that appear to connect to a lake upstream of the railroad. In January 1985, a 4,055 ft (1.24 km) land bridge was selected as a permanent solution to the sinkhole problems for the MOTSU access railroad. Drilled shafts socketed into rock were selected to support the bridge. For areas of known sinkhole activity, it was assumed that 20 ft (6.10 m) of overburden could be lost in two consecutive bents from sinkhole formation (see Fig. 11.10). For areas of no known foundation problems, it was assumed that no more than 15 ft (4.57 m) of overburden could be lost. In the Allen Creek area, it was assumed that there was no overburden available for lateral support but that at least 10 ft (3.05 m) of good rock existed. Cavity locations had a direct impact on the location of shaft size step-downs, cased or uncased shaft design and final tip elevations. The criteria used to offset the effect of cavities was to allow no shaft tip to be founded with less than 5 ft (1.52 m) of rock

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Fig. 11.9 Plan view of MOTSU Railroad in the vicinity of Boiling Springs Dam (after Erwin & Brown, 1988). above an underlying cavity and no cavity larger than 6 in (0.15 m) in height within the next 5 ft (1.52 m). This was to prevent a punching-type failure from end bearing loads. Cavities that were found required stepping-down to next smaller diameter of shaft and casing 1 ft below the cavity (Fig. 11.10). The significant variability in the top rock elevations also greatly affected shaft lengths. Casings were installed and seated 1 ft (0.30 m) into rock using an oscillator prior to excavation of the overburden. A large diameter borehole drill using a bucket auger was used to clean out the overburden in the casings. A large diameter carbide-tipped spiral auger was used to excavate the rock. Final cleanout of the shaft hole was by airlift. Sinkholes developed in only two areas due to cavities encountered during drilling of the shaft holes. These areas were simply backfilled and compacted and shaft installation continued with no further problems. Three sinkholes were triggered during the drilling of the contract borings. Six to eight sinkholes developed in the work area due to the construction activities. Credit for the small number of sinkholes that developed is probably due to that all holes in the overburden were cased at all times and that casings were installed in the dry without use of drilling fluid or water. Totally 124 drilled shafts were successfully installed at a cost of $2,000,000. The dry technique of casing installation used by the contractor worked extremely well and helped avoid potential installation problems that likely would have resulted if other techniques had been used.

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Fig. 11.10 Typical drilled shaft and design assumptions for MOTSU Railroad (after Erwin & Brown, 1988).

12 Loading test of drilled shafts in rock 12.1 INTRODUCTION Drilled shafts cannot be readily inspected once they are constructed. On the other hand, the performance of drilled shafts is highly dependent on the local geology and on the construction procedure followed by the drilled shaft contractor. Hence it is not easy for engineers to be assured that the constructed shafts comply with the design specifications. Loading tests of drilled shafts are therefore highly desirable when it is feasible to perform them. Loading tests of drilled shafts are conducted for two general purposes: 1. to check the integrity of the test shaft and to prove that it is capable of sustaining the applied loading as a structural unit; 2. to gain detailed information on load bearing and deformation characteristics of the soil/rock and shaft system. In the first instance the drilled shaft is constructed in the same manner as the production shafts. The test shaft should sustain a load that is customarily at least twice the working load without excessive displacement. In the second instance, the test shaft is instrumented and usually loaded to failure by an appropriate definition. The instrumentation allows the measurement of load and displacement along the length of the shaft. Such data allows analyses to be made to obtain information on soil or rock resistance as a function of the shaft displacement as well as the structural performance of the drilled shaft itself. Loading tests of drilled shafts are expensive, and the cost should be carefully weighed against the reduction in risk and assurance of satisfactory behavior that the loading test provides. The extent of the test program depends on the availability of experience in designing and constructing drilled shafts in a particular geological environment and the capital cost of the works. A loading test of drilled shafts is most cost-effective when one or more of the following conditions are present: • Many drilled shafts are to be constructed, so even small savings on each shaft will significantly reduce the overall construction cost. • The soil/rock conditions are erratic or unusual. • The structure is especially important or especially sensitive to displacements. • The engineer has little or no experience in the project area. Nearly all large drilled shaft projects should include at least one full-scale loading test. However, it is not practical to test every shaft, even for the largest and most important projects. Therefore, we can only test representative drilled shafts and extrapolate the results to other shafts at the site. Table 12.1 lists the guidelines suggested by Engel (1988) for determining the required number of pile load tests for typical projects.

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Table 12.1 Guidelines for determining the required number of pile load tests (Engel, 1988). Summation of length of all piles at the site Required number of load tests

(ft)

(m)

0–6,000

0–1,800

0

6,000–10,000

1,800–3,000

1

10,000–20,000

3,000–6,000

2

20,000–30,000

6,000–9,000

3

30,000–40,000

9,000–12,000

4

12.2 AXIAL COMPRESSIVE LOADING TEST 12.2.1 General comments The objectives of an axial compressive loading test are usually 1. to determine the ultimate bearing capacity of the shafts, relating this to the design parameters; 2. to separate the total resistance contributed by the side shear and end bearing capacity; 3. to determine the stiffness of the soil/rock and shaft system at design load. A back analysis of the tests data will enable the soil/rock modulus to be evaluated, and hence the deformation of shaft groups may be predicted with greatly increased confidence. These various objectives will necessitate a carefiilly chosen test procedure and instrumentation program. Considering the purposes of loading tests, the test shafts should be representative of the production shafts. It is, therefore, critical that the test shafts be founded in the same formation(s) as the production shafts and the construction procedures that are expected to be used with the production shafts also be used with the test shafts. Since the capacity of a full-scale drilled shaft in rock is usually very large, engineers may be tempted to determine the values of unit shaft and base resistance from tests on small-diameter shafts to reduce the cost of loading tests. It is found, however, that the unit ultimate resistances developed by a small-scale drilled shaft are much higher than those developed by a fullsized drilled shaft (O’Neill et al., 1996). So the unit shaft and base resistances determined from tests on shafts with diameters much smaller than those of the production shafts can be unconcervative. Recent practice in the United States has been to use test shafts in rock that have approximately the same diameter and depth as and are constructed in a manner similar to the production shafts (O’Neill & Reese, 1999).

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12.2.2 Methods of applying loads (a) Conventional loading test The following conventional methods can be used to apply the compressive load on the test shaft: 1. A platform is constructed on the head of the shaft and a mass of heavy material, termed “kentledge”, is placed on the platform. 2. A bridge, carried on temporary supports, is constructed over the test shaft and loaded with enough dead weight. The ram of a hydraulic jack, placed on the shaft head, bears on a cross-head beneath the bridge beams, so that a total reaction equal to the weight of the bridge and its load may be obtained. 3. Reaction shafts capable of withstanding an upward force are constructed on each side of the test shaft, with a beam tied down to the head of the reaction shafts and spanning the test shaft. A hydraulic jack on the head of the test shaft applies the load and obtains a reaction against the underside of the beam (Fig. 12.1). 4. Ground or rock anchors are constructed to transfer the reaction to stiff strata below the level of the shaft base. For methods 1 and 2, more or less any material available in sufficient quantity can be used as the dead weight. Specially-cast concrete blocks or pigs of cast iron may be hired and transported to the site. The cost of transport to and from the site is a significant factor. Regular-shaped blocks have the advantage that they may be stacked securely, and are unlikely to topple unexpectedly. Sheet steel piling, steel rail, bricks, or tanks full of sand or water have been used as the dead weight from time to time. It is important that the mass of material be stable at all times during and after the test. The reaction beams for methods 2 and 3 are subjected to high bending and buckling stresses and they should be designed to carry the maximum load safely. The maximum safe load should be clearly marked on the beam so that it is not inadvertently exceeded during a test. Stresses are transferred from the supports (method 2), reaction shafts (method 3) or ground anchors (method 4) through the soil/rock mass, and such stresses can influence the behavior of the test shaft. Therefore, the supports, reaction shafts or ground anchors must be located well away from the test shaft to minimize this effect. Whitaker (1975) recommends that the supports be more than 1.25 m (4 ft) away from a test pile, to minimize the effect of the supports on pile settlement. He also recommends that any reaction pile should be at least three test-pile diameters from a test pile, center to center, and in no case less than 1.5 m (5 ft). The specifications of the American Society for Testing and Materials for piling, ASTM D-1243, require that a clear distance of at least five times the maximum of the diameters of the reaction or test piles must exist between a test pile and each reaction pile (ASTM, 1995). According to O’Neill and Reese (1999), a 3.5 diameter center-to-center spacing between each reaction shaft and the test shaft is adequate to minimize the reaction shaft and test shaft interaction for loading tests of large diameter drilled shafts in cohesive soils or rocks.

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Fig. 12.1 Compressive loading test using reaction shafts. Because the upper portion of an anchor cable does not usually transfer load to the soil/rock, ground anchors can be placed closer to the test shaft than can reaction shafts. If reaction shafts are constructed in such way that their uplift capacity is only developed in a stratum far below the base of the test shaft (the bond between the reaction shaft and the soil/rock to a depth well below the base of the test shaft is eliminated), the reaction shafts can be as close to the test shaft as feasible. However, the construction disturbance of the soil/rock around the test shaft still should be avoided. The main disadvantages of ground anchors as a reaction system are the axial flexibility of the anchor tendons and the lack of lateral stability of the system. A multiple ground anchor system is to be preferred and each anchor should be proof tested to 130% of the maximum load before use. To reduce the extension of the cables during the test it is usual to pre-stress the anchors to as high a proportion of the maximum load as possible. Hence a suitable reaction frame and foundation must be provided to carry this load safely. (b) Osterberg cell loading test Osterberg cell, named after its inventor, Jorj Osterberg, was first used in an experimental drilled shaft in 1984 (Osterberg, 1984). Because of its advantages over the conventional test systems, Osterberg cell is now widely used in axial loading test of drilled shafts. Figure 12.2 illustrates schematically the difference between a conventional load test and an Osterberg cell load test. A conventional test loads the drilled shaft in compression at its top using an overhead reaction system or dead weight. Side shear Qs and end bearing Qb combine to resist the top load Q and the engineer can only separate

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Fig. 12.2 Comparison of conventional load test and Osterberg cell load test. these components approximately by analysis of strain or compression measurements together with modulus estimates. An Osterberg cell test also loads the drilled shaft in compression, but from its bottom. As the Osterberg cell expands, the end bearing Qb provides reaction for the side shear Qs, and vice versa, until reaching the capacity of one of the two components or until the Osterberg cell reaches its capacity. Tests using the Osterberg cell automatically separate the end bearing and side shear components. When one of the components reaches ultimate capacity at an Osterberg cell load Qb, the required conventional top load Q to reach both side shear and end bearing capacity would have to exceed 2Qb. Thus, an Osterberg cell test load placed at or near the bottom of a drilled shaft has twice the testing effectiveness of that same load placed at the top. Osterberg cells have very large pistons, which makes it possible to apply very large loads with relatively small jack pressures (see Table 12.2). By placing two or more Osterberg cells on the same plane, the test capacity can be significantly increased. On January 30, 2001, Loadtest, Inc. of the USA conducted a load test of 151 MN (17,000 tons) on an 8 ft (2.44 m) diameter, 135.5 ft (41.3 m) deep drilled shaft in Tucson, Arizona, by utilizing three 34 in (870 mm) diameter Osterberg cells on a single plane located approximately 28.5 ft (8.69 m) above the base of the shaft (LOADTEST, 2001). Figure 12.3 shows the typical arrangement for an Osterberg cell load test of a drilled shaft. The load being applied to the drilled shaft is determined by recording the pressure

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and converting it to force from a pre-determined calibration curve. The upward movement of the bottom and top of the shaft, and the downward movement of the bottom of the shaft are measured by telltales and/or gauges. These measurements can be used to obtain the side resistance versus side movement curve and the base resistance versus base movement curve, as illustrated in Figure 12.4 for a test of a rock-socketed shaft in Apalachicola River, Florida. Osterberg (1998) presented a method for constructing the load-displacement curve equivalent to applying the load at the top of the shaft from the side resistance versus side movement curve and the base resistance versus base movement curve, which will be discussed in detail in Section 12.2.5.

Table 12.2 Size and load capacity of Osterberg cells (LOADTEST, 2001). Nominal Diameter (in)

*

(mm)

Nominal Capacity* (kips)

(MN)

9

230

450

2.00

13

330

870

3.87

21

540

2,000

8.90

26

670

3,640

16.2

34

870

6,150

27.4

The Osterberg cell applies a bi-directional load. The total test capacity is twice the nominal Osterberg cell capacity.

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Fig. 12.3 Typical arrangement of Osterberg cell load test (after Ernst, 1995).

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Fig. 12.4 Osterberg cell test loaddisplacement curves of a rock-socketed shaft in Apalachicola River, Florida (after Osterberg, 1998). A loading test with an Osterberg cell at one location as shown in Figures 12.2 and 12.3 is limited by the magnitude of the side shear or base bearing resistance, whichever is smaller. It is possible, however, to place Osterberg cells at two or more locations within the drilled shaft. The first innovative application of this technique was a test for the Alabama Department of Transportation in 1994 (Goodwin et al., 1994; O’Neill et al., 1997). Figure 12.5 is a schematic of that test. The arrangement of the two 26.7 MN (3,000 ton) Osterberg cells was such that it was possible to measure the base bearing resistance, the side shear resistance in the socket in the chalk formation, the side shear resistance in the cased portion of the shaft above the chalk, and the side shear resistance in the chalk upon reversal of the direction of load.

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The Osterberg cell test method offers a number of potential advantages over the conventional loading test methods (Schmertmann & Hayes, 1997): 1. Economy: The Osterberg cell test is usually less expensive to perform than a conventional static load test despite sacrificing the Osterberg cell. Osterberg cell tests are typically 1/3 to 3/2 the cost of conventional tests, with the comparative cost reducing as the test load increases. 2. High load capacity: especially for rock sockets. 3. Separation of side shear and end bearing components: The Osterberg cell test automatically separates the side shear and end bearing components. It also helps determine if construction techniques have adversely affected each component. 4. Improved safety: The test energy lies deeply buried and there is no overhead load. 5. Reduced work area: The work area required to perform an Osterberg cell test is much smaller that that required by a conventional load system. 6. Over-water or battered shafts: Although often difficult to test conventionally, testing over water or on a batter poses no special problems for the Osterberg cell load test method. 7. Static creep and setup (aging) effects: Because the Osterberg cell test is static and the test load can held for any desired length of time, information about the creep behavior of the side shear and end bearing components can be obtained. The aging effects at any time after installation can also be measured conveniently. The Osterberg cell test method also has some limitations compared to the conventional loading test methods. These include (Schmertmann & Hayes, 1997): 1. Advance installation required: The Osterberg cell must be installed prior to construction of the shaft. 2. Balanced component required: An Osterberg cell test usually reaches the ultimate load in only one of the two resistance components. The test shaft capacity demonstrated by the Osterberg cell test is limited to two times the capacity of the component reaching ultimate. Also, once installed the Osterberg cell capacity cannot be increased if inadequate. To use the Osterberg cell efficiently the engineer should first analyze the expected side shear and end bearing components and either attempt to balance the two to get the most information from both or unbalance them to ensure the preferred component reaches ultimate first. The introduction of multi-level Osterberg cell test as discussed earlier can mitigates this limitation, allowing the engineer to obtain both ultimate end bearing and ultimate side shear values in cases where the end bearing is less than the side shear. 3. Equivalent top load curve: Although the equivalent static top load-displacement curve can be estimated with conservatism, it remains an estimate. 4. Sacrificial Osterberg cell: The Osterberg cell is normally considered expendable and not recovered after the test is completed. However, grouting the cell after completion of the test allows using the tested shaft as a load carrying part of the foundation.

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Fig. 12.5 Schematic of two level Osterberg cell test in Alabama (after O’Neill et al., 1997).

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(c) Statnamic loading test Statnamic load test was developed jointly in Canada and the Netherlands in the early 1990s (Bermingham & Janes, 1989; Middendorp et al., 1992). The principle of the Statmanic load test is shown in Figure 12.6(a). Reaction mass is placed on the top of the shaft. Beneath the reaction mass is a small volume of propellant (fast-expanding solid fuel) and a load cell. By burning the propellant and propelling the reaction mass upward off the shaft at accelerations up to 20 g, a load is generated. Since the mass is in contact with the shaft prior to the test, the force associated with propelling of this mass acts equally and oppositely onto the shaft. The reaction mass, usually rings of concrete or steel, needs to be only 5% of the total load to be applied to the shaft. For reasons of safety, the reaction mass is contained within a metal sheath that is also filled with an energy absorber, such as dry gravel, that will cushion the impact of the mass as it fall back upon the head of the shaft. During a test, a high-speed data acquisition system scans and records the load cell, displacement transducers, accelerometers and embedded strain gauges. The test measurements provide a high degree of resolution fully defining the shaft load and displacement response with up to 100,000 data points recorded during a typical ½ second test. Because the measured Statnamic force includes some dynamic forces, some interpretation of the data is necessary, as illustrated in Figure 12.6(b). Since the duration of the axial Statnamic test is adequately longer than the natural period of the drilled shaft, the entire drilled shaft remains in compression and a simple model can be used to determine the static load acting on the shaft as follows (AFT, 2002): Qstatic=QSTN−Ws(as/g)−cνs (12.1) where Qstatic is the derived static load acting on the shaft; QSTN is the measured Statnamic force; Ws(as/g) is the inertia force; cνs is the damping force; Ws is the weight of the drilled shaft; c is the damping coefficient of the drilled shaft and soil/rock system; and as and νs are respectively the measured acceleration and velocity of the drilled shaft. c is the only unknown in Equation (12.1) and can be determined using the principles of the Unloading Point Method (UPM) (Middendorp et al., 1992; Brown, 1994). Statnamic load test appears to offer a number of advantages over other types of tests, including (AFT, 2002):

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Fig. 12.6 (a) Sketch of Statnamic loading test system; and (b) Statnamic load-displacement curve and interpreted static load-displacement curve. 1. Statnamic load test is quick and easily mobilized.

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2. Shaft performance is measured cost-effectively. 3. Statnamic load test is repeatable and the test shaft re-useable. 4. Statnamic load test does not require any reaction system. 5. Statnamic load test requires no special construction procedures so installation is more representative of the actual production shaft construction. This allows Statnamic load test to be used for random quality control testing and for problematic shafts. 6. The system is flexible and adaptable, e.g., single shafts or shaft groups can be tested for compression loading and also lateral loading characteristics. 7. The test is quasi-static, and does not produce harmful compression and tension stresses, which have the potential of damaging the shaft. Statnamic load test also has limitations including (AFT, 2002): 1. Rate of loading precludes long term displacements. 2. Currently maximum Statnamic test load is 32 MN (3,600 tons). 3. Currently Statnamic load test method cannot be used for uplift tests. 12.2.3 Instrumentation (a) Measurement of load Calibrated pressure gauges are sometimes used in conventional load tests to determine the load applied to the test shaft by measuring the hydraulic pressure in the loading ram. However, calibrated pressure gauges may only provide acceptable accuracy of load measurement for increasing load because the unloading cycle is usually nonlinear owing to friction within the hydraulic jack. Therefore, load measurement is preferably carried out using a load cell. The following lists four commonly adopted loadmeasuring devices (Fleming et al., 1992): 1. hydraulic load cell capsules, maximum capacity 4.0 MN (450 tons) and accuracy about±1% 2. load columns, maximum capacity 8.9 MN (1,000 tons) and accuracy about±1% 3. proving rings, maximum capacity 1.8 MN (200 tons) and accuracy about±0.5% 4. strain gauged load cells of various types, maximum capacity 1.8 MN (200 tons) and accuracy about±0.5% For very large loads, an array of load columns or strain gauged load cells may be used. It is important that the load cell is calibrated regularly and that, for those devices sensitive to temperature, suitable corrections are made. When tests are run to obtain information on load transfer in side shear and end bearing resistance, several methods can be used to measure the distribution of load along the length of the drilled shaft, some of which being briefly described below (Fleming et al., 1992; O’Neill & Reese, 1999): Sister bars. The load distribution along the length of a drilled shaft can be obtained using sister bars. A sister bar is a section of reinforcing steel at the middle of which is placed a strain transducer. The sister bar is tied to the rebar cage and its leadwires routed to the surface. The strain transducer can be a vibrating wire gauge or an electrical foil resistance

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strain gauge. The vibrating wire transducer has the advantage that it tends to be stable over longer periods of time than an electrical resistance transducer because the later is quite sensitive to the invasion of moisture. However, electrical resistance transducers are more adaptable to data acquisition systems than vibrating wire transducers. Sister bars of both types are currently the most popular instruments for measuring load distribution along the length of drilled shafts. The electrical output can be converted to strain in the steel rebar through an appropriate calibration factor, which can then be assumed to be equal to the strain in the concrete section. The internal load in the shaft can then be obtained by multiplying the axial stiffness of the test shaft by the strain obtained at the depth of interest. It is important to place sister bars at opposite ends of diagonals at any level so that the averaged readings cancel any bending effects that may occur. It is recommended that two or four gauges be placed at each level at which load is to be measured. Mustran cells. The Mustran cell is mounted on the rebar cage before inserting it into the borehole, in a manner similar to that for sister bars. Because of its electrical circuitry, the Mustran cell indicates strains that are larger than, but proportional to, the actual strains in the concrete. This feature is advantageous for testing large-diameter drilled shafts subjected to relatively small loads. Data from Mustran cells are collected and interpreted in a manner very similar to that for sister bars. As with sister bars, it is good practice to place Mustran cells at opposite ends of diagonals at any level so that the averaged readings cancel any bending effects that may occur. Telltales. Telltales are unstrained metal rods that are inserted into one or more tubes that prevent them from bonding to the concrete in the shaft. The telltales extend to a series of depths along the length of the shaft. The shortening of the test shaft over a particular length can be found by using displacement transducers to measure the difference in the movement of the shaft head and the top of the unstrained rod. Such measurements must be made for each of the telltales and for each of the applied loads. Again, to cancel any unintended bending effects, it is important to install telltales in pairs, at opposite ends of diagonals at each depth. For a particular applied load, the average deformation measured at each depth can be used to plot a compression of the shaft versus depth curve. Differentiation of this deformation versus depth curve with respect to depth will yield the strain of the shaft as a function of depth. The internal load in the shaft can then be obtained by multiplying the axial stiffness of the test shaft by the strain obtained at the depth of interest. Pressure cells. Pressure cells placed at the bottom of the cage can be used to measure the base resistance directly. Pressure cells are accurate because they do not require the assumption of a value for the Young’s modulus of the concrete. Since stresses across drilled shaft bases are generally uniform, the base resistance can be simply obtained by multiplying the average pressure from the cells with the contact area of the shaft base. (b) Measurement of displacement The most commonly used systems to measure the head displacement of the test shaft are dial gauges or electronic displacement transducers such as LVDT’s (linear variable differential transformers) that are held by stable reference beams. Dial gauges are simple, robust, mechanical precision instruments, and as such they should be carefully stored and

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maintained. The reference beams should be long enough so that they can be supported by firm foundations well away from the test shaft and reaction shafts. If possible, adjacent drilled shafts at least four shaft diameters from the test shaft and reaction shafts can be used as the support foundations. It is noted that a compromise must frequently made between a long beam prone to vibration and temperature-induced displacements, and a shorter beam with support foundations in the zone of influence of the test shaft or reaction shafts. Optical leveling may also be used to measure the head displacement of the test shaft. The accuracy of an optical leveling system may be poorer than that of dial gauges by a factor of at least 10. However the absolute accuracy of the system may be of a similar order to that of dial gauges, particularly in situations where it is difficult to establish a stable reference beam. With an optical system it is easy to arrange for the instrument and reference point to be well away the zone of disturbance (Fleming et al., 1992). When tests are run to obtain information on load transfer in side shear and end bearing resistance, telltales as described earlier can be used to measure the displacement of the test shaft from point to point along its length. 12.2.4 Test procedures The procedures for conducting conventional loading test are given in the ASTM Test Designation D-1243 (ASTM, 1995). The U.S. Corps of Engineers Manual on the “Design of Pile Foundations” recommends the following three methods of loading (ASCE, 1993): 1. Slow, maintained load test method. This is the most common test procedure and is referred to as “standard loading procedure” in the ASTM Test Designation D-1243. In this method, the shaft is loaded in eight equal increments up to a maximum load, usually twice a predetermined allowable load. Each of the eight load increments is placed on the shaft very rapidly (as fast as the pump can raise the load, which usually takes about 20 seconds to 2 minutes) and maintained until zero movement is reached, defined as 0.25 mm/h (0.01 in/h). The final load, the 200 percent load, is maintained for a duration of 24 hours. This procedure is very time consuming, requiring from 30 to 70 or even more hours to complete. It should be noted that the phrase “zero movement” is very misleading: the “zero” movement rate is equal to a movement of more than 2 m (7 ft) per year. 2. Constant rate of penetration (CRP) test method. In a CRP test, the load is applied to cause the shaft head to settle at a predetermined constant rate, usually in the vicinity of 0.25 mm (0.01 in) per minute to 2.5 mm (0.1 in) per minute, depending on the soil/rock type. The duration of the test is usually 1 to 4 hours, depending on the variation used. The particular advantage of the CRP test is that it can be conducted in less than one working day. A disadvantage is that ordinary pumps with pressure holding devices like those used for “slow” tests are difficult to use for the CRP test. A more suitable pump is one that can provide a constant, nonpulsing flow of oil. 3. Quick maintained load test method. In this test, the load is applied in increments of about 10 percent of the proposed design load and maintained for a constant time interval, usually about 2 to 15 minutes. The duration of this test will generally be about 45 minutes to 2 hours, again depending on the variation selected. The advantage

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of this test, like the CRP test, is that it can be completed in less than one working day. Unlike the CRP test, however, no special loading equipment is required. For particular projects special loading paths may be called for, to simulate repeated loading for example. Such loading paths my easily be arranged but careful supervision is necessary. Short-period, cyclic or sinusoidal loading requires the use of sophisticated servo-controlled equipment. The loading procedure for the conventional load test method can be easily applied to the Osterberg cell load test method. It is essential to record all the relevant data throughout the test, including the load, displacement, time, problems, and unexpected occurrences. 12.2.5 Interpretation of test data A considerable amount of data may be obtained from an axial loading test, and with more sophisticated instrumentation a greater understanding of the soil/rock and shaft interaction may be achieved. Interpretation of the axial loading test data may be carried out on several levels as described below. (a) Load-displacement curve of shaft head From a conventional load test, the load-displacement curve of the shaft head can be directly obtained. For an Osterberg cell load test, the side resistance versus side movement curve and the base resistance versus base movement curve can be used to construct the load-displacement curve equivalent to applying the load at the top of the shaft (Osterberg, 1998). This is done by determining the side resistance at an arbitrary displacement point on the side resistance versus side movement curve. If the shaft is assumed rigid, the top and bottom of the shaft move the same amount and have the same displacement but different loads. By adding the side resistance and the base resistance at the same displacement, a single point on the top equivalent loaddisplacement curve is obtained. By repeating this process for different displacement points, the top equivalent load-displacement curve can be obtained (see Fig. 12.7). It is very often that the side shear or the end bearing reaches ultimate before the other (In Figure 12.7, the side shear reaches ultimate before the end bearing). In this case, two procedures are possible. The first, which is extremely conservative, is to assume that the component that has not reached ultimate has also reached ultimate and that no further load increase occurs as displacement increases. The other more likely procedure, as shown in Figure 12.7, is to extrapolate the curve that has not reached ultimate. The procedure for constructing the top equivalent load-displacement curve is based on the following assumptions (Osterberg, 1998): 1. The side shear-displacement curve for upward movement of the shaft is the same as the downward side shear-displacement component of a conventional load test. 2. The base resistance-displacement curve obtained from an Osterberg cell load test is the same as the base resistance-displacement component of a conventional load test. 3. The shaft is considered rigid. Typically, the compression of the shaft at ultimate loadis 1–3 mm.

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For a Statnamic test, the Statnamic curve can be corrected to obtain the equivalent static load-displacement curve as described in Section 12.2.2. (b) Estimation of ultimate load The top load-displacement curve is commonly used to determine the ultimate load capacity of the test shaft. A number of criteria for defining the ultimate load capacity have been developed and discussed by different researchers, some of which are listed below (Tomlinson, 1977; Fellenius, 1980; ASCE, 1993): 1. The load at which displacement continues to increase without further increase in load. 2. The load causing a certain amount of total displacement, such as 1 in (25 mm), 10% of the base diameter of the shaft, elastic displacement plus 1/30 of the shaft diameter, or elastic compression of the shaft plus 0.15 in (4 mm) plus 1/120 of the shaft diameter. 3. The load causing a certain amount of plastic displacement, such as 0.25 in (6 mm). 4. The load at a defined plastic to elastic displacement ratio, such as 1.5. 5. The load at a defined slope of the load-displacement curve, such as 0.01 in (0.25 mm) per 1 ton (10 kN). 6. Load-displacement curve interpretation a. Maximum curvature—plot log total displacement versus log load; the ultimate load capacity is at the point of maximum curvature. b. Tangents—plot tangent lines to the initial and failure portions of the loaddisplacement curve; the ultimate load capacity is at the intersection of the two tangent lines. c. Inverse slope—divide each load value by its corresponding displacement values and plot the resulting value against the displacement; the plotted values fall on a straight line and the inverse slope of the line is the ultimate load capacity. Different criteria may result in very different values of the ultimate load capacity. It is important to choose the appropriate criterion/a and check the obtained value(s) of the ultimate load capacity. The method often used by the U.S. Corps of Engineers and presented below is a good example on how to determine the ultimate load capacity: The following method has often been used by the U.S. Corps of Engineers and has merit: determine the load that causes a plastic displacement of 0.25 in (6 mm); determine the load that corresponds to the point at which the load-displacement curve has a significant change in slope; and determine the load that corresponds to the point on the load-displacement curve that has a slope of 0.01 in per ton (0.25 mm per 10 kN). The average of the three loads determined in this manner would be considered the ultimate axial capacity of the pile. If one of these three procedures yields a value that differs significantly from the other two, judgment should be used before including or excluding this value from the average. A suitable factor of safety should be applied to the resulting axial pile capacity.

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Fig. 12.7 Construction of equivalent top load-displacement curve (b) from Osterberg cell load test curves (a) (after Osterberg, 1998).

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(c) Distribution of load along depth With the instruments described in Section 12.2.3, the distribution of load along the length of the shaft can be obtained as shown in Figure 12.8. The load distribution curves clearly show the contribution of the side shear and the end bearing. The shape of the load distribution curves also reflects the distribution of the side shear resistance along the length of the shaft. (d) Load transfer (t-z or τ-w and q-w) curves It is also possible to obtain the load transfer (t-z or τ-w and q-w) curves for instrumented shafts. For example, to obtain the τ-w curve at depth zi, the following procedure can be followed (see Fig. 12.9): 1. For a test load, determine the movement of the shaft relative to the soil or rock at depth zi: The relative movement at depth zi is obtained by subtracting the shaft shortening over the distance zi from the measured displacement at the top of the shaft. The shaft shortening over the distance zi is simply obtained by dividing the cross-hatched area [see Fig. 12.9(a)] by the shaft axial stiffness which is the product of the cross-sectional area and the composite Young’s modulus of the steel and concrete. 2. For a test load, determine the side shear resistance at depth zi: The unit side shear resistance at depth zi is obtained by dividing the slope of the load distribution curve at depth zi [see Fig. 12.9(a)] by the perimeter length of the shaft. 3. Plot the movement of the shaft and the side shear resistance determined in Steps 1 and 2 in the τ versus w plot [see Fig. 12.9(b)]. 4. Repeat Steps 1 to 3 for all other test loads and draw the τ-w curve at depth zi by connecting the points obtained. At other depths, the τ-w curves can be obtained by simply repeating the above steps. For the q-w curve, the base resistance at a test load can be determined by simply taking the base load from the load distribution curve. The base movement of the shaft at a test load can be determined following the method in Step 1 above.

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Fig. 12.8 Typical load distribution curves.

Fig. 12.9 Construction of load transfer curves.

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12.3 AXIAL UPLIFT LOADING TEST Axial uplift loading tests are similar to axial compressive loading tests, and maintained load or constant rate of uplift procedures may be used. Figure 12.10 shows a loading system for uplift loading tests. To avoid bending the test shaft, two jacking points on either side of the test shaft are used and care is required to load the jacks evenly.

Fig. 12.10 Uplift loading test using ground as reaction. Uplift test of shafts may be carried out using a constant rate of uplift procedure which is similar to the constant rate of penetration (CRP) test. However, since the full uplift capacity of a drilled shaft is normally mobilized at a displacement smaller than that required for the full compressive capacity, the imposed rate of uplift should be reduced, say to 0.1 to 0.3 mm per minute. Interpretation of the uplift load test results can follow the same procedures as for the axial compressive load tests. Similarly, load-displacement curves, ultimate uplift load capacity, load distribution along the depth, and load transfer curves (only t-z or τ-w curves for uplift loading tests) can be obtained.

12.4 LATERAL LOADING TEST 12.4.1 General comments Drilled shafts are frequently used to carry lateral loads because of their high moment of inertia. The main purpose of lateral load tests is to verify the behavior of production shafts or to provide data for constructing the load transfer (p-y) curves. The basis for

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conducting a lateral load test can be the ASTM Standard D 3966 (1995) or Eurocode 7 (1994) modified to satisfy the specific project requirements. 12.4.2 Methods of applying loads (a) Conventional loading test A lateral load test is most easily conducted by pulling a pair of shafts together or jacking them apart (Fig. 12.11). In this manner, two lateral load tests can be conducted simultaneously. It is important to note that, however, spacing between the two shafts should be such that the shaft-to-shaft interaction is minimized. Because of the difficulty of applying load exactly at the ground line, load is normally applied at a point above the ground line, resulting in both shear and moment at the ground line. The shaft head is generally unrestrained, giving a free-head shaft boundary condition and making the data easy to analyze. The load is applied with a hydraulic jack and measured by a load cell adjacent to the jack in a manner similar to that for axial loading tests. A spherical bearing head should be used to minimize eccentric loading.

Fig. 12.11 Lateral loading test of a pair of shafts. (b) Osterberg cell loading test Osterberg cells have been used to test rock sockets to duplicate the behavior of laterally loaded drilled shafts in rock (O’Neill et al., 1997). This is done by inserting an Osterberg cell vertically into the socket, casting concrete around the cell and using the cell to jack the two halves of the socket apart (Fig. 12.12). The lateral load applied to the rock per unit socket length is computed by dividing the load in the cell by the length of the socket. Lateral displacement is measured by using LVDT’s that connect between the plates. It

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need be noted that, however, the stress and strain conditions in the rock for the lateral Osterberg cell test, in which the halves of the socket are jacked apart, are different from those for a drilled shaft that translates laterally within the rock without splitting. (c) Statnamic loading test Statnamic devices can also be used to test drilled shafts laterally by assembling them horizontally on a special “sled” (Fig. 12.13). Lateral Statnamic loading test has the advantages of closely modeling lateral loading events, such as wind, seismic load, ship impact and wave action, and of no requirement for the construction of a costly reaction shaft. As for the axial Statnamic loading test, however, analyses need be conducted to consider the effects of shaft and soil/rock inertia and loading rate. The load measurement devices for axial loading tests can be used for the measurement of load in lateral loading tests. If the test is to derive the p-y curves, strain gauges may be installed along the shaft to measure the bending moment. Differentiations of the moment distribution along the depth can yield soil/rock resistance along the depth (see Section 12.4.5 for details).

Fig. 12.12 Lateral Osterberg cell test of rock socket (Modified from O’Neill et al., 1997).

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Fig. 12.13 Lateral Statnamic loading test of drilled shaft (from AFT, 2002). 12.4.3 Instrumentation (a) Measurement of load (b) Measurement of displacement For lateral loading tests, the lateral ground line deflection and the shaft head rotation should, as a minimum, be measured. The ground line deflection can be measured by means of displacement instruments suspended from reference beams, similar to the way settlement or uplift is measured in axial loading tests. Rotation can be measured by measuring the lateral deflections at two points above the shaft head (Fig. 12.14) or the vertical displacements at two points on the shaft head (Fig. 12.15). The shaft head rotation is simply the difference in lateral deflections or vertical displacements at these two points divided by the distance between them. A tiltmeter (Fig. 12.16) or an inclinometer (Fig. 12.17) can also be used for measuring the shaft head rotation. The inclinometer can measure the lateral deflection and rotation along the length of the shaft. Strain gauges installed along the shaft can be used to measure the bending moment distribution along the shaft. Integrations of the moment distribution along the depth can yield lateral deflection along the depth (see Section 12.4.5 for details). 12.4.4 Test procedures

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The procedures for conducting lateral loading tests are given in the ASTM Standard D 3966 (1995) and Eurocode 7 (1994). Like the axial loading tests, two principles should guide the testing procedure: (1) The loading (static, repeated with or without load reversal, sustained, or dynamic) should be consistent with that expected for the production shafts; and (2) The testing arrangement should allow deflection, rotation, bending moment, and shear at the ground line or at the point of load application to be measured or computed. 12.4.5 Interpretation of test data A lateral loading test may produce a considerable amount of data corresponding to the instruments used. Interpretation of the lateral loading test data may be carried out on several levels as described below. (a) Load-deflection curve and load-rotation curve at ground line From a conventional lateral loading test, the load-deflection curve and the load-rotation curve at the ground line can be directly obtained. For a lateral Statnamic loading test, the Statnamic curves can be corrected to obtain the equivalent static load-deflection curve and the load-rotation curve at the ground line. (b) Estimation of lateral load capacity The lateral load-deflection curve can be used to estimate the lateral load capacity. For example, the ultimate load can be taken as the load required to produce a specified deflection at the ground line (e.g., 0.25 in). (c) Distribution of deflection and bending moment along depth Distribution of deflection along the depth can be directly obtained by means of electronic inclinometers running up and down tracks in inclinometer tubing that has been cast in the drilled shaft. The distribution of bending moment along the depth can be obtained from the measurements of strain gauges installed along the shafts. It is important to consider the changes of cross-section and modulus of the shaft along the depth when determining the bending moments from the measurements of strains. Figure 12.18 shows the distribution of bending moment from the lateral loading test of a drilled shaft. There are several ways to determine the load transfer (p-y) curves from the test data, depending the instruments used in the test.

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Fig. 12.14 Measurement of shaft head rotation by measuring lateral deflections at two points above the shaft head.

Fig. 12.15 Measurement of shaft head rotation by measuring vertical displacements at two points on the shaft head.

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(d) Load transfer (p-y) curves

Fig. 12.16 Measurement of shaft head rotation using tiltmeter.

Fig. 12.17 Measurement of shaft deflection and rotation using inclinometer.

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Fig. 12.18 Distribution of bending moment along depth from lateral loading test of a drilled shaft. Using the load-deflection curve and the load-rotation curve at the ground line, computer programs can be executed by varying the form of the p-y curves along the shaft until a match is achieved in both deflection and rotation under all of the loads applied. More accurate definition of p-y curves can be achieved if the distribution of deflection along the depth is available. The bending moment curves can be used to obtain the p-y curves by integrating and differentiating the curves along the length of the shaft. The deflection can be obtained with considerable accuracy by two integrations of the bending moment curves: (12.2) where y is the lateral deflection; Mis the bending moment; EpIp is the flexural rigidity of the shaft; and z is the depth from the ground line. To determine the deflection accurately, the deflection and slope at the ground line have to be measured accurately and it is

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helpful if the shaft is long enough so that there are at least two zero-deflection points along the shaft. The resistance of the soil/rock can be obtained by two differentiations of the bending moment curves: (12.3)

where p is the resistance of the soil/rock. Differentiations can be performed numerically along the shaft by using the data without adjustment (Matlock, 1970) or by fitting analytical curves through the points of measured bending moment and then performing differentiations mathematically. It need be noted that slight errors in the measured values of the bending moment will cause large and unacceptable errors in the values of P.

12.5 INTEGRITY TEST OF DRILLED SHAFTS The presence of any defects in a drilled shaft, such as cracks, waists or voids, may cause a serious decrease of load capacity and an increase of displacement. It is, therefore, important to conduct integrity test to detect the defects in drilled shafts. Many forms of integrity test have been developed to detect flaws in deep foundations (Fleming et al., 1992), two of which are frequently used for drilled shafts: the low strain integrity test using the Pile Integrity Tester (PIT) and the cross-hole sonic logging test. The low strain integrity test utilizes one-dimensional wave propagation. A small hand held hammer impacts the top of the shaft, and an accelerometer attached to the top of the shaft measures the impact and resulting reflections (Fig. 12.19). The measurement is then analyzed for relevant reflections from the shaft toe or major shaft anomalies. A record that shows a clear reflection from the shaft toe and no major reflections from intermediate points indicates a sound shaft. Generally, shafts that contain anomalies show a significant wave reflection from a shorter length and no toe reflection. The low strain integrity test using the PIT can be applied to practically every shaft on site due to its low cost and minimal shaft preparation. It is often the first alternative when questions of shaft acceptability arise after the installation is completed. The low

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Fig. 12.19 Low strain integrity test using Pile Integrity Tester (PIT).

Fig. 12.20 Cross-hole sonic logging test.

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strain integrity test using the PIT is useful for selecting shafts for further testing. If it determines obviously good or obviously bad shafts, the solution is clear. For tests indicating marginal conditions further testing of another type may be desired. The low strain integrity test using the PIT is described in detail in ASTM standard D5882. The cross-hole sonic logging test measures sound velocity along the shaft between an emitting sensor and a receiving sensor lowered down two tubes (Fig. 12.20). If the measured velocity decreases rapidly from the sound velocity of homogeneous concrete (about 4000 m/s), defects such as soil inclusion, cracks or segregation may exist. The cross-hole sonic logging test requires that tubes be installed in the shaft prior to concreting. Table 12.3 lists the advantages and disadvantages of the two integrity test methods.

Table 12.3 Advantages and disadvantages of low strain integrity test using PIT and cross-hole sonic logging test. Test Method

Advantage

Disadvantage

Low strain integrity test using PIT

• No special preparation needed

• Test interpretation limited if toe cannot be seen due to excess length or multiple section changes

• Quick, simple and inexpensive • Yields information on major variations of quality or size

Cross-hole sonic logging test

• Works on drilled shafts of unlimited size or length

• Inspection tubes installed during shaft construction

• Clear identification of defects even at great depth

• Tube debonding sometimes prevents wave transmission • Need to wait for concrete hardening

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Index acoustic televiewer 162 acoustic wave 165 adhesion factor 192, 196, 202 agglomerate 124 air 165, 166 allowable bearing pressure 204, 206, 208, 210 concrete stress 189, 190 design load 6 design stress 190 deformation 8 differential deformation 8 end bearing resistance 206 stress design 6 working load 6 alpite 143 amphibolite 13, 16, 17, 124, 130 andesite 11, 17, 124, 130, 131 angle basic friction 77, 78, 82 dilation 142, 143, 245 dip 36 internal friction 77, 132 plunge 54–56 residual friction 77 roughness 77 trend 54–56 anhydrite 124, 131, 315 anisotropy 71, 93, 132, 150, 298 anorthosite 16 aperture of discontinuity 46 aplite 84 apparent cohesion 77 area of discontinuity 38 argillite 17, 199 asperity 82, 84, 195, 196 augers 1 augite 10, 11 axial deformation influence factor 235, 238 axial displacement 227–250 axial load capacity 189–225 single shafts 189–221 shaft groups 221–225

Index

axial loading test compressive 328–344 uplift 344, 345 Barton model 78, 142 basalt 11, 13, 15, 17, 18, 77, 124, 131, 144, 166 basic friction angle 77, 78, 82 beam 264, 287 bearing capacity 204–220 bedding plane 34 bending moment 251, 264, 275, 282, 284, 347–352 bias in sampling frequency 154 orientation 54, 156 spacing 154 trace length 58–63, 155 bilinear shear strength model 77 biotite 10, 11 body wave 165 bored piles 1 borehole camera 162 borehole core logging 158 sampling 157 borehole dilatometer test 180–183 borehole jack test 182, 183 borehole periscope 162 boring 157 boundary condition 261, 280, 284, 305 boundary element method 270 breccia 12, 124, 218, 233 bridge 3, 307 bridge foundation 3, 307 buckling 187, 329 building codes 191, 210 caissons 1 calcite 10, 12 capacity axial 189–225 lateral 251–262 carbonatite 17 casing 321, 323 cast-in-place pile 1 cave 321, 323 cavity 6, 7, 315, 316, 318, 324 chalk 13, 16, 124 charnockite 131 chert 13, 128 Christensen-Huegel method 159 circular disks 42

424

Index

circular failure 307, 308, 314 classification engineering 14, 15 geological 10–13 intact rock 9–14 rock mass 14–31 weathering 13 clay 131, 166, 175, 248 clay core barrel method 159 clay seam 5 clayshale 217, 218 claystone 12, 13, 17, 124, 131, 146 cleavage 35 closed circuit television (CCTV) 162 coal 12, 124, 146 cohesion 77, 127, 132, 141, 179, 245 compressive wave 18, 165 computer programs COMP624P 264 LPILE 264 UTESAS3 314 XSTABL 314 concrete strength 190, 251–254 conglomerate 12, 17, 124 constant rate of penetration 340, 345 constant rate of uplift 345 continuum approach 234, 269 Iinear 234, 270 nonlinear 245, 278 core boring 157 logging 158 orientation 159 Coulomb model 77, 141 Craelius method 159 cross hole method 171 cyclic loading 340 dacite 11, 124 defects 318, 352 deflection 263, 270 deflection influence factor 270, 271 deflection ratio 300, 302 deformability of discontinuities normal stiffness 73–76 shear stiffness 73–76 deformability of rock mass empirical methods 87–93 equivalent continuum approach 93–116 scale effect 149 deformation

425

Index

426

allowable 8 allowable differential 8 deformation modulus 86–116, 149 density 14, 16, 165 design load 6 design stress 190 diabase 13, 15–18, 130, 217, 233 diamond drilling 157 dielectric property 175 dilatancy 245, 258 dilation 196, 204, 245 dilation law 245 dioritell, 16, 17, 124, 131 dip direction 36 direct shear test 176, 178 direction cosines 54 discontinuity aperture 46, 47 apparent cohesion 77 basic friction angle 77 circular disks 42 cohesion 77 deformability 73–76 ellipse 43 filling 47 frequency 33, 36, 57 internal friction angle 77 orientation 20, 33, 36, 48–56, 60, 64, 67, 68, 159–161 persistence 37–42 persistence ratio 38–42 roughness 44–46 roughness coefficient 78–85 roughness profile 79–82 sampling 153–164 set 54 shape 42–44 size 37, 63–68 spacing 20, 36, 57 stiffness 73–75 strength 73, 76–85 trace length 58–63 wall compressive strength 78, 82–84 discontinuum method 261, 262 discrete element method (DEM) 262, 298 displacement axial 227–250 lateral 263–305 normal 73–76, 142, 245 shear 245 dolerite 11, 124, 128, 130 dolomite 10, 12, 15, 16, 130, 131, 315 dolostone 17, 166

Index

drilled caissons 1 drilled piers 1 drilled shafts 1, 189, 227, 251, 263, 307, 315, 327 drilling clay core barrel 159 diamond 157 directional 157 integral sampling 159 large diameter 157 dunite 11, 16 durability 6, 8 eclogite 16 effective roughness angle 77 elastic continuum 234, 269 elastic modulus 14, 17 electrical resistivity 174, 175 ellipse 43 empirical relations deformation modulus 87–93 end bearing capacity 209–220 side shear resistance 192–198 empirical rock mass strength criterion 123–131 end bearing resistance 191, 204–220 engineering geology 34 equal-angle projection 50 equal-area projection 50 equivalent continuum approach deformability 93–116 strength 132–139 equivalent pier 248–250 exploratory tunnel 164 exponential distribution 57, 59, 64, 65, 67, 68 factor of safety 6, 307, 309, 313 failure criterion 139 failure type circular 307, 308, 314 planar 307–312 toppling 307, 308, 314 wedge 307, 308, 314 fault 34 filling 47 finite difference method 282 finite element method (FEM) 272, 298, 305 flat jack test 187 foliation 35 fracture tensor 68, 69 frequency of discontinuities 33, 36, 57 friction angle 77, 78, 82, 127, 134, 179

427

Index

428

gabbro 11, 13, 16–18, 124, 130, 144 Gamma distribution 59, 64, 65, 67, 68 gauges 331, 336–339, 347, 348 geologic structure 157 geological strength index (GSI) 29–31, 125, 126 geology 151, 152, 164 geomechanics classification 20 geophysical exploration 164–174 gneiss 12, 13, 15–18, 77, 84, 87, 124, 130, 143 Goodman jack 182 granite 11, 13, 15–18, 77, 84, 124, 128, 130, 131, 143, 144 granodiorite 11, 13, 16, 124 granulite 16 great circle 48–50 greywacke 124 gritstone 12 ground penetration radar 175, 318 groundwater 5, 6 group block 222, 223, 249, 261 capacity 221–225, 261 displacement 248–250, 300–305 efficiency 221–224 interaction 221, 249, 303 settlement 248 grout 171 gypstone 124 gypsum 217, 233, 315 halite 315 hardness 10 hardpan 217 hematite 10 hemispherical projection 48 equal-angle 50 equal-area 50 great circle 48, 49 histogram 57 Hoek-Brown strength criterion 123, 211, 258 Hong Kong 206, 209 hornblende 10–12 hornfels 12, 84, 124, 143 hydraulic jack 329, 337, 346 in-plane failure 40–42, 310, 311 in situ shear tset 178, 179 in situ test 177–188 infinite element 247, 248 influence factor axial deformation 235, 238 deflection 270, 271

Index

429

rotation 270–272 intact rock 9 integral sampling method 195 integrity test cross-hole sonic logging test 352, 353 low strain integrity test using PIT 352, 353 interface cohesion 245 interface friction 245 internal friction angle 77, 132 ironstone 12 joint 35 joint element nodal displacement 117 relative nodal displacement 120 joint roughness coefficient (JRC) 78–85, 143 kaolinite 10, 12, 26 karst 317 karst area 317 karst terrain 315 karstic formations 6, 7, 315, 316, 318, 324 kinematic analysis 262 kinetic analysis 262 laboratory testing 175–177 large-diameter boring 162 lateral loading test 345–352 lateral resistance 254–262 limestone 12, 13, 15–17, 77, 87, 124, 128, 130, 131, 144, 166, 175, 194, 218, 222, 233, 315 limit equilibrium approach 262 liparite 131 load distribution 343, 344 load-transfer method linear 228 nonlinear 231 loading test compression 328–344 constant rate of penetration 340, 345 constant rate of uplift 345 lateral 345–352 maintained load 339 uplift 123, 124 loess 131 logging borehole core 158 scanline 153 lognormal distribution 59, 64, 65, 67, 68 low-angle-transition 41, 42 lower hemisphere projection 49, 50 LVDT 339

Index

maintained load test 339 marble 12, 13, 15–18, 124, 130, 131, 144 marl 13, 194, 218, 233 marlstone 17 mean frequency 57 orientation 55, 56 spacing 57 trace length 59–63 migmatite 124 modulus deformation 86 elastic 86, 165 initial tangent 85, 86 recovery 86 shear 165 Young’s 165 monozonite 17 mudstone 12, 128, 130, 131, 190, 194, 198, 217 muscovite 10, 11 Mustran cells 338 mylonites 124 norite 16, 124, 128, 130, 131, 144 normal displacement 73–76, 142, 245 stiffness 73–76 stress 74–79, 141–143 numerical methods discrete element method (DEM) 262, 298 finite difference method 282 finite element method (FEM) 272, 298, 305 obsidian 124 olivine 10, 11 orientation of discontinuities dip 36 distribution 56 mean 55, 56 plunge 54–56 sampling bias 55 strike 36 trend 54–56 polar stereonet 51, 52 orthoclase 11 orthoclase feldspar 10 orthoclase porphyries 11 Osterberg cell 330–335, 346, 347

430

Index

P-wave 165 P-wave velocity 29, 165, 166 p-y curve 263, 264, 266–269, 303, 350–352 pendulum orientation method 159 peridotite 11, 13, 16, persistence of discontinuities 37–42 persistence ratio 38–42 photographic mapping 156 phyllite 12, 13, 17, 124, 294 picrite 11 piers 1 piles 1 Pile Integrity Tester (PIT) 352, 353 plagioclase 11 plagioclase feldspar 10 plagioclase porphyries 11 planar sliding failure 307–312 plane strain 245 plate bearing test 184–186 plunge 54–56 point load index 13, 209 point load test 13, 176 Poisson’s ratio 14, 18, 165 polar stereonet 51, 52 porosity 176 potash 13 pressure cells 339 pressure wave 165 pressuremeter 178, 206 primary wave 165 probability distribution discontinuity orientation 56 discontinuity spacing 57 discontinuity trace length 59 PVC casing 171 pyroxenite 16 Q-system 20, 23–29 quartz 10–13, 16, 26 quartz diorite 17, 128, 130, 144 quartz monozonite 17 quartzite 12, 13, 15, 17, 18, 124, 128, 130, 131, 294 radial displacement 245 radial jacking 188 stress 245 Rayleigh wave 165, 171 reconnaissance 151, 152, 174 reinforcement 189, 190, 253, 254 residual friction angle 77

431

Index

432

resistance end bearing 191, 204–220 lateral 254–262 side shear 191–204 resisting force 308 resistivity survey 174 rhyolite 11, 13, 124, 130 RMR-Q relation 29 rock bridge 40–42, 310, 311 core boring 157 igneous 10, 11, 166, 175 metamorphic 10, 11, 166, 175 sedimentary 10, 11, 166 weathered 166 weathering 11 rock mass 14 rock mass rating (RMR) 20–23, 29, 31, 90–92, 125 rock quality designation (RQD) 14, 16, 18–21, 87–91 rotation 270 rotation influence factor 270–272 roughness angle 77 coefficient 78–85, 143 discontinuity 44–46 factor 195, 196 height 195, 198, 200, 201 interface 196 number 23 profile 79–82, 195, 199 shaft 195–200 S-wave 165 salt 13, 131, 315 sampling bias 54, 58, 59, 154–156 photographic mapping 156 scanline 153–155 window 155, 156 sand 16, 166, 175, 248 sandstone 5, 12, 13, 15–18, 43, 77, 87, 124, 128, 130, 131, 166, 175, 177, 192, 194, 199, 203, 210, 217, 218, 291–294 scale effect 82–84, 144–149 scanline circular 155 straight 153–155 scanline sampling 153–155 schist 12, 13, 15–18, 124, 146 scour 189, 295 seam 319–322 seismic survey 165

Index

433

seismic wave 165 serpentinite 11 settlement ratio 248 shaft group 221, 248, 261, 300 shaft resistance coefficient (SRC) 198–200 shale 5, 12, 13, 15–18, 84, 128, 130, 131, 143, 146, 166, 175, 192, 194, 199, 217, 218, 291–294 shape of discontinuities 42–44 shear displacement 73–76, 142, 245 modulus 165 resistance 191–204 stiffness 73–76 wave 165 wave velocity 165 side shear resistance 191–204 siltstone 3, 4, 13, 17, 18, 77, 124, 128, 130, 131, 192, 210, 218, 220, 277 sink hole 315, 316, 324 sister bars 338 size of discontinuities 37, 63–68 slate 12, 16, 17, 77, 84, 124, 130, 131, 143 sliding force 307 slope sliding 307–314 stability 307–314 soapstone 84 socketed shaft 1 socket 1 soluable rocks 315 sonic logging 352, 353 sound velocity 352 spacing of discontinuities 20, 36, 57 spacing of drilled shafts 248, 303, 329, 330, 346 spacing ratio 148 standard penetration test (SPT) 191, 192 Statnamic loading test 335, 346 steel 166 stereographic projection 48–51 stiffness normal 73–76 shear 73–76 strain gauges 332, 347, 348 strength of discontinuities Barton model 78, 142 bilinear shear strength model 77 Coulomb model 77, 141 strength of rock mass Bieniawski-Yudhbir criterion 127 equivalent continuum approach 132 Hoek-Brown criterion 123 Johnston criterion 128 Ramamurthy criterion 130 scale effect 144

Index

434

strike 36 structural geology 34, 151 subgrade reaction approach linear 265 nonlinear 266 syenite 16, 17 t-z curve 228–234, 249, 343, 344 tangent modulus 85, 86 telltales 338 tensile strength 40, 311 tensor 68 test axial 328–344 borehole dilatometer 180–181 borehole jack 182, 183 flat jack 187 in situ 177–188 in situ shear 178, 179 integrity 352, 353 laboratory 175–177 lateral 345–352 pits 163 plate bearing 184–186 radial jacking 188 tilt 81 uplift 344, 345 thin-layer element 121 tiff 13 till 217 tilt angle 79, 81 tilt test 81 toppling failure 307, 308, 314 trace length mean 59–63 probability distribution 59, 62–68 sampling bias 58, 59 trenches 163 trend 54–56 truncation 69, 60 tuff 124, 128, 131 unconfined compressive strength 14, 17, 20, 91–93, 123, 127–131, 144, 176–178, 192–203, 209– 211, 215–220 uniaxial compressive strength, see unconfined compressive strength universal distinct element code (UDEC) 298 uplift load 222–225 test 344, 345 vector 54, 68

Index

volumetric joint count 19 water 166 weathering 11, 315 wedge sliding failure 307, 308, 314 window sampling 155, 156, yield deflection factor 278 rotation factor 278 strength 190, 251–253 yielding 264, 278 Young’s modulus 165

435