Design of stub girders

P118: Design of Stub Girders Discuss me ...

SCI PUBLICATION 1 18

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Design of Stub Girders

R M Lawson BSc(Eng),PhD,ACGI,CEng,MICE,MlStructE R E McConnel BE(Civil)Hon, ME, DPhil(Oxon),MICE

ISBN 1 870004 80 9 A catalogue record for this book is available from the British Library

0 The Steel Construction Institute 1993

The Steel Construction Institute Silwood Park, A s c o t Berkshire SL5 7QN Telephone: 0344 23345

Fax: 0344

22944

P118: Design of Stub Girders Discuss me ...

This publication has been prepared by Dr R M Lawson of The Steel Construction Institute and Dr R E McConnel of the University of Cambridge. It is one of a series of SCI publications dealing with long span floor solutions in ‘composite’ buildings. These are: Designforopenings in webs ofcompositebeams. Designofhaunchedcomposite beams in buildings. Designoffabricatedcomposite beams in buildings. Parallel beam approach - adesignguide. Design of composite trusses. The design method is consistent with BS 5950: Part 3: Section 3.1 Code of practice for design of simple and continuous composite beams, issued in July 1990. The following representatives of SCI member organisations commented on the draft publication:

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British ConstructionalSteelwork Association Limited Mr P H Allen formerly, WatermanPartnership (now, Kashec ConsultingEngineers) Dr RAllison Mr I C Calder Scott Wilson Kirkpatrick DMr Fung Waterman Partnership Pel1 FrischmannConsultingEngineers Limited MrJ J Nayagam MrJWRackham The Steel ConstructionInstitute. The testwork was carriedout at the University of Cambridgeunderthedirection of Dr R E McConnel, based on preliminary designs by Mr D L Mullett of the SCI. The tests were funded by the British Steel Market Development Fund.

..

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P118: Design of Stub Girders Discuss me ...

Page

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SUMMARY

iv

INTRODUCTION

1

STRUCTURAL OPTIONS FOR LOING SPArN BE,AMS

2

INTRODUCTION TO STUB GIRDER CONSTRUCTION

4

DESIGN CONSIDERATIONS 4.1 Moment capacity due to axial force in the bottom chord 4.2 Longitudinal shear transfer 4.3 Designofthesteelbottomchord 4.4 Localdesignofthestub 4.5Designoftheconcreteflange 4.6Transversereinforcement in slab 4.7 Construction condition 4.8 Deflections 4.9 Secondary beams 4.10Vibrationeffects

7 8 9 10 14 15 16 18 18 19 20

SUMMARY OF TESTS ON STUB GIRDERS 5.1 Description of tests 5.2 Test results 5.3Transversereinforcementoverstubs

21 21 25 27

RESUME OF DESIGN OF STUB GIRDERS 6.1 Recent projects 6.2 Schemedesignofstubgirders 6.3Designexampleforstubgirder

28 28 28 29

DESIGN PROCEDURE FOR STUB GIRDERS 7.1 Construction condition 7.2 Ultimate loads 7.3 Serviceability

30 30 30 31

CONCLUSIONS

32

REFERENCES

33

DESIGN EXAMPLE

35

...

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Design of stub girders The design of ‘stub girders’ is presented in a form consistent with BS 5950: Part 3. The basis of design is simplified by considering that the steel bottom chord resists tension (arising from bending action), vertical shear, and local (Vierendeel) moments across the opening. The design method is comparedwiththeresults of three full-scale stub girder tests and is showntobe conservative but reasonably accurate. Model factors for these tests were in the range of 1.0 to 1.2 when using measured material strengths, increasing to 1.2 to 1.4 when using design strengths. Oneimportantobservation was thattheCoderequirementsfortransversereinforcementare unduly conservative for this form of construction. A design example is included, which covers the important aspects of the design.

Dimensionnement des poutres courtes

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RQumC Lt. dimensionnemvnt des poutres courtes, qui est prksentk dans cette brochure esten conformite‘ avec la norme BS5950: Partie 3. L,PS hypothPses de calcul ont ktk simplijikes en conside‘rant que la membrure inflrieure en acier rksiste a la traction (provenant de la flexion de la poutre), au cisaillement vertical et aux moments locaux (Vierendeel) autour des ouvertures. La mkthode de dimensionnement est cornparkc. a m rksultats d ’essais en vraie grandeurde trois poutres courtes et se re‘v2leCtrc? sck.uitaire et suflsamrnent prkcise. Lxs facteurs de charges, par rapport aux essais, ktaient dc l a l ,2, en utilisant les rcLsistances mtmrkes de 1 ’acier, et de 1,2 d 1,4 en utilisant les rksistances de dimensionnement. On a aussi observk que IPS recommandations de la

norme concernant les renforts transversaux ktaient excessivement conservativespour ce type de construction. Un twmple est donnk dars la brochure qui couvre tous les aspects importants du dimensionnemmt. Berechnung von “Stub Girders” Zusammenfassung

Die Berechnung von “StubGirders” (Trdger mitgroom StegODungen) wird in Ubereinstimmung mit BS 5950, Teil 3, dargmellt. Die Grundlage der Berechnung berucksichtigt vereinfachend, dap der Stahluntergurt durch Zug (infolge Biegung), vertikulen Schub und lokule (Vierendeel) Biegemomente am Randc der Ofiung beansprucht wird. Die Berechnungsmethode wird mit den Ergehnissen aus drei Trligerversuchen verglichen und erweist sich als konservativ aber ziemlich 1,0 bis 1,2 bei genau. Moddlfaktoren f i r dieseTkstswaren in derGrdpenordnungvon gemessenen Materialjktigkeiten und stiegen auf 1,2 bis 1,4 bei Verwendung von rechnerischen Festigkeiten. Eine wichtige Beobachtung war, dap die Anforderungen der Norm beziiglich der Ein Querbewehrung iibermdJig konservativ f i r diese Form der Konstrulction sind. Berechnungsbeispiel, welches die wichtigm Gt.sichtspunkze der Berechnungaufieigt, ist enthalten.

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Proyecto de vigas cortas Resumen

Se presenta el proyecto de vigas cortas en congr4uencia con la norma BS 5950; Parte 3. El modelo de c6lculose simplijka considerandoqueelcord6ninferiordeaceroresiste las tracciones(quesurgen de la Jexixidn) ast como el cortantevertical y 10s momentoslocales (Vierendeel) en la perforacidn. El mktodo se compara con 10s resultados de 10s ensayos de tres modelos flsicos y se muestra que 10s resultados son conservadores per0 precisos. Los factores de 10s modelos se encuentran en la banda I ,O a I ,2 cuando se utilizan resistencias medidas de 10s materiales y llegan desde 1,2 a I ,4 cuando se usan resistencias deproyecto. Otro hallazgo interesante f i e que 10s requisitos de la Norma para armadotransversalson extraordinariamente conservadores para este tip0 de construccidn. Se incluye un ejemplo tipo que abarca todos 10s matices importantes delproyecto. Progettazione di "Stub Girders"

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Sommario

La progettazione di "stub girders" (panicolari travicomposte)vieneprescwtata in questa pubblicazione in forma congruente a quanto conmuto nella normativa BS 5950: Parte 3. I concetti di base della progettazionc. vengono sempli,ficati ipotiuando che il tiranto inferiore resista alla azione assiale di trazione (dovuta alla sollecirazioneJlettente), azione tagliante verticale ed I1 metodo di progetto azioni Jettenti locali attorno alla eventuali aperture (trave Vierendeel). proposto, confrontato con i risultati di tre prove spwimentali eseguito su prototipi a grandma reale di "stub girders", risultaessereconservativo e ragionevolmenteaccurato. I fattori di sicurezza risultano variabili tra 1,0 e 1,2 considerando le eflettive caratteristiche dei materiali, mentre variano da I ,2 a I ,4 usando le caratteristiche nominali. Una importante considerazione e che le prescrizioni dellanormativarelativamenteall'armaturatrasversalerisultano eccessivamenteconservative.Unesempio di progetto, che trattaimportantiaspettidella progettazioni, viene injine incluso in questa pubblicazione.

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1. INTRODUCTION Composite construction in buildings is well established and there is often a strong demand for longercolumn-freespans in buildings.Conventional steel frames with concrete or composite slabs may be used, but often the size of long span beams is such that the floor zone is excessively deep. There is also the need to incorporate a high degree of servicing in modern buildings and coupled with this are the requirements for minimising floor zones and reducing cladding costs where building heights are restricted. Various design solutions are feasible but there are two basic options: either the structure and services are integrated within the same horizontal zone, or the stractural zone is minimized so that theservicesare passed beneath.Thesesolutions are described in simpleterms in the following Section.

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The economics of modern building are such that the small increase in the cost of the steel frame required to integrate the structure and services has a proportionately much smaller effect on the overall cost of the building. Many clients appreciate that these ‘long span solutions’ offer greater flexibility of building use and represent the most ‘economic’ and versatile use of steel frames. One of the potentialsolutionsfor beam spans in theregion of 12 to 20 m is the so-called ‘stub girder’. This is a form of construction first employed if North America where it is often used for regular column grids. It is a relatively easy system to manufacture, the only element of fabrication being the attachment of the ‘stubs’ to the lower chord. Services are contained in a zone between the lower chord and the composite slab which forms the compression element in the structural system. Its main disadvantage is the frequent requirement for temporary propping during construction. This publication describes the important features of stub girder construction and puts forward a method of design consistent with BS 5950: Part l and Part 3. It is one of a series and draws upon guidance offered in other SCI publications (see Foreword). The method may also be used in accordance with Eurocode 4 (to be published in 1993).

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2. STRUCTURAL OPTIONS FOR LONG SPAN BEAMS Composite slabs are usually designed to span 3 to 4 m between support beams and their depth is typically 120 to 150 mm. This dictates the economic layout of the structural grid. The long spanbeamsunderconsideration may be loaded directly by the compositeslab or loaded by secondary beams which support the slab. Therearevariousstructural optionsforachievingthetwinaims of longspans and ready incorporation of services withinnormal floor zones. The 'stub girder' should be consideredalong with the following alternatives:

Beams with web openings

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In this method of construction, the depth of the steel beam is selected so that sufficiently large, usually rectangular-shaped, openings can be cut into the web. For general guidance, it is suggested that the openings should form no more than 70% of the depth of the web, where horizontal stiffeners are welded above and below the opening. Typically, the length of the openings may be up to 2 times the beam depth. The best location of the openings is in the low shear zone of the beams. A step by step method of design is presented in reference(3). A modified form of construction is the notched beam where the lower section of web and method is flange of the section is cut away over a short distance from the support. This not usually practical unless the cut web is stiffened.

Castellated and cellform beams Castellated beams with hexagonal web openings can be used effectively for lightly serviced buildings or for aesthetic reasons where the structure is exposed. Composite action does not significantly increase the strength of the beams but increases their stiffness. Castellated beams have limited shearcapacity and are best used as long span secondarybeams or where loads are relativelylow. The designof castellated beams is covered by an SCI publication(4)which gives design tables for standard non-composite castellated sections. Cellformbeamsaresimilar to castellated beams but havelargecircularopenings.The design method is presented in reference(5).

Fabricated beams with tapered webs The tapered web beam is designed to provide the required moment and shear capacity at all points along the beam, and the voids created adjacent to the columns can be used for modestly sized service runs. Typically, the tapered beam is most economic for spans of 13 to 20 m. The plate size can be selected for optimum structural performance, and the plates welded in an automatic single-sided submergedarcprocess.Thicker webs are welded by double-sided fillet welds. Web stiffeners are often required at the change of section when taper angles exceed approximately 6". Design is covered by reference(6).

Trusses Trusses are frequently used in multi-storey buildings in North America and now in the UK and are best suited for long spans (12 to 20 m), where the truss is designed to occupy the f u l l depth of the floor zone. The cost of fabrication can be high in relation to the material cost but trusses can be cost-effective and have been used in a number of major projects. Little improvement in the moment capacity of the truss is gained from composite action but the stiffness of the truss is increased significantly. The modified Warren truss is the most

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the stiffness of the truss is increased significantly. The modified Warren truss is the most common form as it offers the maximum zone for services between bracing members. The design of composite trusses is presented in reference(7).

Haunched beams Haunched beams are designed by forming a rigid moment connection between the beams and columns. The design method is presented in reference('). The depth of the haunch is selected primarily to provide an economic method of transferring moment into the column. The length of the haunch is selected to reduce the depth of the beam to a practical minimum. The extra service zone created beneath the beam between the haunches offers flexibility in service layout. At edge columns, it would not be normal practice to develop additional continuity through the slab reinforcement, but this is an option at internal columns. This form of construction or concrete shear walls or can also be used for sway frames, i.e. where vertical bracing cores are not provided.

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Parallel beam grillage systems This system is different from the other systems previously described in that continuity can be developed in both the secondary and primary beams. The so-called PBA design method is presented in reference('). The secondary beams are designed to act compositely with the concrete slab, and are made continuous by passing over the primary beams. The primary beams are arranged in pairs and passoneitherside of the columns, to which they are attached by shear-resistingbrackets. These primary beams are non-composite.Parallel beam systemsare ideally suitedtoaccommodatinglargeserviceducts in orthogonal directions.

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3. INTRODUCTION TOSTUBGIRDER CONSTRUCTION The basic structural action of a stub girder is such that the resistance to applied moments is developed by tension in the lower chord and compression in the concrete orcomposite slab. The forces are transferred between these elementsby ‘stubs’ or short sections of beam attached to the lowerchordbywelding orbolting, and by shearconnectors to theconcreteslab. In the orthogonal direction, secondary beams achieve continuity by spanning over the bottom chord. The secondary beams are often designed with pin connections at the quarter span points. In the basic stub girder system the depth of the secondary beams is the same as the depth of the opening (see Figure l(a)). In this system the stubs are formed from the same sectionas that used for the secondary beams. There is therefore an optimum relationship between the size and span of the secondary beams (and hence the depth offered for servicing) and the span of the stub girder.

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Taking, for example, a column grid of 12 m square, the continuous secondary beams could be a 406 UB section. Assuming that the bottom chord is a heavy 254 UC section and the slab is 130 mm deep, it follows that the overall structural depth is approximately 800 mm, corresponding to a span: depth ratio of 15. A more optimumgridmight be a12m x 15m in which thestubgirderspansthelonger distance. Assuming the bottom chord is increased to a 305 UC section, the overall span: depth ratio increases to nearer to 18, which is more typical of ‘regular’ composite construction. In the basicsystem thestubsare sized so thattheshearconnectors needed to developthe appropriateforce in theconcretearedistributed at not less than the minimum spacing recommended in BS 5950: Part 3. This determines the length of the stubs and, consequently the maximum width of the openings available on either side of the stub. Clearly, the length of the stubsdecreases as theforcetransferreddecreases.This means that wider openings can be provided towards the middle of the span. The bottom chord is designed to resist the combined tension, shear and moment developed under compositeaction.It is not usually sufficientlystrong to resist loads developed during construction and, therefore, temporary props are required at normally the mid-span or third-span locations. The basicsystem can be modified, as shown in Figurel(b), to permituseof unpropped construction by introducing a T-section as an upper chord, which is designed to resist compression when the stub-girderis subject to the self weight of the floor slab (i.e.wet concrete) and other construction loads. This T-section is subsequently embedded in the slab. Holes can be drilled in the T-stalk so that reinforcing bars may be passed through and held in position, thus avoiding the need for shear connectors at the stubs. Another possible modification to the basic system is shown in Figure l(c). This addresses the common need for deeper opening zones than are obtainable for efficiently designed secondary beams. In thisapproach,deepdiaphragm‘stubs’arefabricatedfrom welded plate and the secondarybeams can be attached to them by angles or webcleats. The location of the diaphragms can be different from that of conventionalstubs(compareFigures l(a) and (c)). However, in such a design the advantage of continuous secondary beams is lost unless holes are cut in the diaphragms and the beams passed through. This results in buildability problems, and therefore it would probably be more practical to design the secondary beams as simply-supported composite members. Alternatively, the secondarybeams may be attached to the stiffeners welded to the ends of the stubs.

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Shear connectors ,Secondary beams

l Bottom chord /

\ I

Temporary prop (a1 TYPE A : OPENING DEPTH EQUAL TO SECONDARY BEAM

DEPTH : NO TOP CHORD

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SECTION A-A

(b) TYPE B : A S TYPE A, BUT WITH TOP CHORD

TO AVOID TEMPORARY PROPPING

+ l'%--=

---7FFTT7+-T-FFr?TTr-

-

UL

support

t Angle

l (C)

Beam to stiffener

or...

diaphragm c-Plate

I

I

,

11

l

TYPE C : OPENING DEPTH GREATER THAN SECONDARY BEAM DEPTH

: NO TOP CHORD

Figure 1 Different forms of stub girder

This system has been used for stub girder spansof the order of 25 m with openings over 1 metre deep. Secondary beam spans are practical in the range of 8 to 12 m when using this system. Temporary propping would normally be required. Generally, little advantage is gained from trying to achieve moment continuity between the stub girder and the adjacent columns. The main design criterion for stub girders is the longitudinal sheartransferbetweenthechordsviathestubs, which is largely unaffected bycontinuity. Nevertheless, the bottom chord can be easily designed to develop a suitable connection to the columns (e.g. by end plates) and the slab reinforcement designed to resistthe appropriate tension. This can be enhanced by the attachment of any T-section upper chord (as in Figure l@)) to the

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column flange. The column web would usually have to be reinforced locally to resist the forces developed by these connections. The requirementsfor local sheartransfer at the stubsgenerally mean that it is necessary to introduce vertical stiffenersat the ends of the outerstubs (where shear forces are greatest). These stiffeners can often be omitted on the stubs towards mid-span.

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A major series of load tests at the University of Cambridge, covered in references (10,l l ) , has been carried outon aprototypestubgirderdesign of 13.2 m span.This achieved moment continuityby the methodsdescribedabove. The resultsofthisresearch are summarised in Section 5.

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4. DESIGN CONSIDERATIONS The moments and forces in a stub girder can be (and have been) determined by various forms of ‘elastic’ numerical analysis. However, the approach adopted here is a simplified hand-analysis at the ultimate and serviceability limit states. This approach ignores the bending resistance of the concrete slab, and is therefore conservative and leads to slightly greater forces in the chords than in reality. The applied load (including self weight), is assumed to act through the secondary beams. The variation of moment and moment capacity along the beam is illustrated in Figure 2.

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Bottom chord

Bending moment diagram l 1

I

4

l

I

A

partial shear connection

Figure 2 Build-up of moment capacity along stub girder

Most of theresistance of astubgirderto global bending is achieved by compositeaction involving tension in the steel chord and compression in the concrete slab. In addition to this global action, the transfer of vertical shear forces across the ‘web’ openings between the stubs causes local (or ‘Vierendeel’) bending in the chords. The combined moment capacities due to local and composite action should exceed the applied moment at all points along the span. In the simplified hand-analysis presented here,it will be assumed that all the ‘Vierendeel’ bending is resisted by the bottom steel chord. This design approach is adopted as the member forces it predicts are in generally good agreement with the results of three full scale tests which are described in Section 5 . More shear connectors, and hence, longer stubs, are normally required in the higher shear zones towards the outer parts of the span.

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The moment capacity of the stub girder calculated using the simplified approach presented here should exceed the applied moment due to both dead and live loads (using the load factors in BS 5950: Part 1). Alternatively, as suggested above, a computer analysis may be used to determine the moments and forces in the top and bottom chords. The most commonly used is a plane frame approach, in which the various members aremodelled by elements of calculated stiffnesses. This will result in a less conservative distributionof forces than that assumed above, but does mean that the slab has to be designed for the forces that it attracts (Section 4.5).

4.1

Momentcapacitydueto

axial force in thebottom chord

The tensile resistance of the steel section is simply: R, = P y A

where

A

= cross-sectional area of the steel bottom chord

p y = design strength of steel (to BS 5950: Part 1). The compressive resistance of the concrete slab is:

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R, = 0.45fuB , D,, where 0.45f,, represents the compressive strength of a stocky column (or wall) with concrete of cube strengthf,,.

B,

= effectivebreadth of theslabconsidered

D,,

= average depth of the slab in the case where the ribs of the slab run

to act with each beam (discussed below). parallel to the stub

girder, or the minimum depth in other cases. The compressive resistance of a steel top chord may also be included in R,. The effective breadth of the slab is taken as one quarter of the span of the stub girder (or an eighth for edge beams) but not greater than 0.8 times the spacing between the stub girders. This limit of 0.8 is introduced in the design of primary beams because of the influence of combined slab and beam bending in the same direction in such cases. The same effective breadth is also recommended for use in serviceability calculations. lies in theconcrete. If R, < R, then the plastic neutral axis of the compositesection Conservatively, the lever arm is the distance from the mid-depth of the slab to the mid-depth of the steel bottom chord, D , , Hence the moment capacity of the composite section is:

If R, that:

> R, then the plastic neutral axis lies in the steel section, and usually in the top flange, such

M , = R, D,

-k

D R, 2

where D, is the distance from the top of thesteel bottom chord to the mid-depth of the slab such that D, = D , . - D/2, where D is the depth of the steel bottom chord.

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The aboveformulaeassumethat ‘full shear connection’ is provided, so that the full plastic moment capacity of the composite section can be developed. In many cases M, will exceed the mid-span moment by a considerable margin. This is necessary because the steel section should also be able to resist local (Vierendeel) moments across the opening. The moment capacity of the cross-section builds up in stages along the stub girder resulting from the longitudinal force transferred via the shear connectors at the stubs. It is also necessary to check the moment capacity at intermediate locations as shown in Figure 2. These checks are covered in Section 4.2.

4.2

Longitudinal shear transfer

It follows that to achieve the moment capacity calculated in Section 4.1, the longitudinal force to be transferred between the points of zero and maximum moment should exceed the smaller of R, or R,.

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The longitudinal shear transfer is normally achieved by provision of shear connectors between the concrete and the ‘stub’, by shear in the web of the stub, and finally by welds or bolts between the stubs and the steel chord. The number of shear connectors and bolts, and size of welds is chosen to resist this force. The rate of build-up of force in the concrete or steel broadly follows that of the shear force diagramalongthebeam.Consequently,moreshearconnectors, and hencelongerstubs, are required in the high shear zones towards the outer parts of the span. If the design capacity of each of the shear connectors is P d , it follows that the total number of shear connectors needed in the half span is:

The characteristicresistancesof the shearconnectors may beobtainedfrom Table 5 of BS 5950: Part 3. The design capacities are obtained by multiplying these values by a factor of 0.8 in the positive moment region. The above approach applies for ‘full shear connection’ in the composite section at the point of maximum moment. However, there is scope for reducing the total number of shear connectors where the moment capacity exceeds the applied moment. BS 5950: Part 3 permits use of ‘partial shear connection’ for beams up to 16 m span. If the total force transferred by the shear connectors from the adjacent support to the point on the span under consideration is R, (such that R, < R, and < RC),then;

D

M, = R, D, -F R, 2

D, is defined as used in Equation (4). This formula applies when the plastic neutral axis of the section lies in the top flange of the steel bottom chord such that R, > R,, where R,,, is the tensile resistance of the web. The degree of shear connection, K , is defined as K = R, /R, (when R,

< R,) and R, /R, (when

R, < R,). The minimum degree of shear connection at the point of maximum moment is given by:

K 2 (L - 6)/10 and 1.0 2 K 2 0.4 where L is the beam span in metres.

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Therefore theminimum value of R4 needed in mid-span is determined, irrespective of the loading. The same approach may be adopted for the build-up of moment capacity alongthe beam as determined by the magnitude of R, from the adjacent support to any point under consideration. Equation (5) applies when the web is fully in tension. A further equation may be derived when R, < R,, so that the web of the bottom chord is partly in tension as follows:

M,=

R2 D R4 Deff + M , - A . R, 4

where M , is the moment capacity of the steel section. This is an ‘exact’ equation based on ‘stress block’ analysis and represents the maximum available moment capacity of the section. The first term represents the moment due to tension in the bottom chord (as given by R,& and the second term is that due to pure bending in the chord. The final term represents the adverse effect of tension on the moment capacity of the steel section.

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Equation (6) applies where R, is relatively small i.e. close to the supports. In the limit M , tends to M, when R is zero. The effect of high vertical shearshould also beconsidered (see next section) by re;fucing the term R,. The distribution of shear connectors should be such as to ensure that the valueof M , (determined from Equations (5)or (6)) exceeds the global applied moment at all points. Critical cross-sections are at the higher moment side of the openings,shown as points A and B in Figure 2. The moment capacity remains constant between the stubs in the absence of any other means of shear transfer. As a ‘safe’ simplification,the total number of shearconnectors needed at any stub may be distributed in proportion to the shear force diagram along the beam. This determines thenumber of shearconnectorspositionedover each stubrelative to the total numberrequired in the half-span. A nominal numberofshearconnectors (only 1 every 450 mm) is appropriate in zero-shear zones.

The shear connectors may be arranged singly or in pairs along the stubs subject to minimum spacingcriteriaof 44 laterally and 54 longitudinally(where 4 is the stud diameter).These requirements effectively determine theminimum length of stub to be used. Additional transverse reinforcement in the slab is needed toenhancea smooth transfer of shear into theslab (see Section 4.6).

4.3

Design of thesteelbottom

chord

The internal forces developed in a stub girder are presented in Figure 3. Longitudinal forces are transferred discretely at the stubs rather than gradually along the beam. Equilibrium is satisfied by development of moments in the bottom chord.Thereforethe bottom chord is subjectto tensionarisingfrom the global (or primary)bendingaction and also to local (orsecondary) moment and shear arising from the shear forces applied to the girder. The relative magnitude of theseeffectsdependson the ratio of moment and shear at any pointalongthespan (see Figure 3). The steel chord should be a ‘plastic’ or ‘compact’ section in accordance with BS 5950: Part 1. This is to ensure that it can develop its plastic moment capacity. A significant amount of rotation capacity is not required.

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Axial force in slab T + NPd

v

)M+"Lb

T + NPd

Tensile force in bottom chord

Moment due to tension in bottom chord

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Figure 3 Model of load transfer at stub (type A or B)

4.3.1

Vierendeel moments

The shear conditions around an opening demand the greatest consideration. This behaviour is similar to that occurring when a large opening is positioned in the web of a girder (see reference (3)) such that the upper flange and upper part of the web are cut away. The transfer of vertical shear force across the opening is then resisted by local bending of the bottom chord and the slab. This is commonly known as 'Vierendeel bending'. 'The high relative stiffness of the bottom chord to the top chord (the concrete slab) means that most of the Vierendeel bending moment is resisted by the bottom chord and the contribution of the slab can be 'ignored'. For equilibrium, the moment difference between the edges of adjacent stubs is dependentontheshearforcetransferred.Therefore, the Vierendeel moment in the bottom chord, M,, adjacent to the stubs is given by:

M, = V L ,

(7)

where L, is the distance between the edge of the stub and the point of contraflexure in the bottom chord. V is the applied shear force across the opening.

As a first approximation L, may be assumed to be mid-way between the edges of the adjacent stubs. Hence, in Figures l(a) and (b) L, is the length of the opening between the stub and the secondary beam. In Figure l(c) L, is half the distance between the adjacent stubs. At the outer openings, L, is the distance from the outer stub to the adjacent column. The local vertical force applied to the bottom chord by the secondary beam also causes local moments in the bottom chord. However, these additional effects are accounted for if V is defined as being the maximum shear force across each opening, and each opening is checked separately. The bottom chord should be able to maintain equilibrium by ensuring that the moment capacity of the steel section, M,, exceeds M , adjacent to each stub. M , should also take into account the influence of tension and shear, as considered in the following sections.

P118: Design of Stub Girders Discuss me ...

4.3.2

Influenceof

shear

The shear force, V , is considered to be resisted entirely by the web of the bottom chord, because theslenderslabbetween adjacent stubs may not be able to resist signiticantshearforce. Alternatively, a plane frame computer analysis may be used to determine the moments and forces directly in the top and bottom chords. This will result in a less conservative distribution of forces than that assumed, but does mean that the slab is to be designed for the forces that it attracts (Section 4.5). The shear stress applied to the web of the bottom chord reduces the effectiveness of the bottom chord in bending and tension.This may be taken into account by modifying the effective thickness, t,, of the web according to:

t,

= t,

/m

(8)

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where V , is the ultimate shear strength of the web which is equivalent to a shear stress 0 . 6 ~ ~ applied uniformly over the f u l l depth of the section (as in BS 5950: Part 1 Clause 4.2.6), and t , is che actual thickness of the web.

In principle, this formula is rather less conservative for sections subject to high shear than the in BS 5950:Part 1 . It is presented as an alternativefor linearinteractionformulapresented highly stressed sections. However, it is more conservative for sections subject to low shear, and hence no reduction in web thickness need he taken when V < 0.6 V, as in BS 5950: Part 1 (see Figure 4). The effective web thickness, t,, is now used to recalculate the propertiesof the steel bottom chord i.e. R,,,, R, and M,, asdetined in Section 4.2. Theseproperties also intluencethe moment capacity of the composite section, M,.

0

0.5

Figure 4 Effective thickness of web as a function of web shear force

12

P118: Design of Stub Girders Discuss me ...

4.3.3

Influence of tension

The influence of axial tension is already included when calculating M , using Equations (3) to (6). Theseare ‘exact’ equations and representthe maximum capacity of the compositesection. Overall equilibrium should be satisfiedby ensuring that the moment capacity at all points exceeds the applied moment (see Figure 2). However, the moment capacity of the steel section, M,, is influenced by axial tension, T, which in turn influences the resistance to ‘Vierendeelbending. This is best illustrated by considering the combinationofstressblocks in the bottom chord at thelow and high momentsides of the opening, as shown in Figure 5. The proportions of the section not utilized in resisting tension can beused to resist the Vierendeel bending effect (shown shaded). The interaction between moment and tension is then of the form illustrated in Figure 6 . Where the web area is small, a linear interaction is appropriate. This is equivalent to a direct combination of bending and axial stresses. The linear interaction approach the reduced becomes more conservative (by about 10%) as the web area increases. However, may bedetermined with reasonableaccuracyfrom: momentcapacity of the steel section,

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where

T

= applied tensile force at given a location.

This is equivalentto theforce, Rq, transferred via the shear connectors from the support to the point under consideration.

A,ff

Ms

I

te

= effective area of the section (including = moment capacity of thesection(including = effective thickness of the

t,) 5 A t,) 5 M,

web (calculated using equation

(8)).

Therefore,forsatisfactorydesign of the bottom chord, MS red > M , (seeEquation (7)) and T < p y A , , As a first approximation, the web area can be (gnored in calculating the effective section properties.

I

I

7

L 4 I

Low moment side

Mid point between openings

High moment side

Figure 5 Combination of stress blocks in bottom chord between stubs

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P118: Design of Stub Girders Discuss me ...

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T

Figure 6 Interaction between plastic moment capacity o f

4.4

I section and axial tension

Local design of the stub

Equilibrium of forces on the stub dictates that there must be a vertical reaction developed between the base of the stub and the bottom chord. This local behaviour is illustrated in Figure 7. The magnitude of this separation force per unit length is:

where Niis the number of shear connectors of design strength Pd, attached to a stub section of length L, and depth D,. The welds are therefore designed to resist longitudinal shear and uplift. These compression and shear forces may be combined vectorially to give a maximum force per unit length of weld of:

Thisforce is used in thedesign of thefilletweldsattachingthestub to the lowerchord. Similarly,connectingbolts, which can be used as alternatives to welds, must becapableof resisting these combined forces. Friction grip bolts will usually be needed to avoid the effects of slip on deflections.

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P118: Design of Stub Girders Discuss me ...

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Figure 7 Local forces on stub and bottom chord

The web of the stub is designed to resist the compression force, fm, and a longitudinal shear force as transferred via the shear connectors. The web slenderness is 2.5 D, / t when subject to buckling, and its compressive strength can be evaluated as for a strut, according to Table 27 of BS 5950: Part l('). Additionally, shear and compression stresses are combined vectorially using the Von Mises criterion and should be less than py. In many cases it is necessary to stiffen the edge of the stub using a verticalwelded stiffener. This load bearingstiffenershouldbedesignedtoresistaforceequivalent to 4. Pd D, / L, (i.e. ignoring the contribution of the web). It is not economic to stiffen the web of the bottom chord and therefore, it should be checked for its resistance to web bearing or buckling when subject to this force.

4.5

Design of the concreteflange

The concrete slab acts as the compression flange of the stub girder. It behaves effectively as a strut (or more correctly a braced wall) which is restrained at the attachments to the stubs and secondary beams. In theory, the flange behaves as a 'stocky' column or strut provided the ratio of its unsupported length to slab depth does not exceed 12. The real behaviour is rather different in that the slab is notcontinuouslyrestrainedacross its width, and also there is some small flexibility of the attachment of the slab to the shear connectors (see Figure 8). Local moments and shear forces may also develop in the slab due to its stiffness, but these are usually ignored. Local uplift forces on the shear connectors may also be ignored, provided the deformation across the opening is small (see section 4.8). In the absence of other detailed guidance, it is considered necessary to restrict the maximum unsupported length of slab between longitudinal restraints to a span to depth ratio of 10, based D,,. Assuming that theaverage depth is 100 mm,the maximum on the average slab depth, unsupported length of slab when subject to its design compressive stress is therefore lo00 mm. of thumbto be used when sizingstubgirders. This is recommended as areasonablerule Secondarybeamsalso act as effectiverestraints to theslab. This means that the maximum distance between the edges of adjacent stubs in Figure l(a) and (b) is 2000 mm.

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P118: Design of Stub Girders Discuss me ...

S tub

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Figure 8 Stability of compression flange between stubs

This method may be extended to treat slabs with lower axial forces by multiplying the span to depth limit of 10 by RJR, for R, > R, (implying that the slab is not fully stressed to 0.45&,). However,thecoincidentinfluence of sheartends to causeeccentricity of load, and hence exacerbates the instability of the concrete tlange. It is suggested that the slab span-to-depth ratio of 10 is retained for high shear regions irrespective of the force in the slab. The presence of an upper T chord would also have a stabilising effect and may be used to increase the unsupported length of the slab between restraints. Where the analysis determines the moments and forces in the slab (e.g. by plane frame analysis), the slabshouldbedesigned to resisttheseforces. Moment and axial forces may then be combined using the column design charts of BS 81 lo(”). This often necessitates the provision of additional reinforcement in the slab and more shear connectors at the edges of the stubs to resist local uplift forces.

4.6

Transversereinforcement

in slab

In order to develop a smooth transfer of force from the shear connectors into the concrete it is necessary to provide adequate transverse reinforcement (i.e. transverse to the axis of the beam). This can be achieved by straight bars or mesh, but more efiicient detailing arrangements using ‘herring-bone’ reinforcement have been developed (see Figure 9). The resistance to longitudinal shear may be evaluated by considering the potential shear planes oneitherside of the line of shearconnectors. The resistance per unit length is defined in BS 5950: Part 3 (Clause 5.6.3) as:

where

16

A,,

=

cross-sectionalarea of reinforcementperunit shear plane

ACV

=

mean cross-sectional area of concrete per unit length

11

=

1.0 for normal weightconcrete

4

=

designstrength of the reinforcement

and 0.8for

length of the beam for each

lightweightconcrete

P118: Design of Stub Girders Discuss me ...

Forinternalbeams, the resistance V may bedoubledtoaccount for formation of two shear planes. The contribution of the decking can be included as suggested in BS 5950: Part 3, if the deckingrunsperpendiculartothestub girder,orthe longitudinaledges of the decking are fastened together (e.g. by screws or stitch welding). Generally the effect of the decking is not included in stub girder design. The upper limit on V is introduced to prevent local crushing of the concrete (see discussion on tests in Section 5.3 and guidance in reference (13)). The applied force per unit length is conservativelygiven by Nj Pd / L, (see Section 4.4). It follows that this force should not exceed the available longitudinal shear resistance provided over thestubs,as determined by Equation(12). It is usually found thatsignificantamountsof additionalbarreinforcement are needed in the region of thestubs to controlthiseffect. However, the f u l l scale tests showed that the zone of longitudinal shear transfer is longer than the length of the stub and that Equation (12) is very conservative.

Potential shear planes (a-a3

a

a

!

- - - Reinforcement

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(below nead of stub1

Stub

(a> ILLUSTRATION OF LONGITUDINAL SHEAR

FAILURE

reinforcement

Force transfer

AN (b) USE OF 'HERRING-BONE' REINFORCEMENT IN SLABS

Figure 9 Influence of transverse reinforcement in controlling longitudinalshear failure

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P118: Design of Stub Girders Discuss me ...

4.7

Construction condition

In the construction stage, thesteel bottom chord is designed to support theself weight of the floor and other construction loads (taken as equivalent to 0.5 kN/m2 in the design of the beams). The chord may be checked in bending in accordance with BS 5950: Part 1 (using the load factors in Parts 1 and 3). It is usually found that one or two vertical props are needed so that the moments in the chord are reduced. In-built stresses during construction do not affect the final collapse of the stub girder, which is assessed on the basis of factored dead and imposed loads applied to the composite section (see Section 4.1). This implies that signiticant redistributionof internal stresses occurs. Serviceability stresses are not normally calculated in this form of construction because local yielding does not have a major effect on deflections. When the system shown in Figure l(b) is employed, the possibility exists of designing the steel top chord for the compressive forces induced due to the bending moment developed in the girder at the construction stage. The top chord is then designed as a strut acting over the lengths of the openings. This system can avoid the need for temporary propping.

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4.8

Deflections

The deflection of a stub girder system comprises a component due to normal bending action and a component due to the transferof {hear across the openings (Vierendeel action). Detlections are normally calculated for unfactored imposed load and should be limited to the values given in BS 5950 Part 1 . Self weight detlections are oftenvery high and should be limited either by propping, o r by introducing a top chord, or by pre-cambering. The bending deflection is obtained by calculating the effective second moment of area of the section. The area of concrete in the composite section is reduced to an equivalent area of steel by dividing by the modular ratio of steel to concrete, a,. The appropriate values of a, are given in BS 5950: Part 3, but ‘average’ values of 10 for normal weight concrete and 15 for lightweight concrete are generally used for buildings with low permanent imposed loads (i.e. normal usage). The second moment of area of the combined top and bottom chords, ignoring the contribution of the stubs, is:

where

A , = D,, B, plus the contribution of any steel section embedded in theconcrete, and D,g is defined in Section 4.l .

I,

=

second moment ofareaofthe

steel bottomchordofcross-sectionalarea,

A.

Hence, the mid-span bending deflection due to uniform imposed load is obtained directly from:

6, where

5 wiL3 384 E I,

W i = total unfactored imposed load on the beam of span, L. E

18

=

=

elasticmodulus of steel (= 205 kN/mm2).

P118: Design of Stub Girders Discuss me ...

The mid-span‘shear’deflection can beestimated by considering the deflection due to local Vierendeel bending action in the bottom chord such that:

where

v L: -

6,

=

c--

V

=

shear force

3 E I,

W iN L: 24 E I,

per opening.

L, = length of the opening defined as in Section 4.3 or Figure 2. Correctly, L, is the distance from the stub to the point of contraflexure in the chord. The summation is over all the openings in half the beam span. For a regular distribution of openings, the deflection tends to the second formula (where N is the number of openings in the span). In orderto avoidexcessivedistortion of the opening and upliftforces on the shear connectors, it is proposed that 6, / L , <

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The total central deflection is therefore 6, + 6,. It is usually found that the shear term is much less (< 10%) than the bendingterm. This calculation is conservativebecause the bending is propped,thenthe self weight stiffness of theslab has been ignored. If thestubgirder deflection on removal of the props can be calculated in a similar manner (using the long term modular ratio as in BS 5950: Part 3).

4.9

Secondary beams

Secondary beams directlysupportthecompositeslab and can be designedeitherassimply supported or continuous composite beams (see Figure 2). Continuous design is appropriate where the secondarybeams pass over the bottom chord. However, special consideration is required along the column lines where it may not be possible to achieve continuity. In these cases it may be necessary to use a heavier beam. For simply supported beams, the Design Tablesin reference (14) may be used for rapid selection of the beam size. For continuous beams, elastic global analysis or plastic hinge analysis may be used todeterminethe moments and forces in thesection.Thisapproach is presented in BS 5950: Part 3. Plastic hinge analysis is commonly used for ‘plastic’ steel sections, and satisfactory design of beams with one degree of continuity (i.e. end bays) is obtained when:

MP + 0.5 M,, where

[ :,k] 1 -

___

2 M,

M,, = negative (hogging) moment capacity ofthe reinforcement, but ignoring mesh)

composite section (including

MP = positive (sagging) moment capacity of the composite section, and M,, = free bending moment at the ultimate limit state. This method of analysis may result in considerable redistribution of moment at the serviceability limit state due to the early formation of plastic hinges at the supports. The effect is to cause increased deflections under repeated loading. To avoid these so called ‘shakedown’ deflections, the ratio of Mp/Mn should not exceed 1.5. This often dictates the sizing of the steel secondary beams. Typically, when Mp/Mn equals 2.0, the additional deflectiondevelopedasaresult of

19

P118: Design of Stub Girders Discuss me ...

shakedown effects is approximately 30%. Elastic deflections should also take into account the influence of pattern loads.

As a rule of thumb, adequate serviceability performance is achieved when the span/depth ratio of simply-supported beams is less than 20, and that of continuous beams (end bays) is less than 24. The structuraldepth in thiscase is the combined slab and steel beam depth. Detailed analysis of continuous composite sections is covered in BS 5950: Part 3 and in the Commentary on the Code (16).

4.10

Vibration effects

A measure of the adequacy of the floor when subject to occupant induced vibration effects is its natural frequency. Traditionally, a minimum natural frequency of 4 cycledsec has been used as a simple design check. However, the relationship between frequency and response is such that lowerlimits areappropriate, if a full designprocedure isused (refertothe SCI publication

Design guide on the vibration of floor^('^)).

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The natural frequency, f , of the stub girder may be calculated from the simple expression:

where 6,, is th-e instantaneous deflection due to the self weight of the floor and beam and 10% imposed loads (permanent loads in office type buildings) re-applied to the composite section. To avoid lengthy calculations this deflectionmay be determined usingthe same section properties as for the imposed load detlectiol;, but making a 10% reduction for dynamic effects.

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P118: Design of Stub Girders Discuss me ...

5. SUMMARY OF TESTS ON STUB GIRDERS 5.1

Description of tests

A series of three full-scale tests on stub girders were carried outat the University of Cambridge to be representative of modern between 1985 and 1988(")("). The testsweredesigned construction. The stub girder span was 13.2 m with secondary beams at 3.3 m spacing. The slab depth was 125 mmin lightweight concrete. In Tests 1 and 2 the beam to column connections were designed to be moment resisting so that the stub girders could form part of a 'sway' frame. In Test 3 simple connections were used. In all three tests, moments were applied to the columns by horizontal jacks at their ends to keep the columns vertical. The test arrangement and other details are illustrated in Figures 10, 1 1 and 12. Figure 10 also shows the shear connector layout and the deck prior to concreting for Test 3 and the load application via the secondary beams. The method of applying moment to the columns is shown in Figure 11.

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The size of the secondary beam was selected to be representative of a span of about 9 m. Point loads were applied to the secondary beams at 1.83 m from the centre-line of the stub girder. This distance corresponded to the approximate point of contraflexure of the continuous secondary beams, and so representedthe way in which theloads are applied to the stub girder via the secondaries. Other uniform loads were applied to the slab. The deck was orientated parallel to the stub girder in all cases. Brief test details were as follows: Test 1:

The test arrangement was of the form of Figure I(a). The stubs were 1.6 m long creating eight approximately equally sized openings of 850 mm width (see Figure 12). The steel sections (grade 50 steel) were: Bottom chord: Secondary beams: Stubs: Columns:

305 406 406 305

X

X X X

305 X 137 kg/m UC 140 x 39 kg/mUB 140 x 39 kg/m UB 305 x 198 kg/mUC

The depth of the openings was therefore equal to that of the secondary beams. The stub girder was propped during construction. The outer stubs were stiffened vertically. A total of 30 shear connectors (in pairs) were welded to each of the outer stubs. The deck was Holorib (a re-entrant protile). 'Herring-bone' bar reinforcement was provided at the outer stubs positioned below the heads of the shear connectors. Test 2:

The test arrangement was of the form of Figure l(b). The stubs were made from plate section creatinglargeropenings (400 deep X 1600 wide) than in Test 1 . The secondary beams were attached to the stubs by simple connections. The steel sections (grade 50 steel) were: Bottom chord: Secondary beams: Columns: Top chord:

305 x 305 x 97 kg/mUC 356 X 127 X 39 kg/m UB 305 x 305 x 198 kg/mUC 203 x 102 x 43 kg/m T section

21

P118: Design of Stub Girders Discuss me ...

A steel top chord (in the form of a T section)was introduced in order to overcome theneed for propping. Herring-bone reinforcing bars were passed through holes in the T-section and so avoided the use of shear connectors. The deck was Holorib, as in Test 1. Test 3:

The test details and section sizes were basically as in Test 1 except that the deck used was a trapezoidalprofile,Alphalok.This was manufactured from 1.6 mm thicksteel in orderto evaluate the possibility of using the deck to resist the compressive force developed during construction (in the absenceof props or a top chord).

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Thereinforcement in theslab consisted of standardoverlapping mesh ratherthanspecial ‘herring-bone’ bars used in Tests 1 and 2. Theshearconnectorswere welded throughthe decking in Tests 1 and 3.

Figure 10 Overall arrangement of Test 3 prior to concreting the slab

22

P118: Design of Stub Girders

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Discuss me ...

Figure 1 1 Detail at column showing jack at base to apply column moment

Figure 12 Detail at edge of stub and secondary beam attachment

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P118: Design of Stub Girders Discuss me ...

Yielding of outer stubs

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1500

With stiffened

-RI

.o W

1100

L 0,

>

c

1000

0

l-

900

800 700

600 500 400

300 200 Self Weight

100

I

0

I

I

I

I

I

I

I

I

I

I

I

I

10

20

30

40

50

60

70

80

90

1 00

110

120

Mid-span Deflection (mm)

Figure 13 l o a d deflection curve for Test 2

P118: Design of Stub Girders Discuss me ...

5.2

Test results

The load-displacement curves for the three tests were of a similar form,and the curve for Test 2 is shown in Figure 13 as beingrepresentative of all thetests.Deflections at failurewere approximately span/100.Failure correspondedtofactored a load of 12.8kN/m2fora 13.2 m x 9 m bay. All three tests showed that themid-span free bending moment was reduced by about 20% because of the negative moment developed in the columns. In Tests 1 and 2 special reinforcement was positioned in the slab to transfer this moment, but in Test 3 no such reinforcement was provided. Some continuity may have developed due to fixing of the deck to the secondary beams adjacent to the columns. Failure of the tests occurred in the following manner:

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Test 1: Shear yielding of the webs of the stubs Test 2: Tension yielding and plastic bending of the bottom chord Test 3: Longitudinal shear failure of the concrete over the outer stubs. Test 2 initially failed due to web buckling of the stubs but these were later strengthened and the test continued (see Figure 13). The failureloads and general information regarding the three tests is presented in Table l . This is taken from reference (10). The initial failure load of Test 2 is also noted. Measured steel strengths were between 400 and 420 N/mm2 and concrete strengths between 45 andS2 N/mm2.Theforces and moments resisted bythebottomchord and the concrete slab havebeen determined by back-analysis of the strains measured in the bottom chord. Table 1: Summary of failure loads and forces in stub girder tests based measurements~'o' TEST NO.

FAILURE' LOAD (kN)

MID-SPAN* MOMENT (kNm)

1

1 3083

21 97

73

3

1460

Note: 1. 2. 3. 4.

2704

85 1466

15

26

17

63

37

7 2620

68

32

24

83

83

17

27

Concrete Slab 84

10 19

1 5 244

-

Bottom Chord 20

2

Concrete Slab

DEFLECTION AT HALF FAILURE LOAD (mm1

PERCENTAGE OF MOMENT AT MID-SPAN RESISTED BY:

Composite Action

10

74

on strain

-4 2366

7

PERCENTAGE OF SHEARS IN: Bottom Chord

Failure load and moments include 120 kN for self weight plus loading assembly Mid-span moment is for the total load, reduced by moment transfer to columns No verticalstiffener on outerstub Withverticalstiffener on outerstub

The moment resisted by development of composite action in the stub girder (i.e. tension and compression in the steel and concrete respectively) was between 73 and 84% of the mid-span moment, the remaining amounts being transferred directly by the concrete slab and steel beam. The designassumptionthat all the moment is transferred by compositeaction is therefore conservative. The relative proportions of shear resisted by the slab and the bottom chord at the quarter-span points are also indicated. These varied between 63 and 85% in the bottom chord. Again, it is conservative toassume that the slab does not participate in this action. The lower figure occurred in Test 2 where the stiffening effect of the top chord is considerable, apparently because of the presence of the T section.

25

P118: Design of Stub Girders Discuss me ...

The serviceability performance of all the beams was excellent with a deflection at half the failure load ofaboutspan/500.Differentialdeflectionsacrosstheopeningsweresmall.The natural frequency of test beam 1 was measured as 13 cycleshec, well above that of conventional long span construction.

Test 2 also considered the construction stage deflectionof the non-composite section. The central deflection of the girder under the self weight of the beam andwet concrete was 23 mm. In comparison, Test 3 (without the upper chord) also deflected 23 mm. In Test 3 the decking alone appeared to just provide a factor of safety of 1.4 before buckling failure of the deckat mid span and shear failure of the shot-fired pins adjacent to the column occurred. However without further research it is not recommended that the deck is used to resist compression during the construction phase. The test results have been compared to the predicted failure loads according to four modes of failure,aspresented in Table 2. The back-analysis method used was that covered in this publication, using also measured material strengths and setting any partial safety factors to unity. Table 2: Model factors for different modes of failure in tests MODEL FACTORS’

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Test No.

Overall bending in mid-span

Vierendeel at openings2

1

0.68

0.77

0.73

0.98

1.003

0.84

1 .143

0.654

2 3 I

0.84 I

2.4 1 .173

I

Yielding of outer stubs Yielding of outer stubs Yielding of main girder

2.2 1.23

1.13

0.75 I

Mode of failure

Shear failure Longitudinal ofstubsbending failure of shear connectors

I

I

Failure of slab at outer stubs

I

Notes: 1. Model factor = Failure load / failure load determined from analysis of mode consideredand using measured properties. 2. Based on steel section alone. 3. Relevant failure case. 4. Outer stubs stiffened.

The ratio of the actual failure load to that corresponding to the individual failure modes is termed the ‘model factor’. The relevant failure mode is indicated in Table 2. Model factors greater than unity indicate satisfactory performance for the critical mode of failure. Model factors less than unity for the non-critical modes of failure indicate only that the method predicts that this mode is not critical. For example, the model factors for overall bending in mid-span indicate that the f u l l bending capacity of the section had not been developed. The analyses indicated that longitudinal shear failure of the stubs or of the concrete slab was usually the controlling factorin the tests, and that there was a significant marginof failure against overall bending and generally against Vierendeel bending. Only in Test 2 did Vierendeel bending become the dominant effect, because of the longer openings in this case. The tests demonstrated an adequate level of safety with critical model factors in the range of 1.O to l .2. Multiplying these model factors by the ratio of the measured to design yield strength of the steel used increases the model factors to 1.2 to 1.4. However, the tests also highlighted the importance of avoiding local failure mechanisms particularly with regard to the shear transfer at the stubs. Overall and Vierendeel bending effects process a sufficient margin of safety according to the proposed design method and are predictable modes of failure.

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P118: Design of Stub Girders Discuss me ...

5.3 Transverse reinforcement over stubs An importantobservationofthesetests was the large difference between the force actually transferred by the outerstubs, and the predicted longitudinal shear transfer capacity, as given by equation (12) for theamount of transverse reinforcement provided. These results are summarised in Table 3 . The predictions are based on BS 5950: Part 3, and the publication by Chien and R i t ~ h i e ( * ~Other ) . test information is given in reference (17). Test 1 was unusual, in that it had herring-bone reinforcement that dipped down around the base of the shear studs, which almost certainly explains why equation (1 1) from BS 5950: Part 3 and/or Chien and Ritchie under-predicts the trueshear transfer capacity by a considerable degree. Test 2 was also unusual in that as well as herring-bone reinforcement, the top chord helped to transfer the longitudinal shear force uniformly along the slab.In this case it may be assumed that longitudinal shear is transferred uniformly between the secondary beams. Even allowing for this effect, equation (12) appears to under-predict the observed behaviour.

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Only in Test 3 , with simple transverse reinforcement, would BS 5950 and/or the Chien and Ritchie methods appear to he directly applicable. Both methods under-predict the longitudinal shear transfer of the slab. However, calculations (not presented in this report) which included the shear resistance of the slab and reinforcement ahead (that is, on the centre-line side) of the stub, and the transverse tying actionof the secondary beam ahead of the stub, could remove most of the discrepancy shown in Table 3 between predicted and observed behaviour. The longitudinal shear capacity given by equation (12) could therefore be enhanced by including the concrete and reinforcement areas ahead of the stub up to the adjacent secondary beam. This approach is probably valid provided the length of the opening does not exceed the length of the stub. Sufficient beneficial effects are ignored to make this approach a reasonable basis of design to BS 5950: Part 3. Herring-bone reinforcement is also beneficial and this may be conservatively taken into account by increasing the 0.7 term in equation (12) to 1.0. It is thereforeconcludedthatthere is aconsiderablereserve in strictlyinterpretingthe requirements of BS 5950: Part 3 for transverse reinforcement in stub girder construction. The upper limit on the transfer of shear in equation (12) is over-conservative and could be safely omitted when the other beneficial effects are included. Table 3: Comparison of longitudinal shear forces and resistances at outer stub TEST LONGITUDINAL CONCRETE REINFORCEMENT CUBE NO. SHEAR AT OUTER STUB STRENGTH ACROSSSHEAR PLANE (N/mm2) (kN) 1

I

3

Note:

I

2654

I

56

I

REINFORCEMENT STRENGTH (N/mm2)

PERCENTAGE SHEARRESISTANCE (kN) BS 5950: PART 3

Chien Method('31

Excl. Deck Incl. Deck

3.93

45

2480

400

1536

1.16

45

2326

360 2

601

1.21

1.Test 2 benefittedfrom passing reinforcementthrough 2. Shear resistance includes partial safety factors.

280

I

771

I

2204

upper chord.

27

I

851

1030

1054

,

P118: Design of Stub Girders Discuss me ...

6 . RESUME OF DESIGN OF STUB GIRDERS 6.1 Recentprojects A number of projects using stub girder construction have examples of these are:

been carried out in the UK. Three

Sceptre Court, Tower Hill, London: consultant Pel1 Frischmann and Partners. ~O,OOOm2 total floor area - 7 storeys. Stub girder spans of 10 to 15 m, propped during construction. Beaufort House, Petticoat Lane, London: consultant Waterman Partnership. 50,000 m’ total floor area - 12 storeys. Stub girder grillage of 27 m span, propped during construction. Other smaller span stub girders used the plate diaphragm solution to obtain greater opening depths. Dominant House, Queen Victoria Street, London: consultant Waterman Partnership. 20,000 m2 total floor area - 8 storeys. Stub girders as secondary beams spanning 12 m onto primary beams spanning 6 m.

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Many other examples of stub girder design exist in the USA and Canada. Early test work was carried out by Colaco(I8)in 1972, and for consultants Ellisor and Tanner prior to the construction of the First International Building.in Dallas (176,000 m’ floor area - 51 storeys).

6.2 Scheme design of stub girders The proportions of stub girders may be estimated using the following member sizes in grade 50 steel. This guidance may be used at the Scheme Design Stage, and will broadly ensure that a given design is adequate for strength and serviceability performance. It isnot a substitute for precisedesigncalculations.Theguidance is presented for arange of typical gridsizes and conforms to the configuration presented in Figure l(a). The specified imposed load is 5 kNln? (including partitions). The slab span is 3 m and its depth is 130 mm using a standard composite deck.

Grid Size 12 m x 9 m

Secondary beams (9 m span); Bottom chord (stub girder); Overall depth of stub girder; Openings (350 mm deep);

Grid Size 15

m x 12 m

Secondary beams (12 m span); Bottom chord (stub girder); Overall depth of stub girder; Openings (455 mm deep);

28

356 X 127 X 39 kg/m UB 254 X 254 X 132kg/m UC 750 mm deep 4 No. - 500 mm wide 4 No. - 750 mm wide

457 X 152 X 60 kg/m UB 305 X 305 X 198 kg/m C 925 mm deep 4 No. - 400 mm wide 4 No. - 700 mm wide 2 No. - 1000 mm wide

P118: Design of Stub Girders Discuss me ...

Grid Size 18 m x 9 m

Secondary beams (9 m span); Bottom chord (stub girder); Overall depth of stub girder; Openings (400 mm deep);

140 X 46 kg/mUB(Over-design) 406 x 287 kg/m UC 950 mm deep 4 No. - 500 mm wide 4 No. - 600 mm wide 4 No. - 750 mm wide 406 356

X

X

All the above cases require temporary propping at mid-span during construction, forboth strength and deflection reasons. The stubs are the same size as the secondary beams and would normally be stiffened at their ends. Forlongerspans appropriate.

it is proposedthat

the configurations in Figure l@) and l(c)aremore

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6.3 Design example for stub girder A fully workedexampleofastubgirder based ona 15 m X 12mgrid is presented in the Appendix. The design has been carried out in accordance with BS 5950: Part 3 and to a specified imposed load of 5 kN/m2. In thisexample,thedesign is controlled by the combination of bending, shear and tension in the bottom chord adjacent to the most highly stressed opening, and by deflection criteria. The selected members (all grade 50 steel) were as follows: Secondary beams (12 m span); Bottom chord (stub girder); Top chord;

457 x 152 x 60 kg/m UB

305 X 305 x 198kg/m UC T section 102 X 203 X 46 kg/m cut from UC

Shearconnectorsare placed in pairs at 95 mm longitudinalspacing which determines the minimum length of stubs that are needed. Additional transverse reinforcement in the form of 16 mm diameter bars at 95 mm spacing is needed in the slab above the outer stubs.

29

P118: Design of Stub Girders Discuss me ...

7 . DESIGN PROCEDURE FOR STUB GIRDERS The design procedure is set out in three stages representing broadly the sequence used in design. It is assumed that the proportions of the stub girders aretypical of good practice (refer to Scheme Design in Section 6.2). Design should conform to the requirements of BS 5950: Parts 1 and 3, where relevant.

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7.1 Construction condition 1.

Determinethe moment diagramunderfactoredloadsduringconstruction,allowing 0.5 kN/m2 for imposed construction loads.

2.

Select the size of the bottom chord. Determine its moment capacity, M,. Compare with the applied moment in 1. If not adequate (as is generally the case), use temporary props to reduce the moment, or alternatively use a steel top chord. If props are used, compare the moment capacity with the reduced moment.

3.

If a steel top chord is used, determine the compression in the chord when subject to the maximummoment used in 1. Check its adequacy asastrut.Furtherchecksonthe bottom chord are included in Section 7.2.

4.

Calculate the deflection of the stub girderafter construction (ignoring construction loads) based on the steel section (including the top chord if used).

7.2 Ultimate loads 1.

Determine the moment and shear force diagrams at ultimate loads (factored self weight, dead and imposed loads).

2.

For the selected bottom chord, calculate its moment and shear capacity (see construction stage). Check that the shear capacity exceeds the maximum shear force, V. As afirst approximation,the maximum length of theopening L, is suchthat M , > V L, (see Section 4.3.2) and L, / D,, I 10 (see Section 4.5) where D,, is the average slab depth.

3.

Calculate the moment capacity of the composite stub girder section assuming f u l l shear connection (see Section 4.1 equations (3) or (4)). Check that the moment capacity of the stub girder exceeds the maximum moment determined in 1.

4.

Calculate the number of shear connectors needed in the half span for full shear connection (seeSection 4.2). Distribute in accordance with shearforcediagram.Calculatethe number of shearconnectorsthat can be placed along thestubs at the minimum recommended spacing. If this is not greater than the required number, consider partial minimum degree of shear shearconnectiondesign (see 5 below).Determinethe connection for the span of the girder.

5.

For the required degree of shear connection, determine the moment capacity of the stub girder (see Section 4.3 equations (5) or (6)). Check that the moment capacity exceeds the maximum moment determined in 1. If adequate proceed. If not, increase the depth of girder or the sizeof the bottom chord.

6.

Check the adequacy of the stub girder at allDoints along the span. This is based on a sensibledistribution of shearconnectors as in Step 4. Useequations (5) or (6) to determine the moment capacity, inserting the appropriate value of R, at the point under

30

P118: Design of Stub Girders Discuss me ...

consideration.Criticalcross-sectionsare at thehighermomentside of theopenings. Ensure that the moment capacity exceeds the applied moment at all points in the span. 7.

For the selected opening widths, determine the Vierendeel bending moment (see 2 above). Modify the moment capacity of the steel section according to the tension and shear forces existing at the low moment side of the openings (see Section 4.3.2 and 4.3.3 equations (8) and (9)). Re-check the ability of the steel section to resist Vierendeel bending. If not adequate, reduce opening width. Check all openings.

8.

Check the local design of the stub (see Section 4.4equation (1 1)). Use vertical stiffeners for the outer stubs. Increase the length of the stub to reduce theweb stresses or to attach more shear connectors.

9.

Checktheadequacy of transversereinforcement in slab(seeSection

10.

Determine the weld size or number of bolts between the stubs and the bottom chord.

4.6 equation (12)).

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7.3 Serviceability 1.

For theproportions of semi-permanent and permanent imposed loads,determinethe appropriate modular ratio to beused in deflection calculations. Note that no stress checks are made at serviceability as it is assumedthat any local yieldinghaslittleeffecton deflections.

2.

Calculatethesecondmoment of area of thecomposite section(seeSection (13)). Calculate the central bending deflection under imposed load.

3.

Calculatethe‘shear’deflectionduetoVierendeelbendingeffects(fromequation(14)). Sum the bending and shear deflections. Check that the shear deflection acrossan opening does not exceed L, / 1000. Check that the total imposed load deflection does not exceed span /360 or 40 mm (sensible maximum).

4.

Add selfweight and imposed load deflections(forselfweightdeflection,seeconstruction condition step 4). Limit the total deflection to span /200 or 70 mm (sensible maximum). Consider pre-cambering the bottom chord.

5.

Determinethe naturalfrequency of stubgirder(fromequation(17)). If thisexceeds 4 cycles/sec,nofurthercalculationsarerequired. If not, but thenaturalfrequency exceeds a minimum of 3 cycles/sec,furthercalculationsarerequiredtoestablishthe response of the floor(15).

4.8 equation

31

P118: Design of Stub Girders Discuss me ...

8. CONCLUSIONS Stubgirdersareappropriatefor rectangulargrillagesystems with spans of 12 to 20 m and secondary beam spans of 8 to 12 m. Various configurations are possible and the use of an upper chord can in some cases overcome the need for temporary propping. The design proceduregiven in this publication has been verified by analysing the results of three oftenoccurs local to theouterstubs, such as by full scaletests on stub girders.Failure compression or shear failure of the stubs, or longitudinal shear failure of the slabadjacent to the stubs, This emphasises the need to consider local stiffening of the stubs and additional transverse reinforcement in the slab around the shear connectors. In other cases, combined bending and tension on the bottom chord controls the design.

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The requirements for longitudinal shear transfer and for the provisionof transverse reinforcement are over-conservative when strictly interpreting BS 5950: Part 3. In particular the upper limit onlongitudinalshearstress can be safely omitted and theeffectivezone of sheartransfer increased. This is often the critical case for closely spaced shear connectors.

32

P118: Design of Stub Girders Discuss me ...

REFERENCES

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1.

BRITISH STANDARDS INSTITUTION BS 5950: Structural use of steelwork in building Part l : Code of practicefordesign in simple and continuousconstruction:hot sections BSI,1990

2.

BRITISH STANDARDS INSTITUTION BS 5950: Structural use of steelwork in building Part 3: Design in composite construction BSI,1990

3.

LAWSON, R.M. Design for openings in webs of composite beams Steel Construction Institute/CIRIA, 1987

4.

KNOWLES, P.R. Design of castellated beams Steel Construction Institute, 1986

5.

WARD, J.K. Design of composite and non-composite cellular beams Steel Construction Institute, 1990

6.

OWENS, G.W. Design of fabricated composite beams in buildings Steel Construction Institute, 1989

7.

Design of composite trusses Steel Construction Institute, 1992

8.

LAWSON, R.M. & RACKHAM, J.W. Design of haunched composite beams in buildings Steel Construction Institute, 1989

9.

BRETT, P. & RUSHTON, J. Parallel beam approach - a design guide Steel Construction Institute, 1990

rolled

10. MADROS, M.S.Z.B. & MCCONNEL, R.E. Experimental investigation of a stub girder floor system University of Cambridge reports CUED/D-Struct/TR 124 to 127, 1989 11.

MADROS, M.S.Z.B. The structural behaviour of composite stub girder floor systems PhD dissertation, University of Cambridge, 1989

12.

BRITISH STANDARDS INSTITUTION BS 81 10: Structural use of concrete Part l : Code of practice for design and construction BSI,1985

33

P118: Design of Stub Girders Discuss me ...

13.CHIEN,E.Y.L. & RITCHIE,J.K. Design and construction of composite floor system Canadian Institute of Steel Construction, 1984 14. LAWSON, R.M. Design of composite slabs and beams with steel decking Steel Construction Institute, 1989 15. WYATT, T.A. Design guide on the vibration of floors Steel Construction Institute in association with CIRIA, 1989

16. LAWSON, R.M. Commentary on BS 5950: Part 3: Section 3.1, ‘Composite beams’ Steel Construction Institute, 1990

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17.GOSSELIN,G.C. & HOSAIN,M.U. Failure of stub girders due to longitudinal shear IABSE Proceedings P-74/84, May 1984 18. COLACO, J.P. A stub girder system for high rise buildings American Institute of Steel Construction National Engineering Conference July 1972

34

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Stub Girders

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ANALYSIS OF STUB GIRDER ACCORDING TO DESIGN METHOD 15 m SPAN STUB GIRDER WITH 12 m SPAN SECONDARY BEAMS

Column A

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A

Secondary beams span 12 m

L

Deck span

7

Stub girder on A - A

B

.*

15 m (secondary beams @ 3 m c m . )

PLAN

Temporary prop at mid-span

Bottom chord

SECTION A-A

I

Not to Scale

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JWR

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Date

June 92

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DATA - GENERAL Grid size

12 m x 15 m

Slab depth

fire 130 mm (90 min.

Deck

Re-entrant shape (50 mm deep X l mm thick)

Concrete

Lightweight, grade 30

cfcu

Steel

Grade throughout 50

by = 345 N/mm2 f o r t >

resistance)

= 30 N/mm2)

Consider two construction cases: a)

temporary props at 7.5m from supports

b)no

temporary props, but additional T section as top chord

Bottom chord

Choose 305 UC section

Secondary beams

Choose 457 UB section

Columns

Choose 305 UC section

Maximum opening sizes:

1000 mm x 455 mm (2No.)

700 mm x 455 mm (4No.)

400 mm

X

455 mm (4No.)

Total construction depth = 925 mm (spadl6.2).

36

1 4 mm)

P118: Design of Stub Girders Discuss me ...

Design Example

Silwood Park Ascot Berks SL5 7QN Made Telephone:(0344) 23345 Fax:(0344) 22944

Client

by

Date

Apr 92

RML Checked by

CALCULATION SHEET

Date

JWR

June 92

-

DATA LOADING

Imposed loading

4 kN/m2

Partitions

l kN/m2

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Ceiling, Services, etc.Floor Raised

0.7 kN/m2

Sew Weight of Slab and Deck

2.3 kN/m2

Self Weight of Beams

0.4 kN/m2 (assumed)

Construction Load

0.5 kN/m2

(Imposed)

No reduction in imposed load with loaded area is made in this example. However, effect may be taken into account leading to a 14% reduction in imposed loads.

this

Factored Load During Construction WC

=

1.4

-

4.6 kN/m2

X

(2.3

+ 0.4) + 1.6 X

0.5

Factored Load In-service W,

+ 0.4) + 1.4 X

=

1.4 (2.3

-

12.8 kN/m2

0.7

+ 1.6 (1.0 + 4.0)

Point Load due to Secondary Beams p,

-

W,

x 12 x 3

Mid-span Moment:

Mu

-

2 P,,

-

9 P, kNm

X

7.5

-

P,,

X

4.5 - P,

X

1.5

diugram (see

on sheet 11)

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CALCULATION SHEET

Date

A m 92

RML Date

JWR June

92

DESIGN OF SECONDARY BEAMS The secondary beams (except along the column line AB) may be designed as continuous beams. However, a member length of 24 m may be excessive. Therefore, design as simply-supported. Tryspan: depth = 20 (where depth = beam + slab). The beams size may be just#ied by reference to the SCI publication ‘Design of composite slabs and beams with steel decking’(14). For this span and load, choose: 60 kg/m UB grade 50 steel with shear connectors in every trough.

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457

X

152

X

A reduction in beam weight to 52 kg/m is justified if the beams not on the column line are designed as continuous. Choose 457 X 152 X 60 kg/m UB as stubs (i.e. same section as f o r secondary beams).

38

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Stub Girders Subiect

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Design Example

Client

Date Made by

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CALCULATION SHEET

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CONSTRUCTION CONDITION a)

temporary prop at mid-span (i.e. 7.5 m from support) Design bottom chord conserv&’vely as propped cantilever

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Moment

-

W,

=

4.6 x 12

-

388.1 kNm

L2/8 X

7.52/8

Choose 305 x 305 UC X 198 kg/m gmde 50 which is a ‘plastic’ section. Moment capacity (using a design strength of steel of 345 N/mm2 (t > 16 mm)) is: MS

b)

Id

=

3440

-

1186.8 kNm >

X

X

345

X

IOd

388.1 kNm

OK

no temporary props, but use T section as top chord Moment

-

9Pu

-

1490 kNm

=

9 x 4.6 x 3 x 12

(see sheet

3)

But moment capacity of bottom chord = 1186.8 kNm Consider T section of area A, = 55 cm2 = area of web of bottom chord (A, = 54 cm2). plastic neutral axis lies in top flange.

Moment capacity = A, py D,

+ A py D/2

where D, is the height of the T section above the top jlunge.

39

P118: Design of Stub Girders Discuss me ...

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--The ---. Steel Construction a _Institute

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Job No. Sheet

060f 24

Job Title

Stub Girders Subject

Silwood Park Ascot Berks SL5 7QN Telephone:(0344) 23345 Fax:(0344) 22944

Rev.

PUB 118

Design Example M a d e by

Client

Date

Apr 92

RML Checked by

CALCULATION SHEET

Date

JWR June

92

Choose T cut from 203 X 203 X 86 kg/m UC grade 50 so that upstand is less than height of slab minus top cover (in this case the T section is 111 mm high and 43 kg/m weight). Dr

2 :

455

MC

=

55

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-

+ l 0 (allowance f o r heightofcentroidof

X

102

X

882

465

X

345

i1478 =

X

T section) = 465 mm

10-~

2360 k N m > 1490 k N m

This moment capacity exceeds the applied moment by a significant margin. As the bottom chord is assumed to resist the total shear and ‘Vierendeel’ bending effects under factored in-service conditions, no further checks need be done at the construction stage. See sheet 22 f o r deflection check.

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P118: Design of Stub Girders Discuss me ...

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The mmSteel Construction -Institute -

-PIT

~

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0 7 o f 24

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Job Title

Stub Girders Subiect

~~

Design Example

Silwood Park Ascot Berks SL5 7QN Telephone:(0344) 23345 Fax:(0344) 22944

Client

Made by

Date

Apr 92

RML Checked by

CALCULATION SHEET

Date

JWR

June 92

ULTIMATE CONDITION Point load due to secondary beams PU

-

36

X

12.8 = 461 kN

Mid-span moment

MU

-

9 P,,

(see sheet 3)

4147 kNm

=

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Moment capacity of bottom chord MS = 1186.8 kNm Compressive resistance of concrete slab (ignoring contribution of T section, conservatively). Effective breadth of slab =

L/4 z 0.8b

=

1500/4 = 3750

RC

=

0.45fcu B,

DZ lV

=

equivalent depth of solidslab f o r re-entrant deck with ribs parallel to beam = 120 mm

Be

2

12000

X

0.8

Dav

Tensile resistance of bottom chord

=

252

X

102

X

345

X

10-~

41

P118: Design of Stub Girders Discuss me ...

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08 of 24

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Stub Girders Subject

Design Example

Silwood Park Ascot Berks SL5 7QN Telephone:(0344) 23345 Fax:(O344) 22944

I

Client

I Date

Made bv

RML

I

Apr 92

Checked bv

CALCULATION SHEET

Check moment capacity of stub girder assumingfull shear connection. Tensile resistance of web of bottom chord R,

= -

-

277 X 19.2 1835 kN

X

345

X

As R, > R, > R , plastic neutral axis lies in top flange of bottom chord

For simplicity, the plastic neutral axis is taken to be at the top of the top flangeof the bottom chord.

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Moment capacity, (about flange)

M, = R, D,

+ Rs D/2

where D, = height of centre of concrete slab above the top flange of bottom chord = 455 + 130/2 mm

MC

+ 130/2) X

=

6075

+

8694 x 340/2 x

-

3159.0

=

4637.0 k N m > 4147 k N m

X

(455

lop3

+ 1478.0

This is acceptable. An additional conserv&’ve factor is that the span of the stub girder has been taken as fromthe centre of the columns (rather than the face). Shear connection Force to be transferred f o r full shear connection = R, = 6075 k N < R, Design capacity of 19 mm dia shear connectors ‘d

-

0.8 x 0.9 x l00 72 kN

No reduction f o r deck shape in this case.

42

X

95 mm high in LWC (grade 30):

1

P118: Design of Stub Girders Discuss me ...

Design Example

Silwood Park Ascot Berks SL5 7QN Made 23345 Telephone:(0344) Fax:(0344) 22944

.

Client

by

Date

Apr 92

RML Checked by

Date

JWR

CALCULATION SHEET

June 92

Number of shear connectors required in '/z span Nsc

6075/72 = 84.4, say 85

=

Minimum spacing of shear connectors = 54 (= 95 mm) longitudinally and 44 laterally in pairs. Therefore, overall length of stubs needed

-

84 x 5 x 1912 = 3990 mm

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Length available

=

2050

-+ 1600 = 3650 mm

<

3990 mm

(ignoring middle stub) Consider partial shear connection: Minimum degree of shear connection =

(L

-

6)/10

= (15

-

6)/10 = 0.9

Reduce number of shear connectors to 84.4 x 0.9 = 76 Overall length of stubs needed -

75 x 5 x 19/2

= 3563 mm

< 3650 mm

OK

Check moment capacity of stub girder for partial shear connection: Longitudinal force transferred R,

-

Resistance of web

76 x 72

R,,,

= 5472 kN

= 1835 kN

< R,

43

P118: Design of Stub Girders Discuss me ...

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Job No.

Job Title

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1 o o f 24

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Rev.

Stub Girders

Examole Desim " Made by

Client

Telephonc:(0344) '23345 Fax:(0344) 22944

I

PUB 118

Date

RML

I

Checked by

CALCULATION SHEET

Aor 92

I

JWR

Date

June 92

Plastic neutral axis lies in top flange of bottom chord MC

=

+ Rs D/2 5472 (455 + 13012) X

-

2845.4

-

4323 k N m > 4147 k N m

-

R, DC

l 0-3

+ 8694 X

340/2

X

1O-j

+ 1478.0 OK

Therefore partial shear connection is acceptable.

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Check shear capacity of bottom .chord: Maximum shear force at support V

-

2 P,

=2

X

461

= 922 kN

Shear capacity of web VU

V

=

0.6 p-v A,,

-

0.6 x 345 x 19.2

-

1357 kN

X

340 x 1O-j

< Vu at the supports, which is acceptable.

However as V > 0.6 V,, it is necessary to consider the influence of combined moment and shear adjacent to the first stub (see sheet13).

44

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Job No.

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The ---

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Job Title

Stub Girders Subiect

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Client

by

Made

RML Date

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by Checked

CALCULATION SHEET

JuneJWR

92

CHECK MOMENT CAPACITY AT OTHER CRITICAL CROSS-SECTIONS

-

B

A

4 2pu I

/_07

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Check moment capacity of point A (at high moment side of opening). Moment

=

2 P,, (3.0

+ 0.7) - P,,

where P,,

-

12.8

X

-

2

=

3088.7 kNm

MA

X

X

3

461

X

X

0.7

12 = 461 kN 3.7 - 461

X

0.7

Shear resistance of web of bottom chord VU

but

V,

-

1357 kN

-

P,, = 461 kN

V/V, =

461/1357

= 0.34

< 0.6, so no reduction to web f o r influence of shear. Moment capacity due to composite action at point A: Assume 76 shear connectors are distributed in pairs along the stub girder at the minimum spacing of 95 mm. Number of shear connectors over jirst stub of length 2050 mm NI

=

2050/(95/2)

= 44

Longitudinal shear force transfer R4

-

44 x 72

= 3168 kN

> R,,,

45

P118: Design of Stub Girders Discuss me ...

--

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Job No.

-The m = Steel Construction ---=----t --Institute -

Job Title

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Client

PUB 118

Sheet

12 of 24

Stub Girders Subiect

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CALCULATION SHEET

Apr 92

Date

June 92

Moment capacity at point A MC

+ R, D/2

-

R, D,

=

31 68 (455

-

3125.4 k N m

+ 130/2) X

l OV3

+ 1478.0

This just exceeds theapplied moment of 3088.7 kNm.

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Moment capacity at point B is as calculated previously

= 4323.4 k N m

This exceeds the applied moment. Number of shear connectors over second stub N2

-

76 - 44

= 32

Nominal spacing of shear connectors over central stub, singly at 95 mm spacing N3

-

= 10

1000/95

Total number of shear connectors in span -

46

76

X

2

+ 10

= 162

P118: Design of Stub Girders Discuss me ...

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I

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I

PUB 118

= e = e-B = ----

I Sheet

13 of 24

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Job Title

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Design Example

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Client

by

Date Made

RML Apr Checked by

CALCULATION SHEET

92

Date

JWR

June 92

VIERENDEEL MOMENT CAPACITY OF BOITOM CHORD

2pu

1.0

4

Adjacent to support at point C:

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Vertical shear force adjacent to first stub V

-

2 P,, = 2

X

461

= 922 kN

Vierendeel moment at point C

MV

-

922

X

0.55 = 507.l kNm

Adjacent to secondary beam at point D: Vierendeel moment transfer across opening adjacent to first stub, assuming point of contraflexure at location of secondary beam

MV

-

922

X

Moment capacity of steelsection but

V/V, =

= 368.8 kNm

0.4

M,

922/1351

= 1186.8 kNm

= 0.68 > 0.6

By inspection, even ignoring web effectiveness in bending

M,

> > M vatpoint C

Check point D: Tensile force in bottom chord at point D =

T

-

V/V,

=

force transferred by shear connectors over first stub R,

= 3168 kN

(see sheet 11)

0.68, as above 47

P118: Design of Stub Girders Discuss me ...

The SteelConstruction Institute

-----E S = -

---

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Job No.

Rev.

PUB 118

I

Sheet

140f 24

Job Title

Stub Girders Subject Client

Design Example by

Made

92

AprRML Date

by Checked

92

J u nJeW R

CALCULATION SHEET

Effective thickness of web reduced due to influenceof shear

=

19.2 4 1 - 0.6S2 = 14.0 mm

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Effective properties of section

A,ff

=

25200 - (19.2 - 14.0) (340

Ms,eLf

=

M, - (19.2 - 14.0)

-

1152.3 k N m

X

(340

-

2 x 31.4) = 23759 mm2

2

X

31.4)2

X

10d/4

Reduced moment capacity of steel section under the influence of shear and tension

1 - 3168 x Id 23759 x 345 l

-

0.39

0.61 x 1152.3

red

= 0.61 = 707 k N m

> Mv at point D

Check point A: V M V

-

pu

-

461

= 461 kN X

0.7

= 322.7 k N m

As the section is not reduced by shear, it is adequate, by inspection.

48

P118: Design of Stub Girders Discuss me ...

The Steel Construction Institute

I

’--= --

Sheet

15 of 24

I

Subiect

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Client

by

Made

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CALCULATION SHEET

Apr 92 Date

.lune 92

Check point E: Tensile force in bottom chord at point E

T

=

Total force imnsferred by shear connectors across jirst and second stubs

-

76 x 72

= 5472 kN

No reduction f o r shear in this case -

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Ms, red

l - -5472

= 0.37

8694

MS red

MV

-

0.37 x 1186.8

-

461 x 0.7

= 439.1 kNm

= ,322.7 kNm

< 439.1 kNm

OK

Check point B: For openings in ‘zero shear’ zone, assume a ,shearforce of 9i x maximum shear representing the effect of unequal loading on adjacent spans. V

=

2 x 461/4

= 230.5 k:N

Vierendeel moment transfer MV

But

-

MS,red =

230.5 x 1.0 = 230.5 kNm 439.1 kNm (as f o r point E) > Mv

Therefore the width of all the openings is acceptable. The width of the openings could be increased, but it is not possible to locate a#llthe required shear connectors over the stubs.

49

P118: Design of Stub Girders Discuss me ...

1

I

1

Job No.

Rev.

Design Example

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bv

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I

CALCULATION SHEET

1

Made

Checked by

RML

I

JWR

I

Date

Apr 92 Date

June 92

Check conditions at stubs 457 x 152 x 60 kg/m

\

Reaction at stiffened end

,3168 kN

=

3168

X

0.45Y2.05

703 kN

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Shear stress in web of stub =

31 68

-

193 N/mm2

Limiting shear

-

X

l d/(2050

X

8.0)

= 0.6py

stress

0.6

X

355

= 213 justN/mm2

OK

Design web stgfeners for reaction of 703 kN Use 20 mm thick X 150 mm wide stubs by 8 mm fillet welds.

X

=

Compressive stress

455 mm deep plates welded to the ends of the

703

X

ld/(20

X

150)

= 234 N/mm2

The stiffener should be checked as in BS 5950: Part l , but the above size is acceptable. At second stub, shear stress in web

50

=

2304

-

l80 N/mm2

X

ld/(8.0

X

1600)

< 213 N/mm2

OK

P118: Design of Stub Girders Discuss me ...

Design Example

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Checked by

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92

Vettical reaction with no stiffeners

-

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

-

2304 x Id x 455 (l60&/6) 2457 N/mm

Vertical stress

-

2457B.O

Combined stress

=

(3072

=

437 N/mm2 > 355 N/mm2

+3

= 307 N/mm2 X

Therefore use vertical stiffeners as previously. However, it may have been possible to justify the use of a heavier stub (e.g. 457 X 152 x 82 kg/m) although this section is actually 10 mm deeper than the chosen section. Force transfer between the stubs and the bottom chord Longitudinal shear transfer at outer stub = 3,168 kN (see sheet 11) Use 10 mm filletwelds along flange tips. Assume longitudinal shear is resisted by these fillet welds and vertical reactions are resisted by vertical shyfener weMs.

WeM strength

-

l0 X 255 X 10-3 -

6

= -1.8 kN/mm

Id

51

P118: Design of Stub Girders Discuss me ...

The Steel Construction Institute

----

I

118PUB Job Title

E --

---

Sheet

18 of 24

I

Stub Girders Subject

Design Example

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Made by

Client

Date

Apr 92

RML Date

Checked by

.lune 92

.IWR

CALCULATION SHEET

Longitudinal shear resistance 2

=

X

1.8

2050

X

= 7380 kN

Vertical reaction at stiffener Weldresistancebetween

> 3168 kN

= 703 kN (see page 16) = 1.8 x 150

stiffener and bottom chord

= 270 kN

< 703 kN

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

As this isinadequate, provide full penetration butt weld between the stiffener and bottom chord (see below). CriDplinn o f web of bottom chord under stub stiffener

Stub

4 L 2 0 13.3

31.4

+ Bottom chord

t

Bearing length on web of bottom chord (conservatively) *b

=

20

+ 13.3

-

Bearing stress

X

2.5

+ 3.4

X

704 x I d 210 x 19.2

2

X

2.5

= 210 mm

= 175 N/mm2

< py

Slenderness f o r buckling check

x

=

2.5 d/t = 2.5

X

246119.2

= 32

By inspection, web is OK.

An addilional check on localmnge bending due to the tensile force transferred from the stiffener may be necessary. It is not critical in this case.

52

P118: Design of Stub Girders Discuss me ...

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Rev.

Job No.

PUB 118

= -m -

Sheet

19 of 24

---

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CALCULATION SHEET

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JWR

June 92

DEFLECTlONS No checks are made on serviceability stresses in stub girders because a small degree of plasticity is not considered to affect deflection!r significantly.

Calculate imposed load deflections to concrete, cye := 15

Second moment ratio of steel

T h Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

I

I

D ef f

I

=

(130 - 50) x 3750/15 = 20000 mm2

AAc/me

=

25200 mm2

4

=

508

-

130/2

l

C I

-

X

lo6 mm4

+ 455 + 340/2 = 690 mm

Ac . A / I s + D e 2. (Ye

(20

=

508

=

(508 -k 5309)

=

5817 x 106mm4

X

lo6

+

X

X 25.2 X l @ ) x 6902 (20 + 25.2)

lob

Serviceability loading on beam W

=

(4

+ l) x

12

= (60 kN/m

Deflection due to pure bending

-

- 5x 384

60 x 1s' x 109 205 x 581 7 x lo6

53

P118: Design of Stub Girders Discuss me ...

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JOV

I

IYO.

PUB 118

Sheet

20 of 24

I

nuv.

Job Title

Stub Girders Subject

Design Example Client

Date Made by

Apr 92

RML Checked by

CALCULATION SHEET

Date

June 92

JWR

Deflection due t:, Vierendeel bending per opening in half span V

=

shear force at opening

Lo

=

length of opening (to point of contraflexure)

~

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

At first opening LO

6,

-

400 mm

-

-l x 3

6, /L*=

For second L O

6"

V = 2 P

=

2

X

360 x 40d 200 x 508 x lo6

0.2

X

=

3

X

(4

+ l)

= 360 kN

= 0.08 mm

180 kN

-

700 mm

-

-l x 3

180 x 7 0 d 200 x 508 x lo6

0.3

1 0 - ~ OK

X

X

1 0 - ~ OK

opening V = P

6, / L o =

12

= 0.21 mm

Total deflection due to Vierendeel bending over all openings = 0.6 mm Total deflection due to imposed loading

54

+ 0.6

-

33.1

=

span / 445

= 33.7mm

OK

P118: Design of Stub Girders Discuss me ...

Job No.

The Steel Construction Institute

Rev.

PUBI 118

21 of 24

Sheet

---

I

Silwood Park Ascot Berks SL5 7QN

Design Examole

I I1 - " RML ADr 92 I Client

Telephone:(0344) 23345 Fax:(0344) 22944

Made by

Date

I

I

CALCULATION SHEET

I

Checked bv

JWR

I

Date

June 92

TRANSVERSE REINFORCEMENT

Maximum longitudinal shear transfer at outer stubs due to studs in p a i n at 56 spacing =

2 x 72 x

Id

/(5 x 19)

= 1516 kN/m

Shear stress on two shear planes through minimum slab depth V

=

1516/(2

X

(130 - 50))

= 9.47 N/mm2

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Shear resistance per unit length VC

=

0.03

X

0.8&, (D, - D$

+ 0.7 A,,f, + vP

Use T16 bars at 95 mm spacing over stub and A142 mesh. Place bars below head of studs Note: Shear transfer is assumed to occur directly over the stub, which is conservative AS" VC

-

lr x 162 x l000 + 14f2 = 2258 mm2 -

4

95

=

0.03 x 0.8

-

57.6

=

784.7 N/mm (or kN/m)

For 2 shear planes

VC

X

30 (130 -50)

+ 0.7

X

2258

X

460 x l 0-3

+ vp

+ 727.1 + vp -

1569.4 N/mm > 1516 N/mm

This is satisfactory, even ignoring the contribution of the decking, vp. Curtail half of the bars at 1.0 m from the beam, and the remainder at 2 m. Note that the upper limiton shear transfer in equation (12) has been ignored.

55

P118: Design of Stub Girders Discuss me ...

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I

Job No.

I

PUB 118

I Rev. Sheet

22

of

24

I

Job Title

Stub Girders ~

~~

Subiect

Design Example

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I

Client

I

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Made bv '

RML

Checked by

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Date

Apr 92 Date

JWR

June 92

On removal of props re-apply self weight to composite section. Take term loads

a, = 25 for long

CALCULATION SHEET

DEFLECTION DUE TO SELF WEIGHT

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

a)

Propped beam

A, /ae =

(130

50)

X

4

=

508 x lob

i-

=

(508 -k 3870)

=

4378

-

X

= 12000 mm2

3750/25 (12

X

25.2 X Id) x 6902 (12 + 25.2) X

l&

lob mm4

Deflection on removal of props: Self weight loading

=

-

2.7 x 12 - 5x

384

= 32.4 kN/m

32.4 x 1 9 x l @ 205 x 4378 x lob

Vierendeel deflections are small (allow 0.7 mm) Additional dead load deflections applied to composite section are due to a load of 0.7 kN/m2 (ceiling i- services) Total deflection

56

=

23.8 i- 5.715.0

=

62.8 mm (span/239). Thisisjust

X

34.2 acceptable.

P118: Design of Stub Girders Discuss me ...

I B _

_ _ -_

-

Sheet

23 of 24

I

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Des[gn Example June

CALCULATION SHEET

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92

with top chord =

Area of top chord

I

Job Title

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b) Unpropped beam

1

I

Job No.

PUB: 118

The Steel Construction Institute

5500 mm2

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ModiJied second moment of area of bottom an4d top chords is:

%W

=

(508

+ 2149) X

=

2657

X

-

23.8 x 4378/2657

Total deflection

lob mm2

of beam =

lo6

(span/l90) 78.9 mm

+ 0.7 = 39.9 mm = 39.9 + 5.7/5.0 X 34.2 = 39.2

-

acceptable not

However, the top chord will reduce imposed load deflections by about 5%. This calculation nevertheless shows that the total deflection of stub girders is ojlen the limiting factor. The chords would need to be significantly heavier in order to reduce deflections due to self weight.

57

P118: Design of Stub Girders Discuss me ...

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1

Job No.

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PUB 118

Sheet

24of 24

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Rev.

Job Title

Stub Girders Subject

Design Bample

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Date

Made by

Client

Apr 92

RML Checked bv

CALCULATION SHEET

Date

JWR [

June 92

NATURAL FREOUENCY Self weight + dead load (excluding partitions)

+ 0.1

X

-

2.7

+ 0.7 + 0.1 X

=

3.8

X

imposed load 4.0

= 3.8 kN/m2

12 = 45.6 kN/m

Created on 30 March 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Deflection due to these ‘permanent’ loads applied to the composite section (allowing 10% increase in stiffness for dynamic effect)

%W

-

45.6 - 5x 384 205

x 1 9 x 109 x o.9 x 581 7 x lo6

22.7 mm

Natural frequency of stub girder

= 18/6,,0*5

This is greater than the absolute minimum value of 3 cycledsec. Full analysis of the response of the floor may be carried out in accordance with reference (15), but the vibration response will be acceptable f o r normal office usage, given the large area (and hence, mass) of the floor that wouM need to respond to any impulsive action. CONCLUSION The design is limited by serviceability criteria and the minimum spacing of the shear connectors on the stubs.

58

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