Design of stub girders

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Author: Robert Mark Lawson | R. E. McConnel | Steel Construction Institute (Great Britain)

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P118: Design of Stub Girders Discuss me ...

SCI PUBLICATION 1 18

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

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


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:


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.

..

11


Page


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

...

111


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


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.

iv


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"


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.

V


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.


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).

1


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


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

2


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.


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.

3


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.


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.

4


Shear connectors ,Secondary beams

l Bottom chord /

\ I

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

DEPTH : NO TOP CHORD


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

5


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.


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.

6


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.


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.

7


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:


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.

8


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,.


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.

9


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.


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.

10


Axial force in slab T + NPd

v

)M+"Lb

T + NPd

Tensile force in bottom chord

Moment due to tension in bottom chord


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.


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)


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


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,


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

13



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.

14



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.

15


S tub


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


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


(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

17


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.


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).


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 ,
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


T

I

Job No.

I

PUB 118

I Rev. Sheet

22

of

24

I

Job Title

Stub Girders ~

~~

Subiect

Design Example

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

I

Client

I

I

Made bv '

RML

Checked by

I

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


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.


I B _

_ _ -_

-

Sheet

23 of 24

I

Rev.

Stub Girders Subject

Des[gn Example June

CALCULATION SHEET

Checked by

Apr 92

RML JWR

92

with top chord =

Area of top chord

I

Job Title

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

b) Unpropped beam

1

I

Job No.

PUB: 118

The Steel Construction Institute

5500 mm2


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


I

I

1

Job No.

I

PUB 118

Sheet

24of 24

I

Rev.

Job Title

Stub Girders Subject

Design Bample

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

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


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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