2 July 2002 – The Structural Engineer|31

SynopsisBS 5400: Part 5 requires consideration to be given to the non-uniform flow of longitudinal shear and to the slippage betweenthe slab and steel member of composite beams, where thelongitudinal spacing of shear connectors exceeds 600mm.Analysis of such effects is complex and involves finite-elementanalysis with contact logic. This paper presents a simplespreadsheet based alternative solution. The method is suitablefor serviceability limit state (SLS) checks and assumes thatlongitudinal shear is transferred between the steel member andslab only at shear connector locations. This method can be usedfor the assessment of existing bridges with widely spaced shearconnectors. It can also be used for designing the strengtheningmeasures for bridges built before composite construction waswidely recognised.

IntroductionThe full interaction transformed sections method assumes that acontinuous shear connection exists between the concrete slab andsteel section in composite beams. Therefore, it is not suitable forcarrying out the SLS checks required by BS 5400: Part 51 in theassessment of bridges where shear connector spacing exceeds600mm. Such checks are required because of the non-uniformdistribution of longitudinal shear in the connectors and becauseof the potential for slippage between the steel and concrete.

This work presents a method for assessing the effect sheartransfer discontinuities have on bending stresses and shearconnector forces.

The effect of discontinuities in the shear transfer in compositebeams was considered in the pioneering work on composite beamdesign by Caughey and Scott in 19292. Later experimental workshowed that imperfect shear connection leads to slip at thesteel–concrete interface and a method was developed to analysethis incomplete interaction3. More recently a method has beendeveloped for designing composite beams with a relatively lowdegree of shear connection4. Both methods implicitly account forthe slip created by the flexibility of the shear connectors,althoughthe shear connection is assumed to be continuous along the lengthof the beam. By contrast, the method presented here assumeslongitudinal shear is only transferred between the sections atdiscrete locations along the beam. The method may be classifiedas full interaction because shear connectors are assumed to berigid under elastic conditions.

The methodThe method is illustrated in Fig 1. In brief:Step 1: The shear connection between the slab and beam isreleased. The bending stress distributions are then quantifiedassuming no shear is transferred between the slab and the steel.The slip between the two surfaces is calculated once the bendingstresses are quantified. It is assumed that the concrete can resisttensile stresses at this stage,although,in the final state the tensilestresses become compressive stresses, when the neutral axis lieswithin the steel.Step 2: Externally applied loads are removed. Instead, internalshear connector forces are applied to produce a reverse slip ofequal and opposite magnitude to the step 1 slip. The stresses inthe section caused by the shear connector forces are quantified.Step 3: The stress distributions calculated in steps 1 and 2 aresummed to produce the final stress distribution.The force in indi-vidual shear connectors is equal to the change in shear connectorforce.

Shear connection released (Step 1)Fig 2 illustrates the method in more detail. An element of beamlocated between two shear connectors is subdivided to facilitate

analysis of the inter-connector stresses.The design moment (Fig2a) is applied to the combined sections with the shear connectionremoved (Fig 2c). It is assumed that the slab remains in contactwith the section. Thus the radii of curvature of the beam (Rs) isequal to slab (Rc) + the distance between the two neutral axes.Since Rc ≅ Rs the analysis can be simplified by assuming that Rc=Rs= R. Analysis accounting for differences in radii is tedious andalters final stresses by less than 1%.From simple bending theorythe moments in the concrete slab (Mc) and steel beam (Ms) are:

MR

E Ic

c c= ....(1)

and

MR

E Is

s s= ....(2)

Since M = Mc + Ms and R = Ey/σ, the stresses in the bottom ofthe concrete (σc.b) and top of the steel (σs.t) can be easily calculated,i.e.:

paper: byfield

Analysis of composite beams withwidely spaced shear connectors

M. P. ByfieldBEng, PhD, CEng,MICEEngineering SystemsDepartment, CranfieldUniversity, RMCSShrivenham, Swindon,SN6 8LA, UK

Received: 14/09/01Accepted: 19/02/02Keywords: longitudinalstrain, shear, slipconnections, compositebeams, slabs, analysis.spreadsheets,serviceability, bridges

Fig 1. Basis of method

Fig 2. Determining slip atthe shear connectorlocations on release

32|The Structural Engineer – 2 July 2002

E I E I

M E y.c b

c c s s

i c c=

+v ....(3)

and

E I E I

M E y.s t

c c s s

i s s=-

+v ....(4)

Where: Mi is the applied moment at any cross-section,see Fig 2ayc is half the depth of the slabys is half the depth of the steel beam (if beam is symmetric)E and I are elastic modulus and second moment of area respec-

tively (the subscripts c and s represent concrete and steel)Given these stresses the end displacement of the concrete (δc.1)

and steel (δs.1) can be calculated from δ = Lσ/E, i.e.:

E I E I

Ly M.c

c c s s

c1 =

+d ....(5)

E I E I

Ly M.s

c c s s

s1 =-

+d ....(6)

Where the average moment between connector pairs is:

M n M1

i

i

n

0

==

! ....(7)

L is the shear connector spacingn is the number of sections between shear connectors,see Fig 2b.Thus, the total end slip, sm (equal to s1 +s2, Fig 2c) is:

s . .m c s1 1= +d d ....(8)

Shear connector forces applied to produce reverse slip (Step 2)The externally applied loading is removed from the combinedsections and internally applied shear connector forces T areapplied to produce reverse slip, see Fig 3. The longitudinal forceT is the sum of shear connector forces between the section underconsideration and the end of the simply supported beam or pointof contraflexure.

Physically,the centreline of the force T must be near to the baseof the shear connectors.However,the force is also accompanied byan interface normal force to maintain equilibrium5. Thus themoment induced in the slab,Tyc, is reduced by the moment result-ing from the normal force.This reduced moment may be accountedfor by an equivalent eccentricity, e, of the force T. Thus, themoments in the concrete and steel become:

( )M T y ec c= - ....(9)

( )M T y es s= + ....(10)

No gap is allowed to form between the slab and section. Theposition of the equivalent eccentricity can be determined giventhat that the normal forces maintain the interface contact. SinceRc ≅ Rs =R and M/I = E/R the position of the equivalent eccentric-ity, e, can be defined where:

eE I E I

y E I y E I

s s c c

c s s s c c-=

+....(11)

Thus,by ensuring that the curvatures of the steel and concrete

remain the same, the equivalent eccentricity of T is independentof moment and dependent on sectional and material propertiesalone.The method can be adjusted to account for the minor differ-ences in radii of curvature, although results are altered by lessthan 1%. The force T causes stresses σc.b and σs.t to be developedin the bottom of the concrete and top of the steel respectively,where:

( )T

A I

y e y1.c b

c c

c c=- +

-v e o ....(12)

and

( )T

A I

y e y1.s t

s s

s s= +

+v e o ....(13)

These stresses cause the end displacements sketched in Fig.3,where:

( )

E

L

E

LT

A I

y e y1.c

c

c b

c c c

c c2 = =- +

-d

ve o ....(14)

and

( )

E

L

E

LT

A I

y e y1.s

s

s t

s s s

s s2 = = +

+d

ve o ....(15)

The total end slip (2.St, see Fig 3) = δc2 – δs2. It is assumed thatthe stress distributions resulting from the shear connector forcescorrespond to those produced by the application of a moment andaxial force to a long beam.Local stress concentrations at the shearconnectors are ignored and strain distributions are assumed to belinear in both the steel and concrete. In simply supported beams,where the maximum shear connector forces occur remote from themaximum bending forces, this assumption will be approximatelycorrect. It is also assumed that no slip occurs at the connectorsunder elastic conditions and that inter-connector slip produceszero friction.

Combine steps 1 and 2 to determine final composite stresses(Step 3)It is assumed that zero slip occurs at the shear connectors, i.e:step1 end slip + step 2 end slip = 0. Given this the previous formulaecan be combined to produce the total shear connector thrust T, atany section, where:

( ) )( ( )T

E I E I

M y y

E A E I

y e y

E A E I

y e y1 1

c c s s

c s

c c c c

c c

s s s s

s s

1

=+

++

-+ +

+-

e o ....(16)

T is equal to the sum of the shear connector forces between theend of the span and the section under consideration; connectorshear is given by the change in T.This is illustrated in Fig 4,wherethe average moment between the first two shear connectors isdouble that of the adjacent section. Thus, the change in totalthrust (or shear connector force) is 1T.

The stress in the steel and concrete can be determined byadding the stresses calculated in steps 1 and 2 to produce thefollowing formulae:

Final stress in the top of concrete slab:

( )

E I E I

M E yT

A I

y e y1.c t

c c s s

i c c

c c

c c-=-+

--

v e o ....(17)

Final stress in the bottom of concrete slab:

paper: byfield

Fig 3. A section ofcomposite beamlocated betweenshear connectors,shear connector forceapplied and externalloading removed

Fig 4. Calculation of the

shear connector force

2 July 2002 – The Structural Engineer|33

( )

E I E I

M E yT

A I

y e y1.c b

c c s s

i c c

c c

c c+=++

--

v e o ....(18)

Final stress in the top of the steel section:

( )

E I E I

M E yT

A I

y e y1.s t

c c s s

i s s

s s

s s+ ++

=-+

v e o ....(19)

Final stress in the bottom of the steel section:

( )

E I E I

M E yT

A I

y e y1.s b

c c s s

i s s

s s

s s+ -=++

+v e o ....(20)

ResultsFigs 5 and 6 illustrate the model predictions for a beam (Fig 7)subjected to a dead and imposed load of 54.5kN/m run,with shearconnectors at 600mm intervals. The impact of inter-connectorslippage on stress is clearly visible,with stress oscillating about amean value that corresponds exactly to that produced from themethod of transformed sections. Both the top of the concrete andthe bottom of the steel remain largely unaffected by the breaks inshear transfer.As would be expected,slip has its largest influencewhere there is the greatest rate of change of bending moment.Thus, the edges of the beam are greatly affected, with slip creat-ing significant tensile as well as compressive stresses.Conversely,slippage has relatively little effect where bending is approximatelyuniform, as is the case at the centre of the beam. In reality, sharpchanges in stress would be smoothed because of the flexibility ofshear connectors and the effects of local stress concentrations atthe connectors. The shear connector forces predicted from theproposed method correspond to those predicted from a conven-tional analysis of shear flow.

A possible application for the method is the assessment of exist-ing bridges where shear connector spacing is larger than thelimits specified in modern design codes, such as BS 5950: Part 5.Where connector spacing (or groups of connectors) exceeds 600mmthe code states5 that checks are necessary to take account of thenon-uniform flow of longitudinal shear and of the slip between theslab and steel.Previously,no simple method existed for such calcu-lations,so this approach may have application,including those old

structures built before composite construction was widely recog-nised.Such structures,which could be railway or highway rivetedgirder bridges, could be strengthened through the installation ofshear connectors at discrete locations. The potential of suchstrengthening is illustrated in Fig 8 which shows the directbending stresses for the beam sketched in Fig 7. The analysisassumes longitudinal shear is only transferred between the steeland concrete by groups of shear connectors 3m from each end ofthe 12m long,simply supported beam.The figure shows the partialshear connection leads to a substantial reduction in the maximumdirect bending stresses in the steel section.

Calculations show that the position of the effective eccentricityof the shear stud force, e, consistently lies just below the centroidof the slab. Since the centreline of the force must be near to thebase of the shear connectors, the normal forces in the shearconnectors and at the steel concrete interface must be importantin inducing this equivalent eccentricity. As shear connectorspacing is increased, the tendency for the steel and concrete toseparate will increase. Thus, partial shear connection may needto be accompanied by additional connectors along the length of thebeam to prevent separation.

Fatigue failure of shear studs is known to occur in bridges.Thismethod provides a useful tool for analysing the effect that the lossof shear connection through fatigue would have on the loading onremaining studs. Such an analysis could reveal when stud lossthrough fatigue would become critical, causing a beam to ‘unzip’as the studs that remain become progressively overloaded.Finally,the method can also be used for analysing steel and wrought-ironsections built up from separate components riveted together atdiscrete points along their length.

ConclusionsA novel release/restore method is presented for analysing thedirect bending stresses and shear connector forces in compositebeams.The method implicitly recognises the breaks in the shearconnection at the steel–concrete interface. It can be quicklyprogrammed using, for example, spread-sheet software and canbe easily adapted to analyse any type of composite member built-up from components connected together longitudinally along theirlength.The method finds application as a means to carry out theBS 5400:Part 5 SLS checks required for the assessment of bridgeswhere shear connector spacing exceeds the normal code limits.Another application includes the design of strengthening meas-ures for bridge structures built before composite construction waswidely recognised.

paper: byfield

Fig 5. (top)Direct stresses in theconcrete slabFig 6. (above)Direct stresses in steelsectionFig 7. (left)Section properties ofworked example

1. BSI.: BS 5400: Part 5: 1979 – Steel, concrete and composite bridges: Part 5. Code of prac-tice for design of composite bridges. London, 1979.

2. Caughey, R. A. & Scott, W. B.: ‘A practical method for the design of I beams haunchedin concrete’. The Structural Engineer, 7(no. 8), 1929, pp 275-93.

3. Newmark, N. M., Siess, C. P. & Viest, I. M.: ‘Tests and analysis of composite beamswith incomplete interaction’. Proceedings Society for Experimental Stress Analysis, 9, No.1, 1951, pp 75-92.

4. Oehlers, D. J. and Sved, G.: ‘Composite beams with limited-slip-capacity shearconnectors’. ASCE Journal of Struct. Eng. Vol. 121, No. 6, 1995, pp 932-938.

5. Oehlers, D. J. and Bradford, M. A.: Composite steel and concrete structural members:fundamental behaviour. Pergamon Press, Oxford, 1995.

REFERENCES

Fig 8. Direct bendingstresses in a non-composite steel beam(Fig 7) strengthenedthrough the additionof shear connectors3m from each end.