Elastic composite beams

Computers & Slrucrures Vol. 59, No. 3. pp. 437-451. 1996 Copyright 0 1996 Elsetier Science Ltd Printed in Great Britain. All rights reserved 004s7949/% $I 5.00 + 0.00

00457949(95)00275-g

ELASTIC

COMPOSITE

BEAMS

R. Betti and A. Gjelsvik

Department of Civil Engineering

and Engineering Mechanics, Columbia 610 Seeley W. Mudd Building, NY 10027, U.S.A.

University,

(Received 30 September 1994) Abstract-In this paper, an application of the model of the analog-beam for the analysis of composite beams is presented for the case of a simply-supported beam with a uniformly distributed load. These beams are composed of an upper slab and a lower beam, connected at the interface by shear transmitting studs. The sixth-order differential equation of equilibrium for the vertical displacement is solved for two different sets of boundary conditions: (1) with rigid shear restraints at both the end sections of the beam and (2) with no shear restraints. The results (displacements, rotations, normal stresses and internal shear forces) are compared with those obtained using an equivalent classical Navier beam. The importance of the effectiveness of the interaction between the upper slab and the lower beam is emphasized. By imposing perfect bonding between the two subbeams, the analog-beam model presents the same behavior as the equivalent Navier beam.

INTRODUCTION

Composite steel-concrete beams are composed of a concrete slab and a steel beam, connected at their interface by a shear transmitting device such as studs, as shown in Fig. 1. The purpose of the shear studs is to transmit the horizontal shear force between the slab and the beam. The shear interface is of course not completely rigid but has a force-displacement relationship of the type shown in Fig. 2. As a composite beam is loaded, some slip must take place at the beam-slab interface. The composite beam therefore will no longer follow the Navier hypothesis of beam theory, even though the slab and the beam separately can still be expected to do so. Shear deformation of the steel beam can be important for composite beams, but it is not included in this analysis. Most of the studies of composite beams reported in the literature concentrated on the strength of the beam rather than on its elastic behavior (see for example Refs [l, 21). In simple (or continuous) span bridges with steel girders and concrete slabs, the neutral surface for the composite section will generally (if not always) fall in the slab. This results from the fact that the slab is between 7 and 12 in thick, and the attributed slab to a single beam could be as much as 10 ft (and sometimes more). Consequently, the bottom of the slab will be subjected to tensile stresses. If these stresses exceed the tensile strength for the concrete (about 470psi for a 4000 psi compressive strength for the concrete), then the slab will crack and the elastic theory is no longer applicable. The present paper concentrates on the elastic behavior of a composite beam and is an extension of composite beam theory developed in Ref. [3]. Here 437

the slab and beam were assumed to be linearly elastic and connected by a linearly elastic thin shear layer. This leads to a sixth-order differential equation for the deflection of the composite beam, necessitating three boundary conditions at each end of the beam. In particular it becomes possible to suppress the slip at the ends of a beam. A single, very strong and rigid shear transmitting device at the ends can have a large effect on the strength and stiffness of the composite beam. In Ref. [3], only the general theory outlined above was developed; no solutions to particular cases were presented. In this paper, the solution for a simply supported beam is obtained. It is of course realized that neither the flexural behavior of a concrete slab nor the slip behavior of a shear stud is exactly linear elastic. Nevertheless, a linear theory makes possible an analysis which can explain and predict quite accurately the behavior of the more complex system, as well as give an insight into the way the beam works. GENERAL THEORY

Since a general theory for a beam with a shear layer was developed in Ref. [3], only a brief summary of this theory will be given here. Figure 3 shows the three coordinate systems used. The system x, y,, z, is placed at the centroid U of the slab; the system x, y,, z IS placed at the centroid L of the beam while the system x, y, z is located at the global centroid 0. Figure 4 shows the relative displacements of the slab and beam system. The displacement components of the point 0 on the global centroidal axis are V(z) in the y-direction and W(z) in the z-direction. It is assumed that there is no displacement in the x-direction.

R. Betti and A. Gjelsvik

Fig. 1. Composite beam cross-section.

t

Fig. 4. Displacements of the composite beam.

Q’

I

r c

6

Fig. 2. Typical slip behavior of a shear stud.

The slab and the beam separately are considered to obey simple beam theory and to have no shear deformation. The shear layer slips an amount 6(z) producing a rotation, 4(z), of the line UL connecting the centroids of the slab and of the beam. The rotation angle b(z) is positive for a positive rotation about the x-axis (counterclockwise in Fig. 4). The slip is then

s=h(++Y’),

t Fig. 5. Bending moments and axial forces components.

In Ref. [3] it was shown that the active axial forces and bending moments in the slab and beam, N,,, Mu, N, and M,, are given by

(1)

where h is the distance between the centroids of the slab and of the beam and the superscript (‘) indicates the derivative with respect to z. The internal forces acting on the various components of the composite section are shown in Fig. 5.

Y Y

Fig. 3. Coordinate systems.

N,, = (AE),( W’ - h,cj’)

(2)

N, = (AE),(W’ + Iv)‘)

(3)

M” = - (EZ), Y”

(4)

M, = - (EZ), v”,

(5)

Elastic composite beams

439

where (AZ?), and (AE), are the axial stiffnesses of the slab and of the beam, respectively, while @I), and (EZ), represent the bending stiffness of the slab and of the beam. The global axial force N is placed at the global z-axis as shown in Fig. 5 and is given by ([3])

[

M,=

(15)

Mb

6% 1

g

1

Mb

[

and substituting N = (AE) W’ = Nu + N,,

0,

Mu=

b

(16)

eqn (9) into eqn (13) leads to

(6)

-N,=N+

where (AE) is the axial stiffness per unit length for the global cross-section, equal to

The corresponding normal stresses in the slab and in the beam associated with the axial forces NU and N, can now be expressed as

(AE) = (AE), + (AE),.

For a typical composite beam the axial force N is zero. From eqns (2), (3) and (6) it follows that W’ is also zero and eqns (2) and (3) become Nu = -(AE),h,c$’

N, = (AEM, ’

Nu

(7)

4

..=A,=

-j-J,

(8) while the corresponding stresses uUand cr, associated with the combined bending moments on the subbeams are (Fig. 5)

while from eqn (6) -N,

= N,.

(9)

(20)

The position of the centroid of the global section, 0, also follows from eqn (9)

M/ 1,

(21)

CT,=-z,.

(AEM, = (AEM,. The global bending moment M can be decomposed into two components

The shear force is somewhat more complicated. The horizontal shear force in the shear layer, q per unit length, can be expressed as

(11)

M=M,+M,,

q = kc5 = kh(c$ + V’),

where in the while from acting

(22)

M, can be identified as the bending moment

beam from what can be called its ‘truss action’ bending moment the individual beam action of the subbeams independently Mb represents the combined

Mb = - (EZ), V” = Mu + M, M, = (EZ),$’ = N,h, - N,h,

where k is the shear stiffness of the shear layer. Similarly to the total bending moment, the total or global shear force acting on the composite beam can be decomposed into two components

Q = Qb+ Q,,

(12) (13)

Qb = M;, = -(EZ),

with

(Ez), = VW, + W), (EZ), = (AE),h:

(EZ) = (EZ), + (EZ), . (EZ), and (EZ), represent the bending stiffnesses for the beam component and for the truss component, respectively. By using eqns (4), (5) and (12), it is possible to obtain

Mb

V”’

Q, = hq = kh2(4 + v’).

+ (AE),h:

?“’= -(EZ),

(23)

where

(14)

(24) (25)

The shear component Qb represents the combined subbeams’ shear force while Ql is associated with the deformation of the shear layer and is representative of the truss action of the subbeams. By imposing the equilibrium conditions for the analog beam, it is possible to obtain the governing equations for the rotation 4 and displacement V. These equations can be expressed as

6% 6% 4”“’ _ (EZ)$“’ = p kh2

(26)

R. Betti and A. Gjelsvik

440

(27) where p(z) represents the transversal loading acting on the beam. Appropriate boundary conditions need to be chosen according with the end conditions (shear constraint, no shear constraint). SOLUTION

OF THE GOVERNING

c( equal to zero corresponds to complete interaction (completely rigid shear studs k = co), and a equal to infinity corresponds to zero interaction (no shear studs k = 0). The governing eqns (26) and (27) now reduce to

v’= -4 f$’ - p%p

EQUATIONS

To obtain solutions to particular cases of loading and boundary conditions, consider the case of a beam subjected to uniformly distributed loading, i.e. p(z) =p = constant, as shown in Fig. 6. For the solutions of eqns (26) and (27) it is convenient to introduce the following non-dimensional quantities and parameters:

+rx#P

(31)

= p2y,

(32)

where the roman superscripts represent differentiation with respect to the new variable [. The solution to eqns (31) and (32) is

4(i)=

-Y

1 [;+Clr’+C,i+C, +C,sinh(P 0.01 (Fig. 18). Similar conclusions can be l-

7

Shear Restraint

-

h=l c .\ 54 .

No Shear Restraint

I.

.4

Non Dimensional

0.6 Distance <

Fig. 21. Non-dimensional shear.

.I.

I

0.8

1

Elastic composite beams derived for the case of a composite beam with shear restraints at the ends (Figs 19 and 20). For this case, the presence of rigid constraints at both ends precludes the normal stresses associated with the truss effect to approach zero. These curves emphasize the importance of having an effective connection between the beam and the slab. For a non-effective bond between the two sub-elements, the Navier theory underestimates the values of the maximum stresses, with important consequences for a proper design of the beam. Associated with the deformation of the shear layer, the distribution of the shear force Q along the length of the beam is critical in determining the spacing and dimensions of the shear studs. Figure 21 presents the variation of the shear force Q along the beam, normalized with respect to the average value of the shear force for an equivalent Navier beam, qEa,, PL q&v,, = 4h.

The rigid shear restraints at the end sections have the beneficial effect of drastically reducing the magnitude of the shear force along the entire length of the beam.

451 CONCLUSIONS

With the use of the analog-beam model, the solution for composite beams becomes relatively simple. From the results, the importance of having effective shear connectors between the upper slab and the beam is clear. When sliding is allowed, displacements and stresses distributions are strongly affected. The importance of rigid shear constraints at the end sections as stiffeners for the entire beam is emphasized.

REFERENCES 1.

I. M. Viest, Composite steel-concrete construction, Report of the Sub-committee on the state-of-the-art survey of the task committee on composite construction of the committee on metals of the structural division. J. struct. Div. AXE 100, ST5 1085-l 139 (1974). 2. C. W. Roeder (Ed.) Composite and mixed construction. In: Proc. US/Japan Joint Seminar, 18-20 July, New York, ASCE (1984). 3. A. Gjelsvik, Analog-beam method for determining shear-lag effects. J. Engng Mech. ASCE 117, 1575-l 594 (1991).