Analytical modeling of nonlinear behavior of composite beams with deformable connection

Journal of Constructional Steel Research 52 (1999) 195–218
www.elsevier.com/locate/jcsr
Analytical modeling of nonlinear behavior of
composite beams with deformable connection
N. Gattesco
Dipartimento di Ingegneria Civile, Universita’ di Udine, Via delle Scienze N. 208, 33100 Udine, Italy
Received 6 May 1998; received in revised form 25 January 1999; accepted 1 April 1999
Abstract
A numerical procedure for the analysis of steel and concrete composite beams is herein
presented. The program accounts for nonlinear behavior of concrete, steel and shear connec-
tors. In particular the most refined stress–strain constitutive relationships available in the litera-
ture can be used in the procedure. For shear connectors an empirical nonlinear load–slip
relationship is used. The accuracy and reliability of the program are demonstrated by the
analysis of four composite beams over the entire loading range up to failure. The analytical
results are compared with the corresponding experimental data with good agreement between
them. The reported results demonstrate that the numerical approach is a valid tool for extensive
parametric studies on composite beams with complete or partial shear connection.  1999
Elsevier Science Ltd. All rights reserved.
Keywords: Shear connection; Nonlinear analysis; Structural analysis; Composite structures; Stud
connectors
1. Introduction
The composite system considered here consists of a concrete slab connected by
means of shear connectors to a steel beam as normally used in bridge superstructures
and in floor systems for buildings. Very often the concrete slab is realized using
precast elements or a composite slab with profiled steel sheeting. These techniques
have the advantage of avoiding the use of formworks and, moreover, the structure
can be completed very quickly, but normally a wider connector spacing has to be
used. In fact, with solid precast slabs the connectors are welded to the steel beam
0143-974X/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved.
PII: S0143-974X(99)00026-7

196 N. Gattesco/Journal of Constructional Steel Research 52 (1999) 195–218
in groups disposed at a certain distance from each other. When profiled steel sheeting
is used only one connector per trough may normally be adopted because the concrete
rib formed by a trough may not be strong enough to resist the force from two or
more connectors. These situations imply that for these kinds of composite beams,
complete shear connection can rarely be obtained.
For a given beam, loading, and design method, complete shear connection is
defined as the least number of connectors Nf such that the bending resistance of the
beam would not be increased if more connectors were provided. The number Nf can
be defined only in relation to particular methods of design and of testing connectors
(e.g. Eurocode No. 4 [1]). Partial shear connection is when the number of connectors
N used in a beam is lower than Nf.
In the design with complete shear connection, it is normally assumed that failure
of shear connectors does not occur and the influence of connector deformation on
the structural behavior can be neglected. With partial shear connection, the ultimate
resistance of the beam is governed by the connection, in fact it depends on the
ultimate resistance of the single connector and on its ductility. In these cases, the
correct evaluation of the slip at the interface between the concrete slab and the steel
flange, in order to check if the ductility of the connection used allows it, is of great
importance. Moreover, the slip can cause significant redistribution of stresses
between the connectors in both serviceability and ultimate limit states.
In Eurocode No. 4, Part 1 [1], two simplified design methods for partial shear
connections are proposed but only for members with compact steel sections (class
1 and class 2). Using ‘ductile’ connectors (normally headed studs with the shank
diameter ranging between 16 mm and 22 mm, with a shank length greater than four
times the diameter embedded in concrete with characteristic cylinder strength fck ⬍
30 MPa), the methods are the interpolation method and the equilibrium method [1,2]
(Fig. 1).
Fig. 1. Interpolation and equilibrium method of design.

197
N. Gattesco/Journal of Constructional Steel Research 52 (1999) 195–218
The ultimate load capacities Wu and Ws relate to the composite member with full
interaction and to the steel member alone, respectively. They are obtained when
ultimate moments of resistance are reached in the critical sections, provided that
sufficient rotation capacity be allowed by internal support sections, otherwise the
maximum load has to be evaluated with respect to the maximum rotation capacity
allowed.
In the equilibrium design method (curve ABC in Fig. 1), ultimate moments in
critical sections are evaluated by simple equilibrium of stresses and using a com-
pressive force in the slab Nc ⫽ NQu (Fig. 2), where N ⬍ Nf is the number of connec-
tors used and Qu is the design resistance per connector.
Fig. 2. Beam subdivision in small elements of length ⌬l (a); 8 degree of freedom element (b).

The interpolation method is more conservative and is represented by the line AC
in Fig. 1. The maximum load is then immediately evaluated when the ultimate load
capacities of composite Wu and steel Ws members are known, in fact
W ⫽ Ws ⫹ n(Wu ⫺ Ws) (1)
where n ⫽ N/Nf is the degree of connection.
However, for both methods a minimum shear connection ratio nmin is specified.
Some results available in the literature [2,3] have shown that the curve ABC of Fig.
1 cannot be used for degrees of connection lower than nB (corresponding to point
B on the curve) due to a likely rupture of the connectors. In these cases a safe design
can be obtained using line BO; indications for point B can be found in the literature
[1–3].
These simplified methods are very useful for designing composite beams and are
proposed in some codes (e.g. Eurocode No. 4, Part 1 [1]) but they need further
numerical and experimental investigations in order to bring near the scattered results
for point B [2,3].
Therefore, reliable numerical approaches are necessary to increase the number and
quality of numerical results now available. For this purpose in the present paper, the
features of a powerful nonlinear numerical approach for studying the behavior of
steel and concrete composite beams with deformable connections are presented.
2. The proposed approach
The structure is subdivided into n elements of length ⌬li (Fig. 2a). Each single
element is schematically represented by two beam-type members (C concrete, S
steel) laid one upon the other and connected at their interface by two horizontal
springs (Fig. 2(b)) [4]. These elements have four nodal points with three degrees of
freedom per node: horizontal and vertical displacements and the rotation in the x–
y plane.
Some results available in the literature (i.e. [5,6]) show that the uplift of concrete
slab with respect to the steel member has a negligible effect on the distribution of
slip along the beam as well as on the ultimate load capacity of the beam. In the
proposed procedure uplift is not included. In fact, the following hypotheses are
assumed:
1. the distribution of strain is linear over the depth of both concrete and steel
elements;
2. the stress–strain relationship of steel is the same in tension and in compression;
3. the concrete slab and the steel member can slip along the connection without
separation (i.e. uplift neglected);
4. the concrete slab and the steel member sections have equal curvature.
Hypotheses (3) and (4) imply that the vertical displacement and the rotation of
the concrete slab are equal to those of the steel member. The number of degrees of

199
freedom of each element then reduces to eight (Fig. 2(b)). The procedure does not
consider buckling effects on the steel member.
2.1. Linear approach
Assuming a linear behavior for materials, the relationship between forces and
displacements for the general i-th element is
Fi ⫽ Ki Xi ⫺ F̄i (2)
where Fi is the vector of nodal forces, F̄i is the vector of fixed end forces, Xi is the
vector of nodal displacements and Ki is the stiffness matrix of the element. Vectors
Fi, F̄i and Xi have the dimensions [8 ⫻ 1]
Fi ⫽
冤
Nc,i
Ns,i
Vi
Mi
Nc,i ⫹ 1
Ns,i⫹1
Vi⫹1
Mi⫹1
冥F̄i ⫽
冤
N̄c,i
N̄s,i
V̄i
M̄i
N̄c,i⫹1
N̄s,i⫹1
V̄i⫹1
M̄i⫹1
冥Xi ⫽
冤
␰1,i
␰2,i
␩i
␽i
␰1,i⫹1
␰2,i⫹1
␩i⫹1
␽i⫹1
冥
and the stiffness matrix has the following form
Ki⫽





k1,i ⫹
EcAci
⌬1i
⫺ k1,i 0 ⫺ k1,i z ⫺
EcAci
⌬1i
0 0 0
k1,i ⫹
EsAsi
⌬1i
0 k1,iz 0 ⫺
EsAsi
⌬1i
0 0
12(EcIci ⫹ EsIsi)
⌬13
i
⫺
6(EcIci ⫹ EsIsi)
⌬12
i
0 0 ⫺
12(EcIci ⫹ EsIsi)
⌬13
i
⫺
6(EcIci ⫹ EsIsi)
⌬12
i
4(EcIci ⫹ EsIsi)
⌬1i
⫹ k1,iz2
0 0
6(EcIci ⫹ EsIsi)
⌬12
i
2(EcIci ⫹ EsIsi)
⌬1i
Symmetric k2,i ⫹
EcAci
⌬1i
⫺ k2,i 0 ⫺ k2iz
k2,i ⫹
EsAsi
⌬1i
0 k2iz
12(EcIci ⫹ EsIsi)
⌬13
i
6(EcIci ⫹ EsIsi)
⌬12
i
4(EcIci ⫹ EsIsi)
⌬1i
⫹ k2,iz2





where Aci, Asi, Ici, Isi are the area and moment of inertia of concrete slab and steel
beam, respectively, k1,i, k2,i are the stiffnesses of left and right springs of the i-th
element and z ⫽ hcb ⫹ hp ⫹ hst. The terms of the matrix express the forces and the
moments due to unitary displacements applied to any degree of freedom of the
element (Fig. 2(b)).
The equilibrium equations for the composite beam may be written in the form
AY ⫺ ␭B ⫽ 0 (3)

where A is the global stiffness matrix, Y is the vector of nodal displacements, B is
the fixed external loading vector (assembling of fixed-end forces F̄i) related to a
reference load W̄ and the scalar ␭ is a load level parameter that multiplies B. The
latter assumption is valid only in case of proportional loading.
Solving Eq. (3), the nodal displacements corresponding to different values of the
load parameter l, are obtained. Then with Eq. (2) the axial forces (Nc, Ns), the shear
forces (Vc, Vs) and the bending moments (Mc, Ms), acting both on the concrete slab
and on the steel member may be evaluated in any section along the beam. The
moment of the composite section M can be calculated using the equation
Mi ⫽ Mc ⫹ Ms ⫹ Ns·z (4)
and the slip at the steel–concrete interface s may be obtained from the relationship
si ⫽ ␰i,1 ⫺ ␰i,2 ⫺ ␽i·z. (5)
2.2. Nonlinear approach
The constitutive laws of materials (concrete and steel) and the load–slip relation-
ship of shear connectors are not linear so that the equilibrium equations assume the
general form
G(Y,␭) ⫽ R(Y) ⫺ ␭B ⫽ 0 (6)
where G(Y,␭) is the gradient of the total potential energy and R(Y) is the internal
force vector (nonlinearly dependent on the displacements Y). In order to evaluate
the unknown displacements a linearized form of Eq. (6), via a truncated Taylor series,
is needed
G(Y(m)
l⫹1,␭(m)
l⫹1) ⫽ G(Y(m)
l ,␭(m)
l ) ⫹
∂G
∂Y
␦Yl ⫹
∂G
∂␭
␦␭l ⫽ G(Y(m)
l ,␭(m)
l )
⫹ AT␦Yl ⫺ B␦␭l ⫽ 0, (7)
where AT is the tangent stiffness matrix, the superscript m denotes the iterative loop
number and the subscript l denotes the iteration number. Eq. (7) has to be solved
iteratively until convergence to the real solution is reached. Specifically, Eq. (7)
introduces the Newton–Raphson technique, which is the most commonly used in
nonlinear problems.
Such a process needs the evaluation and factorization of the tangent stiffness at
each iteration, which is time consuming, so it is normally preferred to use the modi-
fied Newton–Raphson technique which uses the initial stiffness matrix at each iter-
ation. The stiffness matrix is then only formed and factorized at the beginning of
each load increment.
If the equilibrium curve has limit points, due for example to softening behavior,
the method fails (␭ ⫽ const.; Fig. 3) and it is difficult to evaluate correctly the
ultimate load as well as the corresponding displacements for structures with an
almost perfect plastic behavior. To overcome this problem, the curve bounding the

201
Fig. 3. Solution curve and constraint functions in a one-dimensional problem.
stiffness extrapolation of the solution curve has to be oblique or, alternatively, a
different function f(Y,␭) ⫽ 0 (Fig. 3) [7–9].
The numerical procedure COBENA presented herein uses the modified Newton–
Raphson technique with a linearized form of the arc-length method [10] (piecewise
linear constraint) as illustrated in Fig. 4.
In particular, the constraint function adopted has the form
f(Y(m)
l ,␭(m)
l ) ⫽ ⌬YT
l ␦Yl ⫹ ␦␭l⌬␭lBT
B ⫽ 0, (8)
where the vector ⌬Yl and the scalar ⌬␭l are incremental while the vector ␦Yl and
the scalar ␦␭l are the changes at each iteration (Fig. 5). Eq. (8) ensures the orthog-
onality between the iterative change (␦Yl, ␦␭lB) and the ‘secant change’ (⌬Yl,
⌬␭lB).
Provided that the constraint curve is oblique to ␭ ⫽ const., the value of the load
level varies at each iteration so there is one more unknown, the load magnitude
parameter ␭, besides the n nodal displacements. Then Eq. (8) has to be added to Eq.
(7), leading to an enlarged system, the stiffness matrix of which is neither symmetric
nor banded
冋 A ⫺ B
⌬YT
l ⌬␭lBT
B
册冋␦Yl
␦␭l
册⫽ ⫺ 冋G(Y(m)
l ,␭(m)
l )
0
册. (9)
Fig. 4. Arc-length method with piecewise linear constraint [10].

Fig. 5. Linearized arc-length method procedure and notation for one-dimensional problems.
Instead of solving Eq. (9) directly, the constraint represented in Eq. (8) may be
introduced by following the technique of Batoz and Dhatt [11] for displacement
control. In this case the iterative displacement ␦Yl is split in two parts, hence the
Newton change at the new unknown level, ␭(m)
l⫹1 ⫽ ␭(m)
l ⫹ ␦␭l, becomes
␦Yl ⫽ ⫺ A−1
T G(Y(m)
l ,␭(m)
l⫹1) ⫽ ⫺ A−1
T (R(Y(m)
l ) ⫺ ␭(m)
l⫹1B) ⫽
⫺ A−1
T (G(Y(m)
l ,␭(m)
l ) ⫺ ␦␭lB). (10)
G(Y(m)
l ,␭(m)
l ) represents the out-of-balance force at the l-th iteration (m-th loading
step) and may be obtained from the equation
G(Y(m)
l ,␭(m)
l ) ⫽ R(Y(m)
l ) ⫺ ␭(m)
l B, (11)
where the members on the right-hand side are internal and external forces, respect-
ively. Indicating with
␦Ȳl ⫽ ⫺ A−1
T G(Y(m)
l ,␭(m)
l ), (12)
the iterative change that would stem from the standard load-controlled (␭ ⫽ const.)
Newton–Raphson method (at a fixed load level, ␭(m)
l ), and with
Y(m)
T ⫽ A−1
B (13)
the displacement vector corresponding to the fixed load vector B. Provided that the
modified Newton–Raphson method is used the vector Y(m)
T needs to be computed
only for the initial ‘predictor’ step because it does not change during iterations.
Substituting Eqs. (12) and (13) into Eq. (10) the iterative change expression becomes
␦Yl ⫽ ␦Ȳl ⫹ ␦␭lY(m)
T (14)
and so the new incremental displacements are
⌬Yl⫹1 ⫽ ⌬Yl ⫹ ␦Yl ⫽ ⌬Yl ⫹ ␦Ȳl ⫹ ␦␭lY(m)
T . (15)
On the right-hand side of Eq. (15) the term ␦␭l is still unknown and its value can
be obtained by combining Eq. (8) with Eq. (14)

203
␦␭l ⫽ ⫺
⌬YT
l ␦Ȳl
⌬YT
l Y(m)
T ⫹ ⌬␭lBT
B
. (16)
and consequently the new incremental load parameter
⌬␭l⫹1 ⫽ ⌬␭l ⫹ ␦␭l. (17)
The global displacement vector and the corresponding load parameter are then,
respectively,
Y(m)
l⫹1 ⫽ Y(m)
1 ⫹ ⌬Yl⫹1, ␭(m)
l⫹1 ⫽ ␭(m)
l ⫹ ⌬␭l⫹1. (18)
The iterations will stop when the following inequality is satisfied
|gk|
⌬␭l·|bk|
ⱕ ⑀1 (k ⫽ 1 to 4·n). (19)
where gk is the k-th element of the out-of-balance vector G(Y(m)
l ,␭(m)
l ) and bk is the
k-th element of the initial fixed-end force vector B. The tolerance error ⑀1 is normally
assumed equal to 10−6
.
The procedure continues in this way with other load increments up to the failure
of the beam due to one of the following three causes: crushing of concrete in com-
pression (excessive compression strain), ceasing of structural steel in tension
(excessive tensile strain) and rupture of shear connection (excessive slip).
The ‘predictor’ solution at the beginning of each iterative loop is given by the equ-
ation
⌬Y1 ⫽ ⌬␭1·Y(m)
T (20)
where the load increment factor ⌬␭1 has to be fixed. As a matter of fact only the
increment ⌬␭(1)
1 of the first iterative loop needs to be provided by the user (normally
1/5 of the total load) so as to obtain a starting arc-length increment using the equation
[7] (Fig. 5)
⌬a(1)
⫽ ⌬␭(1)
1 ·√Y(m)T
T Y(m)
T ⫹ BT
B. (21)
For subsequent iterative loops, the lengths may be adjusted so to achieve a nearly
constant number of iterations, hence the following criterion is considered [8,10]
⌬a(m⫹1)
⫽ ⌬a(m)
·冪
Id
Im
, (22)
where Im is the number of iterations required to achieve equilibrium at the m-th loop
and Id is the desired number of iterations for the m⫹1-th loop (normally Id ⫽ 5).
The initial increment loading parameter ⌬␭(m)
1 , for all loops other than the first,
are obtained from (Fig. 5),
⌬␭(m)
1 ⫽ ⫾
⌬a(m)
√Y(m)T
T Y(m)
T ⫹ BT
B
, (23)

the sign follows that of the previous increment unless the determinant of the tangent
stiffness matrix AT has changed sign, in which case, a sign reversal is applied [8].
This procedure is powerful when the aim is to obtain the complete load–deflection
curve up to beam collapse, which is, in contrast, very difficult to obtain with many
procedures available in the literature (e.g. [2,6]) when the structure has either a per-
fectly plastic or a softening behavior. A shortcoming of the proposed procedure
concerns its inadequacy for calculations at specific target loads because the load is
not known a priori. However, in these cases, if the target load is not too close to
the ultimate load, the modified Newton–Raphson technique with a horizontal con-
straint (␭ ⫽ const.) should be used.
2.3. Nonlinear behavior of sections
The internal forces vector R(Y(m)
l ) of Eq. (11) is evaluated by considering the
nonlinear behavior of sections. For simplicity, the superscript m (iterative loop
number) and subscript l (iteration number) are not reported in the following.
Once the displacement vector Y has been evaluated at each iteration, the axial
strains at the centroid fiber of concrete slab and of the steel beam as well as the
curvature at both ends of each element may be drawn by the equations
⑀ci ⫽
␰1,i⫹1 ⫺ ␰1,i
⌬li
⑀si ⫽
␰2,i⫹1 ⫺ ␰2,i
⌬li
␸1,i ⫽
6·(␩i⫹1 ⫺ ␩i)
⌬l2
i
⫺
4␽i ⫹ 2␽i⫹1
⌬li
␸2,i ⫽ ⫺
6·(␩i⫹1 ⫺ ␩i)
⌬l2
i
⫹
2␽i ⫹ 4␽i⫹1
⌬li
(24)
where the subscript i refers to the element number. Considering the sections of Fig.
6(a) (concrete slab) and Fig. 6(b) (steel member) the axial strains in any fiber of
them can be evaluated
⑀ci(y) ⫽ ⑀ci ⫹ ␸i·y
⑀si(y) ⫽ ⑀si ⫹ ␸i·y.
(25)
The constitutive relationships of materials (concrete and steel) allow us to deter-
mine the corresponding normal stresses in the concrete ␴ci(y), in reinforcing steel
␴ri,j and in structural steel ␴si(y). So the axial force in the sections can be calculated
by using the equations

205
Fig. 6. Discretization of concrete (a) and steel (b) sections for numerical integration of stresses.
Nci ⫽ 冕
hcb
⫺ hct
␴ci(y)·b(y)·dy ⫹ 冘
nl
j⫽1
␴ri,j·Asj
Nsi ⫽ 冕
hsb
⫺ hst
␴si(y)·b(y)·dy
(26)
where nl is the number of reinforcing steel layers and Asj is the steel area of each
layer. In the same way the moments in both sections can be evaluated
Mci ⫽ 冕
hcb
⫺ hct
␴ci(y)·y·b(y)·dy ⫹ 冘
nl
j⫽1
␴ri,j·ysj·Asj
Msi ⫽ 冕
hsb
⫺ hst
␴si(y)·y·b(y)·dy.
(27)
The vertical shear forces are obtained by equilibrium.
The integrals in Eqs. (26) and (27) are solved by numerical integration subdividing

the section into n strips, as indicated in Fig. 6, and then using a Gauss–Legendre
quadrature formula in each strip. The general integral of Eqs. (26) and (27) becomes
冕
hb
⫺ ht
S(y)·dy ⫽ 冘
n
p⫽1
冕
yp⫹1
yp
S(y)·dy ⬇ 冘
n
p⫽1
yp⫹1 ⫺ yp
2
·冘
5
q⫽1
wq·S(yq) (28)
where the variable yq is defined as follows
yq ⫽
yp⫹1 ⫹ yp
2
⫹
yp⫹1 ⫺ yp
2
·␨q, (29)
␨q and wq are the Gauss points in the interval [⫺1, ⫹1] and the weight factors,
respectively.
The slip at the steel–concrete interface is evaluated using the equation
sj ⫽ ␰j,1 ⫺ ␰j,2 ⫺ ␽j·z (j⫽1,n⫹1) (30)
and the corresponding longitudinal shear force at the ends of each element
(Qi,1, Qi,2) are evaluated from the load–slip relationship of shear connectors. There-
fore, the actual nonlinear nodal forces can be evaluated. In particular, axial forces are
Nci,1 ⫽ ⫺ Nci ⫺ Qi,1
Nci,2 ⫽ ⫺ Nci ⫹ Qi,2
Nsi,1 ⫽ Nsi ⫺ Qi,1
Nsi,2 ⫽ Nsi ⫹ Qi,2
(31)
the bending moments and vertical shear forces at both ends of the i-th element are
Mi,1 ⫽ (Mci,1 ⫹ Msi,1) ⫹ Qi,1·z
Mi,2 ⫽ (Mci,2 ⫹ Msi,2) ⫺ Qi,2·z
Vi,1 ⫽ Vci,1 ⫹ Vsi,1
Vi,2 ⫽ Vci,2 ⫹ Vsi,2.
(32)
Assembling the forces of each element obtained with Eqs. (31) and (32) the actual
nonlinear nodal force vector of the structure R(Y) is obtained and so can be used in
Eq. (11).
2.4. Constitutive relationships
The numerical procedure allows us to consider various constitutive laws for the
materials of the composite beam. However, for simplicity, herein are presented the
most refined laws available for each material (Fig. 7). The relationship suggested by
the CEB Model Code 90 [12] is adopted for concrete both in compression and in
tension (Fig. 7(a)). In particular for concrete in compression

207
Fig. 7. Constitutive relationships for concrete (a), steel (b), shear connection (c) and interface slip (d).
␴c ⫽
k·⑀o ⫺ ⑀2
o
1 ⫹ (k ⫺ 2)·⑀o
·fck for 0 ⱕ ⑀o ⱕ ⑀u (33)
where k ⫽ Ec·⑀c1/fck, ⑀o ⫽ ⑀c/⑀c1, and ⑀u ⫽ ⑀cu/⑀c1 is the nondimensional strain corre-
sponding to half the compressive strength in the softening branch. Beyond this value
of the strain
␴c ⫽
fck
(␨·⑀u ⫺ 2)·冉⑀o
⑀u
冊2
⫹ (4 ⫺ ␨·⑀u)·冉⑀o
⑀u
冊
(34)
with
␨ ⫽ 4·
⑀2
u·(k ⫺ 2) ⫹ 2·⑀u ⫺ k
(⑀u·(k ⫺ 2) ⫹ 1)2
. (35)
For concrete in tension the following bilinear relationship is considered up to
cracking
␴ct ⫽ Ec·⑀c for 0 ⱕ ⑀c ⱕ 0.9·fctk/Ec (36)

and
␴ct ⫽ 0.9fctk ⫹ 0.1fctk·
⑀c ⫺ 0.9fctk/Ec
⑀ct1 ⫺ 0.9fctk/Ec
for 0.9fctk/Ec ⱕ ⑀c ⱕ ⑀ct1. (37)
The effect of tension stiffening is neglected.
Both for reinforcing steel and for structural steel a stress–strain relationship with
strain hardening is considered (Fig. 7(b)). Specifically, the relationship is linearly
elastic up to yielding, perfectly plastic between the elastic limit and the beginning
of strain hardening and follows the present equation in the strain hardening branch
␴s ⫽ fsy ⫹ Esh·(⑀s ⫺ ⑀sh)·冉1 ⫺ Esh·
⑀s ⫺ ⑀sh
4·(fsu ⫺ fsy)冊. (38)
The load–slip relationship of the connector is represented by the equation (Fig.
7(c))
Q ⫽ Qu·(1 ⫺ e−␤·s
)␣
(39)
where Qu is the ultimate load of the connector, and ␣ and ␤ are coefficients to be
determined from the experimental results [2,4,5,13].
Moreover, in order to consider the bond at the steel–concrete interface, the follow-
ing relationship is assumed (Fig. 7(d)) [12]:
␶ ⫽ ␶1·√s/s1 0 ⱕ s ⱕ s1
␶ ⫽ ␶1 s1 ⱕ s ⱕ s2
␶ ⫽ ␶1·冪
s ⫺ s2
su ⫺ s2
s2 ⱕ s ⱕ su
(40)
where the coefficients ␶1, s1, s2 and su have to be determined from experimental
results. In the procedure, the bond effect is concentrated in the nodes so it is simu-
lated as fictitious additional connectors with an equivalent load–slip relationship,
Qbi ⫽ ␶(s)·b·(⌬li⫺1 ⫹ ⌬li)/2, (41)
where ␶(s) derives from Eq. (40), b is the top flange width and ⌬li is the length of
the ith element (Fig. 2).
3. Comparison with experimental data
Any analytical model for complex nonlinear problems has to be verified by means
of experimental data to ensure its validity and degree of accuracy. In this section
four different beams are analyzed using COBENA and the results are compared with
the corresponding experimental data. The first two examples concern simply sup-
ported composite beams while the other two refer to two span continuous beams.

209
3.1. Simply supported beams
The numerical simulation with COBENA of two simply supported beams, based
on those tested by Chapman and Balakrishnan at Imperial College of London [14]
was carried out. The beams have a span length of 5490 mm, an I-shaped steel mem-
ber 305 mm deep (12⬙ ⫻ 6⬙ ⫻ 44 lb/ft BSB) and a concrete slab 152 mm thick
(1220 mm wide). The first beam (beam E1) is subjected to a midspan point load
and the other (beam U4) is subjected to uniformly distributed loads. Shear connectors
are arranged according to the longitudinal shear: headed studs (12.7 mm diameter)
in pairs at 120 mm pitch for beam E1 and 32 headed studs (19 mm diameter) ‘tri-
angularly’ distributed for beam U4. The details of the beams are reported in Fig. 8
and in Table 1. In Table 2 the material properties as well as the value of the coef-
ficients characterizing the constitutive relationships used in the numerical simulation
are summarized.
The coefficients for the load–slip relationship of shear connectors were derived
from experimental push-out tests carried out by Chapman and Balakrishnan [14] on
the types of studs which were used in the beams. Moreover, provided that bonding
Fig. 8. Geometrical characteristics of simply supported beams: test E1 (a) and test U4 (b) [14].

Table 1
Geometrical characteristics of simply supported test beams
Beam identification E1 U4
Span length (mm) 5490 5490
Loading type Midspan point load Uniformly distrib.
Concrete slab Thickness (mm) 152.4 152.4
Width (mm) 1220 1220
Steel beam Section 12⬙ ⫻ 6⬙ ⫻ 44lb/ft BSB 12⬙ ⫻ 6⬙ ⫻ 44lb/ft BSB
Area (mm2
) 8400 8400
Shear connectors Kind of studs 12.7 ⫻ 50 19 ⫻ 102
Distribution of studs Uniform in pairs Triangular in pairs
Number of studs 100 32
Longitudinal Top (mm2
) 200 200
reinforcement Bottom (mm2
) 200 200
Table 2
Material properties and constitutive coefficient values
Beam identification E1 U4
Concrete Compressive strength fc (MPa) 32.7 33.8
Tensile strength fct (MPa) 3.07 3.14
Peak strain in compression ⑀c1 0.0022 0.0022
Peak strain in tension ⑀ct1 0.00015 0.00015
Steel Yield stress (MPa) Flange 250 269
Web 297 301
Reinforcement 320 320
Ultimate tensile Flange 465 470
stress (MPa) Web 460 479
Strain–harden. Flange 0.00267 0.00196
strain ⑀sh Web 0.00144 0.00146
Elasticity modulus Es (MPa) 206 000 206 000
Strain–harden. modulus Esh (MPa) 3500 3500
Connection Qu (kN) 66 129
␤ (mm−1
) 0.8 1.3
␣ 0.45 0.65
Interface bond ␶1 (MPa) 1.5 1.5
s1 (mm) 0.05 0.05
s2 (mm) 0.15 0.15
su (mm) 0.60 0.60
was not prevented in test beams, account has been taken of its effect on the behavior
of the beam. In the experimental results [14] an average shear stress, calculated on
an elastic basis, at several loading phases was given. From these results it was poss-
ible to roughly estimate the coefficients for a bond relationship.
In the numerical simulation 26 elements per half span were used for both test
beams. Shear connectors were simulated in their actual position along the beam.

211
Figs. 9 and 10 show the comparison between numerical and experimental results
of beams E1 and U4, respectively. In particular the load versus midspan deflection
is plotted in Fig. 9(a) and Fig. 10(a), while the values of slip at the steel–concrete
interface along the beam axis are plotted in Fig. 9(b) and Fig. 10(b) for various
loading levels. The plots show the good agreement between the analytical results
and the experimental data. The small differences in the slip curves are likely due to
the bond relationship. In fact this relationship is extrapolated from the few average
experimental information [14] and is the same along the whole beam.
3.2. Continuous beams
In order to verify the numerical model in the presence of negative moments, two
continuous beams were studied. The beams, tested experimentally by Teraszkiewicz
[15] and by Ansourian [16], were simulated with the numerical model. Teraszkiew-
icz’s beam (beam CBI) has two equal spans of 3354 mm, an I-shaped steel member
Fig. 9. Comparison between numerical and experimental results of beam E1: deflection history (a), and
slip distribution along span at various load levels (b).

Fig. 10. Comparison between numerical and experimental results of beam U4: deflection history (a),
and slip distribution along span at various load levels (b).
152 mm deep (6⬙ ⫻ 3⬙ ⫻ 12 lb/ft BSB) and a concrete slab 60 mm thick (610 mm
wide). Stud shear connectors (9.5 mm diameter) are distributed in pairs at 146 mm
pitch along the beam. Ansourian’s beam (beam CTB4) has two equal spans of 4500
mm, an H-shaped steel member 190 mm deep (HEA 200) and a concrete slab 100
mm thick (800 mm wide). Stud connectors (19 mm diameter) are equally spaced in
trials at 350 mm along the beam except over the internal support (1050 mm both
sides) where the pitch reduces to 300 mm. Both beams are loaded symmetrically
with point loads at midspan. The details of the beams are reported in Fig. 11 and
in Table 3. The material properties as well as the value of the coefficients characteriz-
ing the constitutive relationships used in the numerical simulation are listed in
Table 4.
In these simulations bonding was not considered because the experimental beams
were greased at the steel–concrete interface to prevent bonding. A total number of
25 elements and 29 elements per span were used for beam CBI and beam CTB4,

213
Fig. 11. Geometrical characteristics of simply supported beams: test CBI (a) and test CTB4 (b).
Table 3
Geometrical characteristics of continuous test beams
Beam identification CBI CTB4
Span length (mm) 3354 4500
Loading type Midspan point load Midspan point load
Concrete slab Thickness (mm) 60 100
Width (mm) 610 800
Steel beam Section 6⬙ ⫻ 3⬙ ⫻ 12 lb/ft BSB HEA 200
Area (mm2
) 2276 5380
Shear connectors Kind of studs 9.5 ⫻ 50 19 ⫻ 75
Number of studs 96 84
Pitch of studs (mm) Sag 146 350
Hog 146 300
Longitudinal Hog top (mm2
) 445 804
reinforcement Hog bottom (mm2
) – 767
Sag top (mm2
) – 160
Sag bottom (mm2
) – 160

Table 4
Material properties and constitutive coefficient values for continuous test beams.
Beam identification CBI CTB4
Concrete Compressive strength fc (MPa) 46.7 34.0
Tensile strength fct (MPa) 3.89 3.15
Peak strain in compression ⑀c1 0.0022 0.0022
Peak strain in tension ⑀ct1 0.00015 0.00015
Steel Yield stress (MPa) Flange 301 236
Web 301 238
Ultimate tensile Flange 470 393
stress (MPa) Web 470 401
Strain–harden. Flange 0.012 0.018
strain ⑀sh Web 0.012 0.018
Reinforcement 0.010 0.010
Strain–harden. Steel beam 2500 3000
modulus Esh Reinforcement 2500 3500
Elasticity modulus Es (MPa) 206 000 206 000
Connection Qu (kN) 32.4 110
␤ (mm−1
) 4.72 1.2
␣ 1.0 0.85
respectively. As the beams were symmetric, only one half of the beams was modeled
to save computational time.
The comparison between some results of the simulation of beam CBI and the
corresponding experimental results is shown in Fig. 12. In particular the deflected
shape (Fig. 12(a)), the slip along the steel–concrete interface (Fig. 12(b)) and the
longitudinal strain profile along the lowermost fiber of the beam flange (Fig. 12(c))
are plotted. In compliance with experimental results these quantities were plotted at
W ⫽ 121.6 kN, which corresponds to 80.8% of the experimental ultimate load Wu
⫽ 150.5 kN. The predicted bearing capacity of the beam was 146.8 kN which is
within 2.5% of the corresponding experimental value.
In Fig. 12 the experimental results of the right span are plotted upon the results
of the left span to facilitate comparison with the numerical results. It can be noted
that the curve of the analytical results lies almost always among the experimental
results of the two halves of the beam.
The comparison between some results of the simulation of beam CTB4 and the
corresponding experimental results is shown in Fig. 13. In particular the load versus
sagging curvature (Fig. 13(a)) the load versus hogging curvature (Fig. 13(b)) and
the load versus midspace deflection relationships (Fig. 13(c)) are plotted. The sagging
curvature is evaluated at midspan and the hogging curvature is evaluated at 150 mm
from the support. It can be observed that the numerical curves and the experimental
results match rather closely, verifying the accuracy of the simulation.

215
Fig. 12. Comparison between numerical and experimental results of beam CBI: deflected shape (a), slip
distribution along span (b) and strain profile along the span in the bottom flange (c) at W ⫽ 121.6 kN.
4. Conclusions
The following conclusions can be drawn.
앫 The numerical procedure presented herein allows to consider the actual nonlinear

Fig. 13. Comparison between numerical and experimental results of beam CTB4: curvature at sagging
(a) and hogging (b) sections, and deflection history (c).

217
behavior of the component materials of steel and concrete composite beams as
well as the load–slip relationship of shear connectors.
앫 The shear transfer between concrete slab and steel beam occurs only where con-
nectors are located (in the absence of bonding) and it is then possible to study
beams with partial shear connections which normally means large connector pitch-
es.
앫 The arc-length method, implemented in the procedure, allows the collapse load
to be reached even for those structures whose load–deflection curve has either a
perfectly plastic or a softening behavior. This target is difficult to reach with many
procedures available in the literature (e.g. [2,6]).
앫 The program, in contrast to many other similar procedures, allows consideration
of bonding at the steel–concrete interface which may be useful, in some cases,
to predict its effect on the structural behavior.
앫 The favorable comparisons between the numerical results and the experimental
results available in the literature allows to state that this procedure is capable of
tracing the detailed response of composite beams over the whole loading range
up to failure, provided failure is not initiated by buckling (Figs. 9, 10, 12 and 13).
앫 Programs such the one described can be used to conduct extensive parametric
studies in order to better understand the inelastic response of continuous composite
beams with complete or partial shear connections.
Acknowledgements
The financial support of the Italian Ministry of University and Scientific Research
(M.U.R.S.T.) is gratefully acknowledged.
References
[1] Commission of the European Communities. Eurocode No. 4: Part 1. Design of composite steel and
concrete structures: General rules and rules for buildings. Revised draft, March 1992.
[2] Johnson RP, Molenstra N. Partial shear connection in composite beams for buildings. Proc Inst Civil
Eng, Part 2 1991;91:679–704.
[3] Aribert JM. Design of composite beams with partial shear connection. Mixed structures including
new materials, IABSE Symposium Report, Brussels, 1990;215–220.
[4] Gattesco N, Pittolo A. The influence of connection deformability on the behavior of composite
beams subjected to repeated loads. Studi e Ricerche, Corso di Perfezionamento per le Costruzioni
in Cemento Armato F.lli Pesenti, Politecnico di Milano, Vol. 9, 1987;343–364.
[5] Aribert JM, Aziz KA. Calcul des poutres mixtes jusqu’à l’etat ultime avec un effect de soulèvement
à l’interface Acier-Béton. Construct Métall 1985;4:3–36.
[6] Aribert JM, Aziz KA. Modèle général pour le calcul des poutres mixtes hypertatiques jusqu’à la
ruine. Construct Métall 1986;4:3–41.
[7] Ricks E. An incremental approach to the solution of snapping and buckling problems. Int J Solids
Struct 1979;15:524–51.
[8] Crisfield MA. A fast incremental iterative solution procedure that handles ‘snap-through’. Comput
Struct 1981;13:55–62.

[9] Padovan J, Moscarello R. Locally bound constrained Newton–Raphson solution algorithms. Comput
Struct 1986;23(2):181–97.
[10] Ramm E. Strategies for tracing the nonlinear response near limit points. In: Bathe KJ, Stein E,
Wunderlich W, editors. Europe–US Workshop on Nonlinear Finite Element Analysis in Structural
Mechanics, Ruhr Universitat Bochum, Germany. Berlin: Springer, 1980;63–89.
[11] Batoz JL, Dhatt G. Incremental displacement algorithms for nonlinear problems. Int J Num Meth
Eng 1979;14:1262–6.
[12] C.E.B., Model Code 1990. Final Draft, July 1991, Bull. No. 203.
[13] Gattesco N, Giuriani E. Experimental study on stud shear connectors subjected to cyclic loading. J
Construct Steel Res 1996;38(1):1–21.
[14] Chapman JC, Balakrishnan S. Experiments on composite beams. Struct Eng 1964;42:369–83.
[15] Teraszkiewicz JS. Static and fatigue behavior of simply supported and continuous composite beams
of steel and concrete. PhD thesis, University of London, 1967.
[16] Ansourian P. Experiments on continuous composite beams. Proc Inst Civil Eng, Part 2
1981;71:25–51.

Analytical modeling of nonlinear behavior of composite beams with deformable connection

More Related Content

What's hot (19)

Similar to Analytical modeling of nonlinear behavior of composite beams with deformable connection (20)

Recently uploaded (20)

Analytical modeling of nonlinear behavior of composite beams with deformable connection