Behaviour of composite plate girders with partial interaction.
Yatim, Y. ; Shanmugam, Nandivaram E. ; Badaruzzaman, Wan 等
Introduction
Steel-concrete composite plate girders display greater strength and
stiffness compared to the corresponding bare steel plate girder acting
alone. This can be attributed to the contribution by concrete slab
which, when added to the post-buckling strength of the thin webs results
in larger shear strength (Baskar et al. 2002). Composite action is
achieved when the concrete slab is firmly fixed to the top flange of the
steel girder by mechanical forms of shear connection. The performance of
composite girders is governed by the effectiveness of interaction
between the interconnected elements. The possible loss in shear
connection stiffness is mainly due to the deformation of shear
connectors which permits relative slip at steel-concrete interface (Nie,
Cai 2003; Queiroz et al. 2007). The behaviour of composite members
depends on the type of shear connection between the two materials. Rigid
shear connectors usually develop full composite action between the
individual materials, thus the conventional principle of analysis of
full interaction can be applied. In contrast, flexible shear connectors
permit development of partial composite action due to strain
incompatibility at the interface and therefore, the analysis procedures
require consideration of the interlayer slip between the materials.
Horizontal shear force exerted at the interface is transferred from one
element to another through the deformable connectors.
Owing to the complex mathematical problems in partial interaction
theory (Johnson 2004), composite design is simplified by assuming
perfect interaction between steel and concrete elements. In fact, the
presence of slip induces significant additional curvature where
ignorance of this effect may result in inaccurate predictions of load
carrying capacity and deflection of composite girders. The flexible
nature of shear connectors indicates that imperfect interaction always
exists even in full composite design and reduced levels of shear
connection results in increase of available rotations in the joint
region of a continuous beam (Oehlers et al. 1997; Uy, Nethercot 2005).
Moreover, the need for partial interaction design is essential when the
top flange of girder cannot accommodate the number of shear connectors
required for full interaction.
A considerable amount of research has been directed in the past
towards the study of composite plate girders. Allison et al. (1982)
tested composite plate girders subjected to combined bending and shear.
A procedure to determine the shear strength was developed in accordance
with the Cardiff tension field method (Porter et al. 1975; Evans et al.
1978). Small-scale thin webbed composite girders with diagonal
stiffeners at end panels were tested to failure by Porter and Cherif
(1987). Methods for predicting shear strength of such girders are also
presented. Shanmugam and Baskar (2003) carried out tests on composite
plate girders to investigate the shear strength and concluded that the
tension band width in webs increased due to composite action. A number
of composite girders subjected to negative bending were also tested to
failure (Baskar, Shanmugam 2003).
[FIGURE 1 OMITTED]
Second-order differential equation allowing for slip in composite
beams was first developed by Newmark et al. (1951) by assuming equal
curvature between the interacting elements. Experimental studies were
conducted on composite beams subjected to single concentrated load
applied at the mid-span. Expressions for slip and deflection were
derived accounting for imperfect interaction. Adekola (1968) presented
an interaction theory for composite beams allowing for interface
friction, slip and uplift deformation. Fourth- and second-order coupled
differential equations were derived and solved by finite difference
method. It was found that the uplift deformation is insignificant.
Further works on partial interaction in simply supported composite beams
(Bradford, Gilbert 1992; Xu, Wu 2007) and continuous ones (Seracino et
al. 2004, 2006) have been reported in the past.
An approximate method is developed in this paper to determine the
load-deflection relationships and to predict the ultimate strength of
simply supported composite plate girders with partial interaction. The
girder may be subjected to concentrated loads or uniformly distributed
load on the entire span. The degree of interaction is specified by
varying the longitudinal spacing of shear connectors along the span.
Nevertheless, the uplift deformation or separation between elements is
negligible as it is assumed that, both elements deflect equally with the
same amount of curvature along the length. The long-term effects such as
creep and shrinkage in the concrete are also disregarded herein.
[FIGURE 2 OMITTED]
1. Analytical method
In a composite plate girder with transverse stiffeners spaced at a
distance b along the span, consider a finite length, dx of the web panel
near the support as shown in Figure 1(a). The normal force acting at
steel-concrete interface may be disregarded since the uplift deformation
is not taken into account in this study. Free body diagrams of the
finite length of the web panel and the concrete slab are shown along
with the internal forces in Figure 1(b).
In the above figures, [M.sub.c] and [M.sub.s] are, respectively,
moment carried by concrete slab and steel girder, [V.sub.c] and Vs are
the corresponding shear forces, F is the compressive or tensile force
exerted on concrete or steel, [F.sub.s] is the tensile membrane forces
in the web introduced in the post-buckling stage and q is the horizontal
shear force at the steel-concrete interface. The cross-section A-A'
of the composite girder along with the strain distribution across the
depth is shown in Figure 2. Three different load conditions viz. single
concentrated load applied at mid-span, uniformly distributed load along
the entire span and symmetrically placed two concentrated loads,
considered in the present study, are shown in Figure 3(a)-(c). In this
figure, x refers to distance from the left support of any section
A-A' along the span.
[FIGURE 3 OMITTED]
1.1. Slip expressions
Equilibrium consideration of horizontal forces acting on concrete
slab and steel girder in Figure 1(b) gives:
dF/dx = -q + [[F.sub.s]cos[theta]/b], (1)
where [theta] is the angle of inclination of tensile membrane
forces in the web panel. Assuming that the amount of slip, s, is
directly proportional to q, one may write:
q = ks/p, (2)
where k is shear stiffness of the connectors and p is the
longitudinal pitch between connectors along the span length of the
girder. Distribution of strain along the depth of concrete slab and
steel girder is assumed to be linear as shown in Figure 2. Strain at the
bottom of concrete slab, [[epsilon].sub.cb], and that at the top of
steel, [[epsilon].sub.st], are given as:
[[epsilon].sub.cb] = [M.sub.c][y.sub.c]/[E.sub.c][I.sub.c] -
F/[E.sub.c][A.sub.c]; (3)
[[epsilon].sub.st] = -[M.sub.s][y.sub.s]/[E.sub.s][I.sub.s] +
F/[E.sub.s][A.sub.s], (4)
where [E.sub.c], [I.sub.c] and [A.sub.c] are referred to as
Young's modulus, second moment of area and cross-section area of
concrete, respectively, and Es, Is and As are the respective values for
steel. The rate of change in slip along the steel-concrete interface
(Nie, Cai 2003) may be calculated as:
ds/dx = [[epsilon].sub.cb] - [[epsilon].sub.st]. (5)
Employing the curvature compatibility principle (Newmark et al.
1951), the curvature may be expressed as:
[d.sub.2]Y/[dx.sup.2] = [M.sub.c]/[E.sub.c][I.sub.c] =
[M.sub.s]/[E.sub.s][I.sub.s] = [M - [Fd.sub.c]]/[[E.sub.c][I.sub.c] +
[E.sub.s][I.sub.s]], (6)
where Y is the total deflection of the entire section, M is the
moment carried by the entire section and dc = yc + ys. In view of Eqns
(3), (4) and (6), the relative slip strain in Eqn (5) may be expressed
as:
ds/dx = [alpha](M - [Fd.sub.c]) - [beta]F (7)
and
[d.sup.2]s/[dx.sup.2] = [alpha] [dM/dx - [d.sub.c](dF/dx)] - [beta]
(dF/dx), (8)
where [alpha] = [d.sub.c]/([E.sub.c][I.sub.c] + [E.sub.s][I.sub.s])
and [beta] = ([E.sub.c][A.sub.c] +
[E.sub.s][A.sub.s])/[E.sub.c][A.sub.c][E.sub.s][A.sub.s]. For girders
subjected to a single concentrated load as in Figure 3(a), the general
expressions for moment at any section along the span are:
[M.sub.p/1] = Px/2 for 0 [less than or equal to] x [less than or
equal to] L/2; (9)
[M.sub.p/2] = P/2 (L - x) for x > L/2. (10)
Substitution of Eqns (1), (2) and (9) into Eqn (8) yields:
[d.sup.2]s/[dx.sup.2] = [alpha] [[P/2] -
[d.sub.c]([F.sub.s]cos[theta]/b - ks/p)] - [beta]([F.sub.s]cos[theta]/b
- ks/p). (11)
Integrating Eqn (11) twice, with boundary conditions ds/dx= 0 at x
= 0 and s = 0 at x = L/2, the slip expression may be simplified as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)
Similarly, with Eqn (10) for moment and satisfying boundary
conditions ds/dx = 0 at x = Land s = 0 at x = L/2, slip solution for
right-hand side of the applied load reduces to:
[s.sub.2] = [alpha]P + 2[psi] ([F.sub.s]cos[theta]/b)/(4/[(x -
L/2)(3L/2 - x)]) + 2[psi]K, for x > L/2 (13)
where [psi] = [ad.sub.c] + [beta] and K = k/p.
As for uniformly distributed load case, similar procedures may be
carried out using the respective moment expressions. Satisfying the
boundary conditions ds/dx= 0 at x = 0 and L and s = 0 at x = L/2, the
following slip expressions are obtained for 0 [less than or equal to] x
[less than or equal to] L/2 and L/2 < x [less than or equal to] L,
respectively:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (14)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)
For a girder subjected to two point loads of equal magnitude, P as
shown in Figure 3(c), slip expressions may be derived with appropriate
boundary conditions as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)
1.2. Slip-induced deflection
The interlayer slip induces additional curvature when a composite
girder bends. The slip-induced curvature may be obtained from (Nie, Cai
2003):
[d.sup.2]([DELTA]y)/[dx.sup.2] = 1/H ([d.sub.s]/dx). (18)
For the girder under single concentrated load, substituting the
first derivative of Eqn (12) into Eqn (18) and performing double
integration with boundary conditions d ([DELTA]y)/dx = 0 at x = L/2 and
[DELTA]y = 0 at x = 0, the expression for slip-induced deflection, Ay
can be obtained as:
[DELTA]y = 1/H[psi]LKb[phi] [[alpha]PbL/2 -
[psi][LF.sub.s]cos[theta]] x [2 [tanh.sup.-1] ([psi]Kx/[psi]) + [phi]x],
(19)
where [phi] = [square root of (8 + [psi][KL.sup.2]/4) [psi]K)].
Similarly, the expressions for [DELTA]y derived using the
appropriate boundary conditions for the girders subjected to uniformly
distributed load or two concentrated loads may be given, respectively,
as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)
where [bar.[omega]] = [square root of ((8 +
[psi][KL.sup.2])[psi]K)]
and
[DELTA]y = 1/H[psi]LKb[phi] [[alpha]Pb(2L - [a.sub.1] - [a.sub.2])
- [psi][LF.sub.s]cos[theta]] [2 [tanh.sup.-2] ([psi]Kx/[phi]) - [phi]x],
(21)
where [phi] = [square root of ((2 + [psi][Ka.sup.2.sub.1])[psi]K)].
2. Strength parameters
2.1. Shear strength of concrete slab
One of the approximate methods to predict the strength of composite
plate girders is by simply adding the shear strength of concrete slab to
the shear capacity of steel girder acting alone (Narayanan et al. 1989).
To account for partial shear connection, the following relationship is
introduced assuming that the degree of interaction does not affect the
shear strength of steel plate girder:
[V.sub.cc] = [V.sub.a] + [N/[N.sub.f]] ([V.sub.f] - [V.sub.a]),
(22)
where [V.sub.cc] is the shear resistance of the concrete slab in
girders with partial interaction, [V.sub.a] is the shear resistance of
concrete slab alone (Eurocode 2 2004), [V.sub.f] is the shear resistance
of concrete slab for girders with full interaction, N is the actual
number of shear connectors provided and [N.sub.f] is the number of shear
connectors required to achieve full shear connection (Johnson 2004). It
should be noted that [V.sub.f] is taken equal to the pull-out capacity
of shear connectors (Liang et al. 2004). For full shear connection, i.e.
N/[N.sub.f] = 1, Eqn (22) reduces to [V.sub.cc] = [V.sub.f].
Results from extensive finite element analyses on composite plate
girders with different degrees of interaction showed that approximately
40% of the slab shear strength, denoted as [V.sub.cc,e], occurs within
the elastic stage with remaining percentage, denoted as [V.sub.cc,pb],
in the post-buckling phase. Thus, in the analysis presented herein, the
contribution in the elastic and post-buckling phases, by the concrete
slab to the shear capacity of the girders is assumed in the same
proportion as that predicted by the finite element results.
2.2. Buckling load
At the elastic stage, critical shear strength, [V.sub.cr] that
causes buckling in the web plate is taken as the sum of web panel shear
resistance and the strength contribution by concrete slab in the elastic
phase, [V.sub.cc,e] i.e.:
[V.sub.cr] = [[tau].sub.cr]dt T [V.sub.cc,e]; (23)
where d is the depth of the web panel, t is the web panel thickness
and [[tau].sub.cr] is the critical shear stress of web panel given as:
[[tau].sub.cr] = C [[[pi].sup.2][E.sub.s]/12(1 - [v.sup.2])]
[(t/d).sup.2], (24)
where C is the buckling coefficient and v is the Poisson's
ratio of web material. For a simply supported girder subjected to
concentrated load at mid-span, one can simply determine the load
corresponding to buckling from:
[P.sub.cr]/2 = [[tau].sub.cr]dt + [V.sub.cc,e]. (25)
From equilibrium, the shear carried by each of the supports of a
composite girder at any load beyond elastic buckling may be expressed
as:
[V.sub.C] = [V.sub.cr] + [V.sub.cc,pb] + [F.sub.s] sin [theta],
(26)
where [V.sub.cc,pb] is shear strength contributed by the concrete
slab beyond elastic buckling phase. In view of Eqn (23) and since
[V.sub.cc] = [V.sub.cc] e +[V.sub.cc,pb], Eqn (26) may be written as:
[V.sub.C] = [[tau].sub.cr]dt + [V.sub.cc] + [F.sub.s] sin [theta].
(27)
For the simply supported girder with a concentrated load applied at
the mid-span, the applied load, P may be computed from:
P/2 [[tau].sub.cr]dt + [V.sub.cc] + [F.sub.s] sin [tau]. (28)
2.3. Tension field force in the web panel
In accordance with the tension field theory, once the web plate has
lost its capacity to sustain any further increase in compressive stress,
a new load carrying mechanism is developed. Further loading beyond
buckling is supported by an inclined tensile membrane field in the web.
The resultant tensile force is given as:
[F.sub.s] = [[sigma].sub.t][tb.sub.tf], (29)
where [[sigma].sub.t] is the tensile membrane stress and [b.sub.tf]
is the width of diagonal tension band developed in the web panel
determined in accordance with the Cardiff mechanism (Porter et al. 1975;
Evans et al. 1978). Rearranging Eqn (28), an expression for Fs can be
obtained as
[F.sub.s] P - 2([[tau].sub.cr]dt + [V.sub.cc])/2sin[theta], (30)
where when substituted in Eqn (29) yields at as:
[[sigma].sub.t] = P - 2([[tau].sub.cr]dt + [V.sub.cc])/2[tb.sub.tf]
sin[theta]. (31)
It should be noted that in the elastic stage, [F.sub.s] = 0.
Additionally, the applied load, P beyond elastic limit is sustained by
the tensile membrane stress, [[sigma].sub.t] in the web panel.
3. Load-deflection behaviour
3.1. Effective flexural stiffness
The bending stiffness of a composite section is significantly
governed by the shear connection stiffness. Even in a composite design
with full interaction, the interface shear transfer is not completely
perfect and stiffness may be reduced to some extent due to flexibility
of shear connectors. In this study, effective bending stiffness,
[EI.sub.eff] is employed in accordance with the principle suggested by
Girhammar (2009). The flexural stiffness for composite section with
perfect interaction may be written as:
[EI.sub.[infinity]] = [EI.sub.0] +
[EA.sub.p][d.sup.2.sub.c]/[EA.sub.0], (32)
where [EI.sub.0] is the flexural stiffness of the section with no
interaction, [EA.sub.p] is the product of axial stiffness of the
subelements and [EA.sub.0] is the sum of the axial stiffness of the
sub-elements. To account for partial interaction, the effective bending
stiffness is computed as:
[EI.sub.eff] = [EI.sub.0] + [xi]
([EA.sub.p][d.sup.2.sub.c]/[EA.sub.0]), (33)
where [xi] = [[1 + [[pi].sup.2]
[EA.sub.p]/[KL.sup.2][EA.sub.0]].sup.-1] for simple support conditions.
It should be noted that [EI.sub.eff] remains constant throughout
the elastic phase. Upon buckling, the rigidity changes proportional to
the applied load. The [EI.sub.eff] value at post-buckling state should,
therefore, be computed at every load increment using tangent modulus,
[E.sub.t] (Timoshenko, Gere 1961) in place of constant Es value in Eqn
(32). Tangent modulus may be expressed as:
[E.sub.t] = [E.sub.s] [[sigma].sub.yw] -
[[sigma].sub.[theta]]/[[sigma].sub.yw] - [omega][[sigma].sub.[theta]],
(34)
where [[sigma].sub.yw] is the yield stress of web material, [omega]
is a constant and [[sigma].sub.[theta]] is the resulting tensile stress
given as:
[[sigma].sub.[theta]] = [[tau].sub.cr] sin2[theta] +
[[sigma].sub.t] (35)
[FIGURE 4 OMITTED]
3.2. Total deflection
Deflection of the whole composite section is computed at every load
increment. The total deflection, Y is the sum of bending deflection,
[y.sub.b], shear deflection, [y.sub.v] and slip-induced deflection,
[DELTA]y. For a girder under single point load applied at the mid-span,
deflection at the elastic and post-buckling stages may be expressed,
respectively, as:
[Y.sub.e] = [PL.sup.3]/48[([EI.sub.eff]).sub.e] + PL(1 +
v)/2[E.sub.s][A.sub.w] + [DELTA][y.sub.e]; (36)
[Y.sub.pb] = [PL.sup.3] 48[([EI.sub.eff]).sub.Pb] + PL(1 +
v)/2[E.sub.t][A.sub.w] + [DELTA][y.sub.pb], (37)
where [Y.sub.e], [DELTA][y.sub.e] and [([EI.sub.eff]).sub.e] refer
to total deflection, slip-induced deflection and effective flexural
stiffness, respectively, in the elastic stage, [Y.sub.pb],
[DELTA][y.sub.pb] and [([EI.sub.eff]).sub.pb] are the respective values
in the post-buckling stage, v is the Poisson ratio and [A.sub.w] is the
shear area. The contribution by the concrete slab is ignored for shear
deflection.
Substituting in Eqns (36) and (37), the relevant parameters for the
composite plate girder subjected to a single concentrated load applied
at the mid-span, load-deflection curve OBCD can be plotted as shown in
Figure 4. Point B refers to the upper limit of the elastic buckling
stage given by Eqn (25). Beyond this stage, the load-deflection response
exhibits different behaviour compared to the unbuckled state. The change
in slope shown by curve BC is due to the reduced flexural rigidity
calculated from Eqn (33) which induces larger deflection even for small
increment in P when approaching the yield point. The girder reaches its
capacity at the point C when the resulting stress, [[sigma].sub.[theta]]
obtained from Eqn (35) reaching the yield stress of the web panel,
[[sigma].sub.yw]. Further increase in deflection does not result in
appreciable increase in applied load, P beyond this point. Thus, the
load-deflection curve levelled off as shown by CD. Similarly, the
load-deflection plots for other load conditions may be obtained. The
ultimate load for the girder is obtained as the load corresponding to
the peak point of the load-deflection plot.
4. Accuracy of the proposed method
Four composite plate girders namely CPG 1, CPG 2, CPG 7 and CPG 8
tested earlier by other researchers (Shanmugam, Baskar 2003; Baskar,
Shanmugam 2003) were employed in the present study in order to establish
the accuracy of the proposed method and to assess the influence of
partial interaction. The relevant details of the girders are given in
Table 1. These girders were originally tested under concentrated load
applied at the mid-span. In the current study, all these girders were
analysed by the proposed method and also by finite element modelling
using the finite element package LUSAS. Two different loading conditions
viz., single concentrated load at the mid-span or uniformly distributed
load over the entire span were considered in the analyses. Another
composite girder, namely CPG 1-A was also introduced and analysed under
two symmetrical concentrated loads.
Initially, these analyses were carried out for girders with full
interaction with K = 0.65 kN/[mm.sup.2]. Additionally, analyses were
also carried out on all the girders with four different values of K,
obtained by changing the spacing of shear connectors along the span
length of the girder. Brief description of the finite element modelling
is given in the following section.
5. Finite element analysis
Three-dimensional finite element models were developed using the
finite element package, LUSAS for all the girders. Shell and brick
elements were used for steel part of the girders and concrete slabs,
respectively. Both elements are compatible for nonlinear analysis which
allows buckling and second-order effects. The steel plate girders were
modelled as elastic-perfectly plastic using mild steel material with
Poisson's ratio of 0.3. Young's modulus, [E.sub.s] for S275
steel is taken as 200 GPa while the yield stresses assigned to the
flanges, webs and stiffeners vary from 272 MPa to 300 MPa in accordance
with those reported in the experiments. The geometrical properties are
as per those given in Table 1. Specifically, concrete slab was idealised
by hexahedral isoparametric element with six degrees of freedom at each
node. All material properties assigned for concrete are also based on
the experimental data. Strains corresponding to the maximum uniaxial
compressive stress and that to the softening end where concrete crushes
were adopted as 0.0022 and 0.0035, respectively. Additionally, joint
element with specified spring stiffness was assigned at the unmerged
steel-concrete interface accordingly to allow for horizontal slide.
Total Lagrangian strain formulation along with Crisfield's load
incremental procedure was adopted to account for geometric non-linearity
in the analysis. Basically, the global stiffness matrix of the structure
depends on the displacement increments where the solution of the
equilibrium equations is typically accompanied by an iterative method
through the convergence check. In the present models, the non-linear
Newton-Raphson iterative approach was used by updating the tangent
stiffness matrix for each of the iterations (Zubydan, ElSabbagh 2011).
[FIGURE 5 OMITTED]
A perfectly straight and undeformed model may be stiff and provide
different response compared to a model with imperfect geometry. Thus, an
imperfection model has been built in LUSAS by loading the results from
buckling analysis in which the deformed mesh was considered as initially
imperfect geometry of the girders (Basher et al. 2011; Chen, Jia 2010).
The buckling analysis predicts the possible deformed shapes due to
structural instability. The subspace iteration algorithm approximation
technique, available in LUSAS facilities, was employed for solving the
associated eigenvalue problems. Different deformed shapes were attempted
for non-linear analysis. From extensive trials, a mode shape from the
first eigenvalue was selected as it provides satisfactory results in
terms of ultimate load and behaviour. A typical mesh as shown in Figure
5 with element size of 50x50 mm was adopted in the analyses. The mesh
was chosen based on convergence studies carried out to ascertain the
efficiency and effectiveness to provide accurate solutions within an
acceptable computational time.
6. Discussion on the results
Results in terms of ultimate loads and load-deflection behaviour
were obtained from the proposed method for the girders. The analyses by
finite element modelling provided a detailed output from which the
ultimate loads and load-deflection plots were extracted. The results for
ultimate loads are presented in Tables 2-4 for different load
conditions. The ultimate loads obtained by the proposed method are
compared with the corresponding finite element values, as shown by the
ratios [P.sub.u]/[P.sub.u, LUSAS] or [w.sub.u]/[w.sub.u, LUSAS] in the
tables, in order to establish the accuracy of the proposed method. Also,
ultimate loads for girders with different K values are compared with the
corresponding values for the girder with full interaction viz. K= 0.65
kN/[mm.sup.2], as shown by the ratios [P.sub.u, partiaiint]/[P.sub.u,
fuinnt.], so as to assess the influence of partial interaction on the
ultimate strength. Comparison of the ultimate loads shows that the two
values are close within the acceptable level of accuracy. In Tables 2-4,
the ratio [P.sub.u]/[P.sub.u, LUSAS] varies from 0.90 to 1.10 indicating
that the two values viz. [P.sub.u] and [P.sub.u, LUSAS]l ie within [+ or
-] 10%. It is, therefore, confirmed that the proposed method is capable
of predicting the shear strength with sufficient accuracy.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
Comparison of shear strength values for girders with partial
interaction with those corresponding to the girders of full interaction
is shown in the tables in terms of the ratios [P.sub.u, partial
int.]/[P.sub.u full int]. In Tables 2 and 3, composite girders with K=
0.65 kN/[mm.sup.2] and the one in Table 4 with K= 0.74 kN/[mm.sup.2] are
considered as those with full interaction. The remaining girders in
which K values are less are considered as those with partial
interaction. It is clear from the tables that the ultimate shear
strength drops with reduction in degree of interaction. For example, in
the girder CPG 1 subjected to a concentrated load shown in Table 2, the
girder with K= 0.17 kN/[mm.sup.2], shows 17% drop in the shear strength
compared to the one with K= 0.65 kN/[mm.sup.2]. This girder under
uniformly distributed load also displays same amount of drop in shear
strength. The drop is larger viz. 19% in the case of CPG 7 having longer
span length. Similar reduction in load carrying capacity can be observed
in all the girders with smaller K values. The results presented in the
tables also show that the proposed method is capable of predicting the
shear strength within the acceptable level of accuracy for all the
girders with partial interaction, K value ranging from 0.65 to 0.17
kN/[mm.sup.2]. The average value of the ratio [P.sub.u]/[P.sub.u, LUSAS]
in all cases varies from 0.96 to 0.99.
Additionally, the accuracy of the proposed method has also been
assessed by comparing the predicted load-deflection behaviour with the
corresponding results obtained from the finite element analyses. Typical
results presented in Figure 6 show the variation of mid-span deflection
with the applied load for selected girders with different K values. It
can be seen from the figures that the two results are very close to each
other from the initial stage of loading to the ultimate load condition.
The observation is true for different loading conditions and for
different K values. The results show that the proposed method is also
capable of predicting the behaviour with sufficient accuracy.
The flexural behaviour of the girders obtained by the proposed
method is also illustrated in Figures 7-9, in which load-deflection
plots are shown for girders subjected to single concentrated load
applied at the mid-span, uniformly distributed load and two point loads,
respectively. In each of the figures, load-deflection curves for girders
having different values of K are presented in order to show the extent
of influence of this parameter on the behaviour of the girders. It is
clear from the figures that ultimate load drops with different rate and
magnitude of drop as the value of K is reduced from full degree of
interaction to negligible amount of interaction. Stiffness of the
girders at the initial stages of loading is marginally affected with the
variation in the K values. This observation is true irrespective of the
girders and the loading patterns.
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
Conclusions
An approximate method to determine the behaviour and flexural
capacity of composite plate girders with partial shear connection is
presented in this paper. The girders considered include those subjected
to concentrated or uniformly distributed loads. The accuracy of the
method has been established by comparing the results with the
corresponding results obtained by finite element method using LUSAS
software. Additionally, effect of varying the degree of interaction has
been examined by analysing the girders with different degrees of
interaction. It is found from the results presented herein that the
proposed method is accurate enough to predict the behaviour of composite
plate girders under different types of loading and that the method could
account for variation in degree of interaction. It is observed that the
bending stiffness and load carrying capacities of the girders reduced
with decreasing degree of interaction. The drop in ultimate load with
degree of interaction varies with the loading type, significant in some
cases and negligible in certain cases.
Acknowledgement
The authors acknowledge the support provided by the Department of
Civil and Structural Engineering, Universiti Kebangsaan, Malaysia.
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Md Y. YATIM, Nandivaram E. SHANMUGAM, Wan BADARUZZAMAN Department
of Civil and Structural Engineering, Universiti Kebangsaan Malaysia,
43600 UKM Bangi, Selangor, Malaysia
Received 20 Dec 2011; accepted 8 Feb 2012
Corresponding author: Nandivaram Elumalai Shanmugam
E-mails:
[email protected];
[email protected]
Md Y. YATIM. A PhD candidate at the Department of Civil and
Structural Engineering, Kebangsaan University, Malaysia. His research
interests include steel-concrete composite structures, computational
mechanics, structural concrete and steel, high-rise structures, etc.
Nandivaram E. SHANMUGAM. A Professor at the Department of Civil and
Structural Engineering, Kebangsaan University, Malaysia. He obtained his
PhD degree from the University of Wales (Cardiff) in 1978. He has taught
at undergraduate and graduate levels for more than 45 years. He has
published more than 200 scientific papers in international journals and
conference proceedings. He is a member of the editorial board of Journal
of Constructional Steel Research, Thin-Walled Structures, Journal of
Structural Stability and Dynamics, International Journal of Steel
Composite Structures, International Journal of Steel Structures and IES
Journal Part A: Civil and Structural Engineering. He is a Chartered
Engineer (CEng), Fellow of the Institution of Structural Engineers,
London, (FIStructE), Fellow of the Royal Institution of Naval Architects
(FRINA), Fellow of the American Society of Civil Engineers (FASCE),
Fellow of the Institution of Engineers, Singapore (FIES) and Fellow of
the Institution of Engineers, India (FIEI). His research interests
include steel plated structures, steel-concrete composite construction,
long-span structures and connections, cold-formed steel structures,
elastic and ultimate load behaviour of steel structures, etc.
Wan BADARUZZAMAN. A Professor at the Department of Civil and
Structural Engineering, Kebangsaan University, Malaysia. He received his
PhD degree from the University of Wales (Cardiff) in 1994 and has taught
at undergraduate and graduate levels for more than 25 years. He is the
author and the co-author of many scientific papers in international
journals and conference proceedings. He is a corporate member of the
Institution of Engineers, Malaysia (MIEM). His research interests
include behaviour of lightweight composite structures, industrialised
building system, reinforced concrete design and construction,
computational analysis, etc.
Table 1. Geometrical properties of the girders
Specimens L Panel aspect Slenderness Web,
(mm) ratio (b/d) ratio (d/t) t (mm)
CPG 1 2400 1.5 250 3
CPG 2 2400 1.5 150 5
CPG 7 4800 1.5 250 3
CPG 8 4800 1.5 150 5
CPG 1-A 3655 1.5 250 3
Specimens Flanges (top and bottom) Shear connectors
[t.sub.f] [b.sub.f] [t.sub.s] [d.sub.s]
(mm) (mm) (mm) (mm)
CPG 1 20 200 100 19
CPG 2 20 260 100 19
CPG 7 12 160 100 19
CPG 8 20 235 100 19
CPG 1-A 20 200 100 19
Specimens Reinforced
concrete slab
(a)[b.sub.c]
x [h.sub.c]
(mm)
CPG 1 1000x150
CPG 2 1000x150
CPG 7 1200x150
CPG 8 1200x150
CPG 1-A 1000x150
(a) [b.sub.c] denotes effective width of reinforced concrete slab.
Table 2. Comparison of ultimate loads for girders under single
concentrated load at mid-span
Specimens K Pu [P.sub.u, partialint.]/
(kN/ (kN) [P.sub.u, fullint.]
[mm.sup.2])
0.65 827 1.0
0.51 787 0.95
CPG 1 0.37 746 0.90
0.31 726 0.88
0.17 686 0.83
0.65 1355 1.0
0.51 1315 0.97
CPG 2 0.37 1276 0.94
0.31 1255 0.92
0.17 1216 0.90
0.65 754 1.0
0.51 714 0.94
CPG 7 0.38 675 0.90
0.32 653 0.86
0.19 612 0.81
0.65 1266 1.0
0.51 1238 0.98
CPG 8 0.38 1210 0.96
0.32 1195 0.94
0.19 1166 0.92
Specimens [P.sub.u, LUSAS] [P.sub.u]/
(kN) [P.sub.u, LUSAS]
836 0.99
818 0.96
CPG 1 793 0.94
715 1.02
698 0.98
1371 0.98
1324 0.99
CPG 2 1282 0.99
1226 1.02
1171 1.04
760 0.99
736 0.97
CPG 7 683 0.98
625 1.04
588 1.04
1285 0.98
1281 0.96
CPG 8 1271 0.95
1236 0.96
1134 1.02
Table 3. Comparison of ultimate loads for girders under uniformly
distributed load
Specimens K (kN/ [w.sub.u] [w.sub.u, partialint.]/
[mm.sup.2]) (kN/m) [w.sub.u, fullint.]
CPG 1 0.65 344 1.0
0.51 328 0.95
0.37 311 0.90
0.31 302 0.88
0.17 286 0.83
CPG 2 0.65 565 1.0
0.51 549 0.97
0.37 532 0.94
0.31 524 0.93
0.17 507 0.90
CPG 7 0.65 157 1.0
0.51 148 0.94
0.38 140 0.89
0.32 135 0.86
0.19 127 0.81
CPG 8 0.65 264 1.0
0.51 258 0.98
0.38 252 0.95
0.32 249 0.94
0.19 243 0.92
Specimens [W.sub.u, LUSAS] [w.sub.u]/
(kN/m) [w.sub.u, LUSAS]
CPG 1 368 0.93
335 0.98
299 1.04
276 1.09
270 1.06
CPG 2 614 0.92
589 0.93
578 0.92
483 1.08
470 1.08
CPG 7 174 0.90
162 0.91
150 0.93
144 0.94
116 1.09
CPG 8 291 0.91
280 0.92
269 0.94
262 0.95
221 1.10
Table 4. Comparison of ultimate loads for girders under two
concentrated loads
Specimens K (kN/ Pu (kN) [P.sub.u, partialint.]/
[mm.sup.2]) [P.sub.u, fullint.]
CPG 1-A 0.74 409 1.0
0.60 389 0.95
0.43 369 0.90
0.36 359 0.88
0.21 338 0.83
Specimens [P.sub.u, LUSAS] [P.sub.u]/
(kN) [P.sub.u, LUSAS]
CPG 1-A 430 0.95
401 0.97
385 0.95
370 0.97
346 0.97
