GRILLAGE ANALYSIS OF COMPOSITE CONCRETE SLAB ON
STEEL BEAMS WITH PARTIAL INTERACTION
Prof. Dr. Husain M. Husain1, Dr. Ali N. Attiyah2 and Jenan Ni’amah Yasser3
ABSTRACT: The present study is concerned with the behavior of a composite structure
made up of a concrete slab connected to steel beams in two directions by shear connectors by
taking into consideration the linear action of shear connectors in the force-slip relationship.
The grillage or grid framework method as simplified method of analysis is used in this study
to study slip, deflection and stresses caused by moments from applied normal loads. A
method is suggested to derive the required section rigidities (the flexural and torsional
rigidities) of the grillage members from the composite action of the individual grillage
composite members. Design charts are constructed for estimating the percentage decrease in
flexural rigidity of each composite member with partial shear connection. It was found for a
composite structure analyzed by grillage members, the effective width of each member should
be used to calculate the flexural rigidity of that member. Also Poisson’s ratio effect was
included in the calculation of the flexural rigidities of the grillage members. Effect on
deflections by transverse shearing forces was found to be small and thus it can be neglected
(percentage differences is less than 11.8 %).
1. Introduction
Each different building material has a special prominent quality which distinguishes it
from other materials. There is no material that can provide all the structural
requirements. This is the reason of using different materials that can be arranged in an
optimum geometric configuration, with the aim that only the desirable property of
each material will be utilized by virtue of its designated position. The structure is then
known as a composite structure, and the relevant method of building as composite
construction.
The composite concrete slab-on-steel beam structure consists of three major structural
elements, namely a reinforced concrete slab resting on longitudinal and transverse
steel beams, which interact, compositely with the slab by means of mechanical shear
connectors. The analysis of composite beams and their behavior assuming linear and
nonlinear material and shear connector behavior has been in general based on an
approach initiated by Newmark, Siess and Viest in 1951[1]. The equilibrium and
compatibility equations for an element of the beam were reduced to a single second
1) University of Tikreet , College of Engineering , Civil Department.
2) University of Kufa , College of Engineering , Civil Department.
3) University of Kufa , College of Engineering , Civil Department.
1
order differential equation in terms of either the resultant axial force in the (concrete)
flange or the interface slip. Solution for the axial force or the interface slip was
substituted back into the basic equilibrium and compatibility equations, which could
then be solved to give the displacements and the strains throughout the beam. That
approach was initially based on linear material and shear connector behavior.
In the method suggested for the present study, the composite structure is idealized as a
grillage, the grillage mesh is assumed to be coincident with the center-lines of the
main steel beams. The concrete slab and the steel beams are assumed to behave in the
elastic range and the force- slip behavior of the shear connectors is linear. To use the
T-beam approach, the concept of the effective width is used which refers to a fictitious
width of the slab that when acted on by the actual maximum stress the slab would
have the same static equilibrium effect as the existing variable stress. The effective
width is affected by various factors, such as the type of loading, the boundary
conditions at the supports and the ratio of beam spacing to span B/L [2].
Johnson (1975)[3] proposed a partial interaction theory for simply supported composite
beams, in which the analysis was based on elastic theory. Kennedy, Grace, and
Soliman (1989)[4] presented an experimental study that was conducted on three
composite bridge models each subjected to one- vehicle load. Jasim (1994)[5]
presented a method of analysis which depended on elastic theory. In that analysis he
adopted same assumptions of Newmark[1].
2. Assumptions of the Grillage Analogy
The grillage analogy involves the representation of effectively a three- dimensional
composite structure by a two- dimensional assemblage of discrete one- dimensional
interconnected beams in bending and torsion. In analysis, the following assumptions
are introduced:
1- Concrete and steel are linearly elastic materials. The concrete slab is assumed
to be able to sustain sufficient tension such that no tensile cracks develop in
this part. The distribution of strains through the depth of each component is
linear.
2- The longitudinal and transverse steel beams are assumed rigidly connected
(welded connections).
2
3- The shear connection between the two components is continuous along the
length. The discrete deformable connectors with equal moduli and uniform
spacing are assumed to be replaced by a medium of negligible thickness.
Friction and bond effects between the two components are neglected.
4- The amount of slip permitted by the connector is directly proportional to the
force transmitted through the connector.
5- At every section of the composite beams, each component deflects the same
amount. No separation is assumed to occur.
3. Evaluation of Elastic Section Rigidities of Grillage Members
The idealization of a composite slab–beam structure by an equivalent grillage requires
the evaluation of the elastic section rigidities of the grillage members. The elastic
rigidities of these members should be derived from the section properties of the actual
composite slab–beam structure so that an adequate picture for the composite section
behavior under the applied loadings can be obtained from the equivalent grillage. The
elastic section rigidities required for the sections of the equivalent composite grillage
members are as follows:
1-Bending (or flexural) rigidity (EI).
2-Torsional rigidity (GJ).
3-Shearing
rigidity (GAv).
Herein, suggestions are presented for these quantities and adopted in this work.
3.1 Bending (or Flexural) rigidity:
Flexural rigidities of the equivalent grillage members play an important role in the
calculation of deflections and in the distribution of moments. In analyzing the
composite slab-beam structure by the grillage analogy, the flexural rigidities of the
composite members are derived from partial interaction theory. Generally, two factors
(besides the partial interaction effect) must also be considered in the calculation of the
flexural rigidity of the grillage members. These factors are due to the shear lag and
Poissonُs ratio effects. Shear lag effects can be included by using the effective width
concept. The two–dimensional confining effect of Poisson’s ratio can be considered
by dividing the modulus of elasticity of concrete E1 by (1-υ2).The interaction
phenomenon can be illustrated from the discussion of the lower and upper limits of
behavior of composite beams, i.e., no interaction and complete (or full) interaction,
3
respectively. The analysis and flexural rigidity will be carried out on the basis of
elastic theory.
Usually, the interaction between steel and concrete is incomplete due to the
occurrence of slip. It produces a discontinuity in the strain distribution at the interface
where appreciable strain difference. The neutral axis of the slab is closer to the beam
and that of the beam is closer to the slab, when compared with the no- interaction
case. The result of the partial interaction is the partial development of the compressive
force in the concrete slab and tensile force in the steel beam. This leads to less
ultimate load than that resisted when complete interaction exists. Partial interaction is
the usual practical case in the design and analysis of composite structures.
A large number of research studies have been devoted to calculate the deflections of
composite beams with partial shear interaction.
The solution submitted by Jasim[5] for the final form of the governing equation for a
composite beam by using Fourier series method will be adopted in the present study to
calculate the flexural rigidity of composite sections for simply supported beams under
different loading cases.
Uniformly Distributed Load
For the case of uniformly distributed load on a simply supported beam the solution for
the maximum deflection is:
yp
yf
Where
 1
yf 
24C3  1 1

  1  cosh K  tanh K  sinh K 
2 2 K
5K 

5 w  L2

384 2  
(1)
= the mid-span deflection of composite beam with full shear
connection ‚ w is the displacement in z-direction, L is the span length, E2 is the
modulus of elasticity of steel, I is the moment of inertia of the transformed fully
composite section about the elastic neutral axis assuming uncracked section, yp is the
mid-span deflection of composite beam with partial shear connection,
C3 
h1  h2 2  1 .1. 2 . 2 and
C122


 41 .1   2 . 2  1 .1   2 . 2 
1t   2    1  1 
 1t  2 
4
K
C1  L
2
.
(2)
Where C12 is the depth of center of gravity of steel beam below mid-plane of slab, I1,
I1t and I2 are the moments of inertia of concrete slab about its own centroid,
transformed area of concrete about its own centroid, steel beam about its own
centroid, respectively. A1 , A1t and A2 are the cross sectional area of concrete slab,
transformed area of concrete above interface, cross sectional area of steel beam,
respectively.h1 is the thickness of concrete slab and h2 is the depth of steel beam. 1 is
the effective modulus of elasticity for concrete slab due to lateral confinement of slab
and E2 is the modulus of elasticity of steel. C1 is a factor found from
C1
=
2

.n  1
1
C12




  1 .1  2 . 2 1 .1   2 . 2 
(3)
Where n is the number of connectors per row and p is the spacing of connectors along
the beam.

d2y


Since
is the curvature , then the integration of this equation twice results

dx 2
in
y  
w  L4

(4)
5
where λ is a factor depending on the boundary conditions (   384 for simply
supported beams under uniform load w per unit length).
Thus
yp
yf

  w  L4
 f

 p
  w  L4
(w is the applied load) or
yp
yf

 f
 p
(5)
By substituting Eq. (5) into Eq. (1), then
 f
24  C3  1 1

 1
   1  cosh K  tanh K  sinh K 
2
 p
5K
2 K

Defining D1 
24  C3
5K 2
(6)
1 1

   1  cosh K  tanh K  sinh K  and substituting this into
2 K

Eq.(6) ,then this equation can be written as
 p 
 f
(7)
1  D1 
Point Load at Mid-span
For the case of a point load at mid-span of a simply supported beam, the solution for
the maximum deflection is:
5
yp
yf
 1
3  C3  1

 1   tanh K 
2 
K
 K

(8)
w  L3
48   2  
where yf 
By using the same procedure, the pertinent equation is
 f
3 C  1

 1  23  1   tanh K 
 p
K
 K

Using the notation D2 
 p 
(9)
3  C3
K2
 1

 1   tanh K  Eq. (9) reduces to
 K

 f
(10)
1  D2 
Point Load at ¼ Span
For this loaded case the maximum deflection is
yp
yf
 1
96  C3  sinh K 2  sinh K


 cosh K  

2 
11K
 tanh K

 K
1

4
(11)
11  w  L3
y

Where f 768    
2
Defining D3 
 p 
96  C3  sinh K 2  sinh K
 1

 cosh K    , then

2
11K  K
 tanh K
 4
 f
(12)
1  D3 
Distributed Load of Trapezoidal Shape
For this case of loading the pertinent equation is
yp
yf
where
 1
48  C3  1  sinh K 1  1 


  
5K 2  K 2  sinh 2K 2  4 
yf 
(13)
5 w1  w 2   L4

768
2  
Using notation D4  48  C2 3   12   sinh K  1   1 
5K
K
 sinh 2 K
2  4
Eqs. (5) and (13) are combined to give
 p 
 f
(14)
1  D4 
6
Boundary Conditions
Furthermore, the effect of two types of boundary conditions on the prediction of
flexural rigidity of a composite beam is studied. They are a beam with fixed ends and
a cantilever. The effect of different boundary conditions can be considered by
changing the beam effective length. This effect should be included in Eq.(2) by
replacing the beam span (L)with the beam effective length (Le) . For the fixed –
ended beam , the beam effective length is half its span, Le= 0.5 L. For the cantilever,
Le = 2L. Thus Eq. (2) may be rewritten as:
K
C1  Le
(15)
2
Effect of Load Pattern
The following three load patterns were studied: (1) a concentrated load at the beam
center; (2) a concentrated load at ¼ span; and (3) trapezoidal distributed load.
Comparisons were made between these types of load patterns with the uniformly
distributed load to find the flexural rigidity of composite beam with partial interaction
(EIp). Results are presented for a representative composite beam 8.6m in span with
universal steel section UB 305×127×37 and concrete flange 1500mm in width and
150mm in depth. The Youngُs moduli of steel and concrete were taken as 205000
N/mm2 and 25000 N/mm2, respectively. Connector stiffness k = 180000 N/mm and
spacing P = 520 mm. Tab.(1) shows the maximum difference between the uniformly
distributed load case and other pattern load cases for EIp value. In all cases, the
difference is less than 1.3%, thus Eq. (7) may be used for all loading cases to obtain
the flexural rigidity of a composite beam with partial interaction. This means that for
each value of factor C the values of D1, D2, D3, and D4 are almost equal for the
majority of K2 values. A discrepancy occasionally occurs in D1 and it is about 1%.
This leads to the conclusion that the same chart may be used for all types of loads
which in turn greatly simplifies the calculations needed in design [5].
Thus, Fig.(1) shows such a chart for various values of factor C and in terms of the
percentage increase in flexural rigidity of composite beam with partial shear
7
connection and the parameter K2 in this chart is for simply supported beams, Figs (2)
and (3) are design charts to find D1 for fix-ended beam and cantilevers respectively.
Tab.(1): Maximum difference in EIp between uniformly distributed load case and other load cases
(a) Simply supported beam
(b) Beam with fixed ends
Point
Load
arrangement
Uniform
load
Central
point
load
load
at ¼
Point
Trapezoidal
Uniform
load
load
span
Central
point
load
load
Trapezoidal
at ¼
load
span
Maximum
difference
between
flexural
0
0
(reference
1.20
0.62
0
value)
(reference
value)
1.02
0.54
0
rigidities
(%)
3.2 Torsional Rigidity of a Composite Section
It is hypothesized that the strength and the stiffness of composite sections under
torsion are to be considered as that of an open section consisting of two parts acting
independently, i.e., the upper part consisting of the reinforced concrete section with
the upper flange of the steel I-section firmly attached to it, and the lower part
consisting of the web and the lower flange of the steel I-section, as shown in
Fig.(4). Based on this hypothesis the stiffness of a composite section is evaluated in
the pre – cracked stage as follows [8]:
The upper part of the composite section is divided into three portions, two equal
concrete portions of dimensions (bce×h1) and a central composite portion of
dimensions (bs × (h1+tf)), as shown in Fig. (5). The torsional stiffness of the upper part
may then be estimated from the following Eq. for the interior composite beam
G TP J TP 

1
2b ce  h 13  G 1  b s h 1  t f 3 G eq
2

(16)
and for the edge beam
G TP J TP 

1
b ce  h 13  G 1  b s h 1  t f 3 G eq
2

8
(17)
Where G eq 

 eq
2 1   eq

‚
neq: equivalent Poissonُs ratio of central portion of the upper part, (  eq
 0.15 )
Eeq: equivalent modulus of elasticity of central portion of the upper part of composite
section,  eq 
1  h 1   2  t f
h1  t f
β2 is a coefficient is a function of (b/a)[9] and b is the longer dimension of the
rectangular cross section and a is the shorter dimension of the rectangular cross
section. The torsional stiffness of the lower part may be estimated as follows
G s J sd 
a. Free to warp:


1
b s  t 3f  h  t 3w G 2
3
(18)
1
3
3
b. Warping prevented (or restrained): G s J sd  J s G 2  b s  t f  G 2
Here
Cw
L
J s 
G 2  m
‚ m
 L 



   L  2 tanh 1 
3  1
2 
C w   2  1 
is the warping constant, C w 
 J G
‚ 1   s 2
 Cw 2
(19)
1
2
 ‚


h  t f 2  t f  b 3s
24
In this work, the case of warping being prevented will be used, and the torsional
rigidity of a composite section can be calculated from the following equation
GJ P  G TP J TP  G s J sd
(20)
This hypothesis is giving an experimental to theoretical ratio of (0.95)[10].
3.3 Shearing Rigidity
Distortion by transverse shearing forces is one of the modes of deformation that can
occur in a composite structure when it is subjected to a general loading. The vertical
(or transverse) shearing force across a composite section causes the flanges and webs
to bend independently out of plane (as a result of shearing deformation). It is known
that the transverse shearing deformation is usually small compared with deformation
due to bending. But in some cases, such as in short deep members subjected to high
shearing forces, it is necessary to consider the transverse shearing deformation in
order to obtain a more accurate description of the behavior of the beam. A shearing
9
rigidity (GAV) is assigned to the stiffness matrix of a grillage member to take into
account the effect of transverse shearing forces on the deformation of that member.
In the grillage analogy, the ability of the composite structure to resist distortion can be
approximately achieved by providing the grillage members an equivalent shear area
(AV). The independent bending moments, which are developed in the webs and in the
flanges are caused by the shearing forces generated in these components. However, in
the present work, the transverse shearing rigidity for a composite member will be
computed by two methods as follows
1- Shearing rigidity for the steel component only by calculating the shear area for the
steel web, Fig.(6a), and it can be stated as:
Gv  G2  tw  h2
(21)
2- Shearing rigidity for concrete and steel components together because the depth of
concrete may take into account the shear area especially when it is not small.
Recognizing that the transformed section concept can be applied to the steel web as
shown in Fig. (6b), thus this method can be stated as:
GA v  G 2  m  t w h 1  h 2 
(22)
Where m is the modular ratio = E2/E1
3
2
1.5
1
2.5
2.5
2
2
C=2.
5
1.5
C=2
1
C=1.5
0.5
Value of D1
Value of D1
2.5
C=3.
C=3.5
5
C=3
C=3.5
C=3
C=2
.5
C=
2
C=1.
5
C=1.
1.5
0.5
1
0.5
C=1.25
25
0
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
Log10K2
-0.5 10K2
Log
Fig. (1) Design chart for simply supported
beams.
10
Fig. (2) Design chart for fix- ended beams.
3
C=3.5
Value of D1
2.5
C=3
2
C=2.5
1.5
1
h1
C=2
0.5
tf
C=1.5
20
C=1.25
-2.5
-1.5
-0.5
0.5
1.5
2.5
3.5
Log10 K2
Fig. (4): Shear stress flow in composite sections.
Fig. (3) Design chart for cantilevers.
bce
bs
bce
bce
tf
(a) Composite action (b) Independent action
bs
h1
h1
h1 h1
tf
(a) Interior beam
mtw
tf
tw
h2
(b) Edge beam
(b) transformed area
Fig. (5): Evaluation of pre – cracked stiffness for
upper part division.
(a) Steel area
Fig. (6): transverse shearing rigidity.
4. Applications
A composite slab-beam structure is selected from the available reference to assess the
accuracy of the grillage method. The theoretical results of Kennedy model[4] were
derived by the finite element method using the orthotropic plate element; also an
experimental study was made for this model. The composite slab-beam model
considered here is simply supported at two opposite edges and being free at the
longitudinal edges. This type of construction is used in bridge decks. The structure
dimensions are shown in Fig.(7), and material properties are as follows
Upper Component (concrete slab)
Depth of concrete h1= 48 mm.
Compressive strength of concrete f´c = 35 N/ mm2
Modulus
of
elasticity
of
concrete
E 1=
27806
N/
mm2
(calculated
 c  4700 f c N mm 2 )
Poisson’s ratio of concrete υ1= 0.15
Shear modulus of elasticity of concrete G1= 12090 N/ mm2 (calculated from
G =E/2(1+ υ)).
11
from
h2
Lower Components (Longitudinal and transverse steel Shear Connectors (stud shear connectors)
Depth of steel beam h2 = 152.2 mm
Length of shear connector = 38 mm
beams)
Flange width of steel beam bs = 152.2 mm
Diameter of shear connector = 12 mm
Thickness of flange of steel beam tf = 6.6 mm
According
to
(OHBD)
code
Thickness of web of steel beam tw = 5.84 mm
Number of connectors per row n =2
requirements
Cross sectional area of steel beam A2 = 2858 mm2
Spacing P=180 mm.
Moment of inertia of steel beam I2 =12112334.49
Strength of shear connector = 57000 N.
Modulus of elasticity of steel beam E2 = 200000 MPa
mm4
Poisson’s ratio of steel beam υ2 =
0.3
mmmmmmmmmmmmmmmm2
Shear modulus of elasticity of steel beam G2= 76923 N/ mm2 (calculated from G =E/2(1+υ)).
Connector stiffness may be conservatively estimated as the secant stiffness at the
shear connector design strength with an equivalent slip of 0.8 mm [11], hence k =
57000/ 0.8 = 71250 N/ mm.
Evaluating the elastic rigidities for each grillage member as given in section(3)
1- For longitudinal members:
a-edge beams:
(EIp= 0.6 EIf = 2.9×1012 N.mm2) , (GJ= 2.6 ×1011 N.mm2).
b-interior beams: (EIp= 0.5 EIf = 3.4×1012 N.mm2) , (GJ= 3.0×1011 N.mm2).
2- For transverse members in this model it is assumed that the flexural rigidity is the
average value between fully and zero interaction as follows, taking the effective of
the concrete slab in the longitudinal direction equal 0.5b as shown in Fig.(11)[4]:
EIp = 0.5(EIf + EIo)
But if there are shear connectors between the concrete slab and the transverse steel
beam, the value of the flexural rigidity must be estimated by the same method
represented in section 3.5.1, thus:
a-for edge beams:
(EIp= 0.5( EIf + EIo)= 4 ×1012 N.mm2) ,
(GJ= 3.6 ×1011 N.mm2).
b-for interior beams: (EIp= 0.5( EIf + EIo)= 4 ×1012 N.mm2) ,
(GJ= 3.5 ×1011 N.mm2).
The shearing rigidity is constant for all grid members and it can be calculated as
shown in section 3.5.3, thus
GAv = 101.67 N (for transformed shear area) ,or: GAv = 68.37 N (for steel
shear area)
12
Two different loading conditions are considered. Point load of 89 kN is applied, the
position of this load is given in the following
1-A center load applied over the bridge (point no. 13, Fig.(7)).This is the first
loading condition.
2-An eccentric load applied over the edge of the bridge (point no. 3, Fig.(7)).
This is the second loading condition.
In Fig.(8), the vertical deflections at the mid- span cross- section (section A-A) are
plotted for the first loading condition. The corresponding values of the deflections for
the second loading condition are plotted in Fig. (9).Tab. (2) shows the comparisons of
the maximum deflections in the composite structure as calculated by the suggested
method for the two loading conditions. In the grillage analysis the maximum
deflections in both cases of loading are calculated for:
Case (I): without transverse shear effect. , Case (II): with transformed shear area.
, Case (III): with steel shear area only.
Tab. (2):Comparisons of maximum deflections (composite bridge model) (percentage differences
with respect to experimental results)
1st loading
Method of analysis
Grillage
Case (I)
Case (II)
analogy
Case (III)
Orthotropic plate method
[4]
Experimental
[20] result
2nd loading
Max.
Percentage
Max.
Percentage
Deflection
Difference
Deflection
Difference
(mm)
(%)
(mm)
(%)
3.30
3.57
3.69
3.30
2.80
+17.90
+27.50
+31.80
+17.90
-
7.86
8.27
8.47
7.50
7.13
+10.0
+15.9
+18.8
+5.2
-
From the above comparison, it is clear that when the effect of transverse shear area
(Av) is ignored the deflections obtained by the grillage analogy are rather in
acceptable agreement with the experimental and finite element results (applied to the
equivalent orthotropic plate). Also this effect is shown in Figures (8) and (9), and it is
well known that an eccentric load on a bridge gives rise to twisting moments that are
much greater in magnitude than those caused by the same load applied at the center.
Thus, the concrete deck slab, with its significant torsional resistance, is able to
13
distribute transversely the eccentric load quite effectively in composite bridges.
Comparisons between the results are also given in Tabs. (3) and (4).
Comparisons between the variations of center deflection with an applied central load
shown in Fig. (10).
Tab. (3):Vertical deflections (in mm) at mid- span of bridge model under 1st. loading condition
(percentage differences with respect to experimental results)
Node
no.
23
18
13
8
3
Perce.
Exper.
Ortho.
Diff.
(%)
2.54
2.67
2.8
2.67
2.54
1.91
2.29
3.30
2.29
1.91
-24.8
-14.2
+17.9
-14.2
-24.8
Grill.
case I
2.3
2.9
3.3
2.9
2.3
Perce.
diff.
(%)
-9.5
+8.6
+17.9
+8.6
-9.5
Grill.
case II
2.38
3.06
3.57
3.06
2.38
Perce.
diff.
(%)
-6.3
+14.6
+27.5
+14.6
-6.3
Perce.
Grill.
case III
2.41
3.12
3.69
3.12
2.41
diff.
(%)
-5.10
+16.8
+31.8
+16.8
-5.10
Tab. (4):Vertical deflections (in mm) at mid- span of bridge model under 2nd. loading
condition (percentage differences with respect to experimental results)
Perce.
Node
Exper.
Ortho.
no.
23
18
13
8
3
Diff.
(%)
-1.70
-0.30
2.16
4.33
7.13
-1.50
-0.29
2.30
4.69
7.50
+11.8
-4.0
+6.5
+8.3
+5.2
Grill.
case I
-1.28
0.36
2.30
4.80
7.86
Perce.
diff.
(%)
24.7
20.0
6.5
10.8
10.0
Grill.
case II
-1.32
0.37
2.38
4.98
8.27
Perce.
diff.
(%)
22.4
23.0
10.2
15.0
15.9
Grill.
case III
-1.34
0.38
2.41
5.07
8.47
Perce.
diff.
(%)
21.2
26.7
11.6
17.0
18.8
5. Effect of Degree of Interaction
The degree of interaction between the concrete slab and the steel beams may be
increased by increasing the number of shear connectors or by increasing the connector
stiffness. This increase leads to increase in the (EIp / EIf) ratio. Thus, in this section
various values of this ratio are assumed to study its effect on the same bridge model,
without including the transverse shear effect.
In Figures (11) and (12), the vertical deflections at the mid- span cross- section are
plotted for the first and second loading conditions respectively. It is clear that the
values of the vertical deflection decreased when the degree of interaction increased.
14
This increase is obtained for longitudinal beams. From this result, it is found that the
composite structure resistance is more efficient for applied load when the degree of
interaction is increased. Also a comparison between the results is shown in Tabs. (5)
and (6).
Tab. (5): Influence of degree of interaction on vertical deflections (in mm) for 1st. loading
condition
Node no.
23
18
13
8
3
EI= EIo
2.73
3.38
3.76
3.38
2.73
EIp= 0.7 EIf
1.70
2.26
2.60
2.26
1.70
EIp= 0.9 EIf
1.25
1.78
2.09
1.78
1.25
EI= EIf
1.09
1.61
1.92
1.61
1.09
Tab. (6): Influence of degree of interaction on vertical deflections (in mm) for 2nd.
loading condition
Node no.
23
18
13
8
3
EI= EIo
-1.45
0.49
2.73
5.57
8.99
EIp= 0.7 EIf
-1.33
0.04
1.704
3.95
6.80
EIp= 0.9 EIf
-1.09
-0.06
1.25
3.103
5.55
15
EI= EIf
-0.99
-0.083
1.09
2.799
5.09
2290 mm
.
.
24
.
25
17
.
18
.
19
.
20
.
12
.
13
.
14
.
15
6
.
7
.
8
.
9
.
10
1
.
2
.
3
.
4
.
5
16
.
.
11
.
.
.
0.5b=267.225
0.5b=267.225
A
48
x
3050 mm
(a)
152.2
y
B
254.17
.
.
b=534.45
B
A
23
22
21
(b)
48 mm
152.2 mm
(c)
Deflection (mm)
Fig. (7): Details of composite bridge model.(a) Plan view, (b) Section (A-A), (c) Section (B-B)
23
1.5
Node number
18
8
13
3
Experimental [ 4]
Orthotropic plate [ 4]
Grillage case I
Grillage case II
Grillage case III
2
2.5
3
3.5
4
4.5
76.1
610.55
1145
1679.45
Distance from left end (mm)
2213.9
Fig (8): Vertical deflections at mid-span section of bridge deck model under
1st.loading
condition
Node number
Deflection (mm)
23
-4
18
8
13
3
Experimental [4]
Orthotropic plate[4]
-2
0
2
4
Grillage case I
Grillage case II
Grillage case III
6
8
10
76.1
610.55
1145
1679.45
Distance from left end (mm)
2213.9
Fig. (9): Vertical deflections at mid-span section of bridge deck model under 2nd.loading
condition
16
Load at center ( kN)
100
90
80
70
60
50
40
30
20
10
0
Experimental
Grill. case I
Grill. case II
Grill. case III
0
0.3
0.6
0.9
1.2 1.5 1.8 2.1 2.4 2.7
Deflection at center ( mm)
3
3.3
3.6
3.9
Fig. (10): Load-deflection curve at center of Kennedy’s bridge deck model
Deflection (mm)
Node number
2
03
1
2
3
4
5
6
7
8
76.1
2213.9
1
8
3
8
13
EI=EIo
EIp=0.7 EIf
EIp=0.9 EIf
EI=EIf
Grill. case I
Experimental [4]
610.55
1145
1679.45
Distance from left end (mm)
Fig (11) Influence of degree of interaction on vertical deflections for 1st.loading condition
Node number
Deflection (mm)
23
18
13
8
3-4
-2
0
2
4
6
8
10
EI=EIo
EIp=0.7EIf
EIp=0.9EIf
EI=EIf
Grill. case I
Experimental [4]
76.1
2213.9
610.55
1145
1679.45
Distance from left end (mm)
Fig (12) Influence of degree of interaction on vertical deflections for 2nd.loading condition
17
6. Conclusions
The main concluding remarks that have been achieved in this study may be
summarized as follow
1. Design charts are constructed for estimating the percentage decrease in flexural
rigidity of each composite member with partial shear connection. The charts are in
terms of the parameter k2, and were given for various values of the factors C.
K2 
K  n  C  L2
1

4  P  2
it
, C
2

C12
1
  it
  it  1 it 
2. The loss of interaction between the concrete slab and the steel beams leads to
considerable increase in deflection (as the sum of flexural rigidities of the two
separate components is considerably smaller than the value for the connected
components). Almost fully interacting components give stiffer structure.
3. To calculate the flexural rigidity of the equivalent grillage members the case of
uniformly distributed load can be used in place of any loading case because the
difference between the results from different load patterns is negligible (less than
1.3%).
4. In representing a composite structure by grillage members, the effective width of
each member should be used to calculate the flexural rigidity of that member. Also
Poisson’s ratio effect is to be included in the calculation of the flexural rigidities of
the grillage members.
5. Effect of transverse shearing forces on deflection is found to be small and thus it
can be neglected (percentage differences is less than 11.8 %).
References:
1. Heins,C.P. and Fan,H.M., ”Effective Composite Beam Width at Ultimate
Load”, Journal of the Structural Division, Proc. of the ASCE, Vol.102, ST11,
pp. 2163-2179, Nov.1976.
2. Newmark,N.M.,Siess,C.P. and Viest,I.M., “Tests and Analysis of Composite
Beams with incomplete interaction”, Proc. Soc. Experimental Stress Analysis,
Vol.9, No.1, pp. 75-92 , 1951.
3. Johnson,R.P., “Composite Structures of Steel and Concrete: Vol.1”, Crosby
Lockwood Staples, London , 210pp. , 1975.
4. Kennedy,J.B.,Grace,N.F. and Soliman,M., “Welded- versus Bolted-Steel IDiaphrams in Composite Bridges”, Journal of the Structural Division, Proc. of
the ASCE, Vol.115, ST2, pp. 417, Feb.1989.
18
5. Jasim,N.A., “The Effect of Partial Interaction on Behaviour of Composite
Beams “, Thesis presented for the degree of Ph.D.,Department of Civil
Engineering, College of Engineering, University of Basrah, Iraq,
188pp.,Oct.1994.
6. Hendry,A.W. and Jeager,L.G., “The Analysis of Grid Framework and Related
Structures”, Chatto and Windus , London , 1958.
7. Gere,J.M.and Weaver,W.,”Analysis of Framed Structures”,Van Nostrand Co.,
New York,1958.
8. Hassan,F.M. and Kadhum,D.A.R., “Behaviour and Analysis of Composite
Sections under Pure Torsion”, Engineering and Technology, Vol.7, No.1, pp.
67-97,1989.
9. Timoshenko, S., “Strength of Materials :Part II”, Van Nostrand Co., New York,
1958.
10. Frodin,J.G., Taylor, R. and Stark, J.W.,”A Comparison of Deflection in
Composite Beams Having Full and Partial Shear Connection”, Proc.of Inst.of
Civil Engineers, Part 2,Vol.41,pp. 307-322,June1978.
11. Wang,Y.C., “Deflection of Steel-Concrete Composite Beams with Partial
Shear Interaction”, Journal of Structural Engineer,Vol.124,No.10,pp. 11591165,Oct.1998.
19
‫التحليل بطريقة املشبك للسقوف اخلرسانية والعتبات احلديدية املركبة ابستخدام أسلوب‬
‫التداخل اجلزئي‬
‫الخالصة‬
‫الدراسة احلالية هتتم بسلوك املنشآت املركبة املتكونة من سقف كونكرييت مربوط إىل عتباات دديدياة ابهاا‬
‫بواسطة روابط قص آخذة بنظر االعتبار الفعل اخلطي لروابط القاص‬
‫التاداخل اجلزئاي‬
‫اذد الدراساة‬
‫تقاادل للياال للانشااآت املركبااة ابسااتخدام طريقااة املشاابكات كطريقااة مبسااطة لدراسااة ا طااو وا‬
‫ااا ات‬
‫الناهاة ماان االااا املساالطة واقرتدااط طريقااة حلساااب القاايم املطلوبااة لالااا ة االدنااال ولااا ة اللااي لكاال‬
‫عضو مان ااعضاال ناان املشابك وبناالا علاذ كلاك اتاتقط اداو تالااياية حلسااب مقادار النقالاا‬
‫لا ة االدنال بسبب التداخل اجلزئي وو د‬
‫البحث احلايل إنه البد من اعتبار العرض املاثرر وإ خالاه‬
‫اتتقاق لا ة االدنال لكل عضاو إنشاائي وو اد كاذلك إ قرا قاو القاص علاذ قايم ا طاو قليال ادا‬
‫وال يتجاوز ‪ % 11.8‬وميكن إمهاله‬
‫‪20‬‬