1
GRILLAGE ANALYSIS OF COMPOSITE
CONCRETE
SLAB ON
STEEL
BEAMS WITH PARTIAL INTERACTION
Prof.
Dr.
Husain M. Husain
1
,
Dr. Ali N. Attiyah
2
and
Jenan Ni
’
amah Yasser
3
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 u
sed 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 memb
ers, 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 sh
earing forces was found to be small and thus it can be neglected
(percentage differences is less than 11.8 %).
1.
I
ntroduction
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 c
oncrete 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 connecto
r 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.
2
order differential equation in terms of either the resultant axial force in the (concrete)
flange or the inte
rface 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 initiall
y 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 con
crete 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 o
f 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 foll
owing 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 ea
ch component is
linear.
2
-
The
longitudinal
and transverse steel beams are assumed rigidly connected
(welded connections).
3
3
-
The shear connection between the two components is continuous along the
length.
The discrete deformable connectors with equal moduli an
d 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 th
e 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 equ
ivalent 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 pictur
e 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) ri
gidity (EI).
2
-
Torsional rigidity (GJ).
3
-
Shearing
rigidity (GA
v
).
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 im
portant 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. General
ly
, 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
Poisson
ُ
s ratio
effects. Shear
lag effects can be included by u
sing the effective width
concept. The two
–
dimensional confining effect of Poisson’s ratio can be considered
by dividing the modulus of elasticity of concrete E
1
by (1
-
υ
2
).The interaction
phenomenon can be illustrated from the discussion of the lower and up
per limits of
behavior of composite beams, i.e., no interaction and complete (or full) interaction,
4
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 compa
red 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 interac
tion 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 calculat
e
the deflections of
composite beams with partial shear interaction.
The soluti
on submitted by Jasim
[
5]
for t
he 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
differ
ent loading cases
.
Uniformly Distributed Load
For the case of uniformly distributed load on a simply supported beam the solution for
the maximum deflection is
:
K
sinh
K
tanh
K
cosh
1
K
1
2
1
K
5
C
24
1
y
y
2
3
f
p
(
1
)
W
here
2
2
f
L
w
384
5
y
= the mid
-
span deflection of composite beam with full shear
connection
‚
w is the displacement in z
-
direction, L is the span length, E
2
is the
modulus of elasticity of steel, I is the moment of inertia of the transformed fully
compos
ite section about the elastic neutral axis assuming uncracked section, y
p
is the
mid
-
span deflection of composite beam
with partial shear connection,
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
2
1
2
1
2
12
3
.
.
.
.
.
.
.
4
1
1
h
h
C
C
t
t
and
2
L
C
K
1
.
(2)
5
Where C
12
is the depth of cen
ter of gravity of steel beam below mid
-
plane of slab, I
1
,
I
1t
and I
2
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. A
1
, A
1t
and A
2
are the cross sectional area of concrete slab
,
transformed area of concrete above interface,
cross sectional area of steel beam,
respectively.
h
1
is the thickness of concrete slab and h
2
is the depth of steel beam.
1
is
the effective
modulus of elasticity for concrete slab due to lateral confinement of slab
and E
2
is the modulus of elasticity of steel.
C
1
is a factor found from
C
1
=
2
2
1
1
2
12
2
2
1
1
.
.
C
.
1
.
1
n
.
(
3
)
Where n i
s the n
umber of connectors per row
and p is t
he spacing of connectors along
the beam.
Since
2
2
dx
y
d
is the c
urvature
,
then
the
integration of this equation twice results
i
n
4
L
w
y
(
4
)
where
λ
is a factor depending on the boundary conditions
(
384
5
for simply
supported beams under uniform load w per unit length)
.
Thus
4
f
p
4
f
p
L
w
L
w
y
y
(w is the applie
d load)
or
p
f
f
p
y
y
(
5
)
By substituting Eq. (
5
) in
to
Eq. (
1
), then
K
sinh
K
tanh
K
cosh
1
K
1
2
1
K
5
C
24
1
2
3
p
f
(
6
)
Defining
K
K
K
K
K
C
D
sinh
tanh
cosh
1
1
2
1
5
24
2
3
1
and
substituting this into
Eq.(
6
)
,then
this
equation can be written as
1
1
D
f
p
(
7
)
Point Load at Mid
-
span
For the case of a point load at mid
-
span of a simply supported beam, the sol
ution for
the maximum deflection is
:
6
K
tanh
K
1
1
K
C
3
1
y
y
2
3
f
p
(
8
)
where
2
3
f
48
L
w
y
By using the same
procedure, the
pertinent e
quation is
K
tanh
K
1
1
K
C
3
1
2
3
p
f
(
9
)
Using the notation
K
K
K
C
D
tanh
1
1
3
2
3
2
Eq.
(
9
) reduces to
2
1
D
f
p
(
10
)
Point Load at ¼ Span
For this loaded case
the maximum deflection is
4
1
K
cosh
K
tanh
K
sinh
K
2
K
sinh
K
11
C
96
1
y
y
2
3
f
p
(
11
)
W
here
2
3
f
768
L
w
11
y
Defining
4
1
cosh
tanh
sinh
2
sinh
11
96
2
3
3
K
K
K
K
K
K
C
D
, then
3
1
D
f
p
(
12
)
Distributed Load of Trapezoidal Shape
For this case of loading
the pertinent equation
is
4
1
2
1
K
2
sinh
K
sinh
K
1
K
5
C
48
1
y
y
2
2
3
f
p
(
13
)
where
2
4
2
1
f
L
w
w
768
5
y
Using notation
4
1
2
1
2
sinh
sinh
1
5
48
2
2
3
4
K
K
K
K
C
D
Eq
s.
(
5
) and (
13
)
are combined to give
4
1
D
f
p
(
14
)
7
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 fix
ed
–
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
:
2
L
C
K
e
1
(
15
)
Eff
ect 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 uni
formly
distributed load to find the flexural rigidity of composite beam with partial interaction
(EI
p
). 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 an
d
150mm in depth. The Young
ُ
s modul
i
of steel and concrete were taken as 205000
N/mm
2
and 25000 N/mm
2
, respectively. Connector stiffness k = 180000 N/mm and
spacing P = 520 mm.
Tab
.
(1) shows the maximum difference between the uniformly
distributed load cas
e and other pattern load cases for EI
p
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 mean
s
that for
each value of fac
tor
C
the values of D
1
, D
2
, D
3
, and D
4
are almost equal for the
majority of
K
2
values.
A discrepancy occasionally occurs in D
1
and it is about 1%.
This leads to the conclusion that the same chart may be used for all types of load
s
which in turn greatly sim
plifies 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
8
connection and the parameter
K
2
in this chart i
s for simply supported beams, Figs (
2
)
and (
3
) are design charts to find
D
1
for fix
-
ended beam and cantilevers respectively.
Tab
.
(
1
): Maximum difference in EI
p
between uniformly distributed load case and other load cases
Load
arrangement
(a) Simply support
ed beam
(b) Beam with fixed ends
Uniform
load
Central
point
load
Point
load
at ¼
span
Trapezoidal
load
Uniform
load
Central
point
load
Point
load
at ¼
span
Trapezoidal
load
Maximum
difference
between
flexural
rigidities
(%)
0
(reference
value)
1.20
0.62
0
0
(reference
value)
1.02
0.54
0
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
in
dependently,
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 (b
ce
×
h
1
) and a
central composite portion of
dimensions (b
s
×
(h
1
+t
f
)), 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
eq
3
f
1
s
1
3
1
ce
2
TP
TP
G
t
h
b
G
h
b
2
1
J
G
(
16
)
and for the edge beam
eq
3
f
1
s
1
3
1
ce
2
TP
TP
G
t
h
b
G
h
b
1
J
G
(
17
)
9
W
here
eq
eq
eq
1
2
G
‚
n
eq
: equivalent Poisson
ُ
s ratio of central portion of the upper part
,
(
15
.
0
eq
)
E
eq
: equivalent modulus of elasticity
of central portion of the upper part of composite
section,
f
1
f
2
1
1
eq
t
h
t
h
β
2
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
a. Free to warp:
2
3
w
3
f
s
sd
s
G
t
h
t
b
3
1
J
G
(
18
)
b. Warping prevented (or restrained):
2
3
f
s
2
s
sd
s
G
t
b
3
1
G
J
J
G
(
19
)
Here
m
2
s
G
L
J
‚
2
L
tanh
2
L
C
1
1
3
1
2
w
m
‚
2
1
2
w
2
s
1
C
G
J
‚
C
w
is the
warping constant,
24
b
t
t
h
C
3
s
f
2
f
w
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
sd
s
TP
TP
P
J
G
J
G
GJ
(
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 whe
n 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 de
formation 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 desc
ription of the behavior of the beam.
A shearing
10
rigidity (GA
V
) 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
(A
V
). The independent bending moments, which are developed in the webs and in the
flanges are caused by the shearing forc
es 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
stee
l web,
Fig.(
6
a
), and it can be stated as:
2
2
h
t
G
G
w
v
(
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. (
6
b), thus this method can be
stated as:
2
1
w
2
v
h
h
t
m
G
GA
(
22
)
Where m is the m
odular ratio = E
2
/E
1
Fig. (
1
) Design chart for simply supported
beams.
5.2
5
5.2
5
5.2
Value of D
1
C=3.5
-0.5
0
0.5
1
1.5
2
2.5
3
-3
-2
-1
0
1
2
3
C=3
C=2.
5
C=2
C=1.5
C=1.25
C=
3
.
5
Log
10
K
2
Fig. (
2
) Design chart for fix
-
ende
d beams.
Value of D
1
5.2
5
5.2
5
5.2
-3
-2
-1
0
1
2
3
C=3.5
C=3
C=2
.5
C=1.
5
C=1.
25
C=
2
Log
10
K
2
11
4.
Applications
A composite slab
-
beam structure is selected from the available reference to assess the
accuracy of the grilla
ge 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 t
wo 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 h
1
= 48 mm
.
Compressive strength of concrete f
´
c
= 35 N/ mm
2
Modulus of elasticity of concrete E
1
= 27806 N/ mm
2
(calculated from
2
c
c
mm
N
f
4700
)
Poisson’s ratio of concrete
υ
1
= 0.15
Shear modulus of elasticity of concrete G
1
= 12090 N/ mm
2
(calculated fro
m
G =E/2(1+
υ
)).
(b) Edge beam
h
1
t
f
b
s
b
ce
Fig. (
5
)
: Evaluation of pre
–
cracked stiffness for
upper part division.
(a) Interior beam
t
f
b
ce
b
s
b
ce
h
1
Fig. (
6
): transverse shearing rigidity.
(a) Steel area
h
2
h
1
t
w
(b) transformed area
mt
w
h
2
h
1
Fig. (
3
) Design chart for cantilevers.
Value of D
1
-2.5
-1.5
-0.5
0.5
1.5
2.5
3.5
5.2
5
5.2
5
5.2
Log
10
K
2
C=3.5
C=3
C=2.5
C=2
C=1.5
C=1.25
Fig. (
4
): Shear stress flow in composite sections.
(b) Independent action
t
f
h
1
(a) Composite action
t
f
20
12
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
:
(
EI
p
= 0.6 EI
f
= 2.9
×
10
12
N.mm
2
)
,
(
GJ= 2.6
×
10
11
N.mm
2
).
b
-
interior beams
:
(
E
I
p
= 0.5 EI
f
= 3.4
×
10
12
N.mm
2
) , (
GJ= 3.0
×
10
11
N.mm
2
).
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 lon
gitudinal direction equal 0.5b as shown in Fig.(
11
)
[4]
:
EI
p
= 0.5(EI
f
+ EI
o
)
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
represente
d in section 3.5.1, thus:
a
-
for edge beams
:
(
EI
p
= 0.5( EI
f
+ EI
o
)= 4
×
10
12
N.mm
2
) ,
(
GJ= 3.6
×
10
11
N.mm
2
)
.
b
-
for interior beams
: (
EI
p
= 0.5( EI
f
+ EI
o
)= 4
×
10
12
N.mm
2
) ,
(
GJ= 3.5
×
10
11
N.mm
2
).
The shearing rigidity is const
ant for all grid members and it can be calculated as
shown in section 3.5.3, thus
GA
v
= 101.67 N (for transformed shear area)
,
or
:
GA
v
= 68.37 N (for steel
shear area)
Lower Components (Longitudinal and transverse steel
beams)
Shear Connectors (stud shear connectors)
Depth of steel beam
h
2
= 152.2 mm
Length of shear connector = 38 mm
Flange width of steel beam
b
s
= 152.2 mm
Diameter of shear con
nector = 12 mm
Thickness of flange of steel beam
t
f
= 6.6 mm
According
to (OHBD) code
requirements
Thickness of web of steel beam
t
w
= 5.84 mm
Number of connectors per row
n
=2
Cross sectional area of steel beam
A
2
= 2858 mm
2
Spacing
P
=180 mm.
Moment of inertia of steel beam
I
2
=
12112334.49
mm4
Strength of shear connector = 57000 N.
Modulus of elasticity of steel beam
E
2
= 200000
MPa
mmmmmmmmmmmmmm
mm
2
Poisson’s ratio of steel beam
υ
2
=
0.3
Shear modulus of elasticity of steel beam
G
2
= 76923 N/ mm
2
(calculated from G =E/2(1+
υ
)).
13
Two different loading conditions are considered. Point load of 89 kN is applied, the
po
sition 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 sec
ond 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)
Metho
d of analysis
1
st
loading
2
nd
loading
Max.
Deflection
(mm)
Percentage
Difference
(%)
Max.
Deflection
(mm)
Percentage
Difference
(%)
Grillage
analogy
Case (I)
3.30
+
17.90
7.86
+
10.0
Case (II)
3.57
+
27.50
8.27
+
15.9
Case (III)
3.69
+
31.80
8.47
+
18
.8
Orthotropic plate method
[20]
3.30
+
17.90
7.50
+
5.2
Experimental result
[
4
]
2.80
-
7.13
-
From the above comparison, it is clear that when the effect of transverse shear area
(
A
v
) is ignored the deflections obtained by the grillage analogy are rathe
r 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 momen
ts 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
14
distribute transversely the eccentric load quite effectively in composite br
idges.
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 1
st. loading condition
(percentage differences with respect to experimental results)
Node
no.
Exper.
Ortho.
Perce.
Diff.
(%)
Grill.
case I
Perce.
diff.
(%)
Grill.
case II
Perce.
diff.
(%)
Grill.
case III
Perce.
diff.
(%)
23
2.54
1.91
-
24.8
2.3
-
9.5
2.38
-
6
.3
2.41
-
5.10
18
2.67
2.29
-
14.2
2.9
+
8.6
3.06
+
14.6
3.12
+
16.8
13
2.8
3.30
+
17.9
3.3
+
17.9
3.57
+
27.5
3.69
+
31.8
8
2.67
2.29
-
14.2
2.9
+
8.6
3.06
+
14.6
3.12
+
16.8
3
2.54
1.91
-
24.8
2.3
-
9.5
2.38
-
6.3
2.41
-
5.10
Tab
.
(
4
):Vertical deflections (in mm) a
t mid
-
span of bridge model under 2nd. loading
condition (percentage
differences with respect to experimental results)
Node
no.
Exper.
Ortho.
Perce.
Diff.
(%)
Grill.
case I
Perce.
diff.
(%)
Grill.
case II
Perce.
diff.
(%)
Grill.
case III
Perce.
dif
f.
(%)
23
-
1.70
-
1.50
+
11.8
-
1.28
24.7
-
1.32
22.4
-
1.34
21.2
18
-
0.30
-
0.29
-
4.0
0.36
20.0
0.37
23.0
0.38
26.7
13
2.16
2.30
+
6.5
2.30
6.5
2.38
10.2
2.41
11.6
8
4.33
4.69
+
8.3
4.80
10.8
4.98
15.0
5.07
17.0
3
7.13
7.50
+
5.2
7.86
10.0
8.27
15.9
8.47
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 (EI
p
/ EI
f
) 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 a
re
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.
15
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 deflection
s (in
mm) for 1st. loading
condition
Node no.
EI= EI
o
EI
p
= 0.7 EI
f
EI
p
= 0.9 EI
f
EI= EI
f
23
2.73
1.70
1.25
1.09
18
3.38
2.26
1.78
1.61
13
3.76
2.60
2.09
1.92
8
3.38
2.26
1.78
1.61
3
2.73
1.70
1.25
1.09
Tab
.
(
6
): Influence of degree of interaction on
vertical deflections (in
mm) for 2nd.
loading condition
Node no.
EI= EI
o
EI
p
= 0.7 EI
f
EI
p
= 0.9 EI
f
EI= EI
f
23
-
1.45
-
1.33
-
1.09
-
0.99
18
0.49
0.04
-
0.06
-
0.083
13
2.73
1.704
1.25
1.09
8
5.57
3.95
3.103
2.799
3
8.99
6.80
5.55
5.09
16
Fig (
8
)
:
Vertical deflections at mid
-
span section of bridge deck model under
1st.loading
c
ondition
Fig
.
(
9
)
:
Vertical deflections at mid
-
span section of bridge deck model under 2nd.loading
condition
A
Fig. (
7
): Detai
ls of composite bridge model.(a) Plan view, (b) Section (A
-
A), (c) Section (B
-
B)
5
.
5
.
3
.
4
.
2
.
55
.
9
.
8
.
7
.
6
.
52
.
54
.
53
.
55
.
55
.
55
.
59
.
58
.
57
.
56
.
52
.
54
.
53
55
.
.
55
.
x
y
(a)
3050
mm
2290 mm
A
B
B
(b)
152.2 mm
(c)
.
5b=267.225
5
.
5b=267.225
5
48 mm
48
152.2
254.17
b=534.45
Node number
Distance from left end (mm)
Deflection (mm
)
23
Experimental
[
4
]
Orthotropic
plate
[
4
]
Grillage case I
Grillage case
II
Gr
illage case III
1.5
2
2.5
3
3.5
4
4.5
3
8
13
18
76.1
610.
55 1145
1679.45 2213.9
Node number
Distance from left end (mm)
Deflection (mm)
Expe
rimental
[
4
]
Orthotropic plate
[
4
]
Grillage case I
Grillage case II
Grillage case III
10
23
18
13
8
3
76.1
610.55 1145 1679.45
2213.9
-
4
-
2
0
2
4
6
8
17
Fig
.
(
10
)
:
Load
-
deflection curve at center of Kennedy’s bridge deck model
Fig (
11
) Influence of degree of interaction on vertical deflections for 1st.loading condition
Fig (
12
) Influence of degree of interaction on vertical deflection
s for 2nd.loading condition
Node number
0
1
2
3
4
5
6
7
8
3
8
13
1
8
2
3
Distance from left end (mm)
Deflection (mm)
EI
p
=0.9 EI
f
EI=EI
o
EI
p
=0.7 EI
f
EI=EI
f
Grill. case I
Experimental
[
4
]
76.1
610.55
1145
1679.45
2213.9
EI=EI
o
EI
p
=0.7EI
f
EI
p
=0.9EI
f
EI=EI
f
Grill. case I
Experimental
[
4
]
Node number
-
4
-
2
0
2
4
6
8
10
Distance from left end (mm)
Deflection (mm)
76.1
610.55
1145
1679.45
2213.9
23
18
13
8
3
0
10
20
30
40
50
60
70
80
90
100
0
0.
3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3
3.3
3.6
3.9
Deflection at center ( mm)
Load at center ( kN)
Experimental
Grill. case I
Grill. case II
Grill. case III
18
6.
Conclusions
The main concluding remarks that have been achieved in this st
udy 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
k
2
, and were given for various values of the factors
C
.
it
2
2
2
1
P
4
L
C
n
K
K
,
it
it
2
12
it
1
C
1
C
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 struct
ure.
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 o
f
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 I
-
Diaphrams in Composite Bridges”,
Journal of t
he Structural Division, Proc. of
the ASCE, Vol.115, ST2, pp. 417, Feb.1989.
19
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.
H
assan,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., Tay
lor, 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
Sh
ear Interaction”, Journal of Structural Engineer,Vol.124,No.10,pp. 1159
-
1165,Oct.1998.
20
فوقسلل كبشملا ةقيرطب ليلحتلا
ةيناسرخلا
تابتعلاو
ةيديدحلا
بولسأ مادختساب ةبكرملا
يئزجلا لخادتلا
ةصلاخلا
ا ت ةةش ملا تولةةسب كتةةب ةةةيلاحلا ةةةساردلا
ةةةيديد تاةةبت ةةلي حوةةبرق يتيرننوةةك تقةةس كةةق ةةةنونتملا ةةةبكرمل
يةةي يةئزجلا لخادةةتلا يةي ةةةقلا باورةل يةةطخلا لةعللا راةةبت لا رةب ب ا ةةخب ةةخ ةةباور ةطةساوب كيياةج اب
ةةةةساردل ةطةةةسبق ةةةةقيرطك تانبةةةشملا ةةةةقيرس مادختةةةساب ةةةةبكرملا ت ةةةش ملل لةةةيلح كيدةةةق كةةة ةةةةساردلا ل ةةةي
ةةج ا لا تاااةةباهاو ووةةطبلا
ةطلةةسملا واةةم ما كةةق ة
ةةةقيرس طةة رتخاو
اا ةة ل ةةةبولطملا كيةةقلا باةةسحل
كبشملا كمض ءاض ما كق وض لنل يللا اا صو ءا حنلا
ةيميمة ووادةا طقتةقا كةلل ةل
ً
اءا بو
يةئزجلا لخادةتلا دبةسب ءاة حنلا اا ةص يةي اة ق لا رادقق باسحل
دةبل ةني يلاةحلا لةحبلا يةي دةاوو
رةعلا رابت ا كق
رير ة ي كل ةك دةاوو يئاةشني وةض لةنل ءاة حنلا اا ةص داقتةقا يةي ةلاخايو ررثةملا
زواجتي لو
ً
ادا ليلخ ووطبلا كيخ ل ةقلا ىوخ
..11
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