1. NONLINEAR RESPONSE OF COMPOSITE STEEL-CONCRETE
BOX GIRDER BRIDGES
Sanjay Tiwari
Indian Institute of Technology
(IIT Roorkee)
Roorkee (India)
Summary
Cellular steel section composite with a concrete deck is one of the most suitable
superstructures in resisting torsional and warping effects induced by highway
loading. This type of structure has inherently created new design problems for
engineers in estimating its load distribution when subjected to moving vehicles.
Current composite steel and concrete bridges are designed using full interaction
theory assuming there is no any relative displacement or slip at interface of concrete
and steel. However, in the assessment of existing composite bridges this
simplification may not be warranted as it is often necessary to extract the greatest
capacity and endurance from the structure. This may only be achieved using partial-
interaction theory which truly reflects the behaviour of the structure. This paper
presents a non linear three dimensional finite element model incorporating the slip of
shear connectors using commercially available software ANSYS, capable of
analyzing composite box girder bridges of various geometries. The proposed model
has been validated against the published results from literature. A distribution factor
approach has been suggested as a simplified design method for the preliminary
proportioning of such bridges.
Introduction:
The use of composite bridges in interchanges of modern highway systems has become
increasingly popular for functional, economic as well as aesthetic considerations. This
type of construction leads to an efficient transverse load distribution due to excellent
torsional stiffness of the section. Further utilities and services can be readily provided
within the cells.
Among the refined methods, the FEM is the most general and
comprehensive technique of analysis capturing all aspects affecting the structural
response. But it is too involved and time consuming to be used for routine design
purpose. Practical requirement in the design process necessitate a need for a simpler
design method. This paper presents a three dimensional non linear model using the
Finite element method in which two and four lane bridges of 20m span has been
analyzed using a commercially available package ANSYS. The effect of cross bracings
has been presented in detail. Results from published literature are used to substantiate the
analytical modeling. Based on the parametric study, load distribution factors are deduced
for such bridges subjected to IRC loadings.
Concept of Distribution Factor:
The concept of the distribution factor allows the design engineer to consider
the transverse effect of wheel loads in determining the shear and moments of girders
under the longitudinal as well transverse placement of live loads, thus simplifying the
analysis and design of bridges. According to the approach of the load distribution,
maximum shear and moments in bridge are obtained first as if the wheel loads are
applied directly to bridge as a beam. These values are then multiplied by the

2. appropriate live-load distribution factors to obtain critical live-load shear and
moments in the different girders in a bridge.
M
MDF = max ----------- (1)
M
The above relationship used for calculation of moment distribution factor,
MDF, carried by each girder of the bridge, the maximum moment, M, was calculated
in a simply supported girder subjected to one train of IRC (Indian Road Congress:
Standard Specifications and Code of Practice for Road Bridges) Class 70R wheel
loads. The longitudinal moment carried by each girder of the prototype bridge, Mmax,
was calculated by integrating the normal stresses at midspan, determined from the
finite-element analysis.
Bridge Modeling
A four-node shell element ‘shell43’ with six degrees of freedom at each node was
used to model the concrete deck, steel webs, steel bottom flange, and end diaphragms.
A three dimensional two-node beam element ‘beam188’ was adopted to model the
steel top flanges, cross bracings and top-chords. The modeling of shear connector is
done using three mutually perpendicular nonlinear springs which constitutes for the
stiffness in three directions viz., stiffness parallel to stud longitudinal axis and
stiffness perpendicular to longitudinal axis (one parallel to the bridge axis and one
perpendicular to bridge axis). The shell elements in the top flange were connected
using ‘combin39’, type nonlinear spring elements, with the elements (flanges) of the
web to ensure load-slip relation. Because of their insignificant flexural and torsional
stiffness, cross-bracing and top-chord members are considered as axial members
loaded in tension and compression. Two different constraints were used in the
modeling, namely, the roller support at one end of the bridge, constraining both
vertical and lateral displacements at the lower end nodes of each web, and the hinge
support at the other end of the bridge, restricting all possible translations at the lower
end nodes of each web.
Modeling of Shear Connector:-
Kuan-Chen Fu and Feng Lu (Kuan-Chen 2003) suggested that the shear stud
can be modeled by a bar element, which can be seen as two independent linear springs
with a stiffness K N parallel to the longitudinal axis of the bar and K T perpendicular
to the axis. Note that
EsAs
KN =
hs -------- (2)
Where, Es = elastic modulus; As = area of cross section; and hs = height of stud.
Along the tangent surface, the constitutive behaviour is defined by a typical load-slip
function proposed by Yam and Chapman (Yam 1972), which is
P = a (1 − e −by ) --------- (3)
Where P = load; a and b = constants; and y = interface slip.
By choosing two points on the function such that the relationship y2=2y1 is
maintained, the constants a and b can be determined as
P×P
a=
2 P 2 − P1 ---------- (4)
1  P1 
b = log 
y1  P 2 − P1  ---------- (5)

3. Therefore, the stiffness in the tangential direction is
dP
KT = = abe − by
dy ------------ (6)
In the present problem modeling of shear connector is done using three mutually
perpendicular nonlinear springs which constitutes for the stiffness in three directions
viz., stiffness parallel to stud longitudinal axis and stiffness perpendicular to
longitudinal axis (one parallel to the bridge axis and one perpendicular to bridge axis).
Each bar element provides a dimensionless link between the concrete deck element
and neighboring top flange element of the girder.
Material Modelling
Material nonlinearity is incorporated in the analysis using nonlinear material
model available in ANSYS software. For concrete Drucker-Prager failure criterion is
used while for steel bilinear isotropic hardening is used as yielding criterion.
Concrete modeling:-
The non-linear response of concrete is caused by four major material effects: cracking
of the concrete; aggregate interlock; and time dependent effects such as creep,
shrinkage, temperature, and load history.
In spite of its obvious shortcomings the linear theory of elasticity combined with
criteria defining “failure” of concrete is most commonly used material law for
concrete in reinforced concrete analysis. The linear elastic modeling can be
significantly improved by using the non-linear theory of elasticity.
The Drucker-Prager (DP) option available in ANSYS is applicable to granular
(frictional) material such as soils, rock, and concrete, and uses the outer cone
approximation to the Mohr-Coulomb law. This option uses the Drucker-Prager yield
criterion with either an associated or non-associated flow rule. The yield surface does
not change with progressive yielding, hence there is no hardening rule and the
material is elastic- perfectly plastic.
The equivalent stress for Drucker-Prager is
1
1 T 2
σ e = 3βσ m +  { S } [ M ]{ S } 
2  ------------ (7)
Where,
σ m = (σ x + σ y + σ z )
1
3 = mean or hydrostatic stress. ------------ (8)
{ s} = deviatoric stress
[ M ] = plastic compliance matrix.
2 sin φ
=
β = material constant 3 ( 3 − sin φ ) ------------ (9)
Where, φ = angle of internal friction.
The material yield parameter is defined as
6c × cos φ
σy =
3 ( 3 − sin φ ) ------------ (10)
Where, c = cohesion value.
The yield criterion is then

4. 1
1 T 2
F = 3βσ m +  { S } [ M ]{ S }  − σ y = 0
2  ------------ (11)
This yield surface is cone with material parameters chosen such that it corresponds to
the outer aspices of the hexagonal Mohr-Coulomb yield surface.
Fig.1 Mohr-Coulomb and Drucker-Prager yield surfaces.
Steel modeling:-
Plasticity theory provides a mathematical relationship that characterizes the elasto-
plastic response of materials. The yield criterion determines the stress level at which
yielding is initiated. For multi-component stresses, this is represented as a function of
the individual components, f ({σ}), which can be interpreted as an equivalent stress
σe. The material will develop plastic strains. If σe is less than σy, the material is elastic
and the stresses will develop according to the elastic stress-strain relations. The
equivalent stress can never exceed the material yield since in this case plastic strains
would develop instantaneously, thereby reducing the stress to the material yield.
Fig.2 Stress strain relationship for bilinear isotropic hardening.
This option (bilinear isotropic hardening) uses Vonmises yield criterion with
associated flow rule and isotropic (work) hardening.
The equivalent stress is,

5. 1
3 2
σe =  {S }T [ M ]{S } ------------ (12)
2 
And the yield criterion is,
1
3 2
F =  { S } [ M ]{ S } −σk = 0 ------------ (13)
T
2 
For work hardening σk is a function of the amount of plastic work done. For the case
of isotropic plasticity assumed here, σk can be directly determined from equivalent
plastic strain.
Inputs required for the Software for Material modeling:-
Concrete: - For specifying Drucker-Prager failure criterion only three inputs are
required in ANSYS software 1) angle of internal friction = 45˚; 2) cohesion= 3kN/m2;
3) flow angle 0˚ [Diganta Goswami, 2003] additionally inputs required are modulus
of elasticity E(27000MPa); Poisson’s ratio µ (0.2).
Steel: - For specifying Yield criterion for steel Bilinear Isotropic hardening option is
used inputs required are modulus of elasticity of steel E (200000MPa); Poisson’s ratio
µ (0.3); yield strength of steel fy(250MPa) and tangent modulus (0) i.e. perfectly
elasto-plastic behaviour.
Input for COMBINE39: -
The element requires force Vs deflection relationship as input for accounting its
transverse stiffness. In the present study load slip relation has been taken from
published literature (Dennis 2005).
Validation of Proposed Model
The results from an experimental study on a beam model [L.C.P.Yam 1968] are used
to validate the modeling adopted for current bridge. The details of the above referred
experiment are as below:
A number of simply supported and continuous composite beams were tested at
Imperial College. One of these specimens is analyzed to validate the models. The
simply supported beam selected from the test series which is loaded at midspan. The
beam consist of as 152mm thick concrete slab and I-section steel girder,
304×152mm×0.196kN/m, connected by 100 uniformly distributed head studs,
19×100mm. The geometric configuration of beam is as shown in figure 3 below. The
material properties are steel: Es=2×105MPa, µ=0.3, Concrete: ES=3×104MPa, µ=0.2.

6. a) Elevation and cross section
b) Finite Element Idealization.
Fig.3 Modeling of composite beam for validation .

7. Only one quarter of beam is considered in analysis taking advantage of double
symmetry of the specimen. The finite element mesh is as shown in figure 5.3. The
interface slip values are compared in the following Table 1 for a load of 448kN. And
very good coincidences with experimental values are observed.
Table 1 Comparison of results for interface slip with literature review.
Results from Interface slip (mm)
At 2.0m from
left hand
At midspan At support
support
Test 0 0.139 0.508
ANSYS 0 0.151 0.436
Loading Conditions:
The live load considered is the IRC Class 70R wheeled vehicle loading. These
loads were first applied on a simply supported girder, with a span equal to that of the
bridge prototype, to determine which case produced maximum moment at midspan or
the maximum shear force at the support. Subsequently two loading cases were
considered for each bridge prototype including central and eccentrically placed IRC
Class 70R wheeled loading, and the bridge dead load. The live load was considered as
static patch loads of appropriate contact dimensions as per IRC: 6-2000 in the
analysis. The loads are so placed in accordance with IRC: 6-2000, section: II-Loads
and Stresses.
Description of Bridge Prototypes:
A parametric study has been carried out using the proposed finite element model in
which two and four lane bridges of various geometries has been analyzed.The
parameters considered are number of cells, number of lanes.
For this study, 8 simply supported single-span bridges of different configurations
were used. The basic cross-sectional configurations for the bridges studied are
presented in Table 2. The symbols used in the first column in Table 2 represent
designations of the bridge types considered: l stands for lane, c stands for cell, and the
number at the end of the designation represents the span length in meters. For
example, 2l-3c-20 denotes a simply supported bridge of two-lane, three-cell and of 20
m span. The cross sectional symbols used in Table 2 are shown in Fig. 4. The number
of lanes was taken as 2 and 4. Number of cells ranged from 1 to 4 for two-lane
bridges and 4 to 7 for 4 lane bridges.
When changing the number of cells for the same bridge width, the thicknesses
of the top steel flanges, webs, and bottom flanges were altered to maintain
approximately the same overall flexural stiffness as well as shear rigidity of the cross

8. section. The bridge width was taken as 8.5 m for two lane bridges and 16.7m for four
lane bridges.
Fig. 4: Geometric details of model
Table 2: Geometries of Prototype Bridges in Parametric Study
Bridge Cross Sectional Dimensions (mm)
type A B C D F t1 t2 t3 t4
2l-1c-20 8500 4250 300 800 1050 22 20 16 250
2l-2c-20 8500 2835 300 800 1050 22 14 12 250
2l-3c-20 8500 2125 300 800 1050 16 10 10 250
2l-4c-20 8500 1700 300 800 1050 16 8 10 250
4l-4c-20 16700 3340 300 800 1050 18 16 12 250
4l-5c-20 16700 2785 300 800 1050 18 14 10 250
4l-6c-20 16700 2385 300 800 1050 18 12 10 250
4l-7c-20 16700 2085 300 800 1050 16 10 10 250
The moduli of elasticity of concrete and steel were taken as 27 and 200 GPa,
respectively. Poisson’s ratio was assumed as 0.2 for concrete and 0.3 for steel. End
diaphragms were provided at the supports with minimum thickness and the cross
bracings were provided at some interval along the span. The material for the end
diaphragms and the cross bracings were taken to be the same as those for the webs.
Determination of Load Distribution Factors:
To determine the distribution factors, the bridge deck was loaded with wheel loads
positioned along the longitudinal direction of the bridge that produced the maximum
moment. The wheels were then moved transversely across the width of the bridge for
the maximum response per girder. The maximum interior and exterior girder moments
and web shears were calculated in each loading case.
(a) Cross-section symbols for five-cell bridge

9. (b) Idealized cellular bridge for moment distribution
Fig. 5 Cross-section of five cell bridge prototype
The cellular cross section was divided into I-beam shaped girders as shown in Fig.
5(b). Each idealized girder consisted of the web, steel top flange, concrete deck slab,
and steel bottom flange.
Results and Discussions:
Effects of Cross-Bracing systems:
The torsional stiffness of a box girder results from three components: the Saint-
Venant rigidity, the warping rigidity, and the distortional rigidity. Increasing the
flexibility of any of these components reduces the rigidity of the box girder. Adding
bracings between support lines is generally required for stability purposes at the
construction phase. Tables (2 & 3) show the effect of bracings on the moment
distribution between idealized girders for central lane loadings and eccentric lane
loadings respectively.
Table 3: Effect of cross bracings on moment distribution factors for central lane
loading
Bridge Number Outer
type of cross girder Interior Girders Outer
bracings girder
between I II III IV V VI
supports
2l 3c 20 0 0.12 0.26 0.26 - - - - 0.12
2l 3c 20 1 0.17 0.21 0.21 - - - - 0.17
2l 3c 20 2 0.14 0.24 0.24 - - - - 0.14
2l 3c 20 3 0.16 0.22 0.22 - - - - 0.16
2l 3c 20 5 0.16 0.22 0.22 - - - - 0.16
4l 7c 20 0 0.05 0.17 0.25 0.29 0.29 0.25 0.17 0.05
4l 7c 20 1 0.14 0.22 0.2 0.2 0.2 0.2 0.22 0.14
4l 7c 20 2 0.08 0.19 0.23 0.26 0.26 0.22 0.19 0.08
4l 7c 20 3 0.12 0.22 0.22 0.22 0.22 0.22 0.22 0.1
4l 7c 20 5 0.14 0.2 0.21 0.21 0.21 0.21 0.2 0.14

10. Table 4: Effect of cross bracings on moment distribution factors for eccentric
lane loading
Bridge Number Outer
type of cross girder Interior Girders Outer
bracings girder
between I II III IV V VI
supports
2l 3c 20 0 0.24 0.35 0.22 - - - - 0.07
2l 3c 20 1 0.2 0.25 0.24 - - - - 0.19
2l 3c 20 2 0.2 0.28 0.25 - - - - 0.15
2l 3c 20 3 0.2 0.25 0.25 - - - - 0.18
2l 3c 20 5 0.2 0.26 0.25 - - - - 0.17
4l 7c 20 0 0.25 0.39 0.37 0.31 0.23 0.15 0.07 -0.01
4l 7c 20 1 0.23 0.29 0.26 0.24 0.23 0.21 0.19 0.09
4l 7c 20 2 0.2 0.33 0.32 0.29 0.23 0.17 0.15 0.07
4l 7c 20 3 0.21 0.31 0.29 0.25 0.23 0.2 0.17 0.09
4l 7c 20 5 0.24 0.3 0.28 0.25 0.22 0.19 0.17 0.11
It can be observed that with increase in number of cross bracing system the
bending moment increases in the outer girder and decreases in the central girder for
central lane loading while in the case of bridges with eccentric lane loading (Table 3)
the maximum moment carried by the loaded outer girder is considerably reduced. As
an example, when using five cross- bracing systems the bending moment increases up
to a maximum of 81% in the outer girder and decreases by a maximum of 23% in the
central girder for the bridge type 4l-7c-20, while in the case of bridges with eccentric
lane loading (Table 3) the maximum moment carried by the loaded first intermediate
girder is reduced by more than 21%. Thus adding cross bracings improves the ability
of the cross section to transfer loads from one girder to the adjacent ones.
For the span length of 20m it is revealed that adding more than three cross
bracing systems has an insignificant effect on the distribution factors substantiate the
fact that for the span length of 20m three number of cross bracing systems gives
reasonable load distribution.
Conclusion
Most composite bridges are designed assuming full-interaction between the
concrete deck and steel box girder interface because of the complexities of partial-
interaction analysis techniques. However, in the assessment of existing composite
bridges this simplification may not be warranted as it is often necessary to extract the
greatest capacity and endurance from the structure. This may only be achieved using
partial-interaction theory which truly reflects the behaviour of the structure. The
proposed model can be efficiently utilized incorporating the realistic behavior of shear
connectors.
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