Abstract
This paper presents a modified rigid cross beam method to study the lateral distribution of multiI beam composite curved bridge with slip effect. First, the effective stiffness expression of single composite curved beam with slip effect were established and calculated by the FEM. Secondly, the lateral load distribution of multiI beam composite curved bridge is obtained by substituting the effective stiffness of the main composite curved beam into the rigid cross beam method. Finally, The FEM numerical examples show that this method can accurately describe the load distribution characteristics of multiI beam composite curved bridge with slip effect.
1. Introduction
The application of multiI beam composite curved bridge increases gradually, which shows very complex structural behaviour since the coupling effect of axial, flexural and torsional deformation. In addition, the shear connectors generally permit the development of only partial composite action between the individual components of the member, and their analysis requires the consideration of the interlayer slip between the subcomponents. Therefore, as important parts of bridge design, it is of great significance to study the lateral distribution of this kind of bridge under complex structural behavior.
So far, the lateral distribution researches of multibeam composite curved bridge analysis, correlative research work is relative scarce. Huang et al. [1], conducted experiments and numerical analysis to investigate the lateral distribution behavior of the oblique composite beam with large angle. The experiment results were compared with the finite element results and norm of AASHTO in their study. Samaan et al. [2], using a finite element method, studied the structural responses of twoequalspan continuous curved boxgirder bridges. Barr and Amin. [3] provided a finite element model to investigate the effects of girder spacing, span length, overhang distance and skew angle on the shear liveloud distribution factor. Harris and Gheitasi [4] presented an analytical approach for determining lateral load distribution characteristics of beamslab type bridges, using classical plate theory. A modified method was developed [5] based on the rigidjointed girder method considering the different main girder stiffnesses and the space between the original bridge and the widened bridges. Ma et al. [6] provided a modified prediction method of eccentric lateral load distribution for investigating the lateral load distribution on continuous composite box girder bridge with corrugated steel webs. It shows in their study that an improved plane model which considering the box girder torsional stiffness of the box girder can be introduced to solve the spatial problem by using modified method of eccentric lateral pressure.
The researches have been accomplished in regard to the lateral load distribution of the multiI beam composite curved bridge, however, limited researches have focused on the multibeam composite curved bridge with partial shear interaction. Xiang et al. [7] provided a modified rigid transverse beam method and rigid connected beam method for the multibox composite beam bridges by considering the interface slippage’s effect. The results show that the modified rigid connected beam method considering slippage effect is effective to calculate the lateral load distribution of midspan cross section in multibox composite beam bridge. In the present paper, a modified rigid cross beam method is developed to study the lateral distribution of composite curved bridge with multiI beam with slip effect. The numerical results are compared with the FEM.
2. Effective stiffness of single composite curved Ibeam
2.1. The effective stiffness with partial interaction
In general, for a single composite Ibeam with partial shear connectors, the actual value of effective stiffness is the value between no shear connection and rigid shear connection. So, the interpolation method can be adopt to calculate the effective stiffness of partial shear connectors, namely:
where, ${\left(EI\right)}^{p}$ and ${\left(GJ\right)}^{p}$, ${\left(EI\right)}^{f}$ and ${\left(GJ\right)}^{f}$, ${\left(EI\right)}^{c}$ and ${\left(GJ\right)}_{}^{c}$ are the effective bending stiffness and torsional stiffness of the Ibeam with partial interaction, noshear interaction, rigid shear connection, respectively. ${\eta}_{E}$ and ${\eta}_{G}$ are the interpolating function. According to the Ref. [8], ${\left(EI\right)}^{f}$, ${\left(GJ\right)}^{f}$, ${\left(EI\right)}^{c}$ and ${\left(GJ\right)}_{}^{c}$ can be calculated by the following equations:
where ${E}_{c}$ and ${E}_{s}$, ${I}_{c}$ and ${I}_{s}$, ${G}_{c}$ and ${G}_{s}$, ${J}_{c}$ and ${J}_{s}$, ${A}_{c}$ and ${A}_{s}$, are the elasticity modulus, bending inertia moment, shear modulus, torsional inertia moment, sectional area of the concrete slab and the Isteel girder, respectively. $r$ is distance between the centroid of the concrete slab and the steel girder. ${b}_{c}$ and ${h}_{c}$ are the width and the thickness of the concrete slab. ${t}_{f1}$ is the thickness of the steel girder. ${c}_{I}$ is the coefficient of the tensional stiffness.
2.2. The stiffness interpolating function
According to the Ref. [8], the effective bending stiffness for the straight single composite beam with simply supported ends can be express as:
where $E{I}_{eff}={\left(EI\right)}^{p}$, $E{I}_{0}={\left(EI\right)}^{f}$, $E{I}_{\infty}={\left(EI\right)}^{c}$ and
The Eq. (4) can be rewritten as:
Assume that the interpolating function of the stiffness has the similar expression with the single straight composite beam, which are:
where, ${\epsilon}_{E}^{}$ and ${\epsilon}_{G}^{}$ are the coefficient of the bending stiffness and torsional stiffness interpolating function.
3. The coefficient of the stiffness
A FEM (finite element model shown in Fig. 1) proposed by Erkmen [9] and Wonseok [10] is adopted to determine the coefficient of bending stiffness and torsional stiffness (${\epsilon}_{E}^{}$ and ${\epsilon}_{G}^{}$). Both the concrete slab and steel Igirder are modeled as 4node shell elements in the model. The connections between the slab and Igirder (shown in Fig. 2) are simulated by multiplepoint constraints (MPC). The spring elements are used in the tangential direction and the radial direction to allow for the possibility of movement.
Fig. 1FEM model of single composite curved beam
Fig. 2Element connection of the composite beam
If the influence of curvature radius for the bending stiffness coefficient ${\epsilon}_{E}$ is small, the coefficient of the single curved composite Ibeam can be taken as the coefficient (Eq. (5)) of the single straight composite Ibeam.
Fig. 3Section sizes of the composite beam (unit: m)
To demonstrate the influence of curvature radius for the coefficient of the bending stiffness, consider a simply supported single curved composite Ibeam subjected to uniform load with beam length $L=$ 29.2 m. The other parameters in the model are shown in Table 1 and Fig. 3. The effective stiffness of the composite beam is calculated under different central angle. The results (Fig. 4) show that the coefficient of composite curved beam can be taken as the coefficient of composite straight beam because the influence of central angle on the effective stiffness is very small. So, we take ${\epsilon}_{E}^{}={\pi}^{2}/{L}^{2}$ as the coefficient in Eq. (5).
To quantify the ${\epsilon}_{G}^{}$, the torsional stiffness interpolation function is calculated with different design parameters shown in Table 2. According to the calculation results in Fig. 5, the torsional stiffness interpolation function ${\epsilon}_{G}$ can be taken as ${\epsilon}_{G}={\pi}^{2}{\left({h}_{c}/{h}_{s}\right)}^{1/4}$ through curve fitting.
Table 1Material properties of the composite beam
Parameters  Concrete slab  Steel girder 
Elasticity modulus (MPa)  ${E}_{c}=$ 3.45×10^{4}  ${E}_{s}=$ 2.06×10^{5} 
Poisson’s ratio  ${\mu}_{c}=$ 0.167  ${\mu}_{s}=$0.3 
Table 2Design parameters of composite beam
Parameters  Values (Model 1)  Values (Model 2)  Values (Model 3) 
Thickness of the concrete slab ${h}_{c}$ (m)  0.12  0.24  0.36 
Height of the steel girder ${h}_{s}$ (m)  0.66  1.32  1.98 
Fig. 4Effective stiffness of the composite beam
a) Bending stiffness
b) Torsional stiffness
Fig. 5The curves of torsional stiffness interpolation function
a) The ${\eta}_{G}$ when the thickness is changed
b) The ${\eta}_{G}$ when the height is changed
4. The coefficient of the stiffness
4.1. The stiffness interpolating function
To obtain calculation formula of lateral load distribution for multiI beam composite bridge, the traditional rigid cross beam method is modified the interface slip effect. Substitute the deformation constant of the single composite Ibeam into the formula of rigid cross beam method, the lateral load distribution for multibeam composite bridge considering the interface slip effect can be get, which are:
where ${\eta}_{ij}^{F}$ and ${\eta}_{ij}^{T}$ are the value of lateral load distribution influence line under the vertical load and torque load, respectively. Where other parameters can be refer to the Ref. [11].
4.2. The stiffness interpolating function
According to the relation between midspan bending moment and external load [12], the midspan moment of main beam $i$can be got after obtain the lateral load distribution influence line of main beam $i$:
where ${R}_{i}$ is the curvature radius of main beam $i$, $\theta $ is the central angle of the curved composite beam. The influence line of lateral load distribution for bending moment is:
5. Numerical examples
To validate the accuracy of the lateral load distribution presented in this paper, a multiI beam composite curved bridge with simply supported ends (Fig. 7) is taken as an example. The bridge is composed of five pieces of main beam. The central angle is $\theta =$ 30°. ${R}_{1}=$50 m, ${R}_{2}=$ 52.85 m, ${R}_{3}=$ 55.7 m, ${R}_{4}=$ 58.55 m and ${R}_{5}=$ 61.4 m are the center radius of the main beam, respectively. Three diaphragms are set in middle diaphragm and either side of the bridge, respectively. In this example, the material properties, geometric properties are list in Table 3. The shear stiffness is 10^{6} N/m for a stud, and 61 studs are arranged along the whole span with equal distance. Finite element model (Fig. 7) described here with the same way like single composite curved beam model.
Fig. 7Fivebeam composite curved bridge model
Table 3Material properties and section sizes of main beams
parameters  Elasticity modulus (MPa)  Poisson’s ratio  Sectional dimension (m) 
Concrete slab  ${E}_{c}=$ 3.45×10^{4}  ${\mu}_{c}=$ 0.167  ${b}_{c}=\text{2.85}{h}_{c}=\text{0.24}$ 
Steel girder  ${E}_{s}=$ 2.06×10^{5}  ${\mu}_{s}=$ 0.3  ${b}_{f1}={b}_{f2}=\text{0.475}{t}_{f1}={t}_{f2}=\text{0.029}$ ${t}_{w}=\text{0.019}{h}_{s}=\text{0.66}$ 
Table 4 shows the lateral midspan moment distribution factors of main beams based on the finite element method, this paper solution, rigid cross beam method and the error between them. From the result list in Table 4, we can see:
1) Big errors are obtained from the finite element analysis and the rigid cross beam method with no interface slip. The mean error is found to be 17.4 % happened at the fifth main beam. The results show that the influence of interface slip to the lateral midspan moment distribution factors can’t be ignored for the week composite curved bridge with shear connection.
2) Close agreement is observed from the finite element analysis and this paper solution. The mean error is found to be –3.6 % happened at the first main beam.
Table 4Lateral moment distribution factors of main beams
Beam number  FEM  This paper solution  Error (%)  Rigid cross beam method  Error (%) 
1  0.594  0.573  –3.6  0.630  6.1 
2  0.517  0.512  –1.0  0.500  –3.3 
3  0.552  0.551  –0.2  0.558  1.2 
4  0.598  0.600  0.3  0.654  9.3 
5  0.668  0.666  –0.4  0.785  17.4 
Notice: Error = (Rigid cross beam method or this paper solution)/FEM *100 % 
6. Conclusions
In this paper, a modified rigid cross beam method has been developed and presented for the lateral distribution of multiI beam composite curved bridge with interface slip. The finite element method is used to determine the stiffness interpolation coefficients of single composite curved beam and verify the lateral distribution calculation of multibeam composite curved bridge. The numerical examples show that the influence of interface slip to the lateral midspan moment distribution factors can’t be ignored for the composite curved bridge with week shear connection, and the lateral distribution got by rigid cross beam method considering slip effect is in good agreement with that by finite element static analysis, which indicates that this method can accurately describe the load distribution characteristics of multibeam composite curved bridge.
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