A Finite Segment Method for Skewed Box Girder Analysis

A finite segment method is presented to analyze the mechanical behavior of skewed box girders. By modeling the top and bottom plates of the segments with skew plate beam element under an inclined coordinate system and the webs with normal plate beam element, a spatial elastic displacement model for skewed box girder is constructed, which can satisfy the compatibility condition at the corners of the cross section for box girders.The formulation of the finite segment is developed based on the variational principle. The major advantage of the proposed approach, in comparison with the finite element method, is that it can simplify a threedimensional structure into a one-dimensional structure for structural analysis, which results in significant saving in computational times. At last, the accuracy and efficiency of the proposed finite segment method are verified by a model test.


Introduction
The continued economic development and increasing investment in infrastructure in China has resulted in a marked improvement in construction standards for transportation networks throughout the country. This investment and development include the increased design and implementation of high-speed rail lines, especially in the more populous eastern portion of the country [1]. Skew bridges are commonly used on high-speed railway lines due to their following advantages: (a) maintaining harmony with the surrounding buildings and environment by requiring less land space for the new structure; (b) reducing resistance to flow for piers located in the water; and (c) meeting high-speed rail performance demands. Compared with a bridge having an orthogonal substructure, the behavior of skewed bridges is more complicated due to torsional effects that result from the skew angle [2].
During the past few decades, experimental and computational studies of skewed highway bridges have been performed [3][4][5][6]. Scordelis et al. [7] presented the theoretical and experimental results of a 45 ∘ skew two-span four-cell reinforced concrete box girder bridge model with a 1 : 2.82 scale. Results are given for reactions, deflections, strains, and moments due to working stress point loads applied before and after overload stress levels in the bridge. Evans and Rowlands [8] carried out experimental and theoretical investigations of the behavior of single-cell, single-span box girders on skew supports. The behavior of bridge was analyzed by the FEM. The effect of the skew angle on the behavior of bridge was discussed. Meng et al. [9] described an analytical and experimental study of a skewed bridge model. The FE analyses (FEA) were performed using the commercial software SAP2000. Good agreement between the experimental and FEA results is obtained. Meng and Liu [10] proposed a refined stick model for the preliminary dynamic analysis of skew bridges as well. Badwan and Liang [11] presented an in-site static load test and a FEA on an existing 60 ∘ skew, three-span continuous steel girder bridge. The measured bridge response under the test load was used to develop and calibrate a FEM model by using ANSYS software. The calibrated FEM model was demonstrated that it is realistic to predict the bridge response. Menassa et al. [12] reported that the effect of a skew angle on simplespan reinforced concrete bridges using FEM. The commercial FE software SAP2000 was used to generate the threedimensional finite element models. The results show that the AASHTO Standard Specifications procedure overestimated the maximum longitudinal bending moment when a skew angle is larger than 20 ∘ . Since conventional grillage methods 2 Mathematical Problems in Engineering  cannot account for some important structural actions of thinwalled box girders, the Canadian Highway Bridge Design Code (CHBDC) [13] as well as the American Association of State Highway Transportation Officials (AASHTO) [14,15] has prohibited the use of the conventional grillage method in the case of certain types of box girder bridges. Due to a large number of degrees of freedom needed, a detailed threedimensional FEA of box girders performed using commercial FE programs is too complex and time-consuming. Therefore, it is important to develop a practical FEM to reduce the amount of computational work for the analysis of the skewed box girder bridges. Finite segment method has been used to simplify a three-dimensional thin-walled box girder into a one-dimensional structure for curved box girder analysis [16]. But until now, such method used for skew box girder analysis do not appear to be reported in the open literature. In this study, by modeling the top and bottom plates of a skew box girder with skewed plate beam element and the webs with normal plate beam element, a spatial displacement field for a skewed box girder is constructed. The displacement compatibilities of the subelements at corners are met accordingly. Skewed plate beam element represents that the element is formulated under an inclined coordinate system while the normal plate beam element represents that the element is formulated under an orthogonal coordinate system. The skewed box girder is discretized into finite segments which were assembled by 4-plate beam subelements along the length of the girder. To verify the accuracy of the proposed finite segment method (FSM), the results obtained using this approach were compared with the results obtained by using ANSYS and the tested results from a modal test.

Basic Formulae under Inclined Coordinate System
There is an inclined angle between the central line and the support line of the bridge. Therefore, an inclined coordinate system is chosen for the top and bottom plates of the skewed box girder, as shown in Figure 1.
The inclined angle of the specific plates is ; the relationships between the rectangular coordinates ( ) and the inclined coordinates ( V ) are where , V, and are the transverse (along the direction), vertical (along the direction), and longitudinal displacement (in the direction) of the top or bottom plate, respectively. Using the derivative principle of the multivariable functions, there is

Displacement Model of the Cross Section
For convenience of illustration, a single-cell skewed box girder is used as an example herein. The cross-sectional displacement parameters of a single-cell skewed box girder are shown in Figure 2. To establish the displacement model of the spatial skewed box girder, two assumptions based on Vlasov's thin-walled beam theory [17] are adopted in this paper.
(1) As the skewed box girder is discretized into several skewed box girder segments along the length of bridge, the displacement compatibilities of these elements at the locations of corners are satisfied.
(2) The tensile and compressive deformations along the width and height directions of the four walls of the girder are ignored.
For each segment of the single-cell skewed box girder, the section displacement parameters include four longitudinal displacement parameters , , , for the four corners; two transverse displacement parameters and for the top and bottom plates; and two vertical displacement parameters V and V for the left and right upper corners of the skewed box girder. Hence, there are eight independent displacement parameters for the cross section of the singlecell skewed box girder, that is, , , , , , , V , and V . The cross-sectional parameters of the skewed box girder segment are all defined in the rectangular coordinates ( ); that is, the longitudinal displacements of the segment , , , and are along the axial direction, not along the skewed bridge axial line.

Subelement Displacement Parameters
If the whole skewed beam segment of the skewed box girder is taken as one-beam element, the segment can be discretized into wall subelements, by which a suitable subelement displacement mode can be established. According to the thinwalled beam theory [17], the displacement at the gravity center of the subelement cross section is equal to the average value of displacements at the corresponding two nodes and the subelement displacement parameters are determined by the displacements at the two nodes.
The nodal displacement parameters are shown in Figure 3. The subelement displacement parameters are given as follows.
For the left web, we have the following. The transverse displacement (in the web plane) is The vertical displacement (out of the web plane) is The longitudinal displacement (in the direction) is The torsional angle around axial is For the right web, we have the following. The transverse displacement (in the web plane) is The vertical displacement (out of the web plane) is The longitudinal displacement (in the direction) is The torsional angle around axial is In these above equations, [ ( )] and [ ( )] are cubic shape function matrix and linear shape function matrix about , respectively. The detailed expressions are as follows: For the top and bottom subelements, because of their skew geometry, their displacement parameters are all defined relative to the rectangular coordinate ( ) for convenience in expressing the stiffness matrices of the specific subelements. Hence, the four nodal transverse displacements , , , are along the direction, the nodal vertical displacements V , V , V , V are along the direction, and the longitudinal node displacements , , , are along the direction as shown in Figure 3. The subelement displacement parameters are given as follows: For the top plate, we have the following. The transverse displacement (in the direction) is The vertical displacement (in the direction) is The longitudinal displacement (in the direction) is The torsional angle around axial is For the bottom plate, we have the following. The transverse displacement (in the direction) is The vertical displacement (in the direction) is The longitudinal displacement (in the direction) is The torsional angle around axial is Mathematical Problems in Engineering   5 where

Relationship between the Displacement Parameters of Subelement and the Section
According to the deformation compatibility condition between the part and the whole, the four walls of the skewed box girder considered as subelement beams are connected at the corners [16]. For the top plate, there are = , For the bottom plate, there are = , Sectional geometry and displacement parameters of the left and right webs of the subelement are shown in Figure 4, where section is normal to the center line of the skewed box girder. In view of sectional geometry of skewed box girder and the basic assumptions of the box girder, there are According to the geometric relationship in Figure 4, we obtained = , = , Rearranging (12) and (13) gives the following.
For the left web, ⋅ ( cos + sin ) , 6 For the right web,

Elastic Strain Energy of the Subelements
The transformation relationship between the inclined coordinate and the rectangular coordinate as defined in (1) and (2) was used to calculate the stiffness matrices. Once the displacement model of the subelements is given, the displacements at arbitrary point of the subelement can be determined.
On the top plate, we have the following. The transverse displacement (in the direction) is The vertical displacement (in the direction) is The longitudinal displacement (in the direction) is On the bottom plate, we have the following. Mathematical The transverse displacement (in the direction) is The vertical displacement (in the direction) is The longitudinal displacement (in the direction) is Based on (1) and (2), the strain-displacement relationship of the top plate can be written as Similarly, the strain-displacement relationship of the bottom plate can be written as Thus, the elastic strain energy of the top plate is Similarly, the elastic strain energy of the bottom plate can also be obtained. The shape of the webs has no difference in comparison with conventional beam. Therefore, the expression of the elastic strain energy of webs can be written as (23)

Establishment of Segment Equilibrium Equation
The total elastic strain energy of the skewed box girder segment, which consists of those of its subelements, is given as where , , , and are the elastic strain energies of the top plate, the bottom plate, the left web, and the right web, respectively, and the last item of the right-hand side is the work done by external force. { } is the vector of nodal force; any forces acted not at the corners of the element should be transformed to the corners according to the shape functions [18]. Rearranging gives the elastic strain energy of the top and bottom plates as Substituting (23), (25) into (24) gives Substituting (16) We constructed the following equations: ] is same as that of the stiffness matrix of the conventional spatial beam element. By using the same assembling method as the conventional finite element method, the global stiffness matrices and equilibrium equation can be established and solved.

Loading Description.
Two load cases were considered, as shown in Figure 6 and Table 1. The model box girder was instrumented to record displacements and strains during static testing. Instrumentation consisted of 22 displacement transducers, each having 0.01 mm precision, mounted on the two sides of the box girder's webs near the bottom, and more than 100 resistance strain gauges measuring concrete and steel reinforcement strains. Static loads were applied using nine jacks and spreader beams.

Results and Discussion.
The commercial finite element program ANSYS 14.0 was used to validate the accuracy of the proposed finite segment method (FSM). A linear 3D FE model was established. The girder was modeled with 3D block element Solid45, and the FE model is shown in Figure 7. The modal bridge is discretized into 14712 solid elements. For the proposed method, the skewed box girder structure is divided into 10 skewed box girder segments so that each segment of the bridge is subjected to nodal loads only. The deflection distributions of the webs calculated by the proposed FSM are compared with the experimental results and ANSYS results in Figures 8 and 9. The torsional angle distributions of the section calculated by the proposed FSM and tested results are compared in Figure 10.
From Figures 8 and 9, it can be seen that the results obtained using the proposed finite segment method agree well with that calculated by ANSYS and the tested results which verified the accuracy of the proposed method. For     symmetric loading, the deflection distribution of the webs along the length of the bridge is symmetrical. And the vertical deflection of the left web is almost the same as that of the right web. For torsional loading, the symmetrical characteristic of the deflection distribution of the web along the length of the bridge is not as good as symmetric loading. For the left web, the deflection of the left part of the main span is larger than that of the right part, while, for the right web, the deflection of the right part of the main span is larger than that of the left part, which are consistent with the characteristics of the load. The value of deflections and differences between the results of FSM and test in section IV-IV, V-V and VI-VI are listed in Table 2. It can be seen that the results calculated by the proposed method are close to the tested results. For symmetric loading, the maximum differences of the left web and right web between the FSM results and the tested results are no more than 2.1% and 5.4%, respectively. For torsional loading, the maximum differences of the left web and right web between the FSM results and the tested results are no more than 2.5% and 6.1%, respectively.
From Figure 10, it can be seen that the torsional angle distribution along the length of the bridge is basically antisymmetric for both symmetric loading and torsional loading.
Because the number of the discretized elements in ANSYS model is much larger than the proposed FSM, the computational time of the proposed method is saved largely. In  other words, the proposed FSM is more efficient than the conventional finite element method. In these examples, using the same computer Dell R1308 PC with a four-core Intel i3-3100 2.4 GHz CPU and AMD Radeon HD 4500 graphics card, the solution time of the ANSYS is 436 s while that of the FSM is 2.1 s, from which it can be seen that the computational time is saved significantly.

Conclusions
In this paper, a finite segment method was presented to analyze the mechanical behavior of the skewed box bridges. Using the skewed plate beam element under inclined coordinate system for the skewed top and bottom plates and normal plate beam element for the webs, a spatial displacement field for the skewed box girder is constructed. The displacement function is directly constructed according to the behavior of the skewed box girder. The skewed box girder is discretized into finite segments along the length of the girder with each segment assembled by 4 plate beam subelements. The compatibility condition of the displacement at corners of subelements is satisfied accordingly. The stiffness matrix of the finite segment is established based on the potential variation principle and, by using the same assembling method as the conventional finite element method, the global stiffness matrices and equilibrium equation can be established and solved. A model test and the commercial FEM software ANSYS 14.0 were adopted to verify the accuracy of the proposed method. The agreement among the tested results, results obtained by ANSYS, and the proposed finite segment method is good, which demonstrates that the present method is accurate and efficient. Compared with the conventional finite element method, the major advantage of this approach lies in that it can simplify a three-dimensional structure into a one-dimensional structure for analysis and therefore reduces the computational efforts significantly. The proposed method is especially suitable to analyze the global behavior of the skew box girder bridge such as the global stiffness and deflection, and it is a cost-effective method in preliminary design of bridge.