Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 2465025, 12 pages

http://dx.doi.org/10.1155/2016/2465025

## A Trigonometric Analytical Solution of Simply Supported Horizontally Curved Composite I-Beam considering Tangential Slips

College of Traffic, Jilin University, Changchun 130025, China

Received 27 March 2016; Revised 31 July 2016; Accepted 14 August 2016

Academic Editor: Leonid Shaikhet

Copyright © 2016 Qin Xu-xi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents an analytical solution of the simply supported horizontally composite curved I-beam by trigonometric series considering the effect of partial interaction in the tangential direction. Governing equations and boundary conditions are obtained by using the Vlasov curved beam theory and the principle of minimum potential energy. The deflection functions and the Lagrange multiplier functions are expressed as trigonometric series to satisfy the governing equations and the simply supported constraints at both ends. The numerical results of deflections and forces which are obtained by this method are compared with both FEM results and experimental results, and the inaccuracy between the analytical solutions in this paper and the FEM results is small and reasonable.

#### 1. Introduction

A composite beam can be defined as “partially composite beam” when the number of shear connectors is less than the required number for fully composite design; therefore, the interface shear force is limited by the strength of shear connectors. In contrast to fully composite beam, slip between layers can be significant and results in a decrease in the elastic stiffness of partially composite beam. Composite beam exhibiting partial shear interaction will stand larger deflection than the beam exhibiting full shear interaction. By assuming complete shear interaction, the calculation of deflection for partial shear interaction beam is maybe underestimated. Because serviceability issues often govern the structural design of composite section, the accurate calculation of deflection is critical. The partial interaction is applied not only in steel-concrete composite beam, but also in other types of composite beams, such as layered wooden beams, wood-concrete floor systems, and other multilayered laminated composite structures [1, 2].

Earlier studies on behavior of the partially composite beam are mostly focused on the straight composite beam. The first paper dealing with the analysis of composite beam with partial interaction has been completed by Newmark et al. [3]. After that, Goodman and Popkov [4, 5] have conducted the analytical and numerical research on the relative slip between layers and found that the relative slip between layers has a significant effect on the overall characteristics of the composite beam with the reduction of shear connectors’ stiffness. Girhammar et al. [6, 7] have applied a partial shear interaction theory for composite beam subjected to static loads and dynamic loads. Wang [8] has developed a method to calculate deflection of partially composite beam based on the stiffness of the shear connectors. Dall’Asta [9] has developed a three-dimensional theory for composite beam with partial shear interaction dealing with combination of bending in the symmetry plane, torsion, and transverse bending in the plane parallel to the shear connector interface. Nie and Cai [10] have studied the effects of shear slip on the deflection of steel-concrete composite beam. Ranzi and Bradford [11] have presented analytical solutions for time dependent behavior of partially composite beam. Liu et al. [12] have found out the solution of shearing slip for steel-concrete composite beam under the concentrated load. Campi and Monetto [13] have presented a new formulation to analyze two-layer linearly elastic Timoshenko beam with interlayer slip. In the aspect of numerical simulations studies, different kinds of numerical and finite element formulations for the analysis of composite beam with interlayer slip have been suggested [1, 14–19].

Many significant researches have been accomplished in regard to the behavior of straight partially composite beam. In the same time, various scholars have done researches on the curved beam theory. One of the earliest works dealing with the stability behaviors of curved beam has been put forward by Vlasov [20]. After that, scholars covered different extensions and enhancements to the Vlasov model [21–26]. These researches are helpful to the development of the composite beam theory. However, very little literature has focused on the curved partially composite beam such as horizontally composite curved steel I-beam bridge. Thevendran et al. [27] and Shanmugam et al. [28] have conducted experiments on the steel-concrete composite curved beam to investigate the ultimate load behavior. In their study, the finite element software ABAQUS was used to analyze the behavior of test specimens. Full composite action between steel beam and the concrete slab was assumed. The results of deformations, stress distributions, and ultimate strengths obtained by finite element analysis were found to be in good agreement with the experimental results. After that, Topkaya et al. [29] have conducted experimental and numerical studies to establish the behavior of composite curved bridge during construction. In their study, two FEM models were established by different kinds of software to predict the behavior of the curved steel trapezoidal box-girder, and the authors have drawn a conclusion that the reasonable finite element model is able to accurately capture girder behavior during construction. Giussani and Mola [30] have developed an analytical equation for elastic composite beam curved in-plane with the long term behavior. But the partial interaction between the steel girder and concrete slab was not considered in the study. Erkmen and Bradford [31] have solved the equation of the composite curved beam considering the two-layer partial interaction by providing a highly efficient 3D beam finite element. The results demonstrate that the developed formulation is accurate and effective in capturing the behavior of composite beams curved in-plane.

Even though some previous researches of these evaluation methodologies have been accomplished, there is still lack of the fundamental understanding of the system-level behavior on the overall performance of composite curved beam with partial shear interaction. Besides, although exact solutions can be obtained by the 3D finite element model, it is very complex and time consuming. Therefore, the FEM analysis procedures are not suitable for the initial design. This study aims to provide an analytical theory of the horizontally composite curved beam considering the partial interaction in tangential direction. The beam is assumed to be statically determinate with a constant radius of curvature along the longitudinal axis. Governing equations and boundary conditions are obtained by using both the Vlasov curved beam theory and the energy variation principle. The undetermined vertical deflection, torsional deflection, and Lagrange multipliers will be approximated by Fourier series to solve the governing equations of the partial interaction composite beam theory in the procedures. The numerical results of deflections and forces obtained by using proposed theory are presented and compared with FEM results and experimental results. Comparison results show that the calculation can be easily and accurately handled which is a great advantage of this method.

#### 2. Basic Assumptions and Conditions

This study is on a horizontally composite curved I-beam where the model is shown as in Figure 1, and the noteworthy features of this research are shown as follows:(1)The slab and I-girder are linear-elastic different materials, and the cross-sections made of both materials are rigid in their plane. The effects of shear deformation, warping deformation, and distortion deformation are neglected in this research. Structural analysis of the beam is based on the Vlasov curved beam theory (for each part of the beam).(2)The interlayer connectors between the slab and I-girder are flexible, and they are continuous in tangential direction and rigid in radial direction. The load-slip behavior (per unit length) of connectors in tangential direction is described in a linear-elastic range with a constant slip modulus .(3)The uplift between the slab and I-girder is neglected. The radius of curvature is constant along the beam.