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

Authors: Qin Xu-xi; Liu Han-bing; Wu Chun-li; Gu Zheng-wei
Journal: Mathematical Problems in Engineering
 (2016)

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## 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.


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## Body

## 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 Figure1, 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 modulusK[N/m2].(3)
The uplift between the slab and I-girder is neglected. The radius of curvature is constant along the beam.Figure 1
Model of a horizontally composite curved I-beam.The deflections in tangential direction, vertical direction, and radial direction (z-direction, y-direction, and x-direction) are indicated as u, w, and v, respectively. The I-girder has the same torsional deflection ϕ and vertical deflection w as the slab has. In this paper, subscript “1” refers to the cross-section of the slab and subscript “2” refers to the cross-section of I-girder, such that the deflections of the slab and I-girder can be represented as (w,ϕ,u1,v1) and (w,ϕ,u2,v2), respectively. The geometric relationships between strains and deflections may be written as(1)εzi=ui′-viR,kyi=vi′′+ui′R,kxi=w′′-ϕR,kzi=ϕ′+w′Ri=1,2,where εzi is the axial strain, kyi is the curvature in y-direction, kxi is the curvature in x-direction, and kzi is the curvature in z-direction. Where ()′=d()/dz, ()′′=d2()/dz2.The constitutive equations are as follows:(2)Ni=EiAiεzi,Myi=EiIyikyi,Mxi=EiIxikxi,Ti=GiITikzii=1,2,where Ni, Myi, Mxi, and Ti are internal axial force in z-direction, internal bending moment in y-direction, internal bending moment in x-direction, and internal torsion moment in z-direction, respectively.The microunit’s force diagram of the horizontally composite curved I-beam is shown in Figure2, which is under the external distribution force q0 and torsion moment m0. The balance equations of the slab are (3)dQx1dz+N1R+qx1=0,dN1dz-Qx1R+qz1=0,dMy1dz+Qx1=0,dQy1dz-q0+qy1=0,dMx1dz+T1R-Qy1+mx1=0,dT1dz-Mx1R+mz1+m0=0.The balance equations of the I-girder are(4)dQx2dz+N2R+qx2=0,dN2dz-Qx2R+qz2=0,dMy2dz+Qx2=0,dQy2dz+qy2=0,dMx2dz+T2R-Qy2+mx2=0,dT2dz-Mx2R+mz2=0,where mxi is the distributed bending moment caused by shear force qzi, mzi is distributed torque caused by shear force qxi, Qxi is the shear force in x-direction, and Qyi is the shear force in y-direction. For the shear forces qzi, qxi, and qyi, there are(5)qz1+qz2=0,qx1+qx2=0,qy1+qy2=0.Figure 2
A microunit’s force diagram of the horizontally composite curved I-beam.By eliminatingQx1 and Qx2 from the first three equations of (3) and (4), we can get(6)-My1+My2′′+N1+N2R=0,N1+N2′+My1+My2′R=0.When there are no axial force and bending moment in y-direction of the I-beam caused by external forces, we can get (7)N1=-N2,My1=-My2.


## 3. Equilibrium at the Interface

At the interface, considering the deflections described in Figure3, shear force qz2 described in Figure 2 can be written as(8)qz2=KΔu=Ku2-u1+w′b=-N1′-My1′R=N2′+My2′R,where b=b1+b2 and Δu is the amount of the deflection at the interface in the tangential direction. So, the equilibrium at the interface in the tangential direction can be written as (9)Ku2-u1+w′b+N1′+My1′R=0.Figure 3
Deflections of differential elements for the beam iny-z plane.Considering deflection described in Figure4, equilibrium at the interface in the radial direction can be written as (10)v2-v1+ϕb=0.Figure 4
Deflection of the composite curved I-beam inx-y plane.In order to solve the problem under consideration, it is convenient to rearrange (9) and (10) as detailed in what follows. Firstly, taking the derivative of (9) one time with respect to z component, then eliminating v2 and u2 by inserting the derivative equation and (10) into the first formula of (1), and at last using the first and second formulas of (2), one can obtain the following:(11)fz=E1A1Ku1′-v1R′′+E1Iy1KRv1′′+u1′R′′-SAE1A1u1′-v1R+w′′b-ϕbR=0.Analogously, taking the derivative of differential equation (9) one time and taking the derivative of (10) two times with respect to z component firstly and then eliminating v2 and u2 by inserting these two derivative equations into the second formulas of (1), using (2), one can obtain the following:(12)fy=E1A1KRu1′-v1R′′+E1Iy1KR2v1′′+u1′R′′-SyE1Iy1v1′′+u1′R+ϕ′′b+w′′bR=0,where SA=1/E2A2+1/E1A1 and Sy=1/E1Iy1+1/E2Iy2.


## 4. Problem Formulation

The governing differential equations of the beam will be derived by the principle of minimum potential energy. The potential energy of the beam takes the following form:(13)Π=UMx+UTz+UMy+Ua+Us-W.U M x is the bending strain energy due to the internal bending moments Mx1 and Mx2.(14)UMx=12∫0RθMx1kx1+Mx2kx2dz=12EIx∫0Rθw′′-ϕR2dz,where EIx=E1Ix1+E2Ix2.U T is the torsional strain energy due to the internal torsion moments T1 and T2.(15)UT=12∫0RθT1kz1+T2kz2dz=12GIT∫0Rθϕ′+w′R2dz,where GIT=G1IT1+G2IT2.U a is the axial strain energy due to the internal axial forces N1 and N2.(16)Ua=12∫0RθN1ε1+N2ε2dz=12∫0RθSAE1A1u1′-v1R2dz.U M y is the bending strain energy due to the internal bending moments My1 and My2.(17)UMy=12∫0RθMy1ky1+My2ky2dz=12∫0RθSyE1Iy1v1′′+u1′′R2dz.U s is the strain energy due to the connector deflections. (18)Us=12∫0Rθqz1·Δudz=12∫0Rθ1KN1′+My1′R2dz=12∫0Rθ1KE1A1u1′-v1R′+E1Iy1Rv1′′+u1′R′2dz.W is the potential energy due to the external loading.(19)W=∫0Rθm0ϕdz+∫0Rθq0wdz+Mxw′0Rθ+Qyw0Rθ+Tzϕ0Rθ,where Mx, Qy, and Tz are the bending moment, shear force, and torsion moment of the beam, respectively. Using the Lagrange multiplier method, the principle of minimum potential energy in augmented form now may be written as(20)min⁡Π∗=Π+∫0Rθλz·fzdz+∫0Rθλy·fydz,where the parameters λz and λy are the Lagrange multipliers and fz and fy are the equilibrium conditions at the interface corresponding to (11) and (12), respectively. In (20), ϕ, w, v1, u1, λz, and λz are all independent variables. The variation of Π∗ is(21)δΠ∗=∫0RθΓ1δwdz+∫0RθΓ2δϕdz+∫0RθΓ3δλzdz+∫0RθΓ4δλydz+∫0RθΓ5δv1dz+∫0RθΓ6δu1dz+H1δw0Rθ+H2δw′0Rθ+H3δϕ0Rθ+H4δϕ′0Rθ+H5δu10Rθ+H6δu1′0Rθ+H7δv10Rθ+H8δv1′0Rθ+H9δv1′′0Rθ+H10δE1Iy1RKv1′′+u1′R′+E1A1Ku1′-v1′R′︸Δu0Rθ,where Γi(i=1~6) and Hi(i=1~10) are calculated from the variation of Π∗, which are given in Appendix A. Each term in (21) must be identically zero. It makes Γ5=0, Γ6=0, H5=0, H6=0, H7=0, H8=0, and H9=0; that is, the undermined Lagrange multipliers λz and λy are (22)λz=E1A1u1′-v1R=N1,λy=E1Iy1v1′′+u1′R=My1.Equation (22) notes that the Lagrange multipliers and the deflections (u1 and v1) are related. So, there will be four rather than six independent variables in this problem. We can take ϕ, w, λz, and λy as independent variables. The rest of governing equations are Γ1=0, Γ2=0, Γ3=0, and Γ4=0, which can be rearranged and rendered in terms of matrix form, as shown below.(23)
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.We can get the pertaining boundary conditions:(24)-EIxw′′′-ϕ′R-bλz′-bRλy′+GITϕ′R+w′R2-Qyδw0Rθ=0,EIxw′′-ϕR+bλz+bRλy-Mxδw′0Rθ=0,GITϕ′+w′R-bλy′-Tzδϕ0Rθ=0,bλyδϕ′0Rθ=0,λz+λyRδΔu0Rθ=0.It is evident that (23) is tenth-order ordinary differential equation. We have ten boundary conditions for the beam, that is, five boundary conditions for each end. For the boundary conditions (see (24)) to hold, the following alternatives at each end shown as in Table 1 are conducted.Table 1
Common support conditions.
Type The boundary conditions Q y = 0 M x = 0 T z = 0 λ z = 0 λ y = 0  w = 0 ϕ = 0 w ′ = 0 ϕ ′ = 0 Δ u = 0  w = 0 ϕ = 0 M x = 0 λ z = 0 λ y = 0  w = 0 T z = 0 M x = 0 λ z = 0 λ y = 0


## 5. Analytical Solution by Trigonometric Series

In this section, a closed form solution of the simply supported horizontally composite curved I-beam is obtained. The deflection functions and the Lagrange multiplier functions are expressed as undetermined coefficients and known trigonometric series to satisfy the governing equations and the rigid torsion constraints at each end. The deflection and the Lagrange multiplier are assumed to be the following form:(25)w=∑i=1nwisin⁡iπzL,ϕ=∑i=1nϕisin⁡iπzL,λz=∑i=1nλzisin⁡iπzL,λy=∑i=1nλyisin⁡iπzL,where wi, ϕi, λzi, and λyi are unknown Fourier coefficients to be determined for each i(i=1,…,n) and L is span length. The applied distributed load q0 and m0 are expanded in single trigonometric series as(26)q0=∑i=1nqisin⁡iπzL,m0=∑i=1nmisin⁡iπzL,where qi and mi are the Fourier coefficients. Multiplying both sides of (26) by each trigonometric series and then integrating them over (0,L), finally, the trigonometric coefficients qi and mi can be readily determined with the orthogonality properties. Substituting (25)-(26) into (23) and solving (23), the unknown parameters wi, ϕi, λzi, and λyi can be determined. The unknown Fourier coefficients are given as follows:(27)wi=qiγq1+γq2γ+miγm1γ,ϕi=qiγq3γ+miγm2+γm3γ,λzi=qiγq4γ+miγm4γ,λyi=qiγq5γ+miγm5γ,where γ, γqi, and γmi(i=1,…,5) are the relevant coefficients which are given in Appendix B.


## 6. Numerical Example

A steel-concrete composite curved I-beam calculated by Thevendran et al. [27] is taken as a numerical example in order to show the accuracy and reliability of the present trigonometric solution of the beam. The beam is simply supported at both ends as shown in Figure 1. And the beam is subjected to F=150KN vertical load at the midspan section. The material properties and dimensions of the curved I-beam (SP4 beam) are shown in Table 2.Table 2
Material properties and dimensions of the curved I-beam.
Young’s modulus(GPa) Poisson’s ratio Density(kg/m3) Width(mm) Thickness(mm) Central angle Radius of curvature(m) Steel girder 206 0.3 7850 Flange: 332Web: 172 Flange: 13Web: 8 θ = 14 . 3 ° 24 Concrete slab 26 0.27 2400 1500 100The finite element software ANSYS is used to model the described structure (Figure5). In the FEM model, both the concrete slab and the steel I-girder are modeled as space beam elements and the connection between the slab and I-girder is simulated by multiple-point constraints (MPC) which are modeled by two rigid links connected through interface nodes between the slab and I-girder. The spring elements are used in the tangential direction to allow for the possibility of movement. Coupling degrees of freedom w are used to prevent the uplifting issue.Figure 5
Finite element model diagram of the composite curved I-beam.The vertical deflectionw, the tangential deflection u, the radial deflection v, and the torsional deflection ϕ are restrained to satisfy the simply supported constraints at both ends. Although the BEAM4 element in ANSYS cannot consider the warping effect, the element meets the theory of this paper quite well. So, 120 BEAM4 elements are selected as the spatial beam elements. The element of MPC 184 with the number of 122 is modeled as the multiple-point constraints element. In addition, the element of COMBIN 14 with the number of 61 is used when tangential slip is considered.For the full interaction case, the slip parameters are taken asK=1010N/m2. Figure 6 shows the vertical deflection w based on the FEM, the solution calculated by this paper, and Thevendran et al. experimental results, respectively. For this paper solution, the number of Fourier coefficients n (in (25)) can be chosen as large as we can to achieve a required accuracy. As an illustration of convergence of the results, we show the results from this paper by taking the number n as 1, 5, and 10 terms, respectively. It can be seen in Figure 6 that the vertical deflections results solved from this paper and from Thevendran et al. experiment are in reasonable agreement (the difference is within 12%), and the solved results from this paper and from FEM are in good agreement.Figure 6
Vertical deflection of the beam.Considering only vertical deflections are available from the experiments of Thevendran et al., the other results of the solution calculated by this paper and by the FEM are compared in Figures7–12. The results are the torsional angle ϕ, the bending moment My1 (My1=λy), the axial force N1 (N1=λz), the bending moment Mx, the shear force Qy, and the torsion moment Tz, respectively.Figure 7
Torsional angle of the beam.Figure 8
Bending momentMy1 of the beam.Figure 9
Axial forceN1 of the beam.Figure 10
Bending momentMx of the beam.Figure 11
Shear forceQy of the beam.Figure 12
Torsion momentTz of the beam.It can be seen from Figures7–12 that the results solved by this paper can be quickly convergent, and one can get very good results by taking more than 5 terms. When the stiffness of the shear connectors is changed, the vertical deflection and torsional angle at midspan section are shown in Figures 13 and 14, respectively. In Figure 13, it can be seen that the vertical deflections converge to the minimum when the slip parameter K is greater than 1010N/m2. Moreover, the changes with slip parameter K indicate the shear connecter’s stiffness in experimental model of Thevendran et al. with the range from 109N/m2 to 1010N/m2. As it can be seen from Figures 13 and 14, the performance of the curved composite I-beam becomes significantly rigid when the shear connecter’s stiffness is increased from 107 to 109N/m2. The reduction of shear connectors’ stiffness between layers has a significant effect on the vertical deflection and torsional angle at midspan section.Figure 13
Vertical deflection at midspan section when the stiffness of the shear connecter is changed.Figure 14
Torsional angle at midspan when the stiffness of the shear connecter is changed.The inaccuracy of the vertical deflection and torsional angle between analytical solutions in this paper and the FEM results in Figures6–14 is small and reasonable. Thus, the model created in this paper can be applied sufficiently for practical purposes.


## 7. Conclusions

In this paper, an analytical solution has been developed and presented for the simply supported horizontally composite curved I-beam. It is primarily used to solve a static problem of a two-layered composite curved beam with flexible shear connection in the tangential direction. The trigonometric series are adopted in solution expression in terms of span coordinate. Both governing equations and boundary conditions are obtained by using the Vlasov curved beam theory and the principle of minimum potential energy. The accuracy of calculation results based on the theory presented in this paper is verified by comparing with the FEM and the Thevendran et al. experimental results. Therefore, the analytical expression and model established in this paper possess certain actual engineering significance.



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