^{1, 2, 4} Distinguished Center of Thermoelasticity, Mechanical Engineering Dept.,

 Amirkabir University of Technology, Tehran, Iran

³ Mechanical Engineering Dept., Louisiana State University, USA

This paper presents the finite element solution of a beam based on the first-order shear deformation theory subjected simultaneously to arbitrary time-dependent thermal and uniform mechanical transverse shock loads. The beam is made of homogenous and isotropic materials. The equation of motion and the conventional coupled energy equation are simultaneously solved to obtain the displacement components and temperature distribution in the beam. Damping effect and internal friction are neglected in the beam. The governing partial differential equations of the problem are solved simultaneously using the Galerkin finite element method with C¹- continuous shape function leading to fast convergence of the solution. Results are presented for simply supported boundary conditions.   

Keywords: beam; first-order shear deformation theory; coupled thermoelasticity; Galerkin Method

Introduction

The equations for a coupled, thermoelastic beam, including the effects of shear deformation and rotatory inertia, are derived by Jones [1]. Mcquillen and Brull [2] presented analytical solution for the dynamic, thermoelastic response of cylindrical shells using a variational theorem. Coupled thermally induced vibrations of Euler-Bernoulli and Timoshenko beam with one-dimensional heat conduction are investigated by Seibert and Rice [3].  Coupled thermoelasticity of beams is discussed by Massalas and Kalpakidis [4,5]. The analytical solution of the coupled thermoelasticity of beams with the Euler-Bernoulli assumption is given in [4], and that with Timoshenko assumption is given in [5]. In the treatment of these problems a linear approximation for temperature variation across the thickness direction of the beam is considered. Eslami and Vahedi [6] presented the one-dimensional coupled thermoelasticity problem of rods using the Galerkin finite element method. Manoach and Ribeiro developed a numerical procedure to study the coupled large amplitude thermoelastic vibrations of Timoshenko beams subjected to the thermal and mechanical loads using the finite difference approximation and modal coordinate transformations [7].

This paper presents the behavior of a beam under lateral thermal shock with coupled thermoelastic assumption. The analysis is based on the Galerkin finite element method, using a C¹- continuous shape function. The beam formulations are based on the first-order shear deformation theory.

References

[1]     John, P., Jones, Thermoelastic Vibration of Beams, Journal of Acoustical Society of America, Vol. 39, p. 542-548, 1966.

[2]     Mcquillen, E.J., Brull, M.A., Dynamic Thermoelastic Response of Cylindrical Shell, Journal of Applied Mechanics, Vol. 37, p. 661-670, 1970.

[3]     Seibert, A.G., Rice, J.S., Coupled Thermally Induced Vibrations of Beams, AIAA Journal, Vol. 11, p. 1033-1035, 1973.

[4]     Massalas, C.V., Kalpakidis, V.K., Coupled Thermoelastic Vibration of a Simply Supported Beam, Journal of Sound and Vibration, Vol. 88, p. 425-429, 1983.

[5]     Massalas, C.V., Kalpakidis, V.K., Coupled Thermoelastic Vibration of a Simply Supported Beam, Journal of Sound and Vibration, Vol. 88, p. 425-429, 1983.

[6]     Eslami, M.R., Vahedi, H., Coupled Thermoelasticity Beam Problems, AIAA Journal, Vol. 27, p. 662-665, 1988.

[7]     Manoach, E., Ribeiro, P., Coupled Thermoelastic Large Amplitude Vibrations of Timoshenko Beams, Journal of Mechanical Science, Vol. 46, p. 1589-1606, 2004.

[8]     Durbin, F., Numerical Inversion of Laplace Transform, Computational Journal, Vol. 17, p. 371-376, 1973.