### Transient Thermoelastic Problem of Semi- Infinite Circular Beam with Internal Heat Source

Transient Thermoelastic Problem of Semi- Infinite Circular Beam with Internal Heat Source

Abstract- This paper is concerned with transient thermoelastic problem in which we need to determine the temperature distribution, displacement function and thermal stresses of a semi-infinite circular beam when the boundary conditions are known. Integral transform techniques are used to obtain the solution of the problem.

Key Words: Semi-infinite circular beam, transient problem, Integral transform, heat source

I. INTRODUCTION

In 2003, Noda et al.  have published a book on Thermal Stresses, second edition. Khobragade  studied Thermoelastic analysis of a thick annular disc with radiation conditions and Khobragade  discussed Thermoelastic analysis of a thick circular plate. Pathak et al.  studied Transient Thermo elastic Problems of a Circular Plate with Heat Generation. Love  published a book on a treatise on the mathematical theory of elasticity. Marchi and Zgrablich  studies Vibration in hollow circular membrane with elastic supports. Nowacki  discussed the state of stress in thick circular plate due to temperature field. Wankhede  studied the quasi-static thermal stresses in a circular plate. In this paper, an attempt has been made to determine the temperature distribution, unknown temperature gradient, displacement function and thermal stresses of thick, semiinfinite circular beam due to heat generation. The governing heat conduction equation has been solved by using MarchiZgrablich and Fourier Cosine transform techniques. The result presented here will be more useful in engineering applications. II. STATEMENT OF THE PROBLEM Consider a thick circular beam occupying the space D: a  r b, 0 z< ∞. The material is homogeneous and isotropic. The differential equation governing the displacement potential function  (r ,z ,t) as Noda et al.  is T r r r z t                       1 1 1 2 2 2 2 (1) where  and t are the Poisson’s ratio and the linear coefficient of thermal expansion of the material of the plate