This paper presents a finite element formulation for investigating the free vibration of uniform Timoshenko beams resting on a Vinkler-type elastic foundation and prestressing by axial force. | Vietnam Journal of Mechanics, VAST, Vol. 29, No . 1 (2007), pp. 1 - 12 FREE VIBRATION OF PRESTRESS TIMOSHENKO BEAMS RESTING ON ELASTIC FOUNDATION NGUYEN DINH KIEN Institute of Mechanics Vietnamese Academy of Science and Technology Abstract. This paper presents a finite element formulat ion for investigating the free vibration of uniform Timoshenko beams resting on a V.' inkler-type elastic foundation and prestressing by axial force. Taking the effect of prestress, foundation support and shear deformation into account, a stiffness matrix for Timoshenko-type beam element is formulated using the energy method . The element consistent mass matrix is obtained from the kinetic energy using simple li near shape functions . Employing the formulated element, the natural frequencies of the beams having various boundary conditions are det ermined for different values of the axial force and foundation stiffness . The vibration characteristics of the beams pa rtially supported on the foundation arc a lso st udi ed and highlighted. Specially, the effects of shear deformation on the vibration frequenci es of prestress beams fully and partially supported on the elast ic foundation are investigated in det a il. 1. INTRODUCTION Practical problems like railroad tracks, hight way pavements, continuously pipelines . can be modelled by means of beams on elastic foundation. Static analysis of beams on various types of foundation has been extensively carried out by many researcher [L 2]. In the context of dynamic analysis, in [3] Rosa described an analytical approach for investigating the effect of foundation support on the vibration characteristics of T imoshenko beams resting on a Pasternak foundation. Using the so-called Ray leigh-Ritz method, Rao has investigated the large vibration characteristics of simply supported and cantilever Timoshenko beams resting on a two-parameter foundation [4]. By solving the governing equation, Hung [5] derived the stiffness and mass matrices used
