Free Vibration Analysis of a Sloping-frame: Closed-form Solution versus Finite Element Solution and Modification of the Characteristic Matrices (TECHNICAL NOTE)

This article deals with the free vibration analysis and determination of the seismic parameters of a sloping-frame which consists of three members; a horizontal, a vertical, and an inclined member. The both ends of the frame are clamped, and the members are rigidly connected at joint points. The individual members of the frame are assumed to be governed by the transverse vibration theory of an Euler-Bernoulli beam. To solve this classical problem, a closed-form solution is firstly proposed and then, a numerical analysis is performed for some verification purposes. The closed-form solution is developed by solving the frame equations of motion, directly. For this reason, some mathematical techniques are utilized, such as Fourier transform and the well-known complementary solutions. In this way, some differential equations must be solved, and several boundary conditions should be satisfied. Herein, the more accurate derivation of the last boundary condition is the most important challenge of this paper. This boundary condition is expressed as three distinctive versions, and the free vibration parameters of the frame for the three versions are attained. Moreover, these results are obtained by the use of the finite element method. In this comparison process, some differences are observed between the closed-form and the numerical results. This fact motivated us to propose some modifications in the characteristic matrices of the finite element model of the frame. This modification makes the results of the Finite element method similar to the results of the first version of the closed-form solution. Finally, the natural frequencies and mode shapes are presented for a wide range of angles of the sloping member. Also, two particular cases are discussed and their boundary conditions are presented.