An Alternative Procedure for Longitudinal Vibration Analysis of ‎Bars with Arbitrary Boundary Conditions

The present work aims at generating a systematic way for longitudinal vibration (LV) analysis of bars (or rods) with arbitrary boundary conditions (BCs) by mixed-type finite element (MFE) method using the Gâteaux differential. Both materials and geometrical properties of the bar are uniform along the longitudinal direction. The problem is reduced to solution of the classical eigenvalue problem in dynamic analysis. The axial (normal) load and the displacement along the bar are the basic unknowns of the mixed element. The element formulation for the shape function must satisfy only C0 class continuity since the first derivatives of the variables exist in the functional. The functional governed with proper dynamic and geometric BCs of the problem. Results of the recommended method are benchmarked and verified via numerous problems present in the literature. The unique aspects of this study and the possible contributions of the proposed method to the literature can be summarized as follows: by using this new functional, displacements and internal force values can be obtained directly without any mathematical operation. In addition, geometric and dynamic BCs can be obtained easily and a field variable can be included to the functional systematically. To examine the effects of BCs on the longitudinal vibratory motion of a uniform elastic bar and to give a better insight into LV analysis of bars with arbitrary BCs, a set of numerical examples are presented.