Nonlinear pre and post-buckled analysis of curved beams using differential quadrature element method

This paper studied the in-plane elastic stability including pre and post-buckling analysis of curved beams considering the effects of shear deformations, rotary inertia, and the geometric nonlinearity due to large deformations. Firstly, the governing nonlinear equations of motion were derived. The problem was solved performing both the static and dynamic analysis using the numerical method of differential quadrature element method (DQEM) which is a new and efficient numerical method for rapidly solving linear and nonlinear differential equations. Firstly, the method was applied to the equilibrium equations, leading to a nonlinear algebraic system of equations that would be solved utilizing an arc length strategy. Secondly, the results of the static part were employed to linearize the dynamic differential equations of motion and their corresponding boundary and continuity conditions. Without any loss of generality, a clamped-clamped curved beam under a concentrated load was considered to obtain the buckling loads, natural frequencies, and mode shapes of the beam throughout the method. To validate the proposed method, the beam was modeled using a finite element simulation. A great agreement between the results was seen that showed the accuracy of the proposed method in predicting the pre and post-buckling behavior of the beam. The investigation also included an examination of the curvature parameter influencing the dynamic behavior of the problem. It was shown that the values of buckling loads were completely influenced by the curvature of the beam; also, due to the sharp change of longitudinal stiffness after bucking, the symmetric mode shapes changed more than it was expected.

Free in-plane vibration of cracked curved beams: Experimental, analytical, and numerical analyses

In this study, free vibration of a cracked curved beam utilizing analytical, numerical, and experimental methods is investigated. The differential quadrature element method is used to solve the equations of motion numerically. The governing equations are also solved analytically. The crack, which is considered to be open, is modeled as a rotational spring. Furthermore, the effect of curvature on mode shapes is studied. To verify the validity of the proposed methods of determining frequencies and mode shapes, an experimental modal analysis test is conducted on a sample beam having crack with some different depths. This study revealed that the behavior of curved beams toward the mode transition phenomenon depends greatly on the boundary conditions of the beam. Also, both the location and depth of crack have considerable effects on natural frequencies.