In‐Plane Free Vibration Analysis by Using an Efficient Simple Plane Element

Abstract

This article presents an in‐plane free vibration analysis using an efficient plane element based on strain states. In this element formulation, Taylor’s expansion of the strain field is exploited around the origin of the coordinates. The magnitudes of the strain gradient at the origin are referred to as strain states. The second‐order deformation field is calculated by assuming a linear strain field and using the relationship between displacement and strain. The number of unknowns in the formulation is reduced through the establishment of equilibrium equations. Shape functions are obtained by utilizing a dependency between the strain states and nodal displacement. The mass and stiffness matrices of the element are computed using these shape functions. Several benchmark tests are employed to demonstrate the efficiency of the element in free vibration analysis. The results of well‐established elements by other researchers are available in these tests. In some of these tests, the insensitivity of the elements’ results to distortion is evaluated in free vibration analysis. A comparison of the frequency answers of the elements shows that the element created based on strain state has high accuracy even in coarse distorted meshes. Also, the convergence rate of this element is investigated in the regular and distorted meshes. The accuracy of this element in the coarse mesh is significant. Furthermore, the sensitivity of the element created based on strain state to distortion is less than other elements.