A Lagrange Multiplier-based Technique within the Nonlinear Finite Element Method in Cracked Columns

Abstract

In this research, two energy-based techniques, called Lagrange multiplier and conversion matrix, are applied to involve crack parameters into the non-linear finite element relations of Euler-Bernoulli beams made of functionally graded materials. The two techniques, which divide a cracked element into three parts, are implemented to enrich the secant and tangent stiffness matrices. The Lagrange multiplier technique is originally proposed according to the establishment of a modified total potential energy equation by adding continuity conditions equations of the crack point. The limitation of the conversion matrix in involving the relevant non-linear equations is the main motivation in representing the Lagrange multiplier. The presented Lagrange multiplier is a problem-solving technique in the cracked structures, where both geometrical nonlinearity and material inhomogeneity areas are considered in the analysis like the post-buckling problem of cracked functionally graded material columns. Accordingly, some case-studies regarding the post-buckling analysis of cracked functionally graded material columns under mechanical and thermal loads are used to evaluate the results.

The Graphic Derivation Process of Lagrange Multiplier Method

“…Therefore, there are usually three methods to solve conditional extremum (Reichel, Schrder, & Xu, 2023;Zhao & Wang, 2022;Ambartsumyan, 2018;Chen, Xu, Wang, Huang, & Chen, 2022;Salmalian, Alijani, & Azarboni, 2020):…”