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Primal- and dual-mixed finite element models for geometrically nonlinear shear-deformable beams – a comparative study

Bertóti Edgár

Computer Assisted Methods in Engineering and Science

2020

Vol. 27, no. 4

285--215

The relationships between the system matrices of the displacement-based, a primal-mixed, a dual-mixed and a consistent primal-dual mixed finite element model for geometrically nonlinear shear-deformable beams are investigated. Employing Galerkin-type weak formulations with the lowest possible order, constant and linear, polynomial approximations, the tangent stiffness matrices and the load vectors of the elements are derived and compared to each other in their explicit forms. The main difference between the standard and the dual-mixed element can be characterized by a geometry-, material- and meshdependent constant that can serve not only as a locking indicator but also to transform the displacement-based element into a shear-locking-free dual-mixed beam element. The numerical performances of the four different elements are compared to each other through two simple model problems. The superior performance of the mixed, and especially the dual-mixed, beam elements in the nonlinear case is demonstrated, not only for the deflection, but also for the force and moment computations.

Local and Global Instability of the Slender Geometrically Nonlinear SystemWith Non-Prismatic Element Subjected to the Euler’s Load

Szmidla J., Skrzypczak T., Jurczyńska A.

Machine Dynamics Research

2016

Vol. 40, No. 4

79--88

A theoretical considerations and numerical calculations concerning the issue of the stability of the geometrically nonlinear system with non-prismatic element are presented in this work. The analysed columns were subjected to the Euler’s load. On the basis of the minimum potential energy principle as well as the small parameter method, the differential equations of displacements were formulated and its solutions were obtained. The assumption that the approximation of the non-prismatic rod satisfies the condition of constant total volume resulting from the value of the coefficient of flexural stiffness distribution has been made. The results of the carried out numerical simulations refer to the local and global stability loss. It has been proved that taking into consideration in the geometrically nonlinear system appropriate shaped rod of variable cross-section causes an increase in the value of bifurcation load and in a consequence an „exit” from the area of the local instability (loss of rectilinear form of static equilibrium).