An accurate nonlinear 3d Timoshenko beam element based on Hu-Washizu functional

An accurate 3d mixed beam element that is efficient especially in nonlinear analysis is presented in this paper. The mathematical theory is based on Hu-Washizu principle that uses three-fields in the variational form. The composition of the variational form ensures independent selection of displacement, stress and strain fields. Timoshenko beam theory is extended to three dimensions for deriving strains from displacement field. Numerical integration of stress strain relations along control sections is carried out for numerical analysis. The finite element approximation for the beam uses shape functions for section forces that satisfy equilibrium and discontinuous section deformations along the control sections of the beam. This form of the element permits coupling of the stress resultants and eliminates necessity of displacement components along the beam element except at the nodes. Consequently the element is free from shear-locking. Numerical examples on uniform and tapered structural members with solid and hollow circular sections demonstrate the nonlinear interaction between axial, shear force, bending moments and torsion, where the results compare well with closed form solutions and available data in literature. Furthermore, the proposed element has superior performance in both linear and nonlinear analysis compared to a locking-free higher order displacement based 3d Timoshenko beam element

An accurate nonlinear 3d Timoshenko beam element based on Hu-Washizu functional

An accurate 3d mixed beam element that is efficient especially in nonlinear analysis is presented in this paper. The mathematical theory is based on Hu-Washizu principle that uses three-fields in the variational form. The composition of the variational form ensures independent selection of displacement, stress and strain fields. Timoshenko beam theory is extended to three dimensions for deriving strains from displacement field. Numerical integration of stress strain relations along control sections is carried out for numerical analysis. The finite element approximation for the beam uses shape functions for section forces that satisfy equilibrium and discontinuous section deformations along the control sections of the beam. This form of the element permits coupling of the stress resultants and eliminates necessity of displacement components along the beam element except at the nodes. Consequently the element is free from shear-locking. Numerical examples on uniform and tapered structural members with solid and hollow circular sections demonstrate the nonlinear interaction between axial, shear force, bending moments and torsion, where the results compare well with closed form solutions and available data in literature. Furthermore, the proposed element has superior performance in both linear and nonlinear analysis compared to a locking-free higher order displacement based 3d Timoshenko beam element

Variational base and solution strategies for non-linear force-based beam finite elements

This paper presents the variational bases for the non-linear force-based beam elements. The element state determination of these elements is obtained exactly from a two-field functional with independent stress and strain fields. The variational base of the non-linear force-based beam elements implemented in a general purpose displacement-based finite element program requires the inclusion of independent displacement field in the formulation. For this purpose, a three-field functional is considered with independent displacement, stress, and strain fields. Various local and global solution strategies come out from the mixed formulation of the beam element, and these are shown to yield the algorithms presented for non-linear force formulation beam elements in literature; thus removing any doubts on their variational bases. The presented numerical examples demonstrate the accuracy and robustness of the solution algorithms adapted for mixed formulation elements over popularly used displacement-based beam finite elements even for large structural systems

Free vibration characteristics of a 3d mixed formulation beam element with force-based consistent mass matrix

In this analytical study, free vibration analyses of a 3d mixed formulation beam element are performed by adopting force-based consistent mass matrix that incorporates shear and rotary inertia effects. The force-based approach takes into account the actual distribution of mass of an element in the derivation of the mass matrix. Moreover, the force-based approach enables accurate determination of free vibration frequencies of members with varying geometry and material distribution without any need for specification of different displacement shape functions for each individual case. This phenomenon is justified by comparing free vibration frequencies of cantilever beams that have circular and rectangular cross-sections and various mass distribution configurations. Vibration frequencies of the mixed formulation element are compared with the frequencies obtained from closed-form solutions and finite element analyses. Fundamental frequency is computed with only one element per member span and higher order frequencies are determined with two or four elements with considerable accuracy by employing 3d mixed element and force-based consistent mass matrix

Hybrid finite element for analysis of functionally graded beams

A hybrid finite element model is presented, where stiffness and mass distributions over a beam with functionally graded material (FGM) are accurately modeled for both elastic and inelastic material responses. Von Mises and Drucker-Prager plasticity models are implemented for metallic and ceramic parts of FGM, respectively. Three-dimensional stress-strain relations are solved by a general closest point projection algorithm, and then condensed to the dimensions of the beam element. Numerical examples and verification studies on a proposed element demonstrate accuracy and robustness under inelastic material response as well as capturing fundamental, higher, and mix modes of vibration frequencies and shapes