On simple and accurate finite element models for nonlinear bending analysis of beams and plates
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This study is concerned with the development of simple and accurate alternative finite element models to displacement finite element models for geometrically nonlinear bending analysis of beams and plates. First, a unified corotational beam finite element that incorporates the kinematics of classical as well as refined beam theories, including the Timoshenko and Reddy beam theories, is developed in a single finite element. The governing equations are written in a "corotational" local frame that rotates with the element and with respect to which the standard linear engineering relations between strains and internal forces are valid. The element is based on Lagrange interpolation of the axial displacement, Hermite cubic interpolation of the transverse displacement, and related quadratic interpolation of the rotation, and it does not experience shear locking. The model is verified by comparisons with exact and/or approximate solutions available in the literature. Very good agreement is found in all cases. Next, a finite element model is developed using a mixed formulation of the first-order shear deformation theory of laminated composite plates. A p-type Lagrangian basis is used to approximate the nodal degrees of freedom that consist of three displacements, two rotations, and three moment resultants. The geometric nonlinearity, in the sense of the von KÃÂµarman, is included in the plate theory. The mixed plate element developed herein is employed in the linear and nonlinear bending analysis of a variety of layered composite rectangular plates. The effects of transverse shear deformation, material anisotropy, and bending-stretching coupling on deflections and stresses are investigated. The predictive capability of the present model is demonstrated by comparison with analytical, experimental, and numerical solutions available in the literature. The model provides an accurate prediction of the global bending response of thin and moderately thick plates subjected to moderate and moderately large rotations. The inclusion of the bending moments at the nodes results in increased accuracy in the computation of stresses over those determined by conventional displacement-based finite element models. The many results presented here for geometrically nonlinear bending analysis of beams and plates should serve as reference for future investigations.
Urthaler Lapeira, Yetzirah Yksya (2003). On simple and accurate finite element models for nonlinear bending analysis of beams and plates. Doctoral dissertation, Texas A&M University. Texas A&M University. Available electronically from