| Summary: | 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 Kõarman, 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.
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