ON A SIMPLIFIED STRAIN-ENERGY FUNCTION FOR GEOMETRICALLY NONLINEAR BEHAVIOR OF ANISOTROPIC BEAMS

An asymptotically exact methodology, based on geometrically nonlinear, three-dimensional elasticity, is presented for analysis of prismatic, nonhomogeneous, anisotropic beams. The analysis is subject only to the restrictions that the strain is small relative to unity and that the maximum dimension of the cross-section is small relative to a length parameter which is characteristic of the rapidity with which the deformation varies along the beam; thus, restrained warping effects are not considered. A two-dimensional function is derived which enables the determination of sectional elastic constants, as well as relations between the beam (i.e. one-dimensional) displacement and generalized strain measures and the three-dimensional displacement and strain fields. Since the three-dimensional foundation of the formulation allows for all possible deformations, the complex coupling phenomena associated with shear deformation are correctly accounted for. The final form of the strain energy contains only extensional, bending and torsional deformation measures-identical to the form of classical theory, but with stiffness constants that are numerically quite different from those of a purely classical theory. Indeed, the stiffnesses obtained from classical theory may, in certain extreme cases, be more than twice as stiff in bending as they should be. Stiffness constants which arise from these various models are used to predict beam deformation for different types of composite beams. Predictions from the present reduced stiffness model are essentially identical to those of more sophisticated models and agree very well with experimental data for large deformation.