Finite deformations of curved laminated St. Venant–Kirchhoff beam using layer-wise third order shear and normal deformable beam theory (TSNDT)

A layer-wise third order shear and normal deformable plate/shell theory (TSNDT) incorporating all geometric nonlinearities is used to study finite transient deformations of a curved laminated beam composed of a St. Venant–Kirchhoff material. In the TSNDT all displacement components of a point are expressed as 3rd order polynomials in the thickness coordinate in each layer while maintaining the displacement continuity across adjoining layers. No shear correction factor is used. Transverse shear and transverse normal stresses are found from the computed displacement fields and the constitutive relation (i.e., no stress recovery technique is employed). For the St. Venant–Kirchhoff material the strain energy density is a quadratic function of the Green-St. Venant strain tensor appropriate for finite deformations. The software based on the finite element method (FEM) capable of solving static and transient nonlinear problems has been verified by using the method of manufactured solutions. Furthermore, results computed with the TSNDT have been found to agree well with those obtained using the commercial software ABAQUS, and C3D20 elements. Significant contributions of the work include developing a TSNDT considering all geometric nonlinearities and a materially objective constitutive relation, using the method of manufactured solutions to verify the numerical solution of transient nonlinear problems, and showing that results from the plate theory agree well with those from the analysis of plane strain nonlinear problems using the finite elasticity theory. Plate problems using the TSNDT can be analyzed with piecewise linear basis functions in the FEM.