A Geometrically nonlinear curved beam theory and its finite element formulation

Abstract

This thesis presents a geometrically exact curved beam theory, with the assumption that the cross-section remains rigid, and its finite element formulation/implementation. The theory provides a theoretical view and an exact and efficient means to handle a large range of nonlinear beam problems. A geometrically exact curved/twisted beam theory, which assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal reference frames starting from a 3-D beam theory. The relevant engineering strain measures at any material point on the current beam cross-section with an initial curvature correction term, which are conjugate to the first Piola-Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The Green strains and Eulerian strains are explicitly represented in terms of the engineering strain measures while other stresses, such as the Cauchy stresses and second Piola-Kirchhoff stresses, are explicitly represented in terms of the first Piola-Kirchhoff stresses and engineering strains. The stress resultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimensional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are considered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the initial curvature correction term. For the finite element formulation and implementation of the curved beam theory, some basic concepts associated with finite rotations and their parametrizations are first summarized. In terms of a generalized vector-like parametrization of finite rotations under spatial descriptions (i.e., in spatial forms), a unified formulation is given for the virtual work equations that leads to the load residual and tangent stiffness operators. With a proper explanation, the case of the non-vectorial parametrization can be recovered if the incremental rotation is parametrized using the incremental rotation vector. As an example for static problems, taking advantage of the simplicity in formulation and clear classical meanings of rotations and moments, the non-vectorial parametrization is applied to implement a four-noded 3-D curved beam element, in which the compound rotation is represented by the unit quaternion and the incremental rotation is parametrized using the incremental rotation vector. Conventional Lagrangian interpolation functions are adopted to approximate both the reference curve and incremental rotation of the deformed beam. Reduced integration is used to overcome locking problems. The finite element equations are developed for static structural analyses, including deformations, stress resultants/couples, and linearized/nonlinear bifurcation buckling, as well as post-buckling analyses of arches subjected to conservative and non-conservative loads. Several examples are used to test the formulation and the Fortran implementation of the element.