Ph.D Thesis
Ph.D Student Gal Erez Triangular Shell Element for Geometrically Nonlinear Structural Analysis Department of Civil and Environmental Engineering Professor Robert Levy

Abstract

Research on nonlinear analysis of shell structures is one of the most challenging topics in structural engineering. With the finite element method as a very effective tool, the development of shell finite elements for geometrically nonlinear analysis becomes a major research field. Here a flat triangular finite element for geometrically nonlinear analysis of thin shell structures is derived. The linear shell element is composed of the constant strain triangle (CST) flat triangular membrane element and of the discrete Kirchhoff theory (DKT) flat triangular plate element. In order to upgrade this linear shell element to perform geometrically nonlinear analysis, the nonlinear effects are introduced by load perturbing the global linear equilibrium equations in their discrete form, a method successfully applied elsewhere to trusses, plane frames, space frames and membranes shells. The analysis is performed by applying the load in increments. For each load increment, equilibrium is iteratively achieved using the full Newton Raphson method. Another aspect of paramount importance in nonlinear analysis is the isolation of the pure elastic deformations (rotations and translations) by removing rigid body rotations and the retrieval of internal stresses at each iteration. Pure membrane strains are easily attained by comparing element initial and final geometries whereas a special procedure that uses finite rotations and the pseudo rotation vector of Rodriguez attains the pure bending rotations. Stresses are retrieved using linear constitutive relations. The reliability of the proposed shell finite element was verified by benchmark problems from the literatures. The analysis results showed excellent matching and traced the equilibrium path to the post buckling region for snap through problems. Overall performance was very good as far as numerical stability, convergence, meshing and number of load steps. The approach of using discrete equilibrium equations of the linear shell finite element for deriving the geometrical nonlinear effects was proven correct. The two main contributions of this research have yielded a simple and effective procedure for solving complex nonlinear shell problems. These contributions are the first order complete geometric stiffness matrix for shells and the unique removal of rigid body rotations.