NORMAN F. KNIGHT, JR.
1.1. INTRODUCTION
The design of structures for automotive, naval, aircraft, and space applications exploits the finite element modeling and analysis techniques to ensure reliability and robustness in a given design. More and more often, the use of nonlinear structural analyses are required in order to simulate the complex physical response to extreme loading conditions and to utilize the structural design and material to its fullest extent. Ideally structural collapse should not occur under normal operating conditions; however, an understanding of the collapse phenomena can lead to increased structural integrity and confidence in residual strength predictions.
The finite element method is an approximation method for solving differential equations of mathematical physics. Starting from the differential equations and boundary conditions, a weak form of the problem is developed in a variational sense. This variational statement is used to define elemental properties that may be written as matrices and vectors as well as to identify primary and secondary variables and all possible boundary conditions. Specific equilibrium problems are solved by first discretizing the spatial domain of the problem, evaluating the elemental matrices and vectors, assembling elemental terms to form the global system of algebraic equations, applying boundary conditions, and solving the resulting system of algebraic equations. This system of equations is often nonlinear. In solid mechanics, the types of nonlinearity include geometric nonlinearity (typically from the strain-displacement relations), material nonlinearity (typically from the stress-strain relations), and boundary condition nonlinearity (typically from contact or friction).
Nonlinear finite element analyses are readily performed using any one of several commercially available finite element software systems such as MSC/NASTRAN, HKS/ABAQUS, or ANSYS. Today’s analyst can easily model the spatial geometry of large complex systems and generate finite element models which easily exceed one million active degrees of freedom. Coupling these two facts with the availability of high-performance computing systems provides analysts with simulation capabilities that far exceed the capabilities available less than a decade ago. Several aspects of such computations are described by Hibbitt (1986, 1993).
Common to all of the nonlinear solution techniques is the need to trace out the entire load-displacement response curve. As such, the computational cost of including such a nonlinear analysis approach within a design optimization loop will be high. This feature provides the impetus for the development of robust and efficient nonlinear analysis methods for designing structures. If the structure is designed to exploit its postbuckling stiffness, then effective design methodologies need to be developed and incorporated into the structural optimization procedure. Within the design optimization sequence, the values of the design parameters may vary, and in composite structural design, the number of possible design parameters increases dramatically (e.g., Stroud, 1982). Each change in a design variable typically requires at least one additional nonlinear analysis if gradient-based optimization methods are used. The use of traditional nonlinear analysis methods within such a design optimization sequence is unattractive, or more precisely uneconomical, from a computational standpoint and also from the designer’s standpoint in that only limited design configurations can be assessed in a given amount of computational time.
The objectives of this paper are to present a brief overview of the finite element method, to review selected finite element analysis techniques as applied to determining the nonlinear postbuckling and collapse response of elastic structures, and to describe selected application problems. First, some of the fundamental concepts associated with finite element approximations are described. Second, the basic equations of nonlinear solid mechanics are given for the case of small strain, large elastic deformations including a brief discussion of lamination theory for composite structures. Third, different variational formulations are presented along with their finite element models. Then solution techniques for nonlinear systems (both static and dynamic) are discussed. The traditional approach of nonlinear finite element analysis using a Newton-Raphson approach is reviewed. Reduction methods for nonlinear problems such as the global function approach and...
