An Introduction to Matrix Structural Analysis and Finite Element Methods

The following sections are included:

• Preface
• Acknowledgments
• Contents

Matrix Structural Analysis (MSA) and Finite Element Methods (FEM) are numerical analysis techniques which rely on the reduction of complex physical problems into sets of linear equations solved using computer algorithms. Prior to the 20^th century, numerical analysis could only be executed by human or analog computers, which were slow, expensive, and prone to error. The advent of digital computers in the 1940s offered the possibility of accurately performing calculations that could be reduced to a set of simple tasks. Established matrix techniques, which could be reduced to entry-by-entry manipulation, proved to be particularly well-suited for computer execution. The emergence of the Direct Stiffness Method (DSM), the predecessor of MSA and FEM, in the early 1950s facilitated the use of matrix techniques for the analysis of structural systems…

In Chapter 1, we looked at an example structural problem for which we were able to find the support reactions and element forces using only equilibrium conditions. Because an equilibrium-based analysis can only be used to solve determinate systems, its scope of application is significantly limited. In order to analyze indeterminate structures, it is necessary to incorporate kinematic and constitutive relationships…

In Chapter 2, we used constitutive, kinematic, and equilibrium relationships to derive the matrix stiffness formulation for a truss element. We also demonstrated how to solve a simple structural problem using a stiffness-based approach. While the element stiffness formulation was general and robust, the method for analyzing the example structure was explicit and inflexible. In this chapter, we will formulate a systematic approach for the matrix analysis of structures; we will implement this method as code in Chapter 4. To demonstrate this procedure, we will use a slightly more complex example…

In this chapter, we present computer code for the general matrix analysis method developed in Chapter 3. Although computers are able to perform many simple operations very quickly, they lack the human intuition to discern which calculations are necessary for a given process. Hence, we need to specify the scope and duration of every iterative procedure using numbers and indices. Our general method already accounts for some level of iteration (i.e., iterating through elements to assemble the global stiffness matrix), but we must now ensure that every iterative component of our analysis has a corresponding parameter…

We have used the truss element to demonstrate both how to derive a typical element stiffness equation and how to set up a general structural analysis. In this chapter, we introduce two new elements: the beam element, which resolves loads exclusively through bending, and the frame element, which combines the mechanical characteristics of the beam and truss elements. In order to derive the beam element stiffness matrix, we first need to establish the constitutive, kinematic, and equilibrium relationships for bending. Once we have formulated the beam stiffness matrix, we introduce the axial dofs to generate a frame element. We then demonstrate how to rotate the beam element into two spatial dimensions and the frame element into both two and three spatial dimensions.

In addition to the general stiffness formulations for the two elements, we develop a hinge element, which is able transmit forces but not moments. We provide the code modifications necessary to integrate these new elements into our code and demonstrate their behavior with an example analysis. We conclude the chapter with two supplemental procedures: first, we demonstrate how to account for loads between nodes by using Equivalent Nodal Loads (ENLs), and second, we demonstrate how to reduce the size of the active stiffness matrix using symmetry considerations.

This chapter marks the beginning of our venture into the territory of Finite Element Methods (FEM). While FEM shares the same coding framework as MSA, it follows a different approach to the derivation of elements. In previous chapters, we were able to generate truss, beam, and frame elements using established mechanical principles. Although FEM still relies on classical physical principles, it also requires the use of approximation techniques to transform continuum problems into discretized matrix systems. Since numerical approximation is outside typical academic curricula, we devote this chapter to a general introduction to numerical approximation of boundary value problems…

We begin our investigation of FEM with steady-state heat conduction, a common engineering problem, which allows us to demonstrate the fundamentals of FEM without the complexities inherent to linear elasticity. Whereas the primary unknown in elasticity is the 3D vector field of displacement, heat conduction is concerned with the scalar field of temperature. Heat conduction problems are limited to one temperature dof per node, thus greatly reducing the complexity of the mathematics involved in our stiffness derivations.

In this chapter, we begin by presenting the strong form of the heat conduction BVP. Next, we use the Variational Principle to derive the basic formulation of the stiffness equation, which we then use to formulate rod, triangle, and quad stiffness matrices for 1D and 2D heat conduction problems. To aid in these derivations, we introduce several new concepts including shape functions, mapping, and numerical integration. Following the theoretical derivations, we present code for the element function. We conclude with an example analysis.

Having established a basic framework for the implementation of FEM via steadystate heat conduction, we move on to the more advanced problem of static linear elasticity. This increase in complexity is a consequence of the primary unknown taking the form of a displacement vector field, rather than a scalar temperature distribution. Linear elasticity is governed by the same three relationships (equilibrium, constitutive, and kinematic) that we used in the derivation of the truss and beam elements but articulated using three-dimensional directional derivatives.

While we will present the overarching concepts in three dimensions, we will limit new element derivations to the two-dimensional elastic triangle and quad elements. As a pedagogical aid, we also derive the 1D truss element using FEM to demonstrate equivalence between MSA and FEM. Upon establishing these elements, we also demonstrate how to recover strains and stresses. Since strains and stresses are derivatives of a primary function like heat fluxes, they will be discontinuous across element boundaries. Hence, we develop a technique for generating continuous strain, stress, or flux profiles using stress smoothing. Next, we introduce the necessary code modifications for implementing the new elements, extracting strains and stresses, and stress smoothing. We conclude with an illustrative analysis of a deep cantilever.

In MSA, we investigated truss, beam, and frame elements, which resolve loads axially, through bending, and through hybrid axial/bending action, respectively…

The following sections are included:

• Appendix A: Code Supplement
• Appendix B: Nomenclature
• References and Further Reading
• Index