James M. Gere

Computer Programs for Framed Structures

This chapter contains flow charts of computer programs for analyzing the six basic types of framed structures by the direct stiffness method. The flow charts are sufficiently detailed so that persons who are familiar with the elements of computer programming could write their own programs for structural analysis if desired.

Basic Concepts of Structural Analysis

This book describes matrix methods for the analysis of framed structures with the aid of a digital computer. Both the flexibility and stiffness methods of structural analysis are covered, but emphasis is placed upon the latter because it is more suitable for computer programming. While these methods are applicable to discretized structures of all types, only framed structures will be discussed. After mastering the analysis of framed structures, the reader will be prepared to study the finite element method for analyzing more elaborate discretized continua (see Chapter 7 and textbooks on finite elements listed in General References).

Fundamentals of the Stiffness Method

The stiffness method (also known as the displacement method) is the primary method used in matrix analysis of structures. One of its advantages over the flexibility method is that it is conducive to computer programming. Once the analytical model of a structure has been defined, no further engineering decisions are required in the stiffness method in order to carry out the analysis. In this respect it differs from the flexibility method, although the two approaches have similar mathematical forms. In the flexibility method the unknown quantities are redundant actions that must be arbitrarily chosen; but in the stiffness method the unknowns are the joint displacements in the structure, which are automatically specified. Thus, in the stiffness method the number of unknowns to be calculated is the same as the degree of kinematic indeterminacy of the structure.

Additional Topics for the Stiffness Method

The computer programs given in the preceding chapter apply to the analysis of linearly elastic framed structures of arbitrary configuration but consisting of prismatic members of only one material. These structures are assumed to undergo small displacements when subjected to applied loads, and the principle of superposition is used throughout the analyses. Only actions applied at joints and fixed-end actions due to loads on members are acceptable as input load data for the programs.

Finite-Element Method for Framed Structures

Matrix analysis of framed structures may be considered as a subset of the more general method of finite elements [1–4]. Any continuum can be partitioned into subregions called finite elements. These subregions are of finite size and usually have simpler geometries than the boundaries of the original continuum. Such a partitioning serves to convert a problem involving an infinite number of degrees of freedom to one with a finite number in order to simplify the solution process. Applications in solid mechanics consist of framed structures, two-and three-dimensional solids, plates, shells, and so on. One-, two-, and three-dimensional finite elements may be required for such analyses. However, the members of framed structures are relatively long compared to their cross-sectional dimensions, so only one-dimensional finite elements are needed to model them.

Computer-Oriented Direct Stiffness Method

In the preceding chapter the stiffness method was developed initially by superposition of actions for the free displacement coordinates (Sec. 3.2). Then the method was formalized and extended in Sec. 3.6 using the compatibility matrix CMJ and the virtual work concept. With this second approach the complete joint stiffness matrix S, (for both free and restrained displacements) was assembled by the triple matrix multiplication given as Eq. (3–32). While the formal version is enlightening and well organized, it involves a large and sparse compatibility matrix containing terms that are not easy to evaluate correctly. Neither the generation of this matrix nor the multiplication process for assembly would be suitable for computer programming. A better methodology consists of drawing ideas from both approaches and adding a few computer-oriented techniques to evolve what is known as the direct stiffness method.

Fundamentals of the Flexibility Method

Basic concepts of the flexibility method (also called the force method) are described in this chapter. The flexibility method is a generalization of the Maxwell-Mohr method developed by J. C. Maxwell in 1864 and O. C. Mohr in 1874. In this approach statically indeterminate structures are analyzed by writing compatibility equations in terms of flexibility coefficients and selected redundants. Also involved in such equations are displacements calculated for a statically determinate version of the original structure (with the redundants released). Because the flexibility method requires extensive use of calculated displacements, the material in Appendix A will be referenced frequently.