William Weaver

Inelastic dynamic analysis of tier buildings

Abstract A comprehensive procedure is developed for the inelastic analysis of three-dimensional tier buildings. Incremental member stiffness matrices are derived for the beams and columns of such structures, using Ramberg-Osgood type of functions to analytically model the inelastic action-deformation relationships. These stiffnesses are incorporated into computer programs that analyze two- and three-dimensional frames for static and dynamic loads. The possibilities of reversal of moment at one or both ends of a member and reversal of the axial force are taken into account, and interactions of the bending moments, axial forces, and torques in the columns are included for wide-flange sections. Calculated results for sample problems are compared against both analytical and experimental data, and the dynamic analysis of a three-story tier building is illustrated.

The eigenvalue problem for banded matrices

Abstract The generalized Lanczos method is used for the purpose of calculating frequencies and mode shapes for linearly elastic discretized structures where the energy-consistent stiffness and mass matrices are equally banded. This approach involves reduction of the problem to standard tridiagonal form without expanding the band width of either of the original arrays. Applications of the method to both vibrational and buckling analyses indicate its potential for conserving core storage when solving the eigenvalue problem on a digital computer.

Tier buildings with shear cores, bracing, and setbacks

Abstract The theory of bending and torsion in thin-walled members of open cross section is applied to shear cores included within the analytical model of the tier building. Representative types of bracing arrangements are also considered and their stiffnesses augment those of the beams, columns and shear walls. Changes in floor plan, referred to as “setbacks”, further complicate the analysis; but a relatively simple routine allows automatic handling of this characteristic. Computer programs named STATIER and DYNATIER include all of these features within the linear static and dynamic analyses of three-dimensional multi-story buildings. Results for a twenty-story structure demonstrate the significance of the interaction between a shear core and the skeletal framing. The effect of warping restraints upon the stresses in the shear core is large at the base but diminishes rapidly with height. The normal stress at the base associated with the bimoment is of the same order of magnitude as that due to the combination of biaxial bending and axial force.

Automated design of tier buildings

Abstract A computer program for the automated design of frameworks in steel tier buildings is described and demonstrated. The design process commences with the input of any initial set of member sies for a given geometry: it is not necessary to begin with good estimates. Cyclic analyses of the building frame and revisions of member sizes then follow. The three-dimensional analysis of the building frame includes the effects of the finite dimensions of moment-resisting connections. Within each cycle of design, wide-flange members are selected on the basis of satisfying the dead- and live-load stress requirements of the 1969 AISC Specification for Structural Steel in Buildings. Each member is then checked for the most critical combination of vertical and lateral loading. The lightest satisfactory size within a prespecified range of sizes is ultimately selected. The output consists of beam and column schedules that show the final sizes and the design conditions leading to their selection. Sample three-story and twenty-story building designs show consistent and rapid convergence to a final design without human intervention between consecutive cycles. With highrise buildings becoming increasingly prevalent, an economical automated design procedure should be of great value to the practicing structural engineer.

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.