Matrix analysis of rigid space frames

Abstract

The aim of a structural designer, whether dealing with buildings, bridges, airplanes, or some other type of structure, is to design the most economical structure that will do the job required of it with safety.

In the latter part of the nineteenth century and the early part of the twentieth century, many designers went to great lengths to make their structures statically determinate and often inserted pins and.hinges in the structure in order to achieve this end. With the advent of reinforced concrete and welded steel structures, however, it became evident that continuous, hence statically indeterminate structures were easier and cheaper to build and methods for designing them were devised.

One of the methods of analysis used extensively since its introduction to this country in 1915 by George A. Maney is the slope-deflection method, in which the rotations and translations of the joints are redundants and must be determined before the bending moments can be calculated. This method, as well as the other classical methods, involves the solving of simultaneous linear equations18 - a procedure which can become very tedious in a structure of any size.

Another method, perhaps typical of newer methods of analysis, which has won wide acceptance and a great deal of popularity in its relatively short existence is Hardy Cross's moment distribution. The use of this method gives the analyst a physical picture of joint movements and at the same time saves a great deal of time in the analysis of many structures.

The usual procedure when analysing a building frame or other three-dimensional structure is to consider it as a series of planar structures and ignore the effects of torsion in the members. The use of an appropriate factor of safety, or factor of ignorance, as many insist, makes this an acceptably safe procedure, but can also cause quite a waste of materials. The effect of torsion upon concrete has been found worthy of notice and several papers have been written on analysis of three-dimensional rigid frames, but as yet nothing has been developed that would interest very many practicing design engineers.

If rigid frames are to be analyzed in their true three-dimensional form, a convenient method of analysis must be devised. The classical methods of analysis are much too tedious, and even the most elementary type of three-dimensional frame analyzed using moment distribution is rather a tremendous undertaking, in spite of the saving of time gained by the use of this method.