The collapse load of pin-jointed networks of randomly imperfect rods

Abstract

Twoand three-dimensional trusses are among the most frequently used structures in building practice. One has only to consider the incidence of braced structures in residential and industrial buildings, of pylons for transmission lines and cableways, and also, of structures such as single or multilayered grids and vaults with single or double curvature, that are used to cover large areas. One of the advantages of these structures is that they readily lend themselves to modular construction. The large number of possible different structures, and the repetition of elements within a particular structure, a number of which may be built, require the production of many such modules. This allows the consideration of the mass-production of modules with the associated advantages of improved quality control and economy of materials and labour. Thus, with a sufficiently high number of samples, the geometrical and mechanical characteristics of the units may be statistically determined. In particular, the statistical distribution of geometrical imperfections such as the departure from straightness of individual members, may be determined. Such imperfections can produce a dramatic reduction in the strength of the structure. Previously, the influence on the load capacity of trusses of geometrical imperfections in members has been investi. gated by attributing deterministic values of imperfection to each member. However, although maintaining constant the mean value and the standard deviation of member imperfections over the whole structure, a range of collapse loads is obtained, z's This demonstrates that a deterministic treatment of the problem of member imperfections is unrealistic. As geometrical imperfection is a typical random variable, it only seems natural to suggest a probabilistic formulation to evaluate its effect. The problem may be stated thus: given the statistical parameters of the imperfections in the members of a structure: (i), evaluate the probability of collapse under a particular load system; (ii), evaluate the expected value and the standard deviation of the collapse load of the structure. With the word 'buckling' in place of 'collapse' this becomes the central problem of the theory of elastic stability of imperfections-sensitive structures. This problem has been investigated by many authors and a comprehensive treatment together with a list of references is given elsewhere. More recently it has also been studied by Hansen and Roorda a and Roorda. 6 In this paper truss structures of elastic-perfectly plastic material will be considered. The deterministic treatment of the problem of finding the collapse load for perfect structures also allowing for member's post-buckling behaviour is given in (4) while in (5) the same treatment is extended to structures with geometrically imperfect members and in (1) some numerical results are presented. The problem stated above for the most general truss structures proves to be particularly complex. This paper, which is a first approach to the problem, provides a solution for the class of statically determinate structures.