Optimal point support of shells and plates

Abstract

Optimization by means of appropriate placement of the supports of a structure is the least studied of all the different trends in the optimal design of structures. The reasons why it has not been extensively studied may be explained by the fact that in a real design process, it is possible to control the placement of supports only on comparatively rare occasions. Meanwhile, the effectiveness of such an approach exceeds only slightly that of other approaches. Problems involving point supports occupy a special place among these types of problems. In [1], attempts to create optimal support of a square plate resting on three points and subjected to a uniform transverse load were described. Shells differ from plates by having relatively lesser thicknesses and greater load-carrying capacity. If only a few supports are present, there is a risk of a local failure of the shell at a point of support. Among the different methods used to prevent such failure, we may cite reinforcement in the form of thickening or additional sheathing, attachment of one or more edges in different directions, local variation of the curvature, or fracturing of the median surface created by uneven juncture of two or more comparmaents of the shells. Here we wish to discuss the last of these methods. It may be anticipated that in the case of point supports situated within the horizontal projection at points at which the median surface experiences fracturing (i.e., at places where the curvature is discontinuous), it will not be necessary to incorporate support structures in the form of arched projecting walls with tie beams or trusses, a step which is entirely necessary when a shell is supported at its four corners. To estimate the load-carrying capacity of composite shells with point support, we will be applying the kinematic method of the theory of limit equilibrium [2, 3]. By means of the technique of [2], it is possible to design a shell using an ideal rigidplate material with inhomogeneous properties, determining the effects of extension and compression, and taking into account anisotropy. Let us consider a nonflat roof formed from translation surfaces which, in horizontal projection, are square in shape (Fig. 1). Its central portion is produced by the transit of a convex parabola along a convex parabola (i.e., an elliptic paraboloid). A corner section constitutes an inversion of such a surface. Lateral sections are obtained by the transit of a convex parabola along a concave parabola. The ordinates of the generating surface at characteristic points are indicated in Fig. 1. A transverse load is distributed uniformly over the entire surface. The roof is supported at four points, namely at the places where the central, corner, and side sections are linked together. If we assume that the rigid plate material of the shell obeys the generalized Johansen yield condition, the upper bound Q of the total limit load may be represented by the discrete analog of the functional of [2] thus: