Engineering Transactions, 11, 4, pp. 643-666, 1963

### Pewna Metoda tzw. Parametrycznego Kształtowania Wytrzymałościowego

W. Krzyś
Politechnika Krakowska
Poland

M. Życzkowski
Politechnika Krakowska
Poland

The problem of proper optimum strength design is that of finding of one or several functions determining the form of a structural element and constitutes, in principle, a variational problem. However, for computation reasons or for simpler production methods, the finding of the function is often replaced by that of a few parameters, the types of functions being assumed beforehand. Such an approach is called «parametric optimum strength design». The variational problem is replaced here by an analysis of a function of several variables.
The aim of the present paper is to devise a certain method of parametric optimum design called the «method of transformation linearizing the auxiliary conditions». The problem consists in finding the lower bound of certain function V (2.1) (the weight of the element, for instance)
depending on n parameters ds with m auxiliary conditions expressed in the form of weak iner-qualities (2.2). Thus, we seek for the lower bound of a function on a certain hypersurface in the space of parameters aj or on one side of this hypersurface. The method consists in introducing new parameters x; (2.4), expressing ag in terms of x* (all or some of them) and examining Vin function of the parameters xs. Then, the auxiliary conditions have the form (2.8) that is a linear form. Particular cases of introduction of the parameters x; are discussed, depending above all
on whether m is greater or less than 7. The method is illustrated by means of two examples cf which detailed solution is given. The first example concerns the design of a box profile with constant wall thickness, subject to pure torsion. With no assumption on the fourfold symmetry we have four parameters (wall thickness gd and go height and width of the cross-section, Fig. 1). Four conditions are introduced. Two of them bound the stresses and the other two concern the stability of the walls. The parameters xt are introduced by means of the equations (3.6). In the elastic range the optimum solution is determined by the equations (3.14), (3.15), and (3.16) and corresponds to the case where all the parameters Xi are equal to unity. In addition the problem of elastic-plastic buckling is solved on the basis of the equations of Johnson - Ostenfeld and A. Ylinen. The second case concerns the design of the same profile but subject to pure bending. In addition to the condition of bounded stress and stability of the walls, we have a condition of security against lateral buckling of the bar as a whole. The result of the analysis which is performed for the elastic range only is illustrated by Fig. 2, giving the optimum side ratio of the sought – for profile.

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