The choice of optimal form for a shell of revolution is made within the class of shells of the positive Gaussian curvature. The optimization is considered to be the condition of maximum C-ratio of the ultimate load P to material consumption index G. The P is determined according to the theory of limit analysis [1]. The equation of a shell meridian is given in the form of a polynomial and it is a function of many variables. Given certain restrictions and assuming various numerical values of the polynomial terms, we obtain different forms for shells. C is defined for each shell form. The shell form to which a maximum C value corresponds is considered to be optimal. To estimate the degree of approximation of the empirical optimum thus found to the theoretical optimum, the method of building a confidence interval for the empirical optimum is applied [2]. The C theoretical maximum is a function of many variables. The problem of its estimation is solved on the basis of statistical tests.

The problem to be solved is the selection of the optimal form for the reinforced-concrete shell of revolution having constant thickness and being under a uniformly distributed load along the horizontal projection of the shell. The weight of the shell body is assumed to be a material consumption index. With the ratio of the radius of a shell support ring to the rise and reinforcement defined, C only depends on the meridian contour. In our case it was a function of five variables. A special computer programme yielded over 1,000 shell forms; forty of them were chosen for testing. To reduce the spread of the C values, forty meridian contours were chosen mainly in the vicinity of the parabola of the fourth order that was the best among the twelve contours considered previously [3].

Applying the method of building a confidence interval, the maximum theoretical C value was estimated with the accuracy of up to 95%. It depends on (a) the number of unknown factors of the function being estimated; (b) the number of tests; (c) the maximum value and one of other C values within the spread of the empirically found C values; and (d) the specified level confidence. The difference between this C value and its experimentally found value does not exceed 1%. This proves an effectiveness of the empirical method when determining the optimal form for the shell.

It should be especially stressed that the method of building a confidence interval described in this paper can be used in practically all cases when the optimal factors are determined empirically on the basis of statistical tests.

1

M.A. Danieli (Danielashvili), "Limit Analysis of Reinforced Concrete Shells of Revolution and its Application", In: Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing, B.H.V. Topping (Editor), Civil-Comp Press, Stirling, Scotland, Paper 58, 2003. doi:10.4203/ccp.77.58

2

M.A. Danielashvili, R.Ya. Tchitashvili, "On creating a confidence interval for empirical optimum and estimating optimal form for shells of revolution", Res. into struct. mech., III, Tbilisi, "Metzniereba", 1972.

3

M.A. Danielashvili, "On the choice of optimal factors for reinforced shells of revolution", In: "Structural mechanics of three-dimensional constructions", Tbilisi, "Metzniereba", 1972.

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