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Journal of Theoretical
and Applied Mechanics
6, 1, pp. 79-91, Warsaw 1968
and Applied Mechanics
6, 1, pp. 79-91, Warsaw 1968
Optymalizacja kształtu łuku jako przekroju dźwigarów sklepieniowych
W wielu konstrukcjach budowlanych stosuje się przekrycia dachowe w postaci sklepień o zarysie łukowym. I tak np. dachy obiektów przemysłowych miewają postać dźwigarów sklepieniowych o przekroju poprzecznym złożonym z szeregu łuków. Dźwigary te pracują przede wszystkim na zginanie jako belki o rozpiętości L oparte na podporach w płaszczyznach AB oraz CD. Względy ekonomiczne nakazują optymalizację tych konstrukcji. Jednym z wariantów optymalizacji jest izoperymetryczne zagadnienie rachunku wariacyjnego, które nie jest dotychczas rozwiązane. Praca niniejsza podaje sposób rozwiązania tego problemu przez potraktowanie go jako zagadnienia programowania matematycznego i zastosowanie metody Monte Carlo do jego rozwiązania. Ponadto podano numeryczne rozwiązanie pewnego przypadku tego zagadnienia.
OPTIMUM DESIGN OF THE ARCH SECTION OF SHELL BEAMS
The problem of the optimization of the arch shape (to obtain a maximum of the moment of inertia) with the presence of constraints can be solved neither by using the differential calculus nor by the calculus of variation. In this paper the problem has been solved by using the mathematical programming method. Assuming the arch shape equation in the form of a polynomial, its coefficients were treated as decisive variables. Such values of the said variables have been searched for to obtain the maximum of the moment of inertia, observing at the same time some side conditions, such as the constraints, i.e. the length of the arch and its height should not be longer than the given values. To solve the problem, the Monte-Carlo-method was applied. The computations were made using the GIER computer. The case of the arch by the polynomial of the fourth order was numerically solved. The optimum values of its coefficients were found. The programs and the results of the computations are given. For comparison sake, other arch shapes were also computed, i.e. straight line, broken line, the segment of the circle and the parabola of the second order.
OPTIMUM DESIGN OF THE ARCH SECTION OF SHELL BEAMS
The problem of the optimization of the arch shape (to obtain a maximum of the moment of inertia) with the presence of constraints can be solved neither by using the differential calculus nor by the calculus of variation. In this paper the problem has been solved by using the mathematical programming method. Assuming the arch shape equation in the form of a polynomial, its coefficients were treated as decisive variables. Such values of the said variables have been searched for to obtain the maximum of the moment of inertia, observing at the same time some side conditions, such as the constraints, i.e. the length of the arch and its height should not be longer than the given values. To solve the problem, the Monte-Carlo-method was applied. The computations were made using the GIER computer. The case of the arch by the polynomial of the fourth order was numerically solved. The optimum values of its coefficients were found. The programs and the results of the computations are given. For comparison sake, other arch shapes were also computed, i.e. straight line, broken line, the segment of the circle and the parabola of the second order.