Аналітичне моделювання зосереджених та локалізованих навантажень брусів із криво-лінійною плоскою віссю. Частина 1. Моделювання зосереджених у точці навантажень

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Ковальчук, Станіслав Богданович
Горик, Олексій Володимирович
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Одеська державна академія будівництва та архітектури
In applied mechanics, the common types of load are concentrated force, moment, and also distributed load localized in a certain part of the beam. An effective approach to the analytical modeling of such loads is the use of generalized functions, with the help of which it is possible to avoid considering the many design sections of the beam, within which the load is a continuous function. However, in well-known scientific papers, this approach is developed only for rectilinear rods and partially for circular ones. This paper is devoted to the problem of modeling concentrated loads (force, moment) and loads localized on a surface for bars with a curvilinear plane axis of arbitrary shape in a natural coordinate system. In the first part, the mathematical substantiation of analytical modeling of forces and moments concentrated at a point of a longitudinal cylindrical or end surface of a curvilinear bar in a natural curvilinear coordinate system is illuminated. By limiting the transition from a statically equivalent local load to the boundary case of a load concentrated at a point on the surface of the beam, using the elements of the theory of generalized functions, we obtained general analytical relations for modeling the concentrated force. On the basis of the ratio obtained, by passing to the limit for a pair of concentrated forces as the pair’s arm tends to zero, a mathematical rationale for analytical modeling of the concentrated moment is constructed. The relations obtained are of a general nature and are invariant with respect to the coordinate system under consideration. Based on them, a number of relations have been obtained for modeling concentrated loads in individual cases of circular, elliptical and parabolic bars in natural coordinate systems. The results obtained can be used to solve a wide range of applied problems of deforming curvilinear bars.
криволінійний брус, зосереджена сила, зосереджений момент, природна система координат, узагальнена функція, криволинейный брус, сосредоточенная сила, сосредоточенный момент, естественная система координат, обобщенная функция, curvilinear bar, concentrated force, concentrated moment, local load, natural coordinate system, generalized function