Analytical Modeling of Concentrated and Localized Loads of Bars With a Curvilinear Flat Axis. Part 2. Modeling Localized Loads and Application Examples / Аналітичне моделювання зосереджених та локалізованих навантажень брусів із криволінійною плоскою віссю. Частина 2. Моделювання локалізованих навантажень та приклади застосування

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Ковальчук, Станіслав Богданович
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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. The second part of the work deals with the mathematical justification of the analytical modeling of the distributed load localized on the section of a longitudinal cylindrical surface of a curvilinear bar in the natural for its structure curvilinear coordinate system. Also, the correct derivation of the relations for modeling the concentrated moment applied at the point of the longitudinal surface or the end is given. Some uncertainties in the description of the moment of couple, which in the first part of the work led to incorrect relations that are valid only in the case of the bar with a straight axis, are taken into account. As an approbation of the obtained relations, the authors have given an example of the analytical determination of the internal force factors for a cantilever bar with a parabolic axis and constant sizes of length cross-section, which is under the influence of a complex system of concentrated and distributed loads applied to longitudinal surfaces and the free end. The relations obtained are of a general nature and are invariant with respect to the coordinate system under consideration. The results obtained can be used to solve a wide range of applied problems of deforming curvilinear bars.
curvilinear bar, localized load, concentrated moment, natural coordinate system, generalized function, криволінійний брус, локалізоване навантаження, зосереджений момент, природна система координат, узагальнена функція, криволинейный брус, локализованная нагрузка, сосредоточенный момент, естественная система координат, обобщенная функция