Природна система координат для криволінійних композитних брусів із незмінними лінійними розмірами поперечних перерізів

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Ковальчук, Станіслав Богданович
Горик, Олексій Володимирович
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Луцький національний технічний університет
It is known that there are some solutions of the elasticity theory for the tasks of composite bars deformation that are obtained only for multilayer bars with a rectilinear and circular axis, which impedes the efficient design of composite structures with curvilinear rod elements. The goal of this work is the mathematical justification and determination of parameters of the curvilinear cylindrical orthogonal coordinate system, which is natural for composite bars with stationary linear dimensions of cross sections over the length. The coordinate system is based on a one-parameter family of cylindrical surfaces evenly spaced from the base curve (the axis of a bar) and families of planes that are orthogonal to them. Two ways of parameterization of the proposed coordinate system are considered: by the coordinate of the cross section of a bar and by the angle between the cross section and the axis of the auxiliary Cartesian coordinate system, allowing one to take into account the features of axes of various shapes. General analytical relations for determining the parameters of the natural coordinate system by analytically given equation of the axis of a bar have been obtained for the types of parameterization mentioned above. Using these relations, we received the equations of the elasticity theory in the natural coordinate system for bars with a curvilinear plain axis of an arbitrary shape and stationary linear dimensions of cross sections. Some certain examples of implementation of the equations obtained for the bars with a parabolic, elliptic and cosinusoidal axis are given. The theoretical framework developed in this paper allows us to expand the possibility of applying the equations of the elasticity theory and relations for internal force factors for curvilinear composite bars in the natural coordinate system for solving a wide range of applied problems.
криволінійний брус, криволінійна плоска вісь, природна система координат, сімейство кривих, еквідистанта, рівняння теорії пружності, curvilinear bar, curvilinear flat axis, natural coordinate system, curve family, equidistant, equations of the theory of elasticity, криволинейный брус, криволинейная плоская ось, естественная система координат, семейство кривих, эквидистанта, уравнения теории упругости