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Introduction. The development of models for bent bars with regard to the effect of transverse (tangential) shears began with the solution of the vibration problem for bars suggested by S. P. Timosbenko, who was the first to introduce the concept of a shear-caused rotation angle of a cross section. The models taking into account the transverse shears were called refined, nonclassical, or Timoshenko-type models [I]. The construction of such models and the corresponding theories of bending of bars (beams), plates, and shells is widely described in the literature, including monographs and fundamental overviews [1][2][3][4][5][6][7][8][9]. Especially complicated is the construction of nonclassical models for nonuniform composite objects, particularly for composite bars with a discrete or so-called hybrid structure whose inhomogeneity is manifested both in their structure and cross-sectional shapes. From this it follows that the creation of nonclassical models for such bars is a high-priority task. The present study describes a nonclassical model for bent composite bars, allowing for the influence of transverse shears on the basis of an iterative process.Since we are dealing with transverse (tangential) shears, it is important to take into account not only the normal but also tangential loads, which can significantly affect the stress-strain state of a bar.

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Аннотация: Представлено аналитическое решение задачи растяжения-сжатия многослойного кольца, с произвольным количеством ортотропных слоев, взаимно противоположными нормальными силами, действующими в плоскости кольца вдоль одного диаметра. Решение построено на основе общего решения теории упругости для задачи изгиба многослойного бруса с круговой осью и нагрузкой на торцах. Полученные соотношения, соответствуют точному решению задачи на большей части кольца, а вблизи точек приложения нагрузки дают упрощенное описание напряженно-деформированного состояния (НДС) без учета локального искажения возле сосредоточенной силы. Для демонстрации возможностей и апробации полученных решений приведены результаты определении НДС четырехслойного кольца с отношением среднего радиуса к высоте сечения равным 1,75, а также результаты дополнительных расчетов при увеличении указанного отношения до 5,5 и 10,5.Ключевые слова: многослойное кольцо, ортотропный слой, сосредоточенная сила, напряжения, перемещения Abstract: Closed circular rings are widely used in engineering structures both as individual parts and as reinforcing elements of thin-walled structures. One of the important cases of the load of such elements is tension-compression by mutually opposite forces acting along one diameter. For homogeneous isotropic rings, exact solutions of this plane problem are known, and various applied calculation methods are developed. At the same time, for composite and, in particular, multilayer rings, this problem is studied much worse. The purpose of this work is to develop an analytical solution of the problem of tension-compression of a composite ring with an arbitrary number of orthotropic layers with mutually opposite normal forces acting in the plane of the ring along one diameter. Using the symmetry of the problem, its solution comes to the auxiliary problem of determining the stress-strain state (SSS) of a half ring separated on loaded sections with corresponding static and kinematic conditions at the ends. The SSS of such an element, with the exception of small areas near the ends, is similar to the SSS of a circular multilayer bar with loads on the ends. This allowed us to use the exact solution of the theory of elasticity obtained by the authors for the problem of bending a multilayer circular bar with a load on the ends to develop the solution of the problem under consideration. The relations obtained in this way correspond to the exact solution of the problem for the greater part of the ring and close to the points of application of the load, give a simplified description of the SSS without local distortion near the concentrated force. This enables to determine the main SSS of multilayer rings with different sizes of the inner and outer radius, however, the Механіка та математичні методи / Mechanics and mathematical methods №1, 2019 Стор. 56-71 / Page 56-71 ________________________________________________________________________________ ________________________________________________________________________________ Ковальчук С. Б....

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