The dynamical problem on acting concentrated load on the elastic quarter space

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load acts on the other along a rectangular section at the initial moment of time. Integral Laplace and Fourier transforms are applied sequentially to the equations of motion and boundary conditions in contrast to traditional approaches when integral transforms are applied to solutions' representations through harmonic functions. This leads to a one-dimensional vector homogeneous boundary value problem with respect to unknown displacement's transformants. The problem was solved using matrix differential calculus. The original displacement field was found after applying inverse integral transforms. For the case of stationary vibrations a method of calculating integrals in the solution in the near loading zone was indicated. For the analysis of oscillations in a remote zone the asymptotic formulas were constructed. The amplitude of vertical vibrations was investigated depending on the shape of the load section, natural frequencies of vibrations and the material of the medium.

РЕШЕНИЕ ПРОСТРАНСТВЕННОЙ КОНТАКТНОЙ ЗАДАЧИ ШАРНИРНЫХ УЗЛОВ ОПИРАНИЯ БАЛКИ НА УПРУГИЕ ЧЕТВЕРТЬПРОСТРАНСТВО И ОДНУ ВОСЬМУЮ ПРОСТРАНСТВА

The article discusses the solution of the spatial contact problem arising when calculating a reinforced concrete rafter beam pivotally supported by concrete walls. The walls are modeled by the elastic quarter-space on the left and by one-eighth of the elastic space on the right. This contact problem is solved using the numerical method - the Zhemochkin method. For this purpose, the contact area is divided into fragments. Rigid one-way ties are set in the center of each fragment to implement contact between the beam and the wall. It is assumed that the forces arising in these ties provide uniform distribution of reactive pressures in the appropriate fragment. Then, the system of linear algebraic equations for the mixed method of structural mechanics shall be prepared and solved. Different Green functions are assumed for the left and right wall. The problem under consideration is nonlinear, and it requires an iterative process to calculate the effective area of contact and the values of the related reactive pressures. The iterative process shall be finished when contact stresses at the boundary of separation of the structure from the walls are identically equal to zero, or when there are no stretched Zhemochkin ties. Isolines of contact stresses and vertical displacements of the contact areas of the walls are plotted for the flexibility index corresponding to the real ratio of rigidity of supported structures and the flexibility index corresponding to the support of the absolutely rigid beam. The function is found, describing the torque arising in the beam versus the distance from the edge of one eighth of the elastic space. A beam can be considered as supported on the left and right by the elastic quarter-space when the distance from the beam axis and the edge of one-eighth of the space exceeds the twofold beam width. В статье рассматривается решение пространственной контактной задачи, возникающей при расчете железобетонной стропильной балки, шарнирно опираемой на бетонные стены. Стены моделируются слева упругим четвертьпространством и справа -одной восьмой пространства. Данная контактная задача решается с использованием численного метода -метода Б. Н. Жемочкина. Для этого область контакта разбивается на участки. В центрах каждого участка устанавливаются жесткие односторонние связи, через которые осуществляется контакт балки со стеной. При этом предполагается, что усилия, возникающие в установленных связях, вызывают равномерное распределение реактивных давлений в соответствующем участке. Далее составляется и решается система линейных алгебраических уравнений смешанного метода строительной механики. Для левой и правой стен принимаются различные функции Грина. Рассматриваемая задача является нелинейной и требует итерационного процесса для определения фактической области контакта с величинами соответствующих реактивных давлений. Моментом окончания итерационного процесса служит тождественное равенство нулю контактных напряжений на границе отрыва конструкции от стен либо отсутствие растянутых связей Б. Н. Жемочкина. Построены изолинии контактных напряжений и вертикальных перемещений контактных областей стен при показателе гибкости, соответствующем реальному соотношению жесткостей опираемых конструкций, и показателе гибкости, соответствующем опиранию абсолютно жесткой балки. Установлена зависимость возникающего крутящего момента в балке от расстояния до края одной восьмой упругого пространства. Балку можно считать как опираемую слева и справа на упругое четвертьпространство, когда расстояние от оси балки и края одной восьмой пространства превышает двойную ширину балки.

DEFINITION OF THE INTRA QUARTER SPACES AND THEIR FUNCTIONS IN THE PLANNING STRUCTURE OF HISTORICAL CITIES

The article gives the definition of intra quarter spaces using the example of the historical city of Chernivtsi. The main functions of residential courtyards as elements of the urban planning structure are determined. To define the concept of «intra quarter space», we can turn to several foreign and domestic researchers, architects, city planners. For example, the theorist and architect A. Gutnov characterized the yard as a clearly fixed, closed space. Courtyards were connected by arches or through passages, forming a complex system of intra quarter spaces, which in some cases penetrated large areas of the city [3]. Architect R. Krier in his works on the morphology of urban space defined the interior spaces of historic quarters as protected from the weather and the environment of the territory, which became the appropriate symbols of the private sphere of life [12]. Summarizing all the above statements, we can give a general definition for «intra quarter spaces», that it is an independent, complex-functional structure that provides communication: the house – the courtyard – urban areas. Exploring the intra quarters of the city of Chernivtsi, we can identify the following functions of the courtyards: insulating function, which serves as protection against unplanned intrusion of «strangers» or vehicles into the yard; household function, which includes meeting the needs of residents of nearby houses; trade function, is the creation in the volumes of the first floors of various commercial premises; the function of communication, which plays a socio-psychological role of adaptation of residents to the urban environment; sanitary and hygienic function, including landscaping, the need for insolation, wind and snow protection of the courtyard, etc. aesthetic function that provides visual comfort from objects of small architectural forms, landscaping (lawns, flower beds), murals on empty walls, etc. game function and sports and health function.

Concentrated Force Action on 1/8 Homogeneous Isotropic Space

Using the example of vertical displacements, it is shown that by combining a solution to the problem of determining vertical displacements from the action of four identical concentrated forces symmetrically applied to an elastic half-space and two identical concentrated forces symmetrically applied to an elastic quarter-space, one can obtain a solution about the action of one force on 1/8 of the elastic space with free edges. To find vertical displacements in an elastic half-space, the Boussinesq solution is used, and vertical displacements in an elastic quarter-space – an integral equation obtained by Ya. S. Uflyand to determine vertical displacements in the face of a homogeneous elastic isotropic quarter-space, for which a deformation modulus and Poisson’s ratio are constant. However, an integral equation of Ya. S. Uflyand is very inconvenient for practical use, therefore, in the paper, an approximate expression written in terms of elementary functions is proposed to find vertical displacements in the face of an elastic quarter-space from the action of a concentrated force. To obtain the latter, a special approximation method is used. The desired solution is also expressed in terms of elementary functions. In this case, an accurate calculation is obtained for an incompressible material with Poisson’s ratio 1/8 of the space n = 0.5. Since the solution is obtained in the case of a concentrated force acting on 1/8 of the elastic space, it is easy to find an expression for determining the vertical displacements of the edge of 1/8 of the elastic space from the action of any distributed load by integrating over the area of action of this load from the influence function, which is taken as required decision. Recommendations for improving the accuracy of calculations are offered. The described approach can also be used to determine the stress-strain of 1/8 of the space with both hingedly supported and free edges.