Solving the spatial contact problem for the hinge joints of the beam support by the elastic quarter-space and one eighth of the elastic space

Abstract

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 quarterspace 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 quarterspace when the distance from the beam axis and the edge of one-eighth of the space exceeds the twofold beam width.