Analytical calculation of the frame with an arbitrary number of panels

Building constructions, buildings and structures

A rod model of a statically determinate planar frame with four supports is proposed. The task is to obtain an analytical dependence of the deflection of the truss under the action of various loads on the number of panels in the crossbar and in vertical support trusses. To solve the problem, the computer mathematics system Maple is used. The system of equations of equilibrium is solved in symbolic form. It is shown that for a certain number of panels in the crossbar the determinant of the system of equations of equilibrium turns to zero, which indicates a kinematic changeability of the structure. The corresponding velocity distribution of the nodes of the truss is found. For cases of unchangeable construction, the deflection is found from the Maxwell-Mohr’s formula. A series of solutions for various numbers of panels is generalized by the method of double induction. Using the operators of the Maple system, homogeneous linear recurrence equations are derived and solved for the terms of the coefficients of the desired formula. On the graphs of the dependence of the deflection on the number of panels, significant jumps and extreme points are revealed. To evaluate the strength and stability of the construction, expressions for the forces in the individual most stretched and compressed rods in the middle of the span are found. The cases of loading the truss along the upper belt, the lower belt and the force in the middle of the span are considered. The solutions found can be used to evaluate the operability of the designed design and to optimize it.