Mathematical model of a spatial rectangular contour-type truss deformations
The object of research is spatial structure of a rectangular contour-type cover. A diagram of a statically determinate truss in the form of a closed rectangle with supports along the inner contour is proposed. The truss consists of quadrangular bar pyramids assembled into a square contour with tops connected by a bar belt. Four horizontal braces are located at the corners of the structure. A vertical load is considered, evenly distributed over the nodes of the truss. Method. The design is statically determinate, therefore, to calculate the forces in the rods, it is enough to solve the system of equations for the equilibrium of nodes. The matrix of the system of equilibrium equations consists of the direction cosines of the forces, which are calculated from the coordinates of the nodes. The derivation of the formula for the dependence of the deflection of several characteristic points of the structure on the number of panels in the truss is given. The conclusion is based on an inductive generalization of the decision sequence for structures with an increasing number of panels. The coefficients of the sought formulas are found from the solution of homogeneous linear recurrent equations. Results. The solution of the equilibrium equations of the nodes and all transformations are performed in the Maple symbolic mathematics system. Linear asymptotics of solutions are found. The two main results of the work are the development of a scheme for a regular spatial statically determinate rectangular truss and obtaining an analytical dependence of the deflection of the structure on the number of panels.