The Stress-Strain State of a Rectangular Covering Spatial Truss
The object of research is a model of a statically definable roof truss with supports on the sides of the structure. Spherical and cylindrical supports are located at two corner points of the covering, in other corners and on the sides of the structure at an equal distance from each other — stands. A multi-dome surface shape is set. For the case of loading the truss by the force in the middle of the span by the induction method according to the number of panels, the formula for the deflection is derived. Analytical dependencies of the reactions of supports on the number of panels under the action of a load uniformly distributed over the nodes of the coating surface are found. Method. The calculation of the forces in the rods and the reactions of the supports of an externally statically indeterminate truss is performed in symbolic form using the method of cutting nodes. To compose a system of equilibrium equations and obtain a solution, the operators of the Maple computer mathematics system are used. The deflection is determined by the Maxwell-Mohr's formula. Based on the data on the solution of a series of trusses with a sequentially increasing number of panels by the induction method, formulas for the dependence of the deflection on the load, the size and number of panels of the structure are derived. Operators of the genfunc software package from the Maple computer mathematics system are used to compose and solve homogeneous linear recurrent equations, which are satisfied by the sequences of the coefficients of the sought dependencies. Results. Dependences of deflection and forces on the number of panels have a form polynomial in the number of panels. It is shown that the vertical reactions of the corner supports can have negative signs, which makes it necessary to use retaining supports here. The curves illustrating the solution have a clearly pronounced jump-like shape. The quadratic asymptotics of the solution is found both in terms of the number of panels and linear in height. The dependence of the deflection on the height has a theoretically permissible minimum.
The considered scheme of a spatial statically definable roof truss allows an analytical solution to the deflection problem. The obtained dependences can be used to estimate the accuracy of numerical solutions and in problems of optimization of the structure in terms of rigidity and solutions.