Static Deflection of a Quadrangular Rod Pyramid: An Analytical Solution

Building constructions, buildings and structures

The object of the study is a spatial statically determinate truss in the form of a regular quadrangular pyramid. Along the entire perimeter of the base, the truss has vertical support posts. One corner unit is fixed on spherical support, one on cylindrical support, the others only on vertical posts. The analytical dependence of the deflection of the top of the pyramid on the number of panels at its base is derived. The load distributed over the edges and the vertical concentrated force at the vertex are considered. Method. The deflection is calculated using Mohr's integral. To determine the forces in the rods and the reactions of the supports, a system of equilibrium equations for all nodes in the projection on the coordinate axis is compiled. To generalize a series of partial solutions for trusses with a different number of panels, the induction method and operators of the Maple computer mathematics system are used. Results. A compact formula for the dependence of the deflection on the number of panels is obtained. The two coefficients of the formula have the form of polynomials in the number of panels of degree no higher than the second. The horizontal asymptote of the solution is found. Formulas are derived for the most compressed (in the edges of the structure) and the most stretched (in the base) rods.