Deformations of a three-dimensional model of a trihedral double lattice rod tower

Abstract

Introduction. The calculation of deformations of spatial structures, as a rule, is carried out numerically on the basis of the finite element method. With the development of computer mathematical systems for regular systems an opportunity to obtain analytical solutions appeared. Such solutions can be used as test ones to evaluate numerical solutions and for preliminary calculation of a structure at design stage. The task has been set to derive deformation dependences of a spatial truss under various loads from its size and number of panels.
 
 Materials and methods. The truss of the tower structure is statically determined. The calculation of force values in the bars is performed by cutting out nodes in a Maple symbolic mathematics system. The Maxwell – Mohr formula is used to determine the displacement of a node at the top of the frame, considering only the longitudinal deformations of the rods. From the generalization of a series of analytical solutions for trusses with consecutively increasing number of panels onto an arbitrary number of panels the sought formula is derived by the induction method.
 
 Results. An algorithm of derivation of formulas for the deflections of a building along two mutually perpendicular horizontal axes under the action of lateral uniformly distributed nodal loads is presented. Effort distribution patterns are obtained for the truss rods and analytical stress dependences of some rods on the number of panels are found. Linear asymptotics of the solutions for the number of panels and the points of extremum are found and computed.
 
 Conclusions. The considered model of the spatial statically determined tower truss allows deriving exact formulas for deformations under the action of various loads within the limits of the adopted model. It is possible to use derived formulas for a preliminary estimation of designed structure and to apply as test ones for numerical calculations. The extremum points and analytical expressions for asymptotes allow the derived formulas to be used for solving stiffness optimization problems of a structure.