Deformations and natural vibration frequency of a three-dimensional truss in a multi-storey building

Abstract

Introduction. The analysis of deformations arising under the action of various loads and natural vibration frequencies is an integral part of the structural analysis of a multi-storey building. As a rule, this analysis is performed numerically in various systems using the finite element method and specialized software programmes. Analytical solutions are rare, but they are needed to evaluate the numerical ones and calculate simplified models of structures. The task is to find the analytical dependence of the frame deflection in a multi-storey building and its first natural frequency on the number of panels. The mass of the truss model is assumed to be distributed uniformly over the nodes. Only horizontal displacements of masses are taken into account. Materials and methods. To calculate the forces arising in the rods of a statically determinate truss using the method of joint isolation in the alphameric form, the software compiled in the Maple math system is applied. The calculation of displacements of nodes is performed using the Mohr integral. The Dunkerley method allows finding the analytical form of the lower estimate of the first frequency of natural vibrations for an arbitrary number of storeys in a building. The analysis of a series of solutions, made in the alphameric form for trusses with a successively growing number of panels allows deriving the final vibration frequency formula. When determining the natural vibrations of the characteristic matrix, the Eigenvalues operator from the LinearAlgebra package was used. Results. The resulting formulas used to analyze the deflection (horizontal displacement) of a structure under the action of a horizontal lateral load and the formula for estimating the first vibration frequency have simple forms, convenient for the quick verification of numerical solutions and preliminary evaluation of the behaviour of a designed structure. Conclusions. The proposed spatial statically determinate model of a multi-storey building allows deriving formulas for the analysis of deformations and natural frequencies of a structure with an arbitrary number of storeys. Resulting formulas can be applied to solve optimization tasks and serve as the basis for calculating a more complex statically indeterminate model of a structure.