Design of a beam of variable cross-section on the elastic base by the quasi-analytical method

Abstract

Purpose. Development of the quasi-analytical method of nonlinear differential equation solution and its approbation with reference to beams of variable cross-section on the elastic base with two base factors.Methods of researches. Required function, taking into account its assumpted form, can be approximated by some known function. After its substitution in the nonlinear differential equation, the problem is reduced to an analytical or numerical finding of such function factors at which the estimation of difference between the right and left parts of the differential equation will be minimal at all possible variable values. According to the theory of qualities, the value of a difference estimation allows to conclude, how much successfully the approximating function has been chosen. If power function is used for approximation, the problem is reduced to solution of system of the linear algebraic equations. Solution result accuracy can be improved by means of numerical optimisation at which required factors are varied parameters.Results. Specific solutions in the field of building have been obtained during approbation. The wedge-shaped concrete beam on the elastic base (ground) with rectangular cross-section of variable height has been researched. Solutions for seven various forms of the distributed load have been obtained. In all cases the average solution error had not exceed 0.1%. That confirms quality of the offered solution method and adequacy of the chosen approximating power function.Scientific novelty. The authors had not meet in literature such method of nonlinear differential equation solution.Practical value. The quasi-analytical method that has been offered can be used for solution of differential equations of any order with various types of nonlinearity, including calculations of beams of variable cross-section on the elastic base. Besides, consideration of boundary conditions can be easily realised.