The continuation method for calculation of continuous beams of arbitrary bending rigidity

Andrei Verameichyk,Mikhael Mazyrka,Vitaliy Khvisevich,V Tur,N Matweenko

doi:10.1051/matecconf/202135000017

Abstract

This article presents an effective mathematical continuation method for the numerical implementation of the multipoint boundary value problem, to which the calculation of a beam of arbitrary rigidity at any of its supports is reduced. The problem can be treated as a direct one in the matter of constructing an optimal design based on beam systems. A test example of the calculation is given.

Highlights

In computational practice, it is often necessary to carry out a strength analysis of structural elements whose computational scheme can be reduced to a multispan beam of variable rigidity under a variety of conditions of its support and loading [1, 2]
This paper presents the calculation of irregular beam systems based on the continuation method [3]
Let us consider the range of existence of the variable x (Fig. 1), consisting of any number of intervals of length li, i 1, 2,..., n, while the end of the previous interval coincides with the beginning of the one. Such separation is defined by internal interpolating nodes xi, where it is necessary to determine the solution vector Yi of the differential equation: Ln Y R x at specified boundary conditions

Summary

Introduction

It is often necessary to carry out a strength analysis of structural elements whose computational scheme can be reduced to a multispan beam of variable rigidity under a variety of conditions of its support and loading [1, 2]. Let us consider the range of existence of the variable x (Fig. 1), consisting of any number of intervals of length li , i 1, 2,..., n , while the end of the previous interval coincides with the beginning of the one Such separation is defined by internal interpolating nodes xi , where it is necessary to determine the solution vector Yi of the differential equation: Ln Y R x (2). The Ai li section matrices included in formula (8), as mentioned above, are determined from differential equation (2) written for each specific i -thsection They have the following form: li li2 2!EIi li3 3!EIi qi li4 4!EIi li qi li. Where Wj – deflection at the point j , j – rotation angle at the point j , M j , Pj – bending moment and shear force at the point j respectively

Let us represent one of the indeterminates as a linear dependence

The vectors

Conclusion