Abstract
Weighted residuals method gained a wide popularity during last years especially due to its application in finite element methods. Its goal is in approximate satisfaction of the governing differential equations while boundary conditions are to be fulfilled exactly. This goal is achieved by the proper choice of the sets of so-called trial (basic) functions which give the residuals. Residuals are multiplied by weight functions and minimized by integration over the whole area of task. In fact, they determine the peculiarity and advantages of each particular method. Most popular is the choice of trial and weight (test) function as the trigonometric and polynomial functions. In 2D applications so-called “beam functions” are often used, which are solutions of much simpler 1D problems for beam. In this methodological paper we explore the possibility of using the sets of functions constructed on the consequent exponential functions, which satisfy boundary conditions. The method is investigated on example of very simple 1D axisymmetrical task for shell, where exact solution exists for any loading. For several examples of distributed or concentrated loading the proposed method is compared with similar Navier’s method, which is the expansion on trigonometric functions. Also the proper choice of weight functions is carefully investigated. It is noted, that proposed sets (symmetrical or asymmetrical) of exponential functions has a good perspective in application for more complicated problems in structural mechanics.
Highlights
Consider usual one dimensional differential equation with a right part: G(y) f (x), (1a) where G( y) – operator with respect to the looking for function y, f (x) is a right part, which in structural mechanics is usually an outer loading
Essence of classical approach is in finding the sum of arbitrary particular solution and general solution of homogeneous equation, which consists of several eigenfunctions, and each is factored by unknown coefficient
Special attention is given to choice of test functions, their justification and to consideration of action of concentrated force on the boundary, while the trial functions satisfy to some zeros boundary conditions
Summary
Consider usual one dimensional differential equation with a right part: G(y) f (x) ,. This work is methodological one, introductory to more serious problems, and is aimed on the explanation and extension of WRM capacities on example simple symmetrical and antisymmetrical task for simple one dimensional differential equation of 4th order It has three objectives: First – to introduce exponential functions, especially for more complicated case of infinite body, when only decaying functions can be used, i. As test functions we chose the following sets: k (x) for Galerkin method; k (x) are sets defining angle of rotation; k (x) are expressions for moments; k (x) are transverse force functions; k (x) are the functions for outer loading. Consider procedure of calculation for loading (3a) on example of first case of choice of test functions, namely m m (Galerkin method). Consider particular examples calculation for different loadings and variants of WRM
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
