Application of exponential functions in weighted residuals method in structural mechanics. Part 1: axisymmetrical shell problem.




axisymmetrical shell, distributed loading, concentrated force, Navier method, Galerkin method, sets of exponential functions.


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.

Author Biographies

Igor Orynyak, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"

Applied Mathematics Department, Professor

Yulia Bai, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"

Applied Mathematics Department, Senior Lecturer


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How to Cite

I. Orynyak and Y. Bai, “ Part 1: axisymmetrical shell problem”., Mech. Adv. Technol., no. 3(90), pp. 19–28, Dec. 2020.