Mathematical Problems in Engineering
Volume 2007 (2007), Article ID 78507, 19 pages
Research Article

A Finite Circular Arch Element Based on Trigonometric Shape Functions

H. Saffari1 and R. Tabatabaei2

1Department of Civil Engineering, Shahid Bahonar University of Kerman, P.O. Box 133, Kerman 76169, Iran
2Department of Civil Engineering, Islamic Azad University of Kerman, P.O. Box 7635131167, Kerman 76175-6114, Iran

Received 27 August 2006; Accepted 26 February 2007

Academic Editor: Jan Awrejcewicz

Copyright © 2007 H. Saffari and R. Tabatabaei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The curved-beam finite element formulation by trigonometric function for curvature is presented. Instead of displacement function, trigonometric function is introduced for curvature to avoid the shear and membrane locking phenomena. Element formulation is carried out in polar coordinates. The element with three nodal parameters is chosen on curvature. Then, curvature field in the element is interpolated as the conventional trigonometric functions. Shape functions are obtained as usual by matrix operations. To consider the boundary conditions, a transformation matrix between nodal curvature and nodal displacement vectors is introduced. The equilibrium equation is written by minimizing the total potential energy in terms of the displacement components. In such equilibrium equation, the locking phenomenon is eliminated. The interesting point in this method is that for most problems, it is sufficient to use only one element to obtain the solution. Four examples are presented in order to verify the element formulation and to show the accuracy and efficiency of the method. The results are compared with those of other concepts.