Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 212709, 25 pages
Research Article

Static, Vibration, and Transient Dynamic Analyses by Beam Element with Adaptive Displacement Interpolation Functions

1School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China
2Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China

Received 2 July 2011; Revised 25 September 2011; Accepted 25 September 2011

Academic Editor: G. Rega

Copyright © 2012 Shuang Li et al. 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.


An approach to analyzing structures by using beam elements is developed with adaptive displacement interpolation functions. First, the element stiffness matrix and equivalent nodal loads are derived on the basis of the equilibrium between nodal forces and section forces rather than the compatibility between nodal deformations and section deformations, which avoids discretization errors caused by the limitation of conventional polynomial interpolation functions. Then, six adaptive element displacement interpolation functions are derived and extended to include several cases, such as beams with variable cross-section, variable material properties, and many different steps in cross-section and/or material properties. To make the element usable in dynamic analyses, consistent mass matrix (CMM) and diagonally lumped mass matrix (LMM) are constructed using the presented adaptive displacement interpolation functions. All these features have made the element elegant, which is tested with a number of simple static, vibration, and dynamic examples to show its accuracy.