# A new nine-node element for analysing plates with varying thickness using basic displacement functions

• K. Tayyebi School of Civil Engineering, College of Engineering, University of Tehran 16th Azar St., Enghelab Sq., Tehran, Iran
• A. M. Haghighi Department of Civil Engineering, University of Victoria 3800 Finnerty Road, Victoria, BC, Canada
• R. Attarnejad Centre of Numerical Methods in Engineering,  University of Tehran, Tehran, Iran
Keywords: Basic Displacement Functions, Vibration analysis, Shape functions, Euler–Bernoulli beam theory

### Abstract

The capability of the Finite Element Method in producing accurate and efficient results largely depends on the shape functions adopted to frame the displacement field inside the element. In this paper, a new nine-node Lagrangian element was developed to analyse thin plates with varying cross-sections using the shape functions obtained for non-prismatic straight beams with minimum number of elements. The formulated shape functions, which represent vertical displacements and rotations throughout elements, are rooted from a purely mechanical functions called Basic Displacement Functions (BDFs). These functions are obtained by implementing the force method in Euler–Bernoulli beam theory, which ensures that equilibrium equation is satisfied in all interior points of elements. To verify the competency of the proposed element, solutions for the static analysis of isotropic rectangular plates under various loading conditions, together with free vibration analysis of plates with linear thickness variation were obtained and compared with the previous literature. Results showed that the proposed nine-node Lagrangian element was computationally more cost-effective compared to other competing methods when small number of elements is employed.

Published
2018-12-27
How to Cite
Tayyebi, K., Haghighi, A. M., & Attarnejad, R. (2018). A new nine-node element for analysing plates with varying thickness using basic displacement functions. Journal of Mechanical Engineering and Sciences, 12(4), 4056-4071. https://doi.org/10.15282/jmes.12.4.2018.06.0352
Section
Article