A Unified Method for Modeling of Axially and/or Transversally Functionally Graded Beams with Variable Cross-Section Profile

Abstract

A novel finite element method is presented to solve static and free vibration problems of axially and/or transversally functionally graded (FG) beams with variable cross-section profile. The presented method overcomes the inaccuracy of the conventional finite beam elements for analysis of the FG beams by introduction of exact displacement interpolation functions. The introduced exact displacement interpolation functions have a unified expression for individual case of axially FG beams, transversally FG beams, varied cross-section beams, as well as their coupled cases. Hence, such cases can be modeled with a unified method and by only one beam element, leading to the presented method easy for modeling the FG beams. The element stiffness matrix is derived according to the internal force equilibrium relationship of the beam element. The element mass matrix and element equivalent nodal loads are derived based on the exact displacement interpolation functions. The method has high accuracy and fast convergence, which is tested with verification examples including a stepped beam, a beam with axially varying material properties and a beam with transversally varying material properties.

Keywords:

• Beam element
• Displacement interpolation function
• Equivalent nodal load
• Functionally graded beam
• Mass matrix
• Varied cross-section beam

ACKNOWLEDGMENT

This research project is supported by the National Natural Science Foundation of China (Grant No. 51008101) and the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 201195). The financial supports from the research funds are greatly appreciated by the authors.

Notes

Note: *is calculated without considering rotation mass items.

Note: *is calculated without considering rotation mass items; Error = (Conventional – This paper)/ This paper.

Note: *is calculated without considering rotation mass items.

Note: *is calculated without considering rotation mass items; (·) is calculated using 1st-freq. result with more than two decimals.

^#Communicated by G. Hulbert.