Novel Weak Form Quadrature Element Method for Free Vibration Analysis of Hybrid Nonlocal Euler-Bernoulli Beams with General Boundary Conditions

Year 2017, , 65 - 75, 27.12.2017

Wang Xinwei

https://doi.org/10.24107/ijeas.356197

Cited By: 1

Abstract

A
novel weak form quadrature element method (QEM) is presented
for free vibration analysis
of hybrid nonlocal Euler-Bernoulli beams with general boundary conditions.
For demonstrations, the stiffness and mass matrices of a beam
element with Gauss-Lobatto-Legendre (GLL) nodes are explicitly given by using
the nodal quadrature method together with the differential quadrature (DQ) law.
Convergence studies are performed and comparisons are made with exact solutions
to show the excellent behavior of the proposed beam element. Case studies on hybrid nonlocal
Euler-Bernoulli beams
with different length scale parameters have been conducted.
Accurate
frequencies of the
beams with different combinations of boundary conditions are obtained and
presented.

Keywords

Weak form quadrature element method, hybrid nonlocal Euler-Bernoulli beam, multiple boundary conditions, free vibration

References

• [1] Behera, L., Chakraverty, S., Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models: A review. Archives of Computational Methods in Engineering. 24(3), 481-494, 2017.
• [2] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54 (9), 4703-4710, 1983.
• [3] Civalek, O., Demir, C., A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation, 289, 335-352, 2016.
• [4] Demir, C., Civalek, O., A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Composite Structures, 168, 872-884, 2017.
• [5] Civalek, O., Demir, C., Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Applied Mathematical Modeling, 35, 2053-2067, 2011.
• [6] Mercan, K., Civalek, O., Comparison of small scale effect theories for buckling analysis of nanobeams. International Journal of Engineering & Applied Sciences, 9(3), 87-97, 2017.
• [7] Mindlin, R.D., Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16 (1), 51-78, 1964.
• [8] Akgöz, B., Civalek, O., Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Sciences, 49, 1268-1280, 2011.
• [9] Xu, W., Wang, L.F., Jiang, J.N., Strain gradient finite element analysis on the vibration of double-layered graphene sheets. International Journal of Computational Methods, 13, 1650011, 2016.
• [10] Lim, C.W., Zhang, G., Reddy, J.N.. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298-313, 2015.
• [11] Wei, G.W., Discrete singular convolution for the solution of the Fokker-Planck equations. Journal of Chemical Physics, 110, 8930-8942, 1999.
• [12] Wei, G.W., A new algorithm for solving some mechanical problems. Computer Methods in Applied Mechanics and Engineering, 190, 2017-2030, 2001.
• [13] Wang, X., Duan, G., Discrete singular convolution element method for static，buckling and free vibration analysis of beam structures. Applied Mathematics and Computation, 234, 36-51, 2014.
• [14] Wang, X., Yuan, Z., Discrete singular convolution and Taylor series expansion method for free vibration analysis of beams and rectangular plates with free boundaries. International Journal of Mechanical Sciences, 122, 184-191, 2017.
• [15] Ng, C.H.W., Zhao, Y.B., Xiang, Y., Wei, G.W., On the accuracy and stability of a variety of differential quadrature formulations for the vibration analysis of beams. International Journal of Engineering & Applied Sciences, 1(4), 1-25, 2009.
• [16] Tornabene, F., Fantuzzi, N., Ubertini, F., Viola, E., Strong formulation finite element method: A survey. Applied Mechanics Reviews, 67, 020801,2015.
• [17] Wang, X., Novel differential quadrature element method for vibration analysis of hybrid nonlocal Euler-Bernoulli beams. Applied Mathematics Letters, 77, 94-100, 2018.
• [18] Wang X. Differential Quadrature and Differential Quadrature Based Element Methods: Theory and Applications. Oxford: Butterworth-Heinemann, 2015.
• [19] Jin, C., Wang, X., Ge, L., Novel weak form quadrature element method with expanded Chebyshev nodes. Applied Mathematics Letters, 34, 51-59, 2014.
• [20] Wang, X., Yuan, Z., Jin, C., Weak form quadrature element method and its applications in science and engineering: A state-of-the-art review. Applied Mechanics Reviews, 69, 030801, 2017.
• [21] Jin, C., Wang, X., Accurate free vibration of functionally graded skew plates. Transactions of Nanjing University of Aeronautics & Astronautics, 34(2), 188-194, 2017.
• [22] Jin, C., Wang, X., Weak form quadrature element method for accurate free vibration analysis of thin skew plates. Computers and Mathematics with Applications, 70, 2074-2086, 2015.