ACCURACY EVALUATION OF BOUNDARY CHARACTERISTIC ORTHOGONAL POLYNOMIALS (BCOPs) IN DETERMINING STATIC AND DYNAMIC RESPONSES OF THIN RECTANGULAR PLATES WITH OPENINGS

Document Type : Original Article

Authors

1 Associate Professor, Department of Engineering, University of Science and Culture, Tehran, Iran

2 University of Science and Culture

Abstract

In this article, the efficiency of Boundary Characteristic Orthogonal Polynomials (BCOPs) in analyzing the static and dynamic behavior of thin rectangular plates with openings versus the Finite Element Method (FEM) and the analytical solutions (if they exist), is investigated. Despite this simple procedure, according to the obtained results, the accuracy of BCOPs in most of the studied cases compared to the analytical solutions or those obtained via FEM is acceptable. Besides, different sizes for the openings are assumed and in one case a steel strip is used to stiffen the plate around the opening. Maximum deflection of the plate is the core parameter to be compared seeking the convergence rate of the employed method. Furthermore, natural frequencies of the plate are obtained and compared to assess the capability of BCOPs in dynamic analysis of thin plates. In all of the studied cases, the efficiency of the BCOPs is evident based on its simplicity in comparison with conventional FEM or other competitive methods. Furthermore, one of the main advantages of using these functions in comparison with eigenfunction expansion method in analytical and semi-analytical approaches in the analysis of thin rectangular plates is the existence of all plates’ shape functions for any arbitrary boundary conditions, while this is limited to some special cases in the exact eigen problem solution of the plate.

Keywords

Main Subjects

References

[1] Bhat, R. (1985). Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. Journal of Sound and Vibration, 102(4), 493-499.
[2] Liew, K., Lam, K., & Chow, S. (1990). Free vibration analysis of rectangular plates using orthogonal plate function. Computers & Structures, 34(1), 79-85.
[3] Chakraverty, S. (1992). Numerical solution of vibration of plates. PhD Thesis. University of Roorkee (Now IIT, Roorkee), Roorkee, India.
[4] Singh, B., & Chakraverty, S. (1994). Boundary characteristic orthogonal polynomials in numerical approximation. Communications in numerical methods in engineering, 10(12), 1027-1043.
[5] Chakraverty, S., Bhat, R., & Stiharu, I. (2000). Vibration of annular elliptic orthotropic plates using two dimensional orthogonal polynomials. Applied Mechanics and Engineering, 5(4), 843-866.
[6] Liu, M.-F., & Chang, T.-P. (2005). Vibration analysis of a magneto-elastic beam with general boundary conditions subjected to axial load and external force. Journal of Sound and Vibration, 288(1), 399-411.
[7] Bashmal, S., Bhat, R., & Rakheja, S. (2009). In-plane free vibration of circular annular disks. Journal of Sound and Vibration, 322(1), 216-226.
[8] Lal, R., & Kumar, Y. (2012). Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation. Shock and Vibration, 19(3), 2010-0635.
[9] Niaz, M., & Nikkhoo, A. (2015). Inspection of a rectangular plate dynamics under a moving mass with varying velocity utilizing BCOPs. Latin American Journal of Solids and Structures, 12(2), 317-332.
[10] Jafari, R., Farazandeh, A., Nikkhoo, A., & Rofooei, F. (2015). Vibration Assessment Of The Beams Via Characteristic Orthogonal Polynomials, Sharif Journal of Civil Engineering, 31.2(3.1), 83-92.
[11] Nikkhoo A, Farazandeh A, Ebrahimzadeh Hassanabadi M, Mariani S. (2015). Simplified modeling of beam vibrations induced by a moving mass by regression analysis. Acta Mechanica, 226(7), 2147-2157.
[12] Nikkhoo A, Farazandeh A, Ebrahimzadeh Hassanabadi M. (2016). On the computation of moving mass/beam interaction utilizing a semi-analytical method. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(3), 761-771.
[13] Mahi, A., & Tounsi, A. (2015). A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Applied Mathematical Modelling, 39(9), 2489-2508.
[14] Chakraverty, S., & Behera, L. (2015). Free vibration of non-uniform nanobeams using Rayleigh–Ritz method. Physica E: Low-dimensional Systems and Nanostructures, 67, 38-46.
[15] Behera, L., & Chakraverty, S. (2016). Effect of scaling effect parameters on the vibration characteristics of nanoplates. Journal of Vibration and Control, 22(10), 2389-2399.
[16] Song, X., Han, Q., & Zhai, J. (2015). Vibration analyses of symmetrically laminated composite cylindrical shells with arbitrary boundaries conditions via Rayleigh–Ritz method. Composite Structures, 134, 820-830.
[17] Bhat, R. B. (2015). Vibration of beams using novel boundary characteristic orthogonal polynomials satisfying all boundary conditions. Advances in Mechanical Engineering, 7(4), 1687814015578355.
[18] Reddy, J. N. (1993). An introduction to the finite lement method. Vol. 2. New York: McGraw-Hill.
[19] Pouladkhan, A. R., Emadi, J., Safamehr, M., & Habibolahiyan, H. (2011). The Vibration of Thin Plates by Using Modal Analysis. World Academy of Science, Engineering and Technology, 59.
[20] Timoshenko, S. P., & Woinowsky-Krieger, S. (1959). Theory of plates and shells. McGraw-hill.
[21] Alinia, m. (2000). Theory of plates & shells. Ashtian Publishers.
[22] İmrak, C., & Gerdemeli, İ. (2007). An exact solution for the deflection of a clamped rectangular plate under uniform load. Applied mathematical sciences, 1(43), 2129-2137.
[23] Evans, T. H. (1939). Tables of moments and deflections for a rectangular plate fixed on all edges and carrying a uniformly distributed load. ASME J. Appl. Mech, 6(1).
[24] Taylor, R. L., & Govindjee, S. (2004). Solution of clamped rectangular plate problems. Communications in numerical methods in engineering, 20(10), 757-765.
Journal of Structural and Construction Engineering
Volume 6, Issue 3 - Serial Number 26
November 2019
Pages 175-194

Files

History

  • Receive Date: 11 November 2017
  • Revise Date: 14 April 2018
  • Accept Date: 17 May 2018

Share

How to cite

Statistics