INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 2, 2011 Copyright 2010 All rights reserved Integrated Publishing services Research article ISSN 0976 4399 Buckling analysis of axially compressed SSSS thin rectangular plate using Taylor-Mclaurin shape function Ibearugbulem, Owus M 1., Osadebe, N. N 2., Ezeh, J. C 3., Onwuka, D.O. 4 1,3,4- Department of Civil Engineering, Federal University of Technology, Owerri, Nigeria 2- Department of Civil Engineering, University of Nigeria, Nsukka jcezeh2003@yahoo.com ABSTRACT A comprehensive buckling analysis of axially compressed rectangular flat thin plate with simply supported edges was carried out. The study was accomplished through a theoretical formulation based on Taylor-Mclaurin s series and application of Ritz method. The Taylor Mclaurin s series was truncated at the fifth term, which satisfied all the boundary conditions of the plate and resulted to a particular shape function for SSSS plate. The resulting shape function was substituted into the total potential energy functional, which was subsequently minimized. After minimization, the critical buckling load for the plate was obtained by making N x the subject of the formular. The resulting critical load was found to be a function of a coefficient, H, and the values of H from the present study were compared with the exact values within the range of aspect ratios from o.1 to 1.0 as shown on table 1. The maximum percentage difference was found to be 0.098% for aspect ratio of 0.1, while the least percentage difference was found to be 0.05% for an aspect ratio of 1.0. Hence, the Taylor-Mclaurin series shape function obtained for the SSSS plate is a very good approximation of the exact shape function for the plate. Keywords: Boundary Condition; Critical Buckling Load; In-plane Load; Shape function; Taylor-Mclaurin Series; Total Potential Energy Functional. 1. Introduction Over the years, plate problems have been treated by the use of Fourier series or trigonometric series as the shape function of the deformed plate. Some scholars solve the plate problems from equilibrium approach and others solve the problems from energy and numerical approaches. However, no matter the approach used, the use of trigonometric series (double Fourier series and single Fourier series) has been predominant. Most times, when it is becoming intractable to use the trigonometric series, trial and error means of getting a shape function that would approximate the deformed shape of the plate would be used. Navier (1823) solved a problem of bending SSSS plate using double Fourier series. However, Levy (1899) tried solving plate problems using single Fourier series because the double Fourier series as used by Navier converge slowly. Because of slowness in convergence of the Fourier series, Krylov (1949) proposed an efficient method for sharpening the convergence of the Fourier series. Kantorovich and Krylov (1954) also presented solutions of plate by approximate methods of higher analysis. They made use of Fourier series in these approximate methods. Many other researchers like Nadai (1925), Timoshenko and Woinowsky-Krieger (1959), Iyenger (1988), Ye (1994), Ugural (1999) and Eccher, Rasmussen and Zandonini (2007) also used Fourier series in their work. Received on September, 2011 Published on November 2011 667
There is actually dearth of literature on the use of Taylor-Maclaurin s series in solving plate bending problems. Due to the dearth of literature on the use Taylor-Maclaurin s in solving plate bending problems coupled with the difficulties associated with Fourier series, this paper used Taylor-Mclaurin s series to solve the problem of a plate simply supported at all the four edges (SSSS) and subject to in-plane load in one axis (X - axis) of the principal plane (figure 1.) X b Nx SSSS PLATE Nx Y a P = a / b Figure 1: Schematic representation of in-plane loaded SSSS plate 2. Total potential energy funtional for thin plate buckling Ibearugbulem (2011) derived the total potential energy functional for a rectangular thin isotropic plate subjected to in-plane load in x-direction as follows: Where a and b are plate dimensions in x and y directions. µ is Poisson s ratio. N x is the in-plane load in x direction. D is flexural rigidity. W is the shape function. ; ;. ; If the chosen shape functions is a good approximation of the exact shape function, then (Ventsel and Krauthammer, 2001, Ibearugbulem, 2011). Thus, equation (1) becomes: 668
3. Shape function from Taylor-MCLaurin s series Ibearugbulem (2011) assumed the shape function, w to be continuous and differentiable. He expanded it in Taylor-Mclaurin series and got: is the nth partial derivative of the function w with respect to y. m! and n! are factorials of m and n respectively. x 0 and y 0 are the points at the origin. He took the origin to be zero. After some modification of equation (3) and noting that x = a.r and y = b.q, the infinite series was truncated at m = n = 4 and got: The boundary conditions for SSSS plate are Substituting equations (5) and (7) into equation (4) gave: ; ; ; Also, substituting equation (6) into equation (4) and solving the resulting two simultaneous equations gave: ; Similarly, substituting equation (8) into equation (4) and solving the resulting two simultaneous equations gave: ; Substituting the values of. That is into equation (4) gave 669
4. Application of Ritz method Partial derivatives of equation (9) with respect to either R or Q or both gave the following equation: Integrating the square of these five equations partially with respect to R and Q in a closed domain respectively gave: Substituting equations (15), (16), (18) and (19) into equation (2) gave: Minimizing equation (20) and making N X the subject of the equation gave: That is to say 670
5. Result and discussion The H values from this paper and those from Iyengar (1988) were presented and compared on table 1. Exact solution from Iyengar (1988) was Where M is the buckling mode. In this case it is the first mode. That is M = 1. The values of H for different aspect ratios are shown on table 1 The average percentage difference between the solution from Iyenger (in this case exact) and the present study according to table 1 was 0.069%. It would also be noticed that the closeness of the two solutions improved as the aspect ratio increased from 0.1 to 1.0. This meant that the solution from this present study was a very close approximation of the exact solution. Hence, the Taylor-Mclaurin s shape function for SSSS plate was very close to the exact shape function. However, it was shown that the solution was upper bound solution (that is, the value of the solution is higher than the value of exact solution). Table 1: H values for different aspect ratios for SSSS Thin plate buckling Aspect ratio, P = a/b 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 H from IYEENGAR (1988) 102 27.04 13.2 8.41 6.25 5.138 4.531 4.203 4.045 4 H from present study 102.1 27.07 13.21 8.416 6.254 5.141 4.533 4.205 4.047 4.002 Percentage difference 0.098 0.093 0.085 0.076 0.068 0.061 0.056 0.052 0.051 0.05 6. References 1. Eccher, G., Rasmussen, K.J.R. and Zandonini, R. (2007), Geometric nonlinear isoparametric spline finite strip analysis of perforated thin-walled structures. School of Civil Engineering of Sydrey, Research Report No. R880 (February). 2. Ibearugbulem, O. M. (2011), Application of a direct variational principle in elastic stability of rectangular flat thin plates. Ph. D. thesis submitted to Postgraduate School, Federal University of Technology, Owerri, Nigeria. 3. Iyengar, N. G. (1988), Structural stability of columns and plates. Chichester : Ellis Horwood. 671
4. Kantorovich, L.V., and Krylov, V.I. (1954), Approximate Methods of Higher Analysis, John Wiley and Sons, New York. 5. Krylov, A.N.(1949), Collected Works, 3(1-2), Academy of Sciences of the USSR, Moscow in Russian. 6. Levy, M., (1899), Memoire sur la theorie des plaques elastiques planesjournal de Mathématiques Pures et Appliquées, 3, pp 219. 7. Nadai, A.(1925), Die Elastichen Platten, Springer-Verlag, Berlin. 8. Navier, C.L.M.N., (1823), Bulletin des Science de la Societe Philomarhique de Paris, 9. Timoshenko, S.P. and Woinowsky-Krieger, S.(1959), Theory of Plates and Shells, 2nd edn,mcgraw-hill, New York. 10. Ugural, A. C. (1999), Stresses in plates and shells (2nd Ed.). Singapore: McGraw-hill. 11. Ventsel, E. And Krauthammer, T. (2001), Thin Plates and Shells: Theory, Analysis and Applications. New York: Marcel Dekker. 12. Ye, Jianqiao. (1994), Large deflection of imperfect plates by iterative BE-FE method. Journal of Engineering Mechanics, 120(3), pp 431-445. 672