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Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges
Advances in Engineering Software 41 (2010) 557–560
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges Ömer Civalek a,*, Armag˘an Korkmaz b, Çig˘dem Demir a a b
Akdeniz University, Faculty of Engineering, Civil Engineering Department, Division of Mechanics, Antalya, Turkiye Süleyman Demirel University, Civil Engineering Department, Division of Structures, Isparta, Turkiye
a r t i c l e
i n f o
Article history: Received 12 June 2009 Received in revised form 17 July 2009 Accepted 10 August 2009 Available online 11 December 2009 Keywords: Discrete singular convolution Kirchhoff plate Buckling Rectangular plate Numerical method
a b s t r a c t Buckling analysis of rectangular plates subjected to various in-plane compressive loads using Kirchhoff plate theory is presented. The method of discrete singular convolution has adopted. Linearly varying, uniform and non-uniform distributed load conditions are considered on two-opposite edges for buckling. The results are obtained for different types of boundary conditions and aspect ratios. Comparisons are made with existing numerical and analytical solutions in the literature. The proposed method is suitable for the problem considered due to its simplicity, and potential for further development. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Discrete singular convolution (DSC) method is a new method that was introduced by Wei [1]. Several researchers have applied the DSC method to solve a variety of problems in different fields of science and engineering [2–5]. The pioneer work for the application of the DSC method to the general area of solid mechanics was carried out by Wei [6–8], Wei et al. [9,10], Zhao et al. [11], Lim et al. [12,13] and Civalek [14–17]. New developments, such as the new way to apply the boundary conditions [4] to increase the solution accuracy, have been made on the DSC approach to make the method more attractive for engineering practice. Details on the development of the DSC method and its applications to structural mechanics problems may be found in a recent paper by Wei [6]. The analysis of vibration and buckling of plates has been the subject of the research of structural and mechanical engineering [18–26]. Long list of references and detailed formulation on buckling and vibration of rectangular plates are given, for example, in Refs. [27–50]. A few studies concerning the buckling analysis of plates with various geometries have been carried out, namely by Wang et al. [35,36], Liew and Wang [37,41], Civalek [38], Peng et al. [39], Teo and Liew [40], Liew et al. [42], Setoodeh and Karami [43] and Karami and Malekzadeh [44,45]. In this study, buckling and free vibration analyses of rectangular plates subjected to various in-plane compressive loads are presented. To the author‘s
knowledge, this is the first instance in which the DSC method has been adopted for buckling analysis of rectangular plates. 2. Discrete singular convolution The discrete singular convolution (DSC) method is an efficient and useful approach for the numerical solutions of differential equations. This method introduced by Wei [1] in 1999. Like some other numerical methods, the DSC method discretizes the spatial derivatives and, therefore, reduces the given partial differential equations into a standard eigenvalue problem. The mathematical foundation of the DSC algorithm is the theory of distributions and wavelet analysis. Wei and his co-workers first applied the DSC algorithm to solve solid and fluid mechanics problem [3– 11]. Civalek [14–17] gives numerical solution of free vibration problem of rotating and laminated conical shells, plates on elastic foundation. These studies indicates that the DSC algorithm work very well for the vibration analysis of plates, especially for highfrequency analysis of rectangular plates. Recently, Lim et al. [12,13] presented the DSC–Ritz method for the free vibration analysis of Mindlin plates and thick shallow shells. Consider a distribution, T and g(t) as an element of the space of the test function. A singular convolution can be defined by [7]
FðtÞ ¼ ðT gÞðtÞ ¼
Z
1
Tðt xÞgðxÞdx;
ð1Þ
1
* Corresponding author. Tel.: +90 242 310 6319; fax: +90 242 310 6306. E-mail address: [email protected] (Ö. Civalek). 0965-9978/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2009.11.002
where T(t x) is a singular kernel. For example, singular kernels of delta type [6]
558
Ö. Civalek et al. / Advances in Engineering Software 41 (2010) 557–560
Nomenclature a, b D D, I E h i, j k M Mx, My
n Nx, Ny T w
lengths of rectangular plate flexural rigidity differential and unit matrix modulus of elasticity plate thickness indexes non-dimensional buckling factor computational bandwidth moment resultants
t d D
r k
TðxÞ ¼ dðnÞ ðxÞ ðn ¼ 0; 1; 2; . . .Þ:
ð2Þ
Kernel T(x) = d(x) is important for interpolation of surfaces and curves, and T(x) = d(n)(x) for n > 1 are essential for numerically solving differential equations. The Shannon’s kernel is regularized as [8]
"
dD;r ðx xk Þ ¼
order of the derivative normal forces per unit length of plate singular kernel transverse displacement Poisson’s ratio convolution kernel grid spacing convolution parameter aspect ratio
#
sin½ðp=DÞðx xk Þ ðx xk Þ2 ; exp ðp=DÞðx xk Þ 2r2
r > 0:
ð3Þ
M X
M X
ð4Þ
dD;r ðkDxÞU iþk;j þ 2k2
k¼M
ð2Þ
dD;r ðkDxÞU iþk;j
k¼M M X
ð2Þ
dD;r ðkDyÞU i;kþj þ k4
k¼M
¼N
M X
M X
ð4Þ
dD;r ðkDyÞU i;jþk
k¼M ð2Þ dD;r ðkDxÞU iþk;j ;
ð8Þ
k¼M
where N* is the non-dimensional buckling load given by where D is the grid spacing. Eq. (4) can also be used to provide discrete approximations to the singular convolution kernels of the delta type [10]
f ðnÞ ðxÞ
M X
N ¼
N0 y 1a : b D
ð9Þ
The moment resultants are given
dD ðx xk Þf ðxk Þ;
ð4Þ
k¼M
where dD(x xk) = Dda(x xk) and superscript (n) denotes the nthorder derivative, and 2M + 1 is the computational bandwidth which is centred around x and is usually smaller than the whole computational domain. In the DSC method, the function f(x) and its derivatives with respect to the x coordinate at a grid point xi are approximated by a linear sum of discrete values f(xk) in a narrow bandwidth [x xM, x + xM].
! @2w @2w Mx ¼ D þt 2 ; @x2 @y ! @2w @2w My ¼ D t 2 þ 2 : @x @y
ð10Þ ð11Þ
The boundary conditions are simply supported and clamped. These equations are given as follows [20,22]: Simply supported (SS) edge:
w ¼ 0 and M x ¼ 0 at x ¼ 0; a;
ð12aÞ
3. Governing equations
w ¼ 0 and M y ¼ 0 at y ¼ 0; b:
ð12bÞ
The governing differential equations for buckling of thin elastic plates are given as [27]
Clamped (C) edge:
! @ w @ w @ w @2w @2w @2w Nx 2 Ny 2 2Nxy D þ 2 þ ¼0 4 2 2 4 @x @x @y @y @x @y @x@y
w ¼ 0 and
4
4
4
w ¼ 0 and
@w ¼ 0 at x ¼ 0; a; @x @w ¼ 0 at y ¼ 0; b: @y
ð13aÞ ð13bÞ
ð5Þ where w is transverse displacement; Nx and Ny are normal forces per unit length of plate in the x and y directions, respectively; D is the flexural rigidity of the plate given by 3
Eh D¼ ; 12ð1 t2 Þ
ð6Þ
E is the modulus of elasticity; t is Poisson’s ratio; h is the plate thickness. Eq. (5) can be re-written in non-dimensional form as
! @ w @ w y @ 2 w 2 4@ w ¼ N D þ 2k þ k 1 a : 0 b @X 2 @X 4 @X 2 @Y 2 @Y 4 4
4
After imposing the related boundary conditions using the method of Wei’s et al. [9,10] and Zhao’s et al. [11,48,49] the governing equations of plates for buckling becomes
D4x Iy þ 2k2 D2x D2y þ k4 Ix D4y W ¼ kW:
ð14Þ
In this equation (Eq. (14)), k is the non-dimensional buckling factor. This is defined as 2
k ¼ N0
b ; D
ð15Þ
4
ð7Þ
After the applying the DSC algorithm, the discretized form of Eq. (7) can be given by
in which D is the flexural rigidity of the plate. In order to simplify the results, the boundary conditions for plates are denoted by letters S (simply supported) and C (clamped) as shown in Fig. 1. For example, SCSC denotes that the plate is simply supported at x = 0 and x = a, and clamped at y = 0 and y = b.
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Ö. Civalek et al. / Advances in Engineering Software 41 (2010) 557–560
76
-0.5N0 75.91
b
k
a -N0
Ref.27(38 terms) Ref.21 Present
CASE 1 (α=0.5)
75.5 9x9
b
-N0
11x11
-N0
13x13
15x15
17x17
N Fig. 2. Variation of buckling load versus number of grid of plates under unilateral load (a = 0) for a/b = 0.5.
a
CASE 2 (α=0) Table 1 Comparison of the non-dimensional critical buckling loads for SCSC rectangular plate under uniaxial uniform load. a/b
Wang et al. [21] N = 17
Timoshenko and Gere [30]
Leissa and Kang [27]
Present DSC results (N = 15)
0.5 0.7 0.9
75.910 69.095 77.545
75.9 69.1 77.3
75.910 69.095 77.545
75.913 69.096 77.545
b a -N0
-N0
CASE 3 (α=1) N0
N0
b a
Table 2 Non-dimensional critical buckling loads for SSSS rectangular plate under different uniaxial loads. a/b
a 0.8
1
2
0.4 0.6 0.75 1 1.5
131.452 82.703 70.268 65.236 70.275
149.623 96.131 83.004 77.081 82.708
287.205 238.071 237.952 252.863 237.859
-N0
-N0
CASE 4 (α=2) Fig. 1. Rectangular plate under different type uniaxial edge loading.
Table 3 Non-dimensional critical buckling loads for SCSC rectangular plate under different uniaxial loads (a/b = 0.7).
a
Sources Wang et al. [21]
Leissa and Kang [27]
Present
0 0.5 1 2
69.095 – 134.589 422.465
69.10 – 134.6 422.5
69.088 86.459 134.592 422.473
4. Numerical results As a convergence example and to verify the numerical solution formulated in the preceding sections, application has been made to a numerical example given by Leissa and Kang [27] for a = 0. The convergence rate of the proposed DSC solution is studied in Fig. 2. It is seen that a reasonable accurate results is obtained by the present method when N P 15. It was shown that [2–10] the parameter r is gives more accurate results for the interval 2.2 6 r 6 3.2 in applied mechanics. A comparison of the non-dimensional critical buckling loads for SCSC rectangular plate under uniaxial uniform load is listed in Table 1. The results are obtained for different aspect ratios. As can be seen, the present DSC results compare very well with the analytical solutions [27,30] and results given by Wang et al. [21]. Table 2 summarizes numerical results of non-dimensionalized buckling loads by DSC for rectangular plates with three different load conditions.
Buckling loads obtained for SCSC plates are presented in Table 3 together with the analytical solutions [27], differential quadrature [21]. The DSC results are generally in agreement with the results produced from the analytical [27] and the DQ results [21].
5. Conclusions Solutions of boundary value problems have always been considered as one of the most significant tasks in engineering and
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Ö. Civalek et al. / Advances in Engineering Software 41 (2010) 557–560
physical sciences. A variety of numerical methods are available today for engineering analysis. The discrete singular convolution (DSC) method is a relatively new numerical technique for the numerical solutions of differential equations. The primary objective of this study is to give a numerical solution of buckling analysis of rectangular plates subjected to various in-plane compressive loads. Numerical examples illustrating the accuracy and convergence of the DSC method for rectangular Kirchhoff plates are presented. The results obtained by DSC method were compared with those obtained by the other numerical methods. Acknowledgement The financial support of the Scientific Research Projects Unit of Akdeniz University is gratefully acknowledged. References [1] Wei GW. Discrete singular convolution for the solution of the Fokker–Planck equations. J Chem Phys 1999;110:8930–42. [2] Wei GW. Wavelets generated by using discrete singular convolution kernels. J Phys A Math Gen 2000;33:8577–96. [3] Wei GW, Kouri DJ, Hoffman DK. Wavelets and distributed approximating functionals. Comput Phys Commun 1998;112:1–6. [4] Zhao S, Wei GW, Xiang Y. DSC analysis of free-edged beams by an iteratively matched boundary method. J Sound Vib 2005;284:487–93. [5] Zhao S, Wei GW. Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher’s equation. SIAM J Sci Comput 2003;25:127–47. [6] Wei GW. A new algorithm for solving some mechanical problems. Comput Methods Appl Mech Eng 2001;190:2017–30. [7] Wei GW. Vibration analysis by discrete singular convolution. J Sound Vib 2001;244:535–53. [8] Wei GW. Discrete singular convolution for beam analysis. Eng Struct 2001;23:1045–53. [9] Wei GW, Zhao YB, Xiang Y. Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: theory and algorithm. Int J Numer Methods Eng 2002;55:913–46. [10] Wei GW, Zhao YB, Xiang Y. A novel approach for the analysis of high-frequency vibrations. J Sound Vib 2002;257(2):207–46. [11] Zhao YB, Wei GW, Xiang Y. Discrete singular convolution for the prediction of high frequency vibration of plates. Int J Solids Struct 2002;39:65–88. [12] Lim CW, Li ZR, Xiang Y, Wei GW, Wang CM. On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates. Adv Vib Eng 2005;4:221–48. [13] Lim CW, Li ZR, Wei GW. DSC-Ritz method for high-mode frequency analysis of thick shallow shells. Int J Numer Methods Eng 2005;62:205–32. [14] Civalek Ö. An efficient method for free vibration analysis of rotating truncated conical shells. Int J Pres Ves Pip 2006;83:1–12. [15] Civalek Ö, Demir Ç, Akgöz B. Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory. Int J Eng Appl Sci (IJEAS) 2009;1(2):47–56. [16] Civalek Ö. Free vibration analysis of composite conical shells using the discrete singular convolution algorithm. Steel Compos Struct 2006;6(4):353–66. [17] Civalek Ö. Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods. Appl Math Model 2007;31:606–24. [18] Timoshenko SP. The approximate solution of two-dimensional problems in elasticity. Philos Mag 1924;47:1095–104. [19] Benoy MB. An energy solution for the buckling of rectangular plates under non-uniform in-plane loading. Aeronaut J 1969;73:974–7. [20] Wang X, Wang X, Shi X. Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method. Int J Mech Sci 2007;49(4):447–53. [21] Wang X, Gan L, Wang Y. A differential quadrature analysis of vibration and buckling of an SS–C–SS–C rectangular plate loaded by linearly varying in-plane stresses. J Sound Vib 2006;298(1-2):420–31.
[22] Wang X, Wang X, Shi X. Differential quadrature buckling analyses of rectangular plates subjected to non-uniform distributed in-plane loadings. Thin-Walled Struct 2006;44(8):837–43. [23] Wang X. Differential quadrature for buckling analysis of laminated plates. Comput Struct 1995;57(4):715–9. [24] Bert CW, Wang X, Striz AG. Differential quadrature for static and free vibration analyses of anisotropic plates. Int J Solids Struct 1993;30(13):1737–44. [25] Bert CW, Devarakonda KK. Buckling of rectangular plates subjected to nonlinearly distributed in-plane loading. Int J Solids Struct 2003;40: 4097–106. [26] Devarakonda KK, Bert CW. Buckling of rectangular plate with nonlinearly distributed compressive loading on two opposite sides: comparative analysis and results. Mech Adv Mater Struct 2004;11(4):433–44. [27] Leissa AW, Kang JH. Exact solutions for vibration and buckling of an SS–C–SS–C rectangular plate loaded by linearly varying in-plane stresses. Int J Mech Sci 2002;44:1925–45. [28] Kang JH, Leissa AW. Vibration and buckling of an SS–F–SS–F rectangular plates loaded by in-plane moments. Int J Struct Stabil Dyn 2001;1:527–43. [29] Young WC, Budynas RG. Roark’s formulas for stress and strain. 7th ed. New York: McGraw-Hill; 2002. [30] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. New York: McGraw-Hill Book Company; 1963. [31] Leissa AW. Vibration of plates. NASA, SP-160; 1969. [32] Leissa AW. Recent research in plate vibrations: classical theory. Shock Vib Dig 1977;9(10):13–24. [33] Leissa AW. Plate vibration research: 1976–1980: classical theory. Shock Vib Dig 1981;13(9):11–2. [34] Leissa AW. Recent research in plate vibrations: 1981–1985: Part I. Classical theory. Shock Vib Dig 1987;19(2):11–8. [35] Wang CM, Xiang Y, Chakrabarty J. Elastic/plastic buckling of thick plates. Int J Solids Struct 2001;38(48-49):8617–40. [36] Wang CM, Liew KM, Xiang Y, Kitipornchai S. Buckling of rectangular Mindlin plates with internal line supports. Int J Solids Struct 1993;30(1):1–17. [37] Liew KM, Wang CM. pb-2 Rayleigh–Ritz method for general plate analysis. Eng Struct 1993;15(1):55–60. [38] Civalek Ö. Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Eng Struct 2004;26(2):171–86. [39] Peng LX, Liew KM, Kitipornchai S. Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method. J Sound Vib 2006;289(3):421–49. [40] Teo TM, Liew KM. A differential quadrature procedure for three-dimensional buckling analysis of rectangular plates. Int J Solids Struct 1999;36(8): 1149–68. [41] Liew KM, Wang CM. Elastic buckling of regular polygonal plates. Thin-Walled Struct 1995;21(2):163–73. [42] Liew KM, Kitipornchai S, Wang CM. Research developments in analyses of plates and shells. J Construct Steel Res 1993;26:231–48. [43] Setoodeh AR, Karami G. Static, free vibration and buckling analysis of anisotropic thick laminated composite plates on distributed and point elastic supports using a 3-D layer-wise FEM. Eng Struct 2004;26(2): 211–20. [44] Karami G, Malekzadeh P. Static and stability analyses of arbitrary straightsided quadrilateral thin plates by DQM. Int J Solids Struct 2002;39(19): 4927–5047. [45] Karami G, Malekzadeh P. Application of a new differential quadrature methodology for free vibration analysis of plates. Int J Numer Methods Eng 2003;56:847–68. [46] Wang Y, Wang X, Zhou Y. Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method. Int J Numer Methods Eng 2004;59:1207–26. [47] Wang X, Liu F, Wang XF, Gan LF. New approaches in application of differential quadrature method for fourth-order differential equations. Commun Numer Methods Eng 2005;21:61–71. [48] Zhao YB, Wei GW, Xiang Y. Plate vibration under irregular internal supports. Int J Solids Struct 2002;39:1361–83. [49] Zhao YB, Wei GW. DSC analysis of rectangular plates with non-uniform boundary conditions. J Sound Vib 2002;255(2):203–28. [50] Ng CHW, Zhao YB, Xiang Y, Wei GW. On the accuracy and stability of a variety of differential quadrature formulations for the vibration analysis of beams. Int J Eng Appl Sci (IJEAS) 2009;1(4):1–25.
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges Ömer Civalek a,*, Armag˘an Korkmaz b, Çig˘dem Demir a a b
Akdeniz University, Faculty of Engineering, Civil Engineering Department, Division of Mechanics, Antalya, Turkiye Süleyman Demirel University, Civil Engineering Department, Division of Structures, Isparta, Turkiye
a r t i c l e
i n f o
Article history: Received 12 June 2009 Received in revised form 17 July 2009 Accepted 10 August 2009 Available online 11 December 2009 Keywords: Discrete singular convolution Kirchhoff plate Buckling Rectangular plate Numerical method
a b s t r a c t Buckling analysis of rectangular plates subjected to various in-plane compressive loads using Kirchhoff plate theory is presented. The method of discrete singular convolution has adopted. Linearly varying, uniform and non-uniform distributed load conditions are considered on two-opposite edges for buckling. The results are obtained for different types of boundary conditions and aspect ratios. Comparisons are made with existing numerical and analytical solutions in the literature. The proposed method is suitable for the problem considered due to its simplicity, and potential for further development. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Discrete singular convolution (DSC) method is a new method that was introduced by Wei [1]. Several researchers have applied the DSC method to solve a variety of problems in different fields of science and engineering [2–5]. The pioneer work for the application of the DSC method to the general area of solid mechanics was carried out by Wei [6–8], Wei et al. [9,10], Zhao et al. [11], Lim et al. [12,13] and Civalek [14–17]. New developments, such as the new way to apply the boundary conditions [4] to increase the solution accuracy, have been made on the DSC approach to make the method more attractive for engineering practice. Details on the development of the DSC method and its applications to structural mechanics problems may be found in a recent paper by Wei [6]. The analysis of vibration and buckling of plates has been the subject of the research of structural and mechanical engineering [18–26]. Long list of references and detailed formulation on buckling and vibration of rectangular plates are given, for example, in Refs. [27–50]. A few studies concerning the buckling analysis of plates with various geometries have been carried out, namely by Wang et al. [35,36], Liew and Wang [37,41], Civalek [38], Peng et al. [39], Teo and Liew [40], Liew et al. [42], Setoodeh and Karami [43] and Karami and Malekzadeh [44,45]. In this study, buckling and free vibration analyses of rectangular plates subjected to various in-plane compressive loads are presented. To the author‘s
knowledge, this is the first instance in which the DSC method has been adopted for buckling analysis of rectangular plates. 2. Discrete singular convolution The discrete singular convolution (DSC) method is an efficient and useful approach for the numerical solutions of differential equations. This method introduced by Wei [1] in 1999. Like some other numerical methods, the DSC method discretizes the spatial derivatives and, therefore, reduces the given partial differential equations into a standard eigenvalue problem. The mathematical foundation of the DSC algorithm is the theory of distributions and wavelet analysis. Wei and his co-workers first applied the DSC algorithm to solve solid and fluid mechanics problem [3– 11]. Civalek [14–17] gives numerical solution of free vibration problem of rotating and laminated conical shells, plates on elastic foundation. These studies indicates that the DSC algorithm work very well for the vibration analysis of plates, especially for highfrequency analysis of rectangular plates. Recently, Lim et al. [12,13] presented the DSC–Ritz method for the free vibration analysis of Mindlin plates and thick shallow shells. Consider a distribution, T and g(t) as an element of the space of the test function. A singular convolution can be defined by [7]
FðtÞ ¼ ðT gÞðtÞ ¼
Z
1
Tðt xÞgðxÞdx;
ð1Þ
1
* Corresponding author. Tel.: +90 242 310 6319; fax: +90 242 310 6306. E-mail address: [email protected] (Ö. Civalek). 0965-9978/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2009.11.002
where T(t x) is a singular kernel. For example, singular kernels of delta type [6]
558
Ö. Civalek et al. / Advances in Engineering Software 41 (2010) 557–560
Nomenclature a, b D D, I E h i, j k M Mx, My
n Nx, Ny T w
lengths of rectangular plate flexural rigidity differential and unit matrix modulus of elasticity plate thickness indexes non-dimensional buckling factor computational bandwidth moment resultants
t d D
r k
TðxÞ ¼ dðnÞ ðxÞ ðn ¼ 0; 1; 2; . . .Þ:
ð2Þ
Kernel T(x) = d(x) is important for interpolation of surfaces and curves, and T(x) = d(n)(x) for n > 1 are essential for numerically solving differential equations. The Shannon’s kernel is regularized as [8]
"
dD;r ðx xk Þ ¼
order of the derivative normal forces per unit length of plate singular kernel transverse displacement Poisson’s ratio convolution kernel grid spacing convolution parameter aspect ratio
#
sin½ðp=DÞðx xk Þ ðx xk Þ2 ; exp ðp=DÞðx xk Þ 2r2
r > 0:
ð3Þ
M X
M X
ð4Þ
dD;r ðkDxÞU iþk;j þ 2k2
k¼M
ð2Þ
dD;r ðkDxÞU iþk;j
k¼M M X
ð2Þ
dD;r ðkDyÞU i;kþj þ k4
k¼M
¼N
M X
M X
ð4Þ
dD;r ðkDyÞU i;jþk
k¼M ð2Þ dD;r ðkDxÞU iþk;j ;
ð8Þ
k¼M
where N* is the non-dimensional buckling load given by where D is the grid spacing. Eq. (4) can also be used to provide discrete approximations to the singular convolution kernels of the delta type [10]
f ðnÞ ðxÞ
M X
N ¼
N0 y 1a : b D
ð9Þ
The moment resultants are given
dD ðx xk Þf ðxk Þ;
ð4Þ
k¼M
where dD(x xk) = Dda(x xk) and superscript (n) denotes the nthorder derivative, and 2M + 1 is the computational bandwidth which is centred around x and is usually smaller than the whole computational domain. In the DSC method, the function f(x) and its derivatives with respect to the x coordinate at a grid point xi are approximated by a linear sum of discrete values f(xk) in a narrow bandwidth [x xM, x + xM].
! @2w @2w Mx ¼ D þt 2 ; @x2 @y ! @2w @2w My ¼ D t 2 þ 2 : @x @y
ð10Þ ð11Þ
The boundary conditions are simply supported and clamped. These equations are given as follows [20,22]: Simply supported (SS) edge:
w ¼ 0 and M x ¼ 0 at x ¼ 0; a;
ð12aÞ
3. Governing equations
w ¼ 0 and M y ¼ 0 at y ¼ 0; b:
ð12bÞ
The governing differential equations for buckling of thin elastic plates are given as [27]
Clamped (C) edge:
! @ w @ w @ w @2w @2w @2w Nx 2 Ny 2 2Nxy D þ 2 þ ¼0 4 2 2 4 @x @x @y @y @x @y @x@y
w ¼ 0 and
4
4
4
w ¼ 0 and
@w ¼ 0 at x ¼ 0; a; @x @w ¼ 0 at y ¼ 0; b: @y
ð13aÞ ð13bÞ
ð5Þ where w is transverse displacement; Nx and Ny are normal forces per unit length of plate in the x and y directions, respectively; D is the flexural rigidity of the plate given by 3
Eh D¼ ; 12ð1 t2 Þ
ð6Þ
E is the modulus of elasticity; t is Poisson’s ratio; h is the plate thickness. Eq. (5) can be re-written in non-dimensional form as
! @ w @ w y @ 2 w 2 4@ w ¼ N D þ 2k þ k 1 a : 0 b @X 2 @X 4 @X 2 @Y 2 @Y 4 4
4
After imposing the related boundary conditions using the method of Wei’s et al. [9,10] and Zhao’s et al. [11,48,49] the governing equations of plates for buckling becomes
D4x Iy þ 2k2 D2x D2y þ k4 Ix D4y W ¼ kW:
ð14Þ
In this equation (Eq. (14)), k is the non-dimensional buckling factor. This is defined as 2
k ¼ N0
b ; D
ð15Þ
4
ð7Þ
After the applying the DSC algorithm, the discretized form of Eq. (7) can be given by
in which D is the flexural rigidity of the plate. In order to simplify the results, the boundary conditions for plates are denoted by letters S (simply supported) and C (clamped) as shown in Fig. 1. For example, SCSC denotes that the plate is simply supported at x = 0 and x = a, and clamped at y = 0 and y = b.
559
Ö. Civalek et al. / Advances in Engineering Software 41 (2010) 557–560
76
-0.5N0 75.91
b
k
a -N0
Ref.27(38 terms) Ref.21 Present
CASE 1 (α=0.5)
75.5 9x9
b
-N0
11x11
-N0
13x13
15x15
17x17
N Fig. 2. Variation of buckling load versus number of grid of plates under unilateral load (a = 0) for a/b = 0.5.
a
CASE 2 (α=0) Table 1 Comparison of the non-dimensional critical buckling loads for SCSC rectangular plate under uniaxial uniform load. a/b
Wang et al. [21] N = 17
Timoshenko and Gere [30]
Leissa and Kang [27]
Present DSC results (N = 15)
0.5 0.7 0.9
75.910 69.095 77.545
75.9 69.1 77.3
75.910 69.095 77.545
75.913 69.096 77.545
b a -N0
-N0
CASE 3 (α=1) N0
N0
b a
Table 2 Non-dimensional critical buckling loads for SSSS rectangular plate under different uniaxial loads. a/b
a 0.8
1
2
0.4 0.6 0.75 1 1.5
131.452 82.703 70.268 65.236 70.275
149.623 96.131 83.004 77.081 82.708
287.205 238.071 237.952 252.863 237.859
-N0
-N0
CASE 4 (α=2) Fig. 1. Rectangular plate under different type uniaxial edge loading.
Table 3 Non-dimensional critical buckling loads for SCSC rectangular plate under different uniaxial loads (a/b = 0.7).
a
Sources Wang et al. [21]
Leissa and Kang [27]
Present
0 0.5 1 2
69.095 – 134.589 422.465
69.10 – 134.6 422.5
69.088 86.459 134.592 422.473
4. Numerical results As a convergence example and to verify the numerical solution formulated in the preceding sections, application has been made to a numerical example given by Leissa and Kang [27] for a = 0. The convergence rate of the proposed DSC solution is studied in Fig. 2. It is seen that a reasonable accurate results is obtained by the present method when N P 15. It was shown that [2–10] the parameter r is gives more accurate results for the interval 2.2 6 r 6 3.2 in applied mechanics. A comparison of the non-dimensional critical buckling loads for SCSC rectangular plate under uniaxial uniform load is listed in Table 1. The results are obtained for different aspect ratios. As can be seen, the present DSC results compare very well with the analytical solutions [27,30] and results given by Wang et al. [21]. Table 2 summarizes numerical results of non-dimensionalized buckling loads by DSC for rectangular plates with three different load conditions.
Buckling loads obtained for SCSC plates are presented in Table 3 together with the analytical solutions [27], differential quadrature [21]. The DSC results are generally in agreement with the results produced from the analytical [27] and the DQ results [21].
5. Conclusions Solutions of boundary value problems have always been considered as one of the most significant tasks in engineering and
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