- Email: [email protected]
Stability of Euler's method for evaluating large deformation of shear deformable plates by dual reciprocity boundary element method
ARTICLE IN PRESS Engineering Analysis with Boundary Elements 34 (2010) 819–823
Contents lists available at ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Stability of Euler’s method for evaluating large deformation of shear deformable plates by dual reciprocity boundary element method J. Purbolaksono a,n, M.H. Aliabadi b a b
Department of Mechanical Engineering, Universiti Tenaga Nasional, Km 7 Jalan, Kajang-Puchong, Kajang 43009, Selangor, Malaysia Department of Aeronautics, Faculty of Engineering, Imperial College, Prince Consort Road, London SW7-2BY, United Kingdom
a r t i c l e in fo
abstract
Article history: Received 29 January 2010 Accepted 30 March 2010
In this paper the dual reciprocity boundary element method (DRBEM) is employed to evaluate large deformations of shear deformable plates. Incremental approaches utilizing the Euler method and the total incremental method are used for treating the nonlinear problems. A simple numerical algorithm for analyzing the problems is presented. Two numerical examples of square plates with different boundary conditions subjected to transverse loadings are presented to demonstrate the accuracy of the formulation. The Euler method seems to be a more stable method for treating nonlinear problems of the shear deformable plate if the dual reciprocity method is employed to evaluate the domain integrals that appear in the formulations. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Boundary element method Dual reciprocity Shear deformable plates Large deformation Incremental approach
1. Introduction Understanding on geometrically nonlinear behaviour in solid mechanics is an essential requirement to evaluate problems in engineering practice. Bending analyses of rectangular plates with large deflection by solving von Karman’s equation in term of trigonometric series were studied by Levy [1,2]. An approximate analysis of large deflections for plates was introduced by Berger [3], which has since then become known as the Berger equation. The application of boundary element method (BEM) to geometrically nonlinear problems is relatively new. Kamiya and Sawaki [4] investigated large deflection of elastic plates based on the Berger equation. Tanaka [5] presented coupled boundary and inner domain integral equations in terms of stress and displacement functions based on von Karman’s equation. Ye and Lin [6] analyzed finite deflection of thin plates by the boundary element method. Lei et al. [7] proposed an integral equation formulation for geometrically nonlinear analysis of shear deformable type plates based on general nonlinear differential equations of finite deflection of plates. He and Qin [8] derived an exact boundary equation for the analysis of the nonlinear Reissner plate based on a variational principle. Recently, Purbolaksono and Aliabadi [9] presented a derivation and implementation of domain-boundary integral equation for large deformation of shear deformable plates. The domain integrals that appear in the formulation were treated in two different ways, i.e. evaluating hypersingular
n
Corresponding author. Tel.: + 60 3 89212213. E-mail address: [email protected] (J. Purbolaksono).
0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2010.03.011
integrals and utilizing an approximation function. It was reported that domain cells were used in the analyses. Next, Wen et al. [10] presented geometrically nonlinear analysis of Reissner plates using the boundary element method. They transformed domain integrals resulting from nonlinear terms to boundary integrals by the dual reciprocity method [11]. The total incremental method was adopted for introducing the incremental approach to treat the nonlinear problem. This paper presents the dual reciprocity boundary element method (DRBEM) for evaluating large deformations of shear deformable plates. Euler and the total incremental methods are used for treating the nonlinear problems. In order to show the stability of the approaches, comparison of the results obtained using both methods is made. A numerical algorithm for analyzing the problems is presented. The dual reciprocity method [11] is utilized to evaluate the domain integrals that appear in the formulations. Two numerical examples of square plates subjected to transverse loadings with different boundary conditions are presented to demonstrate the accuracy of the formulation.
2. Governing equations The governing equations for geometrically nonlinear Reissner plates (see Fig. 1) can be written in a compact form using indicial notation as follows: Mab, b þ Qa ¼ 0
ð1Þ
Qa, a þðNab w3, b Þ, a þ q ¼ 0
ð2Þ
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3. Boundary integral equations
X2
X3
1
Boundary integral equations for the large deformation analysis are written as follows: Z Z ðxÞdG Cij wi ðx0 Þ þ Pij ðx0 ,xÞwj ðxÞdG ¼ Wij ðx0 ,xÞplinear j G G Z Z þ Wij ðx0 ,XÞqðXÞdOðXÞ þ Wi3 ðx0 ,XÞðNab w3, w Þ, a ðXÞdOðXÞ ð10Þ O
for plate bending, and the kernel solutions Pij and Wij can be found in Ref. [12]. Cij ¼ dij/2 for x0 on a smooth boundary. Z Z Cya ðx0 Þua ðx0 Þ þ Tya ðx0 ,xÞuðxÞdG ¼ Uya ðx0 ,xÞt linear ðxÞdG G G Z 0 nonlinear þ Uya ðx ,XÞNag, g ðXÞdOðXÞ ð11Þ
X2
O
for two-dimensional plane stress. The fundamental solutions Uya and Tya can be found in Ref. [12]. Cya ¼ dya/2 for x0 on a smooth boundary. Eq. (10) can be written in an alternative form as Z Z ðxÞdG Cij wi ðx0 Þ þ Pij ðx0 ,xÞwj ðxÞdG ¼ Wij ðx0 ,xÞplinear j G G Z Z þ Wij ðx0 ,XÞqðXÞdOðXÞ þ Wi3 ðx0 ,XÞðNab, a w3, w
X3
1
O
O
O
þ Nab w3, wb ÞðXÞdOðXÞ
Fig. 1. Stress resultant equilibrium in geometrically nonlinear plate elements.
Nab, b ¼ 0
ð3Þ
where
To calculate the nonlinear terms, two additional integral equations are required, i.e. for the deflection w3 and the in-plane at domain points as follows: stress resultants Nalinear b Z Z ðxÞdG wi ðX 0 Þ þ Pij ðX 0 ,xÞwj ðxÞdG ¼ Wij ðX 0 ,xÞplinear j G G Z Z þ Wij ðX 0 ,XÞqðXÞdOðXÞ þ Wi3 ðx0 ,XÞðNab, a w3, w O
1n 2n Mab ¼ D wa, b þwb, a þ wg, g dab 2 1n
ð4Þ
Qa ¼ Cðwa þ w3, a Þ
ð5Þ
Nab ¼ Nalinear þ Nanonlinear b b
ð6Þ
ð12Þ
O
þ Nab w3, wb ÞðXÞdOðXÞ
ð13Þ
and
1n 2n B ua, b þ ub, a þ ¼ ug, g dab Nalinear b 2 1n
Nanonlinear ¼ b
1n 2n B w3, b w3, a þ w3, g w3, g dab 2 1n
G
if a ¼ b if a a b
O
ð14Þ ð7Þ
ð8Þ
Here B ( ¼Eh/(1 v2)) is known as the membrane stiffness, D ( ¼Eh3/[12(1 v2)]) is the bending stiffness of the plate, q l2/2 the shear stiffness, E the transverse load, C ( ¼D(1pv) ffiffiffiffiffiffi the modulus of elasticity, l ð ¼ 10=hÞ the shear factor, h the thickness of the plate and n the Poisson’s ratio. Nab are stress resultants for two-dimensional plane stress elasticity, Qa and Mab are stress resultants in plate bending problems, ua and w3 are displacements in x1, x2 (in-plane) and x3 (out of plane), wa are rotations in xa directions and dab is the Kronecker delta function, which has the property 1 dab f ¼ ¼0
Z Z 0 linear Nalinear UD ab ðX 0 ,xÞtD ðxÞdG TD ab ðX 0 ,xÞuD ðxÞdG b ðX Þ ¼ Z G ZG nonlinear nonlinear ng ðxÞ UD ab ðX 0 ,xÞNag ðxÞdG þ UD ab ðX 0 ,XÞNag ðXÞdOðXÞ ,g
where the fundamental solutions UD ab and TD ab can be found in Ref. [12]. The domain integrals that appear in Eqs. (10)–(14) are evaluated by using the dual reciprocity technique as described by Wen et al. [11]. The particular solutions for plate bending and two-dimensional plane stress can also be found in Ref. [11]. After approximation of the integrals, the problem can be written as a nonlinear system Fðw3 ,qÞ ¼ 0
ð15Þ
where w3 describes the deflection (out of plane displacement) and q is the loading applied to the plate.
4. Evaluation of derivative terms
ð9Þ
Indicial notation is used throughout this paper. Greek indices will vary from 1 to 2 and Roman indices from 1 to 3.
Derivatives of deflection w3,g on the boundary and in the domain can bepapproximated by considering a radial basis ffiffiffiffiffiffiffiffiffiffiffiffiffiffi function f ðrÞ ¼ ð c2 þr 2 Þ and the constant c is set to 2. The
ARTICLE IN PRESS J. Purbolaksono, M.H. Aliabadi / Engineering Analysis with Boundary Elements 34 (2010) 819–823
expression may be written as M þN X
w3 ðx1 ,x2 Þ ¼
f ðrÞm Cm
ð16Þ
m¼1
where N and M are the number of selected points x1 and x2 on the boundary and in the domain, respectively, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m 2 ð17Þ r ¼ ðx1 xm 1 Þ þðx2 x2 Þ The Cm are coefficients determined by values at the selected points N and M:
C ¼ F 1 fw3 g
ð18Þ
821
is sufficiently small so that convergence occurs. In the present work, the transverse loading is also provided in a small loading step q3 . Thus, the previous calculated incremental quantities are simply summed up in each loading step. The relaxation procedure shown in Eq. (20) is employed in this method. Once the matrices [Hp], [Gp], [Bp], [Hm], [Gm] and [Bm] in Eqs. (24) and (25) have been computed and the LU-decomposition method is used to solve the system of equations, the matrix coefficients can be stored and used in each increment without any further change. 5.2. Total incremental method
The derivatives may be expressed by w3, g ðx1 ,x2 Þ ¼ f ðrÞ0 g F 1 fw3 g
ð19Þ
nonlinear In a similar way, the nonlinear terms Nag that appear in ,g Eqs. (11) and (14) can also be evaluated. By implementing this approach, there is no need to evaluate the derivatives of the transverse displacement w3,g through the integral equations. The integral equations usually have complicated mathematical terms and may have higher order of singularities. The relaxation procedure is used to improve numerical results. As the nonlinear terms are calculated in each step (k) of increments, the deflection w3 can be modified as follows:
wk3 þ 1 ¼
wk3 þ 1 þ wk3 2
ð20Þ
5. Incremental approaches In this work, two incremental approaches are presented for treating the nonlinear problems. 5.1. Euler’s method In this method, the total load can3 be presented as a cumulative nonlinear are expressed as the of incremental loads. The terms Nab first derivative of Nab as follows: 3 nonlinearðkÞ 3k 3k ð21Þ ¼ B wk3,1 w3,1 þ uwk3,2 w3,2 N11 3 nonlinearðkÞ
N22
3k 3k ¼ B wk3,2 w3,2 þ uwk3,1 w3,1
3 nonlinearðkÞ
N12
¼
ð22Þ
1u 3k 3k B wk3,1 w3,2 þ uwk3,2 w3,1 2
ð23Þ
Hence the system of algebraic equations can be written as follows: ( ) 3k 3 3 kþ1 3 kþ1 p p p k 3k k ½H w þ½G p ¼ ½B Nab w3, a þ Nab w3, a þq
This method treats the transverse load to be divided into small load steps. The boundary integral equations are then transformed to a system of algebraic equations as follows: n o 3 ð27Þ ½Hp fwk þ 1 g þ ½Gp fpk þ 1 g ¼ ½Bp ðNab w3, b Þk, a þðk þ1Þq g ½Hm fuk þ 1 g þ ½Gm ft k þ 1 g ¼ ½Bm fNanonlinearðkÞ b, b
ð28Þ
The LU-decomposition method is also used to solve the system of equations. In order to improve numerical results, the relaxation procedure is also employed.
6. Numerical algorithms A simple numerical algorithm is used to analyze large deformation problems, in which no iterations are required. The algorithm can be summarized as follows: Step 1: Introducing a small loading step q10 with the first step ¼0 and w,a ¼0. k¼1, the final step kfinal and initial values Nalinear b Step 2: Compute the coefficient matrices related to fundamental solutions. They can be stored in the core and used in each increment without any change. Step 3: If ka1and Euler’s method is used, q0k þ 1 ¼ q10 , else if ka1 and the total incremental method is used, qk0 þ 1 ¼ qk0 þ q10 . Solve the linear system equation of the boundary integral equations to obtain boundary values. Then calculate the in-plane and derivative of deflection w,a in the stress resultants Nalinear b domain. Step 4: Apply the relaxation procedure given in Eq. (20). nonlinear ðkÞ Þ and Then calculate the nonlinear terms ðNag ,g [(Nab,aw3,w +Nabw3,wb)](k) using the approximation function as described in Eqs. (16)–(19). The nonlinear terms will be used for the evaluation in the next step k+1. Step 5: Employ the incremental approach in Section 5. Print results in each step. If step k is final step kfinal, go to Step 6, otherwise let step k¼ k+1, go to Step 3. Step 6: The calculation process terminates.
,a
ð24Þ nonlinearðkÞ 3 3k þ 1 3 kþ1 þ ½Gm t ¼ ½Bm Nab, b ½Hm u wk3,þa 1
7. Numerical examples ð25Þ
wk3, a þ w3 k3, a ,
where ¼ k denotes the incremental step and the superscript (:3 ) denotes the incremental value. The cumulative stress resultants can be obtained by
Two numerical examples of square plates with different boundary conditions (Fig. 2) subjected to transverse loadings are
q
q
3k
Nakbþ 1 ¼ Nakb þ Nab
ð26Þ
According to the continuous method as described by Ortega and Rheinboldt [13], use of the solution xk of the kth problem as a starting approximation to solve the (k+ 1)th problem will be a sufficiently good approximation to xk + 1 if the division of interval
Fig. 2. Plates with clamped (left) and simply supported (right) boundary conditions.
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presented to demonstrate the accuracy of the formulation. It is assumed that that the edges of the plate are fixed for clamped condition. For simply supported condition, the plate is free to rotate and the in-plane displacement and deflection of the edge are zero. The dual reciprocity model is shown in Fig. 3. BEM meshes with 25 quadratic elements (50 boundary points) and 25 domain points are used. The incremental approaches utilizing the Euler and total incremental methods are used. Comparisons are made with the analytical results [1,2] and published results [9]. In the following examples, the normalized parameters are defined as follows: Q¼
qa4 Eh4
ð29Þ
Z¼
wmax 3 h
ð30Þ
where a is the width of the square plate. Wen et al. [10] also presented similar numerical examples utilizing the total incremental method, but the results are generated only from relatively small total incremental loads. Here normalized deflections for simply supported and clamped plates are presented in Figs. 4 and 5, respectively. The results obtained using both incremental approaches are in good agreements with the analytical results [1,2]. It can however be seen from Figs. 4 and 5 that the analyses obtained using Euler’s method may go further than those by the total incremental method as the transverse loadings q increase. The total increment method seems to have problems of convergence or stability. Since
a
Fig. 5. Normalized deflections for a clamped square plate.
an incremental approach is used to force nonlinear problems to be linear, increasing transverse loadings in the total incremental method would likely lead to a tendency of divergence when solving Eqs. (27) and (28). The nonlinear terms evaluated based on the total incremental values will pose accumulated divergences as the total loads increase. This is not the case in the Euler method, which uses a small incremental load, leading to less instability. Combination of DRBEM and Euler’s method does not severely weaken the performance of the procedure when computing the stress resultants. However, if domain cells are used in the analysis, the total increment method is slightly more stable for treating nonlinear problems than the Euler method as reported in Ref. [9]. Since the coefficient matrices related to fundamental solutions can be stored in the core and used in each increment without any change, computational efficiency of the Euler and total incremental methods would be the same. However, in terms of stability, this work shows that the Euler method seems more stable for treating nonlinear problems of the shear deformable plate if the dual reciprocity method is employed to evaluate the domain integrals that appear in the formulations.
Fig. 3. Dual reciprocity model for a square plate.
8. Conclusions The dual reciprocity boundary element method was employed to evaluate large deformations of shear deformable plates. Incremental approaches utilizing the Euler method and the total incremental method were used to treat the nonlinear problems. The numerical results obtained by both approaches are in good agreement with the analytical and published results. In the large deformation analyses by DRBEM, the Euler method seems more stable for treating the nonlinear problems of shear deformable plates compared with the total incremental method. References
Fig. 4. Normalized deflections for a simply supported square plate.
[1] Levy S. Bending of rectangular plate with large deflections. NACA TN-846 1942. [2] Levy S. Square plate with clamped edges under normal pressure producing large deflections. NACA TN-847 1942. [3] Berger HM. A new approach to the analysis of large deflections of plates. J Appl Mech 1955;22:465–72. [4] Kamiya N, Sawaki Y. An integral equation approach to finite deflection of elastic plates. Int J Non-linear Mech 1982;17(3):187–94.
ARTICLE IN PRESS J. Purbolaksono, M.H. Aliabadi / Engineering Analysis with Boundary Elements 34 (2010) 819–823
[5] Tanaka M. Large deflection analysis of thin elastic plates. In: Development in Boundary Element Methods, vol. 3. Elsevier Applied Science Publishers; 1984. p. 115–136. [6] Ye TQ, Lin YJ. Finite deflection analysis of elastic plate by the boundary element method. Appl Math Model 1985;9:183–8. [7] Lei XY, Huang MK, Wang XX. Geometrically nonlinear analysis of a Reissner type by the boundary element method. Comput Struct 1990;37(6):911–6. [8] He XQ, Qin QH. Nonlinear analysis of Reissner’s plate by the variational approaches and boundary element methods. Appl Math Model 1993;17: 149–155.
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[9] Purbolaksono J, Aliabadi MH. Large deformation of shear-deformable plates by the boundary element method. J Eng Math 2005;51:211–30. [10] Wen PH, Aliabadi MH, Young A. Large deflection analysis of Reissner plate by boundary element method. Comput Struct 2005;83:870–9. [11] Wen PH, Aliabadi MH, Young A. Application of dual reciprocity method to plates and shells. Eng Anal Bound Elem 2000;24:583–90. [12] Aliabadi MH. The boundary element method. Application in solids and structures, vol. 2. Chicester; New York: Wiley; 2002. [13] Ortega JM, Rheinboldt WC. Iterative solution of nonlinear equations in several variables. Philadelphia: SIAM; 2000.
Contents lists available at ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Stability of Euler’s method for evaluating large deformation of shear deformable plates by dual reciprocity boundary element method J. Purbolaksono a,n, M.H. Aliabadi b a b
Department of Mechanical Engineering, Universiti Tenaga Nasional, Km 7 Jalan, Kajang-Puchong, Kajang 43009, Selangor, Malaysia Department of Aeronautics, Faculty of Engineering, Imperial College, Prince Consort Road, London SW7-2BY, United Kingdom
a r t i c l e in fo
abstract
Article history: Received 29 January 2010 Accepted 30 March 2010
In this paper the dual reciprocity boundary element method (DRBEM) is employed to evaluate large deformations of shear deformable plates. Incremental approaches utilizing the Euler method and the total incremental method are used for treating the nonlinear problems. A simple numerical algorithm for analyzing the problems is presented. Two numerical examples of square plates with different boundary conditions subjected to transverse loadings are presented to demonstrate the accuracy of the formulation. The Euler method seems to be a more stable method for treating nonlinear problems of the shear deformable plate if the dual reciprocity method is employed to evaluate the domain integrals that appear in the formulations. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Boundary element method Dual reciprocity Shear deformable plates Large deformation Incremental approach
1. Introduction Understanding on geometrically nonlinear behaviour in solid mechanics is an essential requirement to evaluate problems in engineering practice. Bending analyses of rectangular plates with large deflection by solving von Karman’s equation in term of trigonometric series were studied by Levy [1,2]. An approximate analysis of large deflections for plates was introduced by Berger [3], which has since then become known as the Berger equation. The application of boundary element method (BEM) to geometrically nonlinear problems is relatively new. Kamiya and Sawaki [4] investigated large deflection of elastic plates based on the Berger equation. Tanaka [5] presented coupled boundary and inner domain integral equations in terms of stress and displacement functions based on von Karman’s equation. Ye and Lin [6] analyzed finite deflection of thin plates by the boundary element method. Lei et al. [7] proposed an integral equation formulation for geometrically nonlinear analysis of shear deformable type plates based on general nonlinear differential equations of finite deflection of plates. He and Qin [8] derived an exact boundary equation for the analysis of the nonlinear Reissner plate based on a variational principle. Recently, Purbolaksono and Aliabadi [9] presented a derivation and implementation of domain-boundary integral equation for large deformation of shear deformable plates. The domain integrals that appear in the formulation were treated in two different ways, i.e. evaluating hypersingular
n
Corresponding author. Tel.: + 60 3 89212213. E-mail address: [email protected] (J. Purbolaksono).
0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2010.03.011
integrals and utilizing an approximation function. It was reported that domain cells were used in the analyses. Next, Wen et al. [10] presented geometrically nonlinear analysis of Reissner plates using the boundary element method. They transformed domain integrals resulting from nonlinear terms to boundary integrals by the dual reciprocity method [11]. The total incremental method was adopted for introducing the incremental approach to treat the nonlinear problem. This paper presents the dual reciprocity boundary element method (DRBEM) for evaluating large deformations of shear deformable plates. Euler and the total incremental methods are used for treating the nonlinear problems. In order to show the stability of the approaches, comparison of the results obtained using both methods is made. A numerical algorithm for analyzing the problems is presented. The dual reciprocity method [11] is utilized to evaluate the domain integrals that appear in the formulations. Two numerical examples of square plates subjected to transverse loadings with different boundary conditions are presented to demonstrate the accuracy of the formulation.
2. Governing equations The governing equations for geometrically nonlinear Reissner plates (see Fig. 1) can be written in a compact form using indicial notation as follows: Mab, b þ Qa ¼ 0
ð1Þ
Qa, a þðNab w3, b Þ, a þ q ¼ 0
ð2Þ
ARTICLE IN PRESS 820
J. Purbolaksono, M.H. Aliabadi / Engineering Analysis with Boundary Elements 34 (2010) 819–823
3. Boundary integral equations
X2
X3
1
Boundary integral equations for the large deformation analysis are written as follows: Z Z ðxÞdG Cij wi ðx0 Þ þ Pij ðx0 ,xÞwj ðxÞdG ¼ Wij ðx0 ,xÞplinear j G G Z Z þ Wij ðx0 ,XÞqðXÞdOðXÞ þ Wi3 ðx0 ,XÞðNab w3, w Þ, a ðXÞdOðXÞ ð10Þ O
for plate bending, and the kernel solutions Pij and Wij can be found in Ref. [12]. Cij ¼ dij/2 for x0 on a smooth boundary. Z Z Cya ðx0 Þua ðx0 Þ þ Tya ðx0 ,xÞuðxÞdG ¼ Uya ðx0 ,xÞt linear ðxÞdG G G Z 0 nonlinear þ Uya ðx ,XÞNag, g ðXÞdOðXÞ ð11Þ
X2
O
for two-dimensional plane stress. The fundamental solutions Uya and Tya can be found in Ref. [12]. Cya ¼ dya/2 for x0 on a smooth boundary. Eq. (10) can be written in an alternative form as Z Z ðxÞdG Cij wi ðx0 Þ þ Pij ðx0 ,xÞwj ðxÞdG ¼ Wij ðx0 ,xÞplinear j G G Z Z þ Wij ðx0 ,XÞqðXÞdOðXÞ þ Wi3 ðx0 ,XÞðNab, a w3, w
X3
1
O
O
O
þ Nab w3, wb ÞðXÞdOðXÞ
Fig. 1. Stress resultant equilibrium in geometrically nonlinear plate elements.
Nab, b ¼ 0
ð3Þ
where
To calculate the nonlinear terms, two additional integral equations are required, i.e. for the deflection w3 and the in-plane at domain points as follows: stress resultants Nalinear b Z Z ðxÞdG wi ðX 0 Þ þ Pij ðX 0 ,xÞwj ðxÞdG ¼ Wij ðX 0 ,xÞplinear j G G Z Z þ Wij ðX 0 ,XÞqðXÞdOðXÞ þ Wi3 ðx0 ,XÞðNab, a w3, w O
1n 2n Mab ¼ D wa, b þwb, a þ wg, g dab 2 1n
ð4Þ
Qa ¼ Cðwa þ w3, a Þ
ð5Þ
Nab ¼ Nalinear þ Nanonlinear b b
ð6Þ
ð12Þ
O
þ Nab w3, wb ÞðXÞdOðXÞ
ð13Þ
and
1n 2n B ua, b þ ub, a þ ¼ ug, g dab Nalinear b 2 1n
Nanonlinear ¼ b
1n 2n B w3, b w3, a þ w3, g w3, g dab 2 1n
G
if a ¼ b if a a b
O
ð14Þ ð7Þ
ð8Þ
Here B ( ¼Eh/(1 v2)) is known as the membrane stiffness, D ( ¼Eh3/[12(1 v2)]) is the bending stiffness of the plate, q l2/2 the shear stiffness, E the transverse load, C ( ¼D(1pv) ffiffiffiffiffiffi the modulus of elasticity, l ð ¼ 10=hÞ the shear factor, h the thickness of the plate and n the Poisson’s ratio. Nab are stress resultants for two-dimensional plane stress elasticity, Qa and Mab are stress resultants in plate bending problems, ua and w3 are displacements in x1, x2 (in-plane) and x3 (out of plane), wa are rotations in xa directions and dab is the Kronecker delta function, which has the property 1 dab f ¼ ¼0
Z Z 0 linear Nalinear UD ab ðX 0 ,xÞtD ðxÞdG TD ab ðX 0 ,xÞuD ðxÞdG b ðX Þ ¼ Z G ZG nonlinear nonlinear ng ðxÞ UD ab ðX 0 ,xÞNag ðxÞdG þ UD ab ðX 0 ,XÞNag ðXÞdOðXÞ ,g
where the fundamental solutions UD ab and TD ab can be found in Ref. [12]. The domain integrals that appear in Eqs. (10)–(14) are evaluated by using the dual reciprocity technique as described by Wen et al. [11]. The particular solutions for plate bending and two-dimensional plane stress can also be found in Ref. [11]. After approximation of the integrals, the problem can be written as a nonlinear system Fðw3 ,qÞ ¼ 0
ð15Þ
where w3 describes the deflection (out of plane displacement) and q is the loading applied to the plate.
4. Evaluation of derivative terms
ð9Þ
Indicial notation is used throughout this paper. Greek indices will vary from 1 to 2 and Roman indices from 1 to 3.
Derivatives of deflection w3,g on the boundary and in the domain can bepapproximated by considering a radial basis ffiffiffiffiffiffiffiffiffiffiffiffiffiffi function f ðrÞ ¼ ð c2 þr 2 Þ and the constant c is set to 2. The
ARTICLE IN PRESS J. Purbolaksono, M.H. Aliabadi / Engineering Analysis with Boundary Elements 34 (2010) 819–823
expression may be written as M þN X
w3 ðx1 ,x2 Þ ¼
f ðrÞm Cm
ð16Þ
m¼1
where N and M are the number of selected points x1 and x2 on the boundary and in the domain, respectively, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m 2 ð17Þ r ¼ ðx1 xm 1 Þ þðx2 x2 Þ The Cm are coefficients determined by values at the selected points N and M:
C ¼ F 1 fw3 g
ð18Þ
821
is sufficiently small so that convergence occurs. In the present work, the transverse loading is also provided in a small loading step q3 . Thus, the previous calculated incremental quantities are simply summed up in each loading step. The relaxation procedure shown in Eq. (20) is employed in this method. Once the matrices [Hp], [Gp], [Bp], [Hm], [Gm] and [Bm] in Eqs. (24) and (25) have been computed and the LU-decomposition method is used to solve the system of equations, the matrix coefficients can be stored and used in each increment without any further change. 5.2. Total incremental method
The derivatives may be expressed by w3, g ðx1 ,x2 Þ ¼ f ðrÞ0 g F 1 fw3 g
ð19Þ
nonlinear In a similar way, the nonlinear terms Nag that appear in ,g Eqs. (11) and (14) can also be evaluated. By implementing this approach, there is no need to evaluate the derivatives of the transverse displacement w3,g through the integral equations. The integral equations usually have complicated mathematical terms and may have higher order of singularities. The relaxation procedure is used to improve numerical results. As the nonlinear terms are calculated in each step (k) of increments, the deflection w3 can be modified as follows:
wk3 þ 1 ¼
wk3 þ 1 þ wk3 2
ð20Þ
5. Incremental approaches In this work, two incremental approaches are presented for treating the nonlinear problems. 5.1. Euler’s method In this method, the total load can3 be presented as a cumulative nonlinear are expressed as the of incremental loads. The terms Nab first derivative of Nab as follows: 3 nonlinearðkÞ 3k 3k ð21Þ ¼ B wk3,1 w3,1 þ uwk3,2 w3,2 N11 3 nonlinearðkÞ
N22
3k 3k ¼ B wk3,2 w3,2 þ uwk3,1 w3,1
3 nonlinearðkÞ
N12
¼
ð22Þ
1u 3k 3k B wk3,1 w3,2 þ uwk3,2 w3,1 2
ð23Þ
Hence the system of algebraic equations can be written as follows: ( ) 3k 3 3 kþ1 3 kþ1 p p p k 3k k ½H w þ½G p ¼ ½B Nab w3, a þ Nab w3, a þq
This method treats the transverse load to be divided into small load steps. The boundary integral equations are then transformed to a system of algebraic equations as follows: n o 3 ð27Þ ½Hp fwk þ 1 g þ ½Gp fpk þ 1 g ¼ ½Bp ðNab w3, b Þk, a þðk þ1Þq g ½Hm fuk þ 1 g þ ½Gm ft k þ 1 g ¼ ½Bm fNanonlinearðkÞ b, b
ð28Þ
The LU-decomposition method is also used to solve the system of equations. In order to improve numerical results, the relaxation procedure is also employed.
6. Numerical algorithms A simple numerical algorithm is used to analyze large deformation problems, in which no iterations are required. The algorithm can be summarized as follows: Step 1: Introducing a small loading step q10 with the first step ¼0 and w,a ¼0. k¼1, the final step kfinal and initial values Nalinear b Step 2: Compute the coefficient matrices related to fundamental solutions. They can be stored in the core and used in each increment without any change. Step 3: If ka1and Euler’s method is used, q0k þ 1 ¼ q10 , else if ka1 and the total incremental method is used, qk0 þ 1 ¼ qk0 þ q10 . Solve the linear system equation of the boundary integral equations to obtain boundary values. Then calculate the in-plane and derivative of deflection w,a in the stress resultants Nalinear b domain. Step 4: Apply the relaxation procedure given in Eq. (20). nonlinear ðkÞ Þ and Then calculate the nonlinear terms ðNag ,g [(Nab,aw3,w +Nabw3,wb)](k) using the approximation function as described in Eqs. (16)–(19). The nonlinear terms will be used for the evaluation in the next step k+1. Step 5: Employ the incremental approach in Section 5. Print results in each step. If step k is final step kfinal, go to Step 6, otherwise let step k¼ k+1, go to Step 3. Step 6: The calculation process terminates.
,a
ð24Þ nonlinearðkÞ 3 3k þ 1 3 kþ1 þ ½Gm t ¼ ½Bm Nab, b ½Hm u wk3,þa 1
7. Numerical examples ð25Þ
wk3, a þ w3 k3, a ,
where ¼ k denotes the incremental step and the superscript (:3 ) denotes the incremental value. The cumulative stress resultants can be obtained by
Two numerical examples of square plates with different boundary conditions (Fig. 2) subjected to transverse loadings are
q
q
3k
Nakbþ 1 ¼ Nakb þ Nab
ð26Þ
According to the continuous method as described by Ortega and Rheinboldt [13], use of the solution xk of the kth problem as a starting approximation to solve the (k+ 1)th problem will be a sufficiently good approximation to xk + 1 if the division of interval
Fig. 2. Plates with clamped (left) and simply supported (right) boundary conditions.
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presented to demonstrate the accuracy of the formulation. It is assumed that that the edges of the plate are fixed for clamped condition. For simply supported condition, the plate is free to rotate and the in-plane displacement and deflection of the edge are zero. The dual reciprocity model is shown in Fig. 3. BEM meshes with 25 quadratic elements (50 boundary points) and 25 domain points are used. The incremental approaches utilizing the Euler and total incremental methods are used. Comparisons are made with the analytical results [1,2] and published results [9]. In the following examples, the normalized parameters are defined as follows: Q¼
qa4 Eh4
ð29Þ
Z¼
wmax 3 h
ð30Þ
where a is the width of the square plate. Wen et al. [10] also presented similar numerical examples utilizing the total incremental method, but the results are generated only from relatively small total incremental loads. Here normalized deflections for simply supported and clamped plates are presented in Figs. 4 and 5, respectively. The results obtained using both incremental approaches are in good agreements with the analytical results [1,2]. It can however be seen from Figs. 4 and 5 that the analyses obtained using Euler’s method may go further than those by the total incremental method as the transverse loadings q increase. The total increment method seems to have problems of convergence or stability. Since
a
Fig. 5. Normalized deflections for a clamped square plate.
an incremental approach is used to force nonlinear problems to be linear, increasing transverse loadings in the total incremental method would likely lead to a tendency of divergence when solving Eqs. (27) and (28). The nonlinear terms evaluated based on the total incremental values will pose accumulated divergences as the total loads increase. This is not the case in the Euler method, which uses a small incremental load, leading to less instability. Combination of DRBEM and Euler’s method does not severely weaken the performance of the procedure when computing the stress resultants. However, if domain cells are used in the analysis, the total increment method is slightly more stable for treating nonlinear problems than the Euler method as reported in Ref. [9]. Since the coefficient matrices related to fundamental solutions can be stored in the core and used in each increment without any change, computational efficiency of the Euler and total incremental methods would be the same. However, in terms of stability, this work shows that the Euler method seems more stable for treating nonlinear problems of the shear deformable plate if the dual reciprocity method is employed to evaluate the domain integrals that appear in the formulations.
Fig. 3. Dual reciprocity model for a square plate.
8. Conclusions The dual reciprocity boundary element method was employed to evaluate large deformations of shear deformable plates. Incremental approaches utilizing the Euler method and the total incremental method were used to treat the nonlinear problems. The numerical results obtained by both approaches are in good agreement with the analytical and published results. In the large deformation analyses by DRBEM, the Euler method seems more stable for treating the nonlinear problems of shear deformable plates compared with the total incremental method. References
Fig. 4. Normalized deflections for a simply supported square plate.
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[5] Tanaka M. Large deflection analysis of thin elastic plates. In: Development in Boundary Element Methods, vol. 3. Elsevier Applied Science Publishers; 1984. p. 115–136. [6] Ye TQ, Lin YJ. Finite deflection analysis of elastic plate by the boundary element method. Appl Math Model 1985;9:183–8. [7] Lei XY, Huang MK, Wang XX. Geometrically nonlinear analysis of a Reissner type by the boundary element method. Comput Struct 1990;37(6):911–6. [8] He XQ, Qin QH. Nonlinear analysis of Reissner’s plate by the variational approaches and boundary element methods. Appl Math Model 1993;17: 149–155.
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