- Email: [email protected]
Non-linear bending analysis of super elliptical thin plates
International Journal of Non-Linear Mechanics 55 (2013) 180–185
Contents lists available at SciVerse ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Non-linear bending analysis of super elliptical thin plates Da-Guang Zhang n School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 September 2012 Received in revised form 11 February 2013 Accepted 8 June 2013 Available online 15 June 2013
In this paper, non-linear bending analysis is first presented for super elliptical thin plates with simply supported edge and clamped edge based on classical plate theory. Approximate solutions of super elliptical thin plates are obtained by Ritz method, convergence studies are discussed, and the validity can be confirmed by comparison with related researchers' results. It can be observed that the characteristics of non-linear bending are significantly influenced by different boundary conditions, ratio of major to minor axis, as well as the power of the super ellipse. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Super elliptical plates Non-linear bending Ritz method
1. Introduction Super elliptical plates which are defined by shapes between an ellipse and a rectangle have a wide range of use in engineering applications, and it is more difficult to analyze non-linear behaviors of super elliptical plates than rectangular, circular and elliptical plates. Some studies for linear behaviors of super elliptical plates are available in the literature, for example, Wang et al. [1] presented accurate frequency and buckling factors for super elliptical plates with simply supported and clamped edges by using Rayleigh–Ritz method. Lim [2] investigated free vibration of doubly connected super-elliptical laminated composite plates. Then Chen et al. [3] reported a free vibration analysis of laminated thick super elliptical plates. Liew and Feng [4] studied three-dimensional free vibration analysis of perforated super elliptical plates. Zhou [5] analyzed three-dimensional free vibration of super elliptical plates based on linear elasticity theory using Chebyshev–Ritz method. Altekin [6] gave out free linear vibration and buckling of super-elliptical plates resting on symmetrically distributed point-supports. Altekin and Altay [7] calculated static analysis of point-supported super-elliptical plates, then Altekin [8,9] discussed free vibration and bending of orthotropic super elliptical plates on intermediate supports. Çeribaşı et al. [10] gave out static linear analysis of super elliptical clamped plates based on the classical plate theory by Galerkin's method. Çeribaşı and Altay [11] investigated free vibration of super elliptical plates with constant and variable thickness by Ritz method, then Çeribaşı [12] investigated static and dynamic linear analyses of thin uniformly loaded
super elliptical clamped functionally graded plates. Jazi and Farhatnia [13] discussed buckling of functionally graded super elliptical plate based on the classical plate theory using Pb-2 Ritz method. As far as known to the author, investigations on non-linear bending of super elliptical plates have not been reported. The present work focuses attention on non-linear bending analysis of super elliptical plates based on classical plate theory, and approximate solutions of super elliptical plates are obtained by Ritz method. 2. Basic formulations of thin plates based on classical plate theory Consider a super elliptical plate of major axis 2a, minor axis 2b and thickness h, and the coordinate system is illustrated in Fig. 1. The boundary shape equation of the super elliptical plates can be represented by x2k y2k þ −1 ¼ 0 a2k b2k
k is the power of the super ellipse, and if k ¼ 1, the shape becomes an ellipse, if k ¼ ∞, the shape becomes a rectangle. According to classical plate theory, the displacement fields are u ¼ u0 −z
∂w ; ∂x
ð2aÞ
v ¼ v0 −z
∂w ; ∂y
ð2bÞ
w ¼ wðx; yÞ n
Tel./fax: +86 213 420 6197. E-mail address: [email protected]
0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.006
ð1Þ
ð2cÞ
in which u, v and w are total displacements, u0 and v0 are midplane displacements in the x and y directions, respectively.
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
181
In Eq. (8) Aij and Dij are the plate stiffnesses, defined by Z h 2 ðAij ; Dij Þ ¼ Q~ ij ð1; z2 Þdz ði; j ¼ 1; 2; 6Þ
ð9Þ
−h2
In the following analysis, all edges of plate are assumed to be simply supported and clamped with no in-plane displacements, i.e. prevented from moving in the x- and y-directions. u0 ¼ v0 ¼ w ¼ M n ¼ 0; ðfor simply supported edge; referred to as Case 1Þ
ð10aÞ u0 ¼ v0 ¼ w ¼
∂w ¼ 0; ðfor clamped edge; referred to as Case 2Þ ∂n ð10bÞ
where n refers to the normal directions of the plate boundary.
3. Ritz method for approximate solutions of non-linear problems of super elliptical thin plates Fig. 1. Geometry and coordinates of a super elliptical plate.
Considering non-linear von Kármán strain–displacement relationships, the strains can be expressed by ½ε ¼ ½εx ; εy ; γ xy T ¼ ½εð0Þ þ z½εð1Þ in which " #T ð0Þ ∂u0 1 ∂w 2 ∂v0 1 ∂w 2 ∂v0 ∂u0 ∂w ∂w þ þ þ þ ε ; ; ¼ 2 ∂x 2 ∂y ∂y ∂x ∂x ∂y ∂x ∂y 2 T ð1Þ ∂ w ∂2 w ∂2 w ε ¼ − 2 ; − 2 ; −2 ∂x∂y ∂x ∂y
21
22
0
Q~ 66
0
ð4aÞ
ð4bÞ
ð6Þ
and E ; 1−ν2
ð7aÞ
νE ; Q~ 12 ¼ Q~ 21 ¼ 1−ν2
ð7bÞ
Q~ 11 ¼ Q~ 22 ¼
Q~ 66 ¼
E 2ð1 þ νÞ
3 2 Mx D11 6 M 7 6D 4 y 5 ¼ 4 21 M xy 0
m ¼ 1;2;⋯
v0 ¼
D12 D22 0
32 ð1Þ 3 εx 6 ð1Þ 7 7 ε 0 7 56 4 y 5 D66 γ ð1Þ xy 0
ð8bÞ
W m;
ð11aÞ
Um;
ð11bÞ
Vm
ð11cÞ
M
∑
m ¼ 1;2;⋯ M
∑
m ¼ 1;2;⋯
where M is total number of series. For symmetrical problems about the plate with Case 1, it can be assumed that x2k y2k 2ðm−1Þ x2i y2j W m ¼ 1− 2k − 2k ð12aÞ ∑ aij 2i 2j a a b b i ¼ 0;2;⋯ Um ¼
x x2k y2k 2ðm−1Þ x2i y2j 1− 2k − 2k ∑ dij 2i 2j a a a b b i ¼ 0;2;⋯
ð12bÞ
Vm ¼
y x2k y2k 2ðm−1Þ x2i y2j 1− 2k − 2k ∑ eij 2i 2j b a a b b i ¼ 0;2;⋯
ð12cÞ
For symmetrical problems about the plate with Case 2, it can be assumed that 2 2ðm−1Þ x2k y2k x2i y2j W m ¼ 1− 2k − 2k ∑ aij 2i 2j ð13aÞ a a b b i ¼ 0;2;⋯ Um ¼
x x2k y2k 2ðm−1Þ x2i y2j 1− 2k − 2k ∑ dij 2i 2j a a a b b i ¼ 0;2;⋯
ð13bÞ
Vm ¼
y x2k y2k 2ðm−1Þ x2i y2j 1− 2k − 2k ∑ eij 2i 2j b a b a b i ¼ 0;2;⋯
ð13cÞ
ð7cÞ
The constitutive equations can be deduced by proper integration. 2 3 2 32 ð0Þ 3 εx Nx A11 A12 0 6 ð0Þ 7 6 N 7 6A 7 0 7 ð8aÞ 4 y 5 ¼ 4 21 A22 56 4 εy 5; ð0Þ N xy 0 0 A66 γ xy 2
M
∑
u0 ¼
ð5Þ
where [Q~ ] is the stiffness matrix transformation, defined by 2 ~ 3 Q 11 Q~ 12 0 6 7 0 5 Q~ ½Q~ ¼ 4 Q~
w¼
ð3Þ
According to Hooke's law, the stresses can be determined as ½s ¼ ½sx ; sy ; τxy T ¼ ½Q~ ½ε
Ritz method is adopted in this section to obtain approximate solutions of super elliptical plates. The key issue is first to assume the deflection and mid-plane displacements of the plate
in which j ¼ 2m−i−2 in Eqs. (12–13). Note that aij , dij and eij are undetermined coefficients, and Eqs. (12–13) satisfy displacement boundary conditions. Non-linear algebraic equations about aij , dij and eij can be obtained by substituting w, u0 and v0 into the following expression. ∂Π ¼ 0; ∂aij
ð14aÞ
∂Π ¼ 0; ∂dij
ð14bÞ
182
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
∂Π ¼0 ∂eij
ð14cÞ
in which Π ¼ U þ V, and the strain energy is Z 1 ðsx εx þ sy εy þ τxy γ xy ÞdΩ U¼ 2 Ω Z 1 ð0Þ ð0Þ ð1Þ ð1Þ ð1Þ ½N x εð0Þ ¼ x þ N y εy þ N xy γ xy þ M x εx þ M y εy þ M xy γ xy dΩ0 2 Ω0
Table 1 Comparisons of results obtained by different series for the dimensionless maximum deflection of aluminum plates with Case 1. Case 1
M
q0 a4 =Eh ¼ 1
q0 a4 =Eh ¼ 3
q0 a4 =Eh ¼ 12
q0 a4 =Eh ¼ 30
k¼1
1 2 3 4 5 6
0.4098 0.4830 0.4808 0.4808 0.4808 0.4808
0.7800 0.8613 0.8512 0.8512 0.8512 0.8512
1.428 1.506 1.473 1.475 1.475 1.475
2.016 2.081 2.025 2.033 2.033 2.033
k¼2
1 2 3 4 5 6
0.1309 0.4164 0.5093 0.5287 0.5322 0.5328
0.3514 0.7843 0.8989 0.9241 0.9290 0.9299
0.8673 1.424 1.558 1.591 1.597 1.599
1.334 2.003 2.147 2.190 2.199 2.201
k¼4
1 2 3 4 5 6
0.0197 0.2226 0.4894 0.5230 0.5229 0.5222
0.0590 0.5377 0.8959 0.9320 0.9287 0.9275
0.2269 1.164 1.586 1.616 1.609 1.607
0.4921 1.730 2.199 2.227 2.218 2.217
k ¼ 10
1 2 3 4 5 6
0.0013 0.0211 0.3423 0.4970 0.5098 0.5134
0.0040 0.0634 0.7557 0.9190 0.9174 0.9224
0.0160 0.2496 1.503 1.620 1.598 1.607
0.0400 0.5823 2.156 2.232 2.206 2.218
ð15aÞ work done by applied forces is V ¼ −∬Ω0 qwdΩ0
ð15bÞ
where q is transverse load, Ω denotes domain of plates and Ω0 denotes mid-plane of plates. As for plates with given loads (like transverse uniformly distributed load q0 ) and other known coefficients, aij , dij and eij can be solved by Newton–Raphson method or other equivalent methods. Take M ¼ 1 for the circular plate with Case 2 as an example, non-linear algebraic equations and the solving process are given out in detail. Work done by applied forces is 2 Z a Z bpffiffiffiffiffiffiffiffiffiffiffiffiffi 1−x2 =a2 x2 y2 πq a2 V ¼− q0 1− 2 − 2 a00 dy dx ¼ − 0 a00 ð16aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a −a −b 1−x2 =a2 b Note that e00 ¼ d00 , and the strain energy is Z Z pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 U¼ þ
a
b
1−x =a
−a
−b
1−x =a
ð0Þ ð0Þ ð1Þ ð1Þ ½N x εð0Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi x þ N y εy þ N xy γ xy þ M x εx þ M y εy 2 2
M xy γ ð1Þ xy dy
¼
dx
ð16bÞ
4
4
Case 2
M
q0 a4 =Eh ¼ 2
q0 a4 =Eh ¼ 6
q0 a4 =Eh ¼ 18
q0 a4 =Eh ¼ 40
k¼1
1 2 3 4 5 6
0.3190 0.3216 0.3229 0.3228 0.3228 0.3228
0.7428 0.7628 0.7709 0.7702 0.7702 0.7702
1.357 1.409 1.427 1.422 1.422 1.422
1.926 1.991 2.025 2.008 2.012 2.012
k¼2
1 2 3 4 5 6
0.1276 0.2865 0.3692 0.3921 0.3968 0.3975
0.3603 0.6998 0.8505 0.8887 0.8964 0.8973
0.8356 1.324 1.537 1.580 1.589 1.590
1.315 1.893 2.163 2.204 2.216 2.217
k¼4
1 2 3 4 5 6
0.0236 0.1508 0.3482 0.4045 0.4071 0.4063
0.0707 0.4295 0.8353 0.9156 0.9156 0.9148
0.2088 1.017 1.558 1.628 1.615 1.609
0.4381 1.615 2.218 2.268 2.242 2.243
k ¼ 10
1 2 3 4 5 6
0.0018 0.0210 0.1948 0.3920 0.4034 0.4051
0.0055 0.0631 0.5518 0.9004 0.9064 0.9241
0.0165 0.1887 1.289 1.628 1.611 1.626
0.0365 0.4146 2.032 2.287 2.243 2.244
Due to Eq. (14b), the following expression can be obtained 1−3ν ða00 Þ2 10a
4
Table 2 Comparisons of results obtained by different series for the dimensionless maximum deflection of aluminum plates with Case 2.
2πA11 32πD11 2πA11 2πνA11 32πA11 ðd00 Þ2 þ − d00 ða00 Þ2 þ ða00 Þ2 − ða00 Þ4 3 15a 5a 3a2 105a2
d00 ¼
4
ð17aÞ
Using Eq. (14a), the relationships between a00 and q0 can be presented 626 12 18 2 A11 64D11 þ ν− ν ða00 Þ3 þ a00 −q0 ¼ 0 ð17bÞ 175 25 25 a4 a4 a00 can be solved by non-linear numerical method, and it is worth noting that linear analytical solution can be obtained if nonlinear term is neglected. For the sake of brevity, non-linear algebraic equations and the solving process of other cases are omitted. Substituting these coefficients back into Eqs. (12–13), w, u0 and v0 may then be completely determined.
4. Results and discussion
4
4
4
4
4.1. Convergence studies In order to discuss about convergence of approximate solutions given by Ritz method, convergence studies are first undertaken in Tables 1 and 2 for dimensionless maximum deflection wmax =h of super elliptical aluminum plates with Case 1 and Case 2 under 4 dimensionless transverse uniform loads q0 a4 =Eh . The material properties are E ¼ 70 GPa, ν ¼ 0:3, and ratio of major to minor axis a=b ¼ 1. Most of the numerical results show that the present method is well converged when M≥5. Thus, for the sake of simplicity, M ¼ 5 is used in all the following calculations.
4.2. Comparison studies To ensure the accuracy and effectiveness of the present method, three examples are solved for linear and non-linear bending analysis of isotropic plates. Example 1. Linear bending solutions of central deflection for isotropic super elliptical thin plates with clamped edge subjected to transverse uniform distributed loads are calculated and
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
183
Table 3 Comparisons of non-dimensional small deflection solutions for isotropic super elliptical thin plates with clamped edge subjected to transverse uniform distributed loads. a/b
1 1.2 1.4 1.6 1.8 2
Çeribaşı et al. [10]
Present
k ¼1
k¼2
k ¼4
k ¼10
k ¼1
k¼2
k¼4
k ¼10
0.01536 0.02142 0.02603 0.02949 0.03203 0.03390
0.01971 0.02688 0.03225 0.03591 0.03827 0.03973
0.02027 0.02763 0.03314 0.03686 0.03922 0.04063
0.02017 0.02751 0.03303 0.03678 0.03919 0.04064
0.01536 0.02142 0.02603 0.02949 0.03203 0.03390
0.01971 0.02688 0.03225 0.03591 0.03827 0.03973
0.02027 0.02763 0.03314 0.03686 0.03922 0.04063
0.02017 0.02751 0.03303 0.03679 0.03919 0.04064
2.0
The present results of super elliptical plates with k=10 Case 1 Case 2
wmax/h
1.5
1.0
0.5 2Q-3 method by Reddy and Chao [15] Case 1 Case 2
0.0 0
100
200
300
q0a4/16Eh4 Fig. 2. Comparisons of central deflection-load curves for isotropic elliptical thin plates with Case 2.
Fig. 3. Comparison of central deflection-load curves for isotropic square plates with Case 1 and Case 2.
2.0
compared with the results of Çeribaşı et al. [10] in Table 3. In this example, non-dimensional central deflection is defined by 3 wmax Eh ð1−ν2 Þ=12q0 .
Example 3. Comparisons of the present results with 2Q-3 method by Reddy and Chao [15] are shown in Fig. 3 for the maximum dimensionless deflection of isotropic thin square plates with Case 1 and Case 2 subjected to transverse uniformly distributed loads. In this example, ν ¼ 0:3, the maximum dimensionless deflection and dimensionless transverse uniformly distributed loads are 4 defined by wmax =h and q0 a4 =16Eh , the super elliptical plates with k ¼ 10, which represents a shape similar to a rectangle. Excellent agreements can be seen from Table 1 and Figs. 2–3, thus the validity can be confirmed. 4.3. Parametric studies A parametric study was undertaken for non-linear bending of isotropic super elliptical plates with ν ¼ 0:3, a=b ¼ 1; 1:5; 2, the power of the super ellipse k ¼ 2; 4; 10, the maximum dimensionless deflection and dimensionless pressure are defined by wmax =h 4 and q0 a4 =Eh , respectively. The maximum dimensionless deflections of isotropic super elliptical plates with Case 1 and Case 2 subjected to transverse uniformly distributed loads are calculated, see Figs. 4–9. The difference between linear and non-linear analysis can be exhibited in Figs. 4 and 7, it make a little difference to linear and
1.5
wmax/h
Example 2. The central deflection—load curves for isotropic elliptical thin plates with Case 2 subjected to transverse uniform distributed loads are calculated and compared in Fig. 2 with perturbation solutions of Chien et al. [14]. In this example, ν ¼ 4 0:3 and non-dimensional pressure is defined by q0 a4 ð1−ν2 Þ=Eh .
1.0 k=2 Linear solutions a/b=1 a/b=1.5 a/b=2
0.5
Nonlinear solutions a/b=1 a/b=1.5 a/b=2
0.0 0
20
40
q0
60
a4/Eh4
Fig. 4. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 2 and Case 1.
non-linear solutions when wmax =ho 1=3, while it make a significant difference when wmax =h 4 1=3. It can be observed that the deflections increase with ratio of major to minor axis subjected to the same mechanical loads. Numerical results confirm that the characteristics of non-linear bending are significantly influenced by different boundary conditions and ratio of major to minor axis, as well as the power of the super ellipse.
5. Conclusions Super elliptical plates are defined by shapes between an ellipse and a rectangle, and this type of plate shape is useful in many applications which require the rectangular plate element to have curved corners so as to avoid high stress concentrations at the
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
2.0
2.0
1.5
1.5
wmax/h
wmax/h
184
1.0
1.0
k=4
k=4
a/b=1 a/b=1.5 a/b=2
0.5
0.5
a/b=1 a/b=1.5 a/b=2
0.0
0.0 0
20
40
0
60
20
q0a4/Eh4
40
60
80
100
q0a4/Eh4
Fig. 5. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 4 and Case 1.
Fig. 8. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 4 and Case 2.
2.0 2.0
1.5
wmax/h
wmax/h
1.5
1.0 k=10 a/b=1 a/b=1.5 a/b=2
0.5
1.0
k=10
0.5
a/b=1 a/b=1.5 a/b=2
0.0 0
20
40
60
0.0
q0a4/Eh4
0
20
40
60
q0a /Eh
Fig. 6. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 10 and Case 1.
4
80
100
4
Fig. 9. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 10 and Case 2.
2.0
conditions, ratio of major to minor axis, as well as the power of the super ellipse.
wmax/h
1.5
References 1.0 k=2
0.5
Nonlinear solutions a/b=1 a/b=1.5 a/b=2
Linear solutions a/b=1 a/b=1.5 a/b=2
0.0 0
20
40
60
80
100
q0a4/Eh4 Fig. 7. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 2 and Case 2.
corners. In this paper, non-linear bending analysis are first presented for isotropic super elliptical plates based on classical plate theory. Ritz method is employed to analyze non-linear bending behaviors, convergence studies are discussed, and the validity can be confirmed by comparison with related researchers' results. Numerical results confirm that the characteristics of nonlinear bending are significantly influenced by different boundary
[1] C.M. Wang, L. Wang, K.M. Liew, Vibration and buckling of super elliptical plates, Journal of Sound and Vibration 171 (1994) 301–314. [2] C.W. Lim, S. Kitipornchai, K.M. Liew, A free-vibration analysis of doubly connected super-elliptical laminated composite plates, Composites Science and Technology 58 (1998) 435–445. [3] C.C. Chen, C.W. Lim, S. Kitipornchai, K.M. Liew, Vibration of symmetrically laminated thick super elliptical plates, Journal of Sound and Vibration 220 (1999) 659–682. [4] K.M. Liew, Z.C. Feng, Three-dimensional free vibration analysis of perforated super elliptical plates via p-Ritz method, International Journal of Mechanical Sciences 43 (2001) 2613–2630. [5] D. Zhou, S.H. Lo, Y.K. Cheung, F.T.K. Au, 3-D vibration analysis of generalized super elliptical plates using Chebyshev-Ritz method, International Journal of Solids and Structures 41 (2004) 4697–4712. [6] M. Altekin, G. Altay, Static analysis of point-supported super-elliptical plates, Archive of Applied Mechanics 78 (2008) 259–266. [7] M. Altekin, Free linear vibration and buckling of super-elliptical plates resting on symmetrically distributed point-supports on the diagonals, Thin-walled Structures 46 (2008) 1066–1086. [8] M. Altekin, Free vibration of orthotropic super-elliptical plates on intermediate supports, Nuclear Engineering and Design 239 (2009) 981–999. [9] M. Altekin, Bending of orthotropic super-elliptical plates on intermediate point supports, Ocean Engineering 37 (2010) 1048–1060. [10] S. Çeribaşı, G. Altay, M. Cengiz-Dökmeci, Static analysis of super elliptical clamped plates by Galerkin's method, Thin-walled Structures 46 (2008) 122–127.
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
[11] S. Çeribaşı, G. Altay, Free vibration of super elliptical plates with constant and variable thickness by Ritz method, Journal of Sound and Vibration 319 (2009) 668–680. [12] S. Çeribaşı, Static and dynamic analyses of thin uniformly loaded super elliptical FGM plates, Mechanics of Advanced Materials and Structures 19 (2012) 323–335. [13] S.R. Jazi, F. Farhatnia, Buckling analysis of functionally graded super elliptical plate using Pb-2 Ritz method, Advanced Materials Research 383–390 (2012) 5387–5391.
185
[14] W.Z. Chien, L.Z. Pan, X.M. Liu, Large deflection problem of a clamped elliptical plate subjected to uniform pressure, Applied Mathematics and Mechanics 13 (1992) 891–908. [15] J N Reddy, W.C. Chao, Non-linear bending of thick rectangular laminated composite plates, International Journal of Non-Linear Mechanics 16 (1981) 291–301.
Contents lists available at SciVerse ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Non-linear bending analysis of super elliptical thin plates Da-Guang Zhang n School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 September 2012 Received in revised form 11 February 2013 Accepted 8 June 2013 Available online 15 June 2013
In this paper, non-linear bending analysis is first presented for super elliptical thin plates with simply supported edge and clamped edge based on classical plate theory. Approximate solutions of super elliptical thin plates are obtained by Ritz method, convergence studies are discussed, and the validity can be confirmed by comparison with related researchers' results. It can be observed that the characteristics of non-linear bending are significantly influenced by different boundary conditions, ratio of major to minor axis, as well as the power of the super ellipse. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Super elliptical plates Non-linear bending Ritz method
1. Introduction Super elliptical plates which are defined by shapes between an ellipse and a rectangle have a wide range of use in engineering applications, and it is more difficult to analyze non-linear behaviors of super elliptical plates than rectangular, circular and elliptical plates. Some studies for linear behaviors of super elliptical plates are available in the literature, for example, Wang et al. [1] presented accurate frequency and buckling factors for super elliptical plates with simply supported and clamped edges by using Rayleigh–Ritz method. Lim [2] investigated free vibration of doubly connected super-elliptical laminated composite plates. Then Chen et al. [3] reported a free vibration analysis of laminated thick super elliptical plates. Liew and Feng [4] studied three-dimensional free vibration analysis of perforated super elliptical plates. Zhou [5] analyzed three-dimensional free vibration of super elliptical plates based on linear elasticity theory using Chebyshev–Ritz method. Altekin [6] gave out free linear vibration and buckling of super-elliptical plates resting on symmetrically distributed point-supports. Altekin and Altay [7] calculated static analysis of point-supported super-elliptical plates, then Altekin [8,9] discussed free vibration and bending of orthotropic super elliptical plates on intermediate supports. Çeribaşı et al. [10] gave out static linear analysis of super elliptical clamped plates based on the classical plate theory by Galerkin's method. Çeribaşı and Altay [11] investigated free vibration of super elliptical plates with constant and variable thickness by Ritz method, then Çeribaşı [12] investigated static and dynamic linear analyses of thin uniformly loaded
super elliptical clamped functionally graded plates. Jazi and Farhatnia [13] discussed buckling of functionally graded super elliptical plate based on the classical plate theory using Pb-2 Ritz method. As far as known to the author, investigations on non-linear bending of super elliptical plates have not been reported. The present work focuses attention on non-linear bending analysis of super elliptical plates based on classical plate theory, and approximate solutions of super elliptical plates are obtained by Ritz method. 2. Basic formulations of thin plates based on classical plate theory Consider a super elliptical plate of major axis 2a, minor axis 2b and thickness h, and the coordinate system is illustrated in Fig. 1. The boundary shape equation of the super elliptical plates can be represented by x2k y2k þ −1 ¼ 0 a2k b2k
k is the power of the super ellipse, and if k ¼ 1, the shape becomes an ellipse, if k ¼ ∞, the shape becomes a rectangle. According to classical plate theory, the displacement fields are u ¼ u0 −z
∂w ; ∂x
ð2aÞ
v ¼ v0 −z
∂w ; ∂y
ð2bÞ
w ¼ wðx; yÞ n
Tel./fax: +86 213 420 6197. E-mail address: [email protected]
0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.006
ð1Þ
ð2cÞ
in which u, v and w are total displacements, u0 and v0 are midplane displacements in the x and y directions, respectively.
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
181
In Eq. (8) Aij and Dij are the plate stiffnesses, defined by Z h 2 ðAij ; Dij Þ ¼ Q~ ij ð1; z2 Þdz ði; j ¼ 1; 2; 6Þ
ð9Þ
−h2
In the following analysis, all edges of plate are assumed to be simply supported and clamped with no in-plane displacements, i.e. prevented from moving in the x- and y-directions. u0 ¼ v0 ¼ w ¼ M n ¼ 0; ðfor simply supported edge; referred to as Case 1Þ
ð10aÞ u0 ¼ v0 ¼ w ¼
∂w ¼ 0; ðfor clamped edge; referred to as Case 2Þ ∂n ð10bÞ
where n refers to the normal directions of the plate boundary.
3. Ritz method for approximate solutions of non-linear problems of super elliptical thin plates Fig. 1. Geometry and coordinates of a super elliptical plate.
Considering non-linear von Kármán strain–displacement relationships, the strains can be expressed by ½ε ¼ ½εx ; εy ; γ xy T ¼ ½εð0Þ þ z½εð1Þ in which " #T ð0Þ ∂u0 1 ∂w 2 ∂v0 1 ∂w 2 ∂v0 ∂u0 ∂w ∂w þ þ þ þ ε ; ; ¼ 2 ∂x 2 ∂y ∂y ∂x ∂x ∂y ∂x ∂y 2 T ð1Þ ∂ w ∂2 w ∂2 w ε ¼ − 2 ; − 2 ; −2 ∂x∂y ∂x ∂y
21
22
0
Q~ 66
0
ð4aÞ
ð4bÞ
ð6Þ
and E ; 1−ν2
ð7aÞ
νE ; Q~ 12 ¼ Q~ 21 ¼ 1−ν2
ð7bÞ
Q~ 11 ¼ Q~ 22 ¼
Q~ 66 ¼
E 2ð1 þ νÞ
3 2 Mx D11 6 M 7 6D 4 y 5 ¼ 4 21 M xy 0
m ¼ 1;2;⋯
v0 ¼
D12 D22 0
32 ð1Þ 3 εx 6 ð1Þ 7 7 ε 0 7 56 4 y 5 D66 γ ð1Þ xy 0
ð8bÞ
W m;
ð11aÞ
Um;
ð11bÞ
Vm
ð11cÞ
M
∑
m ¼ 1;2;⋯ M
∑
m ¼ 1;2;⋯
where M is total number of series. For symmetrical problems about the plate with Case 1, it can be assumed that x2k y2k 2ðm−1Þ x2i y2j W m ¼ 1− 2k − 2k ð12aÞ ∑ aij 2i 2j a a b b i ¼ 0;2;⋯ Um ¼
x x2k y2k 2ðm−1Þ x2i y2j 1− 2k − 2k ∑ dij 2i 2j a a a b b i ¼ 0;2;⋯
ð12bÞ
Vm ¼
y x2k y2k 2ðm−1Þ x2i y2j 1− 2k − 2k ∑ eij 2i 2j b a a b b i ¼ 0;2;⋯
ð12cÞ
For symmetrical problems about the plate with Case 2, it can be assumed that 2 2ðm−1Þ x2k y2k x2i y2j W m ¼ 1− 2k − 2k ∑ aij 2i 2j ð13aÞ a a b b i ¼ 0;2;⋯ Um ¼
x x2k y2k 2ðm−1Þ x2i y2j 1− 2k − 2k ∑ dij 2i 2j a a a b b i ¼ 0;2;⋯
ð13bÞ
Vm ¼
y x2k y2k 2ðm−1Þ x2i y2j 1− 2k − 2k ∑ eij 2i 2j b a b a b i ¼ 0;2;⋯
ð13cÞ
ð7cÞ
The constitutive equations can be deduced by proper integration. 2 3 2 32 ð0Þ 3 εx Nx A11 A12 0 6 ð0Þ 7 6 N 7 6A 7 0 7 ð8aÞ 4 y 5 ¼ 4 21 A22 56 4 εy 5; ð0Þ N xy 0 0 A66 γ xy 2
M
∑
u0 ¼
ð5Þ
where [Q~ ] is the stiffness matrix transformation, defined by 2 ~ 3 Q 11 Q~ 12 0 6 7 0 5 Q~ ½Q~ ¼ 4 Q~
w¼
ð3Þ
According to Hooke's law, the stresses can be determined as ½s ¼ ½sx ; sy ; τxy T ¼ ½Q~ ½ε
Ritz method is adopted in this section to obtain approximate solutions of super elliptical plates. The key issue is first to assume the deflection and mid-plane displacements of the plate
in which j ¼ 2m−i−2 in Eqs. (12–13). Note that aij , dij and eij are undetermined coefficients, and Eqs. (12–13) satisfy displacement boundary conditions. Non-linear algebraic equations about aij , dij and eij can be obtained by substituting w, u0 and v0 into the following expression. ∂Π ¼ 0; ∂aij
ð14aÞ
∂Π ¼ 0; ∂dij
ð14bÞ
182
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
∂Π ¼0 ∂eij
ð14cÞ
in which Π ¼ U þ V, and the strain energy is Z 1 ðsx εx þ sy εy þ τxy γ xy ÞdΩ U¼ 2 Ω Z 1 ð0Þ ð0Þ ð1Þ ð1Þ ð1Þ ½N x εð0Þ ¼ x þ N y εy þ N xy γ xy þ M x εx þ M y εy þ M xy γ xy dΩ0 2 Ω0
Table 1 Comparisons of results obtained by different series for the dimensionless maximum deflection of aluminum plates with Case 1. Case 1
M
q0 a4 =Eh ¼ 1
q0 a4 =Eh ¼ 3
q0 a4 =Eh ¼ 12
q0 a4 =Eh ¼ 30
k¼1
1 2 3 4 5 6
0.4098 0.4830 0.4808 0.4808 0.4808 0.4808
0.7800 0.8613 0.8512 0.8512 0.8512 0.8512
1.428 1.506 1.473 1.475 1.475 1.475
2.016 2.081 2.025 2.033 2.033 2.033
k¼2
1 2 3 4 5 6
0.1309 0.4164 0.5093 0.5287 0.5322 0.5328
0.3514 0.7843 0.8989 0.9241 0.9290 0.9299
0.8673 1.424 1.558 1.591 1.597 1.599
1.334 2.003 2.147 2.190 2.199 2.201
k¼4
1 2 3 4 5 6
0.0197 0.2226 0.4894 0.5230 0.5229 0.5222
0.0590 0.5377 0.8959 0.9320 0.9287 0.9275
0.2269 1.164 1.586 1.616 1.609 1.607
0.4921 1.730 2.199 2.227 2.218 2.217
k ¼ 10
1 2 3 4 5 6
0.0013 0.0211 0.3423 0.4970 0.5098 0.5134
0.0040 0.0634 0.7557 0.9190 0.9174 0.9224
0.0160 0.2496 1.503 1.620 1.598 1.607
0.0400 0.5823 2.156 2.232 2.206 2.218
ð15aÞ work done by applied forces is V ¼ −∬Ω0 qwdΩ0
ð15bÞ
where q is transverse load, Ω denotes domain of plates and Ω0 denotes mid-plane of plates. As for plates with given loads (like transverse uniformly distributed load q0 ) and other known coefficients, aij , dij and eij can be solved by Newton–Raphson method or other equivalent methods. Take M ¼ 1 for the circular plate with Case 2 as an example, non-linear algebraic equations and the solving process are given out in detail. Work done by applied forces is 2 Z a Z bpffiffiffiffiffiffiffiffiffiffiffiffiffi 1−x2 =a2 x2 y2 πq a2 V ¼− q0 1− 2 − 2 a00 dy dx ¼ − 0 a00 ð16aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a −a −b 1−x2 =a2 b Note that e00 ¼ d00 , and the strain energy is Z Z pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 U¼ þ
a
b
1−x =a
−a
−b
1−x =a
ð0Þ ð0Þ ð1Þ ð1Þ ½N x εð0Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi x þ N y εy þ N xy γ xy þ M x εx þ M y εy 2 2
M xy γ ð1Þ xy dy
¼
dx
ð16bÞ
4
4
Case 2
M
q0 a4 =Eh ¼ 2
q0 a4 =Eh ¼ 6
q0 a4 =Eh ¼ 18
q0 a4 =Eh ¼ 40
k¼1
1 2 3 4 5 6
0.3190 0.3216 0.3229 0.3228 0.3228 0.3228
0.7428 0.7628 0.7709 0.7702 0.7702 0.7702
1.357 1.409 1.427 1.422 1.422 1.422
1.926 1.991 2.025 2.008 2.012 2.012
k¼2
1 2 3 4 5 6
0.1276 0.2865 0.3692 0.3921 0.3968 0.3975
0.3603 0.6998 0.8505 0.8887 0.8964 0.8973
0.8356 1.324 1.537 1.580 1.589 1.590
1.315 1.893 2.163 2.204 2.216 2.217
k¼4
1 2 3 4 5 6
0.0236 0.1508 0.3482 0.4045 0.4071 0.4063
0.0707 0.4295 0.8353 0.9156 0.9156 0.9148
0.2088 1.017 1.558 1.628 1.615 1.609
0.4381 1.615 2.218 2.268 2.242 2.243
k ¼ 10
1 2 3 4 5 6
0.0018 0.0210 0.1948 0.3920 0.4034 0.4051
0.0055 0.0631 0.5518 0.9004 0.9064 0.9241
0.0165 0.1887 1.289 1.628 1.611 1.626
0.0365 0.4146 2.032 2.287 2.243 2.244
Due to Eq. (14b), the following expression can be obtained 1−3ν ða00 Þ2 10a
4
Table 2 Comparisons of results obtained by different series for the dimensionless maximum deflection of aluminum plates with Case 2.
2πA11 32πD11 2πA11 2πνA11 32πA11 ðd00 Þ2 þ − d00 ða00 Þ2 þ ða00 Þ2 − ða00 Þ4 3 15a 5a 3a2 105a2
d00 ¼
4
ð17aÞ
Using Eq. (14a), the relationships between a00 and q0 can be presented 626 12 18 2 A11 64D11 þ ν− ν ða00 Þ3 þ a00 −q0 ¼ 0 ð17bÞ 175 25 25 a4 a4 a00 can be solved by non-linear numerical method, and it is worth noting that linear analytical solution can be obtained if nonlinear term is neglected. For the sake of brevity, non-linear algebraic equations and the solving process of other cases are omitted. Substituting these coefficients back into Eqs. (12–13), w, u0 and v0 may then be completely determined.
4. Results and discussion
4
4
4
4
4.1. Convergence studies In order to discuss about convergence of approximate solutions given by Ritz method, convergence studies are first undertaken in Tables 1 and 2 for dimensionless maximum deflection wmax =h of super elliptical aluminum plates with Case 1 and Case 2 under 4 dimensionless transverse uniform loads q0 a4 =Eh . The material properties are E ¼ 70 GPa, ν ¼ 0:3, and ratio of major to minor axis a=b ¼ 1. Most of the numerical results show that the present method is well converged when M≥5. Thus, for the sake of simplicity, M ¼ 5 is used in all the following calculations.
4.2. Comparison studies To ensure the accuracy and effectiveness of the present method, three examples are solved for linear and non-linear bending analysis of isotropic plates. Example 1. Linear bending solutions of central deflection for isotropic super elliptical thin plates with clamped edge subjected to transverse uniform distributed loads are calculated and
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
183
Table 3 Comparisons of non-dimensional small deflection solutions for isotropic super elliptical thin plates with clamped edge subjected to transverse uniform distributed loads. a/b
1 1.2 1.4 1.6 1.8 2
Çeribaşı et al. [10]
Present
k ¼1
k¼2
k ¼4
k ¼10
k ¼1
k¼2
k¼4
k ¼10
0.01536 0.02142 0.02603 0.02949 0.03203 0.03390
0.01971 0.02688 0.03225 0.03591 0.03827 0.03973
0.02027 0.02763 0.03314 0.03686 0.03922 0.04063
0.02017 0.02751 0.03303 0.03678 0.03919 0.04064
0.01536 0.02142 0.02603 0.02949 0.03203 0.03390
0.01971 0.02688 0.03225 0.03591 0.03827 0.03973
0.02027 0.02763 0.03314 0.03686 0.03922 0.04063
0.02017 0.02751 0.03303 0.03679 0.03919 0.04064
2.0
The present results of super elliptical plates with k=10 Case 1 Case 2
wmax/h
1.5
1.0
0.5 2Q-3 method by Reddy and Chao [15] Case 1 Case 2
0.0 0
100
200
300
q0a4/16Eh4 Fig. 2. Comparisons of central deflection-load curves for isotropic elliptical thin plates with Case 2.
Fig. 3. Comparison of central deflection-load curves for isotropic square plates with Case 1 and Case 2.
2.0
compared with the results of Çeribaşı et al. [10] in Table 3. In this example, non-dimensional central deflection is defined by 3 wmax Eh ð1−ν2 Þ=12q0 .
Example 3. Comparisons of the present results with 2Q-3 method by Reddy and Chao [15] are shown in Fig. 3 for the maximum dimensionless deflection of isotropic thin square plates with Case 1 and Case 2 subjected to transverse uniformly distributed loads. In this example, ν ¼ 0:3, the maximum dimensionless deflection and dimensionless transverse uniformly distributed loads are 4 defined by wmax =h and q0 a4 =16Eh , the super elliptical plates with k ¼ 10, which represents a shape similar to a rectangle. Excellent agreements can be seen from Table 1 and Figs. 2–3, thus the validity can be confirmed. 4.3. Parametric studies A parametric study was undertaken for non-linear bending of isotropic super elliptical plates with ν ¼ 0:3, a=b ¼ 1; 1:5; 2, the power of the super ellipse k ¼ 2; 4; 10, the maximum dimensionless deflection and dimensionless pressure are defined by wmax =h 4 and q0 a4 =Eh , respectively. The maximum dimensionless deflections of isotropic super elliptical plates with Case 1 and Case 2 subjected to transverse uniformly distributed loads are calculated, see Figs. 4–9. The difference between linear and non-linear analysis can be exhibited in Figs. 4 and 7, it make a little difference to linear and
1.5
wmax/h
Example 2. The central deflection—load curves for isotropic elliptical thin plates with Case 2 subjected to transverse uniform distributed loads are calculated and compared in Fig. 2 with perturbation solutions of Chien et al. [14]. In this example, ν ¼ 4 0:3 and non-dimensional pressure is defined by q0 a4 ð1−ν2 Þ=Eh .
1.0 k=2 Linear solutions a/b=1 a/b=1.5 a/b=2
0.5
Nonlinear solutions a/b=1 a/b=1.5 a/b=2
0.0 0
20
40
q0
60
a4/Eh4
Fig. 4. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 2 and Case 1.
non-linear solutions when wmax =ho 1=3, while it make a significant difference when wmax =h 4 1=3. It can be observed that the deflections increase with ratio of major to minor axis subjected to the same mechanical loads. Numerical results confirm that the characteristics of non-linear bending are significantly influenced by different boundary conditions and ratio of major to minor axis, as well as the power of the super ellipse.
5. Conclusions Super elliptical plates are defined by shapes between an ellipse and a rectangle, and this type of plate shape is useful in many applications which require the rectangular plate element to have curved corners so as to avoid high stress concentrations at the
D.-G. Zhang / International Journal of Non-Linear Mechanics 55 (2013) 180–185
2.0
2.0
1.5
1.5
wmax/h
wmax/h
184
1.0
1.0
k=4
k=4
a/b=1 a/b=1.5 a/b=2
0.5
0.5
a/b=1 a/b=1.5 a/b=2
0.0
0.0 0
20
40
0
60
20
q0a4/Eh4
40
60
80
100
q0a4/Eh4
Fig. 5. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 4 and Case 1.
Fig. 8. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 4 and Case 2.
2.0 2.0
1.5
wmax/h
wmax/h
1.5
1.0 k=10 a/b=1 a/b=1.5 a/b=2
0.5
1.0
k=10
0.5
a/b=1 a/b=1.5 a/b=2
0.0 0
20
40
60
0.0
q0a4/Eh4
0
20
40
60
q0a /Eh
Fig. 6. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 10 and Case 1.
4
80
100
4
Fig. 9. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 10 and Case 2.
2.0
conditions, ratio of major to minor axis, as well as the power of the super ellipse.
wmax/h
1.5
References 1.0 k=2
0.5
Nonlinear solutions a/b=1 a/b=1.5 a/b=2
Linear solutions a/b=1 a/b=1.5 a/b=2
0.0 0
20
40
60
80
100
q0a4/Eh4 Fig. 7. Central deflection-load curves for isotropic super elliptical thin plates with k ¼ 2 and Case 2.
corners. In this paper, non-linear bending analysis are first presented for isotropic super elliptical plates based on classical plate theory. Ritz method is employed to analyze non-linear bending behaviors, convergence studies are discussed, and the validity can be confirmed by comparison with related researchers' results. Numerical results confirm that the characteristics of nonlinear bending are significantly influenced by different boundary
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