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Buckling Analysis of Rectangular Functionally Graded Material Plates under Uniaxial and Biaxial Compression Load
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 86 (2014) 748 – 757
1st International Conference on Structural Integrity, ICONS-2014
Buckling Analysis of Rectangular Functionally Graded Material Plates under Uniaxial and Biaxial Compression Load I Ramu* and S.C. Mohanty Department of Mechanical Engineering,National Institute of Technology, Rourkela-769008, Odisha, India * E-mail ID:[email protected],
Abstract This article deals with the buckling analysis of rectangular functionally graded material (FGM) plates using classical plate theory (CPT). Finite element method has been applied for the modelling and buckling analysis of FGM plate. Uniaxial and biaxial compression loads along with simply supported boundary conditions on rectangular FGM plates are investigated. Convergence of the solution obtained by MATLAB has also been studied by varying the mesh size. Obtained results are compared with the existing literature, it shows that the buckling characteristics are closer to the reference results. By varying the geometric parameter a/b critical buckling load variation has been calculated and Power law index is also been evaluated. © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2014 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research Keywords: FGM plates,FEM, CPT andbuckling analysis.
1. Introduction The buckling behaviour of rectangular FGM plates subjected to compressive loads has attracted the attention of many researchers working on structural analysis and design. The Functionally graded material can be represented as a non-homogenous material which its mechanical properties vary continuously along the thickness direction from top one surface to the bottom surface. This is achieved by varying the volume fraction of the constituents. Here, it is assumed that the FGM plate is made of ceramic and metal. The ceramic constituent provides high temperature resistance due to its low thermal conductivity. The metal part on the other hand, prevents fracture due to its greater toughness. These are high-performance, heat-resistant materials able to withstand high temperatures and extremely large gradients used in spacecrafts and nuclear plants. FGM are typically designed for a specific function or application. Most of the times they are manufactured to achieve good strength to weight ratios and good thermal or electrical conductivity.
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research doi:10.1016/j.proeng.2014.11.094
749
I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
Hosseini-Hashemi et al. [2008] has studied an exact solution for the buckling of isotropic rectangular Mindlin plates. They considered a combination of six different boundary conditions in which two opposite edges are simply supported. Monoaxial in-plane compressive loads on both directions were considered as well as equal biaxial compressive loads. They presented the non-dimensional critical buckling loads and mode shapes for the six cases analyzed. C.A. Featherston, A. Watson [2005] investigated the behaviour of a number of optimised fibre composite plates of differing geometry, simply supported along two edges and built in along the other two. In their analysis to a varying combination of shear and in-plane bending, for which no theoretical solution exists, and assesses the suitability of analytical techniques and finite element analysis to predict this behaviour. V. Piscopo [2010] investigated the Shimpi theory for buckling analysis of thick rectangular plates and taking into account the shear deformations. The finite element method has long been recognized as one of the most effective numerical method for analyzing the buckling load of thin plate like structures under arbitrary loading and boundary conditions. Chee-Kiong Chin et al. [1993] presented a finite element method using thin-plate elements. This method was capable of predicting the buckling capacity of arbitrarily shaped thin-walled structural members under any general load and boundary conditions. Several researchers have studied buckling of composite plates by, Arthur W. Leissa [1986] purposed to clarify the under what conditions bifurcation buckling loads can exist for generally unsymmetrically laminated plates for arbitrary ply orientation, as well as for the important, special cases of antisymmetric angle-ply and cross-ply laminates. Mahdi Damghani et al. [2011] has studied the critical buckling of composite plates with through the length delaminations by using exact stiffness analysis and the Wittrick–Williams algorithm. MeisamMohammadi, et al. [2010] obtained an exact solution for the buckling analysis of thin functionally graded rectangular plates. Their work based on the classical plate theory and using the principle of minimum total potential energy, the equilibrium equations are obtained. In the present study, the buckling analysis of thin rectangular FGM plates is studied. Based on the classical plate theory and using the principle of minimum total potential energy, the equilibrium equations are obtained. The resulting equations are solved for different loading conditions. Finally, the critical buckling loads for a FGM plate with simply supported boundary conditions and different loading conditions, some aspect ratios and various power law index of FGM, are shown. 2. Materials and Methods 2.1 Functionally Graded Materials (FGM) Consider a case when FGM plate made up of a mixture of ceramic and metal as show in Fig. 1. A rectangular FGM plate of length a, width b, and thickness h, referred to the rectangular Cartesian coordinates (x, y, z). We assume that the effective material properties Pof the plate linearly vary with respect to the thickness coordinate as follow Ref. [8]
P( z ) = PmVm + PV c c
1
Fig. 1 Geometry of the FGM plate where P(z) denotes a material property of FGM plate which maybe substituted by modulus of elasticity E, the coefficient of thermal expansion Į, the conductivity K, Poisson’s ratio ˶and thesubscripts m and c refer to the metal and ceramic constituents, respectively. For power-law FGM, ceramic volume fraction function is expressed as:
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
§ 2z + h · Vc = ¨ ¸ © 2h ¹
n
2
2.2 Physical Neutral Surface of the FGM Plate For plate made of FGM, the neutral surface may not coincide with its geometric mid-surface. The distance of the neutral surface (d) from the geometric mid-surface may be expressed as h/ 2
d=
³
zE ( z )dz
³
E ( z )dz
3
−h / 2 h/2
−h / 2
2.3 Governing Equations Let us refer to the coordinate system of Fig.2 with z axis having the origin on the plate middle plane. The basic assumptions of the classical plate theory are: 1. The displacements of the plate are small in comparison to the plate’s thickness and the strains and mid-surface slopes are much less than unity. 2. The stress σ z is negligible respect to the in-plane stresses σ x and σ y . 3. The transverse shear stresses
σ xz and σ yz are small in comparison with the in-plane components σ x , σ y and σ xy
. In other words, “normals remain normal”.
Fig. 2 Rectangular plate reference system Then the displacement field in the (x, y, z) reference system has the following form ∂w( x, y ) U = u ( x, y ) − z ∂x ∂w( x, y ) 4 V = v ( x, y ) − z ∂y W = w( x, y ) Where u, v are in-plane displacements at a point of the mid-plane and U, V and W are displacement components of a typical point in the plate. The strain-displacement relationship are given as
751
I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757 ∂U ½ ° ∂x ° ° ° ° ε x ½ ° ∂V ° °ε ° ° ∂y ° ° y ° ° ° ° ° ∂U ∂V ° + ®γ xy ¾ = ® ¾ °γ ° ° ∂y ∂x ° ° yz ° ° ∂V ∂W ° + °¯γ xz ¿° ° ° ° ∂z ∂y ° ° ∂U ∂W ° + ° ° ∂x ¿ ¯ ∂z
5
By substituting Eq. (4) into Eq. (5), the strain can be expressed as
∂u ∂2w −z 2 ∂x ∂x ∂v ∂2w εy = − z 2 ∂y ∂y
εx =
6
∂u ∂v ∂2w + − 2z ∂y ∂x ∂x∂y =0
γ xy = γ yz
γ xz = 0 ε x and ε y are the normal strains and γ xy
is shear strain at the middle surface of the plate.
The curvatures are shown by k x = w, xx , k y = w, yy , k xy = 2 w, xy . The stress-strain relationships of the functionally graded plate in the global x-y-z coordinates system can be written as σ xx ½ ª Q11 Q12 0 º ε x ½ °σ ° « »° ° 7 ® yy ¾ = «Q21 Q22 0 » ®ε y ¾ °τ ° « 0 ° ° 0 Q66 »¼ ¯γ xy ¿ ¯ xy ¿ ¬ where σ x and σ y are the normal stresses and τ xy is shear stress of the plate. Q11 = Q66 =
E ( z)
, Q12 = Q21 =
1−υ E (z) 2
υE ( z) , 1−υ 2
2 (1 + υ )
E (ε x + υε y ) 1 −υ 2 E σy = (ε y + υε x ) 1 −υ 2 E τ xy = γ xy 2 (1 + υ )
σx =
The constitutive relations are written as E1 (ε x + υε y ) + 1 −Eυ2 2 ( kx + υ k y ) 1 −υ 2 E E N y = 1 2 ( ε y + υε x ) + 2 2 ( k y + υ k x ) 1−υ 1 −υ E1 E2 N xy = k γ xy + 2 (1 + υ ) (1 + υ ) xy Nx =
8
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
E2 (ε x + υε y ) + 1 −Eυ3 2 ( kx + υ k y ) 1 −υ 2 E2 My = (ε y + υε x ) + 1 −Eυ3 2 ( k y + υ kx ) 1−υ 2 E3 E2 M xy = k γ xy + 2 (1 + υ ) (1 + υ ) xy Mx =
where
E1 = Em h +
( Ec − Em ) h , E n +1
2
9
1 · § 1 = ( Ec − Em ) h 2 ¨ + ¸ © n + 2 2n + 2 ¹
· Em h 1 1 h3 § 1 + ( Ec − Em ) ¨ − + ¸ 12 12 ¨© n + 3 n + 2 4 ( n + 1) ¸¹ 3
E3 =
In the above equations, Ni and Mi are force and moment resultants, respectively. The equilibrium equations of a perfect FGM plate are
N x , xx + 2 N xy , xy + N y , yy = 0 M x , xx + 2 M xy , xy + M y , yy = 0
10
q + N x w, xx + N y w, yy + 2 N xy w, xy = 0 Using Equations 8 and 9, the equilibrium in Equation (10) may be reduced to one equation as
D∇4 w − N x w, xx − 2 N xy w, xy − N y w, yy − q = 0
11
where
D=
E1E3 − E2 2
(
E1 1 − υ 2
)
To establish the stability equations, the critical equilibrium method is used. 3. Finite Element Formulations A four node rectangular element is used for this analysis. Each node of the element has three degrees of freedom, one transverse displacement and two rotations about x and y axis respectively shows in fig.3. Hence, the displacement function of the rectangular element can be approximated by ݓൌ ߙଵ ߙଶ ݔ ߙଷ ݕ ߙସ ݔ ߙହ ݕݔ ߙ ݕଶ ߙ ݔଷ ߙ଼ ݔଶ ݕ ߙଽ ݕݔଶ ߙଵ ݕଷ ߙଵଵ ݔଷ ݕ ߙଵଶ ݕݔଷ 12
w = ª¬1 x
x2
y
xy
y2
x3
x2 y
xy 2
y3
x3 y
xy 3 º¼ {α }
13
ݓൌ ሾܩሺݔǡ ݕሻሿሼߙሽ
14
തതതത ሽ் ൌ ሾܣሿ ሼߙሽ ሼܹ
15
തതതത ሽ ሼߙሽ ൌ ሾܣሿ ିଵ ሼܹ
16
Substituting (15) and (16) becomes തതതത ሽ் ൌ ሾሺݓଵ ሼܹ
ߠ௫ଵ
ߠ௬ଵ ݓ ڮସ
ߠ௫ସ
ߠ௬ସ ሻሿ ൌ ሾܵሺݔǡ ݕሻሿሼݓ ሽ
17 26
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
Fig.3 Geometry of the four node rectangular element Shape function ଵ ଶ ଶ ଼ۍ൫ͳ ݔ ݔ൯൫ͳ ݕ ݕ൯ሺʹ ݔ ݔ ݕ ݕെ ݔെ ݕሻې ێ ۑ ܵ ் ሺݔǡ ݕሻ ൌ ێ ൫ͳ ݔ ݔ൯൫ ݕ ݕ ൯ሺ ݕଶ െ ͳሻ ۑ ଼ ێ ۑ ଶ ൫ݔ ݔ൯൫ͳ ݕݕ ൯ሺݔ െ ͳሻ ۏ ے
18
଼
In deriving this result, it is simpler to use the expression Eq. (18) for substitute w after performing the integration. A typical integral is then the element stiffness matrix and geometric stiffness matrices are derived on the basis of principle of minimum potential energy and work done. The element stiffness matrix is derived as
U( ) = e
{ }
1 (e) q 2
T
{ }
ª K s ( e) º q( e) ¬ ¼
[ ks ] = ³³³ BT DBdV e
where
19
ª ∂2 º « 2 » x ∂ « » « ∂2 » Eh3 [ B] = « 2 » [ S ] , D = y 12(1 ∂ − v2 ) « » « ∂2 » «2 » «¬ ∂x∂y »¼
The element geometric stiffness matrix 2 ª § ∂S ·2 e § ∂S · § ∂S · § ∂S · º ª¬ k g ¼º = ³ « N x ¨ ¸ + N y ¨ ¸ + 2 N xy ¨ ¸ ¨ ¸ »dA ∂x © ∂x ¹ © ∂y ¹ »¼ © ∂y ¹ Ω« ¬ © ¹
20
The overall stiffness and geometric stiffness matrices, Ks and Kg, for a plate structure can be obtained by a standard assembly procedure used in the finite element method after each element stiffness matrix and geometric stiffness matrix has been transformed into global coordinatesystem. Buckling conditions for a structure are obtained when the second variation of the total potential energy vanishes, i.e.
det ( K s + λ K g ) = 0
In which λ is the buckling load.
21
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
4. Results and Discussions 4.1 Validation of the Results In order to validate the accuracy of the present formulations, a comparison has been carried out with the results obtained by Ref. [7], for all edges SSSS and SCSC isotropic plates. The critical buckling load has been listed in Table 2 for a rectangular simply supported plate subjected to uniaxial and biaxial compression and side-thickness ratios. As this table shows, the present results have a good agreement with those reported in Ref. [7]. Table 1 Non-dimensional critical buckling load convergence study of an square plate with t=0.01m Number of elements
Uniaxial compression
Biaxial compression
10X10 15X15 20X20 25X25 30X30 Ref [7]
39.0651 39.2926 39.3734 39.4111 39.428 39.4784
19.5325 19.6463 19.6867 19.7056 19.7081 19.7392
Table 2 2
Comparison of the non-dimensional critical buckling load ( Pcr L / D ) for an isotropic plate Boundary conditions SSSS SCSC
a/b ratio 0.5 1 0.5 1
Uniaxial compression Ref.[7] Present study 15.4212 15.392 39.4784 39.428 18.9775 18.948 75.9099 75.675
Biaxial compression Ref.[7] Present study 12.3370 12.313 19.7392 19.708 14.6174 14.592 37.7996 37.704
4.2 Results After verifying the accuracy of the present solution, in order to obtain the following new results, it is assumed that the FGM plate is made of a mixture of silicon nitride and stainless steel. Figure 4 illustrates the simply supported rectangular plate subjected to uniaxial and biaxial compression loading. To carry out the numerical simulation, following material properties have been considered. SUS304,ߩ ൌ ͺǡͳ݇݃Ȁ݉ଷ , ܧൌ ʹͲǤͺ ൈ ͳͲଽ ܲܽ ,ߴ ൌ ͲǤ͵ͳ ܵ݅ଷ ܰସ , ߩ ൌ ʹǡ͵Ͳ݇݃Ȁ݉ଷ , ܧൌ ͵ʹʹǤʹ ൈ ͳͲଽ ܲܽ , ߴ ൌ ͲǤʹͶ
Fig.4 Rectangular plate subjected to uniaxial and biaxial compression load
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
4.2.1Rectangular FGM Plates Under Uniaxial and Biaxial Compression In table 3 shows the critical buckling loads for different values of the ratio t/b of rectangular FGM plate under uniaxial compression. The convergence of the solution obtained by FEM has been studied and in all cases it was quite quickly achieved. From the analysis it is clear that there is a very good accordance with the proposed analysis. From the obtained results, it seems clear that the FEM buckling analysis results deceases, when the ratio t/b increases, as it would be predictable. Anyway, it is fundamental to note that these obtained results lower if referred to the critical value, the most important parameter in a buckling analysis. In tables 3 and 4 shown for FGM plates under uniaxial and biaxial compression with n= 0, 1, 2, 5 for different values of the ratio t/b.In figs.5 and 6 shown that the critical buckling load parameter variation with respective index value with different thickness values (t=0.01, 0.02 and 0.03) and aspect ratios (a/b=0.25, 1). The FGM plate is subjected to biaxial compression load, the increases index value effects the critical buckling load parameter is shown in fig. 6.Similarly the critical buckling load variation with different aspect ratios (a/b=0.25, 1) shown in figs. 6 (a) and (b). Table 3 The critical buckling load (MN/m) of FGM rectangular plates Aspect ratio a=0.25m,b=1m
a=1m, b=1m
a=4m, b=1m
Power law index n 0 1 2 5 0 1 2 5 0 1 2 5
0.01m 5.0576 4.3156 4.047 3.7379 1.1222 0.95022 0.89103 0.82293 1.0828 0.92289 0.86513 0.79875
Uniaxial compression 0.02m 40.461 34.525 32.376 29.903 8.9104 7.6018 7.1283 6.5834 8.6622 7.3831 6.921 6.39
0.03m 136.56 116.52 109.27 100.92 30.072 25.656 24.058 22.219 29.235 24.918 23.358 21.566
Table 4 The critical buckling load (MN/m) of a FGM rectangular plate Aspect ratio a=0.25m,b=1m
a=1m, b=1m
a=4m, b=1m
Power law index n 0 1 2 5 0 1 2 5 0 1 2 5
0.01m 4.7601 3.9373 3.7899 3.6517 0.55692 0.44605 0.44329 0.42712 0.29751 0.24608 0.26872 0.22823
Biaxial compression 0.02m 38.081 31.499 30.319 29.214 4.4552 3.6845 3.5463 3.4168 2.3801 1.9687 1.8950 1.8259
0.03m 128.52 106.31 102.33 98.597 15.036 12.435 11.969 11.532 8.0327 6.6443 6.3955 6.1623
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
DE
DE
P
P
&ULWLFDOEXFNOLQJORDG
&ULWLFDOEXFNOLQJORDG
P
P
P
P
,QGH[YDOXHQ
,QGH[YDOXHQ
Fig. 5 Critical buckling load of FGM plate verses power law index n with aspect ratio a/b=0.25, 1 under uniaxial compression case. DE
&ULWLFDOEXFNOLQJORDG
P P
P
,QGH[YDOXHQ
(a) DE
&ULWLFDOEXFNOLQJORDG
P
P
P
,QGH[YDOXHQ
(b) Fig. 6 Critical buckling load of FGM plate verses power law index n with aspect ratio (a) a/b=0.25, and (b) a/b= 1 under biaxial compression case.
I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
5. Conclusions The buckling behaviour of rectangular FGM plates under compression is investigated using the finite element method. The compression loadings are assumed to be uniaxial compression and biaxial compression. The effective material properties are computed using the simple power law equation of the volume fraction of the plate constituents. The critical buckling load of the rectangular plate under uniaxial compression is greater than the biaxial compression. From tables 3 and 4 results show that as the aspect ratio a/b (0.25, 1 and 4) increases, the critical buckling load reduces. And the variation of thickness (0.01, 0.02 and 0.03) for each aspect ratios are also shown in tables 3 and 4. The critical buckling load increases by increasing the thickness. In figs. 6 (a) and (b) it is shown that the critical buckling load decreases as the volume fraction index n increase. This is because as volume fraction index increases, the contained quantity of ceramic decreases. References 1. Chee-Kiong Chin, Faris G. A. AI-Bermani, and SritawatKitipornchai [1993], “Finite Element Method for Buckling Analysis of Plate Structures”, Journal of Structural Engineering, Vol.119, No. 4. 2. V. Piscopo [2010], “Refined Buckling Analysis of Rectangular Plates under Uniaxial and Biaxial Compression”, World Academy of Science, engineering and Technology 46, p. 554-561. 3. Hosseini-Hashemi S, Khorshidi K, Amabili M [2008], “Exact solution for linear buckling of rectangular Mindlin plates”, Journal of Sound and Vibrations. 315, 318–342 (2008). 4. Arthur W. Leissa [1986], “Conditions for Laminated Plates to Remain Flat Under Inplane Loading”, Composite Structures 6, 261-270. 5. Mahdi Damghani, David Kennedy, Carol Featherston [2011], “Critical buckling of delaminated composite plates using exact stiffness analysis”, Computers and Structures 89 (2011) 1286–1294 6. C.A. Featherston, A. Watson [], “Buckling of optimised flat composite plates under shear and in-plane bending”, Composites Science and Technology 65 (2005) 839–853 7. MeisamMohammadi, Ali Reza Saidi, EmadJomehzadeh [2010], “Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates”, Appl Compos Mater (2010) 17:81–93 8. Shyang-HoChi,Yen-Ling Chung [2006],“Mechanical behavior of functionally graded materialplates under transverse load—Part I: Analysis”, International Journal of Solids and Structures 43 (2006) 3657–3674.
757
ScienceDirect Procedia Engineering 86 (2014) 748 – 757
1st International Conference on Structural Integrity, ICONS-2014
Buckling Analysis of Rectangular Functionally Graded Material Plates under Uniaxial and Biaxial Compression Load I Ramu* and S.C. Mohanty Department of Mechanical Engineering,National Institute of Technology, Rourkela-769008, Odisha, India * E-mail ID:[email protected],
Abstract This article deals with the buckling analysis of rectangular functionally graded material (FGM) plates using classical plate theory (CPT). Finite element method has been applied for the modelling and buckling analysis of FGM plate. Uniaxial and biaxial compression loads along with simply supported boundary conditions on rectangular FGM plates are investigated. Convergence of the solution obtained by MATLAB has also been studied by varying the mesh size. Obtained results are compared with the existing literature, it shows that the buckling characteristics are closer to the reference results. By varying the geometric parameter a/b critical buckling load variation has been calculated and Power law index is also been evaluated. © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2014 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research Keywords: FGM plates,FEM, CPT andbuckling analysis.
1. Introduction The buckling behaviour of rectangular FGM plates subjected to compressive loads has attracted the attention of many researchers working on structural analysis and design. The Functionally graded material can be represented as a non-homogenous material which its mechanical properties vary continuously along the thickness direction from top one surface to the bottom surface. This is achieved by varying the volume fraction of the constituents. Here, it is assumed that the FGM plate is made of ceramic and metal. The ceramic constituent provides high temperature resistance due to its low thermal conductivity. The metal part on the other hand, prevents fracture due to its greater toughness. These are high-performance, heat-resistant materials able to withstand high temperatures and extremely large gradients used in spacecrafts and nuclear plants. FGM are typically designed for a specific function or application. Most of the times they are manufactured to achieve good strength to weight ratios and good thermal or electrical conductivity.
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research doi:10.1016/j.proeng.2014.11.094
749
I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
Hosseini-Hashemi et al. [2008] has studied an exact solution for the buckling of isotropic rectangular Mindlin plates. They considered a combination of six different boundary conditions in which two opposite edges are simply supported. Monoaxial in-plane compressive loads on both directions were considered as well as equal biaxial compressive loads. They presented the non-dimensional critical buckling loads and mode shapes for the six cases analyzed. C.A. Featherston, A. Watson [2005] investigated the behaviour of a number of optimised fibre composite plates of differing geometry, simply supported along two edges and built in along the other two. In their analysis to a varying combination of shear and in-plane bending, for which no theoretical solution exists, and assesses the suitability of analytical techniques and finite element analysis to predict this behaviour. V. Piscopo [2010] investigated the Shimpi theory for buckling analysis of thick rectangular plates and taking into account the shear deformations. The finite element method has long been recognized as one of the most effective numerical method for analyzing the buckling load of thin plate like structures under arbitrary loading and boundary conditions. Chee-Kiong Chin et al. [1993] presented a finite element method using thin-plate elements. This method was capable of predicting the buckling capacity of arbitrarily shaped thin-walled structural members under any general load and boundary conditions. Several researchers have studied buckling of composite plates by, Arthur W. Leissa [1986] purposed to clarify the under what conditions bifurcation buckling loads can exist for generally unsymmetrically laminated plates for arbitrary ply orientation, as well as for the important, special cases of antisymmetric angle-ply and cross-ply laminates. Mahdi Damghani et al. [2011] has studied the critical buckling of composite plates with through the length delaminations by using exact stiffness analysis and the Wittrick–Williams algorithm. MeisamMohammadi, et al. [2010] obtained an exact solution for the buckling analysis of thin functionally graded rectangular plates. Their work based on the classical plate theory and using the principle of minimum total potential energy, the equilibrium equations are obtained. In the present study, the buckling analysis of thin rectangular FGM plates is studied. Based on the classical plate theory and using the principle of minimum total potential energy, the equilibrium equations are obtained. The resulting equations are solved for different loading conditions. Finally, the critical buckling loads for a FGM plate with simply supported boundary conditions and different loading conditions, some aspect ratios and various power law index of FGM, are shown. 2. Materials and Methods 2.1 Functionally Graded Materials (FGM) Consider a case when FGM plate made up of a mixture of ceramic and metal as show in Fig. 1. A rectangular FGM plate of length a, width b, and thickness h, referred to the rectangular Cartesian coordinates (x, y, z). We assume that the effective material properties Pof the plate linearly vary with respect to the thickness coordinate as follow Ref. [8]
P( z ) = PmVm + PV c c
1
Fig. 1 Geometry of the FGM plate where P(z) denotes a material property of FGM plate which maybe substituted by modulus of elasticity E, the coefficient of thermal expansion Į, the conductivity K, Poisson’s ratio ˶and thesubscripts m and c refer to the metal and ceramic constituents, respectively. For power-law FGM, ceramic volume fraction function is expressed as:
750
I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
§ 2z + h · Vc = ¨ ¸ © 2h ¹
n
2
2.2 Physical Neutral Surface of the FGM Plate For plate made of FGM, the neutral surface may not coincide with its geometric mid-surface. The distance of the neutral surface (d) from the geometric mid-surface may be expressed as h/ 2
d=
³
zE ( z )dz
³
E ( z )dz
3
−h / 2 h/2
−h / 2
2.3 Governing Equations Let us refer to the coordinate system of Fig.2 with z axis having the origin on the plate middle plane. The basic assumptions of the classical plate theory are: 1. The displacements of the plate are small in comparison to the plate’s thickness and the strains and mid-surface slopes are much less than unity. 2. The stress σ z is negligible respect to the in-plane stresses σ x and σ y . 3. The transverse shear stresses
σ xz and σ yz are small in comparison with the in-plane components σ x , σ y and σ xy
. In other words, “normals remain normal”.
Fig. 2 Rectangular plate reference system Then the displacement field in the (x, y, z) reference system has the following form ∂w( x, y ) U = u ( x, y ) − z ∂x ∂w( x, y ) 4 V = v ( x, y ) − z ∂y W = w( x, y ) Where u, v are in-plane displacements at a point of the mid-plane and U, V and W are displacement components of a typical point in the plate. The strain-displacement relationship are given as
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757 ∂U ½ ° ∂x ° ° ° ° ε x ½ ° ∂V ° °ε ° ° ∂y ° ° y ° ° ° ° ° ∂U ∂V ° + ®γ xy ¾ = ® ¾ °γ ° ° ∂y ∂x ° ° yz ° ° ∂V ∂W ° + °¯γ xz ¿° ° ° ° ∂z ∂y ° ° ∂U ∂W ° + ° ° ∂x ¿ ¯ ∂z
5
By substituting Eq. (4) into Eq. (5), the strain can be expressed as
∂u ∂2w −z 2 ∂x ∂x ∂v ∂2w εy = − z 2 ∂y ∂y
εx =
6
∂u ∂v ∂2w + − 2z ∂y ∂x ∂x∂y =0
γ xy = γ yz
γ xz = 0 ε x and ε y are the normal strains and γ xy
is shear strain at the middle surface of the plate.
The curvatures are shown by k x = w, xx , k y = w, yy , k xy = 2 w, xy . The stress-strain relationships of the functionally graded plate in the global x-y-z coordinates system can be written as σ xx ½ ª Q11 Q12 0 º ε x ½ °σ ° « »° ° 7 ® yy ¾ = «Q21 Q22 0 » ®ε y ¾ °τ ° « 0 ° ° 0 Q66 »¼ ¯γ xy ¿ ¯ xy ¿ ¬ where σ x and σ y are the normal stresses and τ xy is shear stress of the plate. Q11 = Q66 =
E ( z)
, Q12 = Q21 =
1−υ E (z) 2
υE ( z) , 1−υ 2
2 (1 + υ )
E (ε x + υε y ) 1 −υ 2 E σy = (ε y + υε x ) 1 −υ 2 E τ xy = γ xy 2 (1 + υ )
σx =
The constitutive relations are written as E1 (ε x + υε y ) + 1 −Eυ2 2 ( kx + υ k y ) 1 −υ 2 E E N y = 1 2 ( ε y + υε x ) + 2 2 ( k y + υ k x ) 1−υ 1 −υ E1 E2 N xy = k γ xy + 2 (1 + υ ) (1 + υ ) xy Nx =
8
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
E2 (ε x + υε y ) + 1 −Eυ3 2 ( kx + υ k y ) 1 −υ 2 E2 My = (ε y + υε x ) + 1 −Eυ3 2 ( k y + υ kx ) 1−υ 2 E3 E2 M xy = k γ xy + 2 (1 + υ ) (1 + υ ) xy Mx =
where
E1 = Em h +
( Ec − Em ) h , E n +1
2
9
1 · § 1 = ( Ec − Em ) h 2 ¨ + ¸ © n + 2 2n + 2 ¹
· Em h 1 1 h3 § 1 + ( Ec − Em ) ¨ − + ¸ 12 12 ¨© n + 3 n + 2 4 ( n + 1) ¸¹ 3
E3 =
In the above equations, Ni and Mi are force and moment resultants, respectively. The equilibrium equations of a perfect FGM plate are
N x , xx + 2 N xy , xy + N y , yy = 0 M x , xx + 2 M xy , xy + M y , yy = 0
10
q + N x w, xx + N y w, yy + 2 N xy w, xy = 0 Using Equations 8 and 9, the equilibrium in Equation (10) may be reduced to one equation as
D∇4 w − N x w, xx − 2 N xy w, xy − N y w, yy − q = 0
11
where
D=
E1E3 − E2 2
(
E1 1 − υ 2
)
To establish the stability equations, the critical equilibrium method is used. 3. Finite Element Formulations A four node rectangular element is used for this analysis. Each node of the element has three degrees of freedom, one transverse displacement and two rotations about x and y axis respectively shows in fig.3. Hence, the displacement function of the rectangular element can be approximated by ݓൌ ߙଵ ߙଶ ݔ ߙଷ ݕ ߙସ ݔ ߙହ ݕݔ ߙ ݕଶ ߙ ݔଷ ߙ଼ ݔଶ ݕ ߙଽ ݕݔଶ ߙଵ ݕଷ ߙଵଵ ݔଷ ݕ ߙଵଶ ݕݔଷ 12
w = ª¬1 x
x2
y
xy
y2
x3
x2 y
xy 2
y3
x3 y
xy 3 º¼ {α }
13
ݓൌ ሾܩሺݔǡ ݕሻሿሼߙሽ
14
തതതത ሽ் ൌ ሾܣሿ ሼߙሽ ሼܹ
15
തതതത ሽ ሼߙሽ ൌ ሾܣሿ ିଵ ሼܹ
16
Substituting (15) and (16) becomes തതതത ሽ் ൌ ሾሺݓଵ ሼܹ
ߠ௫ଵ
ߠ௬ଵ ݓ ڮସ
ߠ௫ସ
ߠ௬ସ ሻሿ ൌ ሾܵሺݔǡ ݕሻሿሼݓ ሽ
17 26
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
Fig.3 Geometry of the four node rectangular element Shape function ଵ ଶ ଶ ଼ۍ൫ͳ ݔ ݔ൯൫ͳ ݕ ݕ൯ሺʹ ݔ ݔ ݕ ݕെ ݔെ ݕሻې ێ ۑ ܵ ் ሺݔǡ ݕሻ ൌ ێ ൫ͳ ݔ ݔ൯൫ ݕ ݕ ൯ሺ ݕଶ െ ͳሻ ۑ ଼ ێ ۑ ଶ ൫ݔ ݔ൯൫ͳ ݕݕ ൯ሺݔ െ ͳሻ ۏ ے
18
଼
In deriving this result, it is simpler to use the expression Eq. (18) for substitute w after performing the integration. A typical integral is then the element stiffness matrix and geometric stiffness matrices are derived on the basis of principle of minimum potential energy and work done. The element stiffness matrix is derived as
U( ) = e
{ }
1 (e) q 2
T
{ }
ª K s ( e) º q( e) ¬ ¼
[ ks ] = ³³³ BT DBdV e
where
19
ª ∂2 º « 2 » x ∂ « » « ∂2 » Eh3 [ B] = « 2 » [ S ] , D = y 12(1 ∂ − v2 ) « » « ∂2 » «2 » «¬ ∂x∂y »¼
The element geometric stiffness matrix 2 ª § ∂S ·2 e § ∂S · § ∂S · § ∂S · º ª¬ k g ¼º = ³ « N x ¨ ¸ + N y ¨ ¸ + 2 N xy ¨ ¸ ¨ ¸ »dA ∂x © ∂x ¹ © ∂y ¹ »¼ © ∂y ¹ Ω« ¬ © ¹
20
The overall stiffness and geometric stiffness matrices, Ks and Kg, for a plate structure can be obtained by a standard assembly procedure used in the finite element method after each element stiffness matrix and geometric stiffness matrix has been transformed into global coordinatesystem. Buckling conditions for a structure are obtained when the second variation of the total potential energy vanishes, i.e.
det ( K s + λ K g ) = 0
In which λ is the buckling load.
21
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
4. Results and Discussions 4.1 Validation of the Results In order to validate the accuracy of the present formulations, a comparison has been carried out with the results obtained by Ref. [7], for all edges SSSS and SCSC isotropic plates. The critical buckling load has been listed in Table 2 for a rectangular simply supported plate subjected to uniaxial and biaxial compression and side-thickness ratios. As this table shows, the present results have a good agreement with those reported in Ref. [7]. Table 1 Non-dimensional critical buckling load convergence study of an square plate with t=0.01m Number of elements
Uniaxial compression
Biaxial compression
10X10 15X15 20X20 25X25 30X30 Ref [7]
39.0651 39.2926 39.3734 39.4111 39.428 39.4784
19.5325 19.6463 19.6867 19.7056 19.7081 19.7392
Table 2 2
Comparison of the non-dimensional critical buckling load ( Pcr L / D ) for an isotropic plate Boundary conditions SSSS SCSC
a/b ratio 0.5 1 0.5 1
Uniaxial compression Ref.[7] Present study 15.4212 15.392 39.4784 39.428 18.9775 18.948 75.9099 75.675
Biaxial compression Ref.[7] Present study 12.3370 12.313 19.7392 19.708 14.6174 14.592 37.7996 37.704
4.2 Results After verifying the accuracy of the present solution, in order to obtain the following new results, it is assumed that the FGM plate is made of a mixture of silicon nitride and stainless steel. Figure 4 illustrates the simply supported rectangular plate subjected to uniaxial and biaxial compression loading. To carry out the numerical simulation, following material properties have been considered. SUS304,ߩ ൌ ͺǡͳ݇݃Ȁ݉ଷ , ܧൌ ʹͲǤͺ ൈ ͳͲଽ ܲܽ ,ߴ ൌ ͲǤ͵ͳ ܵ݅ଷ ܰସ , ߩ ൌ ʹǡ͵Ͳ݇݃Ȁ݉ଷ , ܧൌ ͵ʹʹǤʹ ൈ ͳͲଽ ܲܽ , ߴ ൌ ͲǤʹͶ
Fig.4 Rectangular plate subjected to uniaxial and biaxial compression load
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
4.2.1Rectangular FGM Plates Under Uniaxial and Biaxial Compression In table 3 shows the critical buckling loads for different values of the ratio t/b of rectangular FGM plate under uniaxial compression. The convergence of the solution obtained by FEM has been studied and in all cases it was quite quickly achieved. From the analysis it is clear that there is a very good accordance with the proposed analysis. From the obtained results, it seems clear that the FEM buckling analysis results deceases, when the ratio t/b increases, as it would be predictable. Anyway, it is fundamental to note that these obtained results lower if referred to the critical value, the most important parameter in a buckling analysis. In tables 3 and 4 shown for FGM plates under uniaxial and biaxial compression with n= 0, 1, 2, 5 for different values of the ratio t/b.In figs.5 and 6 shown that the critical buckling load parameter variation with respective index value with different thickness values (t=0.01, 0.02 and 0.03) and aspect ratios (a/b=0.25, 1). The FGM plate is subjected to biaxial compression load, the increases index value effects the critical buckling load parameter is shown in fig. 6.Similarly the critical buckling load variation with different aspect ratios (a/b=0.25, 1) shown in figs. 6 (a) and (b). Table 3 The critical buckling load (MN/m) of FGM rectangular plates Aspect ratio a=0.25m,b=1m
a=1m, b=1m
a=4m, b=1m
Power law index n 0 1 2 5 0 1 2 5 0 1 2 5
0.01m 5.0576 4.3156 4.047 3.7379 1.1222 0.95022 0.89103 0.82293 1.0828 0.92289 0.86513 0.79875
Uniaxial compression 0.02m 40.461 34.525 32.376 29.903 8.9104 7.6018 7.1283 6.5834 8.6622 7.3831 6.921 6.39
0.03m 136.56 116.52 109.27 100.92 30.072 25.656 24.058 22.219 29.235 24.918 23.358 21.566
Table 4 The critical buckling load (MN/m) of a FGM rectangular plate Aspect ratio a=0.25m,b=1m
a=1m, b=1m
a=4m, b=1m
Power law index n 0 1 2 5 0 1 2 5 0 1 2 5
0.01m 4.7601 3.9373 3.7899 3.6517 0.55692 0.44605 0.44329 0.42712 0.29751 0.24608 0.26872 0.22823
Biaxial compression 0.02m 38.081 31.499 30.319 29.214 4.4552 3.6845 3.5463 3.4168 2.3801 1.9687 1.8950 1.8259
0.03m 128.52 106.31 102.33 98.597 15.036 12.435 11.969 11.532 8.0327 6.6443 6.3955 6.1623
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I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
DE
DE
P
P
&ULWLFDOEXFNOLQJORDG
&ULWLFDOEXFNOLQJORDG
P
P
P
P
,QGH[YDOXHQ
,QGH[YDOXHQ
Fig. 5 Critical buckling load of FGM plate verses power law index n with aspect ratio a/b=0.25, 1 under uniaxial compression case. DE
&ULWLFDOEXFNOLQJORDG
P P
P
,QGH[YDOXHQ
(a) DE
&ULWLFDOEXFNOLQJORDG
P
P
P
,QGH[YDOXHQ
(b) Fig. 6 Critical buckling load of FGM plate verses power law index n with aspect ratio (a) a/b=0.25, and (b) a/b= 1 under biaxial compression case.
I. Ramu and S.C. Mohanty / Procedia Engineering 86 (2014) 748 – 757
5. Conclusions The buckling behaviour of rectangular FGM plates under compression is investigated using the finite element method. The compression loadings are assumed to be uniaxial compression and biaxial compression. The effective material properties are computed using the simple power law equation of the volume fraction of the plate constituents. The critical buckling load of the rectangular plate under uniaxial compression is greater than the biaxial compression. From tables 3 and 4 results show that as the aspect ratio a/b (0.25, 1 and 4) increases, the critical buckling load reduces. And the variation of thickness (0.01, 0.02 and 0.03) for each aspect ratios are also shown in tables 3 and 4. The critical buckling load increases by increasing the thickness. In figs. 6 (a) and (b) it is shown that the critical buckling load decreases as the volume fraction index n increase. This is because as volume fraction index increases, the contained quantity of ceramic decreases. References 1. Chee-Kiong Chin, Faris G. A. AI-Bermani, and SritawatKitipornchai [1993], “Finite Element Method for Buckling Analysis of Plate Structures”, Journal of Structural Engineering, Vol.119, No. 4. 2. V. Piscopo [2010], “Refined Buckling Analysis of Rectangular Plates under Uniaxial and Biaxial Compression”, World Academy of Science, engineering and Technology 46, p. 554-561. 3. Hosseini-Hashemi S, Khorshidi K, Amabili M [2008], “Exact solution for linear buckling of rectangular Mindlin plates”, Journal of Sound and Vibrations. 315, 318–342 (2008). 4. Arthur W. Leissa [1986], “Conditions for Laminated Plates to Remain Flat Under Inplane Loading”, Composite Structures 6, 261-270. 5. Mahdi Damghani, David Kennedy, Carol Featherston [2011], “Critical buckling of delaminated composite plates using exact stiffness analysis”, Computers and Structures 89 (2011) 1286–1294 6. C.A. Featherston, A. Watson [], “Buckling of optimised flat composite plates under shear and in-plane bending”, Composites Science and Technology 65 (2005) 839–853 7. MeisamMohammadi, Ali Reza Saidi, EmadJomehzadeh [2010], “Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates”, Appl Compos Mater (2010) 17:81–93 8. Shyang-HoChi,Yen-Ling Chung [2006],“Mechanical behavior of functionally graded materialplates under transverse load—Part I: Analysis”, International Journal of Solids and Structures 43 (2006) 3657–3674.
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