Post-buckling analysis of rectangular plates comprising Functionally Graded Strips in thermal environments

Computers and Structures xxx (2014) xxx–xxx

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Post-buckling analysis of rectangular plates comprising Functionally Graded Strips in thermal environments H.R. Ovesy a,⇑, S.A.M. Ghannadpour b, M. Nassirnia a a b

Dept. of Aerospace Eng. and Centre of Excellence in Computational Aerospace Eng., Amirkabir University of Technology, Tehran, Iran Aerospace Engineering Dept., Faculty of New Technologies and Engineering, Shahid Beheshti University, G.C., Tehran, Iran

a r t i c l e

i n f o

Article history: Accepted 29 September 2014 Available online xxxx Keywords: Post-buckling analysis Thermal loading Classical plate theory Functionally Graded Strips

a b s t r a c t Description is given of a semi-analytical finite strip method for analysing the post-buckling behaviour of functionally graded rectangular plates in thermal environments where plates are under uniform, tent-like or nonlinear temperature change across the thickness. The material properties are assumed to vary through the thickness according to the power law. The formulations are based on the classical plate theory and the concept of the principle of the minimum potential energy. The Newton–Raphson method is used to solve the equilibrium equations. A range of applications are described and the numerical results are compared to the available results, wherever possible. Ó 2014 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials (FGMs) are those in which the volume fractions of two or more constituents are varied continuously as a function of position along certain dimension of the plate to achieve a required function. By gradually varying the volume fraction of constituent materials, their mechanical properties exhibit a smooth and continuous change from one surface to the other one. The ceramic constituent of the material provides the hightemperature resistance due to its low thermal conductivity. Thus FGMs have received considerable attention as one of advanced inhomogeneous composite materials in many engineering applications by eliminating interface problems and alleviating thermal stress concentrations. Most of the researches on FGMs have been restricted to thermal stress analysis, fracture mechanics, and optimization. However, very little work has been done to consider the buckling, post buckling and vibration behaviour of structures constructed of FGM. The Finite Strip Method (FSM) is a special form of the Finite Element Method (FEM). In terms of computational expenses, not only it takes much shorter computing time and smaller amount of core for solution of comparable accuracy, but also it require very small amount of input data due to the small number of mesh lines involved. However, like any other methods, the conventional/ semi-analytical FSM has its own limitations and drawbacks such as boundary conditions limitations induced by shape functions, lacking ability to analyse plates with cut-out and using only ⇑ Corresponding author.

rectangular elements. These issues could be overcome by incorporating other versions of FSM, i.e. B-spline FSM and complex FSM, to name a few. The first two authors of the current paper and their co-workers have made several contributions by developing different versions of finite strip methods, namely full-energy semi-analytical FSM [1,2], full-energy spline FSM [1], semi-energy FSM [2] and exact FSM [3]. Ge et al. investigated post buckling behaviour of composite laminated plates with the aid of the B-spline finite strip method under the combination of temperature load and applied uniaxial mechanical stress [4]. Liew et al. examined the post-buckling behaviour of functionally graded material FGM rectangular plates that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of uniform temperature change, in-plane forces, and constant applied actuator voltage [5]. Sohn et al. worked on static and dynamic stabilities of functionally graded panels based on the first-order shear deformation theory which are subjected to combined thermal and aerodynamic loads. They derived equations of motion by the principle of virtual work and numerical solutions were obtained by a finite element method. In addition, they utilized the Newton–Raphson method to get solutions of the nonlinear governing equations [6]. The authors of the current work developed the semi-analytical finite strip method (S-a FSM) for analyzing the buckling behaviour of rectangular FGM plates under thermal loadings by incorporating the total potential energy minimization and solving the corresponding eigenvalue problem [7]. They also extended the same method to predict post-buckling behaviour of simply-supported FG plates

http://dx.doi.org/10.1016/j.compstruc.2014.09.011 0045-7949/Ó 2014 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

Please cite this article in press as: Ovesy HR et al. Post-buckling analysis of rectangular plates comprising Functionally Graded Strips in thermal environments. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2014.09.011

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subjected to the three types of thermal loadings, i.e. uniform temperature rise, tent-like temperature distribution and nonlinear temperature change across the thickness [8]. Material properties were assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. For simplification, the properties were assumed to be temperature-independent. It is noted that the current paper is an updated and revised version of the conference paper [8]. It is also noted that in the aforementioned conference paper and for that matter in the current paper, it is the first time that the concept of FGS is introduced for thermal post-buckling analysis by comparing two different approaches for solving heat conduction equation. The application and scope of the current paper are strengthened by studying more numerical results corresponding to different boundary conditions and approaches for solving the heat conduction equation.

b h

Fully Metal

A

Fully Ceramic

x

y z Fig. 2. A typical FGS.

2. Theoretical developments Thin plates/shells are generally referred to those structures whose thickness is very small compared with the other two dimensions. Even though there is no precise definition for a thin plate, if the ratio of the thickness to the shorter length of other two dimensions is less than 0.05, the plate can practically be considered thin. Classical Plate Theory (CPT) allows studying thin plates once the Kirchhoff hypothesis holds:  Straight lines perpendicular to the mid-surface before deformation remain straight after deformation.  The transverse normals are inextensible.  The transverse normals rotate such that they remain perpendicular to the mid-surface after deformation. Otherwise, for analysis of thick and moderately thick plate structures, other suitable theories, i.e. higher- and first-order shear deformation theory should be taken into account by relaxing the first and the third assumptions, respectively. Thus, throughout the theoretical developments of this paper which focus on thin plate structures, an initially flat plate based on the classical plate theory is assumed. It is worth mentioning that in this plate theory, all three transverse strain components and subsequently shear stress components are zero by definition, whereas in other two developed theories, approximate distribution of shear stresses through the plate thickness is considered. A rectangular FGP whose longitudinal and transverse dimensions are A and B, respectively, is supposed. This plate can be divided into several strips called Functionally Graded Strips (FGS) while, as illustrated in Fig. 1, they are laid parallel to one another and to the longitudinal edges of the plate. It is noted that a single FGS (shown in Fig. 2) forms part of a rectangular FGP of length A (i.e. the same length as that of a strip) and width b (with B P b). The composition is assumed to vary in

such a way that the upper surface of the strip is completely metal (designated by surface m at z = h/2), whereas the lower surface is fully ceramic (designated by surface c at z = h/2). Thus, it is assumed that the material properties of the FGS such as the modulus of elasticity E, shear modulus G, thermal expansion coefficient a and thermal conduction coefficient K are changed in the thickness direction z by a function #(z) which is introduced by power law variations of material properties distribution as

 #ðzÞ ¼ #cm

z 1 þ h 2

n þ #m

ð1Þ

while Poisson’s ratio v is assumed to be constant. In Eq. (1), #m and #c denote values of the variables at surface m and surface c of the strip, respectively and #cm = #c  #m. Moreover, in the above equation, the term (z/h + 1/2)n is known as the volume fraction of the ceramic phase, and n (volume fraction index) is non-negative real value indicating the material variation profile through the thickness direction. As a result of the CPT assumption, the Kirchhoff normalcy condition is incorporated, and thus:

 ðx; y; zÞ ¼ uðx; yÞ  z @wðx;yÞ u @x

tðx; y; zÞ ¼ tðx; yÞ  z @wðx;yÞ @y

ð2Þ

 y; zÞ ¼ wðx; yÞ wðx;  and w ; t  are components of displacement at a general where u point, whilst u, t and w are similar components at the middle surfaces (z = 0). Using Eq. (2) in the Green’s expressions for the in-plane nonlinear strains and neglecting lower-order terms in a manner consistent with the usual Von-Karman assumptions incorporated with the thermal effects gives the following expressions for strains at a general point:

em ¼ e  eT ¼ ðel þ enl þ zwÞ  eT h FGS

A Nodal Lines

where em and eT are mechanical and thermal strains, respectively, and

el ¼

8 > < > : @u

@u @x @t @y

Fig. 1. Discretization of a rectangular FGP.

eT

9 > = > ;

;

enl ¼

8  2 9 1 @w > > > > > < 2 @x 2 > = 1

@w

2 @y > > > > > : @w @w > ; @x @y 8 9 > <1> = ¼ aðzÞ  DTðx; y; zÞ 1 > : > ; 0

þ @@xt @y

B

ð3-aÞ

;

8 @2 w 9  @x2 > > > > < = 2  @@yw2 w¼ > > > > : @2 w ; 2 @x@y

ð3-bÞ

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where a(z) is thermal expansion coefficient and DT(x, y, z) = T(x, y, z)  Tref is applied temperature change in Kelvin. It is noted that Tref is an initial temperature which is used as a reference. In the case of non-uniform thermal loading for FG plates, Tref is considered to be equal to the temperature at the metal surface of the plate (i.e. Tm) which is kept constant throughout the analysis. In the presence of a temperature loading and on the assumption that the plate is in a state of plane stress, the stress–strain relationship at a general point for the plate becomes:

8 > > <

9

308

2

9

8

91

 rx > Q 11 Q 12 Q 16 > > aðzÞ  DTðx; y; zÞ > > ex > > > > < = 6 = > =C 7B< 6 7 B  ry ¼ 4 Q 21 Q 22 Q 26 5@ ey  aðzÞ  DTðx; y; zÞ C A > > > > > > > > > > : : > : ; ; > ; rxy cxy Q 61 Q 62 Q 66 0 ð4Þ where Qij (i, j = 1, 2, 6) form the reduced stiffness matrix Q. By the use of Eqs. (3) and (4) and appropriate integration through the thickness, the constitutive equations for a FGS can be derived as:

8 Nx > > > > > N > y > > > > < Nxy

8 > > > > > > > > > Z h=2 > <

9 > > > > > > > > > > =

9 > > > > > > > > > > =

rx ry " # ( ) ( T ) sxy feg ½E1  ½E2  fN g   ¼  dz ¼ > > > > Mx > ½E2  ½E3  fwg h=2 > zrx > > fMT g > > > > > > > > > > > > > > > > > > My > zry > > > > > > > > > > > > > : : ; ; M xy zsxy ð5Þ T

T

where {N } and {M } are membrane and bending matrices due to the thermal loading, respectively. In the above equation, Nx, Ny and Nxy are the membrane direct and shearing stress resultants per unit length and Mx, My and Mxy are the bending and twisting stress couples per unit length. In addition, the strip stiffness coefficients are defined as

ðE1ij ; E2ij ; E3ij Þ ¼

Z

h=2

Q ij ð1; z; z2 Þdz;

i; j ¼ 1; 2; 6

ð6Þ

h=2

The strain energy per unit volume is 1/2rTem. By using Eqs. (3) and (4) and integrating with respect to z coordinate through the thickness, the strain energy can be reduced as follows: ZZ

 T  1 Us ¼ el ½E1 el  2eTl ½E2 w þ wT ½E3 w þ 2eTl ½E1 enl  2eTnl ½E2 w 2   þ eTnl ½E1 enl dx dy ZZZ  T 1 þ eT ½Q eT  2eTl ½Q eT  2eTnl ½Q eT þ 2wT ðz½Q ÞeT dx dy dz ð7Þ 2

are transverse polynomial interpolation functions of various types and orders. The crosswise functions appearing in Eq. (8) can be expressed in shape function form, for each longitudinal series term n, as

g u ðyÞ ¼ Nu1 ðyÞ  u1 þ Nu2 ðyÞ  u2 þ N u3 ðyÞ  u3

g v ðyÞ ¼ Nv1 ðyÞ  v 1 þ Nv2 ðyÞ  v 2 þ Nv3 ðyÞ  v 3

ð9Þ

w w w g w ðyÞ ¼ Nw 1 ðyÞ  w1 þ N 2 ðyÞ  h1 þ N 3 ðyÞ  w2 þ N 4 ðyÞ  h2

where h = ow/oy and N(y) are transverse polynomial shape functions. As described in Fig. 3, the second-order Lagrangian shape function with middle node for u, v and cubic Hermitian polynomials for w are utilized to calculate more accurate results. After some algebraic and matrix manipulation, with the development of the finite strip displacement field according to the above equations, the potential energy of a finite strip can ultimately be expressed in the form T

T

T

U s ¼ d T d W þ 1=2d ½K  T d K  d þ 1=6d ½K 1 d T

þ 1=12d ½K 2 d

ð10Þ

where the column matrix d contains the strip degrees of freedom and W is a column matrix of constants. The quantity Td = Tc  Tm is a variable which is changed, whilst Tm is kept constant throughout the analysis, in order to trace the thermal post-buckling characteristics of the plate. It is noted that K, K⁄⁄, K1 and K2 are symmetric square stiffness matrices. Matrix K and K⁄⁄ are associated with linear behaviour and the effects of the temperature change on stiffness, respectively. Besides, the individual entries of matrix K1 and K2 are linear and quadratic functions of the displacements, respectively. In evaluating the various energy contributions shown in Eq. (10), all the integrations are determined analytically. For the whole plate, comprising an assembly of finite strips, the strain energy is simply the summation of the strain energies of all the individual strips, and correspondingly whole-plate matrices are generated by appropriate summations of strip matrices in the standard fashion. Thus, the potential energy for whole plate can be expressed in the same form as that of Eq. (10), as

T T W þ 1=2d T ½K  T K  d  þ 1=6d T ½K 1 d  U s ¼ d d d T ½K 2 d  þ 1=12d

ð11Þ

–

where the overbar ( ) indicates a whole plate quantity. The plate equilibrium equations are obtained by applying the principle of minimum potential energy. In the present study the

Solution of the non-linear problem is sought through the application of the principle of minimum potential energy. This, of course, requires the assumption of a displacement field to represent the variations of u, v and w over the middle surfaces. Here, the displacement field adopted for typical finite strip shown in Fig. 2 is:

u¼

ru X U i ðxÞg ui ðyÞ i¼1

v

rv X ¼ V i ðxÞg vi ðyÞ

w¼

i¼1 rw X

ð8Þ

W i ðxÞg w i ðyÞ

i¼1

where Ui, Vi, and Wi are trigonometric functions satisfying the kinematic conditions prescribed at the strip ends. In addition, ru, rv and rw represent the number of longitudinal terms assumed for the corresponding displacement functions. The gi(y) functions

Fig. 3. Transverse polynomial shape functions for FGS.

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iterative Newton–Raphson procedure is selected for solving the equations. Once the global equilibrium equations are solved and the nodal degrees of freedom are found for a particular thermal loading, it is possible to calculate the displacements u, v and w at any point in any finite strip using Eq. (8), and to determine force and moment quantities through use of Eq. (5). Particularly, the average longitudinal force (Nav) is determined by integrating the membrane stress resultant (Nx) over the strip middle surface area. Eventually, the total longitudinal force acting on a plate, could be exerted by the summation of such strip end forces as

ΔT1 ΔT0

X R b=2 R A b=2

N av ¼

strips

In this section, various results for thermal post-buckling analysis of rectangular FGM plates are obtained. The performance of the present FSM in solving the thermal post-buckling problem is demonstrated by comparing the results obtained by the current method with those presented in the literature. Subsequently, some more data for FG plates are presented. 3.1. Thermal post-buckling of isotropic plates A thin isotropic square plate with the following material properties is considered [4]:

A=B ¼ 1;

Fig. 4. Schematic tent-like temperature distribution.

ð12Þ

A

3. Numerical results and discussion

E ¼ 1;

A=h ¼ 100;

a ¼ 1  106 =K; t ¼ 0:25

The boundary conditions are assumed to be sliding simply supported (i.e. S3 boundary conditions in Ref. [4]); in other words

u ¼ w ¼ 0;

at x ¼ 0; A

v ¼ w ¼ 0;

at y ¼ 0; B

A

B

N ðx; yÞdxdy 0 x

To satisfy the above boundary conditions, the longitudinal functions (i.e. Ui, Vi, Wi) in Eq. (8) are selected as sine, cosine, and sine, respectively. Aforementioned longitudinal trigonometric functions inherently restrict boundary conditions for opposite edges at x = 0, A to be simply supported, notwithstanding this method has no constraint for longitudinal edges at y = 0, B. So similar to other developed forms of FSM, in order to bypass this limitation, B-spline shape functions are proposed to be employed. These alternative shape functions not only favour versatile boundary conditions at x = 0, A, but any support located between these two limits are also feasible to define. The convergence study is examined for two different uniform temperature distributions (i.e. DT = 135 K and DT = 280 K), see Table 1. In the current paper, eight finite strips are selected for calculating the results. The square plate is subjected to the following temperature distribution called tent-like (see Fig. 4):

  1 6 06y 2   1   1Þ 61 6y ¼ DT 0  2  DT 1  ðy 2

DTðx; yÞ ¼ DT 0 þ 2  DT 1  y

It is observed that DT2cr (the critical value of DT2) in the tent-like temperature case (DT2cr = 240.71 K) is about 85% greater than in the uniform loading case (DT2cr = 131.53 K). The calculated results are depicted with those from Ref. [4] in Figs. 5 and 6. In Fig. 5, the tent-like results are calculated for DT0/DT1 = 0.6354 (DT0 = 95.3125 K, DT1 = 150 K). It is clear that the two sets of DT2 versus wmax/h compare very closely to those obtained in previous work. It is worth mentioning that, whether in the post-buckling or pre-buckling regions, the average force Nav for a given DT2 is much greater in the uniform temperature case. 3.2. Thermal post-buckling of FGPs A FG plate made from aluminium (as metal constituent) and alumina (as ceramic constituent) with dimension ratios of A/B = 1 and B/h = 100 is considered. The Young’s modulus, the thermal expansion coefficient, and the heat conduction coefficient of each material are given as follows:

Ec ¼ 380 GPa; ac ¼ 7:4  106 =K; K c ¼ 10:4 W=m K Em ¼ 70 GPa; am ¼ 23  106 =K; K m ¼ 204 W=m K Poisson’s ratio is assumed to be constant (m = 0.3) over the entire plate. Similar to that assumed in Ref. [4], the boundary conditions on four edges are considered to be S1 fixed simply supported (u = v = w = 0). To satisfy the latter boundary conditions, the longitudinal functions (i.e. Ui, Vi, Wi) in Eq. (8) are selected as sine. For the first thermal loading, the behaviour of the above plate is investigated while the temperature uniformly increases from reference temperature (Tref = 300). The maximum deflection variation of plate regarding to temperature change is plotted in Fig. 7. According to the figure, two sets of data related to aluminium and alumina (fully isotropic and homogenous) reveal clear bifurcation points while other series, which are functionally graded through the thickness, deform by any change in temperature. As the second thermal loading case, a nonlinear temperature change across the thickness is assumed. The temperature profile

ð13Þ

 ¼ y=B and coordinate system origin is located on the plate where y corner. In Eq. (13), DT0 is the uniform temperature rise and DT1 is the temperature gradient; therefore, temperature distribution in y-direction of plate varies from DT0 on the longitudinal edges of plate to the maximum value equivalent to (DT0 + DT1) = DT2 at y = B/2. It should be noted that by substituting DT1 = 0, the uniform in-plane temperature distribution will be obtained.

Table 1 Convergence study with regard to the number of strips (uniform temperature distribution). No. of strips

4 6 8 10

DT = 135 K

DT = 280 K

Wmax/h

Nav (kN)

Wmax/h

Nav (kN)

0.1348 0.1452 0.1430 0.1430

0.01769 0.01765 0.01766 0.01766

0.9260 0.9256 0.9250 0.9250

0.02301 0.02248 0.02285 0.02285

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500.0 Uniform Temp [4] Uniform Temp(Present)

400.0

Tent-Like [4] Tent-Like(Present)

ΔT (K)

300.0 200.0 100.0 0.0

0.2

0.0

0.4

0.6

0.8

1.0

Wmax/h Fig. 5. Dimensionless central deflection of plate versus temperature rise.

2.5

related heat conduction equation is derived. Two different approaches are used to solve this differential equation, i.e. approximate and exact solutions. In the approximate approach, the differential equation is solved by means of a polynomial series and the approximate temperature distribution across the thickness is selected by taking the first seven terms of the series [7]. In the second approach, Eq. (14) is solved analytically. The distribution profiles for each approach are clearly depicted and compared in Fig. 8. It is noted that, in the approximate approach, even by taking the first fourteen terms of series (i.e. twice as the number of terms as selected before), the results are still surprisingly different from those obtained by the exact solutions. In Figs. 9 and 10, the maximum dimensionless deflection of some plates under nonlinear temperature change with regards to DTc is shown based on the approximate and exact temperature distributions, respectively. In this loading case, the fully metal surface has a constant temperature value (Tm = 300 K) while fully ceramic surface temperature (Tc) increases monotonously as compared with the reference temperature. Although the isotropic plates (fully made of aluminium or alumina), according to Figs. 9 and 10, have the same post-buckling

2.0

1.5 0.25

1.0 Uniform Temp [4] Uniform Temp (Present)

0.5

z/h

-100Nav [KN]

0.50

Tent-Like [4]

0.00

Tent-Like (Present)

0.0

0

100

200

300

400

500

600

-0.25

ΔT (K) Fig. 6. Longitudinal force versus variation of temperature rise.

-0.50 300

350

400

3.0

n=0,infinity (Exact)

2.5

w/h

n=0,infinity (7-terms)

n=0.5 (7-terms)

500 n=0.5 (Exact)

n=2 (Exact)

n=2 (7-terms)

Fig. 8. Temperature distribution comparison for some FG plates across the plate thickness.

2.0 1.5

2.0

1.0 0.5

1.5 0

25

50

75

100

125

150

175

200

w/h

0.0

ΔT (K) Alumina

n=0.5

n=2

Aluminum

1.0

Fig. 7. Dimensionless central deflection of the FG plate (S1 B.C.) versus DT uniform temperature rise across the thickness.

0.5

is obtained by solving the heat conduction equation for steadystate conditions as:

0.0



450

Temprature (K)

d dT ¼0 KðzÞ  dz dz

0

25

50

100

125

150

175

200

ΔT c (K)

ð14Þ

By substituting Eq. (1) in Eq. (14) and considering temperature boundary conditions (T = Tm at z = h/2 and T = Tc at z = +h/2) the

75

Alumina

n=0.5

n=2

Aluminum

Fig. 9. Dimensionless central deflection of the FG plate (S1 B.C.) versus DTc nonlinear temperature change (approximate temperature distribution).

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H.R. Ovesy et al. / Computers and Structures xxx (2014) xxx–xxx

2.0

2.0

1.5

1.5

w/h

w/h

6

1.0

0.5

0.5

0.0

1.0

0

25

50 Alumina

75

100 ΔT c (K) n=0.5

125

150

175

200

0.0

0

25

50

75

100

ΔT c (K) n=2

Aluminum

Alumina

n=0.5

n=2

Aluminum

Fig. 10. Dimensionless central deflection of the FG plate (S1 B.C.) versus DTc nonlinear temperature change (exact temperature distribution).

Fig. 12. Dimensionless central deflection of the FG plate (SCSC B.C.) versus DTc nonlinear temperature change (exact temperature distribution).

behaviour under nonlinear temperature loadings, whether approximate or exact solution, it is clear that different post-buckling behaviours are obtained for FG plates based on the employed temperature distribution. That is to say that the maximum plate deflection based on the exact temperature distribution is lower than those obtained based on the other distribution. In addition, as volume fraction index is increased, the contained quantity of ceramic decreases. In other words, when volume fraction index n is increased, the central displacement increased. Moreover, the deflection of the plate under uniform temperature rise is greater than that obtained under nonlinear temperature distribution. In the case of non-linear temperature change, on examining post-buckling behaviour of FG plates, in addition to S1 fixed simply supported, a SCSC boundary condition (simply supported at x = 0, A and clamped at y = 0, B) is also considered. The longitudinal functions (i.e. Ui, Vi, Wi) in Eq. (8) are selected as sine. The maximum dimensionless deflections of some plates with latter boundary conditions are illustrated in Figs. 11 and 12. In Fig. 11, the results are extracted from approximate temperature distribution across the plate thickness whilst the data in Fig. 12 is extracted from exact solution (see Table 1). Figs. 11 and 12, again, show that the results extracted from approximate and exact approach differ from each other, and the central deflection of plates decreases by decreasing power law

Table 2 Dimensionless central deflection due to nonlinear temperature change with respect to geometric ratio for two FG plates (B/h = 100, DTc = 100 K).

2.0

w/h

1.5

1.0

0.5

0.0

Boundary conditions

Aspect ratio

n = 0.5

n=2

A/B = 1 A/B = 2

1.170 0.943

1.372 1.110

A/B = 1 A/B = 2

1.014 0.325

1.286 0.318

A/B = 1 A/B = 2

1.134 0.905

1.326 1.066

A/B = 1 A/B = 2

0.889 0.309

1.168 0.388

Approximate solution S1

SCSC

Exact solution S1

SCSC

index. Additionally, it is obvious that, either for approximate or exact solution, the central deflection of plates with SCSC boundary conditions are less than those of the corresponding S1 plates. This is because the boundary conditions on longitudinal edges of SCSC plate are stiffer than those of SSSS plate. On the other hand, in order to compare the effects of different boundary conditions and geometric parameters, considering two presented approach for solving heat conduction equation, on the central deflection, the results are tabulated in Table 2. Table 2 clearly shows that central deflection of FGPs decreases by increasing geometric aspect ratio (A/B). The deflection reduction rate of FGPs with S1 boundary conditions (around 20%) is less than those with SCSC boundary conditions (around 70%). Moreover, as mentioned before, the results extracted from approximate approach are different from those obtained from exact approach. 4. Conclusions

0

25

50

75

100

ΔT c (K) Alumina

n=0.5

n=2

Aluminum

Fig. 11. Dimensionless central deflection of the FG plate (SCSC B.C.) versus DTc nonlinear temperature change (approximate temperature distribution).

The application of the FSM is successfully extended to the analysis of post-buckling behaviour of functionally graded plates subjected to thermal loadings. In the case of solving the heat conduction equation for steady-state conditions, two different approaches were introduced and, depending on the selected solution approach, the results show that post-buckling behaviour of FGPs was different.

Please cite this article in press as: Ovesy HR et al. Post-buckling analysis of rectangular plates comprising Functionally Graded Strips in thermal environments. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2014.09.011

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It is seen that by increasing the volume fraction of ceramic phase, the effects of thermal stresses decreases due to a higher stiffness value of ceramic material. It can be concluded that FG plates under either uniform or nonlinear temperature change across the thickness show better postbuckling characteristics than homogenous metal plates. In other words, maximum out-of-plate deflection within post-buckling regime is lower than those values corresponding to metal plates. Clamped boundary conditions on the longitudinal edges (in SCSC comparing with S1) increase stiffness of plate structures; thus, withstand more temperature change and deform less than the plates whose all boundary conditions are fixed simply supported. References [1] Ovesy HR, Ghannadpour SAM, Morada G. Post-buckling behavior of composite laminated plates under end shortening and pressure loading, using two versions of finite strip method. Compos Struct 2006;75:106–13.

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[2] Ovesy HR, Loughlan J, Ghannadpour SAM. Geometric non-linear analysis of channel sections under end shortening, using different versions of the finite strip method. Comput Struct 2006;84:855–72. [3] Ovesy HR, Ghannadpour SAM. An exact finite strip for the initial postbuckling analysis of channel section struts. Comput Struct 2011;89:1785–96. [4] Ge Y, Yuan W, Dawe DJ. Thermomechanical postbuckling of composite laminated plates by the spline finite strip method. Compos Struct 2005;71: 115–29. [5] Liew KM, Yang J, Kitipornchai S. Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. Int J Solids Struct 2003;40: 3869–92. [6] Sohn K-J, Kim J-H. Structural stability of functionally graded panels subjected to aero-thermal loads. Compos Struct 2008;82:317–25. [7] Ghannadpour SAM, Ovesy HR, Nassirnia M. Buckling analysis of functionally graded plates under thermal loadings using the finite strip method. Comput Struct J 2012. [8] Ovesy HR, Ghannadpour SAM, Nassirnia M. Thermal postbuckling behaviour of rectangular functionally graded plates using the finite strip method. In: Topping BHV, editor. Proceedings of the eleventh international conference on computational structures technology. Stirlingshire, UK: Civil-Comp Press; 2012. http://dx.doi.org/10.4203/ccp.99.51 [Paper 51].

Please cite this article in press as: Ovesy HR et al. Post-buckling analysis of rectangular plates comprising Functionally Graded Strips in thermal environments. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2014.09.011