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MECHANICAL ENGINEERING | RESEARCH ARTICLE
Thermal stability analysis of eccentrically stiffened Sigmoid-FGM plate with metal – ceramic – metal layers based on FSDTPham Hong Cong and Nguyen Dinh Duc
Cogent Engineering (2016), 3: 1182098
z
y
x
k2k1
Metal
Metal
Ceramic
Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
MECHANICAL ENGINEERING | RESEARCH ARTICLE
Thermal stability analysis of eccentrically stiffened Sigmoid-FGM plate with metal – ceramic – metal layers based on FSDTPham Hong Cong1,2 and Nguyen Dinh Duc2*
Abstract: This paper researches the thermal stability of eccentrically stiffened plates made of functionally graded materials (FGM) with metal – ceramic – metal layers subjected to thermal load. The equilibrium and compatibility equations for the plates are derived by using the first-order shear deformation theory of plates, taking into account both the geometrical nonlinearity in the von Karman sense and initial geometrical imperfections with Pasternak type elastic foundations. By apply-ing Galerkin method and using stress function, effects of material and geometrical properties, elastic foundations, temperature-dependent material properties, and stiffeners on the thermal stability of the eccentrically stiffened S-FGM plates in ther-mal environment are analyzed and discussed.
Subjects: Materials Science; Mechanical Engineering; Structural Mechanical Engineering
Keywords: nonlinear stability; eccentrically stiffened thick S-FGM plates; first-order shear deformation; temperature-dependent materials properties
*Corresponding author: Nguyen Dinh Duc, University of Engineering and Technology - Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam E-mail: [email protected]
Reviewing editor:Duc Pham, University of Birmingham, UK
Additional information is available at the end of the article
ABOUT THE AUTHORIn Vietnam, Nguyen Dinh Duc is one of the well-known scientists in mechanical science. He is a full professor of Vietnam National University, Hanoi.
Professor Nguyen Dinh Duc is the Head of Advanced Materials and Structures Laboratory of University of Engineering and Technology - Vietnam National University, Hanoi. He had graduated from Hanoi State University since 1984 and completed his Ph.D degree in 1991 at Moscow State University, Russia. Since 1997, he had become Doctor of Science (Dr. Habilitation) promoted by Russian Academy of Sciences. Webpage: http://uet.vnu.edu.vn/~ducnd/, http://irgamme.uet.vnu.edu.vn/gs-tskh-nguyen-dinh-duc/
PUBLIC INTEREST STATEMENTIn recent years, there has been significant interest in the development of functionally graded materials (FGMs) for engineering applications. FGM materials have been used in aerospace, nuclear, and microelectronics engineering applications, where the materials are required to work in extreme temperature environments. It is also important for these materials to maintain their structural integrity, with minimum failures due to material mismatch. The focus of this manuscript is on a theoretical analysis on the thermal stability of eccentrically stiffened plates made of FGMs with metal – ceramic – metal layers (S-FGM) subjected to thermal load. Both the FGM plate and the outside stiffeners are deformed under temperature and having temperature-dependent properties. The influences of the material and geometrical properties, elastic foundations, temperature-dependent material properties, and outside reinforced stiffeners on the thermal stability of the eccentrically stiffened S-FGM plates in thermal environment are analyzed and discussed. The outcomes from this work are important to composite engineers and designers.
Received: 20 November 2015Accepted: 20 April 2016Published: 14 May 2016
© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.
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Nguyen Dinh Duc
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Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
1. IntroductionSince its first introduction in 1984 by a group of material scientists in Japan (Koizumi, 1997), func-tionally graded materials (FGMs) have attracted considerable attention in many engineering appli-cations such as extremely high-temperature-resistant materials. To date, there have been a number of studies on the stability of eccentrically stiffened FGM plates. However, these studies have only been concentrated on thin structures using classical plate theory. Not much consideration has been given to eccentrically stiffened thick Sigmoid-FGM (S-FGM) plates with shear deformation behaviors, especially when material properties depend on temperature. An overview on studies that apply shear deformation theory to FGM plates is provided in the following part.
Wu (2004) has studied the thermal buckling and post-buckling behavior of simply supported FGM rectangular plates based on the first-order shear deformation plate theory. Ma and Wang (2004) studied the thermoelastic buckling behavior of functionally graded circular/annular plates based on first-order shear deformation plate theory. Duc and Tung studied the mechanical and thermal post-buckling of shear deformable FGM plates with temperature-dependent properties (Duc & Tung, 2010). Shen (1997) studied the thermal post-buckling analysis of imperfect laminated plates using a higher order shear deformation theory. Duc and Tung (2011) studied mechanical and thermal post-buckling of higher order shear deformable functionally graded plates on elastic foundations. Seren Akavci et al. used the first-order shear deformation theory for symmetrically laminated composite plates on elastic foundation (Seren Akavci, Yerli, & Dogan, 2007). Ghiasian et al. studied the thermal buckling of shear deformable temperature-dependent circular/annular FGM plates (Ghiasian, Kiani, Sadighi, & Eslami, 2014). Duc and Cong (2015) studied the nonlinear vibration of thick FGM plates on elastic foundation subjected to thermal and mechanical loads using the first-order shear deformation plate theory. In these studies (Seren Akavci et al., 2007; Duc & Cong, 2015; Duc & Tung, 2010, 2011; Ghiasian et al., 2014; Ma & Wang, 2004; Shen, 1997), the authors used shear deformation theory to study the nonlinear static stability of unstiffened FGM thick plates. Dung and Nga (2013) studied the nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first-order shear deformation theory but without temperature. Shen (2007) studied the thermal post-buckling behavior of shear deformable FGM plates with temper-ature-dependent properties. In Bich, Nam, and Phuong (2011), studied the nonlinear post-buckling of eccentrically stiffened functionally graded plates and shallow shells based on classical shell theory. The nonlinear stability of eccentrically stiffened functionally graded imperfect plates resting on elas-tic foundations has been further studied by Dung and Thiem (2012). In (Duc, 2014; Duc & Cong, 2014), Duc and Cong studied the nonlinear post-buckling of imperfect eccentrically stiffened thin FGM plates with temperature-dependent material properties under temperature while resting on elastic founda-tions using a simple power-law distribution (P-FGM) and the classical plate theory. Swaminathan, Naveenkumar, Zenkour, and Carrera (2015) studied the stress, vibration, and buckling analyses of FGM plates–A state-of-the art review. Thai and Kim (2015) studied a review of theories for the mod-eling and analysis of functionally graded plates and shells. Reddy and Chin (1998) studied the ther-mo-mechanical analysis of functionally graded cylinders and plates.
From the above review, to the best of our knowledge, it has been shown that there are no publica-tions on the nonlinear stability of a thick S-FGM plate (with metal – ceramic – metal layers) reinforced by stiffeners in a thermal environment using the first-order shear deformation plate theory. This paper will focus on studying the buckling and post-buckling of an eccentrically stiffened functionally graded thick plate on elastic foundations under thermal loads with both S-FGM plates and stiffeners having temperature-dependent properties and thermal deformations. The paper also analyzes and discusses the effects of material and geometrical properties, temperature, elastic foundations, and eccentric stiffeners on the buckling and post-buckling loading capacity of the functionally graded plate in thermal environments.
2. Functionally graded plates on elastic foundationsConsider a eccentrically stiffened thick S-FGM plate (metal – ceramic – metal) of length a, width b, and thickness h resting on an elastic foundation. A coordinate system (x, y, z) is established, in which
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Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
(x, y) plane is on the middle surface of the plate and z is the thickness direction (−h/2 ≤ z ≤ h/2), as shown in Figure 1.
By applying a Sigmoid power-law distribution, Young’s modulus and thermal expansion coefficient can be expressed in the form (Duc & Tung, 2010):
where N is volume-fraction index, the subscripts m and c refer to the metal and ceramic constituents, respectively, Poisson’s ratio (v) is assumed to be a constant and Ecm = Ec − Em, �cm = �c − �m.
A material property (Pr), such as the elastic modulus (E), and the thermal expansion coefficient (α) can be expressed as a nonlinear function of temperature (Touloukian, 1967), as:
where T = T0 + ΔT( z ) and T0 = 300 K (room temperature); P−1, P0, P1, P2, P3 are coefficients character-izing the constituent materials; and ΔT is the temperature rise from stress-free initial state. In short, T-D (temperature-dependent) will be used for the cases in which the material properties depend on temperature. Otherwise, T-ID will be used for the temperature-independent cases. The material properties for the latter scenario have been determined by Equation (2) at room temperature, i.e. T0 = 300 K.
3. Theoretical formulationThe present study uses first-order shear deformation plate theory to establish the governing equa-tions and determine the buckling loads and post-buckling paths of the eccentrically stiffened S-FGM plates.
The strains across the plate thickness at a distance z from the middle surface are given by:
where
where �0x , �0y are normal strains, �0xy is shear strain on the mid-plane of the plate, u, v are the displace-
ment components along the x, y directions; and ϕx, ϕy are the rotations in the (x, z) and (y, z) planes, respectively.
(1)[E(z)
�(z)
]
=
[Em�m
]
+
[Ecm�cm
]{ ((2z + h
)∕h
)N, −h∕2 ≤ z ≤ 0
((−2z + h
)∕h
)N, 0 ≤ z ≤ h∕2
(2)Pr = P0
(P−1T
−1 + 1 + P1T1 + P2T
2 + P3T3)
(3)�x = �0x + z�x, �y = �
0y + z�y , �xy = �
0xy + z�xy , �xz = w,x + �x, �yz = w,y + �y ,
(4)�0x = u,x +
(w,x
)2∕2, �0y = v,y +
(w,y
)2∕2, �0xy = u,y + v,x +w,xw,y ,
�x = �x,x, �y = �y,y , �xy = �x,y + �y,x
Figure 1. Geometry and coordinate system of an eccentrically stiffened S-FGM plate on an elastic foundation.
z
y
x
k2k1
Metal
Metal
Ceramic
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Hooke’s law for an S-FGM plate under thermal conditions is defined as:
For stiffeners in thermal environments with temperature-dependent properties, its form proposed adapted from Duc and Cong (2014), is as follows:
where E0, v0, α0 are Young’s modulus, Poisson’s ratio, and thermal expansion coefficient of stiffeners, respectively.
From Equation (4), the geometrical compatibility equation can be written as:
In order to investigate the FGM plates with stiffeners in the thermal environment, the materials’ moduli with temperature-dependent properties have been taken into account. In addition, it can be assumed that all elastic moduli of FGM plates and stiffener are temperature-dependent and they are deformed in the presence of temperature. Hence, the geometric parameters, the plate’s shape and stiffeners are varied through the deforming process due to the temperature change. Assuming that the thermal stress of stiffeners is subtle which distributes uniformly through the whole plate struc-ture, it can be ignored. Lekhnitsky smeared stiffeners technique can be adapted from Bich et al. (2011) for eccentrically stiffened FGM plate under temperatures as follows:
where A1, A2 are cross-section areas of stiffeners; d1, d2 are spacing of the longitudinal and transver-sal stiffeners; I1, I2 are second moments of cross-section areas; z1, z2 are eccentricities of stiffeners with respect to the middle surface of plate; b1, b2 are width of longitudinal and transversal stiffeners; h1, h2 are thickness of longitudinal and transversal stiffeners; and specific expressions of coefficients Aij, Bij, Dij are given in Appendix A and
After the thermal deformation process, the geometric shapes of stiffeners can be determined as follows (Duc & Cong, 2014).
(5)�x= E
(�x+ v�
y− (1 + v)�ΔT
)∕(1 − v2
), �
y= E
(�y+ v�
x− (1 + v)�ΔT
)∕(1 − v2
)
�xy
= E�xy∕(2(1 + v)
), �
xz= E �
xz∕(2(1 + v)
), �
yz= E�
yz∕(2(1 + v)
)
(6)�sx = E0�x − E0�0ΔT∕
(1 − 2v
), �sy = E0�y − E0�0ΔT∕
(1 − 2v
).
(7)�0
x,yy+ �
0
y,xx− �
0
xy,xy=(w,xy
)2−w
,xxw,yy.
(8)
Nx =(A11
+ E0AT1∕dT
1
)�0
x + A12�0
y +(B11
+ CT1
)�x,x + B12�y,y + Φ
1,
Ny = A12�0
x +(A22
+ E0AT2∕dT
2
)�0
y + B12�x,x +(B22
+ CT2
)�y,y + Φ
1,
Nxy = A66�0
xy + B66
(�x,y + �y,x
),
Mx =(B11
+ CT1
)�0
x + B12�0
y +(D11
+ E0IT1∕dT
1
)�x,x + D12�y,y + Φ
2,
My = B12�0
x +(B22
+ CT2
)�0
y + D12�x,x +(D22
+ E0IT2∕dT
2
)�y,y + Φ
2,
Mxy = B66�0
xy + D66
(�x,y + �y,x
),Qx = A44
(w,x + �x
),Qy = A55
(w,y + �y
),
(9)Φ1= −
1
1 − v
h∕2
∫−h∕2
E(z)�(z)ΔTdz, Φ2= −
1
1 − v
h∕2
∫−h∕2
E(z)�(z)ΔTzdz,
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For later use, the reverse relations are obtained from Equation (8), as follows
Substituting Equation (11) into Equation (8), yields:
where coefficients A∗
ij , B∗
ij , C∗
ij, and D∗
ij are given in Appendix A.
The nonlinear equilibrium equations of an eccentrically stiffened S-FGM plate on elastic founda-tions, based on the first-order shear deformation plate theory (Reddy, 2004), are:
Considering Equation (13a), a stress function f(x, y) may be defined as:
The three equations (13b) and (13c) become:
Substituting the expressions of Mx, My, Mxy in Equation (12), and Qx, Qy in Equation (9) into Equation (15), we obtain:
(10)bT1 = b1
(1 + �0ΔT
), bT2 = b2
(1 + �0ΔT
), hT1 = h1
(1 + �0ΔT
), hT2 = h2
(1 + �0ΔT
)
zT1 = z1(1 + �0ΔT
), zT2 = z2
(1 + �0ΔT
),dT1 = d1
(1 + �0ΔT
), dT2 = d2
(1 + �0ΔT
)
(11)
�0
x= A∗
22Nx− A∗
12Ny− B∗
11�x,x
− B∗12�y,y
+ C∗
11Φ1,
�0
y= A∗
11Ny− A∗
12Nx− B∗
21�x,x
− B∗22�y,y
+ C∗
22Φ1,
�0
xy= A∗
66Nxy− B∗
66
(�x,y
+ �y,x
).
(12)
Mx= B∗
11Nx+ B∗
21Ny+ D∗
11�x,x
+ D∗
12�y,y
+ C∗
12Φ1+ Φ
2,
My= B∗
12Nx+ B∗
22Ny+ D∗
21�x,x
+ D∗
22�y,y
+ C∗
21Φ1+ Φ
2,
Mxy
= B∗66Nxy+ D∗
66
(�x,y
+ �y,x
).
(13a)Nx,x + Nxy,y = 0,Nxy,x + Ny,y = 0,
(13b)Qx,x + Qy,y + Nxw,xx + 2Nxyw,xy + Nyw,yy − k1w + k2
(w,xx +w,yy
)= 0,
(13c)Mx,x +Mxy,y − Qx = 0,Mxy,x +My,y − Qy = 0,
(14)Nx = f,yy , Ny = f,xx, Nxy = −f,xy .
(15a)Mx,xx + 2Mxy,xy +My,yy + Nxw,xx + 2Nxyw,xy + Nyw,yy − k1w + k2
(w,xx +w,yy
)= 0,
(15b)Mx,x +Mxy,y − Qx = 0,Mxy,x +My,y − Qy = 0.
(16a)
B∗21f,xxxx +
(B∗11
− 2B∗66
+ B∗22
)f,xxyy + B
∗
12f,yyyy + D
∗
11�x,xxx +
(D∗
12+ 2D∗
66
)�y,xxy
+(2D∗
66+ D∗
21
)�x,xyy + D
∗
22�y,yyy + f,yy
(w,xx +w
∗
,xx
)− 2f
,xy
(w,xy +w
∗
,xy
)
+ f,xx
(w,yy +w
∗
,yy
)− k
1w + k
2
(w,xx +w,yy
)= 0,
(16b)B∗21f,xxx +(B∗11 − B
∗
66
)f,xyy + D
∗
11�x,xx +(D∗
12 + D∗
66
)�y,xy + D
∗
66�x,yy − A44w,x − A44�x = 0,
(16c)B∗12f,yyy +(B∗22 − B
∗
66
)f,xxy + D
∗
22�y,yy +(D∗
66 + D∗
21
)�x,xy + D
∗
66�y,xx − A55w,y − A55�y = 0,
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where w* (x, y) is a known function representing the initial small imperfections of the plate.
The system of Equations (16) include four unknown functions (w, ϕx, ϕy, and f); so it is necessary to find the fourth equation relating to these functions using the compatibility equation (Equation 7). For this purpose, substituting the expressions of �0x , �
0y , �
0xy from Equation (11) into Equation (7), we get:
Equations (16) and (17) are nonlinear equations in terms of the four dependent unknown functions (w, ϕx, ϕy, and f) used to investigate the buckling and post-buckling of an eccentrically stiffened func-tionally graded plate on elastic foundations subjected to compression, thermal and combined loads. The boundary condition (BC), will be considered:
The edges are simply supported and immovable (IM). The associated BCs are
here, Nx0, Ny0 are the pre-buckling force resultants in directions x and y, respectively.
To solve Equations (16) and (17) for unknowns w, ϕx, ϕy, and f, and with consideration of the BC (18), the following approximate solutions (Dung & Nga, 2013) are assumed:
where α = mπ/a, β = nπ/b, m, n = 1, 2, … are the number of half waves in the x, y directions, respec-tively; and W is the amplitude of deflection. Also, λi (i = 1−4) and Fi (i = 1−3) are coefficients to be determined.
Considering the BC (18), the imperfections of the plate are assumed as:
where the coefficient μ, varying between 0 and 1, represents the size of the imperfections.
After substituting Equations (19) and (20) into Equations (16b), (16c), and (17), the coefficients �i
(i = 1 − 4
) and Fi
(i = 1 − 3
) are found as:
and specific expressions of coefficients fi (i = 1−3) and Lj (j = 1−4) are given in Appendix A.
Introduction of Equations (19) and (20) into Equation (16a), and applying the Galerkin method for the resulting equation yields:
(17)A∗
11f,xxxx +(A∗
66 − 2A∗
12
)f,xxyy + A
∗
22f,yyyy − B∗
21�x,xxx −(B∗11 − B
∗
66
)�x,xyy −
(B∗22 − B
∗
66
)�y,yxx
−B∗12�y,yyy −(w,xy
)2− 2w,xyw
∗
,xy +w,xxw,yy +w∗
,xxw,yy +w,xxw∗
,yy = 0.
(18)w = u = �y = Mx = 0, Nx = Nx0 at x = 0and x = a
w = v = �x = My = 0, Ny = Ny0 at y = 0and y = b
(19)
w =W sin �x sin �y,
�x = �1cos �x sin �y + �
2sin 2�x, �y = �
3sin �x cos �y + �
4sin 2�y,
f = F1cos 2�x + F
2cos 2�y + F
3sin �x sin �y + Nx0y
2∕2 + Ny0x2∕2,
(20)w∗ = �h sin �x sin �y,
(21)F1= f
1W(W + 2�h
), F
2= f
2W(W + 2�h
), F
3= f
3W,
�1= L
1W, �
2= L
2W(W + 2�h
), �
3= L
3W, �
4= L
4W(W + 2�h
),
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Equation (22), derived for odd values of m, n, is used to determine the nonlinear buckling and post-buckling response of eccentrically stiffened thick FGM plates in thermal environments. An interesting characteristic of Equation (22) is the temperature-dependent components, which appear in the B∗11, B
∗
21, B∗
22, B∗
66, D∗
11, D∗
12, D∗
21, … coefficients, as shown in Appendix A.
A simply supported FGM plate with immovable edges under thermal loads is considered. The con-dition expressing the immovability on the edges, u = 0 (on x = 0, a) and v = 0 (on y = 0, b), is generally fulfilled as (Shen, 2007):
From Equations (4) and (11), one can obtain the following expressions in which Equation (14) and imperfection have been included:
Substitute Equations (19) and (20) into Equation (24), and then the result into Equation (23), gives:
From Equation (10), we have:
Substitution of Equation (26) into Equation (25), and then the result into Equation (22), gives:
where specific expressions of coefficients e2i(i = 1 − 4
) are given in Appendix A and W =W∕h.
(22)
ab{[
�4B∗21 + �
2�2(B∗11 − 2B
∗
66 + B∗
22
)+ �
4B∗12
]f3 ∕4 +
[�3D∗
11 + ��2(2D∗
66 + D∗
21
)]L1
+[�2�(D∗
12 + 2D∗
66
)+ �
3D∗
22
]L3 − k1 − k2
(�2 + �
2)}W
+8��f3∕3W(W + �h
)+ 32∕
(3��
)(
�3D∗
11L2 + �3D∗
22L4
−2�4B∗21f1 − 2�4B∗12f2
)
W(W + 2�h
)
−�2�2ab∕2(f1 + f2
)W(W + �h
)(W + 2�h
)− ab∕4
(W + �h
)(�2Nx0 + �
2Ny0
)= 0.
(23)
b
∫0
a
∫0
�u
�xdxdy = 0,
a
∫0
b
∫0
�v
�ydydx = 0.
(24)u,x = A
∗
22f,yy − A∗
12f,xx − B∗
11�x,x − B∗
12�y,y −(w,x
)2∕2 −w,xw
∗
,x + C∗
11Φ1,
v,y = A∗
11f,xx − A∗
12f,yy − B∗
21�x,x − B∗
22�y,y −(w,y
)2∕2 −w,yw
∗
,y + C∗
22Φ1.
Nx0
=4
��ab(A∗2
12− A∗
11A∗
22
)[�2(A∗2
12− A∗
11A∗
22
)f3+ �
(B∗11A∗
11+ B∗
21A∗
12
)L1+ �
(B∗12A∗
11+ B∗
22A∗
12
)L3
]
W − �2A∗
12+ �
2A∗
11W(W + 2�h)∕
(8(A∗2
12− A∗
22A∗
11
))+(C∗
22A∗
12+ C∗
11A∗
11
)∕(A∗2
12− A∗
22A∗
11
)Φ1,
(25)
Ny0 =4
��ab(A∗2
12− A∗
11A∗
22
)[�2(A∗2
12− A∗
11A∗
22
)f3+ �
(B∗11A∗
12+ B∗
21A∗
22
)L1+ �
(B∗12A∗
12+ B∗
22A∗
22
)L3
]
W −�2A∗
12+ �
2A∗
22
8(A∗2
12− A∗
11A∗
22
)W(W + 2�h
)+C∗
11A∗
12+ C∗
22A∗
22
A∗2
12− A∗
11A∗
22
Φ1.
(26)Φ1 = HΔT,H = −(Em�m +
(Em�cm + Ecm�m
)∕(N + 1) + Ecm�cm∕
(2N + 1
))∕(1 − v).
(27)ΔT = e21W̄ + e22W̄(W̄ + 𝜇
)+ e23W̄
(W + 2𝜇
)∕(W̄ + 𝜇
)+ e24W
(W + 2𝜇
),
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By setting μ = 0, Equation (27) leads to an equation from which buckling temperature change of the eccentrically stiffened perfect FGM plates may be determined as ΔTb = e
22.
In the case of T-D, both sides of Equation (27) are temperature-dependent, which makes it very difficult to solve. Fortunately, a numerical technique using an iterative algorithm has been applied to determine the buckling loads, as well as to determine the deflection–load relationship in the post-buckling period of the eccentrically stiffened FGM plate. Given more details, including the material parameter N, the geometrical parameter (b/a, b/h), and the value of W/h, we can determine ΔT in Equation (27), as follows. First, the initial step for ΔT1 on the right-hand side in Equation (27) is cho-sen with ΔT = 0 (since T = T0 = 300 K, the initial room temperature). In the next iterative step, the known value of ΔT1 found in the previous step is replaced to determine the right-hand side of Equation (27), ΔT2. This iterative procedure will stop at the kth step if ΔTk satisfies the condition ||ΔT − ΔTk
|| ≤ �. Here, ΔT is a desired solution for the temperature and ξ is a tolerance used in the iterative steps. This is also an interesting point to be solved in this article.
4. Numerical results and discussionIn this section, the components of the material are silicon nitride Si3N4(ceramic) and SUS304 stain-less steel (metal). The material properties (Pr) in Equation (2) are shown in Table 1 and Poisson’s ratio is chosen to be v = 0.3.
In particular, for the case of an S-FGM plate without stiffeners with the conditions: AT1 = AT2 = 0,
IT1 = IT2 = 0 and ceramic – metal – ceramic layers which are compared the numerical results of uns-
tiffened thick S-FGM plates with Duc and Tung (2010). As can be seen, a good agreement is obtained in this comparison (Figure 2).
Figure 3 illustrates the effect of eccentric stiffeners on the nonlinear response of S-FGM plates under thermal loads. It is clear that the stiffeners can enhance the thermal loading capacity for the imperfect and perfect S-FGM plates.
Figure 4 presents the effect of volume fraction index(N) on the post-buckling of eccentrically stiff-ened S-FGM plates under thermal load. These post-buckling curves show that the loading ability of S-FGM plates became worse with the increase of N. It leads the plate’s stiffness to be decreased which results in the decrease of load-carrying of eccentrically stiffener S-FGM plate (the module elastic E of metal is lower than that of ceramic, Ec > Em).
Figure 5 shows the effects of the elastic foundations on the nonlinear response of eccentrically stiffened S-FGM plates with temperature-dependent material properties. Elastic foundations are recognized to have strong impact, as demonstrated by curves (1) and (2), which show that the ability of sustaining compression and thermal load will increase if the effects of elastic foundations en-hance from (K1 = 0, K2 = 0) to (K1 = 100, K2 = 0). Furthermore, Pasternak’s elastic foundation (K2) is more powerful than Winkler’s foundation (K1), which is proven by curve (3) with K1 = 100, K2 = 10, and curve (4) with K1 = 50, K2 = 20.
Figure 6 shows the effects of imperfections on post-buckling response of the eccentrically stiff-ened S-FGM plates under thermal load. In the post-buckling period, the imperfections have actively
Table 1. Temperature-dependent coefficients of silicon nitride and stainless steel
Notes: h1 = 0.08 m, h2 = 0.08 m, b1 = 0.008 m, b2 = 0.008 m, d1 = 0.15 m, d2 = 0.15 m, E0 = Em.
Property Material P−1 P0 P1 P2 P3
E (Pa) Silicon nitride 0 348.43 × 109 −3.070 × 10−4 2.160 × 10−7 −8.946 × 10−11
Stainless steel 0 201.04 × 109 3.079 × 10−4 −6.534 × 10−7 0
α (1/K) Silicon nitride 0 5.8723 × 10−6 9.095 × 10−4 0 0
Stainless steel 0 12.330 × 10−6 8.086 × 10−4 0 0
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Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
affected the load bearing ability of the plate. In other words, the loading ability increases together with μ.
Figure 7 shows the effects of temperature-dependent material properties on the nonlinear stabil-ity of eccentrically stiffened thick S-FGM plates under thermal load. There is a comparison between
Figure 2. Comparisons of thermal post-buckling load-deflection curves for un-stiffened S-FGM.
Figure 3. Effect of eccentric stiffeners on the post-buckling of S-FGM plates under thermal loads.
Figure 4. Effects of volume fraction index on the post-buckling of eccentrically stiffened S-FGM plates under thermal load.
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Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
the thermal post-buckling curves of both perfect and imperfect FGM plates with T-D and T-ID mate-rial properties. It is apparent that T-D material properties make the S-FGM plate considerably weaker under thermal load.
Figure 5. Effects of elastic foundations on the post-buckling of eccentrically stiffened S-FGM plate under thermal load.
Figure 6. Effect of imperfection on post-buckling of eccentrically stiffened thick S-FGM plate under thermal load.
Figure 7. Thermal post-buckling behavior of eccentrically stiffened thick S-FGM plates with T-ID and T-D material properties.
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Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
5. ConclusionsThis paper presents an analysis approach, in conjunction with an iterative procedure, to investigat-ing the buckling and post-buckling behavior of the eccentrically stiffened S-FGM plates under ther-mal load. The formulation is based on the first-order shear deformation theory, accounting for both the von Karman nonlinearity and initial imperfections. The paper also analyzes and discusses the effects of material and geometrical properties, temperature, elastic foundations, and eccentric stiff-eners on the buckling and post-buckling loading capacity of the eccentrically stiffened S-FGM plate in thermal environments.
FundingThis work was supported by the Grant of Newton Fund (UK) [grant number NRCP1516/1/68].
Author detailsPham Hong Cong1,2
E-mail: [email protected] Dinh Duc2
E-mail: [email protected] Centre for Informatics and Computing, Vietnam Academy
of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam.
2 University of Engineering and Technology - Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam.
Citation informationCite this article as: Thermal stability analysis of eccentrically stiffened Sigmoid-FGM plate with metal – ceramic – metal layers based on FSDT, Pham Hong Cong & Nguyen Dinh Duc, Cogent Engineering (2016), 3: 1182098.
Cover imageSource: Author.
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Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
Appendix A
where χ1, χ2 are the shear correction factor, χ1 = χ2 = 4/3.
A11
= A22
= E1∕(1 − v2
), A
12= E
1v∕
(1 − v2
), A
66= E
1∕(2(1 + v)
),A
44= �
1E1
(2(1 + v)
),
A55
= �2E1∕(2(1 + v)
),B
11= B
22= E
2∕(1 − v2
), B
12= E
2v∕
(1 − v2
), B
66= E
2∕(2(1 + v)
),
D11
= D22
= E3∕(1 − v2
), D
12= E
3v∕
(1 − v2
), D
66= E
3∕(2(1 + v)
),
E1= E
mh + E
cmh∕(N + 1), E
2= 0,
E3= E
mh3∕12 + E
cmh3(1∕
(N + 3
)− 1∕
(N + 2
)+ 1∕(4N + 4)
),
Φ1= −
1
1 − v
h∕2
∫−h∕2
E(z)�(z)ΔTdz, Φ2= −
1
1 − v
h∕2
∫−h∕2
E(z)�(z)ΔTzdz,
(11)IT1= bT
1
(hT1
)3∕12 + AT
1
(zT1
)2, IT
2= bT
2
(hT2
)3∕12 + AT
2
(zT2
)2,CT
1= E
0AT1zT1∕dT
1,
CT2= E
0AT2zT2∕dT
2, z1=(h1+ h
)∕2, z
2=(h2+ h
)∕2, �
1= �
2= 4∕3.
Δ =(A11
+ E0AT1∕dT
1
)(A22
+ E0AT2∕dT
2
)− A2
12,A∗
11=(A11
+ E0AT1∕dT
1
)∕Δ,
A∗
22=(A22
+ E0AT2∕dT
2
)∕Δ, A∗
12= A
12∕Δ, A∗
66= 1∕A
66,
B∗11
= A∗
22
(B11
+ CT1
)− A∗
12B12, B∗
22= A∗
11
(B22
+ CT2
)− A∗
12B12,
B∗12
= A∗
22B12
− A∗
12
(B22
+ CT2
), B∗
21= A∗
11B12
− A∗
12
(B11
+ CT1
),
B∗66
= B66∕A
66,D∗
11= D
11+ E
0IT1∕dT
1− B∗
11
(B11
+ CT1
)− B∗
21B12;
D∗
22= D
22+ E
0IT2∕dT
2− B∗
22
(B22
+ CT2
)− B∗
12B12,
D∗
12= D
12− B∗
12
(B11
+ CT1
)− B∗
22B12;D∗
21= D
12− B∗
21
(B22
+ CT2
)− B∗
11B12,
D∗
66= D
66− B∗
66B66,C∗
11=(A12
− A22
− E0AT2∕dT
2
)∕Δ;
C∗
22=(A12
− A11
− E0AT1∕dT
1
)∕Δ,C∗
12= C∗
11
(B11
+ C1
)+ C∗
22B12, C∗
21= C∗
22
(B22
+ C2
)+ C∗
11B12.
f1=(A44B2h+ 4B2
aD∗
11m2
�2)hn2∕
{
32m2B2a
[
A∗
11
(A44B2h+ 4m2
�2B2
aD∗
11
)+ 4m2
�2B2
aB∗21
2]}
,
f2=(4n2�2D∗
22+ A
55B2h
)m2B2
ah∕
{
32n2[
A∗
22
(4n2�2D∗
22+ A
55B2h
)+ 4n2�2B∗
12
2]}
,
f3= h2
(a31∕a
33L1+ a
32∕a
33L3
),
L1= −
[(a12a33
+ a13a32
)L3+ a
33A44
m�Ba
Bh
]
∕((a11a33
+ a13a31
)h),
L2= 8m3
�3B3
aB∗21∕(Bh
(A44B2h+ 4m2
�2B2
aD∗
11
)h2)f1,
L3=
a33
(a11a33
+ a13a31
)A55n� − a
33
(a21a33
+ a23a31
)A44m�B
a
Bh
[(a12a33
+ a13a32
)(a21a33
+ a23a31
)−(a22a33
+ a23a32
)(a11a33
+ a13a31
)]h,
Page 14 of 15
Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
L4=
8n3�3B∗12
Bh
(4n2�2D∗
22+ B2
hA55
) f21
h2, f1=f1
h, f2=f2
h,
f3=f3
h2, L
1= L
1h, L
2= L
2h2, L
3= L
3h, L
4= L
4h2,
a11
=m2
�2B2
a
B2h
D∗
11+n2�2
B2h
D∗
66+ A
44,a
12=Bamn�2
B2h
(D∗
12+ D∗
66
),
a13
=B3am3
�3
B3h
B∗21
+mn2�3B
a
B3h
(B∗11
− B∗66
),
a21
= mn�2Ba
(D∗
21+ D∗
66
)∕B2
h,a
22= n2�2∕B2
hD∗
22+m2
�2B2
a∕B2
hD∗
66+ A
55,
a23
= n3�3∕B3hB∗12
+m2n�3B2a∕B3
h
(B∗22
− B∗66
),
a31
=[m2B2
aB∗21
+ n2(B∗11
− B∗66
)]m�
3Ba∕B3
h, a
32=[B2am2n�3
(B∗22
− B∗66
)+ n3�3B∗
12
]∕B3
h,
a33
=[B4am4
�4A∗
11+ B2
am2n2�4
(A∗
66− 2A∗
12
)+ n4�4A∗
22
]∕B4
h,
A∗
11= A∗
11h, A∗
22= A∗
22h, A∗
12= A∗
12h, A∗
66= A∗
66h
A44
= A44∕h,A
55= A
55∕h,B∗
11= B∗
11∕h, B∗
22= B∗
22∕h, B∗
12= B∗
12∕h,
B∗21
= B∗21∕h,B∗
66= B∗
66∕h,
C∗
11= C∗
11h, C∗
22= C∗
22h,D∗
11= D∗
11∕h3, D∗
22= D∗
22∕h3, D∗
12= D∗
12∕h3,D∗
21= D∗
21∕h3, D∗
66= D∗
66∕h3.
e21=8mn�2B
af3
3B2hP1H
− B2h
�
2m2n2�4B2a∕B4
h
�
A∗
12
2
− A∗
11A∗
22
�
f3
+m�Ba∕B3
h
�B2am2
�2�B∗11A∗
11+ B∗
21A∗
12
�+ n2�2
�B∗11A∗
12+ B∗
21A∗
22
��L1
+ n�∕B3h
⎡⎢⎢⎣
m2�2B2
a
�B∗12A∗
11+ B∗
22A∗
12
�
+n2�2�B∗12A∗
12+ B∗
22A∗
22
�⎤⎥⎥⎦L3
�∕
�
mn�2�
A∗
12
2
− A∗
11A∗
22
�
P1HB
a
�
,
e22= B2
h
{[m4
�4B4
aB∗21
+m2n2�4B2a
(B∗11
− 2B∗66
+ B∗22
)+ n4�4B∗
12
]f3∕B4
h
+[m3
�3B3
aD∗
11+mn2�3B
a
(2D∗
66+ D∗
21
)]L1∕B3
h
+[m2n�3B2
a
(D∗
12+ 2D∗
66
)+ n3�3D∗
22
]L3∕B3
h
− K1D∗
11B4a∕B4
h−(B2am2
�2 + n2�2
)D∗
11B2a∕B4
hK2
}∕(4B
aHP
1
),
e23= 32
(m3
�3B
hB3aD∗
11L2+ n3�3B
hD∗
22L4− 2m4
�4B4
aB∗21f1− 2n4�4B∗
12f2
)∕(3mn�2P
1HB
aB2h
),
e24= �
4(B4am4A∗
11+ 2B2
am2n2A∗
12+ n4A∗
22
)∕
(
32B2hBaP1H
(
A∗
12
2
− A∗
11A∗
22
))
−m2n2�4Ba
(f1+ f
2
)∕(2B2
hP1H),
P1= �
2[(C∗
11A∗
12+ C∗
22A∗
22
)n2 +
(C∗
22A∗
12+ C∗
11A∗
11
)B2am2
]∕
(
4Ba
(
A∗
12
2
− A∗
11A∗
22
))
.
Page 15 of 15
Hong Cong & Dinh Duc, Cogent Engineering (2016), 3: 1182098http://dx.doi.org/10.1080/23311916.2016.1182098
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