Free vibration of functionally graded sandwich plates with stiffeners based on the third-Order shear deformation theory

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 2 (2016), pp. 103 – 122 DOI:10.15625/0866-7136/38/2/6730 FREE VIBRATION OF FUNCTIONALLY GRADED SANDWICH PLATES WITH STIFFENERS BASED ON THE THIRD-ORDER SHEAR DEFORMATION THEORY Pham Tien Dat, Do Van Thom∗, Doan Trac Luat Le Quy Don Technical University, Hanoi, Vietnam ∗E-mail: promotion6699@gmail.com Received August 11, 2015 Abstract. In this paper, the free vibration of functionally sandwich grades plates with stiffeners is investigated by using the finite element method. The material properties are assumed to be graded in the thickness direction by a power-law distribution. Based on the third-order shear deformation theory, the governing equations of motion are derived from the Hamilton’s principle. A parametric study is carried out to highlight the effect of ma- terial distribution, stiffener parameters on the free vibration characteristics of the plates. Keywords: Stiffened plate, sandwich, vibration, functionally graded materials. 1. INTRODUCTION Functionally graded materials (FGMs) made of two constituents, mainly metal and ceramic are widely used. The composition is varied continuously along certain direc- tions according to volume fration from a ceramic-rich surface to a metal-rich surface. Many researchers in past have worked upon to understand behavior of functionally graded plates. Reddy [1] carried out nonliear finite element static and dynamic analyses of functionally graded plates using Navier’s solution, Vel and Batra [2] presented three- dimendional exact solutions for free and forced vibrations of simply supported function- ally graded rectangular plates, Tinh Quoc Bui et al [3] used finite element method to study static bending and vibration of heated FG plates. Dao Huy Bich et al [4,5], Nguyen Dinh Duc et al [6,7] used analytical method to study vibration, nonlinear respones of eccentri- cally stiffened functionally graded cylindrical panels [4, 5] and nonlinear postbucking of FGM plate and eccentrically stiffened thin FGM resting on elastic foundations in thermal environments [6,7]. Several studies have been performed to analyze the behaviour of FG sandwich structures, Zenkour [8], Alipour and Shariyat [9]. In this paper, the free vibartion of stiffened FG sandwich plates is sudied by us- ing finite element method. Here we present FG sandwich plate with material properties c© 2016 Vietnam Academy of Science and Technology 104 Pham Tien Dat, Do Van Thom, Doan Trac Luat symmetric about the mid-plane. The faces of the plate consist of a FGM with properties varying only in the thickness direction. Such faces can be made by mixing two different material phases, for example, a metal and a ceramic. The core material may be homo- geneous and can be made by one of these materials, for example, a ceramic or a metal. Eigenfrequencies and mode shapes of stiffened FG sandwich plates are presented using the higher-order shear deformation theory. 2. GOVERNING EQUATIONS AND FINITE ELEMENT ELEMENT FORMULATION Consider a stiffened FG sandwich plate with thickness, long, wide of the plate are 2h, a, b, and depth, width of the stiffener are hs, bs, respectively (Fig. 1). The xy-plane is the mid-plane of the plate, and the positive z-axis is upward from the mid-plane. The power law distribution is used for describing the volume fraction of the ceramic (V(i)c ) and the metal (V(i)m ) in i-th layer (i = 1, 2, 3) as follow V(i)m +V (i) c = 1. (1) h y x bs Section A-A Type 1 b A A hs z -h Metal Ceramic Metal h bs Section A-A Type 2 hs -h Ceramic Metal Ceramic Y-Stiffener X-Stiffener a h1 -h1 h1 -h1 Layer 1 Layer 2 Layer 3 Fig. 1. A FG plate stiffened by an x-direction and an y-direction stiffeners For type 1  V(1)c = ( z+ h h− h1 )n ; −h ≤ z ≤ −h1 V(2)c = 1 ; −h1 ≤ z ≤ h V(3)c = ( z− h h1−h )n ; h1 ≤ z ≤ h (2) For type 2  V(1)c = ( z+ h1 h1−h )n ; −h ≤ z ≤ −h1 V(2)c = 0 ; −h1 ≤ z ≤ h V(3)c = ( z− h1 h− h1 )n ; h1 ≤ z ≤ h (3) Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 105 Where, 2h-thickness of the plate, 2h1-thickness of second layer; n-the gradient in- dex (n ≥ 0); z-the thickness coordinate variable, and subscripts c and m represent the ceramic and metal constituents, respectively. In this study, the material properties, i.e., Young’s modulus E, Poisson’s ratio ν, and the mass density ρ, can be expressed by the rule of mixture as [1] P(i)(z) = Pm+(Pc−Pm)V(i)c , (4) where P(i) is the material property of i-th layer. In [10] the three-dimensional displacement field (u, v,w) can be expressed in terms of nine unknown variables as follows u (x, y, z) = u0(x, y) + z.ψx(x, y) + z 2.ξx(x, y) + z 3.Φx(x, y), v (x, y, z) = v0(x, y) + z.ψy(x, y) + z 2.ξy(x, y) + z 3.Φy(x, y), w (x, y, z) = w0(x, y), (5) where u0, v0,w0 represent the displacements at the mid-plane of the plate in the x, y and z directions, respectively; ψx,ψy denote the transverse normal rotations of the y and x axes; ξx, ξy denote the higher order displacements and Φx,Φy denote the higher order transverse rotations. 2.1. Plate element We consider a plate element with 4 nodes, each node has 9 degrees of freedom (see Fig. 2). Y X O 1 2 3 4 (a) Plate element in Decaster system s r 1(-1,-1) 2(1,-1) 3(1,1)4(-1,1) (b) Natural coordinate system Fig. 2. Rectangular isoparametric plate element For the plate element, displacement vector of i-th node has the form {qi}= { ui, vi,wi,ψxi,ψyi, ξxi, ξyi,Φxi,Φyi }T ; i = 1, 4. (6) Displacement vector at any point of the plate element {u} is computed using shape function Ni as follows {u}= 4 ∑ i=1 Ni. {qi}, (7) 106 Pham Tien Dat, Do Van Thom, Doan Trac Luat N1 = 0, 25(1− r)(1− s), N2 = 0, 25(1+ r)(1− s), N3 = 0, 25(1+ r)(1+ s), N4 = 0, 25(1− r)(1+ s). (8) The strain vector can be expressed in the form {ε}= {ε0}+z {κ}+z2 {χ}+z3 {η} , {γ}= {γ0}+z {κ′}+z2 {χ′} , (9) where { ε0 } = L1 {u}= L1 4 ∑ i=1 Ni {qi}= [B1] {q}e , (10) {κ}=L2 {u}=L2 4 ∑ i=1 Ni {qi}=[B2] {q}e , {χ}=L3 {u}=L3 4 ∑ i=1 Ni {qi}=[B3] {q}e , (11) {η}=L4 {u}=L4 4 ∑ i=1 Ni {qi}=[B4] {q}e , { γ0 } =L ′ 1 {u}=L′ 1 4 ∑ i=1 Ni {qi}=[B ′ 1 ] {q}e , (12){ κ ′} =L ′ 2 {u}=L′ 2 4 ∑ i=1 Ni {qi}=[B ′ 2] {q}e , { χ ′} =L ′ 3 {u}=L ′ 3 4 ∑ i=1 Ni {qi}=[B ′ 3] {q}e , (13) where [Bj] = Lj 4 ∑ i=1 Ni , [B ′ j] = L ′ j 4 ∑ i=1 Ni , (14) and {q}e= {{q1} , . . . , {q4}}T- displacement vector of plate element. Lj, Lj’- standard strain-displacement matrix of plate element. The stress vector at any point in i-th layer of the plate is expressed as{ {σ}(i) {τ}(i) } = [ [Dmi] [0] [0] [Dsi] ]{ {ε}(i) {γ}(i) } , (15) where {σ}(i)= { σ (i) x σ (i) y τ (i) xy }T , {τ}(i)= { τ (i) xz τ (i) yz } , (16) [Dmi] = E(i)(z) 1− (ν(i))2  1 ν(i) 0ν(i) 1 0 0 0 1−ν (i) 2  , [Dsi] = E(i)(z) 2(1+ ν(i)) diag(2, 2). (17) 2.2. x-Stiffener element The displacement field of x-Stiffened can be expressed as (see Fig. 3) uxs (x, y, z) = u0xs(x, y) + zψxs(x, y) + z 2ξxs(x, y) + z 3Φxs(x, y), vxs (x, y, z) = 0, wxs (x, y, z) = w0xs(x, y). (18) The strain vector for the x-stiffener element has the form {ε}xs= { ε0 } xs+z {κ}xs+z2 {χ}xs+z3 {η}xs , {γ}xs= { γ0 } xs+z { κ′ } xs+z 2 {χ′}xs . (19) Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 107 s r r0 s0 y-Stiffener x-Stiffener 1 2 34 Fig. 3. Plate element and stiffener element where { ε0 } xs= L1xs {u}xs= [B1xs] {q}exs , (20) {κ}xs= L2xs {u}xs= [B2xs] {q}exs , {χ}xs= L3xs {u}xs= [B3xs] {q}exs , (21) {η}xs= L4xs {u}xs= [B4xs] {q}exs , { γ0 } xs= L ′ 1xs {u}xs= [B ′ 1xs ] {q}exs , (22){ κ ′} xs = L ′ 2xs {u}xs= [B ′ 2xs] {q}exs , { χ ′} xs = L ′ 3xs {u}xs= [B ′ 3xs] {q}exs , (23) where [Bjxs] = Ljxs 4 ∑ i=1 Nixs; [B ′ jxs] = L ′ jxs 4 ∑ i=1 Nixs ; Nixs - the shape functions of x-stiffener which can be obtained by substituting s = s0 into Eq. (8); {q}exs - displacement vector of x-stiffener coordinate system; Ljxs, L ′ jxs - standard strain-displacement matrices of x- stiffener element. The stress vector at any point in the stiffener is expressed as{ {σ}xs{τ}xs } = [ [Dmxs] [0] [0] [Dsxs] ]{ {ε}xs{γ}xs } , (24) where {σ}xs= { σxσyτxy }T xs , {τ}xs= { τxzτyz }T xs , (25) [Dmxs] = Exs 1−ν2xs  1 νxs 0νxs 1 0 0 0 1−νxs2  , [Dsxs] = Exs2(1+ νxs)diag(2, 2). (26) 2.3. y-Stiffener element The displacement field of y-stiffened has the form uys (x, y, z) = 0, vys (x, y, z) = v0ys(x, y) + zψys(x, y) + z 2ξys(x, y) + z 3Φys(x, y), wys (x, y, z) = w0ys(x, y). (27) The strain vector for the y-stiffener element is expressed in the form {ε}ys= { ε0 } ys+z {κ}ys+z2 {χ}ys+z3 {η}ys , {γ}ys= { γ0 } ys+z { κ′ } ys+z 2 {χ′}ys , (28) 108 Pham Tien Dat, Do Van Thom, Doan Trac Luat where{ ε0 } ys= L1ys {u}ys= [B1ys] {q}eys , (29) {κ}ys= L2ys {u}ys= [B2ys] {q}eys , {χ}ys= L3ys {u}ys= [B3ys] {q}eys , (30) {η}ys= L4ys {u}ys= [B4ys] {q}eys , { γ0 } ys= L ′ 1ys {u}ys= [B ′ 1ys ] {q}eys , (31){ κ ′} ys = L ′ 2ys {u}ys= [B ′ 2ys] {q}eys , { χ ′} ys = L ′ 3ys {u}ys= [B ′ 3ys] {q}eys , (32) where [Bjys] = Ljys 4 ∑ i=1 Niys; [B ′ jys] = L ′ jys 4 ∑ i=1 Niys; Niys - shape functions of y-stiffener which can be obtained by substituting r = r0 into Eq. (8); {q}eys - displacement vector of y- stiffener coordinate system; Ljys, L ′ jys - standard strain-displacement matrix of y-stiffener element. The stress vector at any point in the stiffener is expressed as{ {σ}ys {τ}ys } = [ [ Dmys ] [0] [0] [ Dsys ] ]{ {ε}ys{γ}ys } , (33) where {σ}ys= { σx σy τxy }T ys , {τ}ys= { τxz τyz }T ys , (34) [ Dmys ] = Eys 1−ν2ys  1 ν 0ν 1 0 0 0 1− νys 2  , [Dsys]= Eys2(1+ νys)diag(2, 2). (35) 2.4. Transformation matrices We consider that the x-Stiffener is attached to the lower side of the plate, the con- ditions of displacement compatibility along the line of connection can be written as {u}z=−0.5h= {u}x|z=0.5hsx . (36) Using Eqs. (5) and (18), the conditions in Eq. (36) thus lead to[ u0x ] i= [ u0 ] i+e 1 x [ψx]i+e 2 x [ξx]i+e 3 x [Φx]i , [ψx]i= [ψxs]i , [ξx]i= [ξxs]i , [Φx]i= [Φxs]i , (37) where e1x= − (h+ hxs) /2, e2x= ( h2−h2xs ) /4, e3x= − ( h3+h3xs ) /8. (38) Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 109 Expressing in matrix form u0x 0 wx ψx−x 0 ξx−x 0 Φx−x 0  i =  1 0 0 e1x 0 e2x e1x 0 e2x 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0   u0 v w0 ψx ψy ξx ξy Φx Φy  i ; i = 1, 4 (39) or {u}xi= Txs {u}i . (40) The nodal displacement vector can be written in the form {q}ex= Tx {q}e , (41) where Tx= Txs.diag(4, 4). (42) The transformation matrix for the y-stiffener element can be derived in a similar way in the form {q}ye= Ty {q}e . (43) 2.5. Weak form of the stiffened plate problem The elastic strain energy of the plate is written as [11] U = 1 2∑Np ∫ Ve {ε}T {σ} dVe+12 ∑Nxs ∫ Vex {ε}Tx {σ}x dVex+ 1 2∑Nys ∫ Vey {ε}Ty {σ}y dVey , (44) or in the matrix form U = 1 2∑Np {q}Te [K]e {q}e+ 1 2 ∑Nxs {q}Tex [K]ex {q}Tex+ 1 2∑Nys {q}Tey [K]ey {q}Tey , (45) where [K]e= ∫ Se  [B1] T[A][B1] + [B1] T[B][B2] + [B1] T[D][B3] + [B1] T[E][B4] + [B2] T[B][B1] +[B2] T[D][B2] + [B2] T[E][B3] + [B2] T[F][B4] + [B3] T[D][B1] + [B3] T[E][B2] +[B3] T[F][B3] + [B3] T[G][B4] + [B4] T[E][B1] + [B4] T[F][B2]+ +[B4] T[G][B3] + [B4] T[H][B4] + [B1 ′]T[A′][B1 ′] + [B1 ′]T[B′][B2 ′]+ +[B1 ′]T[D′][B3 ′] + [B2 ′]T[B′][B1 ′] + [B2 ′]T[D′][B2 ′] + [B2 ′]T[E′][B3 ′]+ +[B3 ′]T[D′][B1 ′] + [B3 ′]T[E′][B2 ′] + [B3 ′]T[F′][B3 ′]  dSe = 110 Pham Tien Dat, Do Van Thom, Doan Trac Luat = 2 ∑ i=1 2 ∑ j=1  [B1] T[A][B1]+[B1] T[B][B2]+[B1] T[D][B3]+[B1] T[E][B4]+[B2] T[B][B1] +[B2] T[D][B2]+[B2] T[E][B3]+[B2] T[F][B4]+[B3] T[D][B1]+[B3] T[E][B2] +[B3] T[F][B3] + [B3] T[G][B4] + [B4] T[E][B1] + [B4] T[F][B2]+ +[B4] T[G][B3] + [B4] T[H][B4] + [B1 ′]T[A′][B1 ′] + [B1 ′]T[B′][B2 ′]+ +[B1 ′]T[D′][B3 ′] + [B2 ′]T[B′][B1 ′] + [B2 ′]T[D′][B2 ′] + [B2 ′]T[E′][B3 ′]+ +[B3 ′]T[D′][B1 ′] + [B3 ′]T[E′][B2 ′] + [B3 ′]T[F′][B3 ′]  |J|wiwj (46) [K]em=bms ∫ lem  [B1m] T[Am][B1m]+[B1m] T[Dm][B3m]+[B2m] T[Dm][B2m]+[B2m] T[Fm][B4m]+ +[B3m] T[Dm][B1m]+[B3m] T[Fm][B3m]+[B4m] T[Fm][B2m]+[B4m] T[Hm][B4m]+ +[B1m ′]T[Am ′][B1m ′] + [B1m ′]T[Dm ′][B3m ′] + [B2m ′]T[Dm ′][B2m ′]+ +[B3m ′]T[Dm ′][B1m ′] + [B3m ′]T[Fm ′][B3m ′] dlem =bms 2 ∑ i=1  [B1m] T[Am][B1m]+[B1m] T[Dm][B3m]+[B2m] T[Dm][B2m]+[B2m] T[Fm][B4m]+ +[B3m] T[Dm][B1m]+[B3m] T[Fm][B3m]+[B4m] T[Fm][B2m]+[B4m] T[Hm][B4m]+ +[B1m ′]T[Am ′][B1m ′] + [B1m ′]T[Dm ′][B3m ′] + [B2m ′]T[Dm ′][B2m ′]+ +[B3m ′]T[Dm ′][B1m ′] + [B3m ′]T[Fm ′][B3m ′] |Jm|wi , (47) with m = x, y, (48) (A, B, D, E, F, H,G) = h/2∫ −h/2 [Dm] ( 1,z,z2,z3,z4,z5,z6 ) dz, (49) ( A′, B′, D′, E′, F′ ) = h/2∫ −h/2 [Ds] ( 1,z,z2 ) dz. (50) |J| is det Jacobian matrix [J] =  ∂N1 ∂r ∂N2 ∂r ∂N3 ∂r ∂N4 ∂r ∂N1 ∂s ∂N2 ∂s ∂N3 ∂s ∂N4 ∂s   x1 y1 x2 y2 x3 y3 x4 y4  , (51) and wi = wj = 1 are Gauss weights for two Gauss points (4 node plate element) and Gauss points = ± 1√ 3 . |Jm| is det Jacobian matrix [Jxs] = [ ∂N1 ∂r ∂N2 ∂r ] [ x1 x2 ] for x-stiffener (52) [ Jys ] = [ ∂N1 ∂s ∂N2 ∂s ] [ y1 y2 ] for y-stiffener (53) Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 111 and wi = 1 is Gauss weight for two Gauss points (2 node beam element) and Gauss points = ± 1√ 3 . The kinetic energy of the stiffened plate is computed by T = 1 2∑Np ∫ Ve {u˙}T ρ {u˙} dVe+12 ∑Nxs ∫ Vex {u˙}x ρx {u˙}x dVex+ 1 2∑Nys ∫ Vey {u˙}y ρy {u˙}y dVey = 1 2∑Np {q˙}Te [ Mp ] e {q˙}e+ 1 2 ∑Nxs {q˙}Te [Mx]e {q˙}e+ 1 2∑Nys {q˙}Te [ My ] e {q˙}e , (54) where [ Mp ] e= ∫ Ve [N]T [ρ] [N] dVe= h/2∫ −h/2 ( 2 ∑ i=1 2 ∑ j=1 [N]T [ρ] [N] |J|wiwj ) dz, [Mx]e=T T x bxs∫ lex [Nx] T[ρx][Nx] dlex  Tx=TTx bxs hxs/2∫ −hxs/2 ( 2 ∑ i=1 [Nx] T [ρx] [Nx] |Jxs|wi ) dz Tx , [ My ] e=T T y bys∫ ley [ Ny ]T[ ρy ][ Ny ] dley Ty=TTy bys hxs/2∫ −hxs/2 ( 2 ∑ j=1 [ Ny ]T[ ρy ][ Ny ] ∣∣Jys∣∣wj ) dz Ty. (55) In this paper, we apply the Hamilton’s principle to find the weak form of the prob- lem (the damping of plate is neglected). The principle is started that t2∫ t1 (δW + δT − δU) dt = 0, (56) where W is the work done by external forces on the stiffened plate. So that Eq. (56) leads to ([M] + [Ms]) {q¨}+ ([K] + [Ks]) {q}= {F} . (57) To obtain natural frequency ({F} = {0}) the following eigenvalue equation must be solved { ([K] + [Ks])−ω2 ([M] + [Ms]) } {q0}= {0} , (58) where ω, {q0} - natural frequency and modal shapes. Integrals in Eqs. (46)-(47) and (55) will be computed by Gauss quadrature [11]. The program for solving Eq. (58) was coded in Matlab. 112 Pham Tien Dat, Do Van Thom, Doan Trac Luat 3. NUMERICAL RESULTS 3.1. Comparison study 3.1.1. Free vibration of a simply supported homogeneous rectangular stiffened plate with two stiffeners A fully simply supported square plate having two centrally placed stiffener has been analyzed by L. X. Peng [12]. The plate and stiffener were made of the same material, with Young’s modulus 3.107 Pa, density 2820 kg/m3, and Poisson’s ratio 0.3 (Fig. 4). The first five natural frequencies of this stiffened plate were calculated by using the present theory. Tab. 1 shows calculation results compared with those by mesh-free method of Peng. A good agreement can be seen in this table. 0.3m x 0.5m 5m Section 1-1 0.3m 0.5m Section 2-2 3m 30 m 60 m 2 2 1 1 Fig. 4. The stiffened rectangular plate with two stiffeners Table 1. Comparisons of frequencies for the simply supported homogeneous stiffened plate with two stiffeners Natural frequency (Hz) 1 2 3 4 5 Present 0.0821 0.0860 0.1043 0.1063 0.1324 L. X. Peng et al. [12] 0.0816 0.0856 0.10003 0.1028 0.1311 3.1.2. A free vibration of square FG sandwich plate A fully simply supported square FG sandwich plate Al/ Al2O3 (type 1) with a = b and 2h is thickness, a/(2h) = 10. The material properties, as given in L. Hadji [11] are Em = 70 GPa, νm = 0.3, ρm = 2707 kg/m3 for Al; Ec = 380 GPa, νc = 0.3, ρc = 3800 kg/m3 for Al2O3; Thickness relation is denoted as the top layer thickness - the core thick- ness - the bottom thickness = 1-1-1. The dimensionless frequenciesv= ( ωa2/(2h) )√ ρ0/E0 (where ρ0 = 1 kg/m3, E0 = 1 GPa) obtained by the present paper are compared with the first-order shear deformation plate theory (FSDT) (analytical method), the third-order shear deformation plate theory (TSDT) (analytical method), and the four-variable refined Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 113 plate theory [11] (analytical method) in Tab. 2. This comparison once again shows clearly that good agreements are obtained. Table 2. Comparisons of the dimensionless frequency v for simply support FG sandwich plate n v (thickness relation = 1-1-1) Present FSDT [11] TSDT [11] Refined plate theory [11] 0 1.856207 1.82442 1.82445 1.82445 0.5 1.540301 1.51695 1.51922 1.51921 Table 3. Dimensionless frequency of stiffened FG sandwich plate with one stiffener (a/b = 1, bs = a/50, hs = 10h, thickness relation =1:8:1) a/(2h) Dimensionless frequency (v) Boundanry SSSS CCCC condition n First type Second type First type Second type 40 3.8447 5.3852 6.6365 8.2091 0.5 3.6780 2.7504 6.3388 4.1175 1 3.5991 2.6715 6.1952 3.9998 2 3.5229 2.5920 6.0551 3.8811 10 3.4174 2.4728 5.8590 3.7032 20 3.3175 4.8090 5.0946 7.4121 0.5 3.1793 2.7206 4.8855 4.0213 1 3.1140 2.6427 4.7859 3.9075 2 3.0512 2.5641 4.6898 3.7929 10 2.9645 2.4464 4.5567 3.6209 10 2.5497 3.7773 3.6246 5.3731 0.5 2.4421 2.1815 3.4710 3.0847 1 2.3914 2.1400 3.3984 3.0244 2 2.3428 2.0985 3.3288 2.9639 10 2.2763 2.0369 3.2337 2.8739 5 0.8770 1.9684 2.5530 3.6931 0.5 0.8726 1.5580 2.4382 2.0938 1 0.8704 1.5268 2.3841 2.0525 2 0.8683 1.4956 2.3323 2.0110 10 0.8654 1.4492 2.2621 1.9489 114 Pham Tien Dat, Do Van Thom, Doan Trac Luat 3.2. Free vibration of square sandwich FG plate 3.2.1. Free vibration of FG plate with one central stiffener - Effect of boundary condition and side-to-thickness ratio A stiffened FG sandwich plate Si3N4/SUS304 with a long, b wide and 2h thick. The material properties, as given in Reddy and Chin [13], are Em = 322.7 GPa, νm = 0.28, ρm = 2370 kg/m3 for Si3N4; Ec = 207.79 GPa, νc = 0.28, ρc = 8166 kg/m3 for SUS304, thickness relation =1:8:1. Tabs. 3 and 4 show the nondimensional natural fre- quencies of different side-to-thickness ratio and volume fraction exponents with four boundary conditions, viz., all edges simply supported (SSSS), all edges clamped (CCCC), two edges opposite simply supported and two edges opposite clamped (CSCS), and two adjacent edges clamped while the other two edges simply supported (CCSS). Table 4. Dimensionless frequency of stiffened FG sandwich plate with one stiffener (a/b = 1, bs = a/50, hs = 10h, thickness relation =1:8:1) a/(2h) Dimensionless frequency (v) Boundary CSCS CCSS condition n First type Second type First type Second type 40 4.2496 6.3919 5.0234 6.0410 0.5 4.0924 3.7179 4.7525 3.0867 1 4.0177 3.6466 4.6195 3.0023 2 3.9456 3.5750 4.4887 2.9171 10 3.8457 3.4682 4.3042 2.7892 20 3.6826 5.6357 4.0892 5.5741 0.5 3.5461 3.3235 3.9068 2.9433 1 3.4813 3.2622 3.8194 2.8680 2 3.4189 3.2009 3.7345 2.7917 10 3.3328 3.1100 3.6164 2.6768 10 2.8807 4.4068 2.9509 4.3743 0.5 2.7681 2.6098 2.8336 2.4699 1 2.7146 2.5619 2.7771 2.4171 2 2.6632 2.5140 2.7224 2.3638 10 2.5928 2.4432 2.6467 2.2836 5 0.8773 1.9696 1.5559 3.0524 0.5 0.8728 1.8426 1.5500 1.7762 1 0.8707 1.8068 1.5467 1.7400 2 0.8685 1.7709 1.5433 1.7036 10 0.8656 1.7172 1.5379 1.6492 Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 115 Remark: The result shows that the natural fundamental frequencies decrease when the power volume index increases. That as n decreases, the ceramic in plate decreases, and so that the rigidity of the plate decreases. In the same conditions (power volume index, side-to-thickness ration), the frequencies are highest for CCCC Stiffened FG sandwich plates followed by CSCS, CCSS, and SSSS stiffened sandwich plate while SSSS stiffened FG sandwich plate has the lowest value of frequencies. This is due to the higher number of constraints introduced in CCCC stiffened plate compared to that of SSSS, CCSS and CSCS plates increasing the stiffness of the plate. The frequencies are to be decreasing with the increasing of the plate’s thickness, which because the mass of plates is increased much more than the stiffness. First four mode shape of stiffened FG sandwich plate (type 1) shows in Fig. 5. (a) Mode shape 1 - 2048.508 Hz (b) Mode shape 2 - 2621.489 Hz (c) Mode shape 3 - 3936.380 Hz (d) Mode shape 4 - 4838.207 Hz Fig. 5. Four first linear mode shape of simply support Si3N4/SUS304 rectangular plate with one stiffener (type 1, a/b = 1, a/(2h) = 10, n = 0.5, a/bs = 50, hs = 10h, thickness relation = 1-8-1) - Effect of core’s thickness Next, we study the effect of core’s thickness for above stiffened plate. Fig. 6 presents the non-dimensionless frequencies with different values of the substrate-to-face sheet thickness ratio h1/h and with different values of the volume fraction index, n = 0 (ce- ramic rich), n = 0.5 (FGM), and n = 10 (metal rich). Remark: The results from Fig. 6 show that the frequencies are to be increasing with the increasing of the core’s thickness of stiffened plate for type 1, but the frequencies of type 2 116 Pham Tien Dat, Do Van Thom, Doan Trac Luat to be decreasing, which because the core’s thickness is increased then the ceramic in type 1 is richer than that in type 2, so that stiffened plate of type 1 becomes stiffener than that of type 2. 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 h1/h D im e n s io n le s s F re q u e n c y n=0 (SSSS) n=0(CCCC) n=0.5 (SSSS) n=0.5 (CCCC) n=10 (SSSS) n=10 (CCCC) (a) Type 1 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 2 2.5 3 3.5 4 4.5 5 5.5 h1/h D im e n s io n le s s F re q u e n c y n=0 (SSSS) n=0(CCCC) n=0.5 (SSSS) n=0.5 (CCCC) n=10 (SSSS) n=10 (CCCC) (b) Type 2 Fig. 6. Frequencies vary to h1/h(bs = a/50, hs = 10h, a/b = 1, a/(2h) = 10) - Effect of stiffener’s position We study the effect of stiffener’s position by varying the ratio x/a, where x is di- mension from one edge of the plate and showed in Fig. 7. x a b Fig. 7. The rectangular plate with one stiffener Fig. 8 show dimensionless frequencies results for two types of the plate with two boundary conditions. The results confirm that when the stiffener is closer to center of the plate, the stiffness of the plate becomes higher, so that the corresponding frequencies are higher. First four mode shapes of simply support (SSSS) stiffened sandwich FGM plate (type 1) for x/a = 0.25 are shown in Fig. 9 Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 117 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1.5 2 2.5 3 3.5 4 x/a D im e n s io n le s s F re q u e n c y n=0 (SSSS) n=0(CCCC) n=0.5 (SSSS) n=0.5 (CCCC) n=10 (SSSS) n=10 (CCCC) (a) Type 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x/a D im e n s io n le s s F re q u e n c y n=0 (SSSS) n=0(CCCC) n=0.5 (SSSS) n=0.5 (CCCC) n=10 (SSSS) n=10 (CCCC) (b) Type 2 Fig. 8. Effects of stiffener’s position on frequencies (bs = a/50, hs = 10h, a/b = 1, a/(2h) = 10) (a) Mode shape 1 - 1895.171 Hz (b) mode shape 2 - 2608.889 Hz (c) Mode shape 3 - 3216.887 Hz (d) Mode shape 4 - 4394.096 Hz Fig. 9. First four linear mode shape of full clamp Si3N4/SUS304 rectangular plate with one stiffener (type 1, a/b = 1, a/(2h) = 10, n = 0.5, a/bs = 50, hs = 10h, thickness relation = 1:8:1) 118 Pham Tien Dat, Do Van Thom, Doan Trac Luat - Effect of stiffener’s depth Next, free vibration analysis of stiffened sandwich plate is carried out for the dif- ference of stiffener’s depth. The results from Fig. 10 show that when the depth of stiffener increases, the frequency of the stiffened plate increases. 1 1.5 2 2.5 3 3.5 4 4.5 5 1.5 2 2.5 3 3.5 4 hs/2h D im e n s io n le s s F re q u e n c y n=0 (SSSS) n=0(CCCC) n=0.5 (SSSS) n=0.5 (CCCC) n=10 (SSSS) n=10 (CCCC) (a) Type 1 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 hs/2h D im e n s io n le s s F re q u e n c y n=0 (SSSS) n=0(CCCC) n=0.5 (SSSS) n=0.5 (CCCC) n=10 (SSSS) n=10 (CCCC) (b) Type 2 Fig. 10. Frequencies vary to hs/2h (type 1, bs = a/50, hs = 10h, a/b = 1, a/(2h) = 10) - Effect of stiffener’s width We study the effect of stiffener’s width on the frequencies of the plate. The results of frequencies depending on value of the width are show on Fig. 11. We can see that, when the width increases, the frequency increases for both two boundary conditions and for both two types plate. 3.2.2. Free vibration of FGM plate with two stiffeners Now, we study the free vibration for sandwich plate with two stiffeners, which have the same width and depth. The results are showed in Tabs. 5 and 6. Remark: The results of plate with two stiffeners have the same signs as that of plate with one stiffener. The natural fundamental frequencies decrease when the power volume index increases. In the same conditions (power volume index, side-tothickness ration), the frequencies are highest for CCCC Stiffened FG sandwich plates followed by CSCS, CCSS, and SSSS stiffened sandwich plate while SSSS stiffened FG sandwich plate has the lowest frequencies. This is due to the higher number of constraints introduced in CCCC stiffened plate compared to that of SSSS, CCSS CSCS plates that increases the stiffness of the plate. First four mode shape of stiffened FG sandwich plate (type 1) are shows in Fig. 12 Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 119 20 25 30 35 40 45 50 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 a/bs D im e n s io n le s s F re q u e n c y n=0 (SSSS) n=0(CCCC) n=0.5 (SSSS) n=0.5 (CCCC) n=10 (SSSS) n=10 (CCCC) (a) Type 1 20 25 30 35 40 45 50 2 2.5 3 3.5 4 4.5 5 5.5 6 a/bs D im e n s io n le s s F re q u e n c y n=0 (SSSS) n=0(CCCC) n=0.5 (SSSS) n=0.5 (CCCC) n=10 (SSSS) n=10 (CCCC) (b) Type 2 Fig. 11. Frequencies vary to a/bs (type 1, bs = a/50, hs = 10h, a/b = 1, a/(2h) = 10) Table 5. Dimensionless frequency of stiffened FGM sandwich plate with two stiffeners (a/b = 1, bs = a/50, hs = 10h, thickness relation =1:8:1 a/(2h) Dimentionless frequency (v) Boundary SSSS CCCCcondition n First type Second type First type Second type 40 4.2478 7.0932 7.2731 11.843 0.5 4.1417 4.3661 7.0859 6.0555 1 4.0904 4.2718 6.9944 5.8825 2 4.040 4.1447 6.9043 5.7082 10 3.9699 3.9541 6.7761 5.4468 20 3.5820 6.0887 5.3453 9.0321 0.5 3.4919 3.8269 5.2030 5.5521 1 3.4485 3.7651 5.1342 5.4528 2 3.4062 3.7030 5.0673 5.3526 10 3.3470 3.6105 4.9737 5.2025 10 2.6471 4.5195 3.5992 6.0461 0.5 2.5758 2.8764 3.4935 3.8003 1 2.5416 2.8319 3.4427 3.7384 2 2.5083 2.7874 3.3935 3.6765 10 2.4621 2.7217 3.3256 3.5845 5 0.8770 1.9684 2.3742 3.8663 0.5 0.8726 1.8574 2.2980 2.3885 1 0.8704 1.8502 2.2614 2.3485 2 0.8683 1.8427 2.2261 2.3084 10 0.8654 1.8025 2.1777 2.2484 120 Pham Tien Dat, Do Van Thom, Doan Trac Luat Table 6. Dimensionless frequency of stiffened FGM sandwich plate with two stiffeners (a/b = 1, bs = a/50, hs = 10h, thickness relation =1:8:1) a/(2h) Dimensionless frequency (v) Boundary CSCS CCSS condition n First type Second type First type Second type 40 5.7643 9.4821 5.6178 8.9041 0.5 5.6155 5.2859 5.4666 4.7606 1 5.5433 5.1347 5.3929 4.6312 2 5.4725 4.9823 5.3204 4.5002 10 5.3723 4.7538 5.2173 4.3026 20 4.4584 7.5528 4.3655 7.3151 0.5 4.3419 4.6961 4.2496 4.3954 1 4.2857 4.6160 4.1936 4.3028 2 4.2310 4.5353 4.1390 4.2071 10 4.1545 4.4147 4.0626 4.0584 10 3.1250 5.2927 2.9684 5.1027 0.5 3.0371 3.3491 2.8925 3.2354 1 2.9948 3.2956 2.8555 3.1828 2 2.9537 3.2422 2.8194 3.1301 10 2.8969 3.1631 2.7690 3.0517 5 0.8773 1.9696 1.5422 3.1982 0.5 0.8728 1.8619 1.5360 2.1050 1 0.8707 1.8550 1.5327 2.0702 2 0.8685 1.8477 1.5293 2.0352 10 0.8656 1.8358 1.5242 1.9830 Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory 121 (a) Mode shape 1 - 2160.708 Hz (b) Mode shape 2 - 2621.426 Hz (c) Mode shape 3 - 2621.551 Hz (d) Mode shape 4 - 4837.733 Hz Fig. 12. First four linear mode shape of simply support Si3N4/SUS304 sandwich rectangular plate (type 1) with two stiffener (a/b = 1, a/h = 20, n = 0.5, a/bs = 50, hs = 10h, thickness relation = 1-8-1) 4. CONCLUSIONS This paper investigates the free vibration behavior of stiffened FG sandwich plates based on the third order shear deformation and using the finite element method. Two plate configurations, i.e., plate with FG face-sheets and the homogenous core are metal and ceramic, respectively, are considered. The present results are compared to analytical and mesh-free method results given by other researchers to demonstrate a good agree- ment. Some problems such as the effects of width, depth, position of stiffener, thickness of layers, power volume index, boundary conditions on the natural frequencies of stiffened FG sandwich plate have been investigated. 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