Nonlinear Analysis of Functionally Graded Material Plates and Shells Subjected to Inplane Loading.

Transcription

1 Volume 118 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Nonlinear Analysis of Functionally Graded Material Plates and Shells Subjected to Inplane Loading. Monslin Sugirtha Singh.J 1 and Dr. Kari Thangaratnam 2 1 Assistant Professor/Civil Engg Velammal Engg College, Chennai-62 jmsugirtha@gmail.com 2 Professor /Civil Engg DMI College of Engineering,Chennai-16 drkariprof@hotmail.com January 9, 2018 Abstract Nonlinear Finite Element formulation based on Green strains and Piola-Kirchhoff stresses, nonlinear terms of transverse and in plane displacement due to bending stretching coupling using Semiloof shell element is reported. The significant effects of prebuckling due to transverse and in plane displacement and curvature terms on buckling loads are studied, with various boundary conditions, volume fraction index and aspect ratios. New results are obtained for square plate, rectangular plate and cylindrical shells. The nonlinear buckling of functionally graded shells are influenced by prebuckling displacement but not in plates. Key Words and Phrases:Functionally Graded Material, prebuckling, finite element, shell, transverse displacement

2 1 INTRODUCTION Functionally graded materials (FGM) are multiphase composites with continuously varying volume fraction and consequently have high thermo-mechanical properties. The ceramic constituent of FGM are able to withstand high temperature environment due to their better thermal resistance characteristics, while the metal constituent provide stronger mechanical performance and reduce the possibility of catastrophic failure and find application in spacecraft structures. In FGM plates and shells, as soon as the in-plane load is applied, it undergoes transverse displacement even though the load applied is much lower than the buckling load. This is termed as prebuckling displacement and the magnitude depends on the bending-stretching coupling of the FGM plate and shell. The buckling analysis of a FGM plates and shells with the prebuckling displacement included become analogous to that of plate and shells with geometric imperfection, which is a nonlinear problem. Hence the question here is buckling of FGM plates and shells should be treated as linear or nonlinear problem. 1.1 Literature Review A.W.Leissa et al. [1] investigated the transverse deflection of unsymmetrical laminated plates subjected to in plane loads subjected to both uniaxial and biaxial loading and concluded that the behaviour is similar to isotropic plates. Metin [2] states that Clamped boundary condition are capable of supplying the necessary bending moments and twisting moments to keep the functionally graded plate flat. Very few literatures are available to study the influence of pre buckling displacement of FGM plates and shells and in this paper prebuckling influence of FGM plates and shells are analysed. The Finite Element formulation using Semiloof shell element by Singh and Thangaratnam [3,4] for stress, buckling and vibration of FGM plate and shell is extended to nonlinear analysis. The nonlinear finite element formulation is based on Green strains and Piola-Kirchhoff stresses, nonlinear terms of transverse and in plane displacement. To investigate the influence of prebuckling, the plates and shells are subjected to inplane loads and treating the problem as (a) Linear buckling without curvature (WOC), (b) Linear buck

3 ling with curvature (WC),(c) Extended Linear Buckling (ELB), (d) Nonlinear Buckling (NLB). 2 FINITE ELEMENT FORMULATION The Finite Element Formulation is based on principle of virtual work. Internal Work done by stresses =External forces due to virtual Displacement. δ e T σdv = δu T.p.da (1) δ = StressV ector, e = Strain vector. u =Displacement component vector, p=externally Applied load, da=elemental area,dv=elemental volume Linear stress strain relation is expressed as σ = [Q](e e T ) (2) [Q] = Transformed Reduced stiffness Matrix, e T = Initial strain due to temperature Rise e T = α T (3) α -Coefficient of thermal Expansion, T-Rise in Temperature The strain at any point in FGM plates is written as [5] e xx = e xx + zk xx [4] e yy = e yy + zk yy [5] e xy = e xy + zk xy [6] Where e xx = U x [U x 2 + V x 2 + W x 2 ] e yy = U y [U y 2 + V y 2 + W y 2 ] e xy = U y + V x + [U x U y + V x V y + W x W y ] K xx = W xx [W xx 2 + W xy 2 ] K yy = W xx [W yy 2 + W xy 2 ] K xy = 2W xy [W xy(w xx + W yy )] Where denotes derivative of U w.r.to x, We can Write 3 979

4 e xx k xx [e] = e yy and [k] = k yy e xy k xy The left-hand side of Eq.(1) may be written as δe T δd v = ([e] + z[k]) T [Q](e e T )dv (6) = ([e] + z[k]) T [Q]([e] + z[k] e T )dv [ = e([e] + z[k])] = ([e] + z[k]) T [Q]([e] + z[k])dv = ([e] + z[k]) T [Q] α T dv (7) For the FGM volume integral is split in to two parts, integrating [3,4] [ T [ ] [ ] e A B e δ da k] [ ] T [ ] e FN δ da (8) B D k k The [A][B] and [D] matrices are called as the extensional stiffness, coupling stiffness, bending stiffness respectively. ([A], [B], [D]) = h 2 h (1, z, z 2 )[Q] (9) 2 And the thermal force F N and the thermal moment M T are given by {F n, M T } = h 2 h [Q]{α(z)} T (1, z)dz (10) 2 We can write [ ] e = [e k L ]+[e NL ] (11) e= plain strain, k =curvature, [e L ] = Linear part, [e N L]=Non Linear part The linear vector is [e L ] = [u x, v x, (u y + v y ), w xx, w yy, 2w xy ] M T The nonlinear part can be written as e NL = 1 2 [R o][φ] (12) Where φ is the vector of slope and defined as [φ] T = [u x, u y, v x, v y.w x.w y, w xx, w yy, w xy ] Where [R o ] = 4 980

5 u x 0 v x 0 w x u y 0 v y 0 w y u y u x v y v x w y w x w xx 0 w xy w yy w xy w xy w xy 0 By taking the variation in Eq. (11) [ ] ɛ δ = δ[ɛ k L ] + 1[R 2 o]δ[φ] + 1δ[R 2 o][φ] = δ[ɛ L ]+[R o ]δ[φ] (14) Where [R o ]δ[φ] = δ[r o ][φ] function matrix of Semiloof shell element is [3,4] [u] = [d][q] (15) Where [q] - Nodal degree of freedom. [d] - Shape function The vector of slope[]can be written as [φ] = [G][q] (16) δ[φ] = [G]δ[q] (17) (13) as The strain energy displacement relation for linear part is given [e l ] = [B L ][q] (18) [e L ] = [B L ]δ[q] Using Eq. (13) and Eq.(17) [e L ] = [R o ][G]δ[q] (19) Therefore the nonlinear strain matrix[b NL ] can be written as [B NL ] = [R o ][G] (20) Substituting Eq.(17), 5 981

6 Eq.(18) and Eq.(19) in Eq. (14) [ ] ɛ δ[ ] = [[B k N ] + [B NL ]]δ[q] = [H]δ[q] (21) H = [B N ] + [B NL ] (22) The finite element representation of FGM plates using the Equations δe T δdv = δu T P da δ = [Q](e e T ) with some simplifications can be written as arbitrary variations in [q] for single element. δ[q] T [H] T [F o ]da = [q] T [ [q] T [P ]]da + [ ] [ ] [H] T [F T ]da (23) A B el Where [F o ] = B D e NL [ ] FN [F T ] = and[f ] = [F o ] [F T ] (24) M T Since [q] is an arbitrary variation of nodal displacement, the non linear equation for FGM plates and shells reduced to ψ = [H] T [F o ]da [F o ] [F T ] = 0 (25) where ψ is the vector residual force. [f m ] = [d] T [p]da (26) [F T ] = [ ] [H] T FN da (27) M T Assuming the solution in the current configuration known as p,[6]the approximation of ψ about q is [ ] ψ(q) = ψ(q+ ) = ψ(q)+ δψ δq+... = 0 (28) δq q Ignoring the higher order terms, a first order approximation relating the vector of residual forces to the displacement increments is obtained at q = (q + δq) = [K T ] q (29) where [K T ] is the tangent stiffness matrix

7 [ ] [K T ] = δψ δq (30) q=q Solution of linear Eq. (29) provides vector of displacement increments and therefore the solution is in the future configuration, Since the linear equation is only a first order approximation to the original nonlinear Eq. (13), iteration must be carried out with an increment to obtain more accurate results. Assuming the solution obtained at the i th iteration is q(i) then the new approximation solution is q(i + 1) = q(i) + q(i) (31) The solution is exact q(i + 1)I exact if q(i + 1) = 0 The explicit expression for tangent stiffness [K T ] in terms of previously determined element matrices can be determined from Eq.(13) δψ = [H] T δ[f o ]da+ δ[h] T [F o ]da δ[h] T [F T ]da = [H] T δ[f o ]da+ δ[h] T [F ]da (32) From Eq.(11) and Eq.(22) delta[h] = δ[b NL ] = δ[]r o [G] (33) Substituting Eq. (17) and Eq. (33) in Eq. (32) δψ = [H] T [E][H]daδq+ [G] T δ[r o ] T F da (34) [ ] A B Where [E] = B D Expanding [[R o ] T ][F ] = P δ[φ] = [p][g]δ[q] (35) N xx N xy N xy N yy W here[r o ] = 0 0 N xx N xy N xy N yy 0 0 (36) N xx N xy N xy N yy Substituting in Eq. (27) δψ = [H] T [E][H]daδ[q]+ [G] T [P ]G]daδ[q] = [K T ]δ[q] (37) Substituting for [H] = [B L ] + [B NL ],From Eq.(14) [[BL ] + [B NL ]] T [E][[B L ]] + [B NL ]daδ[q] + [G] T [P ][G]daδ[q] = [BL ] T [E][B L ]daδ[q]+ [B L ] T [E][B NL ]daδ[q]+ [B NL ] T [E][B L ]daδ[q] [B NL ] T [E][B NL ]daδ[q]

8 [G] T [P ]daδ[q] = [K T ]δ[q] (38) The Tangent stiffness matrix is given by, [K T ] = [K L ] + [K NL ] + [K G ] (39) [K L ] = [B L ] T [E][B L ]da (40) [ KNL ] is initial displacement matrix or Large displacement matrix or nonlinear stiffness matrix. [K NL ] = [B L ] T [E][B NL ]da+ [B NL ] T [E][B L ]da+ [B NL ] T [E][B NL ]da (41) [K G ] = [G] T [P ][G]da (42) [K G ] is geometric stiffness matrix or initial stress matrix. The most common approximation of the nonlinear problem in buckling analysis treating the prebuckling behaviour as linear and taking K NL = 0 δψ = [K L ] + [K G ]δ[q] If the loads are increased by a factorλ we find that a neutral stability exists [3]. That is [[K L ]+λ[k G ]]δ[q] = 0 (43) From this λ can be obtained by solving the typical eign value problem [K L ] + λ[k G ] = 0 (44) In Fig.1 this corresponds to bifurcation on point a The next improvement considers the initial displacements matrix [] as linear (that is, prebuckling deformation is linear) Which leads to the extended Eigen value problem. [K L ]+[K NL ]+λ[k G ] = 0 (45) In Fig. 1 the buckling load corresponds this to point b 8 984

9 Fig:1 Non Linear response of Eigen value Problem If the prebuckling deformation is nonlinear, it will become nonlinear analysis and the buckling load corresponds to point c or d. If it is a case of bifurcation buckling the load corresponds to point c. If it is a limit load case, then buckling load corresponds to point d. 3 CONVERGNCE AND VALIDATION. The program developed using Semiloof shell element by Singh and Thangaratnam [3,4] for thermal stress, vibration and buckling analysis of FGM plates and shells is extended to nonlinear analysis based on the above formulation. The program is validated with results available in the literature and good agreement is observed. The boundary conditions given in Ref. [3,4] are used. SS2 Simply supported u 0, v = 0, w = 0, θxz 0, at x = 0, a and 0, v 0, w = 0, θxz 0, at y = 0, b SS3 Simply supportedu = 0, v 0, w = 0, θxz 0, atx = 0, a and u 0, v =, w = 0, θxz 0, at y = 0, b SS4 Simply supported u 0, v 0, w = 0, θxz 0, at x = 0, a and u 0, v 0, w = 0, θxz 0, aty = 0, b

10 3.1 Central deflection of a Clamped Isotropic plate under uniform Loading. An isotropic square plate subjected to uniform load is analysed. The length of the plate a=300 in and height h=3 in, a/h =100. The material properties are E = 30e 6 psi and = The central deflection of a quarter plate is computed and is validated with Levy [7] and good agreement is observed as shown in Fig. 2 Fig :2 Load versus central deflection of a square plate 3.2 Central deflection of a Clamped cylindrical Shell under normal pressure. A circular cylindrical panel 10 in x 10 in, simply supported is subjected to radial loading is considered. Thickness=0.125 in, R=100 in,e= lb/in 2, poisons ratio =0.3. From Fig 3 it is seen that the present element compares well with other results in Ref.[8]

11 Fig 3: Load versus deflection of a curved pane 4 RESULT AND DISCUSSION. 4.1 Square plate A square plate of size 100mmx100mm and thickness h =1 mm. (a/h=100) is considered. The material considered are FGM1 (Stainless Steel(SUS304), Zirconia(ZrO 2 )) and FGM2 (Alumina (Al 2 O 3 ), Titanium(Ti-6AI-4v)). The material properties used are shown in Table 1. The buckling loads are compared in Table 2, 3, 4 and 5 for the variation of volume fraction index (n= 0.5, 0.7, 1.0, 3.0 and 5.0).The results shows that the nonlinear buckling load is less significant in the case of simply supported boundary conditions and has no effect in clamped clamped boundary conditions. The transverse displacement due to coupling of FGM plate does not affect the critical load carrying capacity of the plate. Load versus displacement to thickness (w/h) curve is studied for linear and nonlinear cases. For FGM2 plate the nonlinear displacement varies significantly as the volume fraction index varies but not much in FGM1, since the Youngs modulus of constituent materials vary much for FGM2 and hence the coupling effect. Linear and Nonlinear analysis are carried out for SS3 plate for FGM1 and FGM2. In Linear analysis the displacement linearly

12 increases and in the nonlinear analysis the displacement increases is nonlinear for FGM2 plates for the load increment as shown in Figure 4 and for FGM1 as shown in Figure 5. For FGM2 the nonlinear displacement varies significantly as the volume fraction index varies but not much in FGM1, since the Youngs modulus of constituent materials vary much for FGM2 and hence the bending stretching coupling effect. Table 1: FGM Material Properties. Table 2: Buckling load versus volume fraction index for square plate with SS2 BC

13 Table3: Buckling load versus volume fraction index for square plate with SS3 BC. Table 4: Buckling load versus volume fraction index for square plate SS4 BC

14 Table 5: Buckling load versus volume fraction index for square plate SS4 BC Figure 4: Load Vs displacement of SS3 plate for FGM

15 Figure 5: Load Vs displacement of SS3 plate for FGM1 4.2 Rectangular plate Rectangular plate of various aspect ratios 2, 2.5 and 3 are analysed for simply supported boundary conditions SS3 for FGM1 and the results are tabulated in Table 6 to Table 8.The results show that there is no influence due to prebuckling effect in the nonlinear buckling as in square plate. Table 6 Buckling loads for SS3 plate subjected to biaxial loading aspect ratio a/b=

16 Table 7 Buckling loads for SS3 plate subjected to biaxial loading aspect ratio a/b=2.5 Table 8 Buckling loads for SS3 plate subjected to biaxial loading aspect Ratio a/b=3 4.3 Cylindrical Shell. The cylindrical shell is analysed for various length to thickness (L/R) ratios (5,10,20,30,40 and 50), Radius R=1cm, thickness h=0.03cm and the material considered are FGM1 and FGM2. The shell is subjected to SSM and CCM boundary conditions and analysed for different volume fraction index n and various Lengths to Radius (L/R) ratios. The buckling loads are given in Table 9 to Table 13. As the L/R ratio increases from 5 to 50 the linear buckling load (LB) decreases and also the nonlinear Buckling load (NLB) but the ratio of nonlinear buckling load to linear buckling load ratio increases from 0.17 to 0.39 for simply supported and for clamped shell increases from 0.2 to For FGM2 shell the nonlinear dis

17 placement varies significantly as the volume fraction index varies but not much in FGM1, since the Youngs modulus of constituent materials vary much for FGM2 and hence the coupling effect as in plates and shells. The results shows that the nonlinear buckling load is more significant in both simply supported and clamped boundary conditions. Linear and Nonlinear analysis are carried out for SSM and CCM shell for FGM1 and for SSM shell for FGM2, and Load versus displacement curve are shown in Figure 6, 7 and 8 respectively. In linear analysis the transverse displacement is linear and not significant but the nonlinear displacement is significant and the displacement curve is linear. For FGM2 shell the nonlinear displacement varies significantly as the volume fraction index varies but not much in FGM1, since the Youngs modulus of constituent materials vary much for FGM2 and hence the coupling effect as in plates and shells. Table 9: Buckling load for various L/R ratios of simply supported FGM1 shell

18 Table 10; Buckling load for various volume fraction index of FGM1Simply Supported shell Table 11: Buckling load for various L/R ratios of FGM1clamped shell Table 12;Buckling load for various volume fraction index of FGM1 clamped shell

19 Table 13; Buckling load for various volume fraction index of FGM2 Simply Supported shell Figure 6 Load Vs displacement of SSM shell for FGM1 Figure 7 Load Vs Displacement of CCM shell for FGM

20 Figure 8. Load Vs Displacement of SSM shell for FGM2 5 CONCLUSION The Finite Element formulation using Semiloof shell element for Functionally Graded Material is presented. The accuracy of the numerical results are verified with the existing results from the literature and the agreement is found good. The influence of pre buckling displacement in the nonlinear buckling of plats and shells are studied and new results are presented. In the case of plate there is not much variations in the buckling load from nonlinear analysis for simply supported plates, but the transverse displacement is more and the behaviour is nonlinear. In the case of shell there is a significant change in the buckling load from nonlinear analysis for simply supported and clamped conditions. The displacement is large for nonlinear analysis even though the behaviour is linear. Hence nonlinear analysis only confirms the buckling is Bifurcation or Limit Point. The behaviour also depends on the youngs modulus of the constituent material which produce bending stretching coupling. References [1] Leissa A.W and Qatu M.S. Buckling or Transverse Deflection of Unsymmetrical Laminated Plates Subjected to in Plane Loads. AIAA Journal,31(1993)

21 [2] Metin Aydogdu.Condition for Functionally Graded Plates to Remain Flat under Inplane Loads by Classical Plate Theory. Composite Structures.82(2008) [3] Singh M. S and Thangaratnam K.R,Analysis of Functionally Graded Plates and Shells: Stress, Buckling, Free Vibration, Journal of Aerospace Sciences and Technologies.66(2014) [4] Singh M. S and Thangaratnam K.R, Thermal Stress and Buckling Analysis of Functionally Graded Plate, Advanced Materials Research (2014) [5] Javaherian,H, Dowling P.J and Lyons L.P.R,Non Linear Finite Element Analysis of Shell Structures Using the Semi-Loof Element, Computers and Structures.12( 1980) [6] Zienkiewicz, The Finite Element Method. McGraw-Hill Publishing Co.Ltd, pp [7] Levy S, Square Plate with Clamped Edges Under Normal Pressure Producing Large Deflection, Technical Report, National advisory committee of Aeronautics [8] Hazim F. Sharhan and Jawad K Al-Bayati,Large Deflection Geometrically Nonlinear Behavior of Cylindrical Shells. The 5th Jordanian International Civil Engineering Conference, Amman,

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