Composite Structures 94 (2012) 2952–2960 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects Dao Huy Bich, Dao Van Dung, Le Kha Hoa ⇑ Vietnam National University, Hanoi, Viet Nam a r t i c l e i n f o Article history: Available online 24 April 2012 Keywords: Functionally graded materials Static and dynamic buckling Shallow spherical shells a b s t r a c t This paper presents an analytical approach to investigate the nonlinear static and dynamic unsymmetrical responses of functionally graded shallow spherical shells under external pressure incorporating the effects of temperature Governing equations for thin FGM spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell geometrical nonlinearity Approximate solutions are assumed and Galerkin procedure is applied to determine explicit expressions of static critical buckling loads of the shells For the dynamical response, motion equations are numerically solved by using Runge–Kutta method and the criterion suggested by Budiansky–Roth A detailed analysis is carried out to show the effects of material and geometrical parameters, boundary conditions and temperature on the stability and dynamical characteristics of FGM shallow spherical shells Ó 2012 Elsevier Ltd All rights reserved Introduction Structures in the form of spherical shells are used widely in many engineering applications Most of these shells are subjected to static and impulsive loads which is cause of instability and strength reduction of the structures As a result, the investigation on the nonlinear static and dynamical buckling of spherical shells is necessary and has attracted attention of many researchers Budiansky and Roth [1] studied axisymmetrical dynamic buckling of clamped shallow isotropic spherical shells Their well-known results have received considerable attention in the literature Huang [2] considered the unsymmetrical buckling of thin shallow spherical shells under external pressure He pointed out that unsymmetrical deformation may be the source of discrepancy in critical pressures between axisymmetrical buckling theory and experiment The static buckling behavior of shallow spherical caps under an uniform pressure loads was analyzed by Tillman [3] Results on the dynamic buckling of clamped shallow spherical shells subjected to axisymmetric and nearly axisymmetric step-pressure loads using a digital computer program were given by Ball and Burt [4] Kao and Perrone [5] reported the dynamic buckling of isotropic axisymmetrical spherical caps with initial imperfection Two types of loading are considered, in this paper, namely, step loading with infinite duration and right triangular pulse Wunderlich and Albertin [6] also studied on the static buckling behavior of isotropic imperfect spherical shells New design rules in their work for these shells were developed, which take into account relevant details like boundary conditions, ⇑ Corresponding author E-mail address: lekhahoa@gmail.com (L.K Hoa) 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compstruct.2012.04.012 material properties and imperfections The nonlinear static and dynamic response of spherical shells has been analyzed by Nath and Alwar [7] using Chebyshev series expansion Based on an assumed two-term mode shape for the lateral displacement, Ganapathi and Varadan [8] investigated the problem of dynamic buckling of orthotropic shallow spherical shells under instantaneously applied uniform step-pressure load of infinite duration The same authors analyzed the dynamical buckling of laminated anisotropic spherical caps using the finite element method [9] Static and dynamic snapthrough buckling of orthotropic spherical caps based on the classical thin shell theory and Reissener’s shallow shell assumptions have been considered by Chao and Lin [10] using finite difference method There were several investigations on the buckling of spherical shells under mechanical or thermal loading taking into account initial imperfection such as studies by Eslami et al [11] and Shahsiah and Eslami [12] Functionally graded materials (FGMs) which are microscopically composites and composed of ceramic and metal constituents or combination of metals have received much interest in recent years Due to essential characteristics such as high stiffness, excellent temperature resistance capacity, functionally graded materials find wide applications in many industries, especially in temperature shielding structures and nuclear plants Shahsiah et al [13] presented an analytical approach to study the instability of FGM shallow spherical shells under three types of thermal loading including uniform temperature rise, linear radial temperature, and nonlinear radial temperature Prakash et al [14] gave results on the nonlinear axisymmetric dynamic buckling behavior of clamped FGM spherical caps Also, the dynamic stability characteristics of FGM shallow spherical shells have been considered by Ganapathi 2953 D.H Bich et al / Composite Structures 94 (2012) 2952–2960 [15] using the finite element method In his study, the geometric nonlinearity is assumed only in the meridional direction in strain– displacement relations Bich [16] studied the nonlinear buckling of FGM shallow spherical shells using an analytical approach and the geometrical nonlinearity was considered in all strain–displacement relations By using Galerkin procedure and Runge–Kutta method, Bich and Hoa [17] analyzed the nonlinear vibration of FGM shallow spherical shells subjected to harmonic uniform external pressures Recently, Bich and Tung [18] reported an analytical investigation on the nonlinear axisymmetrical response of FGM shallow spherical shells under uniform external pressure taking the effects of temperature conditions into consideration Shahsiah et al [19] used an analytical approach to investigate thermal linear instability of FGM deep spherical shells under three types of thermal loads using the first order shell theory basing Sander nonlinear kinematic relations To best of authors’ knowledge, there is no analytical investigation on the nonlinear dynamic stability of FGM shallow spherical shells In the present paper, the nonlinear static and dynamical buckling behavior of clamped FGM shallow spherical shells under uniform external pressure and thermal loads are considered by using an analytical approach Governing equations for thin shallow spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell nonlinear terms Material properties are assumed to be temperature independent and graded in the thickness direction according to a simple power law function Approximate one-term solutions of deflection and stress function that satisfy the boundary conditions are assumed and Galerkin procedure is used to obtain explicit expressions of static critical buckling loads For dynamical analysis, motion equation is solved numerically by applying Runge–Kutta method and the criterion suggested by Budiansky–Roth The effects of material and geometrical properties, temperature and boundary conditions on the response of FGM spherical shells are analyzed and discussed Theoretical formulations k k 2z ỵ h 2z ỵ h ; qzị ẳ qm þ ðqc À qm Þ ; 2h 2h k k 2z ỵ h 2z ỵ h azị ẳ am ỵ ac am ị ; Kzị ẳ K m ỵ K c K m ị ; 2h 2h Ezị ẳ Em ỵ Ec Em ị mzị ¼ m ¼ const; ð1Þ where k P is volume fraction index and E, m, q, a, K are Young’s modulus, Poisson’s ratio, mass density, coefficient of thermal expansion, coefficient of thermal conduction, respectively, and subscripts m and c stand for the metal and ceramic constituents, respectively 2.2 Governing equations It is convenient to introduce an additional variable r defined by the relation r = Rsin u, where r is the radius of the parallel circle with the base of shell If the rise H of the shell is much smaller than the base radius r0 we can take cos u % and Rdu = dr, such that points of the middle surface may be referred to coordinates r and h The strain components on the middle surface of shell based upon the von Karman assumption are of the form e0r ẳ u;r w=R ỵ w2;r =2; e0h ẳ v;h ỵ uị=r w=R ỵ w2;h =2r2 ị; c0rh ẳ rv=rị;r ỵ u;h =r ỵ w;r w;h =r; 2ị vr ẳ w;rr ; vh ẳ w;hh =r2 ị ỵ w;r =r; vrh ẳ w;rh =r w;h =ðr2 Þ; ð3Þ where u, v and w are the displacements of the middle surface points along meridional, circumferential and radial directions, respectively, and vr, vh, vrh are the change of curvatures and twist, respectively Using Eqs (2) and (3), the geometrical compatibility equation is written as Á 1 1 e e0 ỵ r2 e0h;r ị;r rc0rh ;rh ẳ r2 w ỵ v2rh À vr vh ; r2 r;hh r r;r r2 r R ð4Þ @ @ @ where r ẳ @r is a Laplaces operator ỵ r @r ỵ r @h2 The strains across the shell thickness at a distance z from the mid-plane are given by 2.1 Functionally graded shallow spherical shells Consider a clamped FGM shallow spherical shell of thickness h, base radius r0, curvature radius R, rise H as shown in Fig It is defined in coordinate system (u, h, z), where u and h are in the meridional and circumferential directions of the shell, respectively, and z is perpendicular to the middle surface positive inward Assume that the shell is made from a mixture of ceramic and metal constituents and the effective material properties vary continuously along the thickness by the power law distribution er ¼ e0r À zvr ; eh ¼ e0h À zvh ; crh ¼ c0rh À 2zvrh : ð5Þ The stress–strain relationships including temperature effect for an FGM spherical shell are defined by the Hooke law Ezị ẵer ; eh ị ỵ meh ; er ị ỵ mịa DT 1; 1ị; m2 Ezị ẳ c ; 21 ỵ mị rh rr ; rh ị ẳ rrh 6ị where DT is temperature change from stress free initial state The force and moment resultants of an FGM shallow spherical are expressed in terms of the stress components through the thickness as H r ϕ ,u fðNr ; Nh ; Nrh Þ; r0 R Fig Geometry and coordinate system of a spherical cap Z h=2 frr ; rh ; rrh gð1; zÞdz: ð7Þ Àh=2 θ ,v z, w ðM r ; Mh ; M rh ịg ẳ Introduction of Eqs (1), (5) and (6) into Eq (7) gives the constitutive relations Á E1 E / er ỵ me0h 2 vr ỵ mvh ị m ; 1m 1Àm 1Àm Á E1 À E / Nh ¼ e ỵ me0r 2 vh ỵ mvr ị À m ; À m2 h 1Àm 1Àm E1 E c0 v ; Nrh ẳ 21 ỵ mị rh ỵ m rh Nr ẳ 8ị 2954 D.H Bich et al / Composite Structures 94 (2012) 2952–2960 E2 E / er ỵ me0h vr ỵ mvh ị b ; 1Àm 1Àm 1Àm Á E2 À E / Mh ẳ e ỵ me0r vh ỵ mvr Þ À b ; À m2 h 1Àm 1Àm E2 E c0 À v ; M rh ¼ 21 ỵ mị rh ỵ m rh Boundary conditions and solution of problem Mr ẳ 9ị The FGM shallow spherical shell is assumed to be clamped at its base edge and subjected to external pressure uniformly distributed on the outer surface of shell Depending on the in-plane behavior at the edge, two cases of boundary conditions will be considered Ecm h 1 ; ; E2 ẳ Ecm h E1 ẳ Em h ỵ kỵ1 k ỵ 2k ỵ Em h 1 ; E3 ¼ À þ þ Ecm h k þ k þ 4k ỵ 12 k #" k # Z h=2 " 2z ỵ h 2z ỵ h /m ;/b ị ẳ Em ỵ Ecm am ỵ acm DT1;zịdz; 2h 2h Àh=2 Case (i) The base edge of shell is clamped and freely movable (FM) in the meridional direction The associated boundary conditions are where Ecm ¼ Ec À Em ; acm ẳ ac am : 10ị The nonlinear equations of motion of perfect shallow spherical shell according to the classical shell theory are [20] 1 rNr ị;r ỵ Nrh;h Nh ẳ q1 u;tt ; r r r 1 rNrh ị;r ỵ Nh;h þ Nrh ¼ q1 v ;tt ; r r r ! 1 1 ðrM r Þ;rr þ Mrh;rh þ M rh;h þ Mh;hh À M h;r ỵ Nr ỵ N h ị r r r R 1 Nrh w;r ỵ Nh w;h ỵ q ẳ q1 w;tt ; 11ị ỵ rNr w;r ỵ Nrh w;h ị;r ỵ r r r ;h where q is an uniform external pressure acting on the shell outer surface positive inward, and q1 ¼ qm h þ qcm h ; kþ1 qcm ¼ qc À qm : ð12Þ By taking the inertia forces q1u,tt ? andq1v,tt ? into consideration because of u ( w, v ( w [21], two first of Eq (11) are satisfied by introducing the stress function f 1 Nr ẳ f;r ỵ f;hh ; r r Nh ẳ f;rr ; 1 Nrh ¼ f;h À f;rh ; r r ð13Þ and substituting relations (2), (3), (9) and (13) into third of Eq (11) gives 1 1 q1 w;tt ỵ Dr4 w r2 f f;r ỵ f;hh w;rr w;r ỵ w;hh f;rr R r r r r 1 1 ð14Þ w;rh w;h q ẳ 0; ỵ f;rh À f;h r r r r E E E2 where D ẳ m2 ịE1 Eq (14) includes two unknown functions w and f and to find a second equation relating two these functions the geometrical compatibility Eq (4) is used For this aim, from Eq (8) strain components can be expressed through force resultants as à 1 ðNr ; Nh Þ À mðNh ; Nr ị ỵ E2 vr ; vh ị ỵ /m 1; 1ị ; E1 c0rh ẳ ỵ mịNrh ỵ E2 vrh : E1 Nr ẳ 0; Nrh ¼ at r ¼ r ; ð17Þ Case (ii) The base edge of shell is clamped and immovable (IM) For this case, the boundary conditions are u ¼ 0; w ¼ w;r ¼ 0; Nr ¼ N0 ; Nrh ¼ at r ¼ r ; ð18Þ where N0 is fictitious compressive edge load at immovable edge The mentioned boundary conditions can be satisfied, when the deflection w is represented by a single term of a Fourier series This approximate solution is acceptable in the vicinity of the buckling load [22,23] w¼W 16r2 ðr À rÞ2 sin nh; r40 ð19Þ where W = W(t) is a time dependent total amplitude of deflection of shell Regularly, the stress function f should be determined by the substitution of deflection function w into compatibility equation (16) and solving the resulting equation However, such a procedure is very complicated in mathematical treatment because obtained equation is a variable coefficient partial differential equation Accordingly, integration to obtain exact stress function f(r, h) is extremely complex Therefore, the stress function f satisfying boundary conditions (18) is chosen in the same form of deflection w as mentioned in Refs [24,25] f ẳF 16r2 r rị2 r2 sin nh ỵ N0 : r40 ð20Þ Substituting Eqs (19) and (20) into Eqs (14) and (16) and applying Galerkin procedure for the resulting equations yield " # 2E1 Wr 20 32n 2 Fẳ W ; ỵ 2n ị 738 20n2 ỵ 12n4 ị R p 7D38 20n2 ỵ 12n4 ị ỵ 2n2 ị 32nWF Wỵ F pr40 4r 2Rr20 ỵ 2n2 7q ỵ N0 W ẳ : 4pRn 8pn 2r 20 21ị q1 W ;tt ỵ 22ị Eliminating F from Eqs (21) and (22) leads to Á e0r ; e0h ẳ 15ị q1 W ;tt Substituting these equations into Eq (4), taking into account relations (3) and (13) leads to 2 1 1 r f ỵ r w w;rh w;h ỵ w;rr w;hh ỵ w;r ẳ 0: E1 R r r r r w ¼ w;r ¼ 0; ð16Þ Eqs (14) and (16) are governing equations used to investigate the nonlinear static and dynamic buckling of FGM shallow spherical shells À " # 7Dð38 À 20n2 þ 12n4 Þ ð3 þ 2n2 Þ2 E1 W þ þ 4r40 7R ð38 À 20n2 þ 12n4 Þ 96E1 ỵ 2n2 ịn W2 ỵ 7p 7p þ þ 2n2 7q þ N0 WÀ ¼ : 4pRn 8pn 2r 20 Rr20 ð38 20n2 12n4 ị 2048E1 n2 W3 20n2 ỵ 12n4 ị r ð38 ð23Þ Based on this equation, the mechanical and thermal stability analysis of shells are considered below D.H Bich et al / Composite Structures 94 (2012) 2952–2960 Mechanical stability analysis Consider an FGM shallow spherical shell being clamped and freely movable at the edge r = r0 (case (i)) The outer surface of shell is subjected to uniform external pressure q and without the effects of temperature In this case N0 = and Eq (23) reduces to " # 7D38 20n2 ỵ 12n4 ị ỵ 2n2 ị2 E1 W q1 W ;tt þ þ 4r 40 7R2 ð38 À 20n2 þ 12n4 ị 96E1 ỵ 2n2 ịn Rr 20 38 7p 7q ẳ : 8pn 20n2 12n4 ị ỵ W2 ỵ s 2 2 49 38 20n2 ỵ 12n4 h ; D X1 ẳ ỵ 2n2 n4 R sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi 49 à 38 20n2 ỵ 12n4 h X2 ẳ ỵ : D ỵ 2n2 n4 R 24ị By putting E2 ẳ E2 =h ; n ¼ r =R; à EÃ3 ¼ E3 =h ; W à ¼ W=h; à q ¼ q1 =h: D ẳ D=E1 h ị; 25ị Eq (24) is rewritten as (" 4 EÃ1 7Dà ð38 À 20n2 þ 12n4 Þ2 h R ð38 À 20n2 þ 12n4 Þ 4n4 # 3 2 2 ỵ 2n ị h 963 ỵ 2n ịn h W W ỵ R R 7pn2 ) 4 2048n2 h 7q ẳ ỵ W : ð26Þ 8p n 7p2 n4 R qà h2 W ;tt ỵ Note that quantities qupper and qlower in Eq (28) depend on the buckling mode n and the minimum values of buckling loads i.e critical upper and lower buckling loads may be obtained by minimization of these loads with respect to n An FGM shallow spherical shells subjected to external pressure loads varying as linear functions of time, q = st(s – a loading speed), is considered The aim of the problem is to determine the critical dynamic buckling loads In this case of load, Eq (26) is rewritten as (" 4 E1 7D 38 20n2 ỵ 12n4 ị2 h q h2 W ;tt ỵ 4 R 38 20n ỵ 12n ị 4n ) 2 # 3 4 2 2 ỵ 2n ị h 963 ỵ 2n ịn h 2048n h W ỵ W W þ R R 7p2 n4 R 7pn2 À 7st ¼ 0: 8pn 4.1 Static buckling and postbuckling analysis 32ð38 20n2 ỵ 12n4 ị " D 38 20n2 ỵ 12n4 ỵ 2n2 # Thermomechanical stability analysis 2 3 h X1 R n2 2n2 h 2X1 3X21 ỵ X31 ; 49 R " 3 p2 E1 ỵ 2n2 ị3 h X2 38 20n ỵ 12n qlower ẳ D R n2 3238 20n2 ỵ 12n4 ị ỵ 2n2 # 2n2 h ỵ 2X2 3X2 ỵ X2 ; 49 R 32ị 27ị Eq (27) may be used to find static critical buckling load and trace postbuckling load–deflection curves of FGM spherical shells It is evident that q(W⁄) curves originate from the coordinate origin Eq (27) indicates that there is no bifurcation-type buckling for pressure loaded spherical shells and extremum-type buckling only occurs under definite conditions The extremum buckling load of the shell can be found from Eq (27) using the condition dq/dW⁄ = which give p2 E1 ỵ 2n2 ị3 dW =dtjt¼0 ¼ 0: The well-known criterion suggested by Budiansky and Roth [1] is employed herein According to this criterion, for large values of loading speed, the average deflection–time curve (W⁄ À t) of obtained displacement response increases sharply depending on time and this curve obtains a maximum by passing from the slope point, and at the corresponding time t = tcr the stability loss occurs The value t = tcr is called critical time and the load corresponding to this critical time is called dynamic critical buckling load To obtain displacement responses, Eq (31) in conjunction with initial conditions (32) will be solved by using the Runge–Kutta method Omitting the term of inertia force in Eq (26) yields (" 4 8pnE1 7D 38 20n2 ỵ 12n4 ị2 h qẳ R 738 20n2 ỵ 12n4 ị 4n4 # 3 ỵ 2n2 ị h 963 ỵ 2n2 ịn h W W ỵ R R 7pn2 ) 4 2048n2 h ỵ W : 7p2 n4 R 31ị In the present study, initial conditions are assumed as W à jt¼0 ¼ W Ã0 ; A clamped FGM shallow spherical shell with immovable edge (case (ii)) subjected simultaneously to uniform external pressure q and thermal load is considered The condition expressing the immovability of the boundary edge, i.e u = at r = r0, is fulfilled on the average sense as Z pZ r0 @u rdrdh ¼ 0: @r ð33Þ From Eqs (2) and (15) one can obtain the expression of @u/@r and then substituting the result into Eq (33) gives ỵ where 30ị 4.2 Dynamic buckling analysis E1 ẳ E1 =h; qupper ẳ 29ị Providing is 2 2 49 à 38 À 20n2 þ 12n4 h À D > 0: ỵ 2n2 n4 R 2048E1 n2 W3 7p2 r 40 38 20n2 ỵ 12n4 ị 2955 N0 ẳ ð28Þ 16Fn 16E1 W 128E1 W /m À À þ : 3pr0 ð1 À mÞ 15pRnð1 À mÞ 105r 20 ð1 À mÞ ð1 À mÞ ð34Þ Introduction F from Eq (21) into Eq (34) leads to 16E1 266 170n2 ỵ 64n4 ị W 105pRn1 mị38 20n2 ỵ 12n4 ị 128E1 p 38 20n2 ỵ 12n4 ị 40n2 /m W2 : ỵ mị 105p2 r20 mị38 20n2 ỵ 12n4 ị N0 ẳ 35ị 2956 D.H Bich et al / Composite Structures 94 (2012) 2952–2960 5.2 Dynamic stability analysis Substituting this relation N0 into Eq (23) yields (" 4 7Dà ð38 À 20n2 þ 12n4 Þ2 h q R 4n4 ! # ỵ 2n2 ị 28266 170n2 ỵ 64n4 ị h W ỵ ỵ 105p2 n2 mị R " 963 ỵ 2n2 ịn 83 ỵ 2n2 ị266 170n2 ỵ 64n4 ị ỵ 7pn2 105pnð1 À mÞn2  Ã# 3 224 p ð38 20n2 ỵ 12n4 ị 40n2 h W þ R 105p3 nð1 À mÞn2 ) "  Ã# 4 2048n2 643 ỵ 2n2 ị p2 38 20n2 þ 12n4 Þ À 40n2 h Ã3 þ þ W R 7p n 105p2 n4 ð1 À mÞ " # 2 /m ỵ 2n Þ h h 7q Wà À À : 36ị ẳ R 4pn R 8p n mịh 2n2 E1 h W ;tt ỵ 38 20n2 ỵ 12n4 ị Eq (36) is employed to investigate static and dynamic unsymmetric responses of FGM shallow spherical shells under combined mechanical and thermal loads 5.1 Static stability analysis Environment temperature is uniformly raised from initial value Ti, at which the shell is thermal stress free, to final one Tf and temperature change DT = Tf À Ti is independent to thickness variable The thermal parameter /m can be expressed in terms of DT due to Eq (10) as /m ẳ Um0 DTh; 37ị where Um0 ẳ Em am þ Em acm þ Ecm am Ecm acm þ : kỵ1 2k ỵ 38ị Substituting /m from Eq (37) into Eq (36) and neglecting the inertia force, i.e qà h W Ã;tt ¼ 0, yields (" 4 7D 38 20n2 ỵ 12n4 ị2 h R 4n ! # 2 2 ð3 þ 2n Þ 28ð266 À 170n þ 64n Þ h þ þ Wà 105p2 n2 ð1 À mÞ R " 963 ỵ 2n2 ịn 83 ỵ 2n2 ị266 170n2 ỵ 64n4 ị ỵ p n2 105pn1 À mÞn2  à # 224 p 38 20n2 ỵ 12n4 ị 40n2 h ỵ W Ã2 R 105p3 nð1 À mÞn2 ) "  # 4 2048n2 643 ỵ 2n2 ị p2 38 20n2 ỵ 12n4 ị 40n2 h ỵ ỵ W R 105p2 n4 ð1 À mÞ 7p2 n4 " # 2 8pnUm0 ỵ 2n Þ h h À Wà À DT: ð39Þ 7ð1 mị R 4pn R 2n2 8pnE1 qẳ 738 20n2 ỵ 12n4 ị Eq (39) is the explicit expression of external pressure-average deflection curve incorporating the effects of temperature These expressions can be used to consider the nonlinear unsymmetric response of immovable clamped FGM spherical shell subjected to external pressure and exposed to temperature conditions The static critical buckling loads in this case can be obtained from Eq (39) by using condition dq/dW⁄ = Inversely, the temperature difference DT may be obtained in terms of q, W⁄ as well as material and geometric properties due to these expressions Similarly, suppose external pressure depending on time with the law q = st, the motion Eq (36) in conjunction with Eq (37) becomes (" 4 7Dà ð38 À 20n2 þ 12n4 Þ2 h R 4n4 ! 2 # 2 ỵ 2n ị 28266 170n þ 64n Þ h Wà þ þ 105p2 n2 ð1 mị R " 963 ỵ 2n2 ịn 83 þ 2n2 Þð266 À 170n2 þ 64n4 Þ À þ 7pn2 105pnð1 À mÞn2  Ã# 3 224 p 38 20n2 ỵ 12n4 ị 40n2 h W ỵ R 105p3 n1 mịn2 " ) # 4 2048n2 643 ỵ 2n2 ị p2 38 20n2 ỵ 12n4 ị 40n2 h ỵ ỵ W R 7p2 n4 105p2 n4 ð1 À mÞ " # Um0 ỵ 2n2 ị h h 7st ẳ 0: 40ị W À Á DT À R R ð1 À mÞ p n pn 2n qà h2 W ;tt ỵ E1 38 20n2 ỵ 12n4 ị Eq (40) is the governing equation to investigate dynamic behavior of FGM shallow spherical shells under uniform external pressure including temperature effects The analytical solution of Eq (40) is very complicated, so this equation may be solved approximately by applying Runge–Kutta method and the Budiansky–Roth criterion to obtain critical dynamic buckling loads Numerical results and discussion In the following discussions, the FGM spherical shell is made of silicon nitride (Si3N4) and steel (SUS 304) The Young’s modulus, mass densities and the coefficients of thermal expansion for Silicon nitride are Ec = 348.43 GPa, qc = 2370 kg/m3, ac = 5.8723  10À6 1/°C and for steel are Em = 201.04 GPa, qm = 8166 kg/m3, am = 12.33  10À6 1/°C, respectively The Poisson’s ratio is supposed to be m = 0.3 for both constituent materials 6.1 Validation of the proposed formulation To validate the proposed formulation in static and dynamic stability analysis of FGM shallow spherical shell, the present results are compared with results obtained by Ganapathi [15] The nondimensional dynamic pressure Pcr and geometrical parameter k are defined Pcr ẳ qr4 ẵ31 m2 ị1=2 h=Hị2 04 ; Ec h k ẳ 2ẵ31 m2 ị1=4 H=hị1=2 : p where H ẳ R À n2 is the central shell rise and the length of response calculation time is introduced qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R h=2 s ẳ h=r0 ị2 Eef =ẵ121 m2 ịqef h2 t; where Eef ẳ 12 h=2 Ezịdz as in Ref [15] Using Eq (36) with /m = 0, the dynamic buckling study is conducted for step loading of infinite duration Nonlinear dynamic response history with time for the FGM spherical shell parameter k = 6, r0/h = 400 and k = considering different externally applied pressure loads is obtained and illustrated in Fig 2, that is similar with the result in the mentioned paper As can be seen, there is a sudden jump in the value of the average deflection when the external pressure reaches the critical value 2957 D.H Bich et al / Composite Structures 94 (2012) 2952–2960 30 P q 3.5 H τ Applied pressure load r0 t Applied presume load 20 15 1.5 W* W* 2.5 1: P=0.565 2: P=0.566 3: Pcr =0.567 0 0.2 0.4 0.6 0.8 1.2 -5 1.4 Non dimensional time (τ) 0.5 1.5 2.5 Fig Effect of index k on dynamic response (dynamic, FM) Pcr = 0.567.This result is in good agreement with the one of Ref [15] Pcr = 0.6063 obtained by using the finite element method x 10 ξ =0.2, k=1, n=1 6.2 Results for movable clamped FGM shallow spherical shell 1: R/h=1000 2: R/h=1200 3: R/h=1500 q (Pa) To illustrate the proposed formulation a FGM shallow spherical shell of geometric ratios R/h = 1000, n = r0/R = 0.2 and volume fraction index k = under uniform external pressure is considered After calculation of the buckling load according to Eq (28) with various shape modes n, it can see that the smallest buckling load, i.e the static critical buckling load, corresponds to the shape mode n = and receives the value qcr = 2.2167e+005 Pa The effect of material and geometric parameters on the nonlinear unsymmetrical static and dynamic response of the FGM shallow spherical shells with movable clamped edge under uniform external pressure are considered in Figs 3–6 Fig shows the effects of volume fraction index k(=0, and 5) on the nonlinear unsymmetrical static response of FGM spherical shells As can be seen, the load- average deflection curves become lower when k increases The increase in the extremum-type buckling load and load carrying capacity of the shell when k reduces is presented by a bigger difference between upper and lower buckling loads Whereas Fig demonstrates these effects on the nonlinear dynamic response of FGM spherical shells -1 -2 25 q t Applied pressure load W* 25 ξ =0.2, k=1, n=1 20 1: R/h=1000 2: R/h=1200 3: R/h=1500 15 20 40 30 15 45 x 10 10 Fig Effect of R/h on load–average deflection curve (static, FM) 35 W* 2 time (s) Fig Dynamic response W⁄–s with k = 6, k = 1, r0/h = 400 q (Pa) 0.5 10 -1 R/h=1000 ξ =0.2 n=1 0 0.5 1.5 2.5 time (s) 1: k=0 2: k=1 3: k=5 -2 -3 1: k=0 2: k=1 3: k=5 10 1 -0.5 R/h=1000 ξ = 0.2 n=1 25 Fig Effect of R/h on dynamic response (dynamic, FM) 10 12 14 16 18 20 W* Fig Effect of index k on load–average deflection curve (static, FM) It is observed that a sudden jump in the value of the average deflection occurs earlier when k increases, i.e the corresponding dynamic buckling load is smaller This is expected because the 2958 D.H Bich et al / Composite Structures 94 (2012) 2952–2960 Table Comparison between critical loads with the change of index k in case FM 18 q 16 10 Pcr (static) Pcr (dynamic) % 2.8112e+005 2.9995e+005 6.7 2.2167e+005 2.3796e+005 7.3 1.8200e+005 1.9767e+005 8.6 1.7299e+005 1.8826e+005 8.8 14 t Applied pressure load 12 W* k higher value k corresponds to a metal-richer shell which usually has less stiffness than a ceramic-richer shell Figs and consider the effects of curvature radius-to-thickness ratio R/h(=1000, 1200 and 1500) on the nonlinear static and dynamic characteristics respectively of the externally pressurized FGM spherical shells As can be observed, the load bearing capability of the spherical shell is considerably enhanced as R/h ratio decreases Furthermore, the increase of R/h ratio is accompanied by a drop of nonlinear load–deflection curves and more severe snap-through static response and early occurrence of a jump of dynamic response Tables and demonstrate the comparison between nonlinear critical static and dynamic buckling loads with the change of power index k and R/h ratio, respectively Clearly, the dynamic critical load is greater than the static critical load 10 ξ=0.2, R/h=1000, n=1 1: k=0 2: k=1 3: k=5 0 ξ=0.2, k=1, n=1 1: R/h=1000 2: R/h=1200 3: R/h=1500 4: R/h=2000 q (Pa) 3 4 1200 1500 2000 Pcr (static) Pcr (dynamic) % 2.2167e+005 2.3796e+005 7.3 1.8481e+005 1.9565e+005 5.9 1.4789e+005 1.5505e+005 4.8 1.1093e+005 1.1579e+005 4.4 x 10 q (Pa) W* R/h=1000, ξ=0.2, n=1 1: k=0 2: k=1 3: k=5 4: k=10 18 20 q 15 2 Applied pressure load t 10 time (s) 16 ξ=0.2, k=1, n=1 1: R/h=1000 2: R/h=1200 3: R/h=1500 20 14 30 25 12 Fig Effect of ratio R/h on load–average deflection curve (static, IM) 1000 10 W* R/h Table Comparison between critical loads with the change of ratio R/h in case FM 6 x 10 -1 Fig Effect of index k on dynamic response (dynamic, IM) 6.3 Results for immovable clamped FGM spherical shell time (s) Similarly the effects of volume fraction index k and curvature radius-to-thickness ratio R/h on the nonlinear static and dynamic response of the FGM shallow spherical shell with immovable clamped edge subjected to external pressure are illustrated in Figs 7–10, respectively Effects of power index k and R/h ratio on static and dynamic critical loads in IM case are given in Tables and As can be seen, the trend of nonlinear static and dynamic responses of FGM spherical shells in IM case are very similar with re- Fig 10 Effect of ratio R/h on dynamic response (dynamic, IM) 10 15 W* Fig Effect of index k on load–average deflection curve (static, IM) sponses in FM case The dynamic critical load is greater than the static critical load, but the difference of these loads in IM case is smaller than in FM case 2959 D.H Bich et al / Composite Structures 94 (2012) 2952–2960 Table Comparison between the nonlinear static and dynamic buckling loads vs k in case IM k 10 Pcr (static) Pcr (dynamic) % 6.2836e+005 6.4303e+005 2.3 4.9545e+005 5.0813e+005 2.6 4.0688e+005 4.1900e+005 3.0 3.8674e+005 3.9854e+005 3.1 x 10 q (Pa) ξ=0.2 k=1 n=1 FM IM 3 Table Comparison between the nonlinear static and dynamic buckling loads vs R/h s in case IM 1 2 R/h 1000 1200 1500 2000 Pcr (static) Pcr (dynamic) % 4.9545e+005 5.0813e+005 2.6 4.1277e+005 4.2118e+005 2.0 3.3015e+005 3.3550e+005 1.6 2.4757e+005 2.5130e+005 1.5 (1): R/h=1000 (2): R/h=1200 (3): R/h=1500 -1 -2 10 15 20 25 W* Fig 12 Effect of in-plane restraint on nonlinear static response with the change of R/h ratio x 10 R/h=1000 FM ξ=0.2, n=1 IM 30 1: k=0 q (Pa) IM 25 3: k=5 2 q FM 2: k=1 t 20 R/h=1000 ξ=0.2, k=1 n=1 W* Applied pressure load 15 -1 10 -2 -3 10 15 W* Fig 11 Effect of in-plane restraint on nonlinear static response with the change of index k time (s) 6.4 Effect of FM and IM boundary conditions Graph of nonlinear static responses of clamped FGM spherical shells with different boundary conditions are plotted in Figs 11 and 12 As can be observed, the spherical shells with immovable clamped edge have a comparatively higher capability of carrying external pressure than shells with movable clamped edge However, their response is unstable That means the IM shells experience a snap-through with much higher intensity than their movable clamped counterparts Furthermore, these figures also show that the effect of k index and R/h ratio on the critical buckling pressure of shells is very strong Fig 13 shows the comparison of the dynamic response of FGM spherical with FM and IM boundary conditions It also can see that the dynamic critical buckling load of clamped FM shell is smaller than the one of clamped IM shell Table shows the effects of k index and R/h ratio on static critical load in FM case and IM case Once again it is indicated that critical loads in IM case are greater about twice times than the ones in FM case 6.5 Effect of environment temperature The effect of environment temperature on the thermomechanical behavior of FGM shallow spherical shells with immovable clamped edge is considered in this subsection The shells are Fig 13 Effect of in-plane restraint on nonlinear dynamic response exposed to temperature field prior to applying external pressure Figs 14 and 15 analyze the nonlinear unsymmetrical static and dynamic responses of FGM spherical shells for various values of uniformly raised temperature DT(=0, 50, 100 and 150 °C) As shown in Fig 14, the temperature field makes shell to be deflected outward (negative deflection) prior to mechanical load acting on it When the shell is subjected uniform of external pressure, its outward deflection is reduced and when external pressure exceeds bifurcation point of load, an inward deflection occurs Similar behavior occur for dynamic clamped FGM shells, too It is illustrated in Fig 15 Table Effects of k and R/h on static critical load in FM and IM cases R/h 1000 1200 1500 2000 k=0 2.8112e+005FM 6.2836e+005IM 2.3438e+005 5.2350e+005 1.8756e+005 4.1871e+005 1.4068e+005 3.1398e+005 k=1 2.2167e+005 4.9545e+005 1.8481e+005 4.1277e+005 1.4789e+005 3.3015e+005 1.1093e+005 2.4757e+005 k=5 1.8200e+005 4.0688e+005 1.5175e+005 3.3897e+005 1.2144e+005 2.7112e+005 9.1091e+004 2.0330e+005 k = 10 1.7299e+005 3.8674e+005 1.4424e+005 3.2220e+005 1.1543e+005 2.5770e+005 8.6582e+004 1.9324e+005 2960 D.H Bich et al / Composite Structures 94 (2012) 2952–2960 FGM shallow spherical shells under uniform external pressure with and without including the effects of temperature Approximate analytical one-term deflection mode for two types boundary conditions is given and by applying Galerkin procedure explicit expressions of static critical buckling loads and postbuckling load–deflection curves are determined For the nonlinear dynamic buckling analysis, the nonlinear equation of motion of the shell is solved by using Runge–Kutta method The dynamic critical buckling loads are found according to Budiansky–Roth criterion The nonlinear unsymmetric response of the shells is analyzed and the results are illustrated in graphic form and numerical tables The results indicate that the nonlinear response of FGM shallow spherical shells is complex and greatly influenced by the type of loading (static or dynamic), the material and geometric parameters, the in-plane restraint and the preexistent temperature condition x 10 14 R/h=1000, ξ=0.2 k=1, n=1 1: ΔT=0 2: ΔT=50 3: ΔT=100 4: ΔT=150 q (Pa) 12 10 -2 -4 10 12 14 16 18 W* Acknowledgement Fig 14 Effect of temperature on nonlinear static response This paper was supported by the National Foundation for Science and Technology Development of Vietnam – NAFOSTED The authors are grateful for this financial support 25 1: ΔT=0 2: ΔT=50 3: ΔT=100 4: ΔT=150 20 W* 15 R/h=1000, ξ=0.2, k=1, n=1 References q 10 t Applied pressure load -5 10 12 14 time (s) Fig 15 Effect of temperature on nonlinear dynamic response Table Comparison of static and dynamic critical loads DT 50 100 150 Pcr (static) Pcr (dynamic) % 4.9545e+005 5.0813e+005 2.6 7.1219e+005 7.2917e+005 2.4 9.4978e+005 9.7132e+005 2.3 12.0582e+005 12.3231e+005 2.2 The enhancement of temperature difference is accompanied by the increase of bifurcation points, and the intensity of snapthrough behavior of the spherical shells (in static analysis) and the strengtheneth of load bearing capability of the spherical shells under dynamic loading (in dynamic analysis) Comparison between nonlinear static and dynamic critical buckling loads with effect of temperature is given in Table It can see that in this case the dynamic critical buckling load also is greater than static one Concluding remarks This paper presents an analytical approach to investigate the nonlinear unsymmetrical static and dynamic responses of clamped [1] Budiansky B, Roth RS 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cylindrical panels under axial compression Comput Mater Sci 2010;49:313–6 ... investigation on the nonlinear dynamic stability of FGM shallow spherical shells In the present paper, the nonlinear static and dynamical buckling behavior of clamped FGM shallow spherical shells. .. to investigate static and dynamic unsymmetric responses of FGM shallow spherical shells under combined mechanical and thermal loads 5.1 Static stability analysis Environment temperature is uniformly... index k and R/h ratio on static and dynamic critical loads in IM case are given in Tables and As can be seen, the trend of nonlinear static and dynamic responses of FGM spherical shells in IM case