Nội dung

Vietnam Journal of Mechanics, VAST, Vol. 42, No. 2 (2020), pp. 133 – 152 DOI: https://doi.org/10.15625/0866-7136/14749 DYNAMIC ANALYSIS OF FG STEPPED TRUNCATED CONICAL SHELLS SURROUNDED BY PASTERNAK ELASTIC FOUNDATIONS Le Quang Vinh1,∗ , Nguyen Manh Cuong2 1 Viet Tri University of Industry, Phu Tho, Vietnam 2 Hanoi University of Science and Technology, Vietnam E-mail: vinhchc@gmail.com Received: 28 December 2019 / Published online: 28 June 2020 Abstract. This research presents a continuous element model for solving vibration problems of FG stepped truncated conical shells having various material properties and surrounded by Pasternak foundations. Based on the First Order Shear Deformation Theory (FSDT) and the equations of the FGM conical shells, the dynamic stiffness matrix is obtained for each segment of the shell having constant thickness. The interesting assembly procedure of continuous element method (CEM) is employed for joining those segments in order to analyze the dynamic behavior of the FG stepped truncated conical shells an assembly procedure of continuous element method (CEM) is employed for joining those segments. Free vibrations of different configurations of FG stepped truncated conical shells on elastic foundations are examined. Effects of structural parameters, stepped thickness and elastic foundations on the free vibration of FG stepped truncated conical shells are also presented. Keywords: stepped shell, vibration of conical shell, functionally graded shell, continuous element method, Winkler–Pasternak foundation. 1. INTRODUCTION Conical shells are widely used in modern engineering structures such as tunnels, storage tanks, pressure vessels, rockets, missiles, water ducts, pipelines and casing pipes and in other applications. Therefore, static and dynamic analysis of shells in interaction with elastic media is important for the safety and stability of those structures. Most earthen soils can appropriately be represented by the Pasternak model, whereas sandy soils and liquids can be represented by Winkler’s model [1, 2]. The static and dynamic analyses of conical shells on elastic foundations have been studied in recent years. For FGM conical shells resting on elastic foundations, many significant results on the vibration and dynamic buckling of FGM conical shells are obtained. Sofiyev and Kuruoglu [3] c 2020 Vietnam Academy of Science and Technology 134 Le Quang Vinh, Nguyen Manh Cuong studied vibrations of FGM truncated and complete conical shells resting on elastic foundations under various boundary conditions by applying the Galerkin method. The considered elastic foundations include the Winkler- and Pasternak-type elastic foundations. The FGMs are assumed to vary as power and exponential functions through the thickness of the conical shells. Sofiyev and Schnack [4] presented solutions for the vibration analysis of truncated conical shells made of FGM and resting on the Winkler–Pasternak foundations. The governing equations according to the Donnell’s theory are solved by Galerkin’s method and the fundamental frequencies with or without two-parameter elastic foundation have been investigated. Dung et al. [5] presented an analytical approach to investigate the mechanical buckling load of eccentrically stiffened functionally graded truncated conical shells surrounded by elastic medium and subjected to axial compressive load and external uniform pressure based on the classical shell theory and Galerkin method. The stepped conical shells (SCSs) structures offer challenging vibration problems not only due to the degree of complexity of the governing shell equations, but also due to the difficulty associated with matching the continuity conditions between the shell components. Although finite element computer codes (NASTRAN, ANSYS, ABAQUS, etc.) can analyze the vibrations of these SCSs and have been well developed and managed, the disadvantage is that the computation cost is quite expensive. Xie et al. [6] presented a unified approach to determine natural frequencies and forced vibration responses of stepped conical shells with arbitrary boundary conditions. The approach is involved in dividing the stepped shells into narrow segments at the locations of discontinuities of thickness and semi-vertex angle. Flügge theory is used to describe equations of motions of conical segments and displacement functions are expanded as power series. Qu et al. [7] developed an efficient domain decomposition algorithm for free and forced vibration analysis of the uniform and stepped conical shells subjected to classical and nonclassical boundary conditions. Vinh et al. [8] present a new Continuous Element for analyzing dynamic behavior of stepped composite conical shells. In this work, a powerful assembly procedure has been presented for constructing new dynamic stiffness matrix of stepped composite conical shells. The continuous element formulations here are established based on the analytical solution of differential equations for composite conical shells giving high precision results. Nam et al. [9] presented a continuous element model for solving vibration problems of stepped composite cylindrical shells surrounded by Pasternak foundations with various boundary conditions. Based on the First Order Shear Deformation Theory (FSDT), the equations of motion of the circular cylindrical shell are introduced and the dynamic stiffness matrix is obtained for each segment of the uniform shell. The assembly procedure of continuous element method (CEM) is adopted to analyze the dynamic behavior of the stepped composite cylindrical shell surrounded by an elastic foundation. Available vibration study results in the literature for SCSs are few and far between, as it has not received much attention of the researchers, perhaps due to the complexity involving in the modeling and solution procedure. Therefore, a unified method which can be both accurate and efficient to determine the natural frequencies and forced vibration responses of the SCSs would be highly desirable. Dynamic analysis of FG stepped truncated conical shells surrounded by Pasternak elastic foundations 135 The main purpose of this paper is to present a new Continuous Element model to analyze the dynamic behavior of the multi FG stepped truncated conical shells with various material characteristics and surrounded by Winkler-Pasternak foundations. Based on the assembly procedure of single continuous elements, the dynamic stiffness matrix of complex stepped conical shells surrounded by Pasternak foundation is established. In this research, the influences of different parameters are studied in detail such as: stepped thickness, geometrical ratios and elastic foundation stiffness. The achieved numerical results are compared to those calculated by the finite element method and by other researches in some singular cases. The efficiency and accuracy as well as the saving in data storage and computed time of the CE method for complex shells in contact with elastic foundations in medium and high frequencies have been investigated and confirmed in this study. 2. THEORETICAL FORMULATION 2.1. Description of analysis the model Dynamic of FG stepped truncated conical shells with various properties and surrounded by Pasternak elastic foundations 3 Let’s investigate the FGM conical shell with ( x, θ, z) coordinates, as shown in Fig. 1. the FGM conicalthe shell cone with (x,θ,z) coordinates, with as shown in Fig. 1. Theplaced at the The coordinate Let’s x isinvestigate measured along generator the origin coordinate x is measured along the cone generator with the origin placed at the middle of the middle of thegenerators, generators; θ is the circumferential coordinate z is the perpendicular θ is the circumferential coordinate, and z is the perpendicular to theand shell surfaces. R1 and R2 are the are R respectively radiusthe and large cone and surface,large h is the radiuses thickness, the cone length and to the shell surfaces. R2 are small of cone cross sections, 1 and small cone semi-vertex angle of the shell are represented by L and α and the radius coordinate R(x) of a point respectively (see Fig. 1); ishcalculated is the thickness, the cone length and cone semi-vertex angle of M inside the shell as: R(x)=R1+xsinα. This shell is surrounded by a Winkler a foundation R stiffness kw oraby a the shell are represented by L and α andelastic thefoundation radiushaving coordinate ( x ) of point M inside Pasternak foundation with the foundation stiffness kw and shear layer stiffness kp. Such shell is the the shell is calculated as:shell R(element x ) = toRcontribute α.truncated conical shell surrounded by two above basic continuous a FG 1 + x sin types of elastic foundations. Figure 1. Geometry parameters of FG truncated conical shell surrounded by Pasternak elastic foundations Fig. 1. Geometry parameters of a FG truncated conical shell surrounded Typically, FGM shells made from a mixture of two material phases. In this paper, it is assumed by Pasternak elastic foundation that the FGM shells are made of a mixture of ceramic and metal. Young’s modulus E(z), density ρ(z) and Poisson’s ratio (z) are assumed to vary continuously through the shells thickness and can be expressed as a linear combination: This shell is surrounded by a Winkler elastic foundation having a foundation stiffE ( z ) = ( E − E )V + E ness k w or by a Pasternak foundation ( zwith the foundation stiffness k w and (1) shear layer ) = (  −  )V +  stiffness k p . Such shell is the basic continuous  ( z ) = (  −  shell )V +  element to contribute a FG truncated conical shell surrounded byctwo types ofandelastic foundations. in which the subscripts and m above represent the ceramic metallic constituents, respectively, and the fraction Vc follows general four-parameter power-law [3,6,8]: phases. In this paper, Typically,volume FGM shells are two made from a mixture ofdistributions two material  of  ceramic and metal. Young’s it is assumed that the FGM shells are made  1a zmixture   1 z  of : V = 1 − a +  + b +   FGM h ( z ) 2are h  assumed 2 µ modulus E(z), density ρ(z) and Poisson’s ratio to vary(2)continuously  c m c m c m c m c m c m c I (a / b / c / p) p c c  1 z 1 z  FGM II ( a / b / c / p ) : Vc = 1 − a −  + b −   2 2 h h       p in which the power-law exponent p is a positive real number (0  p  ) and the parameters a, b, c dictate the material variation profile through the functionally graded shell thickness. It is assumed that the sum of the volume fractions of the two basis components is equal to unity, i.e., Vc + Vm = 1. 136 Le Quang Vinh, Nguyen Manh Cuong through the shell thickness and can be expressed as a linear combination E(z) = ( Ec − Em )Vc + Em , µ(z) = (µc − µm )Vc + µm , (1) ρ(z) = (ρc − ρm )Vc + ρm , where the subscripts c and m represent the ceramic and metallic constituents, respectively, and the volume fraction Vc follows two general four-parameter power-law distributions [3, 6, 8]      p 1 z 1 z c + +b + , FGM I (a/b/c/p) : Vc = 1 − a 2 h 2 h (2)      p 1 z 1 z c FGM I I (a/b/c/p) : Vc = 1 − a − +b − , 2 h 2 h in which the power-law exponent p is a positive real number (0 ≤ p ≤ ∞) and the parameters a, b, c represent the material variation profile through the functionally graded shell thickness. It is assumed that the sum of the volume fractions of the two basic components is equal to unity, i.e., Vc + Vm = 1. Therefore, according to the relations defined in Eq. (2), when the power-law exponent p is set equal to zero (i.e., p = 0) or equal to infinity (i.e., p = ∞), the FGM material becomes the homogeneous isotropic material, 4expressed as 4 p = 0 → Vc = 1, Vm = 0 → E(z) = Ec , µ(z) = µc , ρ(z) = ρc , (3) p= ∞→ Vto Vdefined 1→ E z(2), )= Emwhen , theµthe (z)power-law = µm ,exponent ρ(z) = c = m = Therefore, according to the relations in Eq. when power-law p isρpm set equal to to Therefore, according the0,relations defined in(Eq. (2), exponent is. set equal zerozero (i.e., p=0) or equal to infinity (i.e., p=  ), the FGM material becomes the homogeneous isotropic (i.e., p=0) equal to infinity p=),(M the1FGM homogeneous Whereas theorcomposition of(i.e., ceramic ) andmaterial metal becomes (M2 ) is the linear for p = isotropic 1. The material, expressed material, expressed as: fraction Vc through the shell thickness for different values of the variations of the as: volume =V1,c =V1m, =V0m → ( z )E=(2. ( z ) =figure,  ,  ( zthe ) = classical volume fraction power-law exponentp =p 0are in Fig. p→ zE) cIn = 0Villustrated = 0E → =, Ethis c→ c ,  ( z ) c= c ,  ( z ) c=  c (3) p= ( z )E=( zE) m=, Em ,( z )=( z) m=,  ( z) =  = m p→ = Vc→=V0c, =V0m, =V1m→ = 1E→ m ,  ( z) m (a) FGM I ( a/b/c/p) (b) FGM I I ( a/b/c/p) Figure 2. Variation ofvolume the volume fraction Vc through the thickness a shell for different values Figure 2. Variation of the fraction Vc through the thickness of aof shell for different values of of power-law exponent p:FGM (a) FGM (b) FGM I (a=1/b=0/c=2/p) II (a=1/b=0/c=2/p) power-law exponent p: (a) ; (b); FGM . . I (a=1/b=0/c=2/p) II (a=1/b=0/c=2/p) (3) Fig. 2. Variation of the volume fraction Vc through the thickness of a shell for different values of power-law exponent p p=1. The variations of the Whereas the composition of ceramic metal is linear 1) and Whereas the composition of ceramic (M1(M ) and metal (M2(M ) is2)linear for for p=1. The variations of the volume fraction Vcthrough thickness different values of the power-law exponent p are volume fraction Vcthrough the the shellshell thickness for for different values of the power-law exponent p are illustrated in Fig. 2. In Fig. 2 the classical volume fraction profiles, such as those reported in literature illustrated in Fig. 2. In Fig. 2 the classical volume fraction profiles, such as those reported in literature [24,25], are presented as special cases of the general distribution by setting b=0. [24,25], are presented as special cases of the general distribution lawslaws by setting a=1a=1 andand b=0. As As cancan be seen from Fig. 2a, for the first distribution FGMI (a=1/b=0/c/p) the material composition be seen from Fig. 2a, for the first distribution FGMI (a=1/b=0/c/p) the material composition is is continuously varied bottom surface = -0.5) of the shell is1 M whereas 1 rich, continuously varied suchsuch that that the the bottom surface (z/h(z/h = -0.5) of the shell is M rich, whereas thethe toptop surface = 0.5) M2 rich. volume fraction Vc decreased from at z/h = -0.5 to zero at z/h surface (z/h (z/h = 0.5) is Mis2 rich. TheThe volume fraction Vc decreased from 1 at1z/h = -0.5 to zero at z/h = = 2b shows for second the second distribution FGMII (a=1/b=0/c/p) surface = 0.5) 0.5. 0.5. Fig. Fig. 2b shows that that for the distribution FGMII (a=1/b=0/c/p) the the toptop surface (z/h(z/h = 0.5) of of Dynamic analysis of FG stepped truncated conical shells surrounded by Pasternak elastic foundations 137 profiles, such as those reported in literature [?, ?], are presented as special cases of the general distribution laws by setting a = 1 and b = 0. As can be seen from Fig. 2(a), for the first distribution FGMI ( a = 1/b = 0/c/p) the material composition is continuously varied such that the bottom surface (z/h = −0.5) of the shell is M1 rich, whereas the top surface (z/h = 0.5) is M2 rich. The volume fraction Vc decreased from 1 at z/h = −0.5 to zero at z/h = 0.5. Fig. 2(b) shows that for the second distribution FGMII ( a = 1/b = 0/c/p) the top surface (z/h = 0.5) of the shell is M1 rich, whereas the bottom surface (z/h = −0.5) is M2 rich, instead. When the volume fraction exponent is increased, the content of M1 in FG layer decreases. So far, all the needed parts of the first-order shear deformation shell theory (FSDT) are presented, and they may be combined to obtain the desired form of the equations of motion. 2.2. Kinematic relations and stress resultants On the basis of the assumptions of moderately thick shell theory, the displacement components of an arbitrary point in the FG shell for the first-order shear deformation theory are expressed in terms of the displacements and rotation components of the middle surface as given below [9] u ( x, θ, z, t) = u0 ( x, θ, t) + zϕ x ( x, θ, t) , v ( x, θ, z, t) = v0 ( x, θ, t) + zϕθ ( x, θ, t) , (4) w ( x, θ, z, t) = w0 ( x, θ, t) , where u, v and w are the displacement components in the x, θ and z directions, respectively; u0 , v0 and w0 are the middle surface displacements of the shell in the axial, circumferential and radial directions, respectively; ϕ x and ϕθ represent the transverse normal rotations of the reference surface about the θ- and x-axis, t is the time variable. The linear strain-displacement relations in the shell space are defined as ∂u0 ∂ϕ x , kx = , ∂x  ∂x  ∂v0 1 ∂ϕ x ∂ϕ sin α 1 u0 sin α + + w0 cos α , k xθ = + θ− ϕ , εθ = R( x ) ∂θ R( x )∂θ ∂x R (x) θ ∂v0 1 ∂u0 sin α 1 ∂ϕθ ε xθ = + − v0 , kθ = ϕ x sin α + , ∂x R( x ) ∂θ R( x ) R( x ) ∂θ ∂w0 − cos α 1 ∂w0 γxz = + ϕx , γθz = v0 + + ϕθ . ∂x R( x ) R( x ) ∂θ εx = Based on Hooke’s law, the stress-strain relations of the shell are written as      Q11 (z) Q12 (z) 0 0 0 εx  σx                0 0 0  σθ    Q12 (z) Q11 (z)   εθ   0 0 Q66 (z) 0 0 τxθ =    γxθ  ,        0 0 0 Q66 (z) 0     τxz    γxz      0 0 0 0 Q66 (z) γθz τθz (5) (6) 138 Le Quang Vinh, Nguyen Manh Cuong where the elastic constant Qij (z) are functions of thickness coordinate z and are defined as E(z) µ(z) E(z) E(z) Q11 (z) = , Q12 (z) = , Q66 (z) = . (7) 2 2 1 − µ (z) 1 − µ (z) 2[1 + µ(z)] The stress and moment resultants are given as ( Nx , Nθ , Nxθ , Q x , Qθ ) = Zh/2 (σxx , σθθ , τxθ , τxz , τθz ) dz, (8) (σxx , σθθ , τxθ ) zdz, (9) −h/2 ( Mx , Mθ , Mxθ ) = Zh/2 −h/2 where Nx , Nθ and Nxθ are the in-plane force resultants, Mx , Mθ and Mxθ are moment resultants, Q x , Qθ are transverse shear force resultants. The shear correction factor f is computed such that the strain energy due to transverse shear stresses in Eq. (10) are equals to the strain energy due to the true transverse stresses predicted by the threedimensional elasticity theory [8]. In this paper, the shear correction factors f = 5/6 [6, 8]. Substituting Eqs. (6)–(7) into Eqs. (8)–(9) following constitutive equations relating the force and moment resultants to strains and curvatures of the reference surface are given in the matrix form      N A A 0 B B 0 0 0 ε     x x 11 12 11 12               N A A 0 B B 0 0 0 ε    12 11 12 11 θ  θ             0  Nxθ  0 A66 0 0 B66 0 0  ε xθ                  Mx B B 0 D D 0 0 0 k x 11 12 11 12  = . (10)  B12 B11 Mθ  0 D12 D11 0 0 0  kθ                 Mxθ  0 B66 0 0 D66 0 0  k xθ        0             Q 0 0 0 0 0 0 f F 0 γ    x  xz  44      Q   0 0 0 0 0 0 0 fF γ  55 θ θz The structure materials employed in the following study are assumed to be functionally graded and linearly elastic. So, the extensional stiffness Aij , the bending stiffness Dij , and the extensional-bending coupling stiffness Bij are respectively expressed as Aij = Dij = Zh/2 −h/2 Zh/2 −h/2 Qij (z)dz, Bij = z2 Qij (z)dz, i, j = 1, 2, 6, Fij = Zh/2 −h/2 Zh/2 −h/2 z.Qij (z)dz, (11) Qij (z)dz, i, j = 4, 5. Dynamic analysis of FG stepped truncated conical shells surrounded by Pasternak elastic foundations 139 2.3. Equations of motion The equilibrium equations of motion for FG truncated conical shell surrounded by Pasternak foundation based on the first-order shear deformation shell theory (FSDT) in terms of the force and moment resultants can be written as [8] ∂Nx sin α 1 ∂Nxθ + ( Nx − Nθ ) + = I0 ü0 + I1 ϕ̈ x , ∂x R( x ) R( x ) ∂θ ∂Nxθ 2 sin α 1 ∂Nθ cos α + Nxθ + + Qθ = I0 v̈0 + I1 ϕ̈θ , ∂x R( x ) R( x ) ∂θ R( x )   2 ∂Q x ∂ w sin α ∂w 1 ∂Qθ sin α cos α 1 ∂2 w + = I0 ẅ0 , + + Qx − Nθ − k w w + k p + ∂x R( x ) ∂θ R( x ) R( x ) ∂x2 R( x ) ∂x R( x )2 ∂θ 2 sin α 1 ∂Mxθ ∂Mx + ( M x − Mθ ) + − Q x = I1 ü0 + I2 ϕ̈ x , ∂x R( x ) R( x ) ∂θ ∂Mxθ 2 sin α 1 ∂Mθ + Mxθ + − Qθ = I1 v̈0 + I2 ϕ̈θ , ∂x R( x ) R( x ) ∂θ (12) where Zh/2 h i ρ(z) 1, z1 , z2 dz, [ I0 , I1 , I2 ] = −h/2 ρ(z) is the density of the shell per unit middle surface area. I0 , I1 and I2 are the mass inertias. 3. DYNAMIC STIFFNESS MATRIX FORMULATION FOR FG TRUNCATED CONICAL SHELL T The chosen state-vector is y = u0 , v0 , w0 , ϕ x , ϕθ , Nx , Nxθ , Q x , Mx , Mxθ . Next, the Fourier series expansion for state variables is written as {u0 ( x, θ, t), w0 ( x, θ, t), ϕθ ( x, θ, t), Nx ( x, θ, t), Q x ( x, θ, t), Mx ( x, θ, t)}T ∞ = ∑ {um ( x ), wm ( x ), ϕθm ( x ), Nxm ( x ), Q xm ( x ), Mxm ( x )}T cos mθeiωt , m =1 {v0 ( x, θ, t), ϕ x ( x, θ, t), Nxθ ( x, θ, t), Mxθ ( x, θ, t)}T ∞ = ∑ (13) {vm ( x ), ϕ xm ( x ), Nxθ m ( x ), Mxθ m ( x )}T sin mθeiωt , m =1 where m is the number of circumferential wave. Substituting (13) in equations (12) and (10), a system of ordinary differential equations in the x-coordinate for the mth mode can be expressed in the matrix form for each circumferential mode m as [9] 140 Le Quang Vinh, Nguyen Manh Cuong dum D B = c4 sin α.um + mc4 vm + c4 cos α.wm + c5 sin α.ϕ xm + mc5 ϕθm + 11 Nxm − 11 Mxm , dx c1 c1 m sin α D66 B66 dvm = um + vm − N + M , dx R( x ) R( x ) c10 xθm c10 xθm 1 dwm Q xm , = − ϕ xm + dx f F55 dϕ xm B A = c2 sin α.um + mc2 vm + c2 cos α.wm + c3 sin α.ϕ xm + mc3 ϕθm − 11 Nxm + 11 Mxm , dx c1 c1 dϕθm m sin α B A = ϕ xm + ϕ + 66 Nxθm − 66 Mxθm , dx R( x ) R( x ) θm c10 c10     dNxm = c6 sin2 α − I0 v 2 um + mc6 sin α.vm + c6 sin α cos α.wm + c7 sin2 α − I1 ω 2 ϕ xm dx   m 1 Nxm − N − c2 sin α.Mxm , + mc7 sin α.ϕθm − sin α c4 + R( x ) R( x ) xθm     dNxθm f F cos2 α f F44 = mc6 sin α.um + m2 c6 + 44 2 − I0 ω 2 vm + m cos α c6 + wm + dx R( x ) R ( x )2   2 sin α f F cos α − I1 ω 2 ϕθm − mc4 Nxm − N − mc2 Mxm , + mc7 sin α.ϕ xm + m2 c7 − 44 R( x ) R( x ) xθm     dQ xm A f F44 A11 vm = c13 c1 sin α + 11 2 cos α + k p c2 sin α um + mc13 cos α + c + cos α + k c p 2 11 dx R( x ) R ( x )2 R ( x )2 ! m2 k p m2 f F44 + c13 wm + c11 cos α + k p c2 cos α − I0 ω 2 + k w + 2 R( x ) R ( x )2   sin α B + c13 c12 sin α + 22 2 cos α + k p + k p c3 sin α ϕ xm R( x ) R( x )   f F44 B22 + mc13 − + c12 + cos α + k c ϕθm p 3 R( x ) R ( x )2   B B B sin α A12 D11 cos α − 12 11 cos α − k p 11 Nxm − Q xm + c13 R ( x ) c1 R ( x ) c1 c1 R( x )   A B B A A + c13 − 12 11 cos α + 12 11 cos α + k p 11 Mxm , R ( x ) c1 R ( x ) c1 c1     dMxm = 2c8 sin2 α − I1 ω 2 um + 2mc8 sin α.vm + 2c8 sin α cos α.wm + + 2c9 sin2 α − I2 ω 2 ϕ xm dx    m 1 + 2mc9 sin α.ϕθm − 2c5 sin α.Nxm + Q x − 2 sin α c3 + Mxm − Mxθm , R( x ) R     dMxθm f F cos α fF = mc8 sin α.um + m2 c8 − 44 − I1 ω 2 vm + m c8 cos α − 44 wm dx R( x ) R( x )   2 sin α + mc9 sin α.ϕ xm + m2 c9 + f F44 − I2 ω 2 ϕθm − mc5 Nxm − mc3 Mxm − M , R( x ) xθm (14) with 2 c1 = A11 D11 − B11 , c2 = ( A12 B11 − A11 B12 ) /R( x )c1 , c3 = ( B11 B12 − A11 D12 ) /R( x )c1 , c4 = ( B11 B12 − A12 D11 ) /R( x )c1 , c5 = ( B11 D12 − B12 D11 ) /R( x )c1 , c6 = ( A12 c4 + B12 c2 + A22 /R( x )) /R( x ), c7 = ( A12 c5 + B12 c3 + B22 /R( x )) /R( x ), c8 = ( B12 c4 + D12 c2 + B22 /R( x )) /R( x ), Dynamic analysis of FG stepped truncated conical shells surrounded by Pasternak elastic foundations 141 2 c9 = ( B12 c5 + D12 c3 + D22 /R( x )) /R( x ), c10 = B66 − A66 D66 , B12 A12 B12 1 A12 c4 cos α + c2 cos α, c12 = c5 cos α + c3 cos α, c13 = c11 = . kp R( x ) R( x ) R( x ) R( x ) 1 + f F55 Eq. (14) can be expressed in the matrix form for each circumferential mode m dym = Am ym , (15) dx with Am is a 10 × 10 matrix (see Appendix). The dynamic transfer matrix Tm is evaluated as RL   Am (ω )dx T T 11 12 Tm (ω ) = e 0 = . (16) T21 T22 Finally, the dynamic stiffness matrix Km (ω ) for FG truncated conical shell is determined by [8]   −1 −1 T12 T11 − T12 Km ( ω ) = . (17) −1 −1 T21 − T22 T12 T11 T22 T12 Natural frequencies will be extracted from the harmonic responses of the structure by using the procedure developed in [8, 9]. 4. CONTINUOUS ELEMENT FOR FG STEPPED TRUNCATED CONICAL SHELLS Let’s investigate a stepped conical shell (SCS) including n segments shown in Fig. 3. The SCS consists of n lengths L1 , L2 , . . . , Li , . . . , Ln and n step thicknesses h1 , h2 , . . . , hi , . . . , hn . Let the coordinate system be chosen as shown in Fig. 2; θ is the circumferential coordinate, R1 and R2 are the respectively small radius and large cone surface, the cone Dynamic analysis angle of FG(α) stepped shells properties and surrounded semi-vertex of thetruncated steps are conical the same; u, v with and various w are the displacement compo- by Pasternak elastic 9 nents in thefoundations x, θ and normal directions, respectively. Figure Geometryofofa aFG FG steppedtruncated truncatedconical conicalshells shells Fig. 3. 3.Geometry stepped The dynamic stiffness matrix Km() for the above FG stepped truncated conical shellss surrounded by elastic foundation will be constructed by assembling the DSM of many sections having different constant thickness and lengths. First, the shell is divided in to n elements. It is necessary to build n separate dynamic stiffness matrices Kseg1, Kseg2,…, Ksegi,…,Ksegn for these segments. Then, Fig. 4 describes the assembly procedure for constructing the DSM for the stepped conical shells. The natural frequencies of the studied structure will be determined from this matrix by using the method detailed in [8]. 142 Le Quang Vinh, Nguyen Manh Cuong The dynamic stiffness matrix Km (ω ) for the above FG stepped truncated conical 3. Geometry of a FG stepped truncated conical shells shells surrounded by Figure elastic foundation will be constructed by assembling the DSM of Thesegments dynamic having stiffness different matrix Kmconstant () for the above FG truncated shellss various thickness andstepped lengths. First, conical the shell is disurrounded by elastic foundation will be constructed by assembling the DSM of many sections having vided into n elements. It is necessary to build n separate dynamic stiffness matrices different thickness lengths. First, the shell is divided in to n elements. It is necessary to Kseg1 , Kseg2constant , . . . , Ksegi , . . . , Kand segn for these segments. Then, Fig. 4 describes the assembly build n separate dynamic stiffness matrices Kseg1, Kseg2,…, Ksegi,…,Ksegn for these segments. Then, Fig. procedure for constructing the DSM for the stepped conical shells. The natural frequen4 describes the assembly procedure for constructing the DSM for the stepped conical shells. The cies of the studiedofstructure be determined from from this matrix bybyusing method natural frequencies the studiedwill structure will be determined this matrix using the the method detailed in [8]. detailed in [8]. Figure 4. Construction of the dynamic stiffness matrix for FG stepped truncated conical shells. Fig. 4. Construction of the dynamic stiffness matrix for FG stepped truncated conical shells The process of joining the dynamic stiffness matrix K(ω)m for stepped conical shell is based on the The procedure of combining the dynamic stiffness matrix K (ω )m forwe stepped conical continuous condition at the joints between the steps of the cone casing. In this study, only shell is basedtheonstepped the continuous at the joints segments of so thetheshell. In investigated conical shellcondition with the conical angle of allbetween stepped being the same, this study, weshells onlyhave investigated conical and shell segments having same stepped of the neutral facesthe overlapping thewith continuous condition at thethe position of conithe calcoupling angle. between Thus, the all steps shellofsegments neutral faces overlapping and the continuous the shells ashave follows: condition at the position of the coupling the segments of the shells as follows ui = ui +1 ; vbetween i = vi +1 ; wi = wi +1 i w i +1 , ui = uNi+i 1= ,N i +1v;i N = vi i=+1N, i+1w; Q i = i +1 xθ xθ x x x = Qx i i +1 i i i +1 i +i1 i +i1 , MQ = Qix+1 , Nx = NxM x,θ =N Mxθxθ =; NMxθx = x x 5. (18) (18) i i +1 Mxθ , Mxi = Mxi+1 . =AND Mxθ NUMERICAL RESULTS DISCUSSION The present exact may be applied to investigate the effects of various geometrical and 5. procedure NUMERICAL RESULTS AND DISCUSSION material properties such as step thickness ratios, the power law index and different boundary The present exact procedure may be applied to investigate the effects of various geometrical and material properties such as step thickness ratios, the power law index and different boundary conditions. Four configuration of functionally graded material are used with the material properties listed in Tab. 1. Table 1. Material properties of functionally graded materials Properties E (GPa) µ ρ (kg/m3 ) FGM1 FGM2 FGM3 FGM4 Al Zirconia Al Alloy Al Al2 O3 Nickel Si3N4 70 0.3 2707 168 0.3 5700 70 0.3 2707 211 0.3 7800 70 0.3 2707 380 0.3 3800 205.098 0.31 8900 322.27 0.24 2370