Effect of Ring Support on Bi-layered FGM Cylindrical Shell

Effect of Ring Support on Bi-layered FGM Cylindrical Shell Mohsin Habib1,Madiha Ghamkhar1 1 Department of Mathematics and Statistics, University of Agriculture, Faisalabad, 38000, Pakistan Corresponding author: [email protected] ORCID IDs: First Author: 0009-0003-9831-8841 Second Author: 0000-0002-1897-6939 DOI : 10.5281/zenodo.18025369 Abstract In this work, the effect of ring support is analyzed on bi-layered FGM cylindrical shell. The bi-layered cylindrical shell are composed of two independent materials. These materials are considered nickel and stainless steel. The characteristics of materials are observed by the volume fraction power law. The fundamental frequencies of cylindrical shell are analyzed under the simply supported end boundary condition. One layer is of functionally graded material and the other layer is of isotropic material. The curvature and strain displacement relations are attained from Sander’s shell theory. The shell fundamental frequency equations are derived by using the Rayleigh Ritz. The vibration frequencies are calculated with Simply supported-simply supported boundary conditions and results are compared with already exists in the literature. Keywords: Ring support, FG Material, Sander’s shell theory, Rayleigh-Ritz method, Simply supported-simply supported. 1 Introduction and Motivation Cylindrical shell are used to transport and store highly pressure gases and liquid for different hydraulic application. They have been applied in pipe flow, chimney design, ships, aircraft fuselages and construction buildings. They are different geometrical shapes such as cylindrical, elliptical, spherical etc. They have been found wide range of engineering application. Those rang from enormous common and mechanical structure to small electrical parts. Modern structures utilizing shells include drilling structure parts pressure vessels oil tanks. In the present work bi-layered FGM cylindrical shell with effect of ring support is well known studied field of research in structural dynamic. Boyd and Rao [ 2 ] have analyzed the without oscillation of ring support or stringer stiffened elliptical cylinders with arbitrary condition. Wilken and Soedel [ 3 ] have analyzed modal characteristic of ring stiffened cylindrical shell. Such methods are based on the method of receptance and use the modal characteristic of the inputs of the un-stiffened cylinder and rings. By entrusting the M29-1 KOSC-2025 Proceedings structure to a cylindrical shell with basic support ends stiffened by similar evenly spaced rings, precision remains in this method. A single non-matrix frequency equation is formed for not uncommon configuration which can be resolved by numerical method with certainty and rapidity. Other approach is general and gives ring stiffeners an insight into impact. Swaddiunhippong et al. [ 4 ] have analyzed independent vibration with middle support around cylindrical shell. They used Rayleigh Ritz method for evaluate the mode shape and natural frequency. Donnell shell theory and the Sander’s shell theory approximation of first order consider. Loy et al. [ 5 ] have studied the vibration of cylindrical shell with ring support. The curvature and strain displacement relations attained from sander’s shell theory. The fundamental frequency examined with Ritz method. Zhang et al. [ 6 ] have studied without oscillation of open annular shell with middle ring support find the frequency of such shell theory is derive from Flugge’s thin shell theory. They investigated the location of ring support for number of middle ring supports and the shell angle variation on natural frequency. Najafizadeh and Isvandzibaei [ 7 ] have analyzed the vibration of functionally graded cylindrical shell with ring support based on higher order shear deformation shell plate theory. Najafizadeh and Isvandzibaei [ 8 ] have analyzed oscillation of thin cylindrical shell made by nickel and stainless. In this research, frequency of the shell with effect of ring support and boundary state were studied. The governing equation of FGM cylindrical shell are taken from different shear deformation theories. Result are instant on the fundamental frequency characteristics effect of boundary condition and effect of ring support. The present result is compatible in literature with that available. Arshad et al. [ 9 ] have discussed the bi-layered of different materials of cylindrical shell. Which one layered is made of isotropic and other layered is made of FGM. Arshad et al. [ 10 ] have studied the bilayered FGM cylindrical shell. Which both layered are made by FGM. Strain and curvature displacement relations are attained from Love’s shell theory. Shell fundamental frequency equation are attained by using R-Ritz method. Rahimi et al. [ 11 ] have examined vibration behavior of FG material cylindrical shell with intermediate ring support governing equation of motion taken from sander’s shell theory. Materials characteristics are considered to be graded in direction of h according to the fraction of the power law value. They used Rayleigh Ritz method for observed vibration analysis. Arshad et al. [ 12 ] examined the without oscillation of sandwich FGM cylindrical shell with intermediate layer of isotropic content in ring support influence. The ring support is used aside from the shell’s radial direction. For curvature and strain displacement relationship are attained from Love’s first order thin shell theory. To form the shell fundamental frequency equation, the Rayleigh Ritz method is applied. Ghamkhar et al. [ 13 ] have studied of three layers of FG material cylindrical shell with additional impact of ring support cylindrical shell internal and external surface isotropic and FGM middle surface. They have analyzed the position of intermediate support is effect the fundamental frequency of cylindrical shell. They have studied fundamental frequency by using R-Ritz method. They compared result of fundamental natural frequency with already exist literature. M29-2 2nd Kocaeli Science Congress, November 19-21, 2025 2 Theoretical Consideration Cylindrical shell is geometrical description in Fig. 1. Consider that R is radius, L is length and h is thickness of thin cylindrical shell. An ( x , θ , z ) are attained at the cylindrical shell central area. The z -coordinate in radial, x -coordinate in axial direction and θ -coordinate is taking along in circumferential direction of the cylindrical shell. The deformational displacement function are represented by U1 ( x, θ, z ), U2 ( x, θ, z )and U3 ( x, θ, z )in axial, circumferential and the radial direction of cylindrical shell respectively. The analytical problem of the 3-D circular cylindrical shell that is translated into a 2-D problem by taking into account the condition of the plane stress, strain and stress component in the z-direction are neglected. The stress and strain relation are stated by Hooke’s law as: {σ}= [Q]{e}.(1) Figure 1: Cylindrical Shell with Ring Support Where [Q] are represent the matrix of reduced stiffness and {e} , {σ} are represent the strain and stress of the shell are defined as: {σ}= (σx, σθ, σxθ)T.(2) Where σxθ denoted shear-stress in xθ -plane, σθ and σx are represented stress in θ and x direction in the plane respectively. The strain are represented as: {e}= (ex, eθ, exθ)T.(3) Where exθ denoted shear strain in the xθ -plane and eθ and ex are represented strain in θ and x direction in the plane respectively. The stiffness matrix is defined as: [Q] =     Q11 Q12 0 Q12 Q22 0 0 0 Q66     .(4) 2nd Kocaeli Science Congress, November 19-21, 2025 M29-3 KOSC-2025 Proceedings Where Q11 =E 1−ν2=Q22, Q66 =E 2(1+ν), Q12 =νE 1−ν2. The component of strain vector {e} are attained as linear function in the form of z and written as: eθ=e2−zκ2 ex=e1−zκ1 exθ =γ−2zξ            (5) Where e2 , e1 and γ are represent of strain surface. κ2 , κ1 and ξ are represent curvatures surface. Curvature and strain displacement relations are attained from sander’s shell theory. Shell theory widely used in analytical shell problems. According to given shell theory e1, e2 and γ are written as: e1=∂U1 ∂x e2=1 R∂U2 ∂θ +U3 γ=∂U2 ∂x +1 R ∂U1 ∂θ            (6) The κ1,κ2and ξare expressed in the form: κ1=−∂2U3 ∂x2 κ2=−1 R2∂2U3 ∂θ2−∂U2 ∂θ  ξ=−1 R∂2U3 ∂x∂θ −3 4 ∂U2 ∂x +1 4R ∂U1 ∂θ             (7) {Uij, Vij, Wij}=Zh 2 −h 2 Qij{1, z, z2}dz. (8) Where Wij, Vij and Uij are the bending, coupling and extensional stiffness. Governing shell motion equation are given by ∂Nx ∂x +1 R ∂Nxθ ∂θ =ρt ∂2u ∂t2.(9) The strain energy are expressed as: S=1 2ZL oZ2π 0 {η}T[I]{η}Rdθdx. (10) Where {η}T={e1, e2, γ, κ1, κ2,2ξ}.(11) [I] = "U V V W#.(12) Where U=    U11 U12 0 U12 U22 0 0 0 U66     ,V=    V11 V12 0 V12 V22 0 0 0 V66     , W =    W11 W12 0 W12 W22 0 0 0 W66     . M29-4 2nd Kocaeli Science Congress, November 19-21, 2025 [I] =             U11 U12 0V11 V12 0 U12 U22 0V12 V22 0 0 0 U66 0 0 V66 V11 V12 0W11 W12 0 V12 V22 0W12 W22 0 0 0 V66 0 0 W66             .(13) Kinetic energy Tare expressed as: T=R 2ZL 0Z2π 0 ρt(∂U1 ∂t 2 +∂U2 ∂t 2 +∂U3 ∂t 2)dθdx. (14) Where ρtare represent mass density per unit length and state as: ρt=Zh 2 −h 2 ρ dz. (15) Eq (10) and (12) put into (9), the strain energy Sare expressed in the following form: S=1 2ZL 0Z2π 0 {U11e2 1+U22e2 2+ 2U12e1e2+U66γ2+ 2V11e1κ1+ 2V12e1κ2+ 2V12e2κ1 +2V22e2κ2+ 4V66γξ +W11κ2 1+W22κ2 2+ 2W12κ1κ2+ 4W66ξ2}Rdθdx. The axial modal displacement associated with trigonometric function of three deformation displacement. For x,θand temporal tvariables, modal displacement equation are attained: U1(x, θ, t) = AmU(x)coswtcos(nθ), U2(x, θ, t) = BmV(x)coswtsin(nθ), U3(x, θ, t) = CmW(x)coswtcos(nθ).            (16) Where W ( x ) , V ( x )and U ( x )are the natural vibrational modes in the radial, circumferential and axial direction respectively. The non-dimensional parameter are introduced for algebraic convenience: W=W(x) R, V =V(x) h, U =U(x) h, X =x L Uij =Uij h, Vij =Vij h2, Wij =Wij h3, a =R L, b =h R,¯ρ=ρt h U ( x ) = dζ(x) dx , V ( x ) = ζ ( x ), W ( x ) = ζ ( x )( x−a )and ζ ( x )represent the axial function. ζ ( x )is defined for the axial deformation function which is the characteristics of beam function as: ζ(x) = b1cosh(amx)+b2cos(amx)−χm(b3sinh(amx)+b4sin(amx)). Where ( bi, i = 1 , .., 4) depend upon of the nature of edge conditions, χm depend on values of 2nd Kocaeli Science Congress, November 19-21, 2025 M29-5 KOSC-2025 Proceedings amand amrepresent the roots of trigonometric equation. U1(x, θ, t) = AmhUcos(nθ)coswt U2(x, θ, t) = BmhV sin(nθ)coswt U3(x, θ, t) = CmRW cos(nθ)coswt            (17) The maximum Lagrangian functional is given by: ψmax =πRLh 2[R2w2¯ρZ1 0b2(AmU)2+b2(BmV)2+ (CmW)2dX −Z1 0 a2b2U11 Am dU dX !2 +U22(nbBmV+CmW)2+ 2abU12 Am dU dX !(nbBmV+CmW) +U66 abBm dV dX −nbAmU!2 −2a3b2V11 Am dU dX ! Cm d2W dX2!+ 2ab2V12 Am dU dX ! (n2CmW+nbBmV)−2a2bV 12(nbBmV+CmW) Cm d2W dX2!+ 2bV 22(nbBmV+CmW) (n2CmW+nbBmV)−4bV 66 abBm dV dX −nbAmU! naCm dW dX +3 4abBm dV dX +n 4bAmU! +a4b2W11 Cm d2W dX2!2 +b2W22(n2CmW+nbBmV)2+2a2b2W12 Cm d2W dX2!(−n2CmW−nbBmV) +4b2W66 −naCm dW dX −3 4abBm dV dX −n 4bAmU!2 ]dX. Where Am, Bm, Cm are the fourier vibrational amplitudes. Apply well known Ritz method to maximum of Lagrangian function ψmax with respect to the vibration amplitudes Am, Bm, Cm to achieve necessary condition for extrema: ∂ψmax ∂Am = 0,∂ψmax ∂Bm = 0,∂ψmax ∂Cm = 0. A system of homogeneous equation in Am, Bm and Cm is derive and transformed into the eigenvalue problem as: {[P]−ω2[L]}X= 0. Where ω2 = R2w2ρ , Xt = [ Am, Bm, Cm ]and [ P ],[ L ]are the stiffness and matrices respectively are describe as: P=    P11 P12 P13 P12 P22 P23 P13 P23 P33     , L =    L11 L12 L13 L12 L22 L23 L13 L23 L33     .(18) 2.1 In an area of highly dominated thermal physical systems, functionally graded material is observed. The expression of an effective material property G of an FGM is of a function temperature T ( K ) M29-6 2nd Kocaeli Science Congress, November 19-21, 2025 2.1 by Touloukian [1] as: G=G0(G−1T−1+1+G1T+G2T2+G3T3).(19) Where G0, G−1, G1, G2 and G3 are the thermal coefficient taken in T ( k )and measure in Kelvin and are specific for the constituent material forming the FG material. The G of FG material depend on the materials characteristics and VF of the constituent materials are represented as: G= k X j=1 GjVfj.(20) Where Vfj and Gj represent the VF and material property of jth integrant materials respectively. The result is unity, when the volume fractions are added. k X j=1 Vj= 1.(21) The VF of the two ingredient materials for a shell having a single FG material can be represented as: V1=2z+h 2hN .(22) Where h is thickness and N is power law of exponent. For cylindrical shell having two layered of different materials one with FGM composed with the constituents M1 and M2 and second layered is isotropic material M3 . M1 is equal to M3 . For bilayered cylindrical shell the effective materials properties are ν poisson’s ratio, ρ mass density and E young’s modulus are expressed as E=Eiso +Efgm ν=νiso +νfgm ρ=ρiso +ρfgm            (23) Where Efgm = (E2−E1)2z+h 2hN+E1 νfgm = (ν2−ν1)2z+h 2hN+ν1 ρfgm = (ρ2−ρ1)2z+h 2hN+ρ1            (24) In type 1 and 2 shell, the FG material constituents layer which vary from 0to h 2 and the thickness of inner layer which vary from −h 2 to 0for isotropic material respectively. In type 1 shell, the property of cylindrical shell material become ν = ν3, ρ = ρ3and E = E3 at z = −h 2 . The material properties are ν = ν2and ν1 , ρ = ρ2and ρ1 , E = E2and E1 at z = 0 and in the same pattern, when z = h 2 these properties are ν = ν1, ρ = ρ1and E = E1 . In type 2 shell, The properties of cylindrical shell material become ν = ν3, ρ = ρ3and E = E3 at z = −h 2 , when z = 0 material properties are ν = ν1and ν2 , ρ = ρ1and ρ2 , E = E1and E2 and in the same pattern, the properties of material are ρ = ρ2, ν = ν2and E = E2 at z = h 2 . The vibration frequencies are calculated under simply supported-simply supported boundary condition. The results obtained by using Matlab program. 2nd Kocaeli Science Congress, November 19-21, 2025 M29-7 KOSC-2025 Proceedings 3 Results and Discussion To check the feasibility of present work. The results of present wok is compared to Zhang et al[ 6 ] results with SS-SS boundary condition for type 1 shell. Table (1) the fundamental natural frequencies are minor increasing against circumferential wave numbers. Table (2) the natural frequency of present work is deceasing with compared to Loy et al[ 5 ] results for SS-SS boundary condition. Table (3) describe the fundamental frequencies of bi-layered FG material cylindrical shell with and without ring for the SS-SS boundary condition. The behavior of fundamental frequencies are increasing against the n for both types of shell. The variation in frequency of cylindrical shell without ring are higher than the fundamental frequencies of cylindrical shell with ring support. Table (4) demonstrates the fundamental frequencies with L to R ratio for type 1 shell for SS-SS boundary condition. The fundamental frequencies increased against the circumferential wave numbers. The natural frequency curve attained highest values 295 . 5160 at n = 5 and the curve taken lowest value 46.1620 at n = 1, m = 1. In type 2 maximum value attained 296.5622 at n= 5 and minimum value taken 46.2160 at n= 1,m= 1. Table (5) represents the fundamental frequencies with h to R ratio for the type 1 shell. The fundamental frequencies slightly increased against the circumferential wave numbers with SSSS boundary condition. The fundamental frequency curve taken maximum value 193 . 6870 at n = 5, m = 1. The fundamental frequency curve attained minimum values 65 . 7217 at n = 1, m = 1. In type 2 shell the frequency curves taken maximum value 194 . 0963 at n= 5,m= 1. The frequency curve attained minimum value 65.8589 at n= 1,m= 1. Fig.2 demonstrates the fundamental frequencies of bi-layered FG material cylindrical shell type 1 with ring support for SS-SS boundary condition. First the fundamental frequencies curves increase gradually from a1 = 0 to a1 = 0 . 5then the frequencies curves decreasing gradually from a1 = 0 . 5 to a1 = 1. The frequency curves attained minimum value 17 . 9655 at a 1 = 0, the curves taken maximum values 199 . 7595 , 395 . 633 , 527 . 464 , 627 . 1379 at a1 = 0 . 5and the curves taken minimum values 17.9655,17.9654,17.9656,17.9657 at a1= 1, m = 1, N = 3.5, h = 0.002, n = 1. Fig.3 describes the fundamental frequencies of bi-layered FG material cylindrical shell type 2. First the fundamental frequencies are increased then decreased against the position of ring with SS-SS boundary condition. The frequency curves attained minimum values 17 . 9702 at a1 = 0, the curves taken maximum values 169 . 2889 , 321 . 0058 , 469 . 4629 , 599 . 1012 at a1 = 0 . 5and the curves attained minimum values 17 . 9704 , 17 . 9702 , 17 . 9702 , 17 . 9703 at a1 = 1 , m = 1 , N = 3.5, h = 0.002, n = 1. Fig.4 represents the fundamental frequencies of bi-layered FG material cylindrical shell with ring support. The fundamental frequency curves swiftly increased against the circumferential wave numbers. The frequency curves attained minimum values 199.8445, 122.9645, 61.434, 46.162 at n = 1 , m = 1 , a 1 = 0 . 1 , h = 0 . 007 , R = 1 , N = 1. The frequency curves taken maximum values 295.516,180.975,94.1696,72.2462 at n= 5 against the various length. Fig.5 shows the fundamental frequencies of bi-layered FG material cylindrical shell with ring support. The fundamental frequency curves gradually increased against the circumferential wave numbers. The frequency curves attained minimum values 65 . 7217 , 92 . 9472 , 113 . 8406 , 131 . 8406 at n = 1 , m = 1 , a 1 = 0 . 1 , L = 9 , R = 1 , N = 1. The frequency curves taken maximum values 96.5982,136.6235 167.4317,193.6870 at n= 5 against the various h. M29-8 2nd Kocaeli Science Congress, November 19-21, 2025 Figure 2: Variation of fundamental frequency bi-layered FG material CS type 1 against the ring position at m= 1, n = 1, h = 0.002, N= 3.5 Figure 3: Variation of fundamental frequency bi-layered FG material CS type 2 against the ring position at m= 1, n = 1, h = 0.002, N = 3.5 Figure 4: Variation of fundamental frequency bi-layered FG material CS type 1 against the nat different L,a1=0.1, m = 1, h= 0.007, N = 1 Figure 5: Variation of fundamental frequency bi-layered FG material CS type 1 against the n at different h,a1=0.1, m = 1, L = 9, N = 1 2nd Kocaeli Science Congress, November 19-21, 2025 M29-9