# Free Vibration of Composite Cylindrical Shells Based on Third-Order Shear Deformation Theory

Authors: Muneerah Saad AL Nuwairan; Saira Javed
Journal: Journal of Mathematics
 (2021)

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## Abstract

The focus of this study is to analyse the free vibration of cylindrical shells under third-order shear deformation theory (TSDT). The constitutive equations of the cylindrical shells are obtained using third-order shear deformation theory (TSDT). The surface and traverse displacements are expected to have cubic and quadratic variation. Spline approximation is used to approximate the displacements and transverse rotations. The resulting generalized eigenvalue problem is solved for the frequency parameter to get as many eigenfrequencies as required starting from the least. From the eigenvectors, the spline coefficients are computed from which the mode shapes are constructed. The frequency of cylindrical shells is analysed by varying circumferential node number, length dimension, layer number, and different materials. The authenticity of the present formulation is established by comparing with the available FEM results.


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## Body

## 1. Introduction

Composite shell structures are source of attraction for marine structures such as submarines [1–6]. Stability of these structures depends upon thickness of shells, materials used, lamination scheme, and ply orientation. The multilayered structures are analysed using classical and shear deformation theories [7–9]. Thin structures have ratio of thickness equal to 1/20 or less and are studied using classical bending theory; however, structures with the ratio greater than 1/20 are examined by shear deformation theories. Love [10] proposed the shell’s classical theory, whereas Naghdi [11] included the shear deformations in kinematics of the shells. In addition, Reddy [12] developed a simple SDT for layered plates, and a simple SDT for layered shells was developed by Reddy and Liu [13]. To avoid the discrepancies of the FSDT, HSDT was developed to precisely estimate the crosswise shear stresses which are significant in thick plates and shells. iHSDT plate theories suggest the expansion of displacements to required degree with respect to thickness coordinates avoiding a shear correction factor (see Vinson [14] and Noor et al. [15]). Further, HSDT calculates more precise interlaminar stress distributions and fulfills the requirements that there are no shear stresses at the top and bottom of the shell surfaces. In TSDT, the displacements are extended to the power of three so that to get crosswise shear stresses and strains variation of fourth power through thickness. This neglects the requirement of shear correction coefficient [16].Different methods incorporating HSDT were adopted by different researchers for analysing free vibration of shells. Among them, Baghlani et al. [17] used Fourier series to examine the free vibration of FGM cylindrical shells, Dehsaraji et al. [18] used the Navier method to analyse free vibration of FG nanoshells, moreover, Sayyad and Ghugal [19] used Navier technique to analyse composite spherical shells, and Rout et al. [20] analysed thermoelastic unrestricted vibration of the multilayered shells considering the Green–Lagrange type of nonlinearity. HSDT along with Navier solutions was used to examine the vibration of shells and plates by Zine et al. [21]. Newton–Raphson iterative technique was used to analyse geometrical nonlinear bending characteristics of shells by Chavan and Lal [22] based on HSDT and Green–Lagrange nonlinearity. FEM was used to observe the unrestricted vibration of FG shells under HSDT (see Zghal et al. [23]). Nonlinear vibration of the layered shells was studied by Hirwani et al. [24] using Fourier transform technique based on HSDT. The Chebyshev–Ritz method was adopted to study the vibration of FG layered beams for HSDT by Shenas et al. [25]. The unrestricted vibration of FG carbon nanotube layered quadrilateral spherical panels was studied using HSDT with DQM by Setoodeh et al. [26]. Frequency variation based on sound radiation of doubly curved layered composite shells was examined by Sharma et al. [27] using HSDT and FEM. The HSDT incorporating with the discrete method was used to study curved structural parts [28]. The Navier method based on HSDT was used by Punera and Kant [29] to investigate FG cylindrical shells. HSDT was used by Hwu et al. [30] to examine the vibration of layered plates and cylindrical shells. Moreover, HSDT and GM were used by Nasihatgozar et al. [31] to analyse the free vibration of doubly curved sheets. Nguyen et al. [32] examined the vibration of functionally graded plates using HSDT. Moreover, nanoplates in the thermal environment were studied by Diakh et al. [33]. Baghlani et al. [34] investigated the free vibration of FGM cylindrical shells on elastic foundation. The FEM method was used to study the free and static vibration of FGM plates by Katili [35]. Cross-ply laminates were studied by Dhari and Singh [36]. Higher-order shear deformation theories with the unified model were studied for composite plates by Li et al. [37]. HSDT was used to analyse wave propagation of a ceramic-metal functionally graded sandwich plates by Tahir et al. [38]. Moreover, hygro-thermo-mechanical bending behavior of advanced functionally graded ceramic-metal plate was studied by Mudhaffar et al. [39]. An original four-variable quasi-3D shear deformation theory for the static and free vibration analysis of new type of sandwich plates was studied by Kouider et al. [40]. Merazka et al. [41] examined the hygro-thermo-mechanical bending response of FG plates. Bending analysis of functionally graded plates was investigated by Hachemi et al. [42].The individuality of the current study is evident that none of the abovementioned researchers used spline approximation to analyse the free vibrational problems except Javed et al. [43]. Moreover, Javed et al. [43] used the spline approximation method to analyse the free vibration of plates whereas in this research spline approximation is used to analyse composite cylindrical shells. In addition to that stress-strain relation of plates and cylindrical shells is different so derivation of equilibrium equation was completely different as compared with Javed et al. [43]. Therefore, current investigation aims to analyse the free vibration of composite cylindrical shells. Spline approximation is used to approximate the displacement and rotational functions. Resulting equations along with the end condition equations obtain a system of equations. The frequency parameter is obtained using eigen solution technique. The mode shapes are created using eigenvector. Cylindrical shells are investigated by varying their lengths, layer sequence, layer constituents, and circumferential node number. Results are depicted using graphs and tables. Autocad software is used to draw some figures.


## 2. Formulation and Method

A composite laminated circular cylindrical shell of having lengthℓ, thickness h, and radius r is shown in Figure 1. The x coordinate of the shell s is taken along the longitudinal direction, θ coordinate along the circumferential direction, and z along the thickness direction.Figure 1
Cylindrical shell (geometry).


### 2.1. Displacement Equations

The displacement field is based on TSDT (see Reddy [16]):(1)ux,y,z,t=u0x,y,t+zϕxx,y,t−4z33h2ϕx+∂w0∂x,vx,y,z,t=v0x,y,t+zϕθx,y,t−4z33h2ϕθ+∂w0∂y,wx,y,z,t=w0x,y,t,where u, v, and w are the displacement functions in x, θ, and z directions, respectively, u0, v0, and w0 are the displacements of the middle surface of the cone, and ψx and ψθ are shear rotations of any point on the middle surface.


### 2.2. Constitutive Equations

Stress, moment, and shear resultantsN, M, and Q and higher-order stress resultants P and R are defined as(2)NxNθNxθMxMθMxθPxPθPxθ=∫−h/2h/2σxσθσxθzσxzσθzσxθz3σxz3σθz3σxθdz,QθQxRθRx=∫−h/2h/2τθzτxzz2τθzz2τxzdz,where σi is the normal stress and τi is the shear stress components.The transformed stress-strain relations are as follows:(3)σxkσθkτxθkτθzkτxzk=Q11kQ12kQ16k00Q12kQ22kQ26k00Q16kQ26kQ66k00000Q44kQ45k000Q45kQ55kεxkεθkγxθkγθzkγxzk,where Qijk are given in Appendix A.The stress-strain relations are obtained as follows:(4)NxNθNxθMxMθMxθPxPθPxθ=A11A12A16B11B12B16E11E12E16A12A22A26B12B22B26E12E22E26A16A26A66B16B26B66E16E26E66B11B12B16D11D12D16F11F12F16B12B22B26D12D22D26F12F22F26B16B26B66D16D26D66F16F26F66E11E12E16F11F12F16H11H12H16E12E22E26F12F22F26H12H22H26E16E26E66F16F26F66H16H26H66εx0εθ0γxθ0εx1εθ1γxθ1εx3εθ3γxθ3,QθQxRθRx=A44A45D44D45A45A55D45D55D44D45F44F45D45D55F45F55γθz0γxz0γθz2γxz2γθzγxz0γθz20γxz2,QθQxRθRx=A44A45D44D45A45A55D45D55D44D45F44F45D45D55F45F55γθz0γxz0γθz2γxz2,where ε is the strain and γ is the shear strain components.Stiffness coefficientsAij, Bij, and Dij (extensional, bending-extensional coupling, and bending stiffnesses) and Eij, Fij, and Hij are the higher-order stiffness coefficients defined in Appendix B.


### 2.3. Cylindrical Shell Equations

The equilibrium equations for cylindrical shells based on TSDT are as follows:(5)∂Nx∂x+1r∂Nxθ∂θ=I0∂2u∂t2+I1−c1I3∂2ϕx∂t2−c1I3∂3w∂x∂t2,∂Nxθ∂x+1r∂Nθ∂θ+1rQθ=I0+2rI1∂2v∂t2+I1+1rI2−c1I3−c11rI4∂2ϕθ∂t2−c1I3+c11rI4∂3w∂θ∂t2,∂Q¯x∂x+1r∂Q¯θ∂θ−1rNθ+c2∂2Px∂x2+21r∂2Pxθ∂x∂θ+1r2∂2Pθ∂θ2=c1I3∂3u∂x∂t2+c1I3+c1rI4∂3v∂θ∂t2+I0∂2w∂t2−c12I6∂4w∂x2∂t2+∂4w∂θ2∂t2+c1I4−c12I6∂3ϕx∂x∂t2+c1I4−c12I6∂3ϕθ∂θ∂t2,∂M¯x∂x+1r∂M¯xθ∂θ−Q¯x=I1−c1I3∂2u∂t2−c1I4−c12I6∂3w∂x∂t2+I2−2c1I4+c12I6∂2ϕx∂t2,∂M¯xθ∂x+1r∂M¯θ∂θ−Q¯θ=I1+1rI2−c1I3−c11rI4∂2v∂t2−c1I4−c12I6∂3w∂θ∂t2+I2−2c1I4+c12I6∂2ϕθ∂t2,where(6)c1=4h2,c2=c13,Ii=∫zρkzdz,i=0,1,2,3,…,6,and ρ is the material density of the k-th layer. M¯x=Mx−c1Px, M¯θ=Mθ−c1Pθ, and M¯xθ=Mxθ−c1Pxθ:(7)Q¯θ=Qθ−c2Rθ,Q¯x=Qx−c2Rx,where Ni,Mi,andQi are stress, moment, and shear resultants, respectively. PiandRi denote higher-order stress resultants.Firstly, stress and strain relations are substituted in equation (5). After that the displacements and rotational functions for shells [44] are used, and resulting differential equation was nondimensionalized by using following parameters:(8)X=xl,  a distant coordinate, andX∈0,1;λ=ωℓI0A11,  a frequency parameter;L=ℓr,a length parameter,H=hr,ratio of total thickness to radius;δk=hkh,  relative layer thickness of thek−th layer.Nondimensional equation is obtained in the matrix form as(9)Y11Y12Y13Y14Y15Y21Y22Y23Y24Y25Y31Y32Y33Y34Y35Y41Y42Y43Y44Y45Y51Y52Y53Y54Y55UVWΦXΦθ=00000,where differential operators Yij′s are given in Appendix C.


### 2.4. Spline Method

Equation (9) consists of UX derivatives of order three, VX derivatives of order two, WX derivatives of order four, ΦXX derivatives of order three, and ΦθX derivatives of order two. These functions are approximated by using cubic and quintic spline functions, in the range of Xε0,1.The displacement functionsUX, VX, and WX and the rotational functions ΦXXandΦθX are approximated by(10)UX=∑i=04aiXi+∑j=0N−1bjX−Xj5HX−Xj,VX=∑i=02ciXi+∑j=0N−1djX−Xj3HX−Xj,WX=∑i=04eiXi+∑j=0N−1fjX−Xj5HX−Xj,ΦXX=∑i=04giXi+∑j=0N−1gjX−Xj5HX−Xj,ΦΘX=∑i=02liXi+∑j=0N−1qjX−Xj3HX−Xj.Here,HX−Xj is the Heaviside step function and N is the number of intervals into which the range 0,1 of X is divided. The pointsX=Xs=s/Ns=0,1,2,…,N are selected as the spline knots. Therefore, differential equation (9) is satisfied by the spline, at all Xs. The resulting expressions contain 5N+5 homogeneous system of equations in the 5N+21 spline coefficients. The S-S boundary condition is considered in order to coincide the number of equations and unknowns.


## 2.1. Displacement Equations

The displacement field is based on TSDT (see Reddy [16]):(1)ux,y,z,t=u0x,y,t+zϕxx,y,t−4z33h2ϕx+∂w0∂x,vx,y,z,t=v0x,y,t+zϕθx,y,t−4z33h2ϕθ+∂w0∂y,wx,y,z,t=w0x,y,t,where u, v, and w are the displacement functions in x, θ, and z directions, respectively, u0, v0, and w0 are the displacements of the middle surface of the cone, and ψx and ψθ are shear rotations of any point on the middle surface.


## 2.2. Constitutive Equations

Stress, moment, and shear resultantsN, M, and Q and higher-order stress resultants P and R are defined as(2)NxNθNxθMxMθMxθPxPθPxθ=∫−h/2h/2σxσθσxθzσxzσθzσxθz3σxz3σθz3σxθdz,QθQxRθRx=∫−h/2h/2τθzτxzz2τθzz2τxzdz,where σi is the normal stress and τi is the shear stress components.The transformed stress-strain relations are as follows:(3)σxkσθkτxθkτθzkτxzk=Q11kQ12kQ16k00Q12kQ22kQ26k00Q16kQ26kQ66k00000Q44kQ45k000Q45kQ55kεxkεθkγxθkγθzkγxzk,where Qijk are given in Appendix A.The stress-strain relations are obtained as follows:(4)NxNθNxθMxMθMxθPxPθPxθ=A11A12A16B11B12B16E11E12E16A12A22A26B12B22B26E12E22E26A16A26A66B16B26B66E16E26E66B11B12B16D11D12D16F11F12F16B12B22B26D12D22D26F12F22F26B16B26B66D16D26D66F16F26F66E11E12E16F11F12F16H11H12H16E12E22E26F12F22F26H12H22H26E16E26E66F16F26F66H16H26H66εx0εθ0γxθ0εx1εθ1γxθ1εx3εθ3γxθ3,QθQxRθRx=A44A45D44D45A45A55D45D55D44D45F44F45D45D55F45F55γθz0γxz0γθz2γxz2γθzγxz0γθz20γxz2,QθQxRθRx=A44A45D44D45A45A55D45D55D44D45F44F45D45D55F45F55γθz0γxz0γθz2γxz2,where ε is the strain and γ is the shear strain components.Stiffness coefficientsAij, Bij, and Dij (extensional, bending-extensional coupling, and bending stiffnesses) and Eij, Fij, and Hij are the higher-order stiffness coefficients defined in Appendix B.


## 2.3. Cylindrical Shell Equations

The equilibrium equations for cylindrical shells based on TSDT are as follows:(5)∂Nx∂x+1r∂Nxθ∂θ=I0∂2u∂t2+I1−c1I3∂2ϕx∂t2−c1I3∂3w∂x∂t2,∂Nxθ∂x+1r∂Nθ∂θ+1rQθ=I0+2rI1∂2v∂t2+I1+1rI2−c1I3−c11rI4∂2ϕθ∂t2−c1I3+c11rI4∂3w∂θ∂t2,∂Q¯x∂x+1r∂Q¯θ∂θ−1rNθ+c2∂2Px∂x2+21r∂2Pxθ∂x∂θ+1r2∂2Pθ∂θ2=c1I3∂3u∂x∂t2+c1I3+c1rI4∂3v∂θ∂t2+I0∂2w∂t2−c12I6∂4w∂x2∂t2+∂4w∂θ2∂t2+c1I4−c12I6∂3ϕx∂x∂t2+c1I4−c12I6∂3ϕθ∂θ∂t2,∂M¯x∂x+1r∂M¯xθ∂θ−Q¯x=I1−c1I3∂2u∂t2−c1I4−c12I6∂3w∂x∂t2+I2−2c1I4+c12I6∂2ϕx∂t2,∂M¯xθ∂x+1r∂M¯θ∂θ−Q¯θ=I1+1rI2−c1I3−c11rI4∂2v∂t2−c1I4−c12I6∂3w∂θ∂t2+I2−2c1I4+c12I6∂2ϕθ∂t2,where(6)c1=4h2,c2=c13,Ii=∫zρkzdz,i=0,1,2,3,…,6,and ρ is the material density of the k-th layer. M¯x=Mx−c1Px, M¯θ=Mθ−c1Pθ, and M¯xθ=Mxθ−c1Pxθ:(7)Q¯θ=Qθ−c2Rθ,Q¯x=Qx−c2Rx,where Ni,Mi,andQi are stress, moment, and shear resultants, respectively. PiandRi denote higher-order stress resultants.Firstly, stress and strain relations are substituted in equation (5). After that the displacements and rotational functions for shells [44] are used, and resulting differential equation was nondimensionalized by using following parameters:(8)X=xl,  a distant coordinate, andX∈0,1;λ=ωℓI0A11,  a frequency parameter;L=ℓr,a length parameter,H=hr,ratio of total thickness to radius;δk=hkh,  relative layer thickness of thek−th layer.Nondimensional equation is obtained in the matrix form as(9)Y11Y12Y13Y14Y15Y21Y22Y23Y24Y25Y31Y32Y33Y34Y35Y41Y42Y43Y44Y45Y51Y52Y53Y54Y55UVWΦXΦθ=00000,where differential operators Yij′s are given in Appendix C.


## 2.4. Spline Method

Equation (9) consists of UX derivatives of order three, VX derivatives of order two, WX derivatives of order four, ΦXX derivatives of order three, and ΦθX derivatives of order two. These functions are approximated by using cubic and quintic spline functions, in the range of Xε0,1.The displacement functionsUX, VX, and WX and the rotational functions ΦXXandΦθX are approximated by(10)UX=∑i=04aiXi+∑j=0N−1bjX−Xj5HX−Xj,VX=∑i=02ciXi+∑j=0N−1djX−Xj3HX−Xj,WX=∑i=04eiXi+∑j=0N−1fjX−Xj5HX−Xj,ΦXX=∑i=04giXi+∑j=0N−1gjX−Xj5HX−Xj,ΦΘX=∑i=02liXi+∑j=0N−1qjX−Xj3HX−Xj.Here,HX−Xj is the Heaviside step function and N is the number of intervals into which the range 0,1 of X is divided. The pointsX=Xs=s/Ns=0,1,2,…,N are selected as the spline knots. Therefore, differential equation (9) is satisfied by the spline, at all Xs. The resulting expressions contain 5N+5 homogeneous system of equations in the 5N+21 spline coefficients. The S-S boundary condition is considered in order to coincide the number of equations and unknowns.


## 3. Results and Discussion

The vibrational response of cylinderical shells under S-S end condition is analysed. Cylindrical shells of cross-ply orientations of two, three, four, five, and six layers are considered. Figure2 shows 6-layered composite with cross-ply orientation.Figure 2
Composite with six layers.Comparative study can be seen in Table1 with Khare et al. [45] (using FSDT and HSDT) and Bhimaraddi [46] (using constant shear deformation theory and thin shell theory) for orthotropic two-layer cross-ply S-S cylindrical shells.Table 1
The value of frequency parameterλ=ωℓI0/A11 of two-layered orthotropic shells under S-S boundary conditions is compared with Khare et al. [45] ω¯=ωa/ρ/E21/2 and Bhimaraddi [46].
LHPresentKhare et al. [45]Bhimaraddi [46]HSDTFOSTHOSTPSDCSD10.050.772140.785670.786270.799930.797980.101.032411.040621.043571.091891.074750.151.282161.295781.303511.381741.3327420.050.551320.565680.566840.580000.577330.100.917340.913670.918010.956640.936530.151.225131.221011.231071.289331.23527Table2 shows the relation between the angular frequency parameter and length of 2-layered cylindrical shells ωm and length of 2-layered cylindrical shells. There is the inverse relation between the angular frequency and length whereas there is a positive relation between frequency and mode number.Table 2
Relation between angular frequencyωm and length L of two-layered 00/900 cylindrical shelln=2.
L00/900GE/KE00/900EGE/KEω1ω2ω3ω1ω2ω30.57.294479.9362512.713207.062738.6007611.0365011.287471.869722.786691.270031.909772.729801.50.238620.700581.250130.273380.736911.2242220.097040.636711.121970.206450.579140.912352.50.086110.537291.050320.028440.524990.85376Table3 depicts the relation between frequency parameter λm and circumferential node number n for three 00/900/00GE/KE/GE and six layered cylindrical shells 00/900/00/900/00/900GE/KE/GE/KE/GE/KE. The length is considered to be L=1.5. The frequency declines till n=5 and slightly rises afterward.Table 3
Relation between frequencyλm and circumferential node number n of three- and six-layered cylindrical shell L=1.5.
n00/900/00GE/KE/GE00/900/00/900/00/900GE/KE/GE/KE/GE/KEλ1λ2λ3λ1λ2λ310.000250.000390.000490.000290.000330.0004130.000150.000220.000300.000200.000260.0003250.000110.000180.000250.000180.000240.0002870.000160.000230.000320.000210.000260.0003290.000230.000350.000470.000290.000330.00041Figures3–5 demonstrate the effect of length on the angular frequency parameter of 3-, 4-, and 5-layered cylindrical shells with the fixed n=2. The frequency declines till L = 1 and remains same afterward. This is because that the membrane longitudinal strain dominates the total strain energy of the system till length of the shell is 1. However, as length increases, the bending deformation becomes dominant in the system.Figure 3
Relation between frequency and length of three-layered shells00/900/00GE/KE/GE; n=2.Figure 4
Relation between frequency and length of four-layered shells00/900/00/900GE/KE/GE/KE; n=2.Figure 5
Relation between frequency and length of five-layered shells00/900/00/900/00EGE/KE/EGE/KE/EGE; n=2.Relation between the frequency and circumferential node number of 3- and 5-layered shells (00/900/00, 00/900/00/900/00) with material combinations (EGE/KE/EGE, EGE/KE/EGE/KE/EGE) and L=0.5 is shown in Figures 6 and 7. The frequency value decreases till n=6 and gradually increases as the circumferential node number increases. Since the decrease in the frequency shows that the rigidity of the structure decreases till n=6 and as the circumferential node number increases, the frequency increases and so the rigidity of the structure also increases. It is also evident that as rigidity of the structure increases, its flexibility decreases.Figure 6
Relation between frequency and circumferential node number of three-layered shells00/900/00EGE/KE/EGE; L=0.5.Figure 7
Relation between frequency and circumferential node number of five-layered shells00/900/00/900/00EGE/KE/EGE/KE/EGE; L=0.5.Figures8 and 9 show the relation between frequency parameter λm and circumferential node number n for 5- and 3-layered cross-ply cylindrical shells, respectively.Figure 8
Relation between frequency and circumferential node number of five-layered shells00/900/00/900/00GE/KE/GE/KE/GE; L=0.5.Figure 9
Relation between frequency and circumferential node number of three-layered shells00/900/00GE/KE/GE; L=0.5.The relation of fundamental frequencyλ and circumferential node number n of 3-layered cylindrical shells with material orientations GE/KE/GE and EGE/KE/EGE is shown in Figure 10. The fundamental frequency value is significantly higher for cylindrical shells with material combination EGE/KE/EGE than GE/KE/GE.


## 4. Conclusion

The present investigation explicates the vibrational response of cylindrical shells under HSDT for SS support condition. It is concluded that length, circumferential node number, stacking sequence, number of layers, and their constituents significantly affect the shell analysis. Moreover, the effect of length on the shell’s angular frequency parameter is that the membrane longitudinal strain dominates the total strain energy of the system till length of the shell is 1, and as the length increases, the bending deformation becomes dominant in the system. The effect of the circumferential node number on frequency shows that the frequency value decreases tilln=6 and gradually increases as the circumferential node number increases. The decrease in the frequency value shows that the rigidity of the structure decreases till n=6, and as the circumferential node number increases, the frequency increases and so the rigidity of the structure also increases. The comparative study confirms the existing results that may contribute its unique part to already existing research of the related field.Figure 10
Relation between frequency and circumferential node number of three-layered shells00/900/00L=1.5.



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