Assessment of a new higher-order shear and normal deformation theory for the static response of functionally graded shallow shells

Bharti M. Shinde; Atteshamuddin S. Sayyad; Nitin S. Naik

doi:10.1515/cls-2024-0014

Article Open Access

Assessment of a new higher-order shear and normal deformation theory for the static response of functionally graded shallow shells

Bharti M. Shinde , Atteshamuddin S. Sayyad and Nitin S. Naik

Published/Copyright: October 18, 2024

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From the journal Curved and Layered Structures Volume 11 Issue 1

Abstract

Static response of simply supported functionally graded (FG) shallow shells using a new higher-order shear and normal deformation theory is focused in this article. The effects of transverse strains and stresses on the bending response of FG shell are considered by the present theory. The current theory considers the impacts of transverse normal and shear deformations that meet the requirements for traction-free boundary conditions. The virtual work principle is applied to the mathematical formulation of the present theory. The simply supported doubly curved shallow shell problems under the static transverse load are analyzed using Navier’s solution technique. To verify the existing theory, the current results are, whenever possible, compared with those that have already been published. Additionally, a few benchmark results are presented in this article that will be helpful to researchers in the future.

Keywords: FG shell; spherical shells; hyperbolic shells; cylindrical shells; elliptic shells; shear and normal deformation theory

1 Introduction

Functionally graded materials (FGM) are formed of ceramic and metal, and their modulus of elasticity varies with thickness. Because of its appealing properties, FGM material has a wide range of applications in industries such as spacecraft, aircraft, the energy sector, and aerospace engineering, which distinguishes it to become a popular study topic. As a result, numerous researchers have given various mathematical and analytical models to study the analysis of functionally graded (FG) shells under the sinusoidal and uniformly distributed load.

In history, Classical shell theory (CST) was developed by Kirchhoff [1] ignoring the shear deformation effect to study the response for thin plates and shells. Further, Mindlin [2] has investigated a first-order shear deformation theory (FSDT). These theories provide more accurate predictions of displacements and stresses by considering variations in transverse shear stresses in the thickness direction of the structure. Higher-order theories (HSDTs) have been created by researchers in response to the inadequacies of the CST and the FSDT. When simulating the behavior of thin-walled structures, HSDTs provide better accuracy and dependability than FSDT, which fails to meet zero transverse shear stress requirements at the top and bottom faces of shells and assumes constant transverse shear stresses throughout the thickness. Thai et al. [3] presented the analysis of an FG sandwich plate based on the FSDT. Thai and Kim [4] used a theory with four degrees-of-freedom (DOF) to study the bending and vibration response of sandwich plates with FG material. Li et al. [5] employed the 3D linear theory of elasticity to study the free vibration of sandwich plates. Using a 2D constitutive model, a nonlinear analysis of FG shells is presented by Daszkiewicz et al. [6]. Arciniega and Reddy [7] describe a non-linear deformation analysis of FG shells based on FSDT. The bending behavior of FG sandwich plates using a four DOF theory based on Levy’s solution is presented by Demirhan and Taskin [8]. A four DOF theory is used by Abdelaziz et al. [9] for the static analysis of FG sandwich plates. Allibeigloo and Noee [10] used a numerical method for the analysis of cylindrical sandwich shells. A four DOF theory is used by Hadji et al. [11] to examine the soft and hardcore FG sandwich plate-free vibration properties.

The hyperbolic plate theory was employed by Rouzegar and Gholami [12] to analyze the thermoelastic bending response of FG sandwich plates. A comparative elastic study of sandwich pipes with and without FG interlayers is carried out by Sburlati and Kashtalyan [13]. A novel three-DOF hyperbolic theory was put forth by Belabed et al. [14] for the free vibration study of FG sandwich plates. An improved sinusoidal HSDT is developed by Mantari and Soares [15] to investigate a static response for FG plates and shells. Thai and Kim [16] give a review article on FG plates and shells research highlighting the research gap and scope of future research. Irfan and Siddiqui [17] recently examined recent developments in sandwich plate FEM. Using FSDT, Tornabene et al. [18] examined the dynamic response of FG shell panels and annular plate structures. Nejati et al. [19] examined the static and free vibrations of FG conical shells using a generalized quadrature technique. FG doubly curved shells and panels of revolution were statically analyzed by Viola and Tornabene [20] using an FSDT-based GDQ approach. A modified Kirchhoff theory was put forth by Nguyen and Do [21] for the free vibration analysis of FGM plates. Using the FSDT, Huan et al. [22] offered an analytical solution for assessing the bending, buckling, and vibration characteristics of FG cylindrical panels. Nevertheless, thermoelastic and vibration analysis of FG cylindrical shells was studied by Zhao et al. [23]. They used the FSDT-based element-free kp Ritz approach. Using Reddy’s parabolic shear deformation theory, Oktem et al. [24] investigated the static response of FG doubly curved shells. Taking into consideration the zig-zag effect, Tornabene et al. [25] used CUF-derived equivalent single-layer theories to study the free vibration of doubly curved FG shells. An FG shell, free vibration behavior was analyzed by Fantuzzi et al. [26] using a 3D shell model and many 2D computational models. The free vibration characteristics of FG sandwich shallow shells supported by two-parameter elastic foundations were analyzed using the FSDT by Wang et al. [27]. They looked into the frequency response under different curvature kinds, geometrical parameters, and boundary conditions. To forecast the bending responses of sandwich doubly curved shells and laminated composites under transverse normal load, Tornabene and Brischetto [28] compared 2D and 3D shell models. They used the layerwise and single-layer theories together in the generalized differential quadrature (GDQ) technique. HSDTs were applied to examine the free vibration behavior of sandwich shells with different thicknesses that were FG in a study by Tornabene et al. [29]. To examine the vibration response numerically, they used the GDQ approach. In the dynamic analysis of FG sandwich plates, Attia [30] suggested four variable refined plate theories that account for non-polynomial distributions of shear strain. The higher-order displacement models for the static and free vibration analysis of laminated and FG cylindrical shells were created by Punera and Kant [31,32]. A third-order shear deformation theory was utilized by Viola et al. [33] to investigate the FG cylindrical shell numerically. Sayyad and Ghugal [34,35,36] have developed a generalized higher-order shell theory for static and free vibration analysis of laminated composite and FGM double curvature shells. Similarly, the fifth-order shell theory has been established by Shinde and Sayyad [37,38,39,40] for the analysis of sandwich, FG, and laminated composite shells. Recently, Jape and Sayyad [41], Shaikh and Sayyad [42] and Tamnar and Sayyad [43] have presented a static and free vibration analysis of double curvature laminated composite and FGM shells with transverse normal strain effect.

2 Novelty of the present theory

This study extends the fifth-order shell theory to analyze doubly curved FG shells under uniformly distributed and sinusoidal loading. Limited research has been done on the static analysis of FG shells, according to a survey of the literature which is presented in this article. The unique aspect of the current theory is that, as suggested by Carrera and Brischetto [44,45,46,47], it accurately predicts the bending response of FG sandwich shells considering the influence of both shear and normal deformations. Additionally, to more precisely forecast bending behavior, the current work makes use of the fifth-order polynomial form function. In brief, the current study presents the static response of the shell structures concerning displacement and stresses, utilizing a higher-order shear and normal deformation theory (HOSNDT). The primary contribution of this study is the availability of the static analysis of doubly curved FG shells subjected to sinusoidal and uniform loadings.

3 Mathematical formulation

The FG shell, simply supported at supports, is considered in the current study’s mathematical formulation, as shown in Figure 1. A shell’s dimensions are length (a), width (b), and thickness (h); R _x and R _y are its fundamental curvature radii, considered as (R ₁ and R ₂) respectively. This work presents the analysis of the following shell types: Elliptic (R ₁ = R, R ₂ = 1.5R), Spherical (R ₁ = R ₂ = R), Hyperbolic (R ₁ = R, R ₂ = −R) and Cylindrical (R ₁ = R and R ₂ = ∞). Eq. (1) is used to explain the material properties of FG shells varying in thickness direction.

(1) E ( z ) = E m + ( E C − E m ) 0.5 + z h p ,

where p is the power-law index. E(z) represents the modulus of elasticity in FG material, where the properties of metal and ceramic are represented by (E _m, E _c), respectively. As the power law varies from p = 0 to ∞, respectively, the characteristics of the FGM shell change from totally metallic to ceramic. At p = 0, a shell will be fully ceramic whereas at p = ∞, a shell will be fully metallic.

Figure 1

Geometry of the FG shell element.

CST serves as the foundation for the development of the present HOSNDT, and is expressed as

(2) u ( x , y , z ) = 1 + z R 1 u 0 − z ∂ w 0 ∂ x + f 1 ( z ) ϕ x + f 2 ( z ) ψ x v ( x , y , z ) = 1 + z R 2 v 0 − z ∂ w 0 ∂ y + f 1 ( z ) ϕ y + f 2 ( z ) ψ y w ( x , y , z ) = w 0 + f 1 ′ ( z ) ϕ z + f 2 ′ ( z ) ψ z ,

where u, v, and w are the displacements in x-, y-, and z- directions and u 0 , v 0 , w 0 , ϕ x , ϕ y , ϕ z , ψ x , ψ y , ψ z are the nine unknowns to be determined which are functions of x and y coordinates. Following are the nonzero strains at any point of the shell domain as given by Reddy [48].

(3) ε x = ∂ u 0 ∂ x + w 0 R 1 − z ∂ 2 w 0 ∂ x 2 + f 1 ( z ) ∂ ϕ x ∂ x + f 2 ( z ) ∂ ψ x ∂ x + f 1 ′ ( z ) R 1 ϕ z + f 2 ′ ( z ) R 1 ψ z ε y = ∂ v 0 ∂ y + w 0 R 2 − z ∂ 2 w 0 ∂ y 2 + f 1 ( z ) ∂ ϕ y ∂ y + f 2 ( z ) ∂ ψ y ∂ y + f 1 ′ ( z ) R 2 ϕ z + f 2 ′ ( z ) R 2 ψ z ε z = f 1 ″ ( z ) ϕ z + f 2 ″ ( z ) ψ z γ x y = ∂ u 0 ∂ y + ∂ v 0 ∂ x − 2 z ∂ 2 w 0 ∂ x ∂ y + f 1 ( z ) ∂ ϕ x ∂ y + ∂ ϕ y ∂ x + f 2 ( z ) ∂ ψ x ∂ y + ∂ ψ y ∂ x

γ x z = f 1 ′ ( z ) ϕ x + ∂ ϕ z ∂ x + f 2 ′ ( z ) ψ x + ∂ ψ z ∂ x γ y z = f 1 ′ ( z ) ϕ y + ∂ ϕ z ∂ y + f 2 ′ ( z ) ψ y + ∂ ψ z ∂ y ,

where

(4) f 1 ( z ) = z 1 − 4 z 2 3 h 2 , f 1 ′ ( z ) = 1 − 4 z 2 h 2 , f 2 ( z ) = z 1 − 16 z 4 5 h 4 , f 2 ′ ( z ) = 1 − 16 z 4 h 4 , f 1 ″ ( z ) = − 8 z h 2 , f 2 ″ ( z ) = − 64 z 3 h 4 .

Similarly, the state of stress at a point in the shell domain is as follows [48]:

(5) σ x σ y σ z τ x y τ x z τ y z = Q 11 ( z ) Q 12 ( z ) Q 13 ( z ) 0 0 0 Q 12 ( z ) Q 22 ( z ) Q 23 ( z ) 0 0 0 Q 13 ( z ) Q 23 ( z ) Q 33 ( z ) 0 0 0 0 0 0 Q 66 ( z ) 0 0 0 0 0 0 Q 55 ( z ) 0 0 0 0 0 0 Q 44 ( z ) × ε x ε y ε z γ x y γ x z γ y z ,

where ( σ x , σ y , σ z ) normal stresses and ( τ x y , τ x z , τ y z ) shear stresses. Q _ij(z) are the reduced stiffness coefficients defined as follows as given by Reddy [48]:

(6) Q 11 ( z ) = Q 22 ( z ) = Q 33 ( z ) = E ( z ) ( 1 − μ ) ( 1 − 2 μ ) ( 1 + μ ) , Q 12 ( z ) = Q 21 ( z ) = Q 13 ( z ) = Q 31 ( z ) = Q 23 ( z ) = Q 32 ( z ) = E ( z ) μ ( 1 − 2 μ ) ( 1 + μ ) , Q 44 ( z ) = Q 55 ( z ) = Q 66 ( z ) = E ( z ) 2 ( 1 + μ ) ,

where µ is Poisson’s ratio; E(z) is the modulus of elasticity. In FG material the modulus of elasticity varies along the thickness accounting for the stiffness across the thickness whereas the Poisson’s ratio constant assumes that the material’s ability to undergo lateral strain relative to axial strain does not change along the thickness, even though the stiffness (Young’s modulus) does.

3.1 Governing equations

To derive the governing equations and boundary conditions, the principle of virtual work done is used.

(7) ∫ 0 a ∫ 0 b ∫ − h / 2 + h / 2 ( σ x δ ε x + σ y δ ε y + σ z δ ε z + τ x y δ γ x y + τ x z δ γ x z + τ y z δ γ y z ) d z d y d x − ∫ 0 a ∫ 0 b q ( x , y ) δ w d y d x = 0 ,

where δ is the variational operator. Using the fundamental lemma of calculus, the following governing equations are derived for the present theory.

(8) δ u 0 : A 11 ∂ 2 u 0 ∂ x 2 + A 11 R 1 ∂ w 0 ∂ x − B 11 ∂ 3 w 0 ∂ x 3 + A S 1 11 ∂ 2 ϕ x ∂ x 2 + A S 2 11 ∂ 2 ψ x ∂ x 2 + A 12 ∂ 2 v 0 ∂ x ∂ y + A 12 R 2 ∂ w 0 ∂ x − B 12 ∂ 2 w 0 ∂ x ∂ y 2 + A S 1 12 ∂ 2 ϕ y ∂ x ∂ y + A S 2 12 ∂ 2 ψ y ∂ x ∂ y + E 13 ∂ ϕ z ∂ x + F 13 ∂ ψ z ∂ x + A 66 ∂ 2 u 0 ∂ y 2 + A 66 ∂ 2 v 0 ∂ x ∂ y − 2 B 66 ∂ 3 w 0 ∂ x ∂ y 2 + A S 1 66 ∂ 2 ϕ x ∂ y 2 + A S 1 66 ∂ 2 ϕ y ∂ x ∂ y + A S 2 66 ∂ 2 ψ x ∂ y 2 + A S 2 66 ∂ 2 ψ y ∂ x ∂ y + Q 111 R 1 + Q 112 R 2 ∂ ϕ z ∂ x + Q 211 R 1 + Q 212 R 2 ∂ ψ z ∂ x = 0 ,

(9) δ v 0 : A 12 ∂ 2 u 0 ∂ x ∂ y + A 12 R 1 ∂ w 0 ∂ y − B 12 ∂ 3 w 0 ∂ x 2 ∂ y + A S 1 12 ∂ 2 ϕ x ∂ x ∂ y + A S 2 12 ∂ 2 ψ x ∂ x ∂ y + A 22 ∂ 2 v 0 ∂ y 2 + A 22 R 2 ∂ w 0 ∂ y − B 22 ∂ 3 w 0 ∂ y 3 + A S 1 22 ∂ 2 ϕ y ∂ y 2 + A S 2 22 ∂ 2 ψ y ∂ y 2 + E 23 ∂ ϕ z ∂ y + F 23 ∂ ψ z ∂ y + A 66 ∂ 2 u 0 ∂ x ∂ y + A 66 ∂ 2 v 0 ∂ x 2 − 2 B 66 ∂ 3 w 0 ∂ x 2 ∂ y + A S 1 66 ∂ 2 ϕ x ∂ x ∂ y + A S 1 66 ∂ 2 ϕ y ∂ x 2 + A S 2 66 ∂ 2 ψ x ∂ x ∂ y + A S 2 66 ∂ 2 ψ y ∂ x 2 + Q 112 R 1 + Q 122 R 2 ∂ ϕ z ∂ y + Q 212 R 1 + Q 222 R 2 ∂ ψ z ∂ y = 0 ,

(10) δ w 0 : B 11 ∂ 3 u 0 ∂ x 3 + 1 R 1 ∂ 2 w 0 ∂ x 2 − D 11 ∂ 4 w 0 ∂ x 4 + B S 1 11 ∂ 3 ϕ x ∂ x 3 + B S 2 11 ∂ 3 ψ x ∂ x 3 + B 12 ∂ 3 v 0 ∂ x 2 ∂ y + 1 R 2 ∂ 2 w 0 ∂ x 2 − D 12 ∂ 3 w 0 ∂ x 2 ∂ y 2 + B S 1 12 ∂ 3 ϕ y ∂ x 2 ∂ y + ∂ 3 ψ y ∂ x 2 ∂ y + J 13 ∂ 3 ϕ z ∂ x 3 + K 13 ∂ 3 ψ z ∂ x 3 + B 12 ∂ 3 u 0 ∂ x ∂ y 2 + 1 R 1 ∂ 2 w 0 ∂ y 2 − D 12 ∂ 3 w 0 ∂ x 2 ∂ y 2 + B S 1 12 ∂ 3 ϕ x ∂ x ∂ y 2 + ∂ 3 ψ x ∂ x ∂ y 2 + B 22 ∂ 3 v 0 ∂ y 3 + 1 R 2 ∂ 2 w 0 ∂ y 2 − D 22 ∂ 4 w 0 ∂ y 4 + B S 1 22 ∂ 3 ϕ y ∂ y 3 + B S 2 22 ∂ 3 ψ y ∂ y 3 + J 23 ∂ 2 ϕ z ∂ y 2 + K 23 ∂ 2 ψ z ∂ y 2 + 2 B 66 ∂ 3 u 0 ∂ x ∂ y 2 + ∂ 3 v 0 ∂ x 2 ∂ y − 4 D 66 ∂ 4 w 0 ∂ x 2 ∂ y 2 + 2 B S 1 66 ∂ 3 ϕ x ∂ x ∂ y 2 + ∂ 3 ϕ y ∂ x 2 ∂ y + 2 B S 2 66 ∂ 3 ψ x ∂ x ∂ y 2 + ∂ 3 ψ y ∂ x 2 ∂ y − A 11 R 1 ∂ u 0 ∂ x + w 0 R 1 + B 11 R 1 ∂ 2 w 0 ∂ x 2 − A S 1 11 R 1 ∂ ϕ x ∂ x − A S 2 11 R 1 ∂ ψ x ∂ x − A 12 R 1 ∂ v 0 ∂ y + w 0 R 2 + B 12 R 1 ∂ 2 w 0 ∂ y 2 − A S 1 12 R 1 ∂ ϕ y ∂ y − A S 2 12 R 1 ∂ ψ y ∂ y − E 13 R 1 ϕ z − F 13 R 1 ψ z − A 12 R 2 ∂ u 0 ∂ x + w 0 R 1 + B 12 R 2 ∂ 2 w 0 ∂ x 2 − A S 1 12 R 2 ∂ ϕ x ∂ x − A S 2 12 R 2 ∂ ψ x ∂ x − A 22 R 2 ∂ v 0 ∂ y + w 0 R 2 + B 22 R 2 ∂ 2 w 0 ∂ y 2 − A S 1 22 R 2 ∂ ϕ y ∂ y − A S 2 22 R 2 ∂ ψ y ∂ y − E 23 R 2 ϕ z − F 23 R 2 ψ z − Q 311 R 1 + Q 312 R 2 ∂ 2 ϕ z ∂ x 2 − Q 411 R 1 + Q 412 R 2 ∂ 2 ψ z ∂ x 2 − Q 312 R 1 + Q 322 R 2 ∂ 2 ϕ z ∂ y 2 − Q 412 R 1 + Q 422 R 2 ∂ 2 ψ z ∂ y 2 + Q 111 R 1 2 + Q 112 R 1 R 2 ϕ z + Q 211 R 1 2 + Q 212 R 1 R 2 ψ z + Q 112 R 1 R 2 + Q 122 R 2 2 ϕ z + Q 212 R 1 R 2 + Q 222 R 2 2 ψ z = − q ,

(11) δ ϕ x : A S 1 11 ∂ 2 u 0 ∂ x 2 + A S 1 11 R 1 ∂ w 0 ∂ x − B S 1 11 ∂ 3 w 0 ∂ x 3 + A S S 1 11 ∂ 2 ϕ x ∂ x 2 + C 11 ∂ 2 ψ x ∂ x 2 + A S 1 12 ∂ 2 v 0 ∂ x ∂ y + A S 1 12 R 2 ∂ w 0 ∂ x − B S 1 12 ∂ 3 w 0 ∂ x ∂ y 2 + A S S 1 12 ∂ 2 ϕ y ∂ x ∂ y + C 12 ∂ 2 ψ y ∂ x ∂ y + L 1 13 ∂ ϕ z ∂ x + L 2 13 ∂ ψ z ∂ x + A S 1 66 ∂ 2 u 0 ∂ y 2 + A S 1 66 ∂ 2 v 0 ∂ x ∂ y − 2 B S 1 66 ∂ 3 w 0 ∂ x ∂ y 2 + A S S 1 66 ∂ 2 ϕ x ∂ y 2 + A S S 1 66 ∂ 2 ϕ y ∂ x ∂ y + C 66 ∂ 2 ψ x ∂ y 2 + C 66 ∂ 2 ψ y ∂ x ∂ y − G 55 ϕ x − I 55 ψ x − G 55 ∂ ϕ z ∂ x − I 55 ∂ ψ z ∂ x + Q 511 R 1 + Q 512 R 2 ϕ z + Q 611 R 1 + Q 612 R 2 ψ z = 0 ,

(12) δ ψ x : A S 2 11 ∂ 2 u 0 ∂ x 2 + A S 2 11 R 1 ∂ w 0 ∂ x − B S 2 11 ∂ 3 w 0 ∂ x 3 + C 11 ∂ 2 ϕ x ∂ x 2 + A S S 2 11 ∂ 2 ψ x ∂ x 2 + A S 2 12 ∂ 2 v 0 ∂ x ∂ y + A S 2 12 R 2 ∂ w 0 ∂ x − B S 2 12 ∂ 3 w 0 ∂ x ∂ y 2 + C 12 ∂ 2 ϕ y ∂ x ∂ y + A S S 2 12 ∂ 2 ψ y ∂ x ∂ y + M 1 13 ∂ ϕ z ∂ x + M 2 13 ∂ ψ z ∂ x + A S 2 66 ∂ 2 u 0 ∂ y 2 + A S 2 66 ∂ 2 v 0 ∂ x ∂ y − 2 B S 2 66 ∂ 3 w 0 ∂ x ∂ y 2 + C 66 ∂ 2 ϕ x ∂ y 2 + C 66 ∂ 2 ϕ y ∂ x ∂ y + A S S 2 66 ∂ 2 ψ x ∂ y 2 + A S S 2 66 ∂ 2 ψ y ∂ x ∂ y − I 55 ϕ x − H 55 ψ x − I 55 ∂ ϕ z ∂ x − H 55 ∂ ψ z ∂ x + Q 711 R 1 + Q 712 R 2 ϕ z + Q 811 R 1 + Q 812 R 2 ψ z = 0 ,

(13) δ ϕ y : A S 1 12 ∂ 2 u 0 ∂ x ∂ y + A S 1 12 R 1 ∂ w 0 ∂ y − B S 1 12 ∂ 3 w 0 ∂ x 2 ∂ y + A S S 1 12 ∂ 2 ϕ x ∂ x ∂ y + C 12 ∂ 2 ψ x ∂ x ∂ y + A S 1 22 ∂ 2 v 0 ∂ y 2 + A S 1 22 R 2 ∂ w 0 ∂ y − B S 1 22 ∂ 3 w 0 ∂ y 3 + A S S 1 22 ∂ 2 ϕ y ∂ y 2 + C 22 ∂ 2 ψ y ∂ y 2 + L 1 23 ∂ ϕ z ∂ y + L 2 23 ∂ ψ z ∂ y + A S 1 66 ∂ 2 u 0 ∂ x ∂ y + A S 1 66 ∂ 2 v 0 ∂ x 2 − 2 B S 1 66 ∂ 3 w 0 ∂ x 2 ∂ y + A S S 1 66 ∂ 2 ϕ x ∂ x ∂ y + A S S 1 66 ∂ 2 ϕ y ∂ x 2 + C 66 ∂ 2 ψ x ∂ x ∂ y + C 66 ∂ 2 ψ y ∂ x 2 − G 44 ϕ y − I 44 ψ y − G 44 ∂ ϕ z ∂ y − I 44 ∂ ψ z ∂ y + Q 512 R 1 + Q 522 R 2 ϕ z + Q 612 R 1 + Q 622 R 2 ψ z = 0 ,

(14) δ ψ y : A S 2 12 ∂ 2 u 0 ∂ x ∂ y + A S 2 12 R 1 ∂ w 0 ∂ y − B S 2 12 ∂ 3 w 0 ∂ x 2 ∂ y + C 12 ∂ 2 ϕ x ∂ x ∂ y + A S S 2 12 ∂ 2 ψ x ∂ x ∂ y + A S 2 22 ∂ 2 v 0 ∂ y 2 + A S 2 22 R 2 ∂ w 0 ∂ y − B S 2 22 ∂ 3 w 0 ∂ y 3 + C 22 ∂ 2 ϕ y ∂ y 2 + A S S 2 22 ∂ 2 ψ y ∂ y 2 + M 1 23 ∂ ϕ z ∂ y + M 2 23 ∂ ψ z ∂ y + A S 2 66 ∂ 2 u 0 ∂ x ∂ y + A S 2 66 ∂ 2 v 0 ∂ x 2 − 2 B S 2 66 ∂ 3 w 0 ∂ x 2 ∂ y + C 66 ∂ 2 ϕ x ∂ x ∂ y + C 66 ∂ 2 ϕ y ∂ x 2 + A S S 2 66 ∂ 2 ψ x ∂ x ∂ y + A S S 2 66 ∂ 2 ψ y ∂ x 2 − I 44 ϕ y − H 44 ψ y − I 44 ∂ ϕ z ∂ y − H 44 ∂ ψ z ∂ y + Q 712 R 1 + Q 422 R 2 ϕ z + Q 812 R 1 + Q 822 R 2 ψ z = 0 ,

(15) δ ϕ z : G 44 ∂ ϕ x ∂ x + I 44 ∂ ψ x ∂ x + G 44 ∂ 2 ϕ z ∂ x 2 + I 44 ∂ 2 ψ z ∂ x 2 + G 55 ∂ ϕ y ∂ y + I 55 ∂ ψ y ∂ y + G 55 ∂ 2 ϕ z ∂ y 2 + I 55 ∂ 2 ψ z ∂ y 2 − E 13 ∂ u 0 ∂ x − E 13 R 1 w 0 + J 13 ∂ 2 w 0 ∂ x 2 − L 1 13 ∂ ϕ x ∂ x − M 1 13 ∂ ψ x ∂ x − E 23 ∂ v 0 ∂ y − E 23 R 2 w 0 + J 23 ∂ 2 w 0 ∂ y 2 − L 1 23 ∂ ϕ y ∂ y − M 1 23 ∂ ψ y ∂ y − N 1 33 ϕ z − N 3 33 ψ z − Q 1 , 613 R 1 + Q 1 , 623 R 2 ϕ z − Q 1 , 813 R 1 + Q 1 , 823 R 2 ψ z − Q 111 R 1 ∂ u 0 ∂ x + w 0 R 1 + Q 311 R 1 ∂ 2 w 0 ∂ x 2 − Q 511 R 1 ∂ ϕ x ∂ x − Q 711 R 1 ∂ ψ x ∂ x − Q 1 , 311 R 1 2 ϕ z − Q 1 , 411 R 1 2 ψ z − Q 112 R 1 ∂ v 0 ∂ y + w 0 R 2 + Q 312 R 1 ∂ 2 w 0 ∂ y 2 − Q 512 R 1 ∂ ϕ y ∂ y − Q 712 R 1 ∂ ψ y ∂ y − Q 1 , 312 R 1 R 2 ϕ z − Q 1 , 412 R 1 R 2 ψ z − Q 1 , 613 R 1 ϕ z − Q 1 , 713 R 1 ψ z − Q 112 R 2 ∂ u 0 ∂ x + w 0 R 1 + Q 312 R 2 ∂ 2 w 0 ∂ x 2 − Q 512 R 2 ∂ ϕ x ∂ x − Q 712 R 2 ∂ ψ x ∂ x − Q 1 , 312 R 1 R 2 ϕ z − Q 1 , 412 R 1 R 2 ψ z − Q 122 R 2 ∂ v 0 ∂ y + w 0 R 2 + Q 322 R 2 ∂ 2 w 0 ∂ y 2 − Q 522 R 2 ∂ ϕ y ∂ y − Q 722 R 2 ∂ ψ y ∂ y − Q 1 , 322 R 2 2 ϕ z − Q 1 , 422 R 2 2 ψ z − Q 1 , 623 R 2 ϕ z − Q 1 , 723 R 2 ψ z = 0 ,

(16) δ ψ z : I 55 ∂ ϕ x ∂ x + H 55 ∂ ψ x ∂ x + I 55 ∂ 2 ϕ z ∂ x 2 + H 55 ∂ 2 ψ z ∂ x 2 + I 44 ∂ ϕ y ∂ y + H 44 ∂ ψ y ∂ y + I 44 ∂ 2 ϕ z ∂ y 2 + H 44 ∂ 2 ψ z ∂ y 2 − F 13 ∂ u 0 ∂ x − F 13 R 1 w 0 + O 13 ∂ 2 w 0 ∂ x 2 − L 2 13 ∂ ϕ x ∂ x − M 2 13 ∂ ψ x ∂ x − F 23 ∂ v 0 ∂ y − F 23 R 2 w 0 + O 23 ∂ 2 w 0 ∂ y 2 − L 2 23 ∂ ϕ y ∂ y − M 2 23 ∂ ψ y ∂ y − N 3 33 ϕ z − N 2 33 ψ z − Q 1 , 713 R 1 + Q 1 , 723 R 2 ϕ z − Q 1 , 913 R 1 + Q 1 , 923 R 2 ψ z − Q 211 R 1 ∂ u 0 ∂ x + w 0 R 1 + Q 411 R 1 ∂ 2 w 0 ∂ x 2 − Q 611 R 1 ∂ ϕ x ∂ x − Q 811 R 1 ∂ ψ x ∂ x − Q 1 , 411 R 1 2 ϕ z − Q 1 , 511 R 1 2 ψ z − Q 212 R 1 ∂ v 0 ∂ y + w 0 R 2 + Q 412 R 1 ∂ 2 w 0 ∂ y 2 − Q 612 R 1 ∂ ϕ y ∂ y − Q 812 R 1 ∂ ψ y ∂ y − Q 1 , 412 R 1 R 2 ϕ z − Q 1 , 512 R 1 R 2 ψ z − Q 1 , 813 R 1 ϕ z − Q 1 , 913 R 1 ψ z − Q 212 R 2 ∂ u 0 ∂ x + w 0 R 1 + Q 412 R 2 ∂ 2 w 0 ∂ x 2 − Q 612 R 2 ∂ ϕ x ∂ x − Q 812 R 2 ∂ ψ x ∂ x − Q 1 , 412 R 1 R 2 ϕ z − Q 1 , 512 R 1 R 2 ψ z − Q 222 R 2 ∂ v 0 ∂ y + w 0 R 2 + Q 422 R 2 ∂ 2 w 0 ∂ y 2 − Q 622 R 2 ∂ ϕ y ∂ y − Q 822 R 2 ∂ ψ y ∂ y − Q 1422 R 2 2 ϕ z − Q 1 , 522 R 2 2 ψ z − Q 1 , 823 R 2 ϕ z − Q 1 , 923 R 2 ψ z = 0 ,

where the stiffness terms are expressed as,

(17) ( A i j , B i j , D i j , A S 1 i j , A S 2 i j , B S 1 i j , B S 2 i j ) = ∫ − h / 2 h / 2 Q i j ( z ) [ 1 , z , z 2 , f 1 ( z ) , f 2 ( z ) , z f 1 ( z ) , z f 2 ( z ) ] d z ( Q 1 i j , Q 2 i j , Q 3 i j , Q 4 i j ) = ∫ − h / 2 h / 2 Q i j ( z ) [ f 1 ' ( z ) , f 2 ' ( z ) , z f 1 ' ( z ) , z f 2 ' ( z ) ] d z ( A S S 1 i j , A S S 2 i j , C i j ) = ∫ − h / 2 h / 2 Q i j ( z ) { [ f 1 ( z ) ] 2 , [ f 2 ( z ) ] 2 , [ f 1 ( z ) f 2 ( z ) ] } d z ( Q 5 i j , Q 6 i j , Q 7 i j , Q 8 i j ) = ∫ − h / 2 h / 2 Q i j ( z ) { f 1 ( z ) [ f 1 ' ( z ) , f 2 ' ( z ) ] , f 2 ( z ) [ f 1 ' ( z ) , f 2 ' ( z ) ] } d z ( Q 13 i j , Q 15 i j , Q 14 i j ) = ∫ − h / 2 h / 2 Q i j ( z ) { [ f 1 ' ( z ) ] 2 , [ f 2 ' ( z ) ] 2 , [ f 1 ' ( z ) f 2 ' ( z ) ] } d z ( G i j , H i j , I i j ) = ∫ − h / 2 h / 2 Q i j ( z ) { [ f 1 ' ( z ) ] 2 , [ f 2 ' ( z ) ] 2 , [ f 1 ' ( z ) f 2 ' ( z ) ] } d z ( E i j , F i j , N 1 i j , N 2 i j , N 3 i j ) = ∫ − h / 2 h / 2 Q i j ( z ) { [ f 1 " ( z ) ] , [ f 2 " ( z ) ] , [ f 1 " ( z ) ] 2 , [ f 2 " ( z ) ] 2 , [ f 1 " ( z ) f 2 ' ( z ) ] } d z ( Q 16 i j , Q 17 i j , Q 18 i j , Q 19 i j ) = ∫ − h / 2 h / 2 Q i j ( z ) { f 1 ' ( z ) [ f 1 " ( z ) , f 2 " ( z ) ] , f 2 ( z ) [ f 1 " ( z ) , f 2 " ( z ) ] , } d z ( J i j , L 1 i j , L 2 i j ) = ∫ − h / 2 h / 2 Q i j ( z ) { f 1 " ( z ) [ z , f 1 ( z ) , f 2 ( z ) ] } d z ( O i j , M 1 i j , M 2 i j ) = ∫ − h / 2 h / 2 Q i j ( z ) { f 2 " ( z ) [ z , f 1 ( z ) , f 2 ( z ) ] } d z .

Based on the mathematical formulation, the associated boundary conditions for the present study are written as follows;

x = 0 and x = a,

(18) u 0 = 0 or N x = 0 ; v 0 = 0 or N x y = 0 w 0 = 0 or M x b = 0; ∂ w 0 ∂ x = 0 or M x y b = 0 ϕ x = 0 or M x s 1 = 0 ; ψ x = 0 or M x s 2 = 0 ϕ y = 0 or M x y s 1 = 0; ψ y = 0 or M x y s 2 = 0 ϕ z = 0 or Q x S 1 = 0; ψ z = 0 or Q x z S 2 = 0,

y = 0 and y = b,

(19) u o = 0 or N x y = 0 ; v 0 = 0 or N y = 0 ; w 0 = 0 or M y b = 0 ; ∂ w 0 ∂ y = 0 or M x y b = 0 ; ϕ x = 0 or M x y s 1 = 0 ; ψ x = 0 or M x y s 2 = 0 ; ϕ y = 0 or M y s 1 = 0 ; ψ y = 0 or M y s 2 = 0 ; ϕ z = 0 or Q y S 1 = 0 ; ψ z = 0 or Q y z S 2 = 0 ;

3.2 Navier-type solution

Navier’s double trigonometric series assumptions are used to find solutions for static analysis for the simply supported shell subjected to static transverse load.

(20) u 0 ϕ x ψ x = ∑ m = 1 , 3 , 5. . ∞ ∑ n = 1 , 3 , 5. . ∞ u m n ϕ x m n ψ x m n cos α x sin β y v 0 ϕ y ψ y = ∑ m = 1 , 3 , 5. . ∞ ∑ n = 1 , 3 , 5. . ∞ u m n ϕ y m n ψ y m n sin α x cos β y w 0 ϕ z ψ y = ∑ m = 1 , 3 , 5. . ∞ ∑ n = 1 , 3 , 5. . ∞ w m n ϕ z m n ψ z m n sin α x sin β y ,

where u m n , v m n , w m n , ϕ x m n , ψ x m n , ϕ y m n , ψ y m n , ϕ z m n , ψ z m n are the unknown coefficients to be determined. α = m π / a , β = n π / b . Similarly, the double trigonometric series for transverse load are also presented as follows:

(21) q ( x , y ) = ∑ m = 1 ∞ ∑ n = 1 ∞ q m n sin α x sin β y ,

where q _mn is the unknown Fourier coefficient for transverse load which is taken as q m n = q 0 ( m = m = 1 ) for sinusoidal load (SL) and q m n = 16 q 0 m n π 2 ( m , n = 1 , 3 , 5 , … ∞ ) for uniform load (UL). Where q ₀ is the maximum intensity of the load. Substituting Eqs. (20) and (21) into the Eqs. (8)–(17), one can get the following equation:

(22) [ K ] { Δ } = { f } ,

where [K] is the stiffness matrix, { Δ } is the vector of unknowns, and { f } is the force vector. Appendix A shows the elements of these matrices.

4 Numerical results and discussion

This article presents and compares the numerical findings from the static analysis of doubly curved FG shells that are spherical, cylindrical, hyperbolic, and elliptical. Wherever possible, the current results are contrasted with those found in the existing literature. A few other findings on deflection and stresses in hyperbolic and elliptic shells are also included in the current work. For metal and ceramic, the modulus of elasticity and density values taken into consideration are as follows:

E _m = 70 GPa, E _c = 380 GPa, µ = 0.3

The following non-dimensional forms are used to present the numerical results:

(23) w ¯ a 2 , b 2 , 0 = 1 , 000 h 3 E 0 q 0 a 4 w , w ˆ a 2 , b 2 , 0 = 1 1 , 000 w h , σ ¯ x a 2 , b 2 , − h 2 = h q 0 a σ x , τ ¯ x z 0 , b 2 , z h = h q 0 a τ x z , E 0 = 1.0 .

5 Discussion

Table 1 displays the transverse displacement for a single-layer FG cylindrical shell under uniform load (UL) for different R/h ratios (R/h = 50, 100, 200) and power law indexes (p = 0.5, 1, 2). To ensure correctness, the current results are compared with those of Huan et al. [23], Zhao et al. [22], and Viola et al. [33]. It is discovered that the current results and the results obtained in the literature agree fairly well. Additionally, Table 1 shows that transverse displacement increases along with increases in the R/h ratio and power law index. Table 2 displays the displacements and stresses in an FG cylindrical shell under SL and UL at various values of (p = 1, 2, 4, 8) and R/a ratios (R/a = 1, 2, 5, 10, 20, 50, and 100). Table 2 shows that up to a ratio of 20, transverse displacement and stresses rise with increasing R/a ratio; after that, they decrease with increasing R/a ratio.

Table 1

Transverse displacement ( w ˆ ) in an FG cylindrical shell under UL (θ = 0.2, b = 1, R ₁ = R, R ₂ = ∞, a = Rθ)

R/h	Models	p
R/h	Models	0.5	1	2
50	Present	0.003799	0.004249	0.004653
	Huan et al. [23]	0.003824	0.004296	0.004702
	Zhao et al. [22]	0.003824	0.004279	0.004693
	Viola et al. [33] (GFSDT)	0.003871	0.004331	0.004739
	Viola et al. [33] (UTSDT)	0.003858	0.004318	0.004729
100	Present	0.054080	0.060620	0.066500
	Huan et al. [23]	0.054410	0.060910	0.066790
	Zhao et al. [22]	0.054250	0.060720	0.066780
200	Present	0.638600	0.718900	0.797300
	Huan et al. [23]	0.650100	0.728100	0.805600
	Zhao et al. [22]	0.650300	0.728300	0.805700
	Viola et al. [33] (GFSDT)	0.654700	0.729900	0.807600
	Viola et al. [33] (UTSDT)	0.651300	0.729400	0.807000

Table 2

Displacement and stresses for different R/a ratios and power law indexes in an FG cylindrical shell (R ₁ = R, R ₂ = ∞), (a/h = 10, a = 1, b = 1)

p	R/a	SL			UL
		w ¯	σ ¯ x	τ ¯ x z	w ¯	σ ¯ x	τ ¯ x z
1	1	0.8291	2.4284	0.1372	1.2822	3.3407	0.3369
	2	1.2687	3.1579	0.2155	1.9921	4.5501	0.4614
	5	1.4710	3.2681	0.2533	2.3194	4.7546	0.5221
	10	1.4981	3.1938	0.2591	2.3632	4.6442	0.5314
	20	1.5018	3.1342	0.2603	2.3693	4.5528	0.5333
	50	1.5008	3.0916	0.2605	2.3677	4.4869	0.5336
	100	1.4998	3.0763	0.2605	2.3663	4.4631	0.5335
2	1	1.0745	2.9665	0.1477	1.6620	4.1074	0.3648
	2	1.6272	3.7568	0.2303	2.5547	5.4268	0.4963
	5	1.8729	3.8125	0.2693	2.9521	5.5515	0.5589
	10	1.9034	3.7008	0.2751	3.0017	5.3836	0.5683
	20	1.9065	3.6196	0.2763	3.0067	2.2587	0.5701
	50	1.9044	3.5632	0.2764	3.0033	5.1714	0.5703
	100	1.9030	3.5432	0.2763	3.0011	5.1403	0.5702
4	1	1.3194	3.5655	0.1505	2.0411	4.9717	0.3633
	2	1.9324	4.3497	0.2264	3.0319	6.2925	0.4841
	5	2.1905	4.3345	0.2604	3.4494	6.3102	0.5387
	10	2.215	4.1937	0.2653	3.4995	6.0979	0.5466
	20	2.2242	4.0977	0.2634	3.5040	5.9504	0.5481
	50	2.2217	4.0324	0.2664	3.5000	5.8495	0.5483
	100	2.2202	4.0094	0.2663	3.4976	5.8139	0.5482
8	1	1.5374	4.2904	0.1308	2.3805	6.0019	0.3057
	2	2.1874	5.1450	0.1898	3.4300	7.4445	0.3994
	5	2.4536	5.1223	0.2152	3.8604	7.4558	04403
	10	2.4867	4.9718	0.2189	3.9140	7.2296	0.4461
	20	2.4905	4.8694	0.2196	3.9203	7.0729	0.4472
	50	2.4887	4.7998	0.2196	3.9174	6.9657	0.4473
	100	2.4874	4.7753	0.2196	3.9153	6.9278	0.4473

Table 3 displays the displacements and stresses in a FG spherical shell under SL and UL for various values of (p = 1, 2, 4, 8) and R/a ratios (R/a = 1, 2, 5, 10, 20, 50, 100). As the R/a ratio increases, Table 3 shows that transverse displacement increases up to R/a = 50 and stresses increase up to R/a = 5, while shear stresses increase as the R/a ratio increases.

Table 3

Displacement and stresses for different R/a ratios and power law indexes in an FG spherical shell (R ₁ = R, R ₂ = R), (a/h = 10, a = 1, b = 1)

p	R/a	SL			UL
p	R/a	w ¯	σ ¯ x	τ ¯ x z	w ¯	σ ¯ x	τ ¯ x z
1	1	0.3424	1.2882	0.0532	0.4981	1.5389	0.1910
	2	0.8291	2.4198	0.1372	1.2806	3.3595	0.3317
	5	1.3510	3.2379	0.2305	2.1242	4.7033	0.4848
	10	1.4710	3.2656	0.2533	2.3193	4.7585	0.5220
	20	1.4981	3.1926	0.2544	2.3632	4.6462	0.5314
	50	1.5017	3.1201	0.2604	2.3692	4.5325	0.5335
	100	1.5008	3.0913	0.2605	2.3677	4.4873	0.5336
2	1	0.4473	1.6240	0.0574	0.6515	1.9794	0.2070
	2	1.0744	2.9563	0.1476	1.6599	4.1329	0.3595
	5	1.7284	3.8280	0.2459	2.7180	5.5727	0.5206
	10	1.8728	3.8097	0.2693	2.9520	5.5571	0.5588
	20	1.9034	3.6994	0.2751	3.0017	5.3864	0.5682
	50	1.9062	3.6009	0.2763	3.0061	5.2317	0.5702
	100	1.9044	3.5629	0.2764	3.0033	5.1720	0.5703
4	1	0.5703	2.0399	0.0611	0.8346	2.5479	0.2081
	2	1.3193	3.5543	0.1504	2.0395	5.0056	0.3585
	5	2.0400	4.4012	0.2401	3.2054	6.4135	0.5055
	10	2.1905	4.3315	0.2604	3.4492	6.3178	0.5386
	20	2.2215	4.1923	0.2653	3.4995	6.1017	0.5466
	50	2.2237	4.0760	0.2664	3.5033	5.9193	0.5483
	100	2.2217	4.0321	0.2664	3.5000	5.8503	0.5483
8	1	0.6904	2.5139	0.0560	1.0151	3.1793	0.1771
	2	1.5373	4.2778	0.1308	2.3779	6.0434	0.3021
	5	2.2984	5.1971	0.2002	3.6090	7.5726	0.4157
	10	2.4536	5.1190	0.2152	3.8603	7.4649	0.4402
	20	2.4867	4.9701	0.2189	3.9140	7.2342	0.4461
	50	2.4903	4.8462	0.2196	3.9199	7.0400	0.4473
	100	2.4887	4.7995	0.2196	3.9174	6.9666	0.4473

Table 4 displays the displacements and stresses in an FG hyperbolic shell under SL and UL for various values of (p = 1, 2, 4, 8) and R/a ratio values (R/a = 1, 2, 5, 10, 20, 50, 100). Table 4 demonstrates how displacement, normal stress, and shear stress increase with increasing R/a ratio up to R/a = 5 and then drop further.

Table 4

Displacement and stresses for different R/a ratios and power law indexes in an FG hyperbolic shell (R ₁ = R, R ₂ = −R), (a/h = 10, a = 1, b = 1)

p	R/a	SL			UL
		w ¯	σ ¯ x	τ ¯ x z	w ¯	σ ¯ x	τ ¯ x z
1	1	0.1888	0.4228	0.0346	0.4015	1.6086	0.0786
	2	1.2212	2.5595	0.2153	2.0507	4.7691	0.2311
	5	2.3275	4.7774	0.4058	3.7071	7.1977	0.7726
	10	1.6452	3.3651	0.2864	2.6016	4.9225	0.5785
	20	1.5329	3.1320	0.2667	2.4197	4.5501	0.5466
	50	1.5042	3.0720	0.2616	2.3731	4.4555	0.5384
	100	1.5001	3.0634	0.2609	2.3666	4.4426	0.5373
2	1	0.2452	0.4889	0.0373	0.5208	1.8709	0.0626
	2	1.6088	3.0437	0.2366	2.6995	5.6533	0.2594
	5	2.9198	5.4328	0.4252	4.6483	8.1753	0.8100
	10	2.0829	3.8647	0.3029	3.2922	5.6505	0.6113
	20	1.9437	3.6031	0.2858	3.0669	5.2332	0.5783
	50	1.9080	3.5355	0.2773	3.0091	5.1272	0.5698
	100	1.9030	3.5259	0.2766	3.0010	5.1127	0.5686
4	1	0.3273	0.6215	0.0403	0.6754	2.2032	0.0663
	2	2.3599	4.3089	0.2826	3.9358	7.8011	0.3416
	5	3.2160	5.7983	0.3820	5.1106	8.6808	0.7327
	10	2.4030	4.3262	0.2853	0.3796	6.3142	0.5756
	20	2.2605	4.0662	0.2683	3.5656	5.9014	0.5480
	50	2.2252	3.9989	0.2639	3.5057	5.7959	0.5409
	100	2.2201	3.9892	0.2633	3.4974	5.7816	0.5399
8	1	0.4222	0.8424	0.0386	0.8455	2.7232	0.0447
	2	3.4615	6.7001	0.3086	5.7373	11.8202	0.4064
	5	3.4274	6.5687	0.3032	5.4379	9.7972	0.5830
	10	2.6691	5.1060	0.2358	4.2059	7.4486	0.4736
	20	2.5229	4.8353	0.2234	3.9830	7.0190	0.4534
	50	2.4926	4.7640	0.2201	3.9238	6.9083	0.4481
	100	2.4875	4.7538	0.2197	3.9154	6.8933	0.4474

Table 5 displays, under a sinusoidal and uniformly distributed load, the transverse deflection, inplane stresses, and shear stresses in a single-layer elliptic shell for various values of (p = 1, 2, 4, 8) and R/a ratios (R/a = 1, 2, 5, 10, 20, 50, 100). Table 5 demonstrates how displacement and normal stress increase with increasing R/a ratio up to R/a = 5 and then drop further, while shear stresses increase as the R/a ratio increases. The distributions of in-plane and transverse shear stresses along the shell thickness at (R/a = 5, p = 5, a/h = 10) are presented in Figures 2–5. From there, it can be noted that the hyperbolic shells have higher values of both stresses than the spherical, cylindrical, and elliptical shells.

Table 5

Displacement and stresses for different R/a ratios and power law indexes in an FG elliptical shell (R ₁ = R, R ₂ = −1.5R), (a/h = 10, a = 1, b = 1)

p	R/a	SL			UL
		w ¯	σ ¯ x	τ ¯ x z	w ¯	σ ¯ x	τ ¯ x z
1	1	0.4952	2.0879	0.0759	0.7419	2.7971	0.2244
	2	1.0278	3.2287	0.1688	1.6017	4.6594	0.3846
	5	1.4298	3.5522	0.2433	2.2524	5.2096	0.5083
	10	1.4965	3.3877	0.2575	2.3607	4.9550	0.5317
	20	1.5062	3.2427	0.2606	2.3764	4.7268	0.5368
	50	1.5038	3.1374	0.2610	2.3725	4.5604	0.5374
	100	1.5016	3.0996	0.2609	2.3691	4.5005	0.5373
2	1	0.6537	2.6690	0.0820	0.9810	3.6245	0.2398
	2	1.3387	3.9838	0.1818	2.0868	5.7835	0.4116
	5	1.8316	4.2197	0.2594	2.8849	6.2033	0.5404
	10	1.9066	3.9631	0.2232	3.0066	5.8039	0.5638
	20	1.9144	3.7633	0.2765	3.0194	5.4890	0.5658
	50	1.9075	3.6233	0.2768	3.0108	5.2677	0.5690
	100	1.9056	3.5736	0.2766	3.0052	5.1891	0.5687
4	1	0.8222	3.3162	0.0855	1.2392	4.5618	0.2396
	2	1.6195	4.7250	0.1805	2.5243	6.8864	0.4026
	5	2.1496	4.8290	0.2489	3.3828	7.1022	0.5163
	10	2.2260	4.5006	0.2609	3.5067	6.5898	0.5360
	20	2.2331	4.2637	0.2632	3.5183	6.2165	0.5399
	50	2.2268	4.1014	0.2634	3.5082	5.9601	0.5402
	100	2.2230	4.0443	0.2633	3.5021	5.8698	0.5399
8	1	0.9741	3.9776	0.0781	1.4690	5.4947	0.2086
	2	1.8483	5.5473	0.1561	2.8802	8.0839	0.3421
	5	2.4050	5.6471	0.2088	3.7814	8.2971	0.4295
	10	2.4875	5.2975	0.2178	3.9151	7.7520	0.4447
	20	2.4975	5.0459	0.2196	3.9316	7.3560	0.4474
	50	2.4931	4.8733	0.2198	3.9245	7.0835	0.4476
	100	2.4899	4.8125	0.2197	3.9193	6.9874	0.4475

Figure 2

Inplane normal stress distributions in thickness direction (a/h = 10) under SL.

Figure 3

Inplane normal stress distributions in thickness direction (a/h = 10) under UL.

Figure 4

Transverse shear stress distributions in thickness direction (a/h = 10) under SL.

Figure 5

Transverse shear stress distributions in thickness direction (a/h = 10) under the UL.

6 Conclusions

A HOSNDT is developed in this study for the static analysis of FG shells, in which the transverse shear deformation impact is taken into consideration using the parabolic function. The non-dimensional displacements and stresses are derived for various materials in spherical, cylindrical, hyperbolic, and elliptic shells subjected to sinusoidal and uniformly distributed forces. To validate the accuracy of the present study current transverse displacement results in cylindrical shells are compared with those found in the literature to forecast the accuracy of the current theory. Displacement and stresses in spherical, hyperbolic, and elliptic shells are anticipated based on the cylindrical shell results; this is a significant contribution of the current study and can be used as a benchmark for further research.

Also, based on the present study, as a concluding remark it is found that as the power law index increases, transverse displacement also tends to increase, indicating that greater material gradation results in a more flexible shell structure. Also, the variation of displacement with R/h and R/a ratio suggests that as the shell becomes thinner (larger R/h or R/a), it becomes more flexible and exhibits greater displacement under load. Hyperbolic shells exhibit higher stress values compared to spherical, cylindrical, and elliptical shells, indicating that they are more prone to stress concentrations. This highlights the importance of careful design considerations when working with hyperbolic shells to prevent failure.

Funding information: Authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. BMS contributed to the analytical formulations, generated the numerical results, and drafted the manuscript. ASS has contributed to reviewing the results and the manuscript. NSK has contributed to the analytical part, finalization of the methodology, and reviewing the manuscript.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: The authors declare that the data supporting the findings of this study are available within the article.

Appendix A A The elements of [K] – stiffness matrix, {Δ} – vector of unknowns, and {f} – the force vector are as follows:

K 11 = − A 11 α 2 − A 66 β 2 , K 12 = K 21 = − A 12 α β − A 66 α β , K 13 = K 31 = A 11 R 1 α + A 12 R R β + B 11 α 3 + B 12 α β 2 + 2 B 66 α β 2 , K 14 = K 41 = − A S 1 11 α 2 − A S 2 66 β 2 , K 15 = K 51 = − A S 2 11 α 2 − A S 2 66 β 2 , K 16 = K 61 = − A S 1 12 α β − A S 1 66 α β , K 17 = K 71 = − A S 2 12 α β − A S 2 66 α β ,

K 18 = K 81 = E 13 + Q 111 R 1 + Q 112 R 2 α , K 19 = K 91 = F 13 + Q 211 R 1 + Q 212 R 2 α , K 22 = − A 22 β 2 − A 66 α 2 , K 23 = K 32 = A 12 R 1 + A 22 R 2 β + B 12 α 2 β + B 22 β 3 + 2 B 66 α 2 β , K 24 = K 42 = − A S 1 12 α β − A S 1 66 α β , K 25 = K 52 = − A S 2 12 α β − A S 2 66 α β ,

K 26 = K 62 = − A S 1 22 β 2 − A S 1 66 α 2 , K 27 = K 72 = − A S 2 22 β 2 − A S 2 66 α 2 , K 28 = K 82 = E 23 − Q 112 R 1 − Q 122 R 2 β , K 29 = K 92 = F 23 − Q 212 R 1 − Q 222 R 2 β K 33 = − D 11 α 4 + D 22 β 4 − 2 α 2 β 2 D 12 + 2 D 66 − 2 α 2 B 11 R 1 + B 12 R 2 − 2 β 2 B 12 R 1 + B 22 R 2 − A 11 R 1 2 + A 22 R 2 2 + 2 A 12 R 1 R 2 ,

K 34 = K 43 = B S 1 11 α 3 + B S 1 12 α β 2 + 2 B S 1 66 α β 2 + A S 1 11 R 1 α + A S 1 12 R 2 α , K 35 = K 53 = B S 2 11 α 3 + B S 2 12 α β 2 + 2 B S 2 66 α β 2 + A S 2 11 R 1 α + A S 2 12 R 2 α , K 36 = K 63 = B S 1 12 α 2 β + B S 1 22 β 3 + 2 B S 1 66 α 2 β + A S 1 12 R 1 β + A S 1 22 R 2 β , K 37 = K 73 = B S 2 12 α 2 β + B S 2 22 β 3 + 2 B S 2 66 α 2 β + A S 2 12 R 1 β + A S 2 22 R 2 β ,

K 38 = K 83 = − J 13 α 2 − J 23 β 2 − E 13 R 1 − E 23 R 2 − Q 311 R 1 + Q 312 R 2 α 2 − Q 312 R 1 + Q 322 R 2 β 2 − Q 111 R 1 2 + Q 112 R 1 R 2 − Q 112 R 1 R 2 + Q 122 R 2 2 , K 39 = K 93 = − O 13 α 2 − O 23 β 2 − F 13 R 1 − F 23 R 2 − Q 411 R 1 + Q 412 R 2 α 2 − Q 412 R 1 + Q 422 R 2 β 2 − Q 211 R 1 2 + Q 212 R 1 R 2 − Q 212 R 1 R 2 + Q 222 R 2 2 ,

(A.1) K 44 = − A S S 1 11 α 2 − A S S 1 66 β 2 − G 44 , K 45 = K 54 = − C 11 α 2 − C 66 β 2 − I 44 , K 46 = K 64 = − A S S 1 12 α β − A S S 1 66 α β , K 47 = K 74 = − C 12 α β − C 66 α β , K 48 = K 84 = L 1 13 α − G 44 α − Q 511 R 1 − Q 512 R 2 , K 49 = K 94 = L 2 13 α − I 44 α − Q 611 R 1 − Q 612 R 2 , K 55 = − A S S 2 11 α 2 − A S S 2 66 β 2 − H 44 , K 56 = K 65 = − C 12 α β − C 66 α β , K 57 = K 75 = − A S 2 12 α β − A S S 2 66 α β , K 58 = K 85 = − I 44 α + M 1 13 α + Q 711 R 1 + Q 712 R 2 ,

K 59 = K 95 = − H 44 α + M 2 13 α + Q 811 R 1 + Q 812 R 2 , K 66 = − A S S 1 22 β 2 − A S S 1 66 α 2 − G 55 , K 67 = K 76 = − C 22 β 2 − C 66 α 2 − I 55 , K 68 = K 86 = L 1 23 β − G 55 β + Q 512 R 1 + Q 522 R 2 , K 69 = K 96 = L 2 23 β − I 55 β + Q 612 R 1 + Q 622 R 2 , K 77 = − A S S 2 22 β 2 − A S S 2 66 α 2 − H 55 , K 78 = K 87 = M 1 23 β − I 55 β + Q 712 R 1 + Q 722 R 2 , K 79 = K 97 = M 2 23 β − H 55 β + Q 812 R 1 + Q 822 R 2 ,

K 88 = − G 44 α 2 − G 55 β 2 − N 1 33 − 2 Q 1 , 613 R 1 − 2 Q 1 , 623 R 2 − Q 1 , 311 R 1 2 − 2 Q 1 , 312 R 1 R 2 − Q 1 , 322 R 2 2 , K 89 = K 98 = − I 44 α 2 − I 55 β 2 − N 3 33 − Q 1 , 613 R 1 − 2 Q 1 , 623 R 2 − Q 1 , 311 R 1 2 − 2 Q 1 , 312 R 1 R 2 − Q 1 , 322 R 2 2 , K 99 = − L 1 44 α 2 − H 55 β 2 − N 2 33 ,

(A.2) F = { 0 , 0 , − q 0 , 0 , 0 , 0 , 0 , 0 , 0 } T ,

(A.3) Δ = { u m n , v m n , w m n , ϕ x m n , ψ x m n , ϕ y m n , ψ y m n , ϕ z m n , ψ z m n } T .

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Received: 2024-02-26

Revised: 2024-08-31

Accepted: 2024-09-21

Published Online: 2024-10-18

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/cls-2024-0014

Keywords for this article

FG shell; spherical shells; hyperbolic shells; cylindrical shells; elliptic shells; shear and normal deformation theory

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