1 Introduction

In recent decades, functionally graded material (FGM) stiffened shells are used more and more widely in modern engineering structures, such as tunnels, pipelines, pressure vessels, and storage tanks. These structures are usually laid on or placed in a soil medium as an elastic foundation^[1]. Thus, their nonlinear stability analysis is an important problem and has received considerable interest by researchers.

Many researches focused on the buckling and postbuckling analysis of un-stiffened shells. Sheng and Wang^[2] considered the effects of the thermal load on the buckling, vibration, and dynamic buckling of the FGM cylindrical un-stiffened shells embedded in a linear elastic medium based on the first-order shear deformation theory (FSDT), taking into account the rotary inertia and transverse shear strains. Shen et al.^[3] studied the postbuckling analysis of the tensionless Pasternak FGM un-stiffened cylindrical shells surrounded by an elastic medium under the internal pressure, using the singular perturbation technique and the higher-order shear deformation shell theory (HDST). Sofiyev and Avcar^[4], Sofiyev^[5], and Sofiyev and Kuruoglu^[6] analyzed the effects of the stability and vibration of the FGM un-stiffened cylindrical shells resting on the Pasternak elastic foundation. Bagherizadeh et al.^[7] investigated the mechanical buckling of the FGM un-stiffened cylindrical shells surrounded by the Pasternak elastic foundation. The equilibrium and stability equations were obtained based on the higher-order shear deformation shell theory. Shen^[8] presented the postbuckling analysis of the pressure-loaded functionally graded cylindrical shells without stiffeners based on the classical shell theory with the von Karman-Donnell-type of kinetic nonlinearity. Huang and Han^[9] presented the nonlinear dynamic buckling of the functionally graded cylindrical shells subjected to the time dependent axial load. Bahadori and Najafizadeh^[10] showed a free vibration analysis of the two-dimensional functionally graded axisymmetric cylindrical shell on the Winkler-Pasternak elastic foundation by the first-order shear deformation theory and the Navier-differential quadrature solution methods. Li and Qiao^[11] used a singular perturbation technique to determine the interactive buckling loads and postbuckling equilibrium paths of the anisotropic laminated cylindrical shells under the combined external pressure and axial compression in thermal environments. Shen and Wang^[12] considered the nonlinear bending and postbuckling of the FGM cylindrical panels subjected to the combined loadings and resting on the elastic foundations in thermal environments based on a singular perturbation technique along with a two-step perturbation approach and the higher-order shear deformation theory. Sofiyev^[13] investigated the effects of the shear stresses on the dynamic instability of the exponentially graded sandwich cylindrical shells by using the shear deformation theory and classical shell theory. Sofiyev^[14] used the first-order shear deformation shell theory to determine the buckling of the heterogeneous orthotropic composite conical shells under the external pressures within the shear deformation theory. Besides, Basing on the first-order shear deformation theory, Sofiyev^[15] presented the closed-form solutions of the thermoelastic stability problem of the un-stiffened functionally graded conical shells subjected to the thermal loading.

Many studies have been carried out with the eccentrically stiffened shells made of homogenous materials. Baruch and Singer^[16] showed the eccentricity effects of stiffeners on the general instability of the stiffened cylindrical shells under hydrostatic pressure, and concluded that the behavior of the eccentricity effect depended very strongly on the geometry of the shell. Ji and Yeh^[17] presented the general solution for the nonlinear buckling of the non-homogeneous axial symmetric ring-stiffened and stringer-stiffened cylindrical shells based on the Donnell equations and the perturbation technique. Reddy and Starnes^[18] studied the buckling of the circumferentially or axially stiffened laminated cylindrical shells subjected to a simply supported end condition by using the layerwise theory and the smeared stiffener approach. Shen et al.^[19] investigated the buckling and postbuckling behavior of the perfect and imperfect stiffened cylindrical shells under combined external pressure and axial compression by using the boundary layer theory, using the singular perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. Tian et al.^[20] extended the Ritz for solving the buckling problem of the ring-stiffened cylindrical shells under pressure loading. Sadeghifar et al.^[21] investigated the buckling of the stringer-stiffened laminated cylindrical shells with non-uniform eccentricity based on Love's first-order shear deformation theory, and calculated the critical loads by using the Rayleigh-Ritz procedure. Stamatelos et al.^[22] presented the results on the local buckling and postbuckling behavior of the isotropic and orthotropic stiffened panels based on the classical lamination plate theory and two-dimensional Ritz displacement function for arbitrary edge supports. Bich et al.^[23-24] studied the nonlinear static and dynamical buckling analysis of the imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression. Dung and Nam^[25] analyzed the nonlinear dynamic behavior of the eccentrically stiffened functionally graded circular cylindrical thin shells under the external pressure and surrounded by an elastic medium. Duc and Quan^[26] analyzed the nonlinear dynamic behavior of the imperfect FGM double curved thin shallow shells with the temperature-dependent properties on the elastic foundation. Najafizadeh et al.^[27] investigated the mechanical buckling behavior of the FGM stiffened cylindrical shells reinforced by rings and stringers based on the classical shell theory (CST), where the stability equation was given in terms of the displacement and the stiffeners and skin were assumed to be made of an FGM whose properties varied continuously through the thickness. By using the classical shell theory, Dung and Hoa^[28] investigated the nonlinear torsional buckling and postbuckling of the eccentrically stiffened FGM cylindrical shells in thermal environments, and assumed that the material properties of the shell and stiffeners were continuously graded in the thickness direction. Dung and Hoa^[29] presented a semi-analytical method for analyzing the nonlinear dynamic behavior of the FGM cylindrical shells surrounded by an elastic medium under the time-dependent torsional loads based on the classical shell theory with the deflection function correctly represented by three terms.

A review of the literature indicates that no study has been presented for the analytical solution of the eccentrically stiffened functionally graded FGM cylindrical shells reinforced by the FGM stiffener system and filled inside by an elastic foundation and in thermal environments based on the higher-order shear deformation shell theory. The present paper attempts to solve the just mentioned problem by an analytical method. The theoretical formulations in terms of the displacement components according to Reddy's third-order shear deformation shell theory (TSDT)^[1] and the smeared stiffeners technique are derived. The thermal elements of the shells and stiffeners are taken into account in two cases, i.e., uniform temperature rise law and nonlinear temperature change. The closed form expression for determining the critical buckling load and postbuckling load-deflection curves are obtained by the Galerkin method. The numerical results are carried out to analyze the effects of the stiffener, the foundation, the material and dimensional parameters, and the pre-existent axial compressive and thermal load on the stability of the stiffened FGM shells.

2 Fundamental equations of eccentrically stiffened FGM shells 2.1 FGM shells

Consider a thin circular cylindrical shell made of cremic and metal with the mean radius R, the thickness h, and the length L subjected to the axial compressive load P, the external uniform pressure q, and the thermal load. Assume that the shell is simply supported at two butt-ends. The middle surface of the shell is referred to the coordinates x, y, and z (see Fig. 1). Assume that the shell is stiffened by closely spaced circular rings and longitudinal stringers. To guarantee the continuity between the stiffeners and the shell, the stiffeners are taken to be pure-metal if they are located at the metal-rich side, while are pure-ceramic if they are located at the ceramic-rich side.

The FGM of the shell in this paper is assumed to be made of a mixture of ceramic and metal. The volume-fraction obeys the following power law:

(1)

where the volume fraction index k≥0 indicates the material variation profile through the FGM shell thickness, and h is the thickness of the shell. z is the thickness coordinate, and varies from -h/2 to h/2. The subscripts m and c refer to the metal and ceramic constituents, respectively.

The effective properties of the FGM shell Pr_eff are determined by the linear rule of the mixture as follows:

(2)

where Pr_m and Pr_c are the temperature-independent material properties of the metal and ceramic constituent, respectively.

According to the mentioned law, the Young moduli E, the thermal expansion coefficient α, and the thermal conductivity coefficient K of the FGM shell can be expressed as follows:

(3)

and the Young moduli and the thermal expansion coefficients of the FGM stiffeners are given by

(4)

where E_s(z), E_r(z); α_s(z), α_r(z), and K_s(z), K_r(z) are the Young moduli, the thermal expansion coefficients, and the thermal conductivity coefficients of the x-direction and the y-direction respectively. k₂ and k₃ are the volume fractions indexes of the stringer and ring, respectively. Since k₂=k₃=1/k, k₂ → ∞ and k₃ → ∞ lead to a homogeneous stiffener. Therefore, the continuity between the shell and the stiffeners is satisfied.

Poisson's ratio ν is assumed to be constant, i.e., ν(z)=ν=const.

2.2 Constitutive equations

According to the third-order shear deformation theory with the von Karman geometrical nonlinearity, the strain components of the shell at a distance z from the middle surface are^[1]

(5)

where

(6)

In the above equations, u=u(x, y), and v=v(x, y), and w=w(x, y) denote the displacements of the middle surface points along the x-, y-and z-directions, respectively. Φ_x and Φ_y represent the transverse normal rotations about the y-and x-axes, respectively. γ_xy is the in-plane shear strain. γ_xz and γ_yz are the transverse shear deformations.

Hooke's law for the shell including temperature effects is defined by

(7)

Hooke's law for the stiffeners taking into account the temperature and shear deformation of the stiffeners is defined by

(8)

In the above equations, the subscripts sh, s, r denote the shell, the stiffener, and the ring, respectively. E_s, E_r and G_s, G_r are Young's moduli and shear moduli of the stringers and ring, respectively. ΔT(z)=T(z) -T₀ is the temperature difference between the surfaces of the FGM cylindrical shell, and T₀=T_m.

Using the smeared stiffener technique in the situation where the stiffeners are equally spaced, omitting the twist of the stiffeners and integrating the stress-strain equations and their moments through the thickness of the shell lead to the expressions for the total in-plane force resultants, the total moment resultants, and the transverse force resultants of the eccentrically stiffened FGM shell in the thermal environments, which are defined similarly by^[30]

(9)

(10)

(11)

(12)

(13)

where a_ij, b_ij, c_ij, d_ij, and e_ij are defined in Appendix A.

The relations (9), (10), and (11) are the most significant contribution found in this work, in which the thermal elements of both the shell and the stiffeners in the equations of N_ij, M_ij, and P_ij are considered.

The nonlinear equilibrium equations of a perfect FGM shell filled by the elastic foundation based on the third-order shear deformation theory are defined by^[1]

(14)

Substituting Eqs.(5), (6), (9), (10), (11), (12), and (13) into Eq.(14), after several transformations, we can obtain the stability equations in terms of the displacement components as follows:

(15a)

(15b)

(15c)

(15d)

(15e)

which are used to analyze the nonlinear stability of the eccentrically stiffened circular cylindrical shell subjected to the combined mechanical and thermal load counting elastic foundations.

3 Temperature 3.1 Uniform temperature rise

When the temperature environments uniformly increases from the initial value T_i to the final one T_f, ΔT=T_f -T_i is a constant. Substituting Eqs.(3) and (4) into Eq.(A1), after calculating the integrals, we can obtain the thermal parameters Φ₁, Φ_1s, and Φ_1r as follows:

(16)

where

(17)

3.2 Nonlinear temperature change across thickness z

Assume that the temperature through the thickness is governed by the one-dimensional Fourier equation of the steady-state heat conduction

(18)

where T_m and T_c are the temperatures at the metal-rich and the ceramic-rich surfaces, respectively. Note that Eq.(18) is established in the coordinate system whose origin is on the middle surface of the shell.

The FGM shell is made of ceramic so as to resist the risen temperature. The temperature is quite different at the two sides of the structure. Assume that the heat transmits from the inside of the shell to the outside of the shell. The temperature through the thickness is governed by the one-dimensional Fourier equation of the steady-state heat conduction in the cylindrical coordinate whose origin is on the symmetric axis of the cylindrical shell. The new coordinate system with is a radial coordinate of the point which is distant from the symmetric axis of the cylinder respect to the point of the shell. Then, the Fourier equation of the steady-state heat conduction (18) becomes

(19)

(i) For shell

According to Eq.(19), the one-dimensional Fourier equation of the steady-state heat conduction for the shell is given by

(20)

By solving Eq.(20) with the mentioned boundary conditions, the solution for the temperature distribution across the shell thickness is obtained as follows:

(21)

Due to the mathematical difficulty in calculating the integral, this section only considers the linear distribution of metal and ceramic, i.e., k=1. Substituting Eq.(3) into Eq.(21) and calculating the integrals, after that substituting z=R -z, we have

(22)

Deduce

(23)

Substituting Eqs.(3) and (23) into Eq.(A1) and accounting, we have

(24)

where

(25)

(ii) For stiffeners

Consider the stringer stiffeners of the shell. According to Eq.(19), the Fourier equation of the steady-state heat conduction can be defined by

(26)

Similar to the shell case, according to Eqs.(4) and (26), we obtain

(27)

where

(28)

Similar to the stringer stiffener case, after some transformations, we obtain

(29)

where

(30)

4 Galerkin method and buckling and postbuckling analysis

In this section, an analytical approach is given to analyze the stability of the eccentrically stiffened FGM shell filled by the elastic foundations. Assume that the shell is only subjected to the axial compressive load p, the external uniform pressure q, and the thermal load. Then, we have

(31)

Let the cylindrical shell be simply supported at the two butt-ends and the corresponding boundary conditions be

(32)

According to Eq.(32), we have

(33)

where m is the number of the half waves in the x-direction, n is the wave number in the circumferential direction, and U, V, W, Φ₁, and Φ₂ are constant coefficients.

Substitute Eq.(33) into Eq.(15), and then use the Galerkin method to obtain the nonlinear algebraic equations for U, V, W, Φ₁, and Φ₂. Then, we have

(34)

where t_ij are defined in Appendix B.

Expressing U and V under W, Φ₁, and Φ₂, after substituting the obtained results into the last two equations of Eq.(34), we obtain Φ₁ and Φ₂ unber W. Combing with the third equation of Eq.(34), through some transformations, we can obtain a relation for q and W as follows:

(35)

where Φ₁, Φ_1s, and Φ_1r are shown in Eq.(16) for the uniform temperature rise case and defined by Eqs.(24), (27), and (29) for the nonlinear temperature change. The coefficients l_i are given in Appendix C.

The closed-form expression (35) is used to determine the buckling load and the postbuckling load-deflection curves in the temperature environments subjected to the combined mechanical and thermal loads.

Special cases

(i) In Eq.(35), taking q=0 and omitting the temperature, we deduce

(36)

where

(37)

(ii) In Eq.(35), taking P=0 with the assumption γ₁ >0 and γ₂² -3γ₁γ₃≥0, and omitting the temperature, we deduce

(38)

where

(39)

(iii) Taking P=q=0, when the uniform temperature increases, from Eq.(35), we deduce

(40)

where

(41)

Similarly, for the nonlinear temperature change, the thermal load ΔT(W) can be determined by Eq.(40), in which Φ₁⁰, Φ_1s⁰, and Φ_1r⁰ in Eq.(41) can be replaced by Φ₁¹, Φ_1s¹, and Φ_1r¹.

5 Numerical results 5.1 Comparison results

To validate the present approach, Table 1 shows the comparison of the critical buckling load of an isotropic thin cylindrical shell under the uniform axial compressive load with the results given by Eq.(5.52) in Ref.[30] by using the adjacent equilibrium criterion and the Donnell shallow shell theory.

Table 2 shows the comparison of the critical buckling loads with the results given by Eq.(5.78) in Ref.[30] for the orthotropic perfect eccentrically stiffened cylindrical shell under the axial compression without elastic foundations and omitting torsion.

In the third comparison (see Table 3), the critical buckling loads are compared with those given by Eq.(23) in Ref.[9] with the data shown in Figs. 2 and 4 in Ref.[9] when k and R/h change, respectively, for the unstiffened FGM shell without elastic foundations. The buckling load of the static-axial loaded FGM cylindrical shell P_scr=P_dcr/τ_cr, in which the non-dimensional parameter τ_cr is the critical parameter and P_dcr is the corresponding dynamic buckling load, is calculated by Huang and Han^[9], and P_upper is given by Eq.(36) in this paper. It is seen that the results agree well with each other.

Table 4 compares the results on the nonlinear response of the FGM cylindrical shell under the external pressure among the results obtained by the TSDT, the first shear deformation theory (FSDT), and the CST, where k=1, k₂=k₃=∞, K₁=K₂=0, and q=0. From the table, we can see that, when the shell is thin, the difference between the results obtained by these theories is not considerable; while when the shell is thick, the difference of the results obtained by the TSDT and the FSDT is bigger than that between the results obtained by the TSDT and the CST. As listed in Table 4, when R/h=10, n₂=50, and n₁=0, the value of P_upperTSDT is 13.522 2MPa, which is smaller than 14.147 2 MPa (see the value of P_upperTSDT when R/h=10, n₁=50, and n₂=0) for about 4.4%. Therefore, the TSDT and FSDT may be used for thick shell. However, the FSDT need the transverse shear correction coefficients, while the TSDT does not need. This is the advantage of the TSDT for thick shell.

In the following subsections, the effects of the input parameters are investigated on the buckling and postbuckling behaviors of the cylindrical shell with the following material properties and geometric properties for the shell:

where n₁ and n₂ are the numbers of the stringer and rings, respectively.

5.2 Effects of inside FGM stiffeners

The effects of the stiffeners on the critical compression load p and the critical external pressure load q are given in Tables 5, 6, and 7. Table 5 shows that in the case of only the compression load p, with the same stiffener number, the critical buckling load of the stiffened shell by stringers is the biggest, then is the stiffened shell by the orthogonal stiffeners, and the critical load of the stiffened shell by the ring stiffeners is the smallest. Moreover, the critical compressive load increases with the increase in the stiffeners number. This increase is considerable. As listed in Table 5, when n₁=n₁₂=50, P_upper=1 421.1 MPa which is larger than 1 375.2 (see the value of P_upper when n₁=n₁₂=20) for about 3.3%. The effects of the stiffeners on the critical pressure q without any foundation in the thermal environments is illustrated in Tables 6 and 7. From the tables, we can see that the values of q_upper in the uniform temperature rise are bigger than those in the nonlinear temperature change. Moreover, the critical pressure q increases when the stiffener number increases. Tables 6 and 7 also show that the critical pressure q of the stiffened shell by the ring stiffeners is the best.

5.3 Effects of temperature

Tables 8 and 9 give the effects of the temperature field on the critical loads. It can be seen that the critical load of the shell raises when Δ_T increases. As listed in Table 8, when n₂=50 and Δ_T=800 K, the value of q_upper is equal to 0.299 9 MPa, which is bigger than those corresponding to Δ_T=0 K and Δ_T=400 K, which are 0.159 8 MPa and 0.229 1 MPa, respectively. Tables 8 and 9 also show that the critical load of the shell in the uniform temperature rise is better than that in the nonlinear temperature change.

The effects of the temperature rise on postbuckling q-W/h curves are shown in Fig. 2 and 3, where h=0.01 m, R/h=100, L/R=2, k=1, k₂=k₃=1/k, K₁=10⁸ N/m³, and K₂=5.10⁵ N/m. The lines 2 and 3 do not start at origin of coordinates. From the figure, we can see that, when the shell is preheated, the temperature field makes the shell to be deflected outward (negative deflection) prior to the mechanical load acting on it. Moreover, when the shell is subjected to the external pressure load, its outward deflection is reduced; and when the compressive load exceeds the bifurcation point of the load, an inward deflection occurs. As expected, the loading carrying capacity of the shell in the uniform temperature rise case is better than that in the nonlinear temperature change case.

5.4 Effects of the volume fraction index k

Tables 10 and 11 consider the effects of the index volume k on the critical buckling external pressure load for a stiffened FGM cylindrical shell in the thermal environments. It is found that, the critical buckling load decreases with the increase in k. The value of the critical load when k=0 is the greatest, while The value of the critical load when k=∞ is the smallest. Moreover, the buckling strength of the FGM shell is larger than the fully metal shell, and is smaller than that of the fully ceramic shell. This property is suitable to the real property of the material, because higher k corresponds to a metal-richer shell which usually has smaller stiffness than a ceramic-richer one. Besides, the critical load of the shell in the uniform temperature rise is higher than that in the nonlinear temperature change.

Figures 4 and 5 show the effects of the volume fraction index k on the nonlinear response of the FGM stiffener of the FGM cylindrical shell, where n₁=n₂=25, R/h=100, L/R=2, k₂=k₃=1/k, h₁=h₂=0.008 m, b₁=b₂=0.004 m, K₁=10⁶ N/m³, and K₂=5 × 10⁵ N/m. As expected, the loading carrying capacity of the shell is reduced considerably when k increases.

5.5 Effect of geometrical parameters

Figures 6 and 7 give the effects of the length-to-width ratio L/R on the postbuckling load-deflection curves, where n₁=n₂=25, R/h=100, L/R=2, k=k₂=k₃=1, (m, n)=(1, 3), h₁=h₂=0.008 m, b₁=b₂=0.004 m, K₁=0 N/m³, and K₂=0 N/m. The obtained results show that the loading carrying capacity of the shell is reduced considerably when L/R increases. Because the shell is too long, the shell is easily instable.

Figures 8 and 9 illustrate the effects of the width-to-thickness ratio R/h on the the postbuckling curves, where n₁=n₂=25, L/R=2, k=k₂=k₃=1, h₁=h₂=0.008 m, b₁=b₂=0.004 m, K₁=0 N/m³, and K₂=0 N/m. The obtained results show that the loading carrying capacity of the shell is reduced considerably when R/h increases. This result agrees with the actual property of the structure, i.e., the shell is thinner when the value of the critical load decreases.

5.6 Effects of stiffeners and foundation

Tables 12-14 consider the effects of the inside stiffeners and foundations on the critical buckling load of the FGM cylindrical shell. The effects of the foundations on the critical buckling compressive load P_upper are given in Table 12. The foundation parameter K₂ in Table 12 affects more strongly than the shear parameter K₁, and inversely in Tables 13 and 14. Especially, the critical buckling load corresponding to the presence of both the foundation parameters K₁ and K₂ is the biggest.

The critical buckling load of the shell without any foundation is the smallest. Moreover, when the shell is reinforced, the critical buckling load of the FGM cylindrical shell increases. Tables 12 and 13 show that the critical buckling load of the stiffened shell by the ring stiffeners is the biggest, then is the critical buckling load of the stiffened shell by the orthogonal stiffeners, and the critical load of the stiffened shell by the stringer stiffeners is the smallest. Furthermore, the values of the critical load q_upper in the uniform temperature rise case are bigger than those in the nonlinear temperature change case.

Figures 10 and 11 show the effects of the volume fraction of the foundation on the nonlinear response of the cylindrical shell, where n₁=n₂=25, L/R=2, k=k₂=k₃=1, h₁=h₂=0.008 m, b₁=b₂=0.004 m, K₁=0 N/m³, and K₂=0 N/m. It is found that, the presence of the foundation parameters K₁ and K₂ make the postbuckling load-deflection curves much higher than those without foundation.

5.7 Effects of pre-existent compressive load P

Figures 12 and 13 consider the effects of the pre-existent compressive load p on the postbuckling q-W/h curves in the thermal environments, where L/R=2, R/h=100, k=k₂=k₃=1, (m, n)=(1, 3), h₁=h₂=0.008 m, b₁=b₂=0.004 m, n₁=n₂=25, ΔT=400℃, K₁=10⁸ N/m³, and K₂=5 × 10⁵ N/m. From the figures, we can see that q decreases when the axial load p increases. As expected, the loading carrying capacity of the shell in the uniform temperature rise case is better than that in the nonlinear temperature change case.

6 Concluding remarks

This study presents an analytical solution to investigate the nonlinear buckling and postbuckling of the eccentrically FGM stiffened the FGM shell filled by the elastic foundation subjected to the axial compressive mechanical loads, the external uniform pressure, and the thermal load. The nonlinear equilibrium and stability equations based on the smeared stiffener technique and the TSDT are derived. The closed-form expression for determining the critical buckling load and analyzing the postbuckling load-deflection curves is obtained by using the Galerkin method. The thermal elements in the shell and stiffeners are considered. Some remarks are deduced from the present study.

(i) The thermal element, elastic foundation, pre-existent axial compressive and thermal load, and geometrical parameters affect strongly the buckling and postbuckling behavior of the eccentrically stiffened FGM shell.

(ii) FGM stiffeners enhance the stability of the shell.

(iii) The values of the critical buckling load calculated by the CST and the TSDT are close to each other for thin shell. However, for thick shell, the difference between the two theories is considerable.

(iv) The third-order shear deformation theory is of great advantage to thicker shell.

(v) The accuracy of the proposed approach is affirmed by comparing the obtained with the previous studies.

Appendix A

The coefficients in Eqs.(9)-(13) are expressed as follows:

where d₁ and d₂ denote the distances between two stringers and rings, respectively. b₁, b₂ and h₁, h₂ are the width and thickness of the stringer and the ring, respectively.

(A1)

In the above equation,

Appendix B

The coefficients in Eqs.(34) are defined by

Appendix C

The coefficients l_i(i=1, 2, ···, 15) in Eq.(34) are given by