1 Introduction

Sandwich-type construction with advantageous properties such as high stiffness, high structural efficiency, strength, and excellent thermal and sound insulation has been widely used in construction of many components in nuclear reactor, aerospace, ship, and building structure. Buckling of cylindrical sandwich shells with metal foam cores subject to axial compression was considered by Hutchinson and He ^[1]. Based on the first-order shear deformation theory (FSDT), Ferreira et al. ^[2] presented the nonlinear analysis of sandwich shell with core plasticity. By different methods, Wang ^[3] and Wang et al. ^[4] investigated vibration frequencies of polygonal sandwich plates and skew sandwich plates with laminated facings. A solution of the problem of thermo-mechanical coupling in sandwich plates for the case of bending caused by different temperatures on the outer surfaces of the plate was considered by Pilipchuk et al ^[5] based on the linear elasticity theory.

However, the sudden change in the material properties from one layer to another can lead to stress concentration, which often results in delamination. In order to overcome this problem, functionally graded material (FGM) sandwich structures were studied. Due to the continuous variation in the properties of FGMs, the stress concentration is eliminated. There were many studies about sandwich plates. By either the shear deformation theories or the classical theory, Zenkour ^[6-7] presented comprehensive analysis of FGM sandwich plates under compressive load, including studies on deflection, stresses, buckling, and free vibration. Based on the sinusoidal shear deformation plate theory, Zenkour and Sobhy ^[8] also studied thermal buckling of FGM sandwich plates. In their paper, the thermal loads were assumed to be uniform, linear, or nonlinear distribution through the thickness. Shen and Li ^[9] and Xia and Shen ^[10] researched post-buckling and vibration of sandwich plates with FGM face sheets in thermal environment by the higher-order shear deformation theory (HSDT) and a general von Kármán-type equation that includes the thermal effect. An investigation on the nonlinear dynamic response of sandwich plates resting on Pasternak elastic foundations under mechanical and thermal loads was presented by Wang and Shen ^[11-12]. Herein, governing equations of plate that include plate foundation interaction were solved by a two-step perturbation technique. Shariyat ^[13] utilized a generalized three-dimensional high-order global-local plate theory for nonlinear dynamic thermo-mechanical buckling analysis of the imperfect sandwich plates. Alipour and Shariyat ^[14] suggested a zigzag-elasticity plate theory for stress and deformation analysis of annular FGM sandwich plates subjected to non-uniform normal and/or shear tractions. By the advanced two-dimensional Ritz-based models based on an entire family of higher-order layer wise and equivalent single-layer theories, natural frequencies of sandwich plates with FGM core were investigated by Dozio ^[15]. Tung ^[16] used the FSDT to research nonlinear post-buckling behavior of FGM sandwich plates on elastic foundations with tangential edge constraints under thermo-mechanical load.

For FGM sandwich shells, vibration and buckling of sandwich cylindrical shells with FGM coatings on elastic foundations subjected to torsional load and hydrostatic pressure were discussed by Sofiyev and Kuruoglu ^[17] and Sofiyev ^[18]. Sofiyev and Kuruoglu ^[19] also investigated the parametric instability of simply-supported sandwich cylindrical shell with a functionally graded core under static and time dependent periodic axial compressive loads based on the FSDT. Besides, Sofiyev and Kuruoglu ^[20] considered the effect of a functionally graded interlayer on nonlinear stability of three-layered truncated conical shells surrounded by an elastic medium with the FSDT. Seidi et al. ^[21] used an improved high-order theory to present temperature-dependent (TD) buckling analysis of sandwich truncated conical shell with functionally graded face sheets. In addition, buckling and post-buckling responses of sandwich panels under non-uniform mechanical edge loadings were reported by Dey and Ramachandra ^[22].

Due to the increased load capacity of the structure, stiffened plates and shells become important in modern engineering. Simultaneously, they also attract attention of many scientists. Barush and Singer ^[23] investigated instability of stiffened cylindrical shells under hydrostatic pressure. Nonlinear dynamic analysis of stiffened plates subjected to transient pressure loading by an explicit central difference/diagonal mass matrix time stepping method was presented by Khalil et al. ^[24]. Based on a spline Gauss collocation method, Shen and Dade ^[25] analyzed the dynamic response of stiffened plates and shells with various constraint conditions. Patel et al. ^[26] reported static and dynamic instability characteristics of stiffened shell panels subjected to a uniform in-plane harmonic edge loading. Recently, the idea of eccentrically stiffened FGM (ES-FGM) structures has been proposed by Najafizadeh and Heydari ^[27]. In this paper, buckling of stiffened FGM cylindrical shells by rings and stringers by Sander's assumption was presented. Subsequently, Bich et al. ^[28] investigated nonlinear dynamical of ES-FGM cylindrical panel with the classical shell theory (CST), the geometrical nonlinearity in the von Kármán-Donnell sense, and the smeared stiffeners technique. Dung and Hoa ^[29] analyzed the nonlinear dynamic torsional buckling of ES-FGM circular cylindrical shells surrounded by an elastic medium using the semi-analytical method.

To the best of the authors' knowledge, there has been no investigation using the TSDT on buckling and post-buckling analysis of FGM sandwich doubly curved shallow shells that are reinforced by FGM stiffeners on elastic foundation under thermo-mechanical loads with the TD material.

This study focuses on highlights as follows: the symmetric sigmoid law distribution and power law distribution are generalized for four different models of the sandwich double curved shallow shells; the shallow shell is reinforced by the FGM stiffeners system; the TD material and stiffener properties are considered in the expression of force and moment resultants; the doubly curved shallow shell is investigated for three different cases of radius of curvature corresponding to three kinds of shells. The explicit expressions for determining the critical buckling load and post-buckling mechanical and thermal load-deflection curves are obtained.

Theoretical formulations are derived based on the TSDT with the von Kármán-type nonlinearity, taking into account the initial geometrical imperfection and the smeared stiffener technique. The influence of the geometrical and material parameters, the initial imperfection, the foundation, the TD properties, the FGM reinforcement stiffeners, and the distribution law of material properties and kind of shells on the nonlinear stability of ES-FGM shells is considered.

2 FGM sandwich shallow shells reinforced by FGM stiffeners on elastic foundations

Consider a sandwich double curved shallow shell of principal radii of the curvature $R_x$ and $R_y $, the in-plane edges $a$ and $b$, and the thickness $h$, reinforced by eccentrically closely spaced FGM longitudinal and transversal and rested on a Pasternak elastic foundation, as shown in Fig. 1.

The reaction-deflection relation of the Pasternak foundation model is ^{[11, 16]}

(1)

where $\nabla ^2=\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}$ is Laplace's operator, $w$ is the deflection of the FGM sandwich shallow shell, $K_1$(N/m$^{3})$ is the Winkler foundation modulus, and $K_2 $ (N/m) is the shear layer foundation stiffness of the Pasternak model.

In this paper, we consider three kinds of shells corresponding to three different radii of curvature as follows (as shown in Fig. 2):

For $R_x =R_y $, it corresponds to the spherical panel (S-panel).

For $R_x \to \infty $, it corresponds to the cylindrical panel (C-panel).

For $R_x =-R_y $, it corresponds to the hyperbolic paraboloidal panel (H-panel).

The sandwich doubly curved shallow shells have three layers, the top face sheet, the core layer, and the bottom face sheet with the thickness $h_{\mathrm t} , h_{\mathrm c}$, and $h_{\mathrm b}$, respectively. Assume that the core and face sheets are perfectly bonded. We consider four cases of FGM sandwich shallow shells with FGM and homogeneous layers, as shown in Fig. 3.

2.1 Sandwich shallow shell of FGM-homogeneous core-FGM model

For the sandwich shallow shell in Case 1, the top skin is graded from a ceramic-rich interface ($z=-h/2)$ to a metal-rich interface ($z=-h/2+h_{\mathrm t} )$, and the bottom skin varies from a metal-rich interface ($z=h/2-h_{\mathrm b} )$ to a ceramic-rich interface ($z=h/2)$. The core is full metal, as illustrated in Fig. 3(a). Young's modulus $E_{\mathrm {sh}} $ and the thermal expansion coefficient $\alpha_{\mathrm{sh}} $ of the shell are expressed by

(2)

Similarly, in Case 2 (see Fig. 3(b)), the effective properties of the shell are given by

(3)

Herein, $E_{\mathrm c} , E_{\mathrm m} , \alpha _{\mathrm c}$, and $\alpha _{\mathrm m} $ are the TD material properties,

$k_{\mathrm t}$ and $k_{\mathrm b} $ are the volume fraction indices of the top FGM face sheet and the bottom FGM face sheet, respectively, and the subscripts m, c, and sh denote the metal, ceramic, shell, respectively.

The laws (2) and (3) are generalized sigmoid laws. Observably, from these laws when $h_{\mathrm t} =h_{\mathrm b} =h/2$ and $k_{\mathrm t} =k_{\mathrm b} $, we obtain known-well symmetric sigmoid-FGM laws.

2.2 Sandwich shallow shell of homogeneous-FGM core-homogeneous model

In this model of sandwich shallow shell, the core is an FGM layer, and the top skin and the bottom skin are made of isotropic homogeneous material (metal or ceramic). For Case 3, the top skin and the bottom skin are full ceramic and metal, respectively, whereas the FGM core layer varies from a ceramic-rich interface ($z=-h/2+h_{\mathrm t} )$ to a metal-rich interface ($z=h/2-h_{\mathrm b} )$ (as shown in Fig. 3(c)). Young's modulus and the thermal expansion coefficient of the shell in this case are

(4)

Likewise, for Case 4 (as illustrated in Fig. 3(d)),

(5)

The laws (4) and (5) are generalized power laws. It can be seen that when $h_{\mathrm t} =h_{\mathrm b} =0$, these laws return to the well-known P-FGM laws.

2.3 FGM longitudinal and transversal stiffeners

In order to guarantee the continuity between the shell and stiffeners, the effective properties in Cases 1 and 4 are given by

(6)

For Cases 2 and 3,

(7)

in which $k_2 $, $k_3 $, $h_x $ and $h_y $ are the volume fraction indices and the thicknesses of the longitudinal and transversal stiffeners, respectively, and the subscript s denotes the stiffener.

3 Theoretical formulations

This study uses the third-order shear deformation theory (TSDT) to establish the governing equations for the FGM sandwich shallow shell reinforced by FGM stiffeners on elastic foundations under thermo-mechanical loads. According to this theory, the strain components across the shell thickness at a distance $z$ from the middle surface are ^{[9-10, 31]}

(8)

where $\varepsilon _x$ and $\varepsilon _y$ are the normal strains, $\gamma _{xy} $ is the in-plane shear strain, $\gamma _{xz}$ and $\gamma _{yz} $ are the transverse shear deformations, and

(9)

in which $\lambda =4/({3h^2})$, $u$ and $v$ are the displacement components along the $x$- and $y$-directions, respectively, and $\varphi _x$ and $\varphi _y $ are the rotations normal to the mid-surface with respect to the $y$- and $x$-axes, respectively. In addition, the von Kármán geometrical nonlinearity and the initial imperfection $w^\ast =w^\ast (x, y)$ are taken into account. Assume that this imperfection is very small in comparison with the shell thickness.

Hooke's law for the FGM sandwich shallow shell with the TD properties can be defined as follows.

For shells,

(10)

For stiffeners,

(11)

The force resultants $N_i $, the moment resultants $M_i $, the higher-order moment resultants $T_i $, the transverse force resultants $Q_i $, and the higher-order shear force resultants $S_i $ of an FGM sandwich doubly curved shallow shells reinforced by FGM stiffeners are defined as follows:

(12)

where $A_i $ is the cross-sectional area of stiffeners in each corresponding axis, and $d_i $ is the spacing of the longitudinal and transversal stiffeners with $i=x, y$.

Substituting Eqs.(10) and (11) into Eq.(12) and using the Lekhnitskii smeared stiffener technique, we obtain

(13)

(14)

(15)

(16)

(17)

where the specific expressions of coefficients $a_{ij} , b_{ij} , c_{ij} , d_{ij}$, and $e_{ij}$ are in Appendix A.

In systems of Eqs.(13)-(17), it is very important contribution to take into account stiffeners and temperature elements in calculation for four cases of sandwich shallow shell.

The nonlinear equilibrium equations of an imperfect double curved shallow shell on elastic foundations based on the TSDT are ^[32-33]

(18)

where $q$ is an external pressure uniformly distributed on the top surface of the shell.

The compatibility equation deduced from Eq.(9) is in the form of

(19)

The strain-force resultant relations reversely are obtained from Eq.(13),

(20)

Substituting Eq.(20) into Eqs.(14) and (15), we have

(21)

(22)

where $a_{{ ij}}^\ast$, $b_{ {ij}}^\ast$, and $c_{{ij}}^\ast $ are given in Appendix B.

The stress function $f=f(x, y)$ is introduced as

(23)

With the stress function $f$, it is observed that the two first equations of Eq.(18) are satisfied automatically. By substituting the expressions of $M_i$, $T_i$, $M_{xy}$, and $T_{xy} $ from Eqs.(21) and (22) and $Q_i$ and $S_i $ from Eqs.(16) and (17) into three remaining equations of Eq.(18), we obtain

(24)

Equation (24) includes four unknown functions $w, \varphi _x , \varphi _y $, and $f$. Therefore, it is necessary to find a fourth equation relating to these functions by the compatible equation.

For this aim, substituting Eq.(23) into Eq.(20) and then into Eq.(19) yields

(25)

Equations (24) and (25) are the governing equations in terms of four dependent unknown functions $w, \varphi _x , \varphi _y $, and $f$ used to investigate the stability of imperfect FGM sandwich double curved shallow shell reinforced by FGM stiffeners on elastic foundations subjected to mechanical, thermal, and thermo-mechanical loads.

4 Boundary conditions and Galerkin method

Three types of boundary conditions are considered in this work ^{[16, 31]}.

(i) Four edges of the shell are simply supported and freely movable. The associated boundary conditions are

(26)

(ii) Four edges of the shell are simply supported and immovable. The associated boundary conditions are

(27)

(iii) Two edges $x=0, a$ are freely movable, and two edges $y=0, b$ are immovable. For this case, the boundary conditions are defined as

(28)

The approximate solution of the system of Eqs.(24) and (25) satisfying the boundary conditions (26)$-$(28) can be expressed by

(29)

where $\alpha =m\pi /a, \;\beta =n\pi /b$, $m$ and $n$ are the numbers of half waves in the $x$- and $y$-directions, respectively, and ${W, }\Phi _x$, and $\Phi _y $ are the amplitudes of deflection and slope rotations, respectively.

The initial imperfection $w^\ast $ is assumed to have the same form of the deflection of the shells

(30)

where the coefficient $\xi \in [{0, 1}]$ is an imperfection size of the shell.

By substituting the expressions (29) and (30) into Eq.(25), we can obtain the equation for determining $f$ as follows:

(31)

The solution of this equation is in the form of

(32)

in which

(33)

Introducing the expressions (29), (30), and (32) into Eq.(24) and then applying the Galerkin method to the resulting equations lead to

(34)

where $\delta _m =({-1})^m-1, \delta _n =({-1})^n-1$, and $l_{ij}$ and $s_i $ are given in Appendix C.

Solving $\Phi _x$ and $\Phi _y $ from the second equation and the third equation of Eq.(34), substituting them into the first equation of Eq.(34), and putting $\overline{W}=W/h, $ we obtain

(35)

Equation (35) is used to analyze the buckling and post-buckling responses of imperfect FGM sandwich doubly curved shallow shells reinforced by FGM stiffeners on elastic foundations under mechanical, thermal, and thermo-mechanical loads.

5 Mechanical buckling and post-buckling analysis

Consider an imperfect ES-FGM sandwich doubly curved shallow shell with all movable edges (see Eq.(26)) and resting on elastic foundation. Two cases of mechanical loads will be analyzed.

5.1 ES-FGMC cylindrical panels only under axial compressive loads

Assume an ES-FGM cylindrical panel resting on elastic foundations and only subjected to axial compressive loads $P_x$ uniformly distributed at two curved edges $x=0, a.$ In this case, $q=0, N_{y0} =-P_y h=0, R_x \to \infty , $ and Eq.(35) leads to

(36)

For a perfect cylindrical panel $({\xi =0})$, taking $\overline {W} \to 0, $ Eq.(36) gives the following upper buckling compressive load:

(37)

The upper static critical buckling compressive load of the perfect panel is determined by the condition $P_{x{\rm upper}}^{\rm cr} =\min P_{x{\rm upper}} $ vs. $( {m, n})$. The lower buckling load of the perfect panel can be found from Eq.(36) using the condition $\frac{{\mathrm d}P_x }{{\mathrm d}\overline{W}}=0$.

5.2 ES-FGM double curved shallow shell only subjected to external pressure

Consider the ES-FGM double curved shallow shell only subjected to an external pressure $q$ (in Pascal). In this case, $N_{x0} =N_{y0} =0$, and Eq.(35) gives us

(38)

The upper and lower buckling pressures of the shell are found from Eq.(38) by the condition $\frac{{\mathrm d}q}{{\mathrm d}\overline{W}}=0$.

6 Thermal stability analysis

Suppose that an imperfect FGM sandwich doubly curved shallow shells reinforced by FGM stiffeners on elastic foundation with all immovable edges (see Eq.(27)) and exposed to a thermal environment uniformly raised from the initial value $T_{\mathrm i} $ to the final one $T_{\mathrm f}$ and $\Delta T=T_{\mathrm f} -T_{\mathrm i} ={\rm const}$. The conditions $u=0$ at $x=0, x=a$ and $v=0$ at $y=0, y=b$ are fulfilled on the average sense as ^{[9-10, 33]}

(39)

From Eqs.(9) and (20), we obtain the expressions of $u_{, x}$ and $v_{, y} $. Substituting Eqs.(29), (30), and (32) into those expressions and then into Eq.(39) yields

where $t_{ij} $ are given in Appendix D.

Introducing the relation (40) into Eq.(35), with $q=0, $ we get

(41)

in which $\overline W =\frac{W}{h}$, $t_i$ and $\overline{\lambda} _i $ are given in Appendix D, and

(42)

Take $\overline W \to 0$. $m$ or $n$ is an even number. Equation (41) gives the upper buckling temperature load for a perfect sandwich shallow shell $({\xi =0})$,

(43)

where ${\lambda }_i $ are given in Appendix D.

Note that Eqs.(41) and (43) are the explicit expressions of the $\Delta T$-$\overline {W}$ relation and the buckling temperature change $\Delta T$, respectively, in the case of temperature-independent (TID) shell material properties. On the contrary, when the material properties of shells and stiffeners depend on the temperature, those equations are implicit expressions. In that case, the post-buckling temperature-deflection curves and the critical buckling temperatures will be determined by two iterative algorithms as follows.

(i) Iterative algorithm 1 for determining the critical buckling temperature

Using Eq.(43) with the following steps.

Step~1 Calculate the TID material properties $E$ and $\alpha $ at the temperature $T_0$ = 300 K. Apply the results for Eq.(43) to get the critical buckling temperature $\Delta T_{\mathrm b}^{\mathrm {cr}(1)} $ for the shell with the TID material.

Step~2 Repeat Step 1 with $\Delta T=T_0 +\Delta T_{\mathrm b}^{\mathrm {cr}(1)} $, and get a new critical buckling temperature $\Delta T_{\mathrm b}^{\mathrm {cr}(2)} $ for the shell of the TD material.

Step~3 Repeat Step 2 until the thermal buckling temperature converges.

(ii) Iterative algorithm 2 for determining the post-buckling temperature-deflection curves

Using Eq.(41) with the following steps:

Step~4 Start with $\overline{W}=0$.

Step~5 Repeat Step 1 to Step 3 for Eq.(41).

Step~6 Specify the new value of $\overline{W}$ and repeat Step 5 until the thermal post-buckling temperature converges.

7 Thermo-mechanical post-buckling analysis

An imperfect ES-FGM sandwich cylindrical panel resting on an elastic foundation under the axial compressive loads uniformly distributed along the edges $x=0$ and $x=a$ and thermal loads is considered. Suppose that the shell is simply supported with the movable edges $x=0, a$ and the immovable edges $y=0, b$ (see Eq.(28)).

Using $N_{x0} =-P_x h$ and the second equation of Eq.(39), we obtain

(44)

Consider the ES-FGM panel under the combined action of axial compressive loads and uniformly raised temperature field (with $q=0)$. Setting Eq.(44) into Eq.(35) yields

(45)

Equation (45) is used to trace the post-buckling mechanical load-deflection curves of the imperfect ES-FGM sandwich shallow shell with the given uniform temperature and the thermal loads.

Inversely, from Eq.(45), we can solve $\Delta T$. That expression will be used to trace the post-buckling thermal load-deflection curves of the imperfect ES-FGM sandwich cylindrical panel with the pre-loaded mechanical loads.

8 Numerical results and discussion

In this section, numerical results are presented for perfect and imperfect FGM sandwich shallow shells reinforced by FGM stiffeners on elastic foundation composed of silicon nitride (Si$_{3}$N$_{4})$ and stainless steel (SUS304). In the FGM layers, the effective value of material properties $P_{\mathrm m} $, such as Young's modulus $E$ and the thermal expansion coefficient $\alpha $, can be expressed as a nonlinear function of the temperature,

where $T=T_0 +\Delta T$, $T_0 =300 {\mathrm K}$ (the room temperature), and $P_0 , P_{-1} , P_1 , P_2$, and $P_3 $ are the coefficients of temperature $T({\mathrm K})$ and are unique to the constituent material. Specific values of these coefficients for $E$ and $\alpha$ of these materials given by Reddy and Chin ^[33] are shown in Table 1. The TID properties are calculated at the room temperature $T_0 =300$ K. Poisson's ratio is assumed to be a constant, and $\nu =0.29$ except in the cases of comparison.

8.1 Verification

To validate the proposed approach of the present study, an un-stiffened square sandwich plate (Case 1) with the FGM face sheets (Si$_{3}$N$_{4}$/ SUS304) and the homogeneous core (SUS304) without elastic foundation under a uniform temperature rise is considered. The critical buckling temperature in this paper is compared with that of Shen and Li ^[9] based on the global HSDT and with Shariyat ^[13] based on a generalized three-dimensional high-order global-local plate theory with TID and TD material properties in Table 2. Herein, $T_{\mathrm {cr}} =T_0 +\Delta T_{\mathrm {cr}} $, where $\Delta T_{\mathrm {cr}} $ is calculated by Eq.(43) (with $R_x \to \infty$ and $R_y \to \infty )$. It can be observed that the present results agree well with those in the literature.

In the second comparison, we still consider a sandwich plate as in comparison previous. The results for the critical buckling compressive loads (Eq.(45) with $\xi =0$, $\overline{W}\to 0$, $R_x \to \infty$, and $R_y \to \infty )$ of an un-stiffened sandwich plate without elastic foundations subjected to simultaneous mechanical and thermal loads are compared with Tung ^[16], as shown in Table 3. It is evident that a very good correlation is achieved.

Finally, Table 4 shows a comparison between the critical buckling compressive load of the FGM cylindrical panel reinforced by homogenous stiffeners without elastic foundation only under the axial compressive load $P_x $ in this paper, and the results were calculated by Bich et al. ^[28] based on the CST. The input parameters of material and geometrical properties are

It can be seen that good agreement can be obtained in this comparison for thin shells $({a/h\ge 50})$. However, for thick shells $({10\le a/h\le 40})$, this difference between two theories is considerable. Therefore, for thick FGM plates and shells, the TSDT is recommended to obtain precise buckling load values and post-buckling behavior. This remark for un-stiffened plates has also been noted by Shariat and Eslamin ^[34].

8.2 Mechanical buckling and post-buckling of ES-FGM sandwich shallow shells

Consider an ES-FGM sandwich shallow shell reinforced by 10 FGM longitudinal stiffeners and 10 FGM transversal stiffeners (unless otherwise). The parameters of geometrical and material properties of the shallow shell and stiffeners are $k_{\mathrm t} =k_{\mathrm b}, h_{\mathrm t} =h_{\mathrm b}, $ $k_2 =k_3 =1/k$, $b_x =b_y =0.006 {\rm m}$, and $h_x =h_y =0.02 {\rm m}$. The shallow shell is rested on elastic foundation with the parameters $K_1 =5\times 10^7 {\rm N/m^3}$ and $K_2 =2\times 10^5 {\rm N/m}$.

Figure 4 describes the external pressure-deflection curves for perfect and imperfect ES-FGM sandwich spherical panel for Case 1 with three different values of $h_{\mathrm t} /h$. Clearly, the post-buckling curve becomes higher when this ratio increases, or in other words, the FGM layer is thicker.

Nonlinear response of three kinds of shell, i.e., spherical panel, cylindrical panel, and hyperbolic paraboloidal panel (Case 1) under the external pressure is investigated in Fig. 5. It is noticeable that the shapes of the post-buckling curves of these shells are significantly different. In other words, the radius of curvature has great influence on the load-deflection curves.

Table 5 points out the influence of the type stiffener and arrangement on the critical load of cylindrical panel (for Case 2) with the input parameters taken as $a/b=1, k_{\mathrm t} =k_{\mathrm b} =2, R_y =3 {\rm m}$, and $b/h=20$. To maintain the continuity between shell and stiffeners, we only consider metal and FGM stiffeners. Clearly, whether the structure is reinforced by metal stiffeners or by FGM stiffeners, they both have the critical load larger than the non-reinforced structures. The critical buckling compressive load is the maximum when the panel is reinforced by FGM stiffeners. Besides, the stiffener position influences the critical load value of shell. This should be noted in fabrication of shell structures with reinforced stiffener for ensuring the best load capacity. Table 5 also considers the influence of the elastic foundation parameter on the critical buckling compressive load of the cylindrical panel in 3 cases: without foundation, Winkler foundation, and Pasternak foundation. Specifically, this value is maximized when the shell is rested on the Pasternak foundation $( K_1 >0$ and $K_2 >0)$ and minimized when it is not rested on the elastic foundation.

Figure 6 compares the load-deflection curves for perfect ES-FGM sandwich spherical panels of Case 1 (FGM/SUS304/FGM) and Case 3 (Si$_{3}$N$_{4}$/FGM/SUS304). Both of the curves are under the influence of the volume fraction indices $k_{\mathrm t} =k_{\mathrm b} =k$. As observed, with the same geometry and material parameter, the external pressure capacity of shells in Case 3 is better than that in Case 1. Besides, the load-deflection curves are higher when $k$ increases.

8.3 Thermal buckling and post-buckling of ES-FGM sandwich shallow shells

We consider an imperfect FGM sandwich shallow shell reinforced by FGM stiffeners on elastic foundation under thermal loads. The geometrical and materials properties of shells used in this section are the same as those in the previous section.

The influence of the thickness of the FGM layer on the thermal post-buckling of ES-FGM sandwich spherical panel in two cases, i.e., TD and TID, is shown in Fig. 7. Similar to the mechanical load case, the thermal load capacity of shells also increases when the thickness of the FGM layer raises. The thermal load capacity is at its best when the ratio $h_{\rm t} /h=0.5$, which means that the shell contain only two symmetric FGM layers and the core layer thickness equals zero. Besides, in the case that the temperature depends on the material parameters, the thermal load-deformation curve is lower than that in the TID case.

The investigation results of the influence of the stiffener type and the number of stiffeners on the critical buckling temperature of three kinds of shells in two cases of TD and TID material properties (Case 2) are shown in Table 6. Clearly, using stiffeners increases significantly the critical buckling temperature of FGM sandwich doubly curved shallow shells. Similar to the case of shell under mechanical loads, the reinforced panel by FGM stiffeners has the best temperature endurance.

Besides, the influence of the radius of curvature on the critical buckling temperature is relatively clear. Specifically, the highest critical buckling temperature load is in the spherical panel case. The following is the case of cylindrical panel, and it is the smallest in the hyperbolic paraboloidal panel. The same thing happens to the thermal load-deflection curves of these shells, as shown in Fig. 8 (with Case 1). Such, in three considered kinds of shells, the spherical panel has the best thermal load capacity.

As shown in Fig. 9, the thermal post-buckling of spherical panel (Case 1) under the influence of thermal load tends to increase when the value of volume fraction indices raises and when there is participation of the Pasternak elastic foundation.

Fig. 10 investigates the thermal load capacity of ES-FGM sandwich spherical panel with five ascending values of initial imperfection. Observably, imperfect panels are deformed by small thermal loads, while perfect panels need a significant thermal load to be deformed. There is an interesting note from this figure that the thermal post-buckling curve decreases when $\xi $ increases with small deflection, but a point seems to exist, where the trend of thermal post-buckling is reversed.

Fig. 11 describes the nonlinear responses of ES-FGM sandwich spherical panel Cases 1 and 3 with $k_{\mathrm t} =k_{\mathrm b} =1$, $a/b=1$, $b/h=20$, $h_{\mathrm t}/h=0.15$, and $a/R_y =0.2$. Visibly, the thermal load-deflection curves in Case 3 are higher, or in other words, the thermal load capacity of shell in Case 3 is higher than that in Case 1. Specially, in the case of TID properties, the difference is more obvious than that in the case of TD properties.

8.4 Thermo-mechanical post-buckling of ES-FGM sandwich cylindrical panel

In what follows, an imperfect FGM sandwich cylindrical panel reinforced by FGM stiffeners in Case 1 resting on elastic foundation exposed to thermal environment under axial compressive load is considered.

The influence of the axial compressive load on the thermal post-buckling of ES-FGM sandwich cylindrical panel under both a uniform temperature field and an axial compressive load is investigated in Fig. 12. It can be observed that the existence of axial compressive load makes the thermal load capacity of cylindrical panel decrease and makes the imperfect panel bend more even when there is no thermal load.

The positive influence of FGM face sheets on the post-buckling loading capacity of ES-FGM sandwich cylindrical panel under uniaxial compression in the thermal environment is described in Fig. 13. 13 Influence of $h_{\mathrm t} /h$ on load-deflection curves for ES-FGM sandwich cylindrical panel under uniaxial compression in thermal environment (TID properties) when $b/h=20$, $k_{\mathrm t}=k_{\mathrm b}$=1, and $R_y$=3 m Simultaneously, the contribution of temperature makes imperfect sandwich cylindrical panel deflect even when the axial compressive load is applied.

Figure 14 compares the influence of temperature field on the load-deflection curve of perfect cylindrical panel in Cases 1 and 3. Clearly, the existence of temperature makes uniaxial compression capacity of shell become weaker. Besides, as there is no temperature field, the shell load capacity in Case 3 is better than that in Case 1. However, an opposite trend happens when there is contribution of the temperature.

9 Conclusions

This paper research highlights as follows: the symmetric sigmoid law distribution and the power law distribution are generalized for four different models of the sandwich double curved shallow shells; these shells are reinforced by FGM stiffeners; the TD material and stiffener properties are taken into account; three kinds of shells corresponding to three different radii of curvature are considered.

The theoretical formulations are established based on the TSDT considering the von Kármán nonlinearity, the initial geometrical imperfection, and the parameters of elastic foundation. The explicit analytical expressions for analysis buckling and post-buckling responses of FGM sandwich reinforced by FGM stiffeners on elastic foundation under mechanical, thermal, and thermo-mechanical loadings are obtained by the Galerkin method. The following conclusions can be drawn from the results.

(i) The FGM layer thickness, the volume fraction index, and the parameters of elastic foundation have significant influence on the critical buckling and load-deflection curves.

(ii) The sandwich shallow shells have better mechanical and thermal load endurance with the participation of FGM stiffeners than those with metal stiffeners or un-stiffened shell. Besides, the numbers of FGM stiffeners and their arrangement also have impact on the critical load of sandwich shallow shells.

(iii) The sandwich spherical panel is the best in load endurance among three types of sandwich shallow shells: sandwich spherical panel, sandwich cylindrical panel, and sandwich hyperbolic paraboloidal panel.

(iv) The mechanical post-buckling and thermal post-buckling of sandwich shallow shells in Case 1 (FGM/SUS304/FGM) are lower than those in Case 3 (Si3N4/FGM/SUS304). The TID case shows the difference of two laws more clearly than the TD case.

Appendix A

in which $E_i$ and $E_{{\mathrm s}i} $ are defined as follows.

For the shell in Cases 1 and 2,

In Case 1, $E_{\mathrm j} =E_{\mathrm c}$ and $E_{\mathrm {kl}} =E_{\mathrm {mc}} $, while in Case 2, $E_{\mathrm j} =E_{\mathrm m}$ and $E_{\mathrm {kl}} =E_{\mathrm {cm}} $.

In Cases 3 and 4,

In Case 3, $E_{\mathrm j} =E_{\mathrm c}$ and $E_{\mathrm {kl}} =E_{\mathrm {mc}}$, while in Case 4, $E_{\mathrm j} =E_{\mathrm m}$ and $E_{\mathrm {kl}} =E_{\mathrm {cm}} $.

For stiffeners,

In Cases 1 and 4, $E_{\mathrm j} =E_{\mathrm c}$ and $E_{\mathrm {kl}} =E_{\mathrm {mc}} $, while in Cases 2 and 3, $E_{\mathrm j} =E_{\mathrm m}$ and $E_{\mathrm {kl}} =E_{\mathrm {cm}} $.

Appendix B Appendix C Appendix D

and

in which