Nonlinear Analysis on Buckling and Postbuckling of Stiffened FGM Imperfect Cylindrical Shells Filled Inside by Elastic Foundations in Thermal Environment Using TSDT

The main aim of this paper is to investigate analytically nonlinear buckling and post-buckling of functionally graded stiffened circular cylindrical shells filled inside by Pasternak two-parameter elastic foundations in thermal environments and under axial compression load and external pressure by analytical approach. Shells are reinforced by closely spaced rings and stringers. The material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. Using the Reddy third order shear deformation shell theory, stress function method and Lekhnitskii smeared stiffeners technique, the governing equations are derived. The closed form to determine critical axial load and post-buckling load–deflection curves are obtained by Galerkin method. The effects of temperature, stiffener, foundation, material and dimensional parameters on the stability behavior of shells are shown. The accuracy of the presented method is affirmed by comparisons with well-known results in references. The results shown for thick cylindrical shells, the use of TSDT for determining their critical buckling load is necessary and more suitable.


INTRODUCTION
Functionally graded materials (FGM) were firstly introduced by a group of scientists in Sendai, Japan, in 1984 (Yamanouchi M, Koizumi M. 1990 andKoizumi M. 1993) and then were rapidly developed by other researchers.Due to essential characteristics such as high stiffness, excellent tem-Latin American Journal of Solids and Structures 14 (2017) 950-977 perature resistance capacity, structures made of functionally graded materials have been found wide applications in many modern industry fields such as space vehicles, aircrafts, nuclear power plants and many other engineering applications.As a result, many researches focused on the buckling and postbuckling analyses of FGM plates and shells.
For un-stiffened shells, many researches are focused on the buckling and postbuckling analysis of shells.Hui and Du (1987) studied initial postbuckling behaviors of imperfect antisymmetric crossply cylindrical shells under torsional load.Shen (2003) investigated the post-buckling analysis of pressure-loaded functionally graded FGM cylindrical shells in thermal environments based on the classical shell theory with von Karman-Donnell-type of kinetic nonlinearity.Also using the Donnell shell theory, Wu et al. (2005) solved the problem on the thermal buckling of FGM cylindrical shells with the linear buckling shape deflection.By the Laplace transform in time domain, the coupled thermoelastic response of FGM circular cylindrical shell was studied by Bahtui and Eslami (2007).Li and Shen (2008) presented the investigation on a post-buckling analysis of 3D braided composite cylindrical shells under combined external pressure and axial compression in thermal environment.They used the higher order shear deformation shell theory and the singular perturbation technique to determine interactive buckling loads and post-buckling equilibrium paths.Using the Ritz method, Huang andHan (2008 and2009), studied the buckling and postbuckling of un-stiffened FGM cylindrical shells under axial compression, radial pressure and combined axial compression and radial pressure according to the Donnell shell theory with the nonlinear strain-displacement relations and the three-term deflection shape.Bagherizadeh et al. (2011) investigated the mechanical buckling of FGM cylindrical shells surrounded by Pasternak elastic foundation using the higher-order shear deformation shell theory.Sofiyev and Kuruoglu (2013) studied the torsional vibration and buckling of cylindrical shell with FGM coatings surrounded by an elastic medium.Shariyat and Asgari (2013), based on the third order shear deformation theory with the von Karman-type kinematic nonlinearity and a nonlinear finite element method, studied the nonlinear thermal buckling and postbuckling analyses of imperfect cylindrical shells made of bidirectional FGM under uniform temperature rises.Tornabene et al. (2015) studied stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory.Sofiyev (2015) investigated the buckling or vibration of FGM truncated conical shells under external pressures or axial load.Sofiyev and Kuruoglu (2016) presented results on the stability of FGM truncated conical shells subject to combined axial and external mechanical loads in the framework of the shear deformation theory.Nejad et al. (2015) presented elastic analyses of FGM rotating thick truncated conical shells with axially-varying properties under non-uniform pressure loading.Ebrahimi and Najafizadeh (2014), by generalized differential quadrature and generalized integral quadrature methods, studied the free vibration of two-dimensional functionally graded cylindrical shells based on the Love first approximation classical shell theory.Wang and Nie (2015) proposed the theoretical model to predict the bi-stable states of initially stressed elastic cylindrical shell structures attached by two piezoelectric surface layers.
Note that the above introduced works only relate to unstiffened FGM structures or stiffened isotropic structures.However, in practice, plates and shells including conical shells, usually reinforced by stiffeners system to provide the benefit of added load carrying capability with a relatively small additional weight.Thus, the study on static and dynamic behavior of these structures are Latin American Journal of Solids and Structures 14 (2017) 950-977 significant practical problem.Singer et al. (1967) analyzed the stability of eccentrically stiffened cylindrical shells under axial compression with stiffeners attached to outside and inside of the shell skin.Ji and Yed (1990), using the Donnell shell theory and the perturbation technique, presented the general solution for nonlinear buckling of non-homogeneous axial symmetric ring-and stringerstiffened cylindrical shells.Reddy and Starnes (1993) studied the buckling of stiffened laminated cylindrical shells according to the layerwise theory and the smeared stiffener technique.Shen et al. (1993) investigated the buckling and post-buckling behavior of perfect and imperfect stiffened cylindrical shells under combined external pressure and axial compression by using the boundary layer theory.The singular perturbation technique to determine the buckling loads and the post-buckling equilibrium paths is applied in their work.By the perturbation technique and smeared stiffener technique, Shen (1997) presented thermal postbuckling analysis of imperfect stiffened laminated cylindrical shell of finite length subjected to uniform or non-uniform parabolic temperature distribution varying in the circumferential or axial direction.Shen (1998) considered the post-buckling of imperfect stiffened laminated cylindrical shell of finite length subjected to combined loading of external pressure and a uniform temperature rise.Also using perturbation method, Zeng and Wu (2003) reported investigation on the post-buckling of stiffened braided thin shells subjected to combined loading of external pressure and axial compression.Sadeghifar et al. (2011) investigated the buckling of stringer-stiffened laminated cylindrical shells with nonuniform eccentricity based on the Love first-order shear deformation theory.
For stiffened FGM shells, Najafizadeh et al. (2009) with FGM stiffener system, investigated the mechanical buckling behavior of functionally graded stiffened cylindrical shells reinforced by rings and stringer subjected to axial compressive loading based on stability equations given in terms of displacement.The stiffeners and skin, in their work, are assumed to be made of functionally graded materials and its properties vary continuously through the thickness direction.Following the direction of FGM stiffener type, Dung andHoa(2013 and2015) obtained results on the static nonlinear buckling and post-buckling analysis of eccentrically stiffened FGM circular cylindrical shells under torsional loads without or with thermal element based on the Donnell shell theory and Galerkin method.Dung and Hoa (2015) presented a semi-analytical approach for analyzing the nonlinear dynamic torsional buckling of stiffened functionally graded material circular cylindrical shells surrounded by an elastic medium.Following homogenous stiffeners, Bich et al. (2013) studied the nonlinear static and dynamic buckling behavior of eccentrically shallow shells and circular cylindrical shells based on the Donnell shell theory by analytical approach.Dung and Nam VH (2014) presented results on the nonlinear dynamic analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium.Duc et al. (2015) reported results on the mechanical and thermal stability of eccentrically stiffened functionally graded conical shell panels resting on elastic foundations and in thermal environment.
As can be observed that the studies (Dung andHoa (2013 and2015), Bich et al. (2013), Dung and Nam VH (2014) and Duc et al. (2015) were carried out by using the classical shell theory, so obtained results only suitable for FGM thin-walled shells.However for FGM thicker shells, it is necessary to use higher order theories.The new novelty of this work is to use the Reddy third order shear deformation theory (TSDT) for investigating the buckling and postbuckling of FGM thick circular cylindrical shells reinforced by stringers and rings and subjected to mechanical load, ther-mal load and filled inside by Pasternak two-parameter elastic foundations.Shells are reinforced by closely spaced rings and stringers.The material properties of shell and stiffeners are assumed to be continuously graded in the thickness direction.Using the Reddy third order shear deformation shell theory, stress function method and Lekhnitskii smeared stiffeners technique, the governing equations are derived.The closed form to determine critical axial load and post-buckling load-deflection curves are obtained by Galerkin method.The effects of temperature, stiffener, foundation, material and dimensional parameters on the stability behavior of shells are shown.

FOUNDATIONS
Consider a thin circular cylindrical shell as shown in Fig. 1, with mean radius R, thickness h and length L subjected to axial compression and external pressure load.Two butt-ends of shell are assumed to be only deformed in their planes and they still are circular.The middle surface of the shells is referred to the coordinates( , , ), x z y R q q = .The coordinate axis x, y, z are chosen in the generatrix, circumferential directions and thickness direction inward of the shell, respectively.In addition, assume that the FGM shell is reinforced by closely spaced FGM rings and stringers attached inside to the shell.The functionally graded materials of shells and stiffeners are assumed to be varied continuously in the thickness direction and made from a mixture of ceramic and metal.So the modulus of elasticity, coefficient of thermal expansion of shells and stiffeners are defined as For shell For stringers For rings The reaction-deflection relation of Pasternak foundation model is given by where

CONSTITUTIVE RELATIONS
Using the Reddy third order shear deformation shell theory, the strain components at the middle surface of imperfect circular cylindrical shells relating to displacements u = u (x,y), v = v(x,y) and w = w(x,y) of the middle surface points along x, y and z, are of the form (Brush DO, Almroth BO. 1975;Reddy JN. 2004;Shen HS. 2009) where ( ) * , w x y is a known function representing initial small imperfection in comparison with the thickness of the shell.
The strains across the shell thickness at a distance z from the middle surface are as ( ) and , x y f f are the rotations of normal to the mid-surface of the shell with respect to y and x axes, respectively.The strains from Eq. ( 5) must be satisfied the deformation compatibility equation as x For stiffeners , .
where T D is temperature rise from stress free initial state.
Latin American Journal of Solids and Structures 14 (2017) 950-977 The middle surface normal force intensities i N , the bending moment intensities i M and higher order bending moment intensities i P , transverse shearing force intensities i Q and the higher order shear force intensities i R of functionally graded shells reinforced by FGM stiffeners are defined as where , , , , , are respective quantities for stiffeners.
Substituting Eqs.(6, 7, 9, 10) into Eq.( 11) and using the Lekhnitskii smeared stiffener technique, after integrating resulting equations we obtain where Latin American Journal of Solids and Structures 14 (2017) 950-977 and the coefficients , , , , a b c d e can be found in Appendix A.
The strain-force reverse relations are found from Eq. ( 12) as .

NONLINEAR EQUILIBRIUM EQUATIONS AND STRESS FUNCTION
According to the Reddy third order shear deformation theory, the nonlinear equilibrium equations of a imperfect circular cylindrical shell filled inside by an elastic foundation and under uniform external pressure of intensity q are of the form (Brush DO, Almroth BO. 1975;Reddy JN. 2004 it is obvious that the Eq. ( 22) are identically satisfied.Replacing Eq. ( 26 Latin American Journal of Solids and Structures 14 (2017) 950-977 Eqs. (27,28,29) and ( 30) are four important governing equations used to investigate the nonlinear buckling of imperfect eccentrically stiffened functionally graded circular cylindrical shells surrounded by elastic foundation.Until now, there are no analytical investigations which have been reported in the literature on the postbuckling analysis of FGM thick cylindrical shells reinforced by FGM stiffeners system using Reddy TSDT.Therefore, the transformations and derivations to Eqs. (27,28,29,30) are one of the most important results in this work.
As can be seen the above system of equations is more complex than the one established by using the classical shell theory or the nonlinear stability analysis of un-stiffened FGM cylindrical shells.However, the higher-order theories (including the Reddy third order shear deformation theory) can represent better the kinematic behavior.This is also the main reason why these theories are used to investigate the nonlinear buckling and postbuckling of thicker FGM shells.

SOLUTION PROCEDURE AND GALERKIN METHOD
Suppose that the FGM cylindrical shell is simply supported, subjected to external pressure uniformly distributed q and axial compression of intensity P. The associated boundary conditions are of the form 0, 0, 0, 0, 0 0; .
The solutions of , , and m is a number of half wave in axial direction, n is a number of wave in circumferential direction of the shell, 0 W are amplitude of the deflection.
(35) we have Eq.( 38) is the general and explicit governing relation used to analyze the nonlinear buckling of ES-FGM imperfect circular cylindrical shells filled inside by elastic foundation in thermal environment under mechanical compressive loads, thermal and thermo-mechanical loads.

POSTBUCKLING ANALYSIS OF ES-FGM SHELLS SUBJECTED TO AXIAL COMPRESSION FORCE
In this case we have 0 0, .
Replacing Eq. ( 39) into Eq.( 38), leads to Eq. ( 40) is explicit expression used to determine postbuckling If the shell is perfect, Eq. ( 40) reduces to Taking 0 0 W  , from Eq.( 41), the critical static compressive load may be obtained as Minimizing Eq. ( 42) with respect to m and n, we will find the upper critical load .cr P

POSTBUCKLING ANALYSIS OF ES-FGM SHELL SUBJECTED TO EXTERNAL PRESSURE AND THERMAL LOADS
Assume that the shell is simply supported and immovable at two edges x = 0, x = L.So the immovable condition, u = 0 at x = 0, L is fulfilled on the average sense as In order to integrate this relation, firstly from Eq. ( 5), yields Suppose that environment temperature is uniformly raised from initial value i T at which the shell is thermal stress free, to final one f T and temperature change is constant and independent to thickness variable.So the thermal parameters of shell, stringer and ring, in this case, can be found respectively in terms of T D as follows If the shell only subjected to thermal loads without external pressure i.e q = 0. Setting Eq. ( 48) into Eq.( 47), after some calculations, leads to Latin American Journal of Solids and Structures 14 (2017) 950-977 Eq. ( 49) is the analytical relationship to determine the temperature-deflection curves for both of the perfect and imperfect circular cylindrical shells under thermal loads.For a perfect shell, Eq. ( 49) reduces to ( ) Taking 0 0 W  , from Eq.( 50) the thermal buckling load may be obtained as Minimizing Eq. ( 51) with respect to m and n, we will find a critical value .
cr T D

Validation of the present study
To verify the accuracy of the present solution, three comparisons are considered below.Table 1, using Eq. ( 42) compares the critical buckling load of un-stiffened isotropic cylindrical shell under axial compression with the results in the monograph of Brush and Almroth (Brush DO, Almroth BO. 1975).
Table 2 compares the critical axial load for un-stiffened FGM cylindrical shell without foundation and under axial load with the results given by Huang and Han (Huang H, Han Q. 2010).The input parameters are taken as Latin American Journal of Solids and Structures 14 (2017) 950-977 Table 3 compares the results on the critical buckling load of stiffened isotropic homogeneous cylindrical shells with the result of Brush and Almroth (Brush DO, Almroth BO. 1975) and with the result of Bich et al. (Bich DH et al. 2013).As can be shown in Tables 1, 2 and 3 that good agreements are obtained in these comparisons.
Latin American Journal of Solids and Structures 14 (2017) 950-977 8.2 Significance of the Use of the Reddy Third Order Shear Deformation Theory for Thicker Shells In order to demonstrate the significance of the use of TSDT, the FGM cylindrical shells under axial load are considered with the following geometric, material properties and foundation parameters as , 20, 30, 40, 50, 80, 100, 200 and 500.Using Eq. ( 42) in this study and Eq. ( 31) in study's Bich et al. (2013) , results of upper critical loads based on the Reddy's third order shear deformation shell theory and classical shell theory, are given in Table 4.
As can be seen, for thin shells, the difference between the upper critical loads found from classical shell theory and TSDT is quite small.However, for the thicker shells, the difference is quite big.For example, from  In subsections below, consider a shell with geometric and material properties as follows

Effects of Reinforcement Stiffener
The effect of reinforcement stiffener on critical buckling axial load are shown in Table 4 in which six cases are considered as: an un-stiffened shell, a stiffened shell with stringers 15 As can be seen that the critical load increases with the increase of the stiffener number.This increase is considerable.For example, the value of Pcr = 2590.16MPa (ns = 15, nr = 0, L = R) in comparison with Pcr = 2704.84MPa (ns = 20, nr = 20, L = R) increases about 4.2%.This is reasonable because the reinforcement stiffeners make the shells to become stiffer, so it has better carrying capacity.
Table 4 also shows effects of the ratio L/R on the critical compressive load.We can see the value of the critical compressive load decreases when the ratio L/R increases.
It is observed that when ratio R/h varies from 50 to 500, the value of critical compressive load decreases from 20704.84MPa to 388.52 MPa (in the case L = R).These characteristics are adequate to true property of shell i.e. the shell is thinner; the load bearing capacity is smaller.

Effects of Elastic Foundation Parameters on Critical Loads
Table 7 illustrate effects of elastic foundation on critical axial loads of shell with k = 1, R = 1m, h = hs = hr = R/50, bs = br = h/2, nr = ns = 20.It is found that the presence of elastic foundations increases the load carrying capacity of shells.In addition, the critical load corresponding to the contribution of the both two foundation parameters is biggest.
Latin American Journal of Solids and Structures 14 (2017) 950-977  8.7 Effects of the Initial Imperfection on Postbuckling P ×Wmax/h Curves Fig. 3 shows effects of initial imperfection on P×Wmax /h postbuckling curves by using Eq. ( 40).
The input parameters are taken as The imperfection parameter varies from 0 to 0.5.It is observed that the postbuckling load carrying capacity is reduced with the increase of imperfection size when the deflection is still small (In this present case max / 2 W h < ) , but an inverse trend occurs when the deflection is sufficiently large (In this present case max / 2 W h < ).Using Eq. ( 41) and the temperature 300 T T = +D , effects of the initial imperfection, stiffener number, ratio R/h and volume fraction indexes on max / T W h ´ temperature-deflection curves.
As can be seen that these parameters affect strongly on temperature-deflection curves.

CONCLUSIONS
This paper presents an analytical method for investigating the buckling and post-buckling of imperfect FGM cylindrical shells reinforced by FGM stiffeners filled inside by elastic foundations and subjected to mechanical loads or thermal loads.The material properties of shells and stiffeners are graded in the thickness direction according to a volume fraction power-law distribution.Using the Reddy TSDT with the von Karman kinematic nonlinearity and Lekhnitskii smeared stiffener tech- N M P Q R , are taken into account.The closed-form expressions for determining the buckling load and analyzing post-buckling load-deflection curves are obtained by Galerkin method.The comparisons results which are in good agreement with the previous known-well results, affirmed the reliability and accuracy of the proposed method.Some remarks are deduced from present results as: For thin shells, the difference between the upper critical loads found from CST and TSDT is quite small, so the classical shell theory can be used to study the stability of thin shells.However, for the thicker shells, the difference is quite big and the use of TSDT to analyze the nonlinear stability of circular cylindrical shells is necessary and more suitable.
The presence of stiffeners enhances the stability of FGM shells.
The thermal element, stiffener, foundation parameters and volume index affect strongly buckling and post-buckling behavior of shells.

For stringers
Latin American Journal of Solids and Structures 14 (2017) 950-977

Figure 1 :
Figure 1: Geometry and coordinate system of a stiffened FGM circular cylindrical shell on elastic foundation.
Winkler foundation modulus and 2 K (N/m) is the shear layer foundation stiffness of Pasternak model, w is the deflection of the shell.

Figure 2 :
Figure 2: Effects of R/h on critical axial loads.

Figure 3 :
Figure 3: Effects of the initial imperfection on max / P W h ´ curves.

Figure 4 :
Figure 4: Effects of the initial imperfection on max / T W h ´ curves.

Figure 5 :Figure 6 :
Figure 5: Effects of stiffener number on max / T W h ´ curves.

Figure 7 :
Figure 7: Effects of volume fraction indexes on max / T W h ´ curves.

Table 1 :
Comparisons of critical buckling load P* for un-stiffened isotropic cylindrical shells under axial compression (E=70 GPa, ν = 0.3) a the numbers in the parentheses denote the buckling mode (m,n)

Table 2 :
Comparisons of critical buckling axial load for un-stiffened FGM cylindrical shells

Table 3 :
Comparisons of critical buckling axial load of stiffened isotropic homogeneous cylindrical Table 4, in comparison cr P = 0.2499 MPa (based on CST) and cr P = 0.2497 MPa (based on TSDT) corresponding to R/h = 500 (thin shell), the percentage error is 0.0864%, but when R/h = 10 (thick shell), the corresponding the percentage error is 4.0985%.It means, when studying the thick shells, should using the Reddy's third order shear deformation shell theory, for higher precision.
a the numbers in the parentheses denote the buckling mode (m,n)

Table 4 :
Upper critical loads MPa crPfound by CST and TSDT.

Table 5 :
Effects of Stiffeners on critical load Pcr MPa

Table 6 :
Effects of R/h on critical load Pcr MPa.8.5 Effects of Volume Fraction Indexes k, k2 and k3 on Critical Axial LoadsTable6and Fig.2describe effects of volume fraction indexes k, k2 and k3 on the critical load of the shell.It is seen that critical axial loads of shells decrease when k increases.This is expected because the higher value of k corresponds to a metal-richer shell which usually has less stiffness, so the load carrying capacity of the shell decreases.This decrease is significant.For example Pcr =4559.36MPa (k = 0.1, L = 2R) in comparison with Pcr = 2081.58MPa (k = 2, L=2R) decreases about 2.2 times.

Table 8 :
Effects of K1 and K2 on static critical loads.

Latin American Journal of Solids and Structures 14 (2017) 950-977 nique
; the nonlinear stability equations for ES-FGM cylindrical shells are derived.Eqs.(12-16) and (27-30) are the most important results found in this study in which the contribution of stiffeners and thermal elements in equations of , , , ,