Analytical and Numerical Investigation of FGM Pressure Vessel Reinforced by Laminated Composite Materials

Transcription

1 Jounal of Solid Mechanics Vol. 6, No. 1 (2014) pp Analytical and Numeical Investigation of FGM Pessue Vessel Reinfoced by Laminated Composite Mateials A.R Ghasemi *, A. Kazemian, M. Moadi Depatment of Mechanical Engineeing, Univesity of Kashan, Kashan, Ian Received 3 Septembe 2013; accepted 8 Decembe 2013 ABSTRACT In this eseach, the analytical and numeical investigation of a cylindical shell made of functionally gaded mateials (FGMs) einfoced by laminated composite subjected to intenal pessue is pesented. Using the infinitesimal theoy of elasticity, the analytical solution of stess and stain in vessels made of FGMs is studied fist. It is assumed that the elasticity modulus follows a powe law distibution in the thickness diection and Poisson's atio consideed to be constant fo simplicity. The esults of the finite element method using ABAQUS softwae fo in-homogeneity constant in the ange of -2 to 2 have been compaed with the analytical esults. The compaison epesents good coincidence between analytical and numeical esults and confims the accuacy of stess and stain solutions pesented fo vessel made of FGMs. The stess and stain solutions in laminated composite vessels ae then investigated. Finally, modeling of FGM vessel einfoced by composite laminates with diffeent lay-up is taken into consideation. The obtained esults demonstate that in the cylindical shell einfoced by laminated composites, the maximum stess is consideably less than the maximum stess in the pessue vessels made of just composites o FGMs.. All ights eseved. Keywods: Stess analysis; Cylindical pessue vessels; FGM; Composite; Finite element 1 INTRODUCTION N OWADAYS with eve-inceasing industial development, pessue vessels and pipes ae widely used in eacto technology, chemical industies, maine and aeospace engineeing. Since in most applications, vessels must opeate unde extemes of themal and mechanical loadings, any failue o factue will be an iepaable disaste. So, adequate stength consideation is so impotant fo these components. Composite mateials ae suitable candidate fo use in pessue vessel. Fo example, Adali et al. pesented an exact solution method fo optimization of laminated composite vessels unde intenal pessue [1]. They obtained a thee-dimensional theoy fo anisotopic composite thick-walled cylindes subjected to symmetic loads. Optimization of pessue vessels showed that the stacking sequence can be employed fo maximum bust pessue, effectively. Mackele pesented a compehensive eview ( ) of theoetical and pactical analysis of pessue vessel stuctue/components and piping. Also, he studied diffeent analysis, applications and components of pessue vessels [2-4]. Kabi pesented intenally pessuized cylindical vessels with ove-wapped metallic line. His analysis esults showed that metallic line causes significant decease in on-axis stess, in both helical and cicula layes [5]. Xia et al. pefomed the analysis of multi-layeed filament-wound composite pipes subjected to intenal * Coesponding autho. Tel.: ; Fax: addess: Ghasemi@kashanu.ac.i (A.R. Ghasemi).. All ights eseved.

2 44 A.R. Ghasemi et al. pessue. They pesented an elastic exact solution fo stess and defomation distibution of these pipes [6]. Futhemoe, using 3D anisotopic elasticity, they showed that the stess and displacement distibutions stongly depend on the stacking sequence. Panas and Nuan developed an analytical pocedue to assess the behavio of a cylindical composite stuctue unde loading conditions paticula to the ocket moto case [7]. In thei study, the classical laminate theoy and the genealized plane stain model wee used to fomulate the elasticity poblem. Also, they studied diffeent loading conditions consist of axial foce and body foce due to otation, tempeatue and moistue vaiations thoughout the body fo a cylindical vessel subjected to intenal pessue. They analyzed a cylindical pessue vessel using two thick-walled and thin-walled solution appoaches. Then they showed that fo composite pessue vessels with a adius atio (oute to inne adius atio) up to 1.1, two appoaches give the same esults in tems of the optimum winding angle and bust pessue. Also, they showed that the thick-walled analysis is equied as this atio inceases [7]. Futhemoe, the esults showed that the maximum angula velocity is stongly influenced by the thickness of the vessel. Theefoe, the thin-walled analysis is said to be an aveage but a safe solution. Hocine et al. studied a vessel made of cabon/epoxy envelope coated on metallic line, expeimentally and analytically. They pesented an exact solution of stesses and stains on cylindical section of vessel unde mechanical loading. Also, they validated the analytical esults by manufactuing and testing some pototype vessels [8]. Stesses, stains and displacements obtained fom the analytical solution [8] ae in good ageement with the numeical esults obtained fom finite element softwae (ANSYS 11.0). Baoping et al. pesented the eliability-based load and esistance facto design of composite pessue vessel subjected to extenal hydostatic pessue. Compaed with the deteministic design methods, the pobabilistic design method povides some advantages such as moe accuate failue pobabilities and avoids costly ove-design and consevative manufactuing of composite vessels [9]. Despite many advantages of full-composite pessue vessels, they have some limitations such as stess concentation, cack at high tempeatue and leakage. In the othe hand, functionally gaded mateials (FGMs) have known advantages such as smooth change of physical and mechanical popeties. Recently, thee has been gowing inteest in mateials delibeately fabicated so that thei mechanical popeties vay continuously in space on the macoscopic scale [10]. Many studies ae done by eseaches on analysis of FGM pessue vessel o disc [10-13]. Fo example, Tutuncu and Muat pesented an exact solution of stesses in FGM pessue vessels. They used infinitesimal elasticity theoy and isotopic mateials assumption to obtain closed-fom solutions fo stesses and displacements in cylindical and spheical FGM vessels subjected to intenal pessue [14]. Poisson s atio was assumed to be constant and adially vaying elastic modulus was appoximated by E() E0, in which is away fom inne to oute adius. Thei esults showed that moe stiffness is needed nea the inne suface fo bette esistance against applied intenal pessue. Thus, by deceasing the stiffness in the adial diection, stess inceases compaed to homogenous case. Using the infinitesimal theoy of elasticity, Dai et al. [10] obtained exact solution of FGM cylindical and spheical vessels subjected to intenal pessue and a unifom magnetic field. Same as [14] the mateial constants follow powe functions in the adial diection. Numeical esults showed that the gadient index has a geat effect on the magneto-elastic stess and petubation of magnetic field vecto. Also, Tutuncu studied the stesses of thick-walled FGM vessels with vaiable exponential popeties in anothe wok and consideed the exponentially-vaying elastic modulus thoughout thickness in the fom of E() E0e [15]. Shaiyat et al. [11] developed analytical and numeical elasto-dynamic solutions fo long thick-walled functionally gaded cylindes subjected to abitay dynamic and shock pessues. Vaiations of the mateial popeties acoss the thickness ae descibed accoding to both polynomial and powe law functions. A numeically consistent tansfinite element fomulation is pesented fo both functions wheeas the exact solution is pesented fo the powe law function. Results obtained fo vaious exponents of the functions of the mateial popeties distibutions, vaious adius atios, and vaious dynamic and shock loads. Rotating disc and pessue vessel have simila steps to fomulation. So using FGMs otating disc is suitable. Ghobanpou et al. [12-13] pesented a semi-analytical solution fo magneto-themo-elasticity poblem and steady state conduction heat tansfe of a thin axisymmetic FG otating disk with vaiable [12] and constant [13] thickness. The mateial popeties of FG disc except Poisson s atio modeled as the powe-law distibution of volume faction. The effects of mateial's gading index on the stesses, the adial displacement, the tempeatue, and the petubation of magnetic field vecto of the FG otating disk had been investigated [12]. Solution fo stesses and petubation of magnetic field vecto in a thin FG otating disk is detemined using infinitesimal theoy of magneto-themoelasticity unde plane stess conditions [13].

3 Analytical and Numeical Investigation of FGM Pessue Vessel Reinfoced by Laminated Composite Mateials 45 These eseaches show that analytical and numeical study of FGMs pessue vessel and otating disc is done extensively but they ae limited only to stess and defomation analysis. Optimization of these mateials is limited to select a pope value of in E() E0e. Despite of the FGMs, in composite pessue vessel using diffeent sequential aangement [1], coating [8] and metal line [5] ae some of optimization methods. Suvey of these aticles show eseaches esticted thei analysis to composite pessue vessels o FGM vessels. Composite mateials cause to pefomance impovement and mateial saving of vessels. Howeve, the majo disadvantage of these mateials is apid changes of inteface popeties of layes, which causes stess concentation, matix cacking and delamination. Additionally cabon/epoxy composite vessels, have a lot of poes that cause leakage of component. As the FGMs ae new mateials with desiable popeties and composite mateial with high stength to weight faction, combination of these two mateials is a new idea to gain FGMs advantages to impove composite disadvantages. Howeve, a eseach on stess and stain distibution in hybid vessels made of FGMs einfoced by laminated composite is a new appoach to find new mateial with pominent popeties. In this study, FGMs ae used to educe the stess concentation, contol cack gowth and to pevent leakage of fluid in composite pessue vessel. Fo these puposes, fist analytical solution of thick-walled FGM pessue vessels is consideed. Then, numeical study is conducted using the ABAQUS commecial softwae. Afte that, numeical study of FGM pessue vessels einfoced by laminated composite subjected to intenal pessue is accomplished. Finally, stess distibution in composite vessel, FGM vessel and hybid vessel ae compaed and evaluated. 2 ANALYTICAL SOLUTION OF FGM VESSELS FGMs vessels ae thick-wall o thin-wall cylindes that contain fluids fo diffeent applications. A schematic of finite element model of the vessel is shown in Fig. 1. Fo a composite vessel o FGM vessel, FGM o composite thickness is zeo, espectively. The adial coodinate and the displacement u ae nomalized as and R u u whee R is the oute adius of the cylinde. Poisson's atio is assumed to be constant and adially vaying R elastic modulus is appoximated by E() E0. Since is away fom zeo and anges in (a,r), by adjusting the constants E 0 and, it is possible to obtain physically popeties esults. Fig. 1 Schematic of FGM pessue vessels einfoced by laminated composites subjected to intenal pessue. Assuming plane stain and axisymmety of cylindical coodinate poblem [14-15] the only non-tivial equilibium equation is expessed in tems of: 0 (1) whee and ae adial and cicumfeential stesses, espectively. Substituting the stain-displacement elation and constitutive equation in equilibium equations, the following equation is obtained [14-15]:

4 46 A.R. Ghasemi et al. (1 ) u u E E u u u u (1 ) (1 ) (1 ) 0 (1 )(12 ) (1 )(12 ) (2) In which v and u ae Poisson s atio and adial displacement, espectively. Applying diffeentiation and simplifying the above equation, the Cauchy-Eule equation (Eq. 3) is obtained [14-15]. 2 v * v u ( 1) u 1u 0 v (1 v) (1 v) (3) Fo the numeical values of 2 2 and * 0.3 consideed in this pape, thee ae only distinct eal oots. Fo distinct eal oots, the exact solution is: m m u A B 1 2 (4) And the oots of the above equation ae: 1 2 4( * m1 1) 2 1 m 2 4( * 2 1) 2 (5) The constants A and B ae detemined fom the bounday conditions. Accoding to Fig. 1, the bounday conditions of the poblem can be defined as pessue on the inne suface equal to -P and zeo pessue on the oute suface. a ( ) P BCs. R R ( ) (1) 0 R (6) Utilizing Eq. (6), these constants ae obtained as follows [14-15]: 0 (1 )(1 2 ) A ( a 1 E (1 ) m P( ) 0 1, R 0 m m 0 (1 )(1 2 ) a ( ) 1 a ( ) 2 B E (1 ) m R R 0 2 (7) Finally the expessions fo the adial and cicumfeential stesses ae deived as follows [14-15]: m 1 m 1 ( 1 2 ) 0 (1 ) 1 (1 ) m m m m ( 1 2 ) 0 (1 ) m (1 ) m 1 2 (8) 3 THICK-WALLED FGM CYLINDRICAL VESSEL MODELING Numeical analysis of thick-walled cylindical vessel is pefomed using the ABAQUS, finite element commecial softwae. Geometic and mateial popeties and bounday conditions ae given in Table 1. Afte simulation, in ode to eliminate bounday conditions effects on stess values, the stess esults ae ead fa fom the vessel edges. As

5 Analytical and Numeical Investigation of FGM Pessue Vessel Reinfoced by Laminated Composite Mateials 47 thee isn t any defined element to model FGMs in the ABAQUS softwae, thick-wall cylinde patitioning method is used in fom of paallel concentic cylindes. Then, accoding to E() E0 elation, elastic popeties of each cylinde is allocated. This model is a pecise appoximation method to simulate the FGM vessel. Table 1 Geometic paametes, elastic constants and bounday condition of the model ν E0 (GPa) R (mm) a (mm) 50 P (MPa) 7 Fig. 2 Stess distibution of FGM vessel modeling unde intenal pessue. Table 2 Radial and cicumfeential stesses of FGM cylinde fo 1 Analytical[14-15] FEM Eo (%) R (mm) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) In Table 1., E, a, R and P ae Poisson s atio, elastic modulus, inne adius, oute adius and intenal pessue of the vessel, espectively. Modeling and simulation is done fo numeical values of 2, 1.5,1,2. Stess distibution of finite element modeling of the FGM vessel is shown in Fig. 2. Radial and cicumfeential stess values fom the analytical solution and the finite element method (FEM), and the diffeence between them fo in-homogeneity constant 1 is given in Table 2. In this table, the maximum diffeence between the analytical and the numeical values of the adial and the cicumfeential stess is about 0.24% and 5%, espectively. This diffeence has maximum value at the inne adius and minimum value at the oute adius. As the inne adius is subjected to pessue, Saint-Venant s pinciple has effected on FEM esults and maximum eo exists in this adius. Incease the adius made this effect decease and the esults ae pecise. The atio of adial stess in the FGM shell (F) to the homogeneous shell (H) is shown in the Fig. 3. These values ae obtained fom the analytical solution [14-15]. The Fig. 4 shows the same stess atio fo the numeical solution using the ABAQUS finite element softwae. The analytical and numeical esults have been depicted in two distinct figues fo bette illustation of diffeences. Also, the figues ae plotted based on dimensionless /R paametes. The similaities and esults consistent between the two figues ae ecognizable, clealy. Thus numeical esults of FEM and pecise mathematical analysis have good ageement and eos always ae less than 5%.

6 48 A.R. Ghasemi et al. Fig. 3 Radial stess distibution in FGM cylinde using analytical method [14-15]. Fig. 4 Radial stess distibution in FGM cylinde using FEM. 4 MODELING FGM CYLINDRICAL SHELL REINFORCED BY COMPOSITES LAMINATES In this section, the hybid cylindical model, that is FGM cylindical shell einfoced by composite laminates, is studied. In this model, oute and inne adius of the vessel is consideed to be 55 mm and 50 mm, espectively. Also, intenal suface is subjected to 10 MPa pessue. The hybid shell consists of FGM and composite laminates with thickness of 3 and 2 mm, espectively. Each FGM laye has 1 mm thickness and the its elastic popeties ae defined using powe law elation E() E0. Futhemoe, the composite pat of the shell consists of fou layes with 0.5 mm thickness, made of cabon/epoxy (T300/934). The elastic popeties of unidiectional cabon/epoxy composite laminates ae mentioned in Table 3. Table 3 Elastic popeties of unidiectional cabon/epoxy ply [16] E xx (GPa) E yy (GPa) G xy (GPa) 3.88 V xy whee E xx is the longitudinal modulus, E yy is tansvese modulus, G xy is the shea modulus, and V xy is the majo Poisson s atio. In ode to veify the esults obtained fom the hybid vessel and have a suitable compaison, FGM and full-composite vessels ae modeled with 5 mm thickness. Fig. 5 shows the maximum on-axis stess distibution in FGM cylindical shell which consists of 5 layes with 1mm thickness and in-homogeneity constant of 2. Also, the maximum on-axis stess distibution in a composite cylindical shell with symmetic stacking sequences [+53/- 53]10S is shown in Fig. 6.

7 Analytical and Numeical Investigation of FGM Pessue Vessel Reinfoced by Laminated Composite Mateials 49 Fig. 5 Maximum pincipal stess distibution in FGM cylinde ( 2 ). Fig. 6 Maximum on-axis stess distibution in symmetic [+53/-53]10S laminated composite cylinde. Fig. 7 Maximum on-axis stess distibution in unsymmetic [+53/-53]10T laminated composite cylinde. The stess of FGMs vessel inceases fom inne laye to oute laye in step fom, egulaly as shown in Fig.5. This step fom stess stated fom 92.5 MPa and inceased to MPa. Also, diffeence between each step is almost simila (3.7 MPa). The minimum and maximum stesses in composite vessel ae 163 and 228 MPa, espectively (Fig. 6). The stess distibution same as stacking sequence is symmetic. The distibution of on-axis stess fo unsymmetic layup [+53 /-53]10T is shown in Fig.7. The minimum and maximum on-axis stesses in fibe diection fo this layup ae almost 190 MPa and MPa, espectively. Clealy, the stess fluctuation in this aangement is much less than symmetic layup. Futhemoe, compaisons of these thee figues show that the stesses in the composite shell ae highe than FGM shells. In the composite cylinde all the laminates have the same othotopic popeties, wheeas in FGM cylinde, mateial popeties aen t constant and impove thoughout the thickness. This mateial changes ae the main eason of decease and contol stess in the FGM vessels. The stess distibution in the hybid FGM-Laminated composite vessel is diffeent. Figs. 8 and 9 show the maximum stess distibution in hybid cylindical FGM shells einfoced by symmetical and unsymmetical laminated composite mateials, espectively. In these models in-homogeneity constant is 2.

8 50 A.R. Ghasemi et al. Fig. 8 Maximum stess distibution in FGM-Composite cylinde with symmetic [+53/-53]4S laminates ( 2 ). Fig. 9 Maximum stess distibution in FGM-Composite cylinde with unsymmetic [+53/-53]4T laminates ( 2 ). The stess distibutions in the FGM pat of both aangements ae the same. The stepped stesses ae about 128 MPa, 135 MPa and 140 MPa. Afte FGM layout, thee is a consideable fall in stess value in both layups. The stess of the symmetic aangement afte this fall is almost constant (93.75 MPa), wheeas, the stess of the unsymmetic composite laminates oscillates between 93.5 and 94 MPa. This stess eduction and oscillation damping in the einfoced shell ae significant. This is one of the benefits of the laminated composite einfocement idea. 5 STACKING SEQUENCES EFFECTS In this section, diffeent stacking sequences implied to the FGM vessel to study optimization effects of these changes. Fist, fo bette compaison with the hybid vessel, numeical analysis fo laminated composite vessel with [+55/-30]10S and [+55/-30]10T layups ae studied. These symmetic and unsymmetic sequences ae chosen to find optimum layups. The esults of this analysis ae pesented in Figs Fig. 10 Maximum on-axis stess distibution in composite cylinde with symmetic [+55/-30]10S layup. Fig. 11 Maximum on-axis stess distibution in composite cylinde with unsymmetic [+55/-30]10T layup.

9 Analytical and Numeical Investigation of FGM Pessue Vessel Reinfoced by Laminated Composite Mateials 51 The Fig.10 shows the maximum on-axis stesses distibution in symmetic composite cylinde with [+55/-30]10S layups. As it is clea fom this figue, the minimum and the maximum stesses ae elied between 122 MPa and 129 MPa and between 203 MPa and 206 MPa, espectively. The stess distibution of unsymmetic [+55/-30]10T layup shows that the stesses ae elied almost between 140 to 170 MPa. The stess oscillation of this layup deceases along the inne plies to the oute plies. So, unsymmetic sequence has less aveage maximum on-axis stess and less stess change than symmetic sequence. Afte the composite vessel study, hybid FGM-laminated composite vessel using these sequences is analyzed. The stess distibutions of the FGM vessel einfoced by [+55/-30]4S and [+55/-30]4T composite layups ae obtained using FEM analysis and shown in Figs Fig. 12 Maximum stess distibution in FGM-composite cylindical shell with symmetic [+55/-30]4S layup ( 2 ). Fig. 13 Maximum stess distibution in FGM-composite cylindical shell with unsymmetic [+55/-30]4T layup ( 2 ). Fig.14 Maximum stess distibution in FGM-composite cylinde with symmetic [+45/-45]4S layup. As shown in Figs , the stess distibution of the FGM layes in both aangements ae equal. The stess steps of the FGMs layes ae 110 MPa, 115 MPa and 119 MPa. Like the hybid vessels studied in section 4, thee is a stess fall afte the FGM layes. Hee, the stess is fallen to 90 MPa on the +55 degee laye; but on the -30 degee laye a significant incease up to 145 MPa can be seen. The eason of the stess distibution diffeences is the symmetic and unsymmetic layups of the two stacking sequences. In the symmetic and unsymmetic layups, the stess distibution is symmetic and unsymmetic, espectively. It should be noted that the maximum stess of the FGM-composite pessue vessel is equal to the minimum stess of the composite pessue vessel. So, the FGM-composite vessel is moe efficient than the composite vessel.

10 52 A.R. Ghasemi et al. Futhemoe, fo bette study of layups, anothe stacking sequence is consideed. As the above sequences used 30 and 55 ply, 45 ply could be anothe choice fo laminates. So, the angle-ply [+45/-45] laminated composite mateial is chosen fo einfocement. The stess distibution in the FGM-composite cylindical shell with symmetic layup [+45/-45]4S is shown in Fig.14. The stesses of the FGM layes fom inne to oute adius of vessel ae 116 MPa, 121 MPa and 125 MPa, espectively. These stesses ae a little highe than [+55/-30]4S and [+55/-30]4T composite layups (about 5.5%). Afte the FGM layes, the stess deceases thoughout the composite layes up to 115 MPa. This stess dop is vey significant and emains nealy constant in the laminated composite mateials. 6 CONCLUSIONS In this study, a new method to einfoce the FGM cylindical pessue vessel by the composite laminates is pesented. Analysis of the stess and the stain in the pessue vessel made of functionally gaded mateial ae studied, analytically and numeically. The esults of the FEM ae in good ageement with the exact solution. This esults show that the FEM is sufficiently accuate fo modeling the FGM vessel. In the FGM vessel the adial stess inceases and the cicumfeential stess deceases with inceasing of in-homogeneity constant between -2 and 2. Also, esults show that in the hybid vessel stess distibution depend on stacking sequences of composite cylindical vessel, diectly, so that if the layup stacking sequence is being symmetic/unsymmetic, the stess distibution is symmetic/unsymmetic, espectively. Also, the stess values of composite layes usually ae vey high. In the hybid cylindical vessel, the stesses at the inteface between the FGM and the composite layes decease fo all studied aangements, significantly. The amount of eduction depends on layups, quietly. Among the study of diffeent stacking sequences, [+53/-53] angles have the maximum stess eduction equal to 33% in some examples. Accoding to the diffeent sample modeling, it seems when the angle of einfocing laminates appoaches to the optimum angle in twisting fibe in filament winding method (almost 54.7 ), the stess eduction will be futhe and einfocement will done bette. ACKNOWLEDGEMENT The authos ae gateful to the Univesity of Kashan fo suppoting this wok by Gant No /06. REFERENCES [1] Adali S., Veijenko V.E., Tabakov P.Y., Walke M., 1995, Optimization of multilayeed composite pessue vessels using exact elasticity solution, ASME-Publications-PVP 302: [2] Mackele J., 1999, Finite elements in the analysis of pessue vessels and piping, an addendum, Intenational Jounal of Pessue Vessels and Piping 76(7): [3] Mackele J., 2002, Finite elements in the analysis of pessue vessels and piping, an addendum: a bibliogaphy, Intenational Jounal of Pessue Vessels and Piping 79(1):1-26. [4] Mackele J., 2005, Finite elements in the analysis of pessue vessels and piping, an addendum: A bibliogaphy, Intenational Jounal of Pessue Vessels and Piping 82(7): [5] Kabi M.Z., 2000, Finite element analysis of composite pessue vessels with a load shaing metallic line, Composite Stuctues 49(3): [6] Xia M., Takayanagi H., Kemmochi K., 2001, Analysis of multi-layeed filament-wound composite pipes unde intenal pessue, Composite Stuctues 53(4): [7] Panas L., Nuan K., 2002, Design of fibe-einfoced composite pessue vessels unde vaious loading conditions, Composite stuctues 58(1): [8] Hocine A., Chapelle D., Boubaka M. L., Benama A., Bezazi A., 2009, Expeimental and analytical investigation of the cylindical pat of a metallic vessel einfoced by filament winding while submitted to intenal pessue, Intenational Jounal of Pessue Vessels and Piping 86(10): [9] Baoping C., Liu Y., Liu Z., Tian X., Ji R., Li H., 2011, Reliability-based load and esistance facto design of composite pessue vessel unde extenal hydostatic pessue, Composite Stuctues 93(11): [10] Dai H.L., Fu Y.M., Dong Z.M., 2006, Exact solution fo functionally gaded pessue vessels in a unifom magnetic field, Intenational Jounal of Solids and Stuctues 43: [11] Shaiyat M., Nikkhah M., Kazemi R., 2011, Exact and numeical elastodynamic solutions fo thick-walled functionally gaded cylindes subjected to pessue shocks, Intenational Jounal of Pessue Vessels and Piping 88(2):

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