Nonlinear vibration of stiffened multilayer FG cylindrical shells with spiral stiffeners rested on damping and elastic foundation in thermal environment

Thin-Walled Structures 145 (2019) 106388

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Nonlinear vibration of stiffened multilayer FG cylindrical shells with spiral stiffeners rested on damping and elastic foundation in thermal environment

T

Habib Ahmadi∗, Kamran Foroutan Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

ARTICLE INFO

ABSTRACT

Keywords: Spiral stiffened Multilayer FG cylindrical shell Elastic foundation Multiple scales method Primary resonance Superharmonic and subharmonic resonances

This paper presents the nonlinear vibration of multilayer FG cylindrical shells reinforced by spiral FG stiffeners surrounded by damping and elastic foundation in thermal environment. The primary, superharmonic and subharmonic resonances of spiral stiffened multilayer FG cylindrical shell are analyzed. It is assumed that the material properties are dependent to temperature. The linear elastic foundation is based on the Winkler and Pasternak model. In order to model the stiffeners, the technique of smeared stiffener is utilized. With regard to classical plate theory of shells, von Kármán equation and Hook law, the relations of stress-strain is derived for shell and stiffeners. According to the Galerkin method, the discretized motion equation is obtained. The primary, superharmonic and subharmonic resonances in presence of thermal environment are analyzed by using the method of multiple scales. The influence of the material parameters, temperature, elastic foundation and various geometrical characteristics on the primary, superharmonic and subharmonic resonances is investigated.

1. Introduction The FG materials are extensively utilized in high thermal environments and the stiffeners are widely used to reinforce the system structural. Also, in order to better performance of systems, the multilayers form is considered. So, the stiffened multilayers FG systems are impressively utilized in the various engineering industrial such as offshore, bridges, submarines, ships, satellites, and aircrafts structures. Some studies are concentrated in the field of plate resonance. The nonlinear resonance analysis of shear-deformable plates with or without initial curvature by utilizing the method of multiple scales was addressed in Ref. [1]. In Ref. [2], the resonance behavior of FG piezoelectric porous plates in the translation state was presented. The resonance analysis of the plate consist of the primary, subharmonic, and superharmonic response was investigated in Ref. [3]. Some studies are paid attention to resonance behavior of the shells, the resonance analysis of cylindrical shells with axially moving was presented in Ref. [4] by means of the method of harmonic balance. In Ref. [5], the internal resonance of imperfect circular cylindrical shell under transversally excitation was studied. In Refs. [6,7], the nonlinear resonance response (three dimensional analysis) for thin shells was analyzed. In these reported works, the nonlinear resonance equations of motion have been solved through the FETM method and the procedures of the direct time integration, respectively. In Ref. [8], the vibration

∗

analysis of the cylindrical shells by means of the method of perturbation was investigated. The nonlinear vibration of the cylindrical shell consist of fractionally damped with a three-to-one internal resonance was considered in Ref. [9]. For other types of cylindrical shells from the viewpoint of the composition of material, the internal resonance response of clamped shallow shells was analyzed in Ref. [10]. In Ref. [11], the resonant analysis of composite laminated circular cylindrical shell was investigated. In this work, the steady-state response was obtained by using the shooting method. In the reported works mentioned above, the researchers have not addressed the resonance behaviors of cylindrical shells with FG material. However, these structures have been utilized in many industrial applications. Then, some researches have been performed the nonlinear dynamic response of cylindrical shells with FG material. In Ref. [12], the resonant response of the functionally graded shallow shells via the method of multiple scales was presented. In Ref. [13], the parametric resonance behavior of the functionally graded cylindrical shell in thermal environment was studied. The cylindrical shells vibrations with FG material under external and parametric excitations was analyzed in Ref. [14]. In Ref. [15], the cylindrical shells resonance with FG material in thermal environment was addressed. Primary resonance of the FG porous cylindrical shells was analyzed in Ref. [16]. In this research, the governing equations according to the von Kármán equation and

Corresponding author. E-mail address: [email protected] (H. Ahmadi).

https://doi.org/10.1016/j.tws.2019.106388 Received 2 March 2019; Received in revised form 7 August 2019; Accepted 30 August 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Nomenclature

u, v Wmn (t ) x , y, z

c damping coefficient d width of stiffeners Em, Ec Young's modulus of metal and ceramic, respectively Esh , Est Young's modulus of shell and stiffeners, respectively F (t ) harmonic excitation stiffener loads in the X Y -axis and X Y -axis F, F h, hs thickness of shell and stiffeners, respectively hu, hm, hb thickness of upper, middle and lower layers, respectively Ksh, Kst volume-fraction index of shell and stiffeners, respectively Km, K c thermal conductivity of metal and ceramic, respectively ks shear layer stiffness of Pasternak foundation kw Winkler foundation L , ls length of the shell and stiffener grid, respectively Mx , My , Mxy bending moment and twisting moment intensities, respectively m number of half wave in axial direction Nx , Ny , Nxy in-plane normal force and shearing force intensities, respectively n number of full wave in circumferential direction R radius of the shell s stiffener spacing t Time

sh,

st

x, y, 0 0 x, y 0 xy

m,

sh , sh x , st x , sh xy ,

c

st sh y st y s xy

mn

Gf f

displacements along x , y axes, respectively deflection amplitude axial, circumferential and radial direction of the cylindrical shell thermal expansion coefficient of shell and stiffeners, respectively perturbation parameter angle of the stiffeners xy change of curvatures and twist of shell normal strains at the mid-surface shear strain at the mid-surface angle of the stiffeners Poisson's ratio density of metal and ceramic, respectively density of shell and stiffeners, respectively normal stress of shell in x , y coordinates, respectively normal stress of stiffeners in x , y coordinates, respectively shear stress of shell and stiffeners, respectively stress function natural frequency of cylindrical shell frequency of the excitation effective shear modulus local bulk modulus

z and y = R are the axial, radial and circumferential direction, respectively. It is considered that the shell has three layers, consist of metal, FG and ceramic. Also, internal spiral stiffeners are composed of ceramic and metal. Considering the power law, the constituent volume fractions are defined as

Donnell's shell theory were derived. In Ref. [17], the dynamic stability and primary resonant response of stiffened cylindrical shells with FG material including ring and stringer stiffeners was studied. The nonlinear vibration analysis of shear deformable FG cylindrical shells surrounded by elastic medium was examined in Ref. [18]. In Ref. [19], primary resonant behavior of functionally graded cylindrical shells with spiral stiffeners by utilizing the multiple scales method was addressed. Also, in Ref. [20], the primary resonance behavior of imperfect spiral stiffened FG cylindrical shells rested on the nonlinear elastic foundation was investigated. Review of the literature shows that there is no work on the resonance analysis of spiral stiffened multilayer FG cylindrical shells surrounded by damping and elastic foundation under external excitation and thermal environment. This subject is the main reason to motivate us for defining present study. Therefore, the main contributions of this paper are: (a) primary, superharmonic and subharmonic resonances in thermal environment are obtained via the method of multiple scales for spiral stiffened multilayer FG cylindrical shell system, (b) the material composition of the stiffeners and multilayer shell are considered functionally graded (c) the temperature is dependent to thickness (d) the effect of elastic foundation and thickness of shell layers is investigated (e) the effect of various angles is analyzed for the spiral stiffeners, (f) the effect of harmonic excitation and volume fraction is considered. Using the technique of smeared stiffeners and theory of classical shell, the equations of motion are derived. Then, the Galekin method is utilized to discretize the equations of motion.

Vc (z ) =

2z + h 2H

N

; H = h , hs ; N = Ksh, Kst ;

(1)

0 are the material power law index of the In Eq. (1), Kst , Ksh stiffeners and shell, respectively. The subscripts m and c refer to the metal and ceramic constituents. Vc and Vm are the ceramic and metal volume fractions. A material coefficient P is defined as a temperature nonlinear function [21]. P= P0 (P 1T

1

+ 1 + P1T + P2 T 2 + P3T 3)

(2)

In this work, the multilayer FG cylindrical shell is considered with two cases. (Metal-FGM-Ceramic (MFC) and Ceramic-FGM-Metal (CFM)) which is shown in Fig. 2. In some cases or applications, the external surface of the shell needs less heat resistance or more flexibility, whereas the internal surface of the shell, needs more heat resistance or less flexibility, therefore, the layout of layers must be considered as the MFC. For example: rockets, airplanes and etc. Also, in some cases or applications, the external surface of the shell needs more heat resistance or less flexibility and the internal surface of the shell needs less heat resistance or more flexibility, therefore, the layout of layers must be considered as the CFM. In the next sub-section, the effective material properties are expressed according to both Voigt and Mori–Tanaka model.

2. Spiral stiffened multilayer FG cylindrical shell Fig. 1 shows the schematic of multilayer FG cylindrical shell reinforced by internal spiral stiffeners rested on the elastic foundation. x,

2

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 1. Schematic of spiral stiffened multilayer FG cylindrical shell with elastic foundation.

2.1. Voigt model

(Em (T ),

The effective properties Peff can be determined as [22].

Peff = Pm (z ) Vm (z ) + Pc (z ) Vc (z )

(Em (T ), =

(3)

With regard to the power law, Young's modulus, thermal expansion and mass density coefficients are obtained as [23]. Case I: Metal-FGM-Ceramic Shell:

[Esh (z , T ),

sh (z ,

T ),

sh (z ,

cm (T ),

m (T ),

m (T ), cm (T ))

(Ec (T ),

m (T ))

(

h 2

m (T ))

+ hu

+ (Ecm (T ), h 2

)

2z + h 2hu Ksh 2hm

c (T ),

h 2

z

h 2

c (T ))

+ hu

hb

z h 2

hb

z

h 2

(4a)

Internal stiffeners:

= (Ec (T ),

T )]

mc (T ),

c (T ),

mc (T ))

c (T ))

2z + h 2hs

+ (Emc (T ),

Kst

Fig. 2. The material distribution of multilayer cylindrical shell and stiffeners.

3

h + hs 2

z

h 2

(4b)

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Case II: Ceramic-FGM-Metal Shell:

[Esh (z , T ),

sh (z ,

(Ec (T ), (Ec (T ),

T ),

sh (z ,

c (T ),

mc (T ))

T )]

h 2

c (T ),

(

h 2

z

+ hu

+ hu h 2

z

sh f (z ,

Vc (z )( m (T )

T) =

c (T )) 3[ (T ) c (T )] Vc (z )] m 3 c (T ) + 4Gc (T )

1 + [1

hb

)

m (T ),

st (z ,

h 2

m (T ))

= (Em (T ), h 2

z

T ),

hb

h 2

z

Vc (z )(Gm (T )

h 2

Gfsh (z , T ) =

(5a)

Gc (T ) Gc (T ) + G (T )

T )]

m (T ))

+ (Ecm (T ),

cm (T ),

cm (T ))

(

)

1

sh f (z ,

+ hs

1 c (T )

sh (z, T ) f 1 m (T )

T) =

1

Vc (z )( c (T ) 1 + [1

Vc (z )]

m (T )) 3[ c (T ) m (T )] 3 m (T ) + 4Gm (T )

+

h 2

m (T )

+ hu h 2

c (T )

st f (z ,

Gfsh (z , T ) =

Vc (z )(Gc (T ) 1 + [1

Gm (T )) G (T ) Gm (T ) Vc (z )] c

h 2

+ Gm (T )

st f (z ,

1 + [1

h 2

1 sh (z , T ) f 1 c (T )

+

hb

h 2

h 2

z

1 m (T )

(

c (T )

h 2

m (T ))

+ hu

h 2

h 2

hb

z

Gfst (z, T ) =

Vc (z )( m (T ) 1 + [1

Vc (z )]

c (T )) 3[ m (T ) c (T )] 3 c (T ) + 4Gc (T )

Vc (z )(Gm (T ) 1 + [1

Gc (T )) G (T ) Gc (T ) Vc (z )] m Gc (T ) + G

(T )

+

c (T )

+ Gc (T )

( (

h 2

h 2

)

z

)

z

+ hs + hs

+

m (T )

hb

1 st f (z, T ) 1 c (T )

(

h 2

+ Gm (T )

(

m (T ))

m (T )

Gm (T ) + G (T )

hb

h 2

z

h 2

+ hs + hs

)

)

h 2

z h 2

z

1 m (T ) 1

(

c (T )

+

m (T )

(

h 2

+ hs

)

h 2

Gm (T )[9 m (T ) + 8Gm (T )] 6[ m (T ) + 2Gm (T )]

G (T ) =

Gc (T )[9 c (T ) + 8Gc (T )] 6[ c (T ) + 2Gc (T )]

h 2

9 f Gf 3 f + Gf

0 x 0 y

x y

=

;

j = sh, st

(9)

where

h 2

x

z

y

2

0 xy

xy

Internal stiffeners:

T) =

Gm (T )) G (T ) Gm (T ) Vc (z )] c

G (T ) =

(7a)

st f (z ,

h 2

z

The strain components are obtained across the shell thickness at the middle surface according to the assumption of von Kármán geometrical nonlinearity as follows

hb

m (T ) c (T )

+ hu

+ hu

3. The theoretical formulation

+ hu h 2

z

h 2

z

It should be noted that the mass density coefficients ( j (z, T ); j = sh, st ) in Voigt model are also valid for the Mori–Tanaka model.

hb

1 m (T )

Vc (z )(Gc (T ) 1 + [1

T) =

Ej (z , T ) =

+ hu

z

h 2

z

where

hb

h 2

z

hb

(8b)

Gm (T ) + G (T )

m (T )

T) =

h 2

m (T )) 3[ c (T ) m (T )] 3 m (T ) + 4Gm (T )

Vc (z )]

z

h 2

z

z

+ hu

Gc (T )

sh f (z ,

h 2

hb

h 2

Vc (z )( c (T )

T) =

Gfst (z, T ) =

+ hu

z

hb

h 2

Gm (T )

h 2

z

z

(8a)

Considering the Mori–Tanaka scheme, the effective shear modulus Gf and the local bulk modulus f can be expressed as [18]. Case I: Metal-FGM-Ceramic Shell: h 2

h 2

c (T ))

+ hu h 2

Internal stiffeners:

2.2. Mori–Tanaka model

m (T )

m (T )

m (T )

(6)

Ec

(

h 2

h 2

+ hu

h 2

hb

z

z

h 2

c (T )

h 2

c (T )

+

sh (z , T ) , st (z , T ) , st (z , T ) and Esh (z , T ) , Est (z , T ) , sh (z , T ) are the Young's modulus, mass density and thermal expansion coefficient of the FG stiffeners and shell, respectively. Also, the subscripts st and sh refer to the stiffeners and shell, respectively.

T) =

+ Gc (T )

c (T ) 2z + h Kst 2hs

where

sh f (z ,

h 2

+ hu

z

hb

h 2

G (T ) Vc (z )] m

1 + [1

(5b)

Emc = Em

h 2

Gc (T ))

h 2

z

+ hu

Gm (T ) st (z ,

m (T ),

c (T )

Gc (T )

Internal stiffeners:

[Est (z , T ),

+

h 2

m (T )

mc (T ),

2z + h 2hu Ksh 2hm

(Em (T ),

h 2

c (T )

h 2

c (T ))

c (T )) + (Emc (T ),

=

Case II: Ceramic-FGM-Metal Shell:

x,

y,

xy

;

x

= w , xx ,

y

= w , yy ,

xy

= w , xy

xy

(10)

are the shell curvatures change and twist.

0 xy

is the

shear strain, and y0 , x0 are the normal strains. Also, according to strain-displacement von Kármán relations [24], the strain components on the middle surface of shells are given by

h 2

0 x

(7b)

4

= u, x +

1 2 w,x , 2

0 y

= v,y

w 1 + w,2y, R 2

0 xy

= u, y + v , x + w , x w , y

(11)

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

where w = w (x , y ) , u = u (x , y ) and v = v (x , y ) are the displacement components along z , x and y axes, respectively. In addition, cylindrical shell must be satisfy the circumferential close conditions as 2 R L

2 R L 0 y

v , y dxdy = 0

0

0

+

0

w R

1 2 w , y dxdy = 0 2

(12)

The compatibility equation according to Eq. (11) can be written as follows 0 x , yy

+

0 y , xx

0 xy , xy

w , xx + w ,2xy R

=

w , xx w , yy

(13)

The stress-strain relations based on the Hooke low for FG cylindrical shell are defined as sh x sh y

Esh (z , T )

=

2 1 Esh (z , T )

sh xy

2

1

Esh (z, T )

0

2 1 Esh (z, T )

x

2

1

0

0

0

y

Esh (z, T ) 2(1 + )

xy

+

1 T (z ) 1 0

Fig. 4. View a rhombic stiffener grid.

(14)

For an isotropic material, the thermal distribution in Eqs. (16) and (17) is assumed as [25].

sh where is the Poisson's ratio. xsh, ysh are the normal stress and xy is the shear stress of cylindrical shell. The temperature distribution across the thickness is also determined by solving the following steady-state one-dimensional heat conduction equation:

d dT K (z ) dz dz

h 2

= 0;

z

h 2

where in this analysis, for uniform temperature distribution (UTD): a = 1 and b = 0 and consequently, Eq. (18) reduced as follows

(15)

T (z ) = Tm +

(Tc

Tm)

(

2z + h 2hu 2hm

)

K sh

i=0

The relations of stress-strain for the spiral stiffeners are obtained by rotation of the strain tensor. The transformation of strain components from the XY -axis to the X Y -axis (Fig. 3.), can be done by the following relations [26].

i

K cm Km

iK sh + 1 i

K cm Km

Case II: Ceramic-FGM-Metal

T (z ) = Tc +

(Tm

Tc )

(

2z + h 2hu 2hm

)

x

=

x cos

2

+2

xy sin

cos +

2 y sin

y

=

2 x sin

2

xy sin

cos +

y cos

=

x 2z + h 2hu 2hm

i=0 K cm Km

K sh

K cm Km

i

=

y

(20)

2

2

xy sin

cos +

2 y sin

sin2

+2

xy sin

cos +

y cos

x cos x

2

(21)

In order to find the stress components of the stiffener, a rhombic stiffener grid of sides ls and spacing s is considered (Fig. 4). The associated stiffener loads are defined by F and F in the X Y -axis and X Y -axis, respectively. According to the uniaxial Hooke's law

iK sh + 1 i

i = 0 iK sh + 1

2

Also, a similar transformation can be done from the XY-axis to the X Y -axis as follows

(16)

i = 0 iK sh + 1

(19)

T (z ) = T

where K (z ) is the thermal conductivity and the effect of curvature is neglected due to high radius to thickness ratio. Eq. (13) admits the following polynomial series solution for the linear temperature distribution (LTD) [18]: Case I: Metal-FGM-Ceramic 2z + h 2hu 2hm

(18)

T (z ) = T (a + bz )

(17)

x

F ; hs dEs

=

x

=

F hs dEs

(22)

According to Fig. 4, the length of the stiffener grid can be calculated

as

ls =

s sin( + )

(23)

The distributed edge stresses of the FG spiral stiffener grid can be written in terms of the discrete stiffener loads as st x

=

F cos F cos + F cos F cos + = ls hs (sin + sin ) ls hs (sin + sin ) ls hs (sin + sin ) (24)

By substituting Eqs. 14–23 in Eq. (24) and applying the similarly method, the relations of stress-strain for spiral stiffeners with FG material are obtained by Eq. (25) [19,27].

Fig. 3. Rectangular coordinates rotation.

5

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan st x st y st xy

=

h1 h2 h3

H11 H12 H13 H21 H22 H23 H31 H32 H33

is substituted into the compatibility Eq. (13), then, by rewriting the resultant forces based on the stress function (from Eq. (31)), the compatibility equation are as follows [20,27].

x y

(25)

xy

J11

st where Hij (i , j = 1,2,3) and hi (i = 1,2,3) are defined in Appendix. xy is st st the shear stress and x , y are the normal stress for spiral stiffener. The thermal stress of stiffeners is assumed subtle which distributes uniformly through the whole shell structure. So, one can ignore it [28]. The geometrical characteristics of stiffeners subjected to thermal environment can be determined as the follows [28].

(dT , sT , hsT ) = (d, s, hs )[1 +

st (z ,

0 x

(Nx , Ny ) = [(J11, J21) Nxy = J33

0 xy

2J36

+ (J12 , J22)

0 y

(J14 , J24)

x

(J15, J25)

y

+

1 (1,

1 w , tt

Mxy = J36

0 xy

2J63

+2

(J41, J51)

(J42 , J52)

x

y

+

k w w + ks (w , xx + w , yy ) =

1 w , tt

cm (T )

hm + 2

c (T )

+

mc (T )

Kst + 1

c (T )

+

+

mc (T )

Ksh + 1

hm + 2

m (T )

+

dT hsT + Kst + 1 sT cm (T )

+

x˜) (y

y˜)cos t

4

+ (J33

Bmn = J21 m4

4

+ (J11 + J22

2 R L

, xx ;

Nxy =

, xy

To find the compatibility equation in terms of the

(34)

sin

c (T ) h u

ny R

Nˆ W (t ) l = 1 mn

J21) m2n2

J12

2 2

2J36 ) m2n2

sin

ny R

+ J22 n4

4

m x L

2 2

cos

m x L

cos

k x L

sin

p x L

cos

sin

m x L

sin

k x L

sin

p x L

sin

0 2 R L 0

+ Ny 0

x2 2

(35)

0

ny R

ny R

+ J12 n4 cos sin

ly R

ly R

4

sin

sin

qy R

qy R

L2 2 2 mn R

dxdy dxdy (36)

By substituting Eqs. ((27), (31), (34) and (35) into Eq. (12), it can be obtained as

+

Ny 0 =

m2 2 Wmn (t ) 8L2I22

(I12

I22)

1

I22

(37)

Eqs. (34) and (35) is substituted into Eq. (33), and the Galerkin method is utilized to discretize the equation of motion as follows

(30c)

With regard to the first two Eq. (29), stress function ( ) is defined as

Ny =

ˆ M k=1

2L2 R2Amn

Amn = J11 m4

2mnklpq =

(30b)

m (T ) hb

, yy ;

Nˆ n=1

(30a)

F (t ) = Q˜ (x

Nx =

ˆ M m=1

1mnklpq =

Case II: Ceramic -FGM- Metal

=

m x ny sin L R

Nˆ Bmn m x W (t )sin L n = 1 Amn mn

0

m (T ) h u

(33)

where.

(29)

+

ks (w, xx + w, yy ) = 0

Wkl (t )( mnkl 1mnklpq + n2k 2 2mnklpq) sin

+ c (T ) h b

1

ˆ M m=1

(x , y , t ) =

where c is damping coefficient, ks and k w are the shear layer stiffness of Pasternak and Winkler foundations, respectively, and the harmonic excitation F (t ) and mass density 1 are Case I: Metal-FGM-Ceramic

Ksh + 1

, yy w , xx

where n and m are the number of full and half wave in the circumferential and axial directions, respectively. Also, Wmn (t ) represent the deflection amplitude. Eq. (34) is substituted into Eq. (32), and solving the resultant equation, is obtained as

)

+ 2 1 cw, t

dT hsT sT

Wmn (t )sin m =1 n=1

+ 1 R

1 R , xx

, yyyy

F (t ) + k w w

Nˆ

w (x , y , t ) =

2 (1, 1)]

xy

(

+

, xx w , yy

ˆ M 0 y

Mx, xx + 2Mxy, xy + My, yy + Nx w , xx + 2Nxy w , xy + Ny w, yy +

m (T )

, xy w , xy

A12

, xxyy

The approximate solution for the simply supported stiffened multilayer system in order to solve Eq. (33) can be considered as follows [30,31].

Nx , x + Nxy, y = 0 Nxy, x + Ny, y = 0

=

2A36 )

4. Discretization of the equation of motion

1)]

where Jij are the components of the coupling, bending and extensional stiffness and 1, 2 are the thermal components of stiffened multilayer system that are presented in Appendix. Considering the classical shell theory, the nonlinear equilibrium equations for cylindrical shells are given by Refs. [21,29,30].

1

, xxxx

where the coefficients Aij and Aij are presented in Appendix.

(28)

+ F (t )

A21

(A11 + A22

(27) + (J15, J25)

w , xx w , yy]

ks (w , xx + w , yy ) = 0

+ 2 1 cw , t + A11 w, xxxx + (A12 + A21 + 4A36 ) w , xxyy + A22

w, yyyy

Resultant moments: 0 x

1

2J36 ) w , xxyy + J12 w, yyyy + R w, xx + [w ,2xy

where the coefficients Jij , Jij are presented in Appendix. Similar to above procedure, by substituting Eq. (28) into the third part of Eq. (29), and using Eq. (11), yields [20,27].

xy

(Mx , My ) = [(J14 , J24 )

+ J21 w , xxxx

, yyyy

(32)

hsT ,

sT

+ J22

, xxyy

2w, xy + w, xx + w, yy + k w w

so, instead of d , s and hs , one can consider , and respectively. st is thermal expansion coefficient of stiffeners. Using the technique of smeared stiffeners, the effect of stiffeners on the shell is considered. To derive the resultant moments (Mx , My , Mxy ) and forces (Nx , Ny , Nxy ) for spiral stiffened multilayer FG cylindrical shell, the stress-strain equations (Eq. (25)) are integrated through the thickness leads to Resultant forces:

dT

J12 + J21)

+ (J11 + J22

(26)

T ) T (z )]

+ (J33

, xxxx

¨mn + 2cWmn + W

(31)

+

and w , Eq. (27)

p

q

k

2 mn Wmn l

i

j

+ r

p s

q

k

l

1mnklpq Wkl (t ) Wpq (t )

2mnpqklijrs Wpq (t ) Wij (t ) Wrs (t ) = Qmn cos t (38)

6

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

where

where the new time independent variable T0, T1 and T2 are defined as 1 L4 1

=

mn

(D

2 2 Bmn [ Amn

1mnklpq =

4 4

2mnpqklijrs = Qmn =

1 L4

B = A21

+

mn

Amn 2 R L

1 0

m4 4

Bmn Bmn Amn

)

+ L4k w + L2ks [( n) 2 + (m )2] +

2mnklpq (k 2q2 + l 2p2 )

[ 2mnklpq (k 2q2 + l 2p2 )

Q˜ (x

x˜) (y

y˜)sin

0

+ (A11 + A22

2J36

n2 (J22 J12 ) H J22 R2 1

j 2 r 2 2pqrsij]

klpq 1mnklpq ][ijrs 1pqrsij

) m2n2 2 2

0:

ny dxdy R

sin

+ J12

2W (0) 11

=0

(1) D02 W11 +

2W (1) 11

=

(1) D02 Wmn

+

2 (1) mn Wmn

2

(0) 2D0 D1 W11

(0) 1111111 (W11 ) +

(0) 2 1mn1111 (W11 )

=

(0) 1 W11

+

(0) 1 W11

; m and n

1 47a,b

(39)

2:

In Eq. (39), mn is cylindrical shell natural frequency and the H parameter is obtained as Case I: Metal-FGM-Ceramic for UTD: H = hu Em

m

+ hm Em

m

+

for LTD:

K cm Km

1 H = hu Em 2

m

+ hm

Em cm + Ecm m K sh + 1

+

Ecm cm 2K sh + 1

+ hb Ec

(2) D02 W11 +

1 + hb Ec 2

i = 0 iK sh + 1

Em m iK sh + 2

+

Em cm + Ecm m (i + 1) K sh + 2

ˆ M Nˆ k=1 l =1 (0) 3 211111111 (W11 ) + Q11

+

1 hu Ec 2

(

+ hm Ec

K cm Km

Dn =

c

+

Ec mc + Emc c K sh + 1

+

Emc mc 2K sh + 1

+ hb Em

m

+

Ec mc + Emc c (i + 1) K sh + 2

+

)

as

=

1mnklpq Wkl Wpq +

2

=

2

1

2

2W (2) (T , mn 0

T1, T2 ) + …

(

=

1 Ae

1mn1111 A2 e 2i 4 2

1

T0

+ AA ]

(52)

=0

T0

+

1 2 A e 3

2i T0

1mn1111 [A2 e 2i

i T0

T0

2 mn

2W (2) 11

3A2 A ( 2mn

+

(T0, T1, T2) + … ; m and n

1111111 [A2 e 2i

i T0

(51)

1 2 2i Ae 3

2AA +

2 (1) mn Wmn

+

(2) D02 W11 +

+

(43) (1) Wmn (t , ) = Wmn (T0, T1, T2 ) +

1 A) e

(53)

T0

+ AA ] + c. c.

(54)

+

8

2

2 mn

AA + 2AA

+ c. c.

(55)

Eqs. (50), (53) and (55) are substituted in Eq. (48) leads to

In order to solve Eq. (41), with assumption that there is no internal resonance between the first mode and other modes, according to the method of multiple scales (MMS) an expansion form is considered as follows 2W (2) 11

= ( 2i D1 A +

A = A (T2) and

2

(1) Wmn =

(42)

(0) (1) W11 (t , ) = W11 (T0, T1, T2 ) + W11 (T0, T1, T2) +

(50)

i T0

+ A (T1, T2) e

Similar to above procedure, the general solution of Eq. (54) can be written as

are

2)

2

T0

Eq. (50) is substituted in Eq. (47b) leads to

(41)

The following definition of the detuning parameters ( 1, presented as

2W (1) 11

+

1111111

(1) D02 Wmn

2mnpqklijrs Wpq Wij Wrs

cos t

(49)

Considering Eq. (52), the general solution of Eq. (51) can be written

(1) W11 =

For analyzing the primary resonance (i.e., mn ), it is assumed 2mnpqklijrs = 2 2mnpqklijrs , c = 2c and 1mnklpq = 1mnklpq , that Qmn = 2Qmn . These new parameters are substituted in Eq. (38). Therefore, we yields

+

+ Q11 sin( T0 + )sin

n = 0,1,2

D1 A = 0

4.1. Primary resonances

2Q mn

(0) 11111kl W11 Wkl(1)

where c.c. is the complex conjugate. For removing the secular terms in Eq. (51), the coefficient of exp(i T0) is considered zero as

(40b)

¨mn + 2 2cWmn + W

cos( T0 + )cos

Nˆ l =1

+ c. c.

Emc mc (i + 2) K sh + 2

m

2 mn Wmn

ˆ M k=1

(0) 111kl11 W11 Wkl(1)

Eq. (50) is substituted in Eq. (47a) as follows

i

K cm Km

;

(0) W11 = A (T1, T2) e i

i = 0 iK sh + 1

1 + hb Em 2

Tn

(1) D02 W11

Ec c iK sh + 2

D12

The solution of Eq. (46) is obtained as

i

i = 0 iK sh + 1

+ hm

c

(0) 2D0 D2 W11

where

Case II: Ceramic-FGM-Metal for LTD:

(1) 2D0 D1 W11

(48)

Ecm cm (i + 2) K sh + 2

(40a)

c

(1) 1 W11

+

(0) 2cD0 W11

c

i

K cm Km

hu Ec

(0) 2 W11

=

i

c

for UTD: H =

2W (2) 11

(1) W11

i = 0 iK sh + 1

H=

0,

(46)

L R

=

2 11

(0) D02 W11 +

:

L2 2 2 mn R

n4 4

(45)

n = 0,1,2

Eqs. 42–44 are substituted in Eq. (41) and the coefficients of and 2 are set to zero as follows

n2m2 2 8L2R2 1 J22

klpq 1mnklpq ] +

m x L

T

nt;

Tn =

1 2

4 2)

= [ 2i A + ( 1mn1111 111mn11 + 11111mn 1mn1111) 8A2 A 2 2 2 mn ( mn

1 i Q cos Q 2 11 2 11 1 1111111 2 2 i T (A e 0 3 2

4 2)

sin

)

3 211111111A2 A

+

A] ei T0

2

2iA c

211111111A3 e 3i

T0

2AA ) + c. c. (56)

1

To remove the secular terms in Eq. (56), the coefficient of exp(i T0) is considered zero as

(44)

7

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

2i A + ( 1mn1111 111mn11 + 11111mn 1mn1111)

(

3A2 A 2 ( mn

8A2 A 2 2 2 mn ( mn

4 2)

2iA c +

1 Q 2 11

4 2)

)

i Q 2 11

cos

(1) : D02 W11 +

m and n

3 211111111A2 A sin

+

2

A= 0

(0) W11

(58)

a

Q11 sin( ) 2

3 ea

=

+

=

8 2 2 2 mn ( mn

4 2)

3 ea

2

+

(1) 2 11 W11

2e i (2

+ 3A

+ A3 e3i

4 2)

)

[2i

=

11 (A

2

+ cA) + 6 2mn111111A

+ 3 2mn111111

11 2 ) T0

11) T0

3e i T0

+

+ 3A2 ei (2

) T0

11

+ 3A 2

2 11) T0

+ 6AA e i

T0

+

3e 3i T0

+ 3A 2ei (2

+ 11) T0

+ 3A2

sin +

m and n

11 t

2

1 2 ) 4 11

cos(2

(61)

3

)sin

Nˆ 2 n=1

1mn1111 a

+2

2

2t

2i

2 ) sin

1mnklpq fkl f pq +

m x L

sin

ny R

+ O ( 3) ;

(T0, T1) +

(1) Wmn (t , ) = Wmn (T0, T1) + … ;

(0) 2 11 W11

a

2

+ 3 2mn111111A2 A + 2mn111111 3ei

m and n

1 i ae 2

T1

=0

3

2mn111111

ca

sin( T1

=

3 2mn111111

a

2

)

1 3 a + 8

+

(75)

2mn111111 11

3

cos( T1

)

(76)

In continue, we define a new phase angle ( ) as follows (77)

= T1

therefore, Eqs. (75) and (76) are transformed in the following form as the autonomous system

a =

ca

2mn111111

3

sin

(78)

11

a

(65)

1

(74)

11

2mnpqklijrs fpq fij frs

(T0, T1) + …

= Q11 cos( T0)

+ cA) + 6 2mn111111A

11

=

3 2mn111111 11

2

a

3 2mn111111 3 a 8 11

2mn111111

3

cos

11

(79)

(66)

When a and are equal to zero, the steady-state motion occurs. From this situation in Eqs. (78) and (79), singular points are obtained. In continue, frequency-response relation for superharmonic resonance are calculated by summing the squares of resultant equations in steady state situation as

Eqs. (65) and (66) are substituted in Eq. (64) and the coefficients of and are set to zero as follows (0) D02 W11 +

(72)

+

where a (T1) and (T1) are real. Eq. (74) is substituted in Eq. (73), and then real part and imaginary part are separated as

In order to solve Eq. (64), with assumption that there is no internal resonance between the first mode and other modes, according to the method of multiple scales (MMS) an expansion form is considered as follows

W11 (t , ) =

11 (A

A=

(64)

(1) W11

11

a =

= Qmn cos t

(0) W11

=

To solve Eq. (73), A is considered in polar form as

For analyzing the non-resonant hard excitations (i.e., is away from mn ), we use the multiple scales method. Therefore, it is assumed that 1mnklpq = 2 1mnklpq , 2mnpqklijrs = 2mnpqklijrs and c = c , and then these new parameters are substituted in Eq. (38). Therefore, we yields 2

11)

(73)

5. Non-resonant hard excitations

+

1 3

Eq. (72) is substituted in Eq. (71), and then secular terms set to zero as follows

(63)

2 mn fmn

+ c. c.

According to Eq. (71) when the frequency of the excitation ( ) is 1 close to 3 11, a new secular term is created which this case is called superharmonic resonance. In order to frequency analysis, a detuning parameter is introduced that this parameter expresses the nearness 1 to 3 11 Therefore, the excitation frequency can be written as

1

¨mn + 2 cfmn + W

T0

5.1. 3/1 superharmonic resonance (

m x L

2t

11 t

2ic ei

(71)

(62)

ˆ M m=1

ny R

2 2( mn

+

11T0 }

3 211111111

2 Q11 4 2

=

w (x , y, t ) = a cos(

0:

(70)

ei (

According to Eq. (56) and elimination of secular term at the final step, the following solution is obtained.

0

(69)

+ c. c.

2)

2mn111111 {3A 2ei (

When a and are equal to zero, the steady-state motion occurs. From this situation in Eqs. (59) and (60), singular points are obtained. In continue, frequency-response relation for primary resonance are calculated by summing the squares of resultant equations in steady state situation as

1 2 2 mn

T0

A2 A ] ei 11T0

(60)

2

c 2a2 +

ei

+

ei ( +2 11) T0 3

2 ( mn

2 11

2(

(1) D02 W11 +

= ( 1mn1111 111mn11 + 11111mn 1mn1111)

(

11T0

Substituting Eq. (69) into Eq. (68) leads to

where e

(68)

1

= A ( T1) ei

(59)

Q + 11 cos( ) 2

2

(0) 2mn111111 (W11 ) ;

Q11

=

where a (T2) and (T2) are real. Eq. (58) is substituted in Eq. (57), and then real part and imaginary part are separated as

ca

3

(0) 2cD0 W11

where c.c. is stands for complex conjugate and

1 i ae 2

a =

(0) 2D0 D1 W11

=

The general solution of Eq. (67) is given by

(57)

To solve Eq. (57), A is considered in polar form as

A=

(1) 2 11 W11

(67) 8

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Table 1 Comparison of the natural frequencies of cylindrical shell (L = 0.2 m, R = 0.1 m, h = 0.247 × 10 m

Present

n

1 1 1 1 1 2 2

7 8 9 6 10 10 11

c 2a2 +

3 2mn111111

a

486.0 490.3 545.8 555.8 634.8 962.3 976.6

2

3 2mn111111 3 a 8 11

a

11

2

=

3

Errors (%)

Errors (%) 0.2 0.1 0.07 0.4 0.3 0.5 0.6

(m,n)

11)

11 (A

A2 ei

T1

2

(82)

Eq. (74) is substituted in Eq. (82), and then real part and imaginary part are separated as

a =

3 2mn111111 2 a sin( T1 4 11

ca

3 2mn111111

a

2

In this case, the lowing form

1

a

=

+

11

1

= T1

3 )

3 2mn111111 2 1 3 a + a cos( T1 8 4 11

Sofiyev et al. [35]

Dung and Nam [36]

0.67480 0.36223 0.20670

0.67921 0.36463 0.20804

0.65 0.66 0.65

Errors (%) 0.67480 0.36223 0.20670

0.00 0.00 0.00

Table 4 Results of the natural frequencies of the stiffened cylindrical shell h = 0.65 × 10 3 m , R = 0.242 m , E = 68.95 × 109N/m2 , (L = 0.6096 m , = 2714kg/m3 , = 0.3, ns = 60 ).

+ 3 2mn111111A2 A + 3 2mn111111

=0

Present

(1,1) (1,2) (1,3)

(81)

+ cA) + 6 2mn111111A

0.2 0.1 0.07 0.4 0.3 0.5 0.6

Errors (%)

Eq. (81) is substituted in Eq. (71), and then secular terms in Eq. (71) must be zero as follows

2i

484.6 489.6 546.2 553.3 636.8 968.1 983.4

Table 3 A comparison of the dimensionless natural frequencies of the cylindrical shell resting on the Winkler foundation (L / R = 2 , R / h = 100 , k w = 10 4 N/m3 ).

11

+

11

= 0.31).

3 2

According to Eq. (71) when the excitation frequency ( ) is close to 3 11, a new secular term is created wich this case is called subharmonic resonance. In order to frequency analysis, a detuning parameter is introduced that this parameter expresses the nearness to 3 11 Therefore, the excitation frequency can be written as

=3

= 2796kg/m3,

Qin et al. [32]

(80) 5.2. 1/3 subharmonic resonance (

m, m = 1, E = 7.12 × 1010N/m2,

Pellicano [33]

484.6 489.6 546.2 553.3 636.8 968.1 983.4

2mn111111

3

(m,n)

Present

Mustafa and Ali [37]

Errors (%)

(1,5) (1,6) (1,7)

228.1 189.4 174.0

226 191 179

0.9 0.8 2.4

(83)

3 )

(84)

is presented as a new phase variable in the fol(85)

3

Considering Eq. (85), Eqs. (83) and (84) are transformed in the following form as the autonomous system

a = a

1

ca

= a

3 2mn111111 2 a sin( 1) 4 11 3 2mn111111 11

a

2

+

1 3 a 8

(86)

3 2mn111111 2 a cos( 1 ) 4 11

(87)

Table 2 Comparison of the natural frequencies of stiffened FG cylindrical shell R / h = 250 , Em = 7 × 1010 N/m2 , R = 0.5 m , m = 1, (L = 0.75 m , 3 3 Ec = 38 × 1010N/m2 , = 0.3, c = 3800kg/m , m = 2702kg/m , ds = dr = 0.0025 m , hs = hr = 0.01 m ). Present Un-stiffened

1654.05 Internal stiffeners 2539.43 External stiffeners 2518.90

Van Dung and Nam [34]

Errors (%)

1654.05

0.00

2539.43

0.00

2518.90

0.00

Fig. 5. Comparison of the natural frequencies of cylindrical shell (m = 1).

Similar to previous sub-section, when a and are equal to zero, the steady-state motion occurs. The frequency-response relation for subharmonic resonance are obtained as

c 2a2 +

a

3 2mn111111 11

9

a

2

+

1 3 a 8

2

=

3 2mn111111 2 a 4 11

2

(88)

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Table 5 Material properties of the constituent materials of the considered FG cylindrical shells [18,28]. Material

Properties

P0

P

Si3 N4 (Ceramic)

E (Pa)

3.48e11 2370

(Kg/m3)

13.723 2.01e11 8166

(K 1) (W/mK)

15.379

(Kg/m3)

SUS304 (Metal)

(K 1) (W/mK) E (Pa)

5.87e-6

1.23e-5

P1

P2

P3

0 0

−3.07e-4 0

2.16e-7 0

−8.95e-11 0

0 0 0

−1.03e-3 3.08e-4 0

5.47e-7 −6.53e-7 0

−7.88e-11 0 0

2.09e-6

−7.22e-10

1

0

9.10e-4

0

8.09e-4

0

−1.26e-3

0

0

0

0

Table 6 The geometrical characteristics of system. Fig. 6. Comparison of the natural frequencies of isotropic internal stiffened cylindrical shell (m = 1).

6. Numerical results 6.1. Validation of the present work

Geometrical characteristics

value

Geometrical characteristics

value

hs d m Ksh = Kst

0.01 m 0.0025 m 1 1

R L hc hu = hb

0.5 m 0.75 m 0.003 m 0.001 m

In the following, the results of nonlinear primary, superharmonic and subharmonic resonances for spiral stiffened multilayer FG cylindrical shell are presented, respectively.

For validation, the presented work is compared with the other related works and then a numerical method is utilized to comparison. Table 1 compares the natural frequencies for cylindrical shells that are obtained in this study with similar results which investigated by Qin et al. [32] and Pellicano [33]. Furthermore, Table 2 lists the natural frequencies are obtained in this work with similar results which presented by Van Dung and Nam [34] for functionally graded cylindrical shell that it is stiffened by stringer and ring. For more validation of approach, present results of the dimension2) less natural frequencies ( mn = mn R (1 ) are compared in E Table 3 with those of the nonlinear vibration behavior of a homogeneous cylindrical shell rested on a Winkler foundation accomplished already by Sofiyev et al. [35] and Dung and Nam [36]. Also, results of a stiffened cylindrical shell are compared with the experimentally validated results of Mustafa and Ali [37] in Table 4. It is observed that there is a good agreement for the results of present study. Figs. 5 and 6 show the obtained natural frequencies with and without stiffeners for cylindrical shell to compare with the

experimentally results reported in Sewall and Naumann [38] and Sewall et al. [39]. Also, these comparisons confirm a good agreement of the results obtained in this paper. In continue the numerical method is utilized to assessment of analytical method for primary and superharmonic resonances. In numerical validation, Eq. (31) is solved by means of the Runge-Kutta algorithm (fourth-order). In this method, for various excitations, the maximum amplitude is extracted from the responses (I. C.: W (0) = 0 and W (0) = 0 ). The curves of numerical and analytical frequency-response are illustrated in Fig. 7. It can be seen, analytical results are almost similar to the numerical ones. For purpose formulation, the results are obtained for m = 1 and n = 5. Therefore, this result shows that the numerical simulation verify the analytical method in this paper.

Fig. 7. Analytical and numerical frequency-response curve ( = 0°,

10

= 90°).

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 8. Frequency-response curves for primary resonances of spiral stiffened CFM cylindrical shells.

Fig. 9. Frequency-response curves for primary resonances of spiral stiffened MFC cylindrical shells. 11

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 10. Effect of elastic foundations on the frequency-response curves for primary resonance spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

Fig. 13. Frequency-response curves for primary resonance of spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

6.2. Primary, superharmonic and subharmonic resonances based on the voigt model Here, nonlinear primary, superharmonic and subharmonic resonant responses of multilayer FG cylindrical shell with FG spiral stiffeners rested on elastic foundation in thermal environment based on the Voigt model are presented. The influence of material parameters, elastic foundation, temperature and different geometrical characteristics are considered. In this work, ns = 40 , T = 300 K and = c = m = 0.3. The rest of material parameters and geometrical characteristics of shell are presented in Tables 5 and 6, respectively. 6.2.1. Primary resonance The influence of various angles of stiffeners on the response of amplitude-frequency for primary resonances of spiral stiffened multilayer FG cylindrical shell with ceramic-FGM-metal (CFM) and metalFGM-ceramic (MFC) shells is illustrated in Figs. 8 and 9, respectively. With regard to these figures, the hardening nonlinearity behavior of system is more than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the hardening nonlinearity behavior of system with CFM and MFC shells is less than others, when the angle of stiffeners is 90° and 60°, respectively.

Fig. 11. Effect of thickness of shell layers on the frequency-response curves for primary resonance spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

Fig. 12. Frequency-response curves for primary resonances of spiral stiffened CFM cylindrical shells under uniform and linear temperature distribution ( = 0°, = 90°).

12

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 14. The response amplitude versus excitation amplitude for primary resonances of spiral stiffened CFM cylindrical shells.

Fig. 15. The response amplitude versus excitation amplitude for primary resonances of spiral stiffened MFC cylindrical shells. 13

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 11 shows the influence of thickness of shell layers on the primary resonance of spiral stiffened CFM cylindrical shell. With regard to Fig. 11, by increasing the thickness of ceramic and metal layers, the hardening nonlinearity behavior of system increases and decreases, respectively. The primary resonant response of CFM cylindrical shells with spiral stiffeners under uniform and linear temperature distribution is shown in Fig. 12. According to this figure, increasing the temperature leads to decreasing the hardening nonlinearity behavior of system. But, the hardening nonlinearity behavior of system under LTD is more than the system under UTD. Fig. 13 shows the influence of the amplitude of external force on the frequency-response for the case of system primary resonance. According to this figure, increasing the amplitude of external force leads to increasing the jump value. Figs. 14 and 15 show the influence of various and (stiffeners angles) on the response amplitude versus excitation amplitude curve for primary resonances of spiral stiffened multilayer system with CFM and MFC shells, respectively. It is observed that in low excitation amplitudes (Q11), a jump phenomenon is occurred. Considering these figures, the jump values of system is less than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the jump values of system with CFM and MFC shells is more than others, when the angle of stiffeners is 90° and 60°, respectively. The detuning parameters ( ) influence on the response amplitudeexcitation amplitude curve for primary resonances of spiral stiffened

Fig. 16. The response amplitude versus excitation amplitude for primary resonances of spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

In Fig. 10, the influence of elastic foundation parameters on the primary resonance of system is shown. As can be seen, the hardening nonlinearity behavior of system with Pasternak elastic foundation coefficient is less than the Winkler elastic foundation coefficient.

Fig. 17. Frequency-response curves for superharmonic resonances of spiral stiffened CFM cylindrical shells.

14

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 18. Frequency-response curves for superharmonic resonances of spiral stiffened MFC cylindrical shells.

Fig. 20. Effect of thickness of shell layers on the frequency-response curves for superharmonic resonance spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

Fig. 19. Effect of elastic foundations on the frequency-response curves for superharmonic resonance spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

15

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 21. Frequency-response curves for superharmonic resonances of spiral stiffened CFM cylindrical shells under uniform and linear temperature distribution ( = 0°, = 90°).

the angle of stiffeners is 90° and 60°, respectively. In Fig. 19, the influence of elastic foundation parameters on the superharmonic resonance of system is shown. As can be seen, the hardening nonlinearity behavior of system with Pasternak elastic foundation coefficient is less than the Winkler elastic foundation coefficient. Fig. 20 shows the influence of thickness of shell layers on the superharmonic resonance of spiral stiffened CFM cylindrical shell. With regard to Fig. 20, by increasing the thickness of metal and ceramic layers, the hardening nonlinearity behavior of system decreases and increases, respectively. The superharmonic resonance of spiral stiffened CFM cylindrical shells under uniform and linear temperature distribution is shown in Fig. 21. According to this figure, increasing the temperature leads to decreasing the hardening nonlinearity behavior of system. But, the hardening nonlinearity behavior of system under LTD is more than the system under UTD. Fig. 22 shows the influence of the amplitude of external force on the frequency-response for superharmonic resonance of system. According to this figure, increasing the amplitude of external force leads to increasing the jump value. Figs. 23 and 24 show the influence of various and (stiffeners angles) on the response amplitude versus excitation amplitude curve for superharmonic resonances of spiral stiffened multilayer system with CFM and MFC shells, respectively. It is observed that in low excitation amplitudes ( ), a jump phenomenon is occurred. Considering these figures, the jump values of system is less than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the jump values of system with CFM and MFC shells is more than others, when the angle of stiffeners is 90° and 60°, respectively. The detuning parameters ( ) influence on the response amplitudeexcitation amplitude curve for superharmonic resonances of spiral stiffened CFM cylindrical shell is illustrated in Fig. 25. It is observed, for positive detuning parameters, multivalued phenomenon is occurred, whereas for others are single-valued. Also, increasing leads to increasing the jump value.

Fig. 22. Frequency-response curves for superharmonic resonance of spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

CFM cylindrical shell is illustrated in Fig. 16. It is observed, for positive detuning parameters, multivalued phenomenon is occurred, whereas for others are single-valued. Also, increasing leads to increasing the jump value. 6.2.2. Superharmonic resonance The influence of various angles of stiffeners on the response of amplitude-frequency for superharmonic resonances of spiral stiffened multilayer FG cylindrical shell with ceramic-FGM-metal (CFM) and metal-FGM-ceramic (MFC) shells is illustrated in Figs. 17 and 18, respectively. With regard to these figures, the hardening nonlinearity behavior of system is more than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the hardening nonlinearity behavior of system with CFM and MFC shells is less than others, when

16

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H. Ahmadi and K. Foroutan

Fig. 23. The response amplitude versus excitation amplitude for superharmonic resonances of spiral stiffened CFM cylindrical shells.

Fig. 24. The response amplitude versus excitation amplitude for superharmonic resonances of spiral stiffened MFC cylindrical shells. 17

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

shells, respectively. With regard to these figures, the hardening nonlinearity behavior of system is more than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the hardening nonlinearity behavior of system with CFM and MFC shells is less than others, when the angle of stiffeners is 90° and 60°, respectively. In Fig. 28, the influence of elastic foundation parameters on the subharmonic resonance of system is shown. As can be seen, the hardening nonlinearity behavior of system with Pasternak elastic foundation coefficient is less than the Winkler elastic foundation coefficient. Fig. 29 shows the influence of thickness of shell layers on the subharmonic resonance of spiral stiffened CFM cylindrical shell. With regard to Fig. 29, by increasing the thickness of metal and ceramic layers, the hardening nonlinearity behavior of system decrease and increases, respectively. The subharmonic resonance response of spiral stiffened CFM cylindrical shells under uniform and linear temperature distribution is shown in Fig. 30. According to this figure, increasing the temperature leads to decreasing the hardening nonlinearity behavior of system. But, the hardening nonlinearity behavior of system under LTD is more than the system under UTD. Fig. 31 shows the influence of the amplitude of external force on the frequency-response for subharmonic resonance of system. According to this figure, increasing the amplitude of external force leads to increasing the jump value. Figs. 32 and 33 show the influence of various and (stiffeners angles) on the response amplitude versus excitation amplitude curve for subharmonic resonances of spiral stiffened multilayer system with CFM

Fig. 25. The response amplitude versus excitation amplitude for superharmonic resonances of spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

6.2.3. Subharmonic resonance Figs. 26 and 27 show the influence of various angles of stiffeners on the response of amplitude-frequency for subharmonic resonances of spiral stiffened multilayer FG cylindrical shell with CFM and MFC

Fig. 26. Frequency-response curves for subharmonic resonances of spiral stiffened CFM cylindrical shells.

18

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 27. Frequency-response curves for subharmonic resonances of spiral stiffened CFM cylindrical shells.

Fig. 29. Effect of thickness of shell layers on the frequency-response curves for subharmonic resonance spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

Fig. 28. Effect of elastic foundations on the frequency-response curves for subharmonic resonance spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

19

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

Fig. 30. Frequency-response curves for subharmonic resonances of spiral stiffened CFM cylindrical shells under uniform and linear temperature distribution ( = 0°, = 90°).

rested on elastic foundation in thermal environment based on the Mori–Tanaka model are presented. 6.3.1. Primary resonance The influence of various angles of stiffeners on the response of amplitude-frequency for primary resonances of spiral stiffened multilayer FG cylindrical shell with metal-FGM-ceramic (MFC) shell is illustrated in Fig. 35. With regard to these figures, the hardening nonlinearity behavior of system is more than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the hardening nonlinearity behavior of system with MFC shell less than others, when the angle of stiffeners is 60°. 6.3.2. Superharmonic resonance The influence of various angles of stiffeners on the response of amplitude-frequency for superharmonic resonances of spiral stiffened multilayer FG cylindrical shell with metal-FGM-ceramic (MFC) shell is illustrated in Fig. 36. With regard to these figures, the hardening nonlinearity behavior of system is more than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the hardening nonlinearity behavior of system with MFC shell is less than others, when the angle of stiffeners is 60° .

Fig. 31. Frequency-response curves for subharmonic resonance of spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

and MFC shells, respectively. Considering these figures, the jump values of system is less than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the jump values of system with CFM and MFC shells is more than others, when the angle of stiffeners is 90° and 60° , respectively. The detuning parameters ( ) influence on the response amplitudeexcitation amplitude curve for subharmonic resonances of spiral stiffened CFM cylindrical shell is illustrated in Fig. 34. It is observed increasing leads to increasing the jump value.

6.3.3. Subharmonic resonance Fig. 37 show the influence of various angles of stiffeners on the response of amplitude-frequency for subharmonic resonances of spiral stiffened multilayer FG cylindrical shell with metal-FGM-ceramic (MFC) shell. With regard to these figures, the hardening nonlinearity behavior of system is more than others, when the angle of stiffeners according to Fig. 1 ( and ) is 0° . Whereas, the hardening nonlinearity behavior of system with MFC shell is less than others, when the angle of stiffeners is 60°.

6.3. Primary, superharmonic and subharmonic resonances based on the Mori–Tanaka model

7. Conclusions An analytical approach was investigated to analyze the primary, superharmonic and subharmonic resonances of the spiral stiffened multilayer FG cylindrical shells rested on damping and elastic

Here, nonlinear primary, superharmonic and subharmonic resonant responses of multilayer FG cylindrical shell with FG spiral stiffeners

20

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H. Ahmadi and K. Foroutan

Fig. 32. The response amplitude versus excitation amplitude for subharmonic resonances of spiral stiffened CFM cylindrical shells.

Fig. 33. The response amplitude versus excitation amplitude for subharmonic resonances of spiral stiffened MFC cylindrical shells. 21

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

foundation in thermal environment. The material properties are dependent to temperature. For modeling the elastic foundation, two linear elastic foundations consist of the Winkler and Pasternak is considered. Whit regard to classical plate theory of shells, von Kármán equation and Hook law, the relations of stress-strain is derived for shell and stiffeners. Then, the non-linear problem was solved using Galerkin method and technique of smeared stiffeners. For obtaining the response of system for primary, superharmonic and subharmonic resonance, the method of multiple scales was used. The effects of different geometrical, material parameters, temperature, elastic foundation and harmonic excitation amplitude were investigated. The principal conclusions can be summarized as follows:

• The hardening nonlinearity behavior and jump value of spiral stif• Fig. 34. The response amplitude versus excitation amplitude for subharmonic resonances of spiral stiffened CFM cylindrical shells ( = 0°, = 90°).

•

fened multilayer cylindrical shell is more than others, when the angle of stiffeners ( and ) is 0° . The hardening nonlinearity behavior and jump value of spiral stiffened CFM and MFC cylindrical shell is less than others, when the angle of stiffeners ( and ) is 90° and 60°, respectively. The hardening nonlinearity behavior of system with Pasternak elastic foundation coefficient is less than the Winkler elastic foundation coefficient.

Fig. 35. Frequency-response curves for primary resonances of spiral stiffened MFC cylindrical shells.

22

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H. Ahmadi and K. Foroutan

Fig. 36. Frequency-response curves for superharmonic resonances of spiral stiffened MFC cylindrical shells.

Fig. 37. Frequency-response curves for subharmonic resonances of spiral stiffened MFC cylindrical shells. 23

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H. Ahmadi and K. Foroutan

• By increasing the thickness of ceramic and metal layers, the hard• •

• By increasing the excitation amplitude, the jump value increases. • Increasing the values of leads to increasing the jump value.

ening nonlinearity behavior of system increases and decreases, respectively. Increasing the temperature leads to decreasing the hardening nonlinearity behavior of system. The hardening nonlinearity behavior of system under LTD is more than the system under UTD.

Declaration of interest Declarations of interest: none.

Appendix

H11 = cos3 + cos3 ; H21 = sin cos2 + sin cos2 ; H31 = cos2 cos2 H12 = sin2 cos + sin2 cos ; H22 = sin3 + sin3 ; H32 = sin cos + sin cos H13 = 2(sin cos2 sin cos2 ); H23 = 2(sin2 cos sin2 cos ); H33 = sin2 sin2

A.1

1/(sin + sin ) h1 h dE h2 = s s sin( + ) 1/(cos + cos ) shs h3 1/2 J11 = J14 = J21 =

E1 1 E2 1 E1

2

+ Z1 E1s (cos3

+ cos3 ) ,

J12 =

2

+ Z1 E2s (cos3

+ cos3 ) ,

J15 =

2

1 E2

+ Z2 E1s (sin cos2

1

J33 =

E1 2(1 + )

J41 =

2

1 E3

2

+ Z1 E3s (cos3

1

J63 =

E3 2(1 + )

h /2 2

= h /2

cos

+ sin

+ cos3 ),

+ 2Z3 E3s (sin

E3

+ sin

+ sin2

cos )

+ Z1 E2s (sin2

cos

+ sin2

cos )

E1 1

2

+ Z2 E1s (sin3

+ sin3 )

2

+ Z2 E2s (sin3

+ sin3 )

E2 1

E2 2(1 + )

J36 = 2

1

+ sin cos2 ), cos

cos

J25 =

cos ), J42 =

+ Z1 E1s (sin2

J22 =

+ sin cos2 ),

+ Z2 E3s (sin cos2

J51 =

2

+ 2Z3 E1s (sin

2 1 E2 2 1

+ sin cos2 ),

+ Z2 E2s (sin cos2

J24 =

E3

A.2 E1

+ 2Z3 E2s (sin

+ Z1 E3s (sin2

J55 =

E3 1

2

cos

+ sin2

cos

+ Z2 E3s (sin3

+ sin

cos )

cos )

+ sin3 )

cos )

A.3

E ( z ) (z ) T (z ) zdz = 0 1

Case I: Metal-FGM-Ceramic h 2 1

= h 2

E (z ) (z ) T (z ) dz= 1

if UTD: hu Em

m

1 hu Em 2

if LTD :

+ hm Em

m

+

Em

K cm Km

m

cm + Ecm Ksh + 1

Ecm cm 2Ksh + 1

+

+ hb Ec

T

c

i

i = 0 iK sh + 1

+ hm

m

Em m iK sh + 2

Em cm + Ecm m (i + 1) K sh + 2

+

K cm Km

+

Ecm cm (i + 2) K sh + 2

+

i

1 hb Ec 2

c

T

i = 0 iK sh + 1

Case II: Ceramic-FGM-Metal h 2 1

= h 2

E (z ) (z ) T (z ) dz= 1

if UTD: hu Ec

c

+ hm Ec

1 if LTD: T H = hu Ec 2

c

+

Ec

+ Emc Ksh + 1

mc

K cm Km

c

+ hm

c

+

Emc mc 2Ksh + 1

+ hb Em

T

m

i

Ec c iK sh + 2

i = 0 iK sh + 1

+

Ec mc + Emc c (i + 1) K sh + 2 K cm Km

+

Emc mc (i + 2) K sh + 2

i

+

1 hb Em 2

J11 = J22 J14

J12 J21, J22 = J12 J24 ,

J22

, J12 =

J12 = J22 J15

J12

,

J11 =

J12 J25,

J11

,

J21 =

J21 = J11 J24

J21

,

T A.4

i = 0 iK sh + 1

= J11 J22

m

J33 =

J21 J14 ,

1 , J33

J36 =

J22 = J11 J25 24

J36 J33

J21 J15

Thin-Walled Structures 145 (2019) 106388

H. Ahmadi and K. Foroutan

A11 = J22 J14 J21 J15 , A21 = J11 J15 J12 J14 , A11 = J11 J14 J21 J15 J41, A12 = J12 J14 A22 = J12 J24 J22 J25 J52 , A36 = J36 J36

A12 = J22 J24 J22 J15 J42, J63

J21 J25, A22 = J11 J25 J12 J24 A21 = J11 J24 J21 J25 J51 A.5

Case I: Metal-FGM-Ceramic

E1 = E2 =

h /2 h /2 h /2

(

zEsh (z ) dz =

h /2

E3 =

h /2 h /2

Ecm K sh + 1

Esh (z ) dz = Em hu + Em +

)h

+ Ec hb

m

2 (Ecm) K sh hm 2(K sh + 1)(K sh + 2)

(

E

z 2Esh (z ) dz = Em hu3 + [ 12m + Ecm

1 K sh + 3

+

1 K sh + 2

+

1 4K sh + 4

)

hm3 + Ec hb3

Case II: Ceramic-FGM-Metal

E1 = E2 =

h /2 h /2 h /2

(

Esh (z ) dz = Ec hu + Ec + zEsh (z ) dz =

h /2

E3 =

h /2 h /2

Emc K sh + 1

)h

m

+ Em hb

2 (Emc ) K sh hm 2(K sh + 1)(K sh + 2)

E

z 2Esh (z ) dz = Ec hu3 + [ 12c +Emc

(

1 K sh + 3

+

1 K sh + 2

+

)

1 4K sh + 4

hm3 + Em hb3

A.6

Case I: Metal-FGM-Ceramic h /2

E1s =

h /2

E2s =

(

zEs (z ) dz =

(h /2 + hs )

E3s =

h 2

(

h +h s 2

Emc K st + 1

Es (z ) dz = Ec +

(h /2 + hs )

)

z 2Es (z ) dz =

Ec 2

hhs

(

hs h

(

3 4

hs2

Ec 3 h 3 s

h2

)h

s

)

+ 1 + (Emc ) hhs +

3 h 2 hs

(

1 hs K st + 2 h

)

+ 1 + (Emc ) hs3

+

1 2K st + 2

1 K st + 3

+

)

1 h K st + 2 hs

+

h2

1 4(K st + 1)

hs2

1 4(K st + 1)

hs2

Case II: Ceramic-FGM-Metal

E1s =

h /2

E2s =

(

Es (z ) dz = Em +

(h /2 + hs ) h /2

zEs (z ) dz =

(h /2 + hs )

E3s =

h 2

( h2 +hs )

z 2Es (z ) dz =

Em 2

Ecm K st + 1

hhs

(

hs h

(

3 4

hs2

Em 3 h 3 s

h2

)h

s

)

+ 1 + (Ecm) hhs +

3 h 2 hs

)

(

1 hs K st + 2 h

+ 1 + (Ecm) hs3

+

1 2K st + 2

1 K st + 3

+

)

1 h K st + 2 hs

+

h2

A.7

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