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Composite Structures 89 (2009) 275¨C284

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Composite Structures

journal homepage: www.elsevier.com/locate/compstruct

Transient dynamic response of initially stressed composite circular cylindrical shells under radial impulse load

S.M.R. Khalili a,b,*, R. Azarafza c, A. Davar a

a

Centre of Excellence for Research in Advanced Materials and Structures, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Pardis Street, Molasadra Avenue, Vanak Square, Tehran, Iran b Faculty of Engineering, Kingston University, London, UK c Mechanical Engineering Department, Islamic Azad University, Sanandaj Branch, Sanandaj, Iran

a r t i c l e

i n f o

a b s t r a c t

Free and forced vibration of multilayer composite circular cylindrical shells under transverse impulse load as well as combined static axial loads and internal pressure was investigated based on ?rst order shear deformation theory. The boundary condition was considered to be simply supported. Displacement components are the product of functions of position and time. The function of position components of displacement was obtained in the form of double Fourier series. Solution to the equilibrium equations of free vibration of the shell was obtained using Galerkin method. In the analysis of transient dynamic response, the impulse load was in the form of sine pulse, which is applied on a rectangular area. The function of time is obtained using the results of free vibration and convolution integral. Finally time response of displacement components is derived by mode superposition method. The effect of ?ber orientation, axial load, internal pressure and some of the geometrical parameters on the time response of the shells has been shown. The results show that the dynamic responses are governed primarily by the natural period of the structure which is affected by applied axial load and internal pressure. The accuracy of the analysis has been examined by comparing the results with those available in the literature. ? 2008 Elsevier Ltd. All rights reserved.

Article history: Available online 20 August 2008 Keywords: Forced vibration Composite Cylindrical shell Axial load Internal pressure Dynamic response

1. Introduction Composite shells due to some advantages such as, high speci?c strength, high speci?c stiffness, and corrosion resistance are greatly used in the industries such as automobile, aircraft, and space system. Most of composite cylindrical shells are used under impulse load and application of this loading may cause a large deformation and strength reduction. Therefore, considering the dynamic response of the structure is necessary in design process. Many researches have been done on the buckling, free vibration and dynamic response of shells. For example Xavier et al. [1] investigated the buckling and vibration of multilayered orthotropic composite cylindrical shells using simple higher order theory and ?rst order shear deformation theory. Geier et al. [2] studied the effect of stacking sequence on the buckling of simply supported cylindrical shell under axially compressive loads. Dym [3] studied the buckling stresses of cylindrical shell under axially compressive loads. Croll James [4] used the reduced stiffness method to analyse

* Corresponding author. Address: Centre of Excellence for Research in Advanced Materials and Structures, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Pardis Street, Molasadra Avenue, Vanak Square, Tehran, Iran. Tel.: +98 2188674747; fax: +98 2188674748. E-mail address: smrkhalili2005@gmail.com (S.M.R. Khalili). 0263-8223/$ - see front matter ? 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.08.002

the buckling of cylindrical shells. Ng et al. [5] presented the dynamic stability analysis of cylindrical panels taking into account the transverse shear effects. Singer et al. [6] investigated the stability of stiffened shells subjected to axial compressive loads. Li and Chen [7] investigated transient dynamic response analysis of orthotropic circular cylindrical shell subjected to external hydrostatic pressure. They used classical shell theory and considered simply supported boundary conditions. Bardell et al. [8] investigated the free and forced vibration analysis for laminated cylindrical shell. Lam and Loy [9] studied the in?uence of boundary conditions for thin laminated rotating cylindrical shell using ?rst Love¡¯s approximation and Galerkin method. Lee and Lee [10] studied the free vibration and dynamic response for cross-ply composite circular cylindrical shell. They considered simply supported boundary conditions. Matemilola and Stronge [11] developed an analytical solution for the impact response of simply supported anisotropic composite cylinder. Christoforou and Swanson [12] investigated the analysis of simply supported orthotropic cylindrical shells subject to lateral impact load. Sheinman and Greif [13] developed a general analytical and numerical method used for free and forced vibration of multilayer thin shells of revolution made of elastic orthotropic materials with arbitrary stacking sequence. Huang and Soedel [14] studied the effect of coriolis acceleration on the forced vibration of rotating cylindrical shells under a ?xed

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or harmonic concentrated load. Jafari et al. [15] studied the free and forced vibration of clamped-free composite circular cylindrical shells. The dynamic response was studied under transverse impulse and axial compressive loads by using modal technique. In this paper, free vibration and dynamic response of multilayered composite circular cylindrical shells under transverse impulse load as well as combined static axial loads and internal pressure was investigated based on ?rst order shear deformation theory. The boundary condition was considered to be simply supported. Classical laminate plate theory was used for the analysis of composite material. Displacement components are the product of functions of position and time. The function of position components of displacement was obtained in the form of double Fourier series consist of a modal beam function in the axial direction and trigonometric functions in the tangential direction. The equilibrium equations of free vibration of the shell are solved using the Galerkin method. In the analysis of transient dynamic response, the impulse load was in the form of sine pulse which is applied on a rectangular area. The area of applied load is variable and can be everywhere on the shell. The function of time is obtained using the results of free vibration and convolution integral. Finally, the time response of displacement components is derived using mode superposition method. The effect of ?ber orientation, axial load, internal pressure and the geometry parameters (such as length to radius and thickness to radius ratios) on the time response of the shells were investigated. 2. Governing equations A circular cylindrical shell with mean radius of R, thickness h and length L is shown in Fig. 1. u, v and w are the displacement components in the axial, tangential and radial directions respectively, and that the deformations are assumed to be small. The definition of stacking sequence of each layer in multilayered composite circular cylindrical shell is shown in Fig. 2. Based on ?rst order shear deformation theory, the equilibrium equations for a shell under axial loads and internal pressure are as follows [5,10]:

Fig. 1. Cylindrical shell geometry and coordinate system.

! oN x oNxu o2 u ow ? qx ?x; u; t? ? ?P ? Rou2 ox ox Rou o2 u o2 b ? I2 2x 2 ot ot ! oN xu oNu Q u o2 v o2 v ow ? ? ? ? Na 2 ? P ox Rou2 ou ox Rou R 2 2I2 o2 v I3 o bu ? I2 ? ? qu ?x; u; t? ? I1 ? 2 R ot R ot2 ! oQ x oQ u Nu o2 w ou ov o2 w ? ? ? ? ? Na 2 ? P ox ox Rou Rou2 ox Rou R ? I1 ? qr ?x; u; t? ? I1 o2 w ot 2 ?I1 ; I2 ; I3 ? ? Z

h 2 h ?2

Fig. 2. Stacking sequence of the cross-section.

?1; z; z2 ?qk dz;

?2?

where qk is the density for each layer. Constitutive equations of composite shell based on classical laminate theory are de?ned by the following relations [17]:

?1?

oM x oMxu o2 b o2 u ? Q x ? mx ?x; u; t? ? I3 2x ? I2 2 ? ox Rou ot ot oM xu oM u ? Q u ? mu ?x; u; t? ? ox Rou 2 o bu I3 o2 v ? I3 2 ? I 2 ? : R ot 2 ot

In the above equations, bx and bu are the slope in the plane of x ? z and u ? z, respectively, Na is axial force and P is internal pressure. Also qx, qu and qr are the external forces, mx and mu are the external moments that excited the shell. I1, I2 and I3 are de?ned by the following relation [16]:

9 2 9 8 38 A11 A12 A16 > e > > Nx > = < < x = 6 7 Nu ? 4 A12 A22 A26 5 e > > > u > ; : :c ; N xu A16 A26 A66 xu 2 38 9 B11 B12 B16 > kx > < = 6 7 ? 4 B12 B22 B26 5 ku > > ; : B16 B26 B66 k xu 9 2 8 38 9 B11 B12 B16 > ex > > Mx > = = < < 6 7 Mu ? 4 B12 B22 B26 5 e u > > > > ; : :c ; M xu B16 B26 B66 xu 2 38 9 D11 D12 D16 > kx > < = 6 7 ? 4 D12 D22 D26 5 ku > > ; : D16 D26 D66 k xu ( ) ( ) cxz Qx H55 H45 ? ; c z Qu H45 H44 u

?3?

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277

where A, B, D and H are the extensional, coupling, bending and thickness shear stiffness matrices respectively and they are de?ned as follows:

?Aij ; Bij ; Dij ? ? Hij ? k Z

h 2

Z

h 2

h ?2

?1; z; z2 ?Q ij dz ?i; j ? 1; 2; 6? ?4?

In Eq. (8-a), Tmn(t) is function of time. Also Amn, Bmn, Cmn, Dmn and Emn are the constant coef?cients of the natural mode shapes associated with the free vibration problems, m is the axial half wave number and n is the circumferential wave number. Values of ai, rm and km in Eq. (8-b) could be obtained from corresponding boundary conditions:

h ?2

?1; z; z2 ?Q ij dz ?i; j ? 4; 5?;

a1 ? a2 ? a3 ? 0; a4 ? 1; rm ? 1; km ? mp:

4. Buckling analysis

?9?

where k0 is the shear correction factor introduced by Mindlin and is 2 equal to p , [10]. Q ij is the transformed reduced stiffness matrix. e , x 12 eu and exu are the mid-plane engineering strains, c and c z are the u xz 0 transverse shear strains and kx , ku and kxu are curvatures and twist of the shell, so that [10]:

8 > < (

> e > > ou > > k > > ox > ox x x = = = < = > < < 1 ov w 1 obu ku e ; ? R ou ? R ; ? u R ou > > > > ; : ; > :c ; : 1 ou ov > > > > ob ; : 1 x obu > kxu ? ox xu ? R ou

9

8

9

8

9

8

obx

9

In other to obtain the critical axial buckling load, the static solution must be considered, so that the function of time in Eq. (8) must be neglected (Tmn(t) = 0). Substituting relations (8) into Eq. (7) and using Galerkin method in solving differential equations, yields a set of ?ve equations in the following form:

Z 2p Z

0 0

L

c c

xz uz

)

(

?

bx ? ow ox v bu ? 1 ow ? R R ou

)

R ou

ox

?L11 u ? L12 v ? L13 w ? L14 bx ? L15 bu ?

L

:

?5?

Z 2p Z

0

cos nudxdu ? 0 ?L21 u ? L22 v ? L23 w ? L24 bx ? L25 bu ?

L

0

3. Boundary conditions The boundary conditions for the cylindrical shell which is simply supported along its curve edges at x = 0 and x = L are considered as [18]:

Z 2p Z

0

sin nudxdu ? 0 ?L31 u ? L32 v ? L33 w ? L34 bx ? L35 bu ?

L

0

?10?

Nx ?0; u; t? ? Nx ?L; u; t? ? 0 Mx ?0; u; t? ? M x ?L; u; t? ? 0 w?0; u; t? ? w?L; u; t? ? 0 v?0; u; t? ? v?L; u; t? ? 0 bu ?0; u; t? ? bu ?L; u; t? ? 0:

In order to solve buckling and free vibration problem, external excitations are taken to be zero. After substituting Eqs. (3) and (5) into equations of motion (1), the results can be simpli?ed to the following equation:

Z 2p Z

0

cos nudxdu ? 0 ?L41 u ? L42 v ? L43 w ? L44 bx ? L45 bu ?

L

0

?6?

Z 2p Z

0

cos nudxdu ? 0 ?L51 u ? L52 v ? L53 w ? L54 bx ? L55 bu ?

0

sin nudxdu ? 0;

and after some simpli?cations, the Eq. (11) was obtained:

?C ij ?fAmn Bmn C mn Dmn Emn gT ? 0 ?i; j ? 1; . . . ; 5?:

?11?

?L?fUg ? f0g;

where

?7-a? 3 9 > > > > > > = fUg ? w?x; u; t? : > > > > > b ?x; u; t? > > > x > > > > ; : bh ?x; u; t? 8 > u?x; u; t? > > > > v?x; u; t? > <

By setting determinant of coef?cients Cij equal to zero, the buckling loads equation is derived as:

2

c1 N2 ? c2 Na ? c3 ? 0; a

?12?

7 6 6 L21 L22 L23 L24 L25 7 7 6 6 L31 L32 L33 L34 L35 7; L?6 7 7 6 4 L41 L42 L43 L44 L45 5 L51 L52 L53 L54 L55

L11 L12 L13 L14 L15

?7-b?

where ci are constant coef?cient and Na is the axial buckling load. After obtaining the solution of Eq. (12), the critical value of axial buckling load, that is Ncr, was obtained in terms of m and n. 5. Free vibration analysis To solve the free vibration analysis, the function of time is treated as T mn ?t? ? eixmn t , where xmn is the natural frequency. By considering the axial compressive load equal to the fraction of critical buckling load and internal pressure, natural frequencies and mode shapes are obtained. By applying Galerkin method, similar to the buckling analysis, the following set of equations are derived as follows:

Lij are the differential operators and are shown in Appendix. In order to satisfy the boundary conditions, u, v, w, bx and bu are de?ned by the following double Fourier series [9]:

u? v?

XX

m n XX

Amn T mn ?t? ? Bmn T mn ?t? ?

XX

m n XX

Amn

dgu ?x? cos nuT mn ?t? dx

Bmn gv ?x? sin nuT mn ?t? C mn gw ?x? cos nuT mn ?t? Dmn gbx ?x? cos nuT mn ?t? Emn gbu ?x? sin nuT mn ?t? ?8-a?

w? bx ?

m n XX

C mn T mn ?t? ? Dmn T mn ?t? ? Emn T mn ?t? ?

m n XX m n

XX

m

n

??K ij ? ? x2 ?M ij ??fAmn Bmn C mn Dmn Emn gT ? 0 ?i; j ? 1; . . . ; 5?; mn

?13?

XX

m n XX m n

bu ?

m n XX m n

where Kij and Mij are stiffness and mass matrices. By setting determinant of coef?cients equal to zero, the characteristic frequency equation is derived as:

k x k x gi ?x? ? a1 cosh m ? a2 cos m L L km x km x ? rm a3 sinh ? a4 sin ?i ? u; v; w; bx ; bu ?: L L

d1 x10 ? d2 x8 ? d3 x6 ? d4 x4 ? d5 x2 ? d6 ? 0;

?14?

?8-b?

where di are constant coef?cients. By solving Eq. (14), natural frequencies are calculated, and by substituting these frequencies in Eq. (13), the constant coef?cients of mode shapes are obtained.

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6. Dynamic response analysis Fig. 3 shows the lateral applied impulsive load on a small rectangular area. The applied loads are de?ned as:

Jmn ?

qx ?x; u; t? ? Q x ?x; u?f ?t? ? ? XX X mn

XX

m n

X mn f ?t?

Z p I 1 A2 ? B 2 ? C 2 mn mn mn 0 ?p 2 ? I2 2Dmn Amn ? 2Emn Bmn ? B2 R mn 2 ?I3 D2 ? E2 ? Emn Bmn dx ? du: mn mn R Z

L

?19?

dgu ?x? cos nuf ?t? dx m n XX Y mn f ?t? qu ?x; u; t? ? Q u ?x; u?f ?t? ? m n XX Y mn gv ?x? sin nuf ?t? ? m n XX Pmn f ?t? qr ?x; u; t? ? Q r ?x; u?f ?t? ? m n XX Pmn gw ?x? cos nuf ?t? ? m n XX Z mn f ?t? mx ?x; u; t? ? M x ?x; u?f ?t? ? m n XX Z mn gbx ?x? cos nuf ?t? ? m n XX W mn f ?t? mu ?x; u; t? ? M u ?x; u?f ?t? ? m n XX W mn gbh ?x? sin nuf ?t?: ?

m n

Jmn is the normalized masses and Gmn is the generalized forces, [10]. The applied load is assumed to be only in the radial direction over a small rectangular area (2l1 ? 2l2) and the other external excitations are neglected. Regarding to Fig. 3, the area of applied load is variable and the center of this area can be every where on the shell as:

?15?

x2 ? x1 ? 2l2 and R?w2 ? w1 ? ? 2l1 x 1 ? x2 w ? w2 xL ? and uL ? 1 ; 2 2

?20?

where xL and uL are coordinates indicating the center point of the applied load area. So the constant Fourier coef?cients Pmn in Eq. (15) can be determined as follows:

n ? 0; n > 0;

P m0 ? P mn

2 pL

Z Z

L

Z p Z p

0 L

?p

Q r ?x; u? sin Q r ?x; u? sin

mpx dxdu L mpx cos nudxdu: L

2 ? pL

?21?

0

?p

In Eq. (15), f(t) is function of time. Xmn,Ymn, Pmn, Zmn and Wmn are the constant coef?cients that can be calculated from the pro?le of the applied loads. Substituting the displacements and assumed exciting forces and moments in the equilibrium Eq. (1), and using the results of free vibration, we have:

For the load considered in Fig. 3:

n ? 0; n > 0;

P m0 ? P mn

€ ? x2 T mn ?t??I1 Amn ? I2 Dmn ? ? T mn ?t??I1 Amn ? I2 Dmn ? ? qx mn 2I2 ? x2 T mn ?t? I1 ? Bmn mn R I3 2I2 I3 € Emn ? T mn ?t? I1 ? Bmn ? I2 ? Emn ? qu ? I2 ? R R R € ? x2 T mn ?t?I1 C mn ? T mn ?t?I1 C mn ? q

mn r

mpx mpx i ?2 h 2 1 cos ? cos ?w2 ? w1 ? mp2 L L h mpx mpx i ?2 2 1 ? cos ? cos ?sin?nw2 ? mnp2 L L ? sin?nw1 ??:

?22?

By substituting Eqs. (22) and (18) into Eq. (17) and some simpli?cations, the following relation is obtained:

Pmn C mn f ?t? € T mn ?t? ? x2 T mn ?t? ? : mn Nmn

In Eq. (23), the term Nmn is introduced by:

?23?

€ ?x ? I3 Dmn ? ? T mn ?t??I2 Amn ? I3 Dmn ? ? mx I3 ? x2 T mn ?t? I2 ? Bmn ? I3 Emn mn R € mn ?t? I2 ? I3 Bmn ? I3 Emn ? mu : ?T R

2 mn T mn ?t??I2 Amn

?16?

After summation of two sides of the above equations and simplifying, we ?nd a second order ordinary differential equation as follows:

Nmn ? I1 A2 ? B2 ? C 2 mn mn mn 1 ? 2I2 Amn Dmn ? Bmn Emn ? B2 R mn 2 ? I3 D2 ? E2 ? Bmn Emn : mn mn R

?24?

For zero initial conditions, the solution of Eq. (17) is obtained using Laplace transformation:

€ T mn ?t? ? x2 T mn ?t? ? Gmn ?t?; mn

where

?17?

T mn ?t? ?

Gmn ?

o RL Rp n Amn qx ? Bmn qu ? C mn qr ? Dmn mx ? Emn mu dxdu 0 ?p J mn ?18?

Pmn C mn Nmn xmn

Z

t

f ?s? sin xmn ?t ? s?ds:

?25?

0

Finally, the time response of the cylindrical shell base on mode superposition theory is:

Z t X X Pmn C mn dg ?x? Amn u f ?s? sin xmn ?t ? s?ds cos nu dx Nmn xmn 0 m n Z t X X Pmn C mn v? Bmn gv ?x? sin nu f ?s? sin xmn ?t ? s?ds Nmn xmn 0 m n Z t X X P mn C 2 mn w? gw ?x? cos nu f ?s? sin xmn ?t ? s?ds N mn xmn 0 m n Z t X X Pmn C mn bx ? Dmn gbx ?x? cos nu f ?s? sin xmn ?t ? s?ds Nmn xmn 0 m n Z t X X Pmn C mn bu ? Emn gbu ?x? sin nu f ?s? sin xmn ?t ? s?ds: Nmn xmn 0 m n u? ?26?

Fig. 3. Load applied laterally on a small rectangular area.

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279

It is to be noted that the mode shape coef?cients (Amn, Bmn, Cmn, Dmn and Emn) are calculated by the use of Eq. (13) together with the property of mode shape orthogonality with respect to mass matrix. 7. De?nition of impulse loads Time duration of applying impulse load begins from t = 0 to t = t1 (t1 is equal to the natural period of the shell). For a half sine impulse load, the solution of convolution integrals in Eq. (26) is obtained in the following form:

Table 2 Ply properties of the composites used in Ref. [19] E11 (GPa) Graphite/epoxy Kevlar/epoxy E-glass/epoxy 132 76.8 38.6 E22 (GPa) 10.8 5.50 8.27 G12 (GPa) 5.65 2.07 4.14

m12

0.24 0.34 0.26

Ply thickness (mm) 0.127 0.127 0.127

( Z

f ?t? ? F 0 sin p1t ; t f ?t? ? 0;

t

Table 3 Comparison between the results for buckling loads of a simply supported anisotropic cross-ply cylindrical shell with lay up [0/90]2S, R = 0.2 m Con?guration Material Graphite/epoxy Ratio (L/R) 1 3 1 2 1 3 Buckling loads (kN/m) Present 83.02(3,11) 83.02(9,11) 38.69(3,10) 38.25(5,10) 42.48(3,12) 42.48(9,12) Ref. [19] 84.23(3,11) 84.23(9,11) 39.38(3,10) 38.96(5,10) 42.97(3,12) 42.97(9,12) ?1.44 ?1.44 ?1.75 ?1.82 ?1.14 ?1.14 Discrepancy (%)

0 6 t 6 t1 t > t1 ?27? 0 6 t 6 t1 t > t1 :

0

f ?s? sin xmn ?t ? s?ds 8 > F 0 t 1 ?p sin?xmn t? ? xmn t1 sin?pt=t1 ?? > ; > < ?p2 ? t 2 x2 ? 1 mn ? > F 0 t 1 p?sin?xmn t? ? sin xmn ?t ? t1 ?? > > ; : ? p 2 ? t 2 x2 ? 1 mn

Kevlar/epoxy E-glass/epoxy

(The numbers (m, n) in the parenthesis represent the buckling modes, i.e. number of axial half waves (m) and number of circumferential waves (n)).

Radial Displacement (m)

To obtain the numerical results, a computer code was written, which includes three subroutines (including buckling, free vibrations and forced vibrations). This program was prepared generally with respect to composite construction, shells geometry, boundary conditions, type of loading (static and dynamic), size and location of applied impulse load, and the location of the point where the dynamic responses (de?ections, stresses and strains vs. time) must be calculated. There is no limitation on the stacking sequence and ?ber orientation. Also the mentioned code was developed based on both classical (CST) and ?rst order shear deformation (FSDT) shell theories. 8. Veri?cation of the model To ensure the accuracy of the model, the buckling load obtained by Ref. [6] for isotropic cylindrical shell and also the analysis obtained by ANSYS software for the same case was compared with the results obtained by the present method. Table 1 shows the discrepancy in the results. The major difference between the present results with that of [6] is due to the application of second order terms in the strain¨Cdisplacement relations used by Ref. [6]. Also to ensure the accuracy of the model for anisotropic cylindrical shells, the buckling loads obtained by Ref. [19] was compared with the same results obtained by the present method. Table 2 shows the ply properties of the composites used in Ref. [19] and Table 3 shows the discrepancy between the results. Comparison of the results of the time response of a simply supported composite shell obtained by Ref. [10] and the present model is also shown in Fig. 4. Fig. 5 also shows the comparison of the results of the natural frequencies of a simply supported composite shell obtained by Ref. [9], ANSYS software analysis and the present method. The good agreement between the results in Tables 1 and 3, Figs. 4 and 5 depict the validity of the proposed method.

Table 1 Comparison between the results obtained by the present model and other methods for critical buckling load of the simply supported isotropic cylindrical shell with the following material and geometrical properties: E11 = E22 = 200 GPa, G12 = G13 = G23 = 76.9 GPa, m12 = 0.3, q = 7800 kg/m3 R = 0.304 m, L = 0.750 m, h = 0.001 m Method of solution Present Ref. [6] ANSYS Number of circumferential waves (n) 12 12 12 Number of axial half waves (m) 4 4 4 Critical buckling load (N/mm) 392 410 419 Discrepancy % ¨C 4.4 6.9

4

x 10

-4

Present Ref.[10] 2

0

-2

-4

-6

-8

0

0.2

0.4

0.6

0.8

1 1.2 Time(sec)

1.4

1.6

1.8 x 10

2

-3

Fig. 4. Comparison of the central de?ection of simply supported composite (CFRP) cylindrical shell with Ref. [10]. Lay up [0/90/0/90/0/90], R = 0.2 m, L = 0.2 m, h = 1.2 mm, 2l1 ? 2l2 = 6.2 ? 2 cm2.

Fig. 5. Comparison of the natural frequencies for simply supported composite shell with the lay up [90/0/90], m = 1, h/R = 0.002, L/R = 20.

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9. Numerical results and discussions In the present research, geometric and material property of the laminated cylindrical shell is given by:

E11 ? 14:34 GPa; E22 ? 5:1 GPa; G12 ? 1:86 GPa; G13 ? 1:86 GPa; G23 ? 0:89 GPa;

m12 ? 0:35; q ? 1390 kg=m3

L ? 2 m; R ? 0:5 m; h ? 0:003 m:

In the analysis of transient dynamic response, the impulse load was considered as sine pulse. The magnitude of peak load (F0) is equal to 6000 Pa. The load is applied at the mid span of the shell. The time response in all of the ?gures is obtained at the mid point of the area of the applied load. The number of modes (m ? n) to attain suf?cient convergence is obtained to be (20 ? 30). The thicknesses of all layers are taken to be the same. Hereinafter, everywhere otherwise stated, the area of applied load is considered to be 2l1 ? 2l2 = 7.85 ? 10 cm2 and the stacking sequence is taken to be [30/?30/?30/30/30]. In Fig. 6, buckling loads vs. circumferential wave number (n) are presented for different values of m. The value of critical buckling load is equal to Ncr = 75,700 N/m which was occured at m = 1 and n = 5. By increasing the value of m, the axial buckling load increases for the considered shell. Natural frequencies vs. circumferential wave number for different values of m are studied in Fig. 7. As it is obvious, the fundamental frequency occurs at m = 1 and n = 5. As the value of m increases, frequency is increased. Variation of frequency due to variation of m is considerable especially in small values of n, but this variation is not so much in large values of n. Effect of axial compressive load on the natural frequencies is presented in Fig. 8. As it is clear, increasing the amount of axial load, all of the natural frequencies decrease, especially those around the fundamental frequency (m = 1, n = 5) corresponding to the mode shape in which critical axial buckling load was occurred. As the value of axial compressive load approaches the critical buckling load, fundamental frequency, corresponding to (m = 1, n = 5) approaches zero. Effect of axial compressive load on the time response of radial displacement, axial and tangential strains is studied in Figs. 9¨C11, respectively. In these ?gures, it is showed that by increasing the value of axial load, amplitudes (peak values) of time responses for displacement and strains and time to happen these peak values increase. Figs. 12¨C16 indicate the effect of internal pressure on the buckling load, natural frequency, time response of radial displacement, axial and tangential strains, respectively. By increasing the value of internal pressure, both critical buckling load and fundamental frequency increase. In addition, by increasing the

Fig. 7. Natural frequencies vs. circumferential wave number.

Fig. 8. Effect of axial load on the natural frequencies for m = 1.

Fig. 9. Effect of axial load on the radial displacement time response of the shell.

Fig. 6. Buckling loads vs. circumferential wave number.

amount of internal pressure, amplitudes of time responses and time elapsed to reach these peak values decrease. Geometric parameters are discussed in Figs. 17 and 18, that is the effect of L/R and h/R ratios on the time response of radial displacement of the shell is investigated. Increasing the value of L/R ratio causes the amplitude

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281

Fig. 10. Effect of axial load on longitudinal strain response.

Fig. 13. Effect of internal pressure on the natural frequencies corresponding to m = 1 (no axial load).

Fig. 11. Effect of axial load on tangential strain response.

Fig. 14. Effect of internal pressure on the radial displacement time response (no axial load).

Fig. 12. Effect of internal pressure on the buckling loads corresponding to m = 1 (no axial load).

Fig. 15. Effect of internal pressure on the longitudinal strain time response (no axial load).

and time elapsed to peak value of radial displacement to increase. Conversely, by increasing the value of h/R ratio, the amplitude and time elapsed to peak value of radial displacement decrease. Effect of ?ber orientation on the dynamic response is depicted in

Fig. 19. Among the stacking sequences considered, the greatest radial displacement is corresponds to 0¡ã and the lowest value is relates to 90¡ã. Deformation of the shell in axial cross-section

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(u = 0¡ã, x = 0 $ L) and transverse cross-section (u = 0¡ã¨C360¡ã, x = L/2) in different times after applying the impulse load are presented in Figs. 20 and 21, respectively. During the presence of pressure load

on the outer surface of the shell, deformation is extended in the axial direction until the applied load is removed, and then the curve of the shell in axial direction becomes more smooth and starts its free

Fig. 16. Effect of internal pressure on the tangential strain time response (no axial load).

Fig. 19. Effect of ?ber orientation on the radial displacement time response.

Fig. 17. Effect of L/R ratio on the radial displacement time response.

Fig. 20. Radial displacement of longitudinal cross-section at u = 0¡ã in different times after applying impulse load.

Fig. 18. Effect of h/R ratio on the radial displacement time response.

Fig. 21. Radial displacement of transverse cross-section at x = 1 m in different times after applying impulse load (deformations are scaled 170 times greater than the actual state).

S.M.R. Khalili et al. / Composite Structures 89 (2009) 275¨C284

283

vibrations, Fig. 20. As it is obvious in Fig. 21, the deformation of the shell at the points with their position tangentially far from the center point of the applied load happens later than the points with their position tangentially near to the center point. This delay is due to wave propagation among the continuous medium of the shell. In Figs. 22¨C24, natural mode shape corresponding to fundamental fre-

quency of the composite cylindrical shell at (m = 1, n = 5) is demonstrated. 10. Conclusions 1. As the axial compressive load increases up to the critical buckling load, the natural frequencies decrease. If the axial compressive load is equal to critical buckling load, the frequency corresponding to the mode shape in which the critical load occurs becomes zero. 2. With increasing the axial compressive load, the peak values of time responses increase. So that if the axial compressive load become equal to eighty percent of the critical buckling load, the maximum value of radial displacement, longitudinal strain and tangential strain increase about 3.5, 3 and 1.5 times, respectively. Also the corresponding times to reach the peak values increase. 3. By increasing the internal pressure, the peak value of time responses decrease. So that if the internal pressure increase four times, maximum value of radial displacement decrease about 2 times and the longitudinal and tangential strains decrease about 1.5 times. Also the corresponding times to reach the peak values decrease. 4. If the h/R ratio increases, the peak value of time responses (radial displacement, longitudinal strain and tangential strain) will decrease and the corresponding times to reach the peak values will decrease too. 5. If the L/R ratio increases, the peak value of time responses will increase too and also the corresponding times to reach the peak values will increase. 6. Time response of radial displacement of the shell is more affected by the variation in the thickness in comparison with the variation in the length. 7. Among different lamination stacking sequence considered, [0/ 0/0/0/0] depicts the lowest, while [90/90/90/90/90] shows the highest radial displacement. Appendix Differential operators Lij are as follows:

Fig. 22. Isometric view of mode shape corresponding to the fundamental frequency (m = 1, n = 5).

L11 ? ?A11 R? L12 L13 L14 L15 L22 L23 L24 L25

Fig. 23. Left view of mode shape corresponding to the fundamental frequency (m = 1, n = 5).

Fig. 24. Top view of mode shape corresponding to the fundamental frequency (m = 1, n = 5).

2 # o2 o2 A66 o ? ?2A16 ? ? ?P ox2 oxou R ou2 " 2 # o2 o2 A26 o ? L21 ? ?A16 R? 2 ? ?A12 ? A66 ? ? ox oxou R ou2 o A26 o ? ?L31 ? ?A12 ? RP? ? ox R ou " 2 # 2 o o2 B66 o ? L41 ? ?B11 R? 2 ? ?2B16 ? ? ox oxou R ou2 " 2 # o2 o2 B26 o ? L51 ? ?B16 R? 2 ? ?B12 ? B66 ? ? ox oxou R ou2 " # 2 o2 o2 A22 o H44 ?P ? ?A66 R ? Na R? 2 ? ?2A26 ? ? ? ox oxou R ou2 R o A22 H44 o ? ?L32 ? ?A26 ? H45 ? ? ? ? RP ox ou R R " # 2 o2 o2 B26 o ? L42 ? ?B16 R? 2 ? ?B12 ? B66 ? ? ? H45 ox oxou R ou2 " # 2 o2 o2 B22 o ? L52 ? ?B66 R? 2 ? ?2B26 ? ? ? H44 ox oxou R ou2

"

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L33 L34 L35 L44 L45 L55

# 2 o2 o2 H44 o A22 ? ?N a R ? H55 R? 2 ? ?2H45 ? ? ? ?P ox oxou R ou2 R o B26 o ? L43 ? ?H55 R ? B12 ? ? H45 ? ox R ou o B22 o ? ?L53 ? ?H45 R ? B26 ? ? H44 ? ox R ou " # 2 o2 o2 D66 o ? ?D11 R? 2 ? ?2D16 ? ? ? H55 R ox oxou R ou2 " # 2 o2 o2 D26 o ? L54 ? ?D16 R? 2 ? ?D12 ? D66 ? ? ? H45 R ox oxou R ou2 " # 2 o2 o2 D22 o ? ?D66 R? 2 ? ?2D26 ? ? ? H44 R : ox oxou R ou2

"

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