International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014
Free Vibration Analysis of Thick Cylindrical Composite Shells Using Higher Order Shear Deformation Theory MOHAMMAD ZANNON, MOHAMAD QATU
Abstract— This paper presents a free vibration analysis of laminated cylindrical shells using higher order shear deformation theory recently developed by Zannon et al., (2014). Equilibrium equations of motion, stress resultants and strain displacement relations are developed using Hamilton’s principles. These equations are based on the theory of elasticity with terms truncated to a third order (hence, third order theory). The equations can be used to characterize the system under various boundary conditions. For the present analysis, the boundary conditions considered are simply supported with a cross-ply lamination sequence. Here, we specifically developed mathematical formulation that considers transverse normal stress, shear deformation and rotary inertia for the shell system. The free vibrational analysis using the third -order shears deformation shell theory led to a system of generalized eigenvalue problem. This eigenvalue problem is then solved numerically using commercial Matlab software to obtain the free undamped vibrational frequencies. Since the higher frequencies are often damped, the first five natural frequency parameters are reported and compared with previously published first order approximation and three dimensional finite element analyses.
structures. Thus, three dimensional analyses of shells is considered to be the most accurate, but is complex and time consuming when the shell system has many graded layers due to the incorporation of composite materials. Even for today’s availability of computational facilities, three dimensional finite element analyses for most practical problems are not feasible. Lately, many shell structures are making use of the functionality of composite materials, which are often energy efficient and durable. Composite shells often consist of many layers of varied strength materials; therefore, each layer has different material characteristics (Naghdi & Berry, 1964; Soldatos, 1999; Noor et al., 1996; Koiter, 1969).
The Literature on shell vibrations is vast. Hundreds of papers were published on free vibration especially for cylindrical shells (Reddy, 2004; Reissner, 1945; Librescu & Frederick, 1989; Love, 1892; Leissa, 1993; Timoshenko & Woinowsky 1959; Koiter, 1969). Various engineering applications and developments are underway for laminated composite shells. For example, aerospace industry, material technology, marine, petroleum, construction and automotive engineering (Qatu et al., 2013; Qatu, 2004). The vibration of thick shells has conventionally been solved using the 1st order shear deformation shell theory (Qatu et al., 2013; Asadia et al., 2012; Chen et al., 1997; Love, 1892). Often three dimensional theory of elasticity is used for solving theories of shell
A comprehensive summary and discussion of shell theories using first order deformation and various mode shape characteristics has been done by researches (Leissa, 1993; Reissner, 1945; Kurylov & Amabili 2010; Qatu 2002; Amabili & Paidoussis 2003) over the past 40 years. The difficulties of composite materials make shell structure hard to explore three dimensional theory of elasticity, which is important for the development of various shell and plate theories. Even though, researchers have done an extensive study on various thin and thick shell vibrations using lower order approximation in displacements, which is one of drawback of many practical applications. Zannon et. al. (2014) remediated some of these drawbacks by incorporating higher order approximations and a non-zero mid-surface displacement in the shell theory. However, any new theory in vibrational analysis has to establish the basic fundamentals concerning free vibrations before we proceed to further analyses. Different plate theories were classified into classical theories; two of the most important ones are first order shear deformation theories and higher-order shear deformation theories based on the thickness ratio. In the case of thick shell theory, the effect of the thickness of shells and the depth ratio of shells should not be neglected (Librescu & Frederick, 1989; Leissa, 1993; Timoshenko & Woinowsky 1959; Zannon et al., 2014; Qatu, 2010; Asadi et al., 2012; Qatu et al., 2013). The applications of shells vibration analysis are very crucial for structural design. Furthermore, experimental design of such structures can be rather cumbersome and costly. Therefore, theoretical model becomes viable and should elucidate various modes of vibrations during the impact period. Hence, composite shell theories are used widely in many shell structural applications.
Manuscript received Oct 06, 2014 MOHAMMAD ZANNON, Department of Mathematics, Tafila Technical University, Tafila, Jordan MOHAMAD QATU, School of Engineering and Technology, Central Michigan University, Mount Pleasant
The characteristic properties of vibrational behavior of composite laminated shells are used to understand the stability criteria of the system as a whole. Over the past years many researchers have analyzed various aspects of vibrations analysis of the system.
Index Terms— Free Vibration, Hamilton Principles, Thick Shell, Natural Frequency, Third order shear deformation, Cross-ply, Eigenvalue I. INTRODUCTION
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Free Vibration Analysis of Thick Cylindrical Composite Shells Using Higher Order Shear Deformation Theory Nomenclature Aij , Aˆij , Aij stretching and shearing stiffness parameters
the
E
Bij , Bij , Bij stiffness parameters E
procedures
z
R
often
encounters
) in both the stress resultant
and strain displacement equations, especially for thick shell. However, some researchers have developed higher order shell
n
Bij , Bˆij , Bij coupling stiffness parameters D
solution
difficulties due to the term (1
Aij , Aij , Aij stiffness parameters D
respective
theories by neglecting the (1
n
Dij , Dˆ ij , Dij bending and twisting stiffness parameters
z
R
) term (Reddy, 2004;
H D , H E , H z normal strains
Librescu & Frederick, 1989; Koiter, 1969; Noor et al., 1996), which is not applicable for thick shell structures. More specifically, Zannon et.al (2014) incorporated this term in the mathematical formulation of a third-order shear deformation shell thick theory by Zannon (TSDTZ) without violating the classical shell theory assumptions. The solution of the TSDTZ is used to obtain the free vibrational characteristics of composite thick shells with specified boundary conditions of simply supported cross-ply laminates. The computational results obtained from TSDTZ are encouraging. It showed that TSDTZ improved the results in comparison to the first order shear deformation theory suggested by several other researchers (Asadi et al., 2012; Qatu et al., 2012; Asadi & Qatu, 2012; Qatu, 1994, 2004).
J DE , J D z , J E z shear strains
II. THEORIES AND FORMULATION
Dij , Dij , Dij stiffness parameters D
E
n
Eij , Fij , Lij , Eij , Eij , Eij
½° ž higher order stiffness parameters Fij , Fij , Fij , Lij , Lij , Lij ° ¿ D
D
E
n
E
D
E
n
n
( D , E , z ) shell coordinates I i rotary inertia
U ( k ) mass density of kth layer V D , V E , V z normal stress V DE , V E z , V D z shear stress
Qij ( k ) , Qij elastic stiffness parameters for layer k
Basic equations of TSDTZ have been developed Zannon et al., (2014) to study the free vibrational analysis of simply supported cross-ply cylindrical shell. The analysis of shell type elastic body is based on three fundamental principles such as equilibrium, continuity (displacement) and constitutive relationship (stress-strain). We used Hamilton’s principle to include various physical mechanisms such as extensional motion (stress, strain, transvers shear), and rotatory inertia of complex shells which undergo resonance (Deana & Werbya, 1992; Naghdi & Berry, 1964). The Hamilton’s principle (Khare & Rode, 2005; Qatu, 2004; Zannon et al., 2014) for the equations of motion of a body with surface S between two arbitrary time intervals t0 and t1 requires that
QD , QE transverse shear force mD(1) , mE(1) , mD( 2 ) , mE( 2 ) distributed couples qz , qE , qD distributed forces A, B Lamec parameters
MD , M E higher order terms rotation of transverse normal
RD , RE principle radii of curvature
ND , N E curvature values of the D and E curves NDE twist curvature For example, Qatu (1994) studied the vibration of symmetrically laminated composite shells with different boundary conditions. Also, Noor (1990) and Leissa (1993) presented the solution and the vibrational analysis for symmetrically laminated thin cantilevered composite shells. Furthermore, Qatu (1994) analyzed the free vibrations of composite laminated thin shells. These researchers have utilized the classical shell theory in their analysis. Chen et al. (1997) studied the vibration of symmetrically laminated composite shells using the higher order shear deformation theory. However, not many researchers have undertaken the concepts of isotropic unsymmetrical laminated composite shells of free, undamped vibration problem (Aydo˘gdu & Timarci, 2007; Khare & Rode, 2005). Furthermore, there are some studies on unsymmetrical cross ply shells with very limited boundary conditions (Lee & Reddy, 2004; Zhou et al., 2002; Asadi et al., 2012; Naghdi & Berry, 1964; Reddy, 2 0 0 4 ). In order to fully understand the various vibrational characteristics of the composite cross- ply thick shell dynamics, the theory of first order shear deformation has to be modified. Towards this attempt, this part of the research has undertaken by incorporating the thickness-depth ratio of the thick shell. Therefore, basic equations derived for shells and
43
G Âłtt ( K W U )dt 1
0
0,
( 2 .1 )
where U is the shell strain energy, K the kinetic energy and W the external work by the system are given in appendix A. However, the approximation of displacement components using the third-order shear deformation shell theory can be written as Zannon et al., (2014)
u (D , E , z ) u0 (D , E ) z\ D (D , E ) z 3 MD (D , E ) v(D , E , z ) v0 (D , E ) z \ E (D , E ) z 3 ME (D , E ) ‌(2.2) w(D , E , z ) w0 (D , E ) z\ z (D , E ). Where h is the shell thickness h h and d z d . u0 , v0 , w0 are midsurface displacements 2 2 of the shell and\ D ,\ ,\ z are midsurface rotations and E MD , M E are higher order terms rotation of transverse normal. The corresponding strain-displacement equations are Zannon et al., (2014):
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 1 (1 z
HD
(H D zND (1) z 2ND ( 2 ) ) 0 )
RD 1 (H E zN E (1) z 2N E ( 2 ) ) 0 (1 z ) RE \ z (D , E ) z 0
HE Hz
1 (1 z
HDE
)
(H 0DE zNDE (1) z 2NDE ( 2 ) )
RD 1 (H zN E D (1) z 2N E D ( 2 ) ). E D 0 z (1 ) RE 1 zG (1) z 2 G ( 2 ) ) (J 0 z D z (1 ) RD 1 (J 0 E zE (1) z 2 E ( 2 ) ) z (1 z ) RE
H ED JD z JEz and where
G (1)
\E 1 w\ z 2ID , G ( 2) A wD R DE
E (1)
\ 1 w\ z 2IE D , B wE RDE
E ( 2)
( 2 .3 )
IE
I D . RD R DE IE I D . RE RDE
The functions on the right-hand side of Eq. (2.3) are v wA w 1 wu0 0 0 , H 0E A wD AB wE RD
H 0D H 0DE
1 wv0 A wD
J 0D z
1 ww0 A wD
(1) ND
u wB w 1 wv0 0 0 , B wE AB wD RE u wA w0 v wB w0 1 wu0 ,H , 0 0 0ED AB wE RDE B wE AB wD RDE u0 v0 v u0 1 ww0 \ D , J 0 \ , E 0E z RD RDE B wE RE RDE
1 w\ D \ E wA \ z (1) , NE A wD AB wE RD
( 2 .4 )
1 w\ E \ D wB \ z , A wE AB wD RE
IE wA I wB 1 wID 1 wIE , N ( 2) , D E A wD AB wE A wE AB wD I wA \z 1 w\ E \ D wA 1 wIE (1) ND , N ( 2) . D E D E A wD AB wE RDE A wD AB wE
( 2) ND
After integrating the stress over the thickness of shell by incorporating the term (1 force resultants:  N D ° ° ® NDE ° ° QD ¯
½ ° ° ¾ ° ° ¿
ÂV D ° ° ³ ®V DE h/2 ° °V D z ¯
 N ° E ° ® N ED ° ° QE ¯
½ ° ° ¾ ° ° ¿
ÂV ½ ° E ° ° °§ · ³ ®V DE ¾ ¨ 1 z R ¸ dz. D¹ h/2 ° °© °V E z ° ¯ ¿
h/2
½ ° °§ ¾ ¨¨ 1 °© ° ¿
z
· dz , RE ¸¸ ¹
z ) , we get the following moment and R
( 2 .5 )
h/2
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Free Vibration Analysis of Thick Cylindrical Composite Shells Using Higher Order Shear Deformation Theory
The bending and twisting moment resultants defined as Zannon et al., (2014):  M (1) D ° ° (1) ® M DE ° (1) ° PD ¯
½ ° ° ¾ ° ° ¿
 M (1) E ° ° (1) ® M ED ° (1) ° PE ¯
ÂV D ° ° ³ ®V DE h/2 ° °V D z ¯ h/2
½ ° °§ 1 ¾¨ ¨ °© ° ¿
z
· z dz , R ¸ E ¸¹
½ ° ° ¾ ° ° ¿  M ( 2) ½ ° D ° ° ° ( 2) ® M DE ¾ ° ( 2) ° ° ¯ PD ° ¿
ÂV ° D ° ³ ®V DE h/2 ° ° ¯V D z
 M ( 2) E ° ° ( 2) ® M ED ° (1) ° PE ¯
ÂV ½ ° E ° ° °§ · 2 ³ ®V DE ¾ ¨ 1 z R ¸ z dz . D¹ h/2 ° °© °V E z ° ¯ ¿
½ ° ° ¾ ° ° ¿
ÂV ½ ° E ° ° °§ · ³ ®V DE ¾ ¨ 1 z R ¸ z dz , D¹ h/2 ° °© °V E z ° ¯ ¿ h/2
h/2
½ °§ ° 1 ¾¨ ¨ °© ° ¿
z
· 2 z dz , R ¸ E ¸¹
h/2
( 2 .6 )
Where PD(1) , PD( 2 ) , P (1) and P ( 2 ) are higher order shear resultants terms. The stress resultant or the stiffness matrices obtained
E
E
from the above equations are given in Appendix B. Substituting these stiffness parameters in the Hamilton equation (2.1), and simplifying the resulting equations, we get the following equations of w wB wA w AB AB NE N DE QD QE ABqD ( BN D ) ( AN ED ) wD wD wE wE RD RDE
AB ( I1u 0 I 2\ D ). w wD
wB
( BN DE )
N ED
wD
wA wE
ND
w
AB
( AN E )
wE
RDE
AB
QD
QE ABqE
RE
AB ( I1v 0 I 2\ E ). w wD w wD
( BQD )
N N N ED N w ) ABqn ( AQE ) AB ( D E DE RD RE RDE wE
(1 )
( BM D )
wB wD
(1 )
ME
wA wE
w
(1 )
M DE
wE
AB
(1 )
( AM ED ) ABQD
RDE
AB ( I1 w0 ). (1 )
(1 )
PE ABmD
AB ( I 2 u0 I 3\ D ). w wE
(1 )
( AM E )
wA wE
(1 )
MD
w wD
(1 )
( BM DE )
wB wD
AB
(1 )
M ED ABQE
RDE
(1 )
(1 )
PD ABmE
AB ( I 2 v0 I 3\ E ). w wD w
motion:
wD
w
(1 )
( BPD )
wE
(2)
( BM D )
(1 )
(1 )
(1 )
( APE ) AB ( N z
wB wD
(2)
ME
w wE
MD
RD
ME
(1 )
RE
( AM ED ) ( 2 ABPD (2)
(1 )
M DE RDE
AB RDE
(1 )
M ED RDE
) ABmz
AB ( I 3\ z ).
(2)
(2)
PD
ABPE
RDE
(2)
) ABmD
( 2 .7 )
AB ( I 3u0 I 4MD ). w wE
(2)
( AM E )
wA wE
(2)
MD
w wD
(2)
( BM DE ) (
AB RE
(2)
PE
AB RD
(2)
PD
2 ABPE(1) ) ABmE( 2 )
AB ( I 3 v0 I 4M E ).
Where the rotary inertia is given as Zannon et al., (2014):
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014
Ii
§ § ¡ I ¡ ¨ I I ¨ 1 1 ¸ i 2 ¸ , i 1, 2, 3, 4 ¨ i i 1 ¨ RD R ¸ RD R ¸ E š E š Š Š
The boundary conditions for S2 are given below, where
N D ND 0 0 either N DE NDE 0 0 either Q D QD 0 0 ) either M 0(1D MD(1) 0
D is a constant
or
u0
0
or
v0
0
or
w0
0
or
\D
0
0
or
\E
0 .
P (1D) PD(1) 0 0 either M ( 2D) MD( 2 ) 0 0 ) either M 0( 2D) E MD( 2E 0
or
\z
0
or
MD
0
or
ME
either
) either M (1D MD(1)E 0 E
either
( 2 .8 )
( 2 .9 )
0
Similar boundary conditions are obtained by taking E , a constant. 3. EXACT SOLUTIONS Consider the following Figure 1, which is a section of a cylindrical shell having in-plane axis, ι in the direction of the x-axis of shell, β, circumferential in-plane along the y-axis, which is normal to the middle-plane axis z and radius R.
Fig.1. Coordinates of a cylindrical shell (Qatu, 1994)
Let us consider the cylindrical laminated shell as shown in figure 1with length a
b
1 under load per unit area, h is the
thickness of the shell. If the load is orthogonal to the surface, then Lame' parameters (elastic and shear modulus) of middle surface A
B 1 and 1
RD
1
R DE
0, R are substituted in equations (2.5-2.8) to formulate the cylindrical shell E
equations for TSDTZ. Therefore, the resulting equation (2.4) for the displacement mid surface cylindrical thick shells is
wv0 w0 wv0 , H 0DE , wE RE wD v wu0 ww0 ww0 \ D , J 0 \ , H 0ED ,J 0D z wD 0 E z wE E wE RE rewritten as: w\ w\ D wID E \z ND(1) , N E(1) , ND( 2 ) , RE wD wE wD wI w\ wI E E E N E( 2 ) , ND(1)E , ND( 2E) . wE wD wD
H 0D
wu0 , wD
H 0E
46
( 3 .1 )
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Free Vibration Analysis of Thick Cylindrical Composite Shells Using Higher Order Shear Deformation Theory Thus, the equations of motion (7) for the cylindrical shell reduces
w w (N ) ( N ) qD ( I1u 0 I 2\ D ). wD D wE ED w ( N ) w ( N ) 1 Q q ( I v I \ ). 1 0 2 E wD DE wE E RE E E w (Q ) w (Q ) N E q ( I w ). 1 0 n RE wD D wE E
w ( M (1) ) w ( M (1) ) Q m(1) ( I u I \ ). 2 0 3 D D D ED wD D wE to w ( M (1) ) w ( M (1) ) Q m(1) ( I v I \ ). 2 0 E 3 E wD DE E E wE
( 3 .2 )
M (1 ) w ( P (1) ) w ( P (1) ) ( N E ) m ( I \ ). 3 z R z z wD D wE E E w ( M ( 2 ) ) w ( M ( 2 ) ) 2 P (1) m( 2 ) ( I u I M ). 3 0 4 D D D ED wD D wE w ( M ( 2 ) ) w ( M ( 2 ) ) 2 P (1) m( 2 ) ( I v I M ). 3 0 4 E wD DE E E E wE
I § ¨ Ii i 1 R E Š
Ii
Where
¡ ¸ , i 1, 2, 3, 4 . š
The above cylindrical shell equation has no exact solution having general boundary conditions. Using the following Fourier series expansion for the midsurface displacements (0th order) field equation (3.3) is used to approximate the solution of the above partial differential equations (3.2):
u0
* * iZt , ÂŚ u0,mn Cos ( A D ) Sin( B E ) e m,n 1
v0
* * iZt , ÂŚ v0,mn Sin( A D ) Cos ( B E ) e m,n 1
N
w0
N
* * iZt , ÂŚ w0,mn Sin( A D ) Sin( B E ) e m,n 1 N
N
ÂŚ \ D ,mn Cos ( A D ) Sin( B E ) e
\D
*
*
N
ÂŚ \ E ,mn Sin( A D ) Cos ( B E ) e
\E
m,n 1
\z
ÂŚ \ z ,mn Sin( A D ) Sin( B E ) e
ID IE
iZt
m,n 1
*
N
*
*
*
( 3 .3 )
iZt ,
iZt ,
m,n 1 N
ÂŚ ID ,mn Cos ( A D ) Sin( B E ) e *
*
iZt ,
m,n 1 N
ÂŚ IE ,mn Sin( A D ) Cos ( B E ) e *
*
iZt .
m,n 1
Where A
*
mS * ,B a
nS in which a and b are the dimensions of the mid-shell along the D and E -axes, respectively. b
For the free vibration system has no external force on the structure and is freely vibrating. Substituting these equations (3.3) into the Partial differential equations of motion (3.2) yields a set of eight homogenous algebraic systems in terms of its respective components. By collecting the coefficients of the system and by taking the external force vector { f } as zero, the resulting equation reduces to an eigenvalue problem. Therefore, the resulting equilibrium equation of motion under free vibration system is written in the following matrix form ( 3 .4 ) {[ L] O [ M ]}{'} {0}.
Z 2 , Z is the circular frequency of vibration, {'} is the unknown displacement vector. The dynamic shell system stiffness parameters L and structure mass matrices M are given in appendix C. ij ij
Where O
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 4. NUMERICAL RESULTS AND DISCUSSION The above equation (3.4) in the matrix form is solved using MATLAB commercial code. This code is specifically modified for free vibration problems of simply supported cross-ply laminated composite shells. A standard MATLAB routine is used to find the eigenvalues of the matrices. Moreover, the eigenvalues are determined numerically because the stiffness matrix elements contain transcendental functions. The solution is also valid for cylindrical shells having principle radii RD RDE f . For the numerical computation we compared the results with the orthotropic material properties of the cylindrical shells having length-to-arc ratio of one unit (i.e. a / b 1 ), and the Poisson ratio of 0.25. The shear correction factors ( K ) for both directions are taken as 5/6. The natural frequency, extensional, bending and coupling stiffness parameters for free vibrations are calculated using MATLAB algorithm for laminated composite cylindrical thick shells. The first five non-dimensional natural frequency parameters ( :i ) has been calculated for two-ply unsymmetrical [90/0] shells and three-ply symmetric [0/90/0] laminated orthotropic composite cylindrical shells for fixed thickness ratio ( a / h 10 ) and various values of curvature ( a / R ) ratios by third order shear deformation theory. The results obtained by TSDTZ are then compared with earlier available results first order shear deformation theory by Qatu (FSDTQ), and three dimensional elasticity from finite element method (FEM). This supports us to evaluate the validity of the present TSDTZ theory. Moreover the results obtained by the present method are in good agreement with the existing theory and given in Tables 1-5. Table 1 and Table 2 show the comparison of present results and those of existing literatures (Asadi et al., 2012; Qatu et al., 2010) for the non-dimensional natural frequency parameters ( :i ). TABLE 1
Comparison of first five non-dimensional natural frequency parameters :
[90 / 0] orthotropic cylindrical shells G12 a 1, K 2 5 , E1 E2 25, E2 6 b a
2
1
0 .5
R
i
0.5, G23 E 2
0.2, G13
Zi a 2 U E2 / h 2 of 2-ply unsymmetrical
G12 , Q 12
0.25, a h 10
Method
:1
:2
:3
:4
:5
FSDTQ (Asadi et al., 2012)
1 3 .7 7 1
2 1 .0 3 7
2 9 .5 7 4
3 1 .2 0 0
3 8 .0 7 3
TSDTZ (Present)
1 3 .7 7 0
2 1 .0 3 9
2 9 .5 8 7
3 1 .2 2 2
3 7 .9 8 1
3D (Asadi et al., 2012; Qatu et al., 2010)
1 3 .7 7 2
2 1 .0 4 0
2 9 .6 3 9
3 1 .4 1 1
3 8 .2 6 6
FSDTQ (Asadi et al., 2012)
1 0 .6 6 6
2 1 .7 0 5
2 4 .0 9 0
3 0 .3 6 8
3 8 .7 2 2
TSDTZ (Present)
1 0 .6 7 1
2 1 .7 5 6
2 4 .1 1 7
3 0 .4 3 0
3 8 .7 1 3
3D (Asadi et al., 2012; Qatu et al., 2010)
1 0 .6 8 6
2 1 .7 6 7
2 4 .1 9 1
3 0 .6 1 4
3 8 .8 9 6
FSDTQ (Asadi et al., 2012)
7 .5 9 1 8
1 5 .2 8 4
1 5 .3 1 0
2 0 .3 0 0
2 4 .0 7 3
TSDTZ (Present)
7 .6 0 7 6
1 5 .3 2 2
1 5 .3 5 9
2 0 .4 2 3
2 4 .1 5 7
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Free Vibration Analysis of Thick Cylindrical Composite Shells Using Higher Order Shear Deformation Theory 3D (Asadi et al., 2012; Qatu et al., 2010)
7 .6 5 3 4
1 5 .4 3 7
1 5 .4 7 3
2 0 .6 4 5
2 4 .3 6 4
TABLE 2 Comparison of first five Non-dimensional natural frequency parameters :
i
symmetric [0 / 90 / 0] orthotropic cylindrical shells
a
2 b 1, K
a
0 .5
1
2
R
5 , E1 E2 6
25, G12 E 2
0.5, G23 E 2
0.2, G13
Zi a 2 U E2 / h 2 of 3-ply
G12 , Q 12
0.25, a h 10
Method
:1
:2
:3
:4
:5
FSDTQ (Asadi et al., 2012)
8 .6 3 0 9
1 3 .5 8 1
1 7 .8 7 8
2 0 .7 3 8
2 0 .9 7 6
TSDTZ (Present)
8 .3 2 3 1
1 3 .3 8 7
1 7 .3 9 9
2 0 .3 4 8
2 0 .8 9 4
3D (Asadi et al., 2012; Qatu et al., 2010)
8 .0 0 9 5
1 3 .1 9 2
1 6 .9 0 5
1 9 .9 4 9
2 0 .8 1 1
FSDTQ (Asadi et al., 2012)
1 3 .1 8 7
1 8 .5 2 4
3 0 .5 6 4
3 2 .2 3 2
3 4 .5 2 3
TSDTZ (Present)
1 2 .8 9 1
1 8 .2 6 7
3 0 .1 4 7
3 1 .2 2 4
3 3 .2 9 2
3D (Asadi et al., 2012; Qatu et al., 2010)
1 2 .5 9 0
1 8 .0 0 5
2 9 .7 3 2
3 0 .1 8 9
3 2 .0 3 7
FSDTQ (Asadi et al., 2012)
1 5 .2 5 0
1 7 .9 8 9
2 9 .4 9 1
3 4 .7 9 5
3 4 .9 1 3
TSDTZ (Present)
1 5 .0 4 0
1 7 .7 3 2
2 9 .2 9 0
3 3 .6 6 5
3 3 .9 9 8
3D (Asadi et al., 2012; Qatu et al., 2010)
1 4 .8 4 0
1 7 .4 6 8
2 9 .0 9 4
3 2 .4 6 4
3 3 .0 4 6
In order to validate the third order shear deformation theory, the extensional
( Aij ) , bending ( Dij ) and coupling ( Bij )
stiffness parameters (Zannon et. al., 2014; Qatu, 1994) are given in Table 3-5 using the Matlab algorithm for laminated composite cylindrical thick shells. It is then compared with the first order shear deformation theory from the literature. There are small discrepancies are seen in the tables 3-5, which is due to the third order shear deformation and the tolerance limitations. Tables 3, 4 and 5 show the extensional stress, coupling, and bending stiffness parameters for [0/90] laminated cylindrical thick shells. While comparing the various stiffness parameters with the existing literature and the present theory, we see that the TSDTZ approximation is more accurate in in comparison with first order shear deformation theory (see Tables1-5 for details).
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 TABLE 3 Non-dimensional extensional stiffness matrix for [0 / 90] laminated cylindrical thick shells E1 E2
15,
G12 E2
0.5 ,
G13 E2
0.5 , X12
Plate Approx.
(i, j )
Aij
E2 a
0.3 ,
a b
a R E
1,
FSDTQ Qatu, (1994)
2
Aij
E2 a
0,
RD R E
0,
TSDTZ (Present) Exact Integration
Aˆij
2
a R DE
2,
E2 a
Aij
2
E2 a
Aˆij
2
a h
10, U
TSDTZ (Present) Third order
E2 a
2
Aij
E2 a
(1, 1)
0 .8 0 4 8 2 9
0 .7 6 9 6 3 4
NA
0 .7 1 6 8 0
NA
0 .7 1 8 0 6
( 2, 2)
0 .8 0 4 8 2 9
NA
0 .7 7 2 1 5 6
NA
0 .7 6 9 6 2
NA
(6, 6)
0 .0 5 0 0 0 0
0 .0 4 9 9 9 9
0 .0 5 0 1 6 7
0 .0 4 7 5 0
0 .0 5 2 5 0
0 .0 4 7 5 8
(i, j )
Bij E2 a
FSDTQ Qatu, (1994)
2
Bij E2 a
2
TSDTZ (Present) Exact Integration
Bˆ ij E2 a
2
Bij E2 a
2
Aˆij
2
E2 a
2
NA
0 .7 7 0 6
0 .0 5 2 5 8
TABLE 4 Non-dimensional coupling stiffness matrix for [0 / 90] laminated cylindrical thick shells R G E1 G a a a a 15, 12 0.5 , 13 0.5 , X12 0.3 , 1, 2, 0, D 0, E2 E2 E2 b R R R h E DE E Plate Approx.
1.
10, U
1.
TSDTZ (Present) Third order
Bˆij E2 a
Bij
2
E2 a
Bˆij
2
E2 a
2
(1, 1)
- 1 .7 6 0 5 6 3
- 1 .6 2 6 4 8 8
NA
- 1 .6 3 4 8
NA
- 1 .6 3 4 8
NA
( 2, 2)
1 .7 6 0 5 6 3
NA
1 .6 3 4 5 4 0
NA
1 .6 2 6 4
NA
1 .6 2 6 4
(6, 6)
0
0 .0 0 8 3 2 9
- 0 .0 0 8 3 7 9
0 .0 0 7 6 3 9
- 0 .0 0 7 9 4 5
0 .0 0 8 2 1 8
- 0 .0 0 8 2 5 7
a h
1.
TABLE 5 Non-dimensional bending stiffness matrix for [0 / 90] laminated cylindrical thick shells E1 E2
15,
G12 E2
0.5 ,
G13 E2
0.5 , X12
0.3 ,
a b
1,
a R E
50
2,
a R DE
0,
RD R E
0,
10, U
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Free Vibration Analysis of Thick Cylindrical Composite Shells Using Higher Order Shear Deformation Theory Plate Approx. Dij
(i, j )
E2 a
2
FSDTQ Qatu, (1994) Dij E2 a
2
TSDTZ (Present) Exact Integration
Dˆ ij E2 a
Dij
2
E2 a
2
Dˆ ij E2 a
TSDTZ (Present) Third order
2
Dij E2 a
2
Dˆ ij E2 a
2
(1, 1)
0 .6 7 0 6 9 1
0 .6 2 4 1 7 8
NA
0 .6 7 0 7
NA
0 .6 7 0 6 9
NA
( 2, 2)
0 .6 9 7 0 9 1
NA
0 .6 3 0 4 5 4
NA
0 .6 7 0 6 9
NA
0 .6 7 0 6 9
(6, 6)
0 .0 4 1 6 6 7
0 .0 4 2 8 9 2
0 .0 4 1 9 1 8
0 .0 4 1 6 5
0 .0 4 1 6 6
0 .0 4 1 6 7
0 .0 4 1 6 7
CONCLUSIONS This paper presented a third order shear deformation theory for laminated thick shells by Zannon et al. (TSDTZ) and solved for free vibrational characteristics of cylindrical thick shells with simply supported boundary conditions and cross-ply laminates. The present results are compared with the 3D theory of elasticity and first order shear deformation theory available in the literature. The natural frequencies of the first five parameters for the cylindrical shells using TSDTZ are in good agreement with the 3D theory of elasticity. The results obtained here show that the present theory (TSDTZ) is more accurate than the (FSDTQ) when compared to three dimensional theory of elasticity. Also, TSDTZ offers many other advantages in the accurate representations such as extensional , coupling, and stress stiffness parameters, as shown in Tables 3 to 5 and is mainly due to the inclusion of the term ( 1
z ) in the mathematical formulation of third order shear deformation theory. We anticipate that R
this theory would be useful for many researchers in the development of complex geometry of shell deformation theory for further future applications. The total shell strain energy
U
APPENDIX A
U can be written in terms of the midsurface strains and stress resultants and is defined as (Zannon et al., 2014)
1 Âł {VD HD V E H E V z H z VDE HDE VD z J D z V E z J E z }dV 2V
1 (1) (1) ( 2) ( 2) Âł Âł {ND H 0D N E H 0E N z H 0 z NDE H 0DE N ED H 0ED MD ND MD ND 2DE M E(1)N E(1) M E( 2 )N E( 2 ) MD(1)E ND(1)E MD( 2E) ND( 2E) M E(1)D N E(1)D M E( 2 )D N E( 2 )D QD J D z 0 QE J 0E z PD(1)G (1) PD( 2 )G ( 2 ) PE(1) E (1) PE( 2 ) E ( 2 ) } ABdD d E . The total external work
W
W of the thick shell can be written as (Zannon et al., 2014):
(1) (1) ( 2) ( 2) Âł Âł {qD u0 qE v0 qn w0 mD G mD G mz\ z
D E
m(1) E (1) m( 2 ) E ( 2 ) } AB dD d E . E E The kinetic energy
T
of the thick shell can be written as (Zannon et al., 2014):
51
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 1 2 2 2 ³ {u v w }dV 2V
T
§ I3 1 1 1 2 2 2 ) ³ ³ {(u0 v0 w0 ) ¨ I1 I 2 ( ¨ 2D E RD RE RD RE © § I5 2 \ 2 ) ¨ I I ( 1 1 ) (\ D z ¨ 3 4 R R R D D RE E ©
· ¸ ¸ ¹
§ I4 1 1 ) 2(u0\ D v0\ E w0\ z ) ¨ I 2 I3 ( ¨ RD RE RD RE © § 2 ) ¨ I I ( 1 1 ) I6 2(u0MD v0M E \ E 5 R ¨ 4 RE RD RE D © § I7 1 1 2(MD\ D M E\ E ) ¨ I5 I 6 ( ) ¨ RD RE RD RE © § 2 M 2 ) ¨ I I ( 1 1 ) I9 (MD 8 R E ¨ 7 RE RD RE D ©
ª ND º «N » « E » « Nz » « » « N DE » « N ED » « (1) » « MD » « M (1) » « E(1) » « M DE » « M (1) » « E( 2D) » «MD » « ( 2) » «M E » « M D( E2 ) » « ( 2) » ¬ M ED ¼
ª A11 « « A12 «A « 13 « A16 « « A16 « B11 « « B12 « B16 « « B16 «D « 11 « D12 «D « 16 «¬ D16
· ¸ ¸ ¹
· ¸ ¸ ¹
· ¸ ¸ ¹
· ¸ ¸ ¹
· ¸} AB dz dD d E ¸ ¹
APPENDIX B
A12 Aˆ
A13
A16 A26
A16 Aˆ
22
A23
26
A23
A33
A36
A26 Aˆ
A36
A66
A36
A66
B12 Bˆ
B13
B16
B23
B26
B36
B66
B36
B66
D13
D16
D23
D26
D36
D66
D36
D66
26
22
B26 Bˆ 26
D12 Dˆ
22
D26 Dˆ
26
ª QD º «Q » « E» « PD(1) » « (1) » « PE » « PD( 2 ) » « ( 2) » ¬« PE ¼»
ª A55 «A « 45 « B55 « « B45 « D55 « ¬ D45
B11 B12
B12 Bˆ
22
A36
B13
A66 Aˆ
B16
B16 Bˆ
D11
B16
66
B66 Bˆ
D16
D16 Dˆ
E11
D66 Dˆ
B36
B26 Bˆ
B66
D12 Dˆ
D16
B66
D26 Dˆ
D66
E12 Eˆ
E16
E16
E66
D55 D45 E55 D45 Dˆ 44 E45
D45 E55 Dˆ 44 E45
E45 Eˆ 44
D11
D36
B66 Bˆ
D16
D26 Dˆ
D66
D16 Dˆ
E11
E12 Eˆ
E16
D16 E12
D66 Dˆ
E16
E16 Eˆ
F11
E16 F12 F16 F16
26
22
D16
D66 E 26
E26 Eˆ
E66
F12 Fˆ
F16
26
22
F26 Fˆ 26
E66 F26 F66 F66
D16 º ª H 0D º »« H » Dˆ
» « 0E » D36 » « H 0 z » »« » D66 » «H 0DE » » Dˆ 66 » «H 0 ED » « (1) » E16 » « N D » » « (1) » Eˆ 26 » N E « (1) » E66 » « N DE » » « (1) » Eˆ 66 » « N ED » ( 2) F16 » « N D » » « ( 2) » Fˆ26 » « N E » » ( 2) F66 « N DE » » « ( 2) » Fˆ66 »¼ ¬ N ED ¼ 26
ª J 0D z º «J » 44 » « 0E z » E45 » « G (1) » » « (1) » Eˆ 44 » « E » F45 » «G ( 2 ) » »« » Fˆ44 ¼ ¬ E ( 2 ) ¼
D55 D45 º » D Dˆ
B45 Bˆ 44
D23
66
B45
44
D13
E66 Eˆ
E66
B45 Bˆ
B36
26
A45 Aˆ 44
B55
D26
66
E26
26
22
26
D66
E 26 Eˆ
D12
D12 Dˆ
66
D26
22
E16
66
B23
26
E12
26
26
22
D16
66
B26
B16 Bˆ
26
D12
26
B16
45
F55 F45
ˆ , Eˆ and Fˆ are given in Zannon et al., (2014) Where Aij , Bij , Dij , Eij , Fij , Aij , Bij , Dij , Eij , Fij , Aˆij , Bˆij , D ij ij ij
52
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Free Vibration Analysis of Thick Cylindrical Composite Shells Using Higher Order Shear Deformation Theory APPENDIX C 2
*
L11
A11 ˜ A
2
ˆ ˜ B* A 66
*
A12 ˜ A
L13
2
*
, L14
RE
*
, L12
B11 ˜ A
*
*
A12 ˜ A ˜ B A66 ˜ A ˜ B ˆ ˜ B* B 66
*
2
*
*
*
*
*
*
2
ˆ ˜B D 66
2
*
, L18
D11 ˜ A
L21
A12 ˜ A ˜ B A66 ˜ A ˜ B , L22
*
*
*
ˆ ˜B A ˆ ˜B A 44 22 *
L23
ˆ ˜ B* B 22
2
*
*
2
ˆ A 44
RE
ˆ ˜ B* A 44
*
L36
A
2
, L35
RE
˜ B55 B
*
2
ˆ ˜B D 22 *
*
*
L41
L14 , L42
L45
D12 ˜ A ˜ B D66 ˜ A ˜ B
L47
E11 ˜ A Eˆ 66 ˜ B
L
L15 , L52
L56
, L38
RE
L34 , L44
L24 , L43 *
*
*
*
2
ˆ 2˜ B 44 RE
ˆ D 44 ( RE )
2
2
*
*
*
2
*
*
*
*
E12 ˜ A ˜ B E66 ˜ A ˜ B ,
L66
D55 ˜ A
L67
2 ˜ D55 ˜ A
L68
* * Eˆ ˜ B Eˆ ˜ B * 2 ˜ Dˆ 55 ˜ B 44 22 , L71 RE RE
L73
L37 , L74
L77
F11 ˜ A
L78
F12 ˜ A ˜ B F66 ˜ A ˜ B
2
*
B55 ˜ A ,
RE
* * Dˆ 22 ˜ B D66 ˜ A Aˆ 44 ,
L45 , L55
L36 , L64 *
D12 ˜ A
*
L16 , L62
Dˆ 44 ˜ B
2
* Dˆ 66 ˜ B A55 ,
E12 ˜ A ˜ B E66 ˜ A ˜ B ,
L61
L26 , L63
2
RE
*
2 2 Dˆ * * Eˆ 22 ˜ B E66 ˜ A 2 ˜ Bˆ 44 44 RE
2
RE 2
* Dˆ 22 ˜ B
B13 ˜ A
L58
*
D66 ˜ A
ˆ ˜ B* B 44
*
, L46
L35 , L54
* Bˆ 44 ˜ B , L57
RE
*
*
D11 ˜ A
2 ˜ B55 , L48
L25 , L53
* Dˆ 22 ˜ B
*
ˆ ˜ B* A 44
2 ˜ B ˜ Bˆ 44
2 ˜ A ˜ B55
51
2
RE
2
*
*
ˆ ˜ B* B 22
*
A55 ˜ A
L37
*
2
ˆ ˆ ˆ A22 B22 B 44 2 RE ( RE )
D12 ˜ A
*
( RE )
* ˆ ˆ ˜ B * B22 ˜ B A 44 RE
*
A55 ˜ A
*
A23 ˜ B
*
, L33
RE B12 ˜ A
*
L34
ˆ A 44
2
*
, L26
RE
*
, L32
*
˜ B * A66 ˜ A*
B12 ˜ A ˜ B B66 ˜ A ˜ B
*
*
A12 ˜ A
L31
2
ˆ A 22
D12 ˜ A ˜ B D66 ˜ A ˜ B , L28
L27
*
D12 ˜ A ˜ B D66 ˜ A ˜ B
*
, L24
B66 ˜ A
*
RE
*
*
RE
L25
A13 ˜ A *
L17
B12 ˜ A
*
B12 ˜ A ˜ B B66 ˜ A ˜ B , L16
L15
,
L46 , L65
A33
B23 RE
L56 Dˆ 22
( RE )
2
*
L87
*
*
2
,
RE
L47 , L75 *
L78 , L88
E12 ˜ A
Fˆ66 ˜ B
L57 , L76 *
2
*
L67 , L81
L27 ,
L18 , L84
L48 , L85
L58
4 ˜ D55 , *
2
L17 , L72
*
, L82
L28 , L83 2
* * Fˆ22 ˜ B F66 ˜ A
Eˆ 44 RE
L38 , , L86 Fˆ44
( RE )
2
L68
4 ˜ Dˆ 44
The mass parameters M ij for the shell is given by Zannon et al., (2014)
53
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 M 11
I1 , M 14
I 2 , M 12
M 13
M 15
M 16
M 22
I1 , M 25
I 2 , M 21
M 23
M 24
M 26
M 33
I1 , M 31
M 32
M 41
I 2 , M 44
I 3 , M 42
M 43
M 45
M 46
M 47
M 48
0,
M 52
I 2 , M 55
I 3 , M 51
M 53
M 54
M 56
M 57
M 58
0,
M 66
I 3 , M 61
M 62
M 63
M 64
M 65
M 67
M 68
0,
M 77
I 4 , M 71
M 72
M 73
M 74
M 75
M 76
M 78
0,
M 88
I 4 , M 82
M 81
M 83
M 84
M 85
M 86
M 87
0.
M 34
M 35
M 36
M 37
M 17 M 27 M 38
M 18 M 28
0, 0,
0,
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