International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014
Mathematical Modeling of a Transverse Shear Deformation Thick Shell Theory MOHAMMAD ZANNON, MOHAMAD QATU
Abstract— Three-dimensional theory (3D) of elasticity in curvilinear coordinates is employed to understand the stress and strain distribution in the middle surface of a thick composite shell under various operating conditions. The equations of motion are derived by making use of the relationships between forces, moments and stress displacements of shell using Hamilton’s principle of minimum energy. The necessary theoretical assumptions are discussed to simplify the three dimensions to a set of two-dimensional (2D) shell equations without violating the theory of elasticity. Displacement through the thickness is of third order in this analysis as compared with the first order approximation as previously published research. Equilibrium equations are formulated using these equations to achieve a set of linear partial differential equations and solved for exact solutions using Fourier series expansion for simply supported laminated cross-ply boundaries. This solution can be further used for various vibration analysis and optimal design of thick shell structures. Finally, the new additional parameters using the third order shear deformation of thick shell theory obtained from Fourier expansion is compared with the first order shear deformation shell theory from the literature. Index Terms—Shells, Thick, Shear Deformation, Laminated, Exact Solution, Cross-ply
1892; Ventsel & Krauthammer, 2001). Love was able to unitize Kirchhoff hypotheses to simplify the relationship between strain displacement and constitutive equations, in the case of the plate bending theory having small deflection and thinness (Ventsel & Krauthammer, 2001). These are the well-known Kirchhoff–Love’s assumptions (Koiter, 1969; Love, 1892 ). Love’s assumptions of thin elasticity shells are referred to as the classical laminated shell theory. Love’s theory had certain shortcomings even its well accepted popularity and success in the applications of shell structures, due to the incorporation of infinitesimal strains of bending (small terms) and extension. However, in Love’s theory, only few of these terms in the model equations were sustained. Thus, in Love’s theory of differential operator matrix, the displacement of equilibrium equations of certain shells became asymmetric. However, later the inconsistencies of Love’s theory (Timoshenko & Woinowsky, 1959) were modified by Reissner, (1945) mathematical model for two-dimensional linear thin shell theory using Love-Kirchhoff hypotheses. “Reissner derived the equations of equilibrium, strain–displacement relations, and stress resultants expressions for thin shells directly from the 3D theory of elasticity by applying the Love–Kirchhoff hypotheses and neglecting small terms of order
z
1, 2 ) (Vlasov, 1964)”.
( i
Ri
Nomenclature
Aij , Aˆij , Aij stretching and shearing stiffness parameters Aij , Aij , Aij stiffness parameters
1. INTRODUCTION
D
A shell is a three dimensional body bounded by two close surfaces. Its thickness is constant if these surfaces are parallel to each other. In general, we can assume that the thickness (i.e. the distance between these two surfaces) is much smaller in comparison to its length, width and radii of the curvature. Considerable amount of works been done on shell theory (Koiter, 1969; Love, 1892; Reissner, 1945; Ye & Soldatos, 1994; Qatu et al., 2012). There are some differences in the results obtained by various researchers depending upon the assumptions made, the resulting theory derived, and the size of the parameters. The solution procedure used in solving the equations of motion incorporates the stress–strain, the strain–displacement and the associated boundary conditions of the shell. Based on the classical linear elasticity, Love was the first person to present a successful shell theory (Qatu, 1999; Love, 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
29
E
n
Bij , Bˆij , Bij coupling stiffness parameters Bij , Bij , Bij stiffness parameters D
E
n
c0 , c1 tracer Dij , Dˆ ij , Dij bending and twisting stiffness parameters Dij , Dij , Dij stiffness parameters D
E
n
Eij , Fij , Lij , Eij , Eij , Eij , Fij , Fij , Fij , Lij , Lij , Lij higher order stiffness parameters D
E
n
D
E
n
D
E
n
t time 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 H D , H E , H z normal strains
J DE , J D z , J E z shear strains Qij ( k ) , Qij elastic stiffness parameters for layer k QD , QE transverse shear force mD(1) , mE(1) , mD( 2 ) , mE( 2 ) distributed couples qz , qE , qD distributed forces
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Mathematical Modeling of a Transverse Shear Deformation Thick Shell Theory
In order to develop a better thick shell theory, we re-examined Love’s first order approximation theory of thin shell. Various studies (Qatu, 2004; Noor, 1990; Leissa, 1973) concluded that even for thick shells, in comparison to other stresses and strains, the transverse normal strain and stress remains small enough to be neglected (Love, 1892). Many shell theories and associated problems were derived based on Love’s ďŹ rst approximation as mentioned above. However, during the middle of the 20th century (Leissa. 1973; Qatu, 2002), many of these theories found to be inconsistent, as reported in the literature on shell theory e.g. Timoshenko & Woinowsky (1959). The introduction of unsymmetrical differential operators by rigid body motion contradicts the theory of reciprocity thus in free vibration analysis, which yields natural frequencies in complex domain. Moreover, other inconsistencies were introduced due to the assumption of small thickness (
z 1 and h ) and also R R
due to symmetric stress resultants (i.e., NDE and M DE
N ED
M ED ) of the shell. In the case of non-spherical
geometrical structures, the stress resultants are not always equal. In order to overcome such deficiencies, various theoretical treatments have been proposed by several researchers. However, Vlasov (1964) resolved these discrepancies by extending the strain displacement term. Usually the terms in stress resultant equation expansion using Taylor series have negligible terms in the denominator (Noor, 1990; Qatu, 1999, 2002). Various survey articles on these topics were implemented on the behavior of homogeneous and laminated composite shells (Ambartsumian, 1962; Gol’denveizer, 1961; Timoshenko & Woinowsky 1959; Love, 1892). Earlier theories and analyses about the effects of laminated shear deformation composite materials allowed its importance in shell theory when compared with isotropic materials (Qatu, 2004).Often, higher order shear shell deformation theory included higher order approximation, arising from stress-strain and rotational inertia. However, those who worked earlier, on shear deformation shell theories, failed to consider (1+z/R) term in the equations of stress resultant as discussed above. This was reported by Bert, (1967) and followed by Qatu et al. (2004), since it increased the inaccuracies in the stress-strain equations for the thick laminated shells. These results were obtained either by integrating the above equations exactly or by Taylor series expansion, which showed good agreement with the 3D elasticity theory (Qatu et al., 2012). Higher order terms of the shear strain, and rotary inertia approximation were used by many others in the shear deformation theories but again neglected (1+z/R) in the stress resultant equations (Kapania, 1989; Khare et al., 2005; Qatu et al., 2010), which is only applicable to shallow shells. These inaccuracies led to the development of constitutive equations and were observed by many researchers (Ye, JQ. & Soldatos, 1994; Kant & Swaminathan, 2001; Qatu, 1994; Qatu et al., 2010). However, Leissa (1973) used the geometric series to truncate these terms and Qatu et al (2010, 2012) integrated this term exactly and the resulting results were in good agreement with 3D theory of elasticity. Others who had worked on the same problem, although included higher order terms of shear strains, yet once again neglecting the contribution of the term
30
(1+z/R) and mostly applied to shallow shells rather than deep shells. Therefore, the inclusion of this term in our present article would add value to the existing theories and can present a better approximation for thick shell under various operating conditions. Due to the importance of the shells of revolution in the development of thin shell theory, contribution of (1+z/R) term cannot be waved aside. This has various applications in engineering. However, Reissner (1941, 1945) developed a spherical shell theory and obtained a classical derivation of the bending problem for shells of revolution. He used asymptotic method for integration after reducing the resulting differential equations of the spherical shell. Flßgge (1962) worked on the spherical and conical shells and, was able to derive general solutions and the result was based on the classical displacement method. In short, a complete and general thin elastic shell theory from the theory of linear elasticity was first developed almost one hundred years ago by Love (1892). Prior to this development, Aron (1874) presented a model equation for bending of thin shells using the Kirchhoff and Clebsch’s small strain and finite displacement theory. However, the erroneous mathematical treatment and the justification given in Aron (1874) model led to Love’s theory of thin shells. Since then, many new developments in the treatment of shell theory for varied shape, size, and method of solution and its interpretation aroused from various researchers in this field (Ventsel & Krauthammer, 2001; Reddy, 1984; Ye, & Soldatos, 1994; Qatu et al., 2013). Towards this end, in this paper, we propose our contribution towards the mathematical theory of third order shear deformation thick shell theory by Zannon (TSDTZ) and its stress-strain deformation at the mid thick shell surface. 2. MATHEMATICAL AND THEORETICAL ASSUMPTIONS 2.1. Displacement Models. In the present work we consider a thick shell having smaller thickness in comparison with other shell parameters such as width, shape length and curvature radii, which is usually taken as (1/10) of its measure. In any vibrational analysis, thick shells often includes rotational inertia factors and shear deformation (Qatu, 2004, 2012). Middle plane displacements are stretched in terms of shell thickness in shell deformation theory and can make use of first or higher order approximation. Therefore, three dimensional elasticity theories are reduced to two dimensional theories by neglecting the normal strain in comparison with other components of the strain, which are acting on the plane parallel to the middle surface. Generally, such assumption can be justified outside of the neighborhood of highly rigorous force, hence the stretching in the z-direction can be ignored, which leads to zero strain in the azimuthal (z) direction. Hence the displacements are written in the following polynomial form in variable thickness (z)
u (D , E , z ) u0 (D , E ) z\ D (D , E ) z 3MD (D , E ) v(D , E , z ) v0 (D , E ) z\ E (D , E ) z 3M E (D , E ) w(D , E , z ) w0 (D , E ) z\ z (D , E ).
( 2 .1 )
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 h
Where
2
dzd
h
, h is the shell thickness, u0 , v0 and w0 are middle surface displacements of the shell and\ D ,\ E ,\ z are
2
middle surface rotations and MD , M E are higher order terms rotation of transverse normal. In the third part of the equations (2.1), we are assuming H z z 0 . By substituting equation (2.1) into the strain-displacement equation of the elasticity theory, the following strain-displacement equations are obtained: 1 HD (H 0D zN D (1) z 2N D ( 2 ) ), H E (1 z ) RD Hz
\ z (D , E ) z 0, 1 (1 z
H ED
1 (1 z
J Ez
RE
)
RD
1 wu0 A wD
1 wv0
H 0 DE
A wD 1 ww0
J 0D z
A wD
v0
wA
AB wE
1 (1 z
u0 wA AB wE
u0 RD
w0
RD
v0 RDE
1 wv0
, H 0E
w0
B wE
AB wD
1 wu0
, H 0 ED
RDE
u0 wB
B wE
1 ww0
\ D , J 0 E z
v0
1 wID A wD
1 w\ D
N E(1D)
wE
B
He r e , G (1) E (1)
1 w\ z A wD 1 w\ z B wE
)
( 2 .2 )
(J 0D z zG (1) z 2 G ( 2 ) )
IE
wA
RE
2ID 2IE
\E RDE
\D RDE
\E
wB
AB wD
, ,
IE
G ( 2)
RD
E ( 2)
IE RE
\
A
A wE
z
RDE
ID RDE
w0
RDE
u0 RDE
1 w\ E
1 wIE
, N E( 2 )
AB wE
v0
B wE
,
RE
wB
\ D wA \z 1 w\ E ) , N D( 2 E A wD AB wE RDE
N D(1E)
RD
w0
AB wD
The curvature and twist changes of the shells can be written as \ E wA \z 1 w\ D , N E(1) N D(1) A wD AB wE RD N D( 2 )
RE
(H 0 E zN E (1) z 2N E ( 2 ) )
(J 0 E z zE (1) z 2 E ( 2 ) ).
Where the middle surface strains are H 0D
)
(H 0DE zN DE (1) z 2N DE ( 2 ) )
)
(H 0 ED zN ED (1) z 2N ED ( 2 ) ), J D z
)
RE
1 (1 z
H DE
1 (1 z
wE
ID
\ E .
\D
wB
AB wD
wB
AB wD
\
z
RE
,
, ( 2 .4 )
ID wA 1 wIE . A wD AB wE
1 wID
, N E( 2D)
( 2 .3 )
,
B wE
IE
wB
AB wD
.
.
ID RDE
. 2
( 2)
The contribution of the terms zG , z G , zE and z E to the overall accuracy of the theory is presented in the later section. Equations (2.2) - (2.4) establish the stress-strain displacement relationship required for 3rd order shear deformation shell theory, which are different from previous researchers (Qatu et al., 2010, 2012; Ye & Soldatos, 1994; Reddy, 1984; Ventsel & Krauthammer, 2001). (1)
2
( 2)
(1)
After integrating the stress over the thickness of shell by incorporating the term (1 resultant equations.
 ND ½ ° ° Ž NDE ž °Q ° ¯ D ¿
 VD ½ ° °§ ¡ ³ ŽV DE ž ¨Š 1 z RE š¸ dz , h/2 ° ° ¯V D z ¿ h/2
 NE ½ ° ° Ž N ED ž ° ° ¯ QE ¿
31
z
R
) , we get the following moment and force
ÂVE ½ ° ° Âł ÂŽV DE ž 1 z RD dz. h/2 ° ° ÂŻV E z Âż h/2
( 2 .5 )
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Mathematical Modeling of a Transverse Shear Deformation Thick Shell Theory Similarly, the bending and twisting moment resultants are defined as
 M D(1) ½ ° (1) ° ® M DE ¾ ° (1) ° ¯ PD ¿  M D( 2 ) ½ ° ( 2) ° ® M DE ¾ ° P ( 2) ° ¯ D ¿ (1)
 VD ½ ° °§ · ³ ®V DE ¾ ¨© 1 z RE ¹¸ z dz , h/2 ° ° ¯V D z ¿
 M E(1) ½ ° (1) ° ® M ED ¾ ° (1) ° ¯ PE ¿
ÂVE ½ ° ° ³ ®V DE ¾ 1 z RD h/2 ° ° ¯V E z ¿
 VD ½ ° °§ · 2 ³ ®V DE ¾ ¨© 1 z RE ¸¹ z dz , h/2 ° ° ¯V D z ¿
ÂM ° ®M °P ¯
ÂVE ½ ° ° ³ ®V DE ¾ 1 z RD h/2 ° ° ¯V E z ¿
h/2
h/2
( 2)
(1)
Where PD , PD , PE
( 2)
and PE
( 2) E ( 2) ED (1) E
½ ° ¾ ° ¿
h/2
h/2
z dz
( 2 .6 ) z 2 dz .
are higher order shear resultants terms and
VD
Q11 H D Q12 H E Q13 H z Q16 J DE .
VE
Q12 H D Q22 H E Q23 H z Q26 J DE .
Vz
Q13 H D Q23 H E Q33 H z Q36 J DE .
VEz
Q44 J E z Q45 J D z .
VD z
Q45 J E z Q55 J D z .
V DE
Q16 H D Q26 H E Q36 H z Q66 J DE .
It is noteworthy that, though the stresses V DE and V ED are the same, but the stress resultants NDE and N ED are not (Qatu, 2004). These stress resultants are only equal ( NDE
NDE ; M D(1E)
M E(1D) ; M D( E2 )
M E( 2D) ) if the radius of curvature, RD
RE ,
which is the conditions of spherical shells. Therefore, the stress resultants obtained from the above equations (2.5)-(2.6) can be rewritten in the following matrix form:
ª A11 « « A12 «A « 13 « A16 « « A16 « B11 « « B12 « B16 « « B16 «D « 11 « D12 «D « 16 «¬ D16 ª QD º «Q » « E » « PD(1) » « (1) » « PE » « PD( 2 ) » « ( 2) » « ¬ PE » ¼
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
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 A55 «A « 45 « B55 « « B45 « D55 « ¬ D45
A45 ˆ A 44
B45 ˆ B 44
D45 ˆ D
B12
B12 Bˆ
22
A36
B13
A66 Aˆ
B16
B16 Bˆ
D11
66
26
B11
B16 D12
B66 Bˆ
D16
D16 Dˆ
E11
66
26
D66 Dˆ 66
D16 E12 E16 E16
B55 B45 D55 D45 E55
B16 B26
B16 Bˆ
26
D12
B23
B36
B36
D13
B26 Bˆ
B66
B66 Bˆ
D16
D12 Dˆ
D16
D16 Dˆ
E11
26
22
B66 D26
D26 Dˆ
D66
E12 Eˆ
E16
26
22
E26 Eˆ
26
D66 E 26 E66 E66
B45 ˆ B 44
D45 ˆ D 44
E45 ˆ E
66
D11
D16
26
E12
D66 Dˆ
E16
E16 Eˆ
F11
66
26
E66 Eˆ 66
E16 F12 F16 F16
D55 D45 E55 E45 F55
D12 Dˆ
D16 D26
D16 º ª H 0D º »« H » Dˆ
» « 0E » D23 D36 D36 » « H 0 z » »« » D26 D66 D66 » «H 0DE » » Dˆ 26 D66 Dˆ 66 » «H 0 ED » « (1) » E12 E16 E16 » « N D » » « (1) » (2.7) Eˆ 22 E26 Eˆ 26 » N E « (1) » E26 E66 E66 » « N DE » » « (1) » Eˆ 26 E66 Eˆ 66 » « N ED » ( 2) F12 F16 F16 » « N D » » « ( 2) » Fˆ22 F26 Fˆ26 » « N E » » ( 2) F26 F66 F66 « N DE » » « ( 2) » Fˆ26 F66 Fˆ66 »¼ ¬ N ED ¼ D45 º ª J 0D z º » ˆ » «J D 44 » « 0E z » (1) » E45 » « G » « » . ( 2 .8 ) ( 1 ) ˆ E E 44 » « » F45 » « G ( 2 ) » » « ( 2) » ˆ E F ¼ 44 ¼ ¬ 22
26
E 45 F45 44 ˆ , Eˆ and Fˆ are defined below. Where Aij , Bij , Dij , Eij , Fij , Aij , Bij , Dij , Eij , Fij , Aˆij , Bˆij , D ij ij ij 44
32
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 N 1 N (k ) (k ) 2 h2 (h h ), B (h ), Œ Q Œ Q k k 1 ij k k 1 ij ij 2 k 1 k 1 1 N 1 N (k ) 3 (k ) 4 3 4 (h ), E (h ), (2.9) D h h Œ Q Œ Q ij 3 ij k k 1 ij 4 ij k k 1 k 1 k 1 1 N (k ) 5 5 (h ) F h Œ Q ij 5 ij k k 1 k 1 BijD Bij E DijD ˆ Aij AijD A B B , A , , ij ij E ij ijD RE RD RE Dij E EijD Eij E ˆ ˆ , , , B B D D D D ( 2 .1 0 ) ij ij E ij ijD ij ij E RD RE RD FijD Fij E LijD Lij E Eij EijD , Eˆij Eij E , Fij FijD , Fˆij Fij E . RE RD RE RD A ij
Where Aij , Bij , Dij , Eij , Fij are defined as follows: N
ÂŚ
Aij
K i K j Qij
(k )
N
ÂŚK K Q 3
Dij
i
j
(k )
ij
(h
3 k
h
ÂŚK K Q i
2
k 1
1
N
1
( hk hk 1 ), Bij 3 k 1
ÂŚK K Q 5 i
j
(h
2 k
h
2 k 1
)
N
1
ÂŚK K Q 4
), Eij
i
j
(k )
ij
(h
4 k
h
4 k 1
)
( 2 .1 1 )
k 1
N
1
(k )
ij
k 1
k 1
Fij
j
(k )
(h
ij
5 k
h
5 k 1
).
i, j
4, 5.
k 1
j directions, hk hk 1 is the thickness of the k th layer of the
Where K i and K j are shear correction coefficients in i and composite shell and N
(k )
Q ij
hk
ÂŚ Âłh
Aijn
k 1
Dijn
k 1
N
k 1
k 1
k 1
k 1
(k )
1 z 4
dz , Eijn
(k )
1 z
k 1
N
dz , Lijn
N
dz ,
Rn
3
(k )
z ˜ Q ij
ÂŚ Âłh
k 1
1 z 5
k 1
1 z
dz , ( 2 .1 2 )
Rn (k )
z ˜ Q ij
hk
ÂŚ Âłh k 1
Rn
1 z
hk
k 1
Rn
z ˜ Q ij
hk
ÂŚ Âłh
Fijn
k 1
Rn
2
(k )
z ˜ Q ij
hk
ÂŚ Âłh
dz , Bijn
z ˜ Q ij
hk
ÂŚ Âłh N
1 z
N
dz . n
D, E
Rn
In order to obtain better numerical stability, we truncated the equations described in (2.12) and the term 1 / (1 ( z / Rn )) in equation (2.12), which can be written in geometric series as
1 1 z / Rn
1
z Rn
The term higher than O (
z Rn
z2 Rn 2
z3 Rn 3
Higher order terms .
(2.13)
)3 are neglected to reduce the mathematical complexity, which is used in equations (2.9)-(2.12) and
obtained the following form and is similar to the derivation given in the references (Qatu et al., 1999, 2010, 2012) for comparison.
33
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Mathematical Modeling of a Transverse Shear Deformation Thick Shell Theory
Aij
ˆ Aij c0 Bij c1 Dij , A ij
Aij c0 Bij c1 Dij ,
Bij
ˆ Bij c0 Dij c1 Eij , B ij
Bij c0 Dij c1 Eij ,
Dij
ˆ Dij c0 Eij c1 Fij , D ij
Dij c0 Eij c1 Fij ,
Eij
ˆ Eij c0 Fij c1 Lij , E ij
Eij c0 Fij c1 Lij .
( 2 .1 4 )
1, 2, 3, 4, 5, 6.
i
And where
§ 1 § 1 1 ¡ 1 ¡ ¨ ¸ and c1 ¨ 2 ¸. Š RD RE š Š RD RD RE š
c0
2.2. Hamilton’s Principle: Equations of Motion and Boundary Conditions. The behavior of elastic vibrations, excitation and its inherent properties on arbitrary bounded shells are often difficult to solve from the elastodynamic theory. Making various dynamics assumptions about the motion of the shell surfaces under numerous conditions can be used to solve these problems, one such method is the vibrational or Hamiltonian principles of minimum energy. Hamiltonian in some approximate form can be used to construct surface vibrations. With the help of Hamilton’s principle various physical mechanisms such as perturbation in motion, rotary inertia, twist and shear distortion can be analyzed (Bert & Baker, 1969; Ames & van der Houwen, 1992; Love, 1892). The Hamilton’s principle (Ye & Soldatos, 1994Khare et al., 2005) of minimum energy of the governing equation is
G
Âł
t1
t0
0.
( K W U ) dt
( 2 .1 5 )
Where U is the strain energy, K the kinetic energy and W the external work by the system. The total strain energy and the middle surface stress and strains resultants of the shell have the following relationship: 1 {V D H D V E H E V z H z V DE H DE V D z J D z V E z J E z }dV U 2 VÂł 1 2
Âł Âł {N
D
H 0D N E H 0 E N z H 0 z NDE H 0DE N ED H 0 ED M D(1)N D(1) M D( 2 )N D( 2 )
D E
( 2 .1 6 )
M E(1)N E(1) M E( 2 )N E( 2 ) M D(1E)N D(1E) M D( E2 )N D( E2 ) M E(1D) N E(1D) M E( 2D)N E( 2D) QD J 0D z QE J 0 E z PD(1) G (1) PD( 2 ) G ( 2 ) PE(1) E (1) PE( 2 ) E ( 2 ) } ABdD d E .
The total external work and the total kinetic energy of the thick shell
Âł Âł {q
W
D
D
(1) ( 2) u0 qE v0 qn w0 mD G (1) mD G ( 2 ) mz\
z
E
(1) ( 2) mE E (1) mE E ( 2 ) } AB d D d E .
T
1 2 1 2
Âł {u
³³
D
2
v 2 w 2 }dV
V
§
1
Š
RD
{(u0 2 v0 2 w0 2 ) ¨ I1 I 2 (
E
2 (\ D \
2 z
§
1
Š
RD
) ¨ I3 I 4 (
2(u0\ D v0\
E
w0\
1 RE
)
1 RE
)
I3
¡ ¸
RD RE š
¡ ¸
I5
RD RE š
§
z
) ¨ I 2 I3 (
1 RD
1 RE
)
I4
¡ ¸
RD RE š
Š § ¡ I6 1 1 ) ¨ I 4 I5 ( ) 2(u0MD v0M E \ ¸ RD RE RD RE š Š § ¡ I7 1 1 2(MD\ D M E\ E ) ¨ I 5 I 6 ( ) ¸ R R R R D E D E Š š § ¡ I9 1 1 2 2 ) (MD M E ) ¨ I 7 I 8 ( ¸} AB dz d D d E RD RE RD RE š Š 2 E
.
( 2 .1 7 )
Where the inertia terms are
34
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 N
hk
> I1 , I 2 , I 3 , I 4 , I 5 , I 6 , I 7 , I8 , I 9 @ ÂŚ Âł k 1
hk 1
U ( k ) ª1, z , z 2 , z 3 , z 4 , z 5 , z 6 , z 7 , z 8 ºŸ dz
( 2 .1 8 )
In the above Hamilton’s principle derivation, equations of motion and boundary conditions are derived by substituting the equations (2.16) and (2.17) in equation (2.15). The resulting equations of motion are (Reddy, 1984; Qatu et al., 2013) 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
ND
wE
w
AB
( AN E )
wE
RDE
AB
QD
QE ABqE
RE
AB ( I1v 0 I 2\ E ). w wD w wD
N N N ED N w ( AQE ) AB ( D E DE ) ABqn RD RE RDE wE
( BQD )
wB
(1 )
( BM D )
wD
(1 )
ME
wA wE
w
(1 )
M DE
wE
AB
(1 )
( AM ED ) ABQD
AB ( I1 w0 ). (1 )
PE
RDE
(1 )
ABmD
AB ( I 2 u0 I 3\ D ). w wE
wA
(1 )
( AM E )
wE
w
(1 )
MD
(1 )
wD
( BM DE )
wB wD
AB
(1 )
M ED ABQE
(1 )
PD
RDE
(1 )
ABmE
AB ( I 2 v0 I 3\ E ). w wD w wD
( BPD )
(1 )
w
(1 )
(1 )
wE
(2)
( BM D )
(1 )
( APE ) AB ( N z
wB wD
w
(2)
ME
wE
MD
RD
(1 )
ME
RE
(2)
(1 )
( AM ED ) ( 2 ABPD
(1 )
M DE
RDE AB
RDE
M ED RDE
(2)
PD
) ABmz ( 2) ABPE RDE
AB ( I 3\ z ).
(2)
) ABmD
.
( 2 .1 9 )
AB ( I 3u0 I 4MD ). w wE
(2)
( AM E )
wA wE
w
(2)
MD
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 I i
§ § 1 Ii 2 ¡ 1 ¡ ¨ I i I i 1 ¨ ¸ ¸, i Š Š RD RE š RD RE š
1, 2, 3, 4 .
( 2 .2 0 )
The corresponding boundary conditions are given below for D , a constant
either
0
N 0D N D
either
N 0 DE N DE
either
Q0D QD
0
M
(1) 0D
either
M
(1) 0 DE
either
P0(D1) PD(1)
either
M
(1) D
M
either M
M
0
(1) DE
0 0
2) either M 0(D M D( 2 ) ( 2) 0 DE
0
( 2) DE
0 0
or
u0
0
or
v0
0
or
w0
0
or
\D
0
or
\E
0
or
\z
0
or
MD
0
ME
0
or
.
( 2 .2 1 )
Similar boundary conditions are obtained by taking E , a constant.
35
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Mathematical Modeling of a Transverse Shear Deformation Thick Shell Theory 3. MATHEMATICAL ANALYSIS It is known that there is no exact solution for a general lamination structure (shell, plate) with boundary conditions and/or lamination having series of sequence and layers. The exact solution of the equations of motion and boundary conditions for simply supported cross-ply thick shell (FlĂźgge, 1962; Leissa, 1973; Ye & Soldatos, 1994) is formulated in the section. These partial differential equations of the motions (2.19) have solutions of the form (Librescu et al., 1989; Reddy, 1984; Reissner, 1945; Qatu et al., 2012, 2013)
u0
N
ÂŚu
*
0 , mn
*
Cos ( A D ) Sin ( B E ) e
iZ t
N
ÂŚv
, v0
m.n 1
w0
ÂŚw
*
*
iZ t
Sin ( A D ) Sin ( B E ) e
0 , mn
N
ÂŚ\
, \D
iZ t
ÂŚ\
*
D , mn
*
C os ( A D ) Si n ( B E ) e
iZ t
m.n 1
N
*
*
Sin ( A D ) Cos ( B E ) e
E , mn
iZ t
( 3 .1 )
N
ÂŚ\
,\z
m.n 1
ID
*
m.n 1
N
m.n 1
\E
*
Sin ( A D ) Cos ( B E ) e
0 , mn
*
*
iZ t
z , mn
Sin ( A D ) Sin ( B E ) e
E , mn
Sin ( A D ) Cos ( B E ) e
m.n 1
N
ÂŚ ID ,mn Cos ( A D ) Sin( B E ) e *
m.n 1
*
mS * ,B a
*
Where A
iZ t
N
ÂŚI
, IE
*
*
iZ t
.
m.n 1
nS in which a and b are the dimensions of the middle part of the shell along the D - and b
E -axes, respectively. Substituting these equations (3.1) into the differential equations of motion (2.19) yields a set of eight linear algebraic systems in terms of its respective components and collecting the coefficients, which can be written as an eigenvalue problem. Therefore, the resulting equations can be written in the following matrix form:
{[ L] O [ M ]}{'} { f }.
Where O
Z , Z is the natural frequency and {'} is the displacement vector. The stiffness parameters Lij and the mass M ij of the thick shell can be defined
parameter L11
2
*
A11 ˜ A
ˆ ˜ B* A 66
*
L13
( 3 .2 )
2
A11 ˜ A
A55 ( RD )
A55 ˜ A
RE
2
ˆ A 44 ( RDE )
2
*
, L12 *
, L14
RD
B11 ˜ A
2
ˆ ˜ B* B 66 *
L15
*
*
*
*
2
*
ˆ ˜ B* D 66
2
2 B55
L17
D11 ˜ A
L21
A12 ˜ A ˜ B A66 ˜ A ˜ B , L22
L23
*
B12 ˜ A ˜ B B66 ˜ A ˜ B , L16
*
A12 ˜ B
*
*
RD
2
RD
*
ˆ ˜ B* A 44 RE
A13 ˜ A
*
ˆ ˜ B* , L A 24 22 2
B55
ˆ ˜ B* B 22
L27
D12 ˜ A ˜ B D66 ˜ A ˜ B , L28 *
L31 L33
*
B66 ˜ A
A55 ˜ A RD
*
*
A55 ˜ A
2
A11 ˜ A RD
ˆ ˜ B* A 44
L36
*
A55 ˜ A *
A
2
B11 ˜ A
˜ B55 B
RD *
2
2
RE 2
A11 2
B12 ˜ A RE
A55
*
*
A66 ˜ A
2
*
ˆ ˜ B* A 44 RE A12 ( RD )
, L35
2
2
( RDE )
B12 ˜ B RD *
D66 ˜ A
A12 RD ˜ RE
RD
A55
*
A12 ˜ B
B55 ˜ A
*
*
D12 ˜ A ˜ B D66 ˜ A ˜ B
A23 ˜ B
ˆ ˜ B* D 22
( RDE )
*
, L18
*
ˆ B 44
RD
RE
*
, L26
RE
*
2
B12 ˜ A
2
RE
ˆ A 44
, L32
( RD )
( RDE )
2
*
RD ˆ D 44
*
*
A12 ˜ A
*
B12 ˜ A ˜ B B66 ˜ A ˜ B
*
*
L34
( RDE )
*
*
B11 ˜ A
ˆ ˜ B* A 22
L25
*
D55 RD ˜ RDE
*
A12 ˜ A ˜ B A66 ˜ A ˜ B
*
*
A12 ˜ A
RD
2
2
2
*
ˆ A 44 ( RE )
2
*
*
ˆ ˜ B* B 22 RE
D55 RD ˜ RDE
ˆ ˜ B* B 44
ˆ 2˜ B 44 RE
RE
ˆ D 44 ( RE )
2
*
( 3 .3 )
RD
( A66 2 ˜ A66 ( RDE )
ˆ ) A 66
2
* * ˆ ˆ ˜ B * B12 ˜ B B22 ˜ B A 44 RD RE
ˆ ˆ ˆ A13 B11 A22 2 ˜ B12 ( A36 A36 ) ( 2 ˜ B66 2 ˜ B66 ) B22 B 44 2 2 2 ( RDE ) ( RE ) RD ( RD ) RE RD ˜ RE RDE
36
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 *
2 ˜ A ˜ B55
L38
2˜ B
L41
L14 , L42
*
*
D11 ˜ A
L45
D12 ˜ A ˜ B
L46
B13 ˜ A
L47
E11 ˜ A
L48
E12 ˜ A ˜ B
*
*
RD
D12 ˜ B
A55 *
D66 ˜ A ˜ B
L35 , L54
L53
D12 ˜ B
L56
* ˆ E ˜B 22
L61
L16 , L62 D55 ˜ A B13 RD
B23 RE
D11 ( RD )
2
RDE
2
( RDE )
*
D66 ˜ A
2
2
L25 ,
*
*
( RE ) *
2
( B36 B36 ) RDE *
E55 ˜ A
E11 ˜ A
*
E12 ˜ A
L73
L37 , L74
L77
F11 ˜ A
L78 L87
*
F12 ˜ A ˜ B
*
2
RD
L57 , L76
F66 ˜ A ˜ B 2
*
*
F66 ˜ A
RDE
F55 ( RDE )
L28 , L83 2
L17 , L72
L18 , L84
4 ˜ E55
, L82
,
RE
L67 , L81
4 ˜ D55 *
ˆ ˜ B* F 22
L78 , L88
RDE
ˆ ˜ B* F 66
*
2
( RDE )
* * * ˆ ˆ ˜B ˜B E E12 ˜ B E * 44 22 ˆ , L71 2˜ D ˜B 55 RE RD RE
2
*
E 66 ˜ A ˜ B ,
ˆ ) ( D66 2 ˜ D66 D 66
L68
*
,
2
A33
2 ˜ D55 ˜ A
L47 , L75
( RDE )
L56
L67
D44
ˆ A 44
E12 ˜ A ˜ B
L46 , L65
ˆ D 22
ˆ E 44
L15 , L52
51
*
L36 , L64
ˆ D ˜B 44
*
D55
ˆ D E 44 44 ˆ 2˜ B 44 RE RD ˜ RDE
2
*
,
RD
* ˆ B ˜ B , L57 44
*
,
2
D11 ˜ A
*
ˆ D ˜B 22
, L55
E66 ˜ A
2
2 ˜ B55
L26 , L63 *
2
*
RE
2
ˆ D 44 ( RDE )
E66 ˜ A ˜ B , L
* ˆ ˜B D 22
*
L58
L66
*
L45
RD
*
* ˆ E ˜B 66
*
RE
*
B55 ˜ A
RE
*
* ˆ ˜B D 22
*
D12 ˜ A
,
RD
*
RD
*
*
D11 ˜ A
L34 2
* ˆ D ˜B 66
L44
RE
˜ D44
L24 , L43 2
*
D12 ˜ A
RDE
B ˆ ˜B 44
*
*
A ˜ D55
*
L37
F66 ( RD )
2
2
L48 , L85
RE
L58
,
( 3 .4 )
L38 , , L86 ˆ E 44
L27 ,
L68
ˆ F 44 ( RE )
ˆ 4˜ D 44
2
The mass parameters M ij is given by M 11
I1 ,
I2 ,
M 14
M 12
I2 ,
M 13
M 15
M 16
M 17
M 18
0
M 22
I1 ,
M 33
I1 ,
M 41
I2
,
M 44
I3 ,
M 42
M 43
M 45
M 46
M 47
M 48
0
M 52
I2
,
M 55
I3 ,
M 51
M 53
M 54
M 56
M 57
M 58
0
M 66
I3
,
M 61
M 62
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 25 M 31
M 21
M 32
M 23 M 34
M 63
M 24 M 35
M 64
M 26 M 36
M 65
M 27 M 37
M 67
0
M 28 0
M 38
M 68
0
( 3 .5 )
The force vector { fij } is given by:
37
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Mathematical Modeling of a Transverse Shear Deformation Thick Shell Theory N
¦q
qD
*
*
Cos ( A D ) Sin ( B E ) e
D , kl
iZ t
, qE
k ,l 1
qn
*
*
Sin ( A D ) Sin ( B E ) e
n , kl
, mD(1)
iZ t
k ,l 1
N
mE
(1 )
E , kl
*
*
Sin ( A D ) Cos ( B E ) e
, mz
iZ t
k ,l 1
iZ t
N
N
¦m
(1 )
*
*
iZ t
*
*
iZ t
Cos ( A D ) Sin ( B E ) e
D , kl
( 3 .6 )
N
¦m
z , kl
Sin ( A D ) Sin ( B E ) e
k ,l 1
¦m
(2)
mD
*
k ,l 1
¦m
(1 )
*
Sin ( A D ) Cos ( B E ) e
E , kl
k ,l 1
N
¦q
N
¦q
(2)
D , kl
*
*
Cos ( A D ) Sin ( B E ) e
iZ t
, mE( 2 )
k ,l 1
N
¦m
(2)
E , kl
*
*
Sin ( A D ) Cos ( B E ) e
iZ t
k ,l 1
4. Numerical Results and Discussion In the above section we have given the mathematical formulation of thick shell, and an extension of first order shear deformation shell theory. The additional new parameters using the third order shear deformation shell theory by Zannon (TSDTZ)obtained from Fourier expansion, which is given in the table 4.1 and compared with the first order shear deformation shell theory from the literature. TABLE 4.1: Comparison of parameters of first order versus third order shear deformation shell theory First order SDST (Literature ) Condition Unknowns Displacements/rotation
Hz
TSDTZ ( Present study)
0
Hz z 0
u0 , v0 , w0 ,\ D ,\ E
u0 , v0 , w0 ,\ D ,\ E ,\ z , MD , M E
Force resultants
ND , N E , NDE , N ED , QD , QE
ND , N E , NDE , N z , N ED , QD , QE
Moment resultants
M D , M E , M DE , M ED , PD , PE
M D(1) , M D( 2 ) , M E(1) , M E( 2 ) , M D(1E) , M D( 2E) , M E(1D) , M E( 2D) , PD(1) , PD( 2 ) , PE(1) , PE( 2 )
Strains at a point (3D)
H 0D , H 0 E , H 0DE , H 0 ED , J 0D z , J 0 E z
ND(1) , ND( 2 ) , N E(1) , N E( 2 ) , ND(1E) , ND( 2E) , N E(1D) , N E( 2D)
Strains at the middle surface
N D , N E , N DE , N E D
ND(1) , ND( 2 ) , N E(1) , N E( 2 ) , ND(1E) , ND( 2E) , N E(1D) , N E( 2D)
Equations Motion Strain-displacement Stress-strain
5 10 12
8 15 19
The above equation (3.2) in the matrix form is solved using MATLAB commercial code. This code is specifically modified for static vibration problems of simply supported cross-ply laminated composite shells. 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. In static analyses, shells are under uniformly distributed load q. Thus, using a Fourier analysis, one finds the coefficients of a Fourier transform as
qmn
16q / mnS 2 in Eqs.
(3.3-3.4). Numerical investigation showed that the terms m and n did not need to exceed fifty for convergence of the results. Dimensionless transverse displacement, moment and force resultants
w*
M i*
103 E2 h 3 w / qn a 4 103 M i / qa 2 ,
N i*
103 N i / qa 3 , Where i D , E , at the center of the shells are calculated based upon both FSDTQ and TSDTZ. Table 4.2 and 4.3 shows
dimensionless displacement and force and moment resultants at the center of isotropic shells with different thickness ratios a/h
38
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-01, Issue-07, October 2014 = 10, and 20 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 ) 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. TABLE 4.2.Comparison of dimensionless displacement and force and moment resultants of 2-ply unsymmetrical [90 / 0] orthotropic cylindrical shells
a
2 b 1, K
a
R
5 , E1 E2 6 Method
w*
FSDTQ (Qatu et al., 2010, 2012)
0 .5
1
0.5, G23 E 2
25, G12 E 2
1 1 .6 3 2
0.2, G13
G12 , Q 12
0.25,
a
h
20
M D*
M E*
ND*
N E*
5 2 .9 5 3
2 8 .8 0 1
1 0 8 6 .1
9 9 4 .6 4
TSDTZ (Present theory)
1 1 .6 2 4
5 3 .0 5 5
2 8 .9 9 2
1 0 8 5 .2
9 9 3 .7 8
3D (Qatu et al., 2010, 2012)
1 1 .6 1 2
5 3 .6 8 9
2 9 .6 4 3
1 0 8 4 .7
9 9 0 .7 8
FSDTQ (Qatu et al., 2010, 2012)
5 .6 7 8 2
3 1 .2 5 5
5 .5 4 8 0
1 0 7 0 .5
9 5 9 .4 5
TSDTZ (Present theory)
5 .6 7 4 1
3 1 .3 4 1
5 .5 8 8 7
1 0 7 0 .5
9 5 9 .8 7
3D (Qatu et al., 2010, 2012)
5 .7 6 4 6
3 1 .6 3 8
5 .9 6 6 9
1 0 7 0 .5
9 5 7 .9 2
TABLE 4.3.Comparison of dimensionless displacement and force and moment resultants of 3-ply symmetric [0 / 90 / 0] orthotropic cylindrical shells
a a
0 .5
R
25, G12 E 2
0.5, G23 E 2
Method
w*
M D*
M E*
ND*
N E*
FSDTQ (Qatu et al., 2010, 2012)
9 .5 1 5 9
1 1 5 .5 7
1 1 .3 9 1
2 2 4 .2 7
2 0 1 .8 9
2 b 1, K
5 , E1 E2 6
39
0.2, G13
G12 , Q 12
0.25,
a
h 10
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Mathematical Modeling of a Transverse Shear Deformation Thick Shell Theory
1
TSDTZ (Present theory)
9 .5 8 5 1
1 1 4 .0 0 5
1 1 .5 9 4
2 2 6 .0 2
2 0 2 .3 2
3D (Qatu et al., 2010, 2012)
1 0 .6 6 1
1 1 2 .5 5
1 3 .8 4 7
2 4 9 .7 8
2 2 2 .5 2
FSDTQ (Qatu et al., 2010, 2012)
7 .8 5 8 9
9 5 .4 1 1
7 .9 3 3 5
3 7 0 .8 2
3 3 5 .5 9
TSDTZ (Present theory)
7 .9 4 0 9
9 3 .2 2 9
8 .0 0 0 1
3 7 1 .2 4
3 3 6 .0 5
3D (Qatu et al., 2010, 2012)
8 .6 3 9 9
9 1 .3 4 0
9 .5 5 6 1
4 0 5 .2 9
3 6 2 .8 8
5. Conclusion In this article we derived the mathematical formulation of a thick shell theory, which is an extension of the most popular first order deformation theory for shell structures. The present approximation of third order shear deformation theory Zannon (TSDTZ) by for simply supported cross-ply thick shell is developed using Fourier series expansion. Also, developed a simplified three dimension to two dimensional theories with a third order shear deformation theory for shells. In a forthcoming article we will use this solution to evaluate the results for free vibration analysis of the cross-ply thick shells and will assess the accuracy obtained from the numerical computation using the proposed theory. 6. References [1] AMBARTSUMIAN, S. (1962) Contribution to the theory of anisotropic laminated shells. Applied Mechanics Review, 15, 245–249. [2] ARON, H. (1874) Das Gleichgewicht und die Bewegung einer unendlich dunnen, beliebig gekrummten elastischen Schale. J. Reine u. Angew. Math, 78, 136-174. [3] BERT, W. (1967) Structural theory of laminated anisotropic elastic shells. J. Compos. Mater, 14, 4 1 4 – 4 2 3 .9 4 4 . [4] BERT, W. & BAKER, L. (1969) Egle Dynamics of composite, sandwich and stiffened shell-type structures. J. S p a c e Ro c k e ts . 6 , 1 3 4 5 – 1 3 6 1 . [5] AMES, W. & VAN DER HOUWEN, PJ. (1992) Hamilton's principle and the equations of motion of an elastic shell with and without fluid loading. Computational and Applied Mathematics. Elsevier Science Publishers B.V, IMACS, 131-140. [6] FLÜGGE, W. (1962) Stresses in Shells. Springer: Berlin. [7] GOL’DENVEIZER, L. (1961) Theory of Elastic Thin Shells. Pergamon Press, New York.
40
[8] KANT, T. & SWAMINATHAN, K. (2001) Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher-order refined theory. Composite Structures. 53, 73-85. [9] KAPANIA, PK. (1989) Review on the Analysis of Laminated Shells. Journal of Pressure Vessel Technology. 111, 88-96. [10] KHARE, RK., RODE V. & JOHN, SP. (2005) Higher-order Closed-form Solutions for Thick Laminated Sandwich Shells. J. Sandwich Structures and Materials. Vol.7-july. [11] KOITER, W. (1969) Theory of Thin Shells. F.L, New York, pp. 93–105. [12] LEISSA, W. (1973) Vibration of Shells. The Acoustical Society of America, NASA SP288. [13] LIBRESCU, L., KHDEIR, A. & FREDERICK, D. (1989) a shear-deformable theory for laminated composite shallow shell-type panels and their response analysis I: free vibration and buckling. Acta Mech.76, 1–33. [14] LOVE, E. (1892) A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge 4th edition, Dover Publishing, New York. [15] QATU, MS. (2004) Vibration of Laminated Shells and Plates. Elsevier, Technology & Engineering. [16] NOOR, AK. (1990) Assessment of computational models for multilayered composite shells. Appl. Mech. Rev. 43, 67–96. [17] QATU, MS. (1994) On the validity of nonlinear shear deformation theories for laminated composite plates and shells. Composite Structure. 27, 395–401. [18] QATU, MS. (2002) Recent research advances in the dynamic behavior of shells: 1989–2000.Part 1 & 2: laminated composite shells. Part II: homogeneous shells. Appl. Mech. Rev. 55, 325–351 & 415–435, and references therein. [19] KAPANIA, PK. (1989) Review on the analysis of laminated shells. J Press Vessel Tech. 111, 88-96.
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