1 Exact 3D Stress Analysis Exact 3D Stress Analysis of of Laminated Composite Plates and Laminated Composite Plates and Shells by Sampling Surfaces Shells by Sampling Surfaces Method Method G.M. Kulikov and S.V. Plotnikova G.M. Kulikov and S.V. Plotnikova Speaker: Gennady Kulikov Speaker: Gennady Kulikov Department of Applied Mathematics & Mechanics Department of Applied Mathematics & Mechanics
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1 Exact 3D Stress Analysis of Laminated Composite Plates and Shells by Sampling Surfaces Method G.M. Kulikov and S.V. Plotnikova Speaker: Gennady Kulikov.
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Exact 3D Stress AnalysisExact 3D Stress Analysis ofofLaminated Composite Plates and Laminated Composite Plates and
Shells by Sampling Surfaces MethodShells by Sampling Surfaces Method
G.M. Kulikov and S.V. PlotnikovaG.M. Kulikov and S.V. Plotnikova
Speaker: Gennady KulikovSpeaker: Gennady Kulikov
Department of Applied Mathematics & MechanicsDepartment of Applied Mathematics & Mechanics
Exact 3D Stress AnalysisExact 3D Stress Analysis ofofLaminated Composite Plates and Laminated Composite Plates and
Shells by Sampling Surfaces MethodShells by Sampling Surfaces Method
G.M. Kulikov and S.V. PlotnikovaG.M. Kulikov and S.V. Plotnikova
Speaker: Gennady KulikovSpeaker: Gennady Kulikov
Department of Applied Mathematics & MechanicsDepartment of Applied Mathematics & Mechanics
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Kinematic Description of Undeformed ShellKinematic Description of Undeformed ShellKinematic Description of Undeformed ShellKinematic Description of Undeformed Shell
(1)(1)
(2)(2)
(3)(3)3)(
3)(
3)(
3)()(
,)( ,, erRegeRg nnnnnn inininininin cA
Figure 1. Geometry of laminated shellFigure 1. Geometry of laminated shellFigure 1. Geometry of laminated shellFigure 1. Geometry of laminated shell
33 eer aa ,, ABase Vectors of Midsurface and SaSBase Vectors of Midsurface and SaS Base Vectors of Midsurface and SaSBase Vectors of Midsurface and SaS
NN - number of layers; - number of layers; IInn - number of SaS of the - number of SaS of the nnth layerth layer
)2(232
cos21
)(21
,
][3
]1[3
)(3
][3
)(3
]1[3
1)(3
n
nn
nnmn
nInnn
Im
hn
n
rr((11, , 22) - position vector of midsurface ) - position vector of midsurface ; ; RR((nn))ii - position vectors of SaS of the - position vectors of SaS of the nnth layerth layer
eeii - orthonormal vectors; - orthonormal vectors; AA, , kk - - Lamé coefficients and principal curvatures of midsurfaceLamé coefficients and principal curvatures of midsurface
cc = 1+k = 1+k3 3 - components of shifter tensor at SaS - components of shifter tensor at SaS((nn))ii nn((nn))ii nn
((n)in)i - thickness coordinates of SaS - thickness coordinates of SaS
[[nn-1]-1], , [[nn]] - thickness coordinates of interfaces - thickness coordinates of interfaces
nn
33 33
33nn
3
Kinematic Description of Deformed ShellKinematic Description of Deformed ShellKinematic Description of Deformed ShellKinematic Description of Deformed Shell
(4)(4)
(5)(5)
(6)(6)
Figure 2. Initial and current configurations of shellFigure 2. Initial and current configurations of shellFigure 2. Initial and current configurations of shellFigure 2. Initial and current configurations of shell
Base Vectors of DeformedBase Vectors of Deformed SaSSaSBase Vectors of DeformedBase Vectors of Deformed SaSSaS
Position Vectors of Deformed SaSPosition Vectors of Deformed SaSPosition Vectors of Deformed SaSPosition Vectors of Deformed SaS
nnn ininin )()()( uRR
)( )(3
)( nn inin uu
u u ((11, , 22) - displacement vectors of SaS) - displacement vectors of SaS((nn))ii nn
)(,, )(33,
)()(3
)(3
)(,
)()(,
)( nnnnnnnn inininininininin uegugRg
((11, , 22) - derivatives of 3D displacement vector at SaS) - derivatives of 3D displacement vector at SaS((nn))ii nn
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Green-Lagrange Strain Tensor at SaSGreen-Lagrange Strain Tensor at SaS
Presentation of Displacement Vectors of SaSPresentation of Displacement Vectors of SaS
(7)(7)
(8)(8)
(9)(9)
)(1
2 )()()()()()(
)( nnnn
nn
n inj
ini
inj
iniin
jin
iji
inij
ccAAgggg
3)()(
333)(
,)()()(
3
)(,)(
)(,)(
)(
,1
2
112
eeue
eueu
nnnn
nn
n
nn
n
n
inininin
inin
inin
inin
in
cA
cAcA
i
iin
iin
ii
ini
in nnnn u ee )()()()( , u
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Presentation of Derivatives of Displacement Vectors of SaSPresentation of Derivatives of Displacement Vectors of SaS
Strain ParametersStrain Parameters
Component Form of Component Form of StrainStrainss of S of SaSaS
Presentation of Derivatives of Displacement Vectors of SaSPresentation of Derivatives of Displacement Vectors of SaS
Strain ParametersStrain Parameters
Component Form of Component Form of StrainStrainss of S of SaSaS
Remark.Remark. Strains (12) exactly represent all rigid-body shell motions in any convected curvilinear Strains (12) exactly represent all rigid-body shell motions in any convected curvilinear coordinate system. It can be proved through Kulikov and Carrera (2008)coordinate system. It can be proved through Kulikov and Carrera (2008)
(10)(10)
(11)(11)
(12)(12)
i
iin
iin nn
Aeu )()(
,1
nnn
nnn
n
n
n
n
n
inininin
inin
inin
inin
in
c
cc
)(3
)(33
)(3)(
)()(3
)()(
)()(
)(
,1
2
112
,)()(
,3)(
3
)()(,
)()(3
)()(,
)(
1,
1
1,
1
AAA
BukuA
uBuA
ukuBuA
nnn
nnnnnnn
ininin
ininininininin
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Displacement Distribution in Thickness DirectionDisplacement Distribution in Thickness Direction
Presentation of Derivatives of 3D Displacement Vector Presentation of Derivatives of 3D Displacement Vector
Strain Distribution in Thickness DirectionStrain Distribution in Thickness Direction
Displacement Distribution in Thickness DirectionDisplacement Distribution in Thickness Direction
Presentation of Derivatives of 3D Displacement Vector Presentation of Derivatives of 3D Displacement Vector
Strain Distribution in Thickness DirectionStrain Distribution in Thickness Direction
Table 1. Results for thick square sandwich plate withTable 1. Results for thick square sandwich plate with a / h = 2 a / h = 2 Table 1. Results for thick square sandwich plate withTable 1. Results for thick square sandwich plate with a / h = 2 a / h = 2
Figure Figure 55. Accuracy of satisfying the boundary conditions . Accuracy of satisfying the boundary conditions ii (-h / 2) and (-h / 2) and ii (h / 2)(h / 2) on the bottom (on the bottom () and ) and
top (top () surfaces of the sandwich plate: (a) a / h = 2 and (b) a / h = 4 , where ) surfaces of the sandwich plate: (a) a / h = 2 and (b) a / h = 4 , where ii = = lglg SSi3i3 – – SSi3i3 Figure Figure 55. Accuracy of satisfying the boundary conditions . Accuracy of satisfying the boundary conditions ii (-h / 2) and (-h / 2) and ii (h / 2)(h / 2) on the bottom (on the bottom () and ) and
top (top () surfaces of the sandwich plate: (a) a / h = 2 and (b) a / h = 4 , where ) surfaces of the sandwich plate: (a) a / h = 2 and (b) a / h = 4 , where ii = = lglg SSi3i3 – – SSi3i3 3D3D
Figure 4. Distribution of transverse shear stresses SFigure 4. Distribution of transverse shear stresses S1313 and S and S2323 through the thickness of through the thickness of
the sandwich plate for Ithe sandwich plate for I11 = I = I22 = I = I33 =7: present analysis ( ) and Pagano’s solution ( =7: present analysis ( ) and Pagano’s solution ())
Figure 4. Distribution of transverse shear stresses SFigure 4. Distribution of transverse shear stresses S1313 and S and S2323 through the thickness of through the thickness of
the sandwich plate for Ithe sandwich plate for I11 = I = I22 = I = I33 =7: present analysis ( ) and Pagano’s solution ( =7: present analysis ( ) and Pagano’s solution ())
Table 2. Results for square (b = a) two-layer angle-ply plate with Table 2. Results for square (b = a) two-layer angle-ply plate with hh11 = h = h22 = h / 2 , a / h = 4 and stacking sequence [-15 = h / 2 , a / h = 4 and stacking sequence [-15/ 15/ 15]]
Table 2. Results for square (b = a) two-layer angle-ply plate with Table 2. Results for square (b = a) two-layer angle-ply plate with hh11 = h = h22 = h / 2 , a / h = 4 and stacking sequence [-15 = h / 2 , a / h = 4 and stacking sequence [-15/ 15/ 15]]
2. Antisymmetric Angle-Ply Plate under Sinusoidal Loading2. Antisymmetric Angle-Ply Plate under Sinusoidal Loading2. Antisymmetric Angle-Ply Plate under Sinusoidal Loading2. Antisymmetric Angle-Ply Plate under Sinusoidal Loading
Table 3. Results for rectangular (b = 3a) two-layer angle-ply plate Table 3. Results for rectangular (b = 3a) two-layer angle-ply plate with hwith h11 = h = h22 = h / 2 , a / h = 4 and stacking sequence [-15 = h / 2 , a / h = 4 and stacking sequence [-15/ 15/ 15]]
Table 3. Results for rectangular (b = 3a) two-layer angle-ply plate Table 3. Results for rectangular (b = 3a) two-layer angle-ply plate with hwith h11 = h = h22 = h / 2 , a / h = 4 and stacking sequence [-15 = h / 2 , a / h = 4 and stacking sequence [-15/ 15/ 15]]
Figure 6. Distribution of transverse shear stresses SFigure 6. Distribution of transverse shear stresses S1313 and S and S2323 through the through the
thickness of square unsymmetric two-layer angle-ply plate for Ithickness of square unsymmetric two-layer angle-ply plate for I11 = I = I22 = 7 = 7 Figure 6. Distribution of transverse shear stresses SFigure 6. Distribution of transverse shear stresses S1313 and S and S2323 through the through the
thickness of square unsymmetric two-layer angle-ply plate for Ithickness of square unsymmetric two-layer angle-ply plate for I11 = I = I22 = 7 = 7 �
Analytical solutionAnalytical solutionAnalytical solutionAnalytical solution3. Cylindrical Composite Shell under Sinusoidal Loading3. Cylindrical Composite Shell under Sinusoidal Loading3. Cylindrical Composite Shell under Sinusoidal Loading3. Cylindrical Composite Shell under Sinusoidal Loading
Table 4. Results for thick two-ply cylindrical shell with R / h = 2 and stacking sequence [0Table 4. Results for thick two-ply cylindrical shell with R / h = 2 and stacking sequence [0 /90/90]]Table 4. Results for thick two-ply cylindrical shell with R / h = 2 and stacking sequence [0Table 4. Results for thick two-ply cylindrical shell with R / h = 2 and stacking sequence [0 /90/90]]
Figure 7. Simply supported cylindrical Figure 7. Simply supported cylindrical composite shell with L / R = 4 composite shell with L / R = 4
Figure 7. Simply supported cylindrical Figure 7. Simply supported cylindrical composite shell with L / R = 4 composite shell with L / R = 4
Figure 9. Distribution of transverse shear stresses SFigure 9. Distribution of transverse shear stresses S1313 and S and S2323 through the thickness of three-ply cylindrical shell with through the thickness of three-ply cylindrical shell with
stacking sequence [90stacking sequence [90/ 0/ 0/ 90/ 90] for I] for I11 = I = I22 = I = I33 = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution ( = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution ())
Figure 9. Distribution of transverse shear stresses SFigure 9. Distribution of transverse shear stresses S1313 and S and S2323 through the thickness of three-ply cylindrical shell with through the thickness of three-ply cylindrical shell with
stacking sequence [90stacking sequence [90/ 0/ 0/ 90/ 90] for I] for I11 = I = I22 = I = I33 = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution ( = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution ())
Figure 8. Distribution of transverse shear stresses SFigure 8. Distribution of transverse shear stresses S1313 and S and S2323 through the thickness of two-ply cylindrical shell with through the thickness of two-ply cylindrical shell with
stacking sequence [0stacking sequence [0/ 90/ 90] for I] for I11 = I = I22 = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution ( = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution ())
Figure 8. Distribution of transverse shear stresses SFigure 8. Distribution of transverse shear stresses S1313 and S and S2323 through the thickness of two-ply cylindrical shell with through the thickness of two-ply cylindrical shell with
stacking sequence [0stacking sequence [0/ 90/ 90] for I] for I11 = I = I22 = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution ( = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution ())
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ConclusionsConclusionsConclusionsConclusions
A simple and efficient method of SaS inside the shell body has been A simple and efficient method of SaS inside the shell body has been proposed. This method permits the use of 3D constitutive equations proposed. This method permits the use of 3D constitutive equations and leads to exact 3D solutions of elasticity for thick and thin and leads to exact 3D solutions of elasticity for thick and thin laminated plates and shells with a prescribed accuracylaminated plates and shells with a prescribed accuracy
A new higher-order layer-wise theory of shells has been developed A new higher-order layer-wise theory of shells has been developed through the use of only displacement degrees of freedom, i.e., through the use of only displacement degrees of freedom, i.e., displacements of SaS. This is straightforward for finite element displacements of SaS. This is straightforward for finite element developmentsdevelopments
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Thanks for your attention!Thanks for your attention!Thanks for your attention!Thanks for your attention!