1
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
2
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
Indices: Indices: nn = 1, 2, …, = 1, 2, …, NN; ; iinn = 1, 2, …, = 1, 2, …, IInn; ; mmnn = 2, 3, …, = 2, 3, …, IInn-1-1
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
nn
((nn)1)1, , ((nn)2)2, …, , …, ((nn))II - sampling surfaces ( - sampling surfaces (SaSSaS))
((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
4
Green-Lagrange Strain Tensor at SaSGreen-Lagrange Strain Tensor at SaS
Linearized Strain-Displacement RelationshipsLinearized Strain-Displacement Relationships
Presentation of Displacement Vectors of SaSPresentation of Displacement Vectors of SaS
Green-Lagrange Strain Tensor at SaSGreen-Lagrange Strain Tensor at SaS
Linearized Strain-Displacement RelationshipsLinearized Strain-Displacement Relationships
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
5
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
6
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
Higher-Order Layer-Wise Shell FormulationHigher-Order Layer-Wise Shell FormulationHigher-Order Layer-Wise Shell FormulationHigher-Order Layer-Wise Shell Formulation
(13)(13)
(14)(14)
(15)(15)
(16)(16)
][33
]1[3
)()()( , nn
i
ini
inni
n
nnuLu
nnnn
nn
ijjnin
jninL
)(3
)(3
)(33)(
nn
n
nnnn jnjn
j
jni
injnini LMuM )(
3,)()()(
3)()( ,)(
n
nn
i
nninij
innij L ][
33]1[
3)()()( ,
LL ((33) -) - Lagrange polynomials of degree Lagrange polynomials of degree IInn - 1- 1 ((nn))ii nn
7
Stress ResultantsStress Resultants
Variational EquationVariational Equation
Constitutive EquationsConstitutive Equations
Presentation of Stress ResultantsPresentation of Stress Resultants
Stress ResultantsStress Resultants
Variational EquationVariational Equation
Constitutive EquationsConstitutive Equations
Presentation of Stress ResultantsPresentation of Stress Resultants
(17)(17)
(18)(18)
(19)(19)
(20)(20)
(21)(21)
CCijkmijkm - components of material tensor of the - components of material tensor of the nnth layerth layer((nn))
ppii , p, pi i - surface loads acting on bottom and top surfaces- surface loads acting on bottom and top surfaces and and [0] [[NN]] [0] [[NN]]
][3
]1[3
321)()()(
n
n
nn dccLH innij
inij
WddAAupccupccH
n i ji iii
Ni
Ni
NNinij
inij
n
nn2121
,
]0[]0[]0[2
]0[1
][][][2
][1
)()(
][33
]1[3
,
)()()( , nn
mk
nkm
nijkm
nij C
n
nnnn
j mk
jnkm
jinijkm
inij DH
,
)()()(
][3
]1[3
321)()()()(
n
n
nnnn dccLLCD jninnijkm
jinijkm
8
Numerical ExamplesNumerical ExamplesNumerical ExamplesNumerical Examples1. Simply Supported Sandwich Plate under Sinusoidal Loading1. Simply Supported Sandwich Plate under Sinusoidal Loading1. Simply Supported Sandwich Plate under Sinusoidal Loading1. Simply Supported Sandwich Plate under Sinusoidal Loading
Analytical solutionAnalytical solutionAnalytical solutionAnalytical solution
Figure 3. Sandwich plate Figure 3. Sandwich plate Figure 3. Sandwich plate Figure 3. Sandwich plate
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
bauu
bauu
bauu
nn
nnnn
inin
inininin
21)(30
)(3
21)(20
)(2
21)(10
)(1
sinsin
cossin,sincos
hzpzbaS
apzahSapzbhS
apzhSapzbahS
apzbahSapzbauhEU
/,/),2/,2/(
/),0,2/(10,/),2/,0(10
/),0,0(10,/),2/,2/(10
/),2/,2/(,/),2/,2/(100
303333
0232301313
2012
212
2022
222
2011
211
403
3T3
IInn SS1111(-0.5)(-0.5) SS1111(0.5)(0.5) SS2222(-0.5)(-0.5) SS2222(0.5)(0.5) SS1212(-0.5)(-0.5) SS1212(0.5)(0.5) SS1313(0)(0) SS2323(0)(0) SS3333(0.5)(0.5)
33 -2-2..64386438 3.26823.2682 -3.9177-3.9177 4.51674.5167 2.33082.3308 -2.3955-2.3955 1.81741.8174 1.35391.3539 1.01701.0170
44 -2.6531-2.6531 3.27873.2787 -3.9232-3.9232 4.52274.5227 2.33782.3378 -2.4028-2.4028 1.84841.8484 1.39901.3990 1.00851.0085
55 -2.6525-2.6525 3.27813.2781 -3.9190-3.9190 4.51744.5174 2.33782.3378 -2.4028-2.4028 1.84831.8483 1.39901.3990 1.00011.0001
66 -2-2..65256525 3.27813.2781 -3.9190-3.9190 4.51734.5173 2.33782.3378 -2.4028-2.4028 1.84801.8480 1.39861.3986 1.00001.0000
77 -2-2..65256525 3.27813.2781 -3.9190-3.9190 4.51734.5173 2.33782.3378 -2.4028-2.4028 1.84801.8480 1.39861.3986 1.00001.0000
Pagano Pagano -2.653-2.653 3.2783.278 -3.919-3.919 44..517517 2.3382.338 -2-2.4.40303 11..8585 11..399399 1.00001.0000
9
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 ())
10
� �
IInn UU33(0)(0) SS1111(0.5)(0.5) SS2222(0.5)(0.5) SS1212(0.5)(0.5) SS1313(0.145)(0.145) SS2323(0.125)(0.125) SS1313(-0.280)(-0.280) SS2323(-0.280)(-0.280)
33 1.65991.6599 6.21136.2113 1.09391.0939 1.37941.3794 2.92332.9233 0.923250.92325 0.644200.64420 0.295460.29546
55 11.7057.7057 6.64416.6441 1.13541.1354 1.48831.4883 33..13313333 0.955590.95559 0.857780.85778 0.404600.40460
77 11..70597059 66.6.6448448 11..13501350 1.1.48848899 33..14531453 0.960510.96051 0.840540.84054 0.398720.39872
99 1.70591.7059 6.64476.6447 1.13501.1350 1.48861.4886 3.14473.1447 0.960370.96037 0.840910.84091 0.398830.39883
1111 1.70591.7059 6.64476.6447 1.13501.1350 1.48861.4886 3.14473.1447 0.960370.96037 0.840910.84091 0.398830.39883
3D Savoia3D Savoia 11..77005959 66.6.64545 11..135135 1.1.489489 3.3.145145 0.0.960960 0.0.841841 0.0.399399
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
Analytical solutionAnalytical solutionAnalytical solutionAnalytical solutionba
ppba
pp N 210
][3
210
]0[3 sinsin
21
,sinsin21
bau
bauu
bau
bauu nnnnnn inininininin 21)(
2021)(
20)(
221)(
1021)(
10)(
1 sincos~cossin,cossin~sincos
bau
bauu nnn ininin 21)(
3021)(
30)(
3 coscos~sinsin
hzapzbhSapzahS
pzbaSapzahSapzbhS
apzbahSapzbauhEU
/,/),2/,0(10~
,/),0,2/(10
/),2/,2/(,/),0,2/(10~
,/),2/,0(10
/),2/,2/(10,/),2/,2/(100
30232302323
033330131301313
20
2403
3T3
11
� �
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 �
IInn UU33(0)(0) SS1111(0.5)(0.5) SS2222(0.5)(0.5) SS1212(0.5)(0.5) SS1313(0.175)(0.175) SS2323(0.145)(0.145) SS1313(-0.270)(-0.270) SS2323(-0.270)(-0.270)
33 2.2.41644164 9.01909.0190 0.873250.87325 1.89271.8927 3.46293.4629 0.336570.33657 0.344990.34499 0.427010.42701
55 2.4902.49022 9.5809.58011 0.9110.9117878 2.0182.01822 3.93.9911911 0.360.36344344 0.40.465676567 0.590.59813813
77 2.49032.4903 9.58049.5804 0.911420.91142 2.01842.0184 3.98773.9877 0.364760.36476 0.456410.45641 0.591100.59110
99 2.49032.4903 9.58049.5804 0.91140.911400 2.01842.0184 3.9873.98755 0.36470.364700 0.4560.4566060 0.5910.5912121
1111 2.49032.4903 9.58049.5804 0.91140.911400 2.01842.0184 3.9873.98755 0.36470.364700 0.4560.4566060 0.5910.5912121
3D Savoia3D Savoia 22..49034903 9.5819.581 0.9110.911 2.0182.018 3.9883.988 0.3650.365 0.4570.457 0.5910.591
12
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
21)(
30)(
3
21)(
20)(
221)(
10)(
1
4cossin
4sinsin,4coscos
Luu
Luu
Luu
nn
nnnn
inin
inininin
hzRpzLuhEU
pzLSRpzLhS
RpzhSRpzhS
RpzLhSRpzLhS
/,/),0,2/(
/),0,2/(,/),8/,2/(10
/),0,0(100,/),8/,0(100
/),0,2/(,/),0,2/(10
34
033
L3
0333302323
013132
0122
12
2022
222
2011
211
IInn UU33(0)(0) SS1111(-(-00..55)) SS1111((00..55)) SS2222(-0.5)(-0.5) SS2222(0.5)(0.5) SS1212(-(-00.5).5) SS1212(0.5)(0.5) SS1313(-0.25)(-0.25) SS2323(0.25)(0.25) SS3333(0.25)(0.25)
33 11..33603360 -2.5808-2.5808 0.198340.19834 -0.28358-0.28358 0.750760.75076 -4.6345-4.6345 2.47082.4708 3.66263.6626 -2.4146-2.4146 -0.31793-0.31793
55 1.401.402626 -2-2..67016701 0.239110.23911 -0.30457-0.30457 0.970320.97032 -5.0167-5.0167 2.68132.6813 4.82184.8218 -3.0165-3.0165 -0.31763-0.31763
77 11..40344034 -2.6600-2.6600 0.249610.24961 -0.30363-0.30363 0.977340.97734 -5.0159-5.0159 2.68502.6850 4.78544.7854 -2.9234-2.9234 -0.31236-0.31236
99 11..44034034 -2.6596-2.6596 0.251090.25109 -0.30359-0.30359 0.977540.97754 -5.0159-5.0159 2.68502.6850 4.78584.7858 -2.9311-2.9311 -0.31296-0.31296
1111 11..40344034 -2.6595-2.6595 0.251110.25111 -0.30359-0.30359 0.977540.97754 -5.0159-5.0159 2.68502.6850 4.78584.7858 -2.9307-2.9307 -0.31292-0.31292
1313 11..40344034 -2.6595-2.6595 0.251110.25111 -0.30359-0.30359 0.977540.97754 -5.0159-5.0159 2.68502.6850 4.78584.7858 -2.9307-2.9307 -0.31292-0.31292
VaradanVaradan 1.40341.4034 -2.660-2.660 0.25110.2511 -0.3036-0.3036 0.97750.9775 -5.016-5.016 2.6852.685 4.7864.786 -2.931-2.931 -0.31-0.31
13
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 ())
14
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
15
Thanks for your attention!Thanks for your attention!Thanks for your attention!Thanks for your attention!