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!