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A New Concept of Sampling A New Concept of Sampling Surfaces in Shell TheorySurfaces in Shell Theory

S.V. PlotnikovaS.V. Plotnikova andand G.M. KulikovG.M. Kulikov

Speaker: Professor Gennady M. KulikovSpeaker: Professor Gennady M. Kulikov

Department of Applied Mathematics & MechanicsDepartment of Applied Mathematics & Mechanics

A New Concept of Sampling A New Concept of Sampling Surfaces in Shell TheorySurfaces in Shell Theory

S.V. PlotnikovaS.V. Plotnikova andand G.M. KulikovG.M. Kulikov

Speaker: Professor Gennady M. KulikovSpeaker: Professor Gennady M. 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

Figure 1. Geometry of the shellFigure 1. Geometry of the shellFigure 1. Geometry of the shellFigure 1. Geometry of the shell

33 eer aa ,, A

33 egeRg IIII cA ,,

Base Vectors of Midsurface and S-SurfacesBase Vectors of Midsurface and S-Surfaces Base Vectors of Midsurface and S-SurfacesBase Vectors of Midsurface and S-Surfaces

eeii - orthonormal vectors; - orthonormal vectors; AA, , kk - - Lamé coefficients and principal curvatures of midsurfaceLamé coefficients and principal curvatures of midsurface

cc = 1+k = 1+k33 - shifter tensor at S-surfaces; - shifter tensor at S-surfaces; 33 - transverse coordinates of S-surfaces (- transverse coordinates of S-surfaces (I I = 1, 2, …, = 1, 2, …, N)N)II II II

(1)(1)

(2)(2)

11, , 22, …, , …, NN - sampling surfaces (S-surfaces) - sampling surfaces (S-surfaces)

rr((11, , 22) - position vector of midsurface ) - position vector of midsurface

RR = = rr++33ee33 - position vectors of S-surfaces - position vectors of S-surfaces

I I = 1, 2, …, = 1, 2, …, NN

II II II

3

Kinematic Description of Deformed ShellKinematic Description of Deformed ShellKinematic Description of Deformed ShellKinematic Description of Deformed Shell

Figure 2. Initial and current configurations of the shellFigure 2. Initial and current configurations of the shellFigure 2. Initial and current configurations of the shellFigure 2. Initial and current configurations of the shell

Base Vectors of DeformedBase Vectors of Deformed S-SurfacesS-SurfacesBase Vectors of DeformedBase Vectors of Deformed S-SurfacesS-Surfaces

)(,, ,,,IIIIIIII3333 uegugRg

((11, , 22) - derivatives of 3D displacement vector at S-surfaces () - derivatives of 3D displacement vector at S-surfaces ( I I = 1, 2, …, = 1, 2, …, N)N)I

III uRR

Position Vectors of Deformed S-SurfacesPosition Vectors of Deformed S-SurfacesPosition Vectors of Deformed S-SurfacesPosition Vectors of Deformed S-Surfaces

(3)(3)

(4)(4)

u u ((11, , 22) - displacement vectors of S-surfaces) - displacement vectors of S-surfaces

II = 1, 2, …, = 1, 2, …, NN

I

4

Green-Lagrange Strain Tensor at S-SurfacesGreen-Lagrange Strain Tensor at S-Surfaces

Linearized Strain-Displacement RelationshipsLinearized Strain-Displacement Relationships

Representation for Displacement Vectors in Surface Representation for Displacement Vectors in Surface FrameFrame

Green-Lagrange Strain Tensor at S-SurfacesGreen-Lagrange Strain Tensor at S-Surfaces

Linearized Strain-Displacement RelationshipsLinearized Strain-Displacement Relationships

Representation for Displacement Vectors in Surface Representation for Displacement Vectors in Surface FrameFrame

)( Ij

Ii

Ij

IiI

jIiji

Iij

ccAAgggg

12

333331

2

112

eeue

eueu

IIII

II

II

II

I

cA

cAcA

,,

,,

i

iIi

I

ii

Ii

I u ee ,u

(5)(5)

(6)(6)

(7)(7)

5

Representation for Derivatives of Displacement VectorsRepresentation for Derivatives of Displacement Vectors

Strain ParametersStrain Parameters

StrainStrainss of S- of S-SSurfacesurfaces

Representation for Derivatives of Displacement VectorsRepresentation for Derivatives of Displacement Vectors

Strain ParametersStrain Parameters

StrainStrainss of S- of S-SSurfacesurfaces

i

iIi

I

Aeu,

1

III

IIIIIII

ukuA

uBuA

ukuBuA

,

,, )(,

33

3

1

11

IIII

II

II

II

I

c

cc

333331

2

112

,

Remark 1.Remark 1. Strains (10) exactly represent all rigid-body shell motions in any convected curvilinear Strains (10) exactly represent all rigid-body shell motions in any convected curvilinear coordinate systemcoordinate system

(8)(8)

(9)(9)

(10)(10)

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Displacement Distribution in Thickness DirectionDisplacement Distribution in Thickness Direction

Presentation for Derivatives of 3D Displacement Vector Presentation for Derivatives of 3D Displacement Vector

Strain Distribution in Thickness DirectionStrain Distribution in Thickness Direction

Displacement Distribution in Thickness DirectionDisplacement Distribution in Thickness Direction

Presentation for Derivatives of 3D Displacement Vector Presentation for Derivatives of 3D Displacement Vector

Strain Distribution in Thickness DirectionStrain Distribution in Thickness Direction

Higher-Order Shell TheoryHigher-Order Shell TheoryHigher-Order Shell TheoryHigher-Order Shell Theory

II

J

Ji

IJIi LMuM 33 ,,)(

I

Iij

Iij L

IJJI

JI

I

Ii

Ii

L

uLu

33

33

LL ( (33) -) - Lagrange polynomials of degree Lagrange polynomials of degree N N - 1- 1 ((I I = 1, 2, …, = 1, 2, …, N)N) I

(11)(11)

(12)(12)

(13)(13)

(14)(14)

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Stress ResultantsStress Resultants

Variational EquationVariational Equation

Constitutive EquationsConstitutive Equations

Presentation for Stress ResultantsPresentation for Stress Resultants

Stress ResultantsStress Resultants

Variational EquationVariational Equation

Constitutive EquationsConstitutive Equations

Presentation for Stress ResultantsPresentation for Stress Resultants

2

2321

/

/

h

h

Iij

Iij dccLH

WddAAupccupccH

I ji iii

Nii

NNIij

Iij 2121

112

1121

,

mk

kmijkmij C,

J mk

Jkm

IJijkm

Iij DH

,

2

2321

/

/

h

h

JIijkm

IJijkm dccLLCD

Remark 2.Remark 2. It is possible to carry out exact integration in (19) using the n-point Gaussian quadrature It is possible to carry out exact integration in (19) using the n-point Gaussian quadraturerule with rule with nn = = NN+1+1

ppi i , p, pi i - surface loads acting on bottom and top surfaces- surface loads acting on bottom and top surfaces

(15)(15)

(16)(16)

(17)(17)

(18)(18)

(19)(19)

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Finite Element FormulationFinite Element FormulationFinite Element FormulationFinite Element Formulation

Displacement InterpolationDisplacement Interpolation

Assumed Strain InterpolationAssumed Strain Interpolation

Displacement InterpolationDisplacement Interpolation

Assumed Strain InterpolationAssumed Strain Interpolation

Figure 3. Biunit square in (Figure 3. Biunit square in (11, , 22)-space)-space

mapped into the exact geometry mapped into the exact geometry four-nodefour-nodeshell element in shell element in (x(x11, x, x22, x, x33))-space-space

Figure 3. Biunit square in (Figure 3. Biunit square in (11, , 22)-space)-space

mapped into the exact geometry mapped into the exact geometry four-nodefour-nodeshell element in shell element in (x(x11, x, x22, x, x33))-space-space

rIi

Iir

r

Iirr

Ii uuuNu P~,

4

1

rIij

Iijr

r

Iijrr

Iij N P~,

4

1

NNr r ((11, , 22)) - bilinear shape functions- bilinear shape functions

= (= ( - - cc)/)/ - - normalized coordinates normalized coordinates

(20)(20)

(21)(21)

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VariantVariant UU33(0)(0) SS1111((––0.5)0.5) SS1212((––0.5)0.5) SS1313(0)(0) SS3333((––0.5)0.5)

NN = 3 = 3 5.6105.610 ––2.6832.683 0.8300.830 1.5961.596 ––1.0661.066

NN = 5 = 5 6.0426.042 ––3.0273.027 1.0451.045 2.3062.306 ––1.0131.013

NN = 7 = 7 6.0466.046 ––3.0133.013 1.0451.045 2.2762.276 ––1.0001.000

ExactExact 6.0476.047 ––3.0143.014 1.0461.046 2.2772.277 ––1.0001.000

Numerical ExamplesNumerical ExamplesNumerical ExamplesNumerical Examples1. Square Plate under Sinusoidal Loading1. Square Plate under Sinusoidal Loading1. Square Plate under Sinusoidal Loading1. Square Plate under Sinusoidal Loading

Figure 4. Simply supported square plate Figure 4. Simply supported square plate with a = b =1 , E = 10with a = b =1 , E = 1077 and and = 0.3 = 0.3 Figure 4. Simply supported square plate Figure 4. Simply supported square plate with a = b =1 , E = 10with a = b =1 , E = 1077 and and = 0.3 = 0.3

Table 1. Results for a thick square plate withTable 1. Results for a thick square plate with a / h = a / h = 22 Table 1. Results for a thick square plate withTable 1. Results for a thick square plate with a / h = a / h = 22

hzapzaauEhU

pzaaSapzahS

apzhSapzaahS

/,/),/,/(

/),/,/(,/),/,(

/),,(,/),/,/(

34

033

3

0333301313

2012

212

2011

211

22100

222010

00102210

10

aa / / hh NN = 5 = 5 Exact Vlasov’s solutionExact Vlasov’s solution

UU33(0)(0) SS1111((––0.5)0.5) SS1212((––0.5)0.5) SS1313(0)(0) UU33(0)(0) SS1111((––0.5)0.5) SS1212((––0.5)0.5) SS1313(0)(0)

44 3.6633.663 ––2.1742.174 1.0261.026 2.3692.369 3.6633.663 ––2.1752.175 1.0271.027 2.3622.362

1010 2.9422.942 ––2.0042.004 1.0561.056 2.3842.384 2.9422.942 ––2.0042.004 1.0561.056 2.3832.383

100100 2.8042.804 ––1.9751.975 1.0631.063 2.3872.387 2.8042.804 ––1.9761.976 1.0641.064 2.3872.387

Table 2. Results for thick and thin square plates with five equally located S-surfaces Table 2. Results for thick and thin square plates with five equally located S-surfaces Table 2. Results for thick and thin square plates with five equally located S-surfaces Table 2. Results for thick and thin square plates with five equally located S-surfaces

Figure Figure 55. Distribution of stresses S. Distribution of stresses S1313 and S and S3333 through the plate thickness: through the plate thickness:

Vlasov’s solution ( ) and present higher-order shell theory for N = 7 ( )Vlasov’s solution ( ) and present higher-order shell theory for N = 7 ( )

Figure Figure 55. Distribution of stresses S. Distribution of stresses S1313 and S and S3333 through the plate thickness: through the plate thickness:

Vlasov’s solution ( ) and present higher-order shell theory for N = 7 ( )Vlasov’s solution ( ) and present higher-order shell theory for N = 7 ( )

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2. Cylindrical Composite Shell under Sinusoidal Loading2. Cylindrical Composite Shell under Sinusoidal Loading2. Cylindrical Composite Shell under Sinusoidal Loading2. Cylindrical Composite Shell under Sinusoidal Loading

25010

205025

40210

028210

0010080100

021002100

6

34

033

3

0333302323

013132

0122

12

2022

222

2011

211

.,

.,.,

/,/,/),,/(

/),,/(,/),/,/(

/),,(,/),/,(

/),,/(,/),,/(

TTLTT

TTTTLTTL

L

E

EGEGEE

RLhzRpzLuhEU

pzLSRpzLhS

RpzhSRpzhS

RpzLhSRpzLhS

Figure 6. Simply supported cylindrical Figure 6. Simply supported cylindrical composite shell (modeled by 32composite shell (modeled by 32128 128 mesh) mesh)

Figure 6. Simply supported cylindrical Figure 6. Simply supported cylindrical composite shell (modeled by 32composite shell (modeled by 32128 128 mesh) mesh)

Table 3. Results for a thick cylindrical shell with R / h = 2Table 3. Results for a thick cylindrical shell with R / h = 2Table 3. Results for a thick cylindrical shell with R / h = 2Table 3. Results for a thick cylindrical shell with R / h = 2

VariantVariant UU33(0)(0) SS1111(0.5)(0.5) SS2222(0.5)(0.5) SS1212((––0.5)0.5) SS1313(0)(0) SS2323(0)(0) SS3333(0)(0)

NN = 3 = 3 6.6936.693 1.1511.151 1.4331.433 ––0.9620.962 0.9930.993 ––1.6741.674 ––0.42160.4216

NN = 5 = 5 7.2487.248 0.9360.936 4.4104.410 ––1.5821.582 1.5081.508 ––2.1232.123 ––0.37620.3762

NN = 7 = 7 7.4667.466 1.2011.201 5.0615.061 ––1.7291.729 1.4951.495 ––1.9811.981 ––0.36490.3649

NN = 9 = 9 7.4977.497 1.3531.353 5.1625.162 ––1.7551.755 1.4971.497 ––2.0632.063 ––0.37550.3755

ExactExact 7.5037.503 1.3321.332 5.1635.163 ––1.7611.761 1.504 1.504 ––2.056 2.056 ––0.370.37

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RR / / hh NN = 7 = 7 Exact Varadan-Bhaskar’s solutionExact Varadan-Bhaskar’s solution

UU33(0)(0) SS2222(0.5)(0.5) SS1313(0)(0) SS2323(0)(0) UU33(0)(0) SS2222(0.5)(0.5) SS1313(0)(0) SS2323(0)(0)

44 2.7822.782 4.8544.854 0.98630.9863 ––2.9702.970 2.7832.783 4.8594.859 0.9870.987 ––2.9902.990

1010 0.91880.9188 4.0484.048 0.51990.5199 ––3.6653.665 0.91890.9189 4.0514.051 0.5200.520 ––3.6693.669

100100 0.51690.5169 3.8403.840 0.39270.3927 ––3.8563.856 0.51700.5170 3.8433.843 0.3930.393 ––3.8593.859

Table 4. Results for thick and thin cylindrical shells with seven S-surfacesTable 4. Results for thick and thin cylindrical shells with seven S-surfacesTable 4. Results for thick and thin cylindrical shells with seven S-surfacesTable 4. Results for thick and thin cylindrical shells with seven S-surfaces

Figure 7. Distribution of stresses SFigure 7. Distribution of stresses S3333 through the shell thickness: through the shell thickness:

exact solution ( ) and present higher-order shell theory for N = 7 ( )exact solution ( ) and present higher-order shell theory for N = 7 ( )

Figure 7. Distribution of stresses SFigure 7. Distribution of stresses S3333 through the shell thickness: through the shell thickness:

exact solution ( ) and present higher-order shell theory for N = 7 ( )exact solution ( ) and present higher-order shell theory for N = 7 ( )

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VariantVariant UU33(0)(0) SS1111((––0.5)0.5) SS1111(0.5)(0.5) SS3333((––0.5)0.5) SS3333(0)(0)

NN = 3 = 3 2.2812.281 5.3455.345 2.4382.438 ––0.58820.5882 ––0.37590.3759

NN = 5 = 5 2.3002.300 4.6164.616 2.0872.087 ––0.97700.9770 ––0.25760.2576

NN = 7 = 7 2.3002.300 4.5724.572 2.0662.066 ––0.99780.9978 ––0.26260.2626

ExactExact 2.3002.300 4.5664.566 2.0662.066 ––1.0001.000 ––0.26260.2626

Table 5. Results for a thick spherical shell with R / h = 2Table 5. Results for a thick spherical shell with R / h = 2Table 5. Results for a thick spherical shell with R / h = 2Table 5. Results for a thick spherical shell with R / h = 2

33. Spherical Shell under Inner Pressure. Spherical Shell under Inner Pressure33. Spherical Shell under Inner Pressure. Spherical Shell under Inner Pressure

hzRpzEhuU

pzSRpzhS

/,/),,(

/),,(,/),,(

32

033

0333301111

0010

000010

Figure 8. Spherical shell under inner pressure with R = 10, Figure 8. Spherical shell under inner pressure with R = 10, = 89.98= 89.98, E = 10, E = 1077 and and = 0.3 = 0.3

(modeled by 64(modeled by 641 mesh)1 mesh)

Figure 8. Spherical shell under inner pressure with R = 10, Figure 8. Spherical shell under inner pressure with R = 10, = 89.98= 89.98, E = 10, E = 1077 and and = 0.3 = 0.3

(modeled by 64(modeled by 641 mesh)1 mesh)

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RR / / hh NN = 7 = 7 Exact LamExact Laméé’s solution’s solution

UU33(0)(0) SS1111((––0.5)0.5) SS1111(0.5)(0.5) SS3333(0)(0) UU33(0)(0) SS1111((––0.5)0.5) SS1111(0.5)(0.5) SS3333(0)(0)

44 2.9452.945 4.5834.583 3.3323.332 ––0.37660.3766 2.9452.945 4.5824.582 3.3323.332 ––0.37660.3766

1010 3.2913.291 4.7844.784 4.2844.284 ––0.45010.4501 3.2913.291 4.7834.783 4.2824.282 ––0.45010.4501

100100 3.4803.480 4.9764.976 4.9264.926 ––0.49500.4950 3.4803.480 4.9754.975 4.9254.925 ––0.49500.4950

Table 6. Results for thick and thin spherical shells with seven S-surfacesTable 6. Results for thick and thin spherical shells with seven S-surfacesTable 6. Results for thick and thin spherical shells with seven S-surfacesTable 6. Results for thick and thin spherical shells with seven S-surfaces

Figure 9. Distribution of stresses SFigure 9. Distribution of stresses S1111 and S and S3333 through the shell thickness: through the shell thickness:

LamLaméé’s solution ( ) and present higher-order shell theory for N = 7 ( )’s solution ( ) and present higher-order shell theory for N = 7 ( )

Figure 9. Distribution of stresses SFigure 9. Distribution of stresses S1111 and S and S3333 through the shell thickness: through the shell thickness:

LamLaméé’s solution ( ) and present higher-order shell theory for N = 7 ( )’s solution ( ) and present higher-order shell theory for N = 7 ( )

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ConclusionsConclusionsConclusionsConclusions

A simple and efficient concept of S-surfaces inside the shell body has A simple and efficient concept of S-surfaces inside the shell body has been proposed. This concept permits the use of 3D constitutive been proposed. This concept permits the use of 3D constitutive equations and leads for the sufficient number of S-surfaces to the equations and leads for the sufficient number of S-surfaces to the numerically exact solutions of 3D elasticity problems for thick and thin numerically exact solutions of 3D elasticity problems for thick and thin shellsshells

A new higher-order theory of shells has been developed which permits A new higher-order theory of shells has been developed which permits the use, in contrast with a classic shell theory, only displacement the use, in contrast with a classic shell theory, only displacement degrees of freedomdegrees of freedom

A robust exact geometry four-node solid-shell element has been built A robust exact geometry four-node solid-shell element has been built which allows the solution of 3D elasticity problems for thick and thin which allows the solution of 3D elasticity problems for thick and thin shells of arbitrary geometryshells of arbitrary geometry

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Thanks for your attention!Thanks for your attention!Thanks for your attention!Thanks for your attention!