Research Article Approximation of Linear Elastic Shells by ... · today s availability of greatly...

Research ArticleApproximation of Linear Elastic Shells by Curved TriangularFinite Elements Based on Elastic Thick Shells Theory

Joseph Nkongho Anyi123 Robert Nzengwa13

Jean Chills Amba3 and Claude Valery Abbe Ngayihi13

1Department of Mechanical Engineering National Advanced School Polytechnics University of Yaounde IPO Box 8390 Yaounde Cameroon2Department of Mechanical Engineering Higher Technical Teachers Training College University of BueaPO Box 249 Buea Road Kumba Cameroon3Laboratory E3M Faculty of Industrial Engineering University of Douala PO Box 2107 Douala Cameroon

Correspondence should be addressed to Joseph Nkongho Anyi nkonghojosephgmailcom

Received 1 June 2016 Accepted 10 July 2016

Academic Editor Francesco Tornabene

Copyright copy 2016 Joseph Nkongho Anyi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We have developed a curved finite element for a cylindrical thick shell based on the thick shell equations established in 1999by Nzengwa and Tagne (N-T) The displacement field of the shell is interpolated from nodal displacements only and strainsassumption Numerical results on a cylindrical thin shell are compared with those of other well-known benchmarks withsatisfaction Convergence is rapidly obtained with very few elements A scaling was processed on the cylindrical thin shell byincreasing the ratio 120594 = ℎ2119877 (half the thickness over the smallest radius in absolute value) and comparing results with thoseobtained with the classical Kirchhoff-Love thin shell theory it appears that results diverge at 2120594 = radic110 = 0316 because of thesignificant energy contribution of the change of the third fundamental form found in N-T model This limit value of the thicknessratio which characterizes the limit between thin and thick cylindrical shells differs from the ratio 04 proposed by Leissa and 05proposed by Narita and Leissa

1 Introduction

In the field of structural mechanics the word shell refers to aspatial curved structural member The very high structuraland architectural potential of shell structures is used invarious fields of civil architectural mechanical aeronauticaland marine engineering The strength of the double-curvedstructure is efficiently and economically used for example tocover large areas without supporting columns [1] In additionto the mechanical advantages the use of shell structuresleads to aesthetic architectural appearance [2 3] Examplesof shells used in civil and architectural engineering are shellroofs liquid storage tanks silos cooling towers containmentshells of nuclear power plants and arch dams Piping systemscurved panels pressure vessels bottles buckets and parts ofcars are examples of shells used in mechanical engineering

[4] In aeronautical and marine engineering shells are usedin aircrafts space crafts missiles ships and submarinesBecause of the spatial shape of the structure the behavior ofshell structures is different from the behavior of beam andplate structures [4]

The considerable effort in the development of rigorousshell theories dates back to the early twentieth century [5ndash7] These shell theories reduce a basically three-dimensionalproblem to a two-dimensional one Nevertheless the analysisof shells with the aid of such theories involves complicateddifferential equations which either cannot be solved at all [8ndash10] or whose solution requires the use of high-level math-ematics unfamiliar to structural engineers Therefore manyapproximate shell theories have been developed mainly onthe assumption that the shell is thin and to obtain genericanalysis tools obviously some accuracy had to be traded for

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8936075 12 pageshttpdxdoiorg10115520168936075

2 Mathematical Problems in Engineering

convenience and simplicity [4 11] Hence it is not surprisingthat the development of the numerical formulations sincethe 1950s has led to a gradual cessation of attempts to findclosed-form solutions to rigorous formulations [4] But withtodayrsquos availability of greatly increased computing power(also since the mid-twentieth century) completeness ratherthan simplicity is given more emphasis

In computationalmechanics many numerical techniquesare developed to solve governing equations of shells We canidentify for this purpose the generalized differential quadra-ture finite elements (DGQFEM) [11ndash13] discrete singularconvolution method (DSC) [14] the finite difference (FD)the discrete volume finite element (DVFEM) and the finiteelements method (FEM) under which solid-shells finite ele-ments (SSFEM) and mixt finite element (MFEM) are found[15ndash18] Our approach is found under the MFEM category inwhich three-dimensional shell parameters are reduced to themidsurface of the shell of thickness ℎ Some finite elements inthis category describewell the behavior of shells under severalloads All these models of finite elements are link with eitherKirchhoff-Love (K-L) or Reissner-Mindlin (R-M) hypothesiswhich neglected the effect of curvature in elastic stiffness[11 19] This is one of the main reasons why K-L and R-Mtheories of shells are incomplete and unsuitable to handlevarious types of shells [20] Structure engineers emphasismore on simplifying assumptions than limit analysis Becauseof the third fundamental form that has been neglected intheir theoretical approach of shells governing equations [21]and severe assumption on shell thickness somemathematicalterms without evident mechanic signification are introducedin the stiffness matrix in order to improve their efficiency[6 22]

After a brief presentation of Nzengwa and Tagne (N-T) kinematic equations of elastic thick shells we develop acurved cylindrical 3-node finite element based on both K-L and N-Trsquos shell models using strain tensor interpolationassumption [23] Next through numerical implementationon classical benchmark convergence of N-Trsquos shell modelis investigated The deviation and the limit thickness ratiobetween the two theoretical approaches of thin shells andthick shells are established and discussed by applying succes-sive scaling of a sample of thin shell

2 Materials and Methods

21 N-Trsquos Two-Dimensional ElasticThick ShellsModel TheN-Trsquos two-dimensional model for linear elastic thick shells hasbeen deduced from the three-dimensional problem withoutany ad hoc assumption whether of geometrical or mechan-ical nature The two-dimensional equations are deducedby applying asymptotic analysis on a family of variationalequations obtained from an abstract scaled shell throughmultiple scaling of the initial three-dimensional equationsThe theoretical elastic thick shells governing equations andlimits displacements obtained from this approach (at the limitanalysis) are more general because they contain additionalterms to those found in Kirchhoff-Love model The modelalso completes the thick shells theory of Reissner-Mindlin[24]

For a three-dimensional shell the following bases (1198921 11989221198923) (119892

1 119892

2 119892

3) are dual bases and (119886

1 119886

2 119886

3) (119886

1 119886

2 119886

3)

defined on the midsurface also constitute dual bases ofshell if 120594 lt 1 Let 119880

119894(119909

1 119909

2 119911) be the three-dimensional

displacement field vector it is defined as follows in the localcoordinate system

119880 = 119880120572119892120572+ 119880

31198923= 119906

120572119886120572+ 119906

31198863

119906120588= 119906

120588(119909

1 119909

2 119911)

1199063= 119906

3(119909

1 119909

2 119911)

(1)

The strain tensor for a three-dimensional shell reads

120576120572120573(119880) =

1

2(119880

120572120573+ 119880

120573120572) = 120576

120572120573(119906)

=[120583

]120572(nabla

120573119906] minus 1198871205721205731199063) + 120583

]120573(nabla

120572119906] minus 119887]1205731199063)]

2

(2)

1205761205723(119880) =

1

2(119880

1205723+ 119880

3120572) = 120576

1205723(119906)

=[120583

]120572119906]3 + (1199063120572 + 119887

]120572119906])]

2

(3)

12057633(119880) = 119880

33= 119880

33= 119906

33= 120576

33(119906) (4)

From the limit analysis N-T demonstrated that the displace-ments field satisfies the equation 120576

1198943(119880) = 0 and the unique

solution is

119906120588(119909

1 119909

2 119911) = 120583

120574

120588119906120574(119909

1 119909

2) minus 119911120597

120588119908

119882 = 119908 = 1199063

119880120572(119909

1 119909

2 119911) = 120583

120588

120572119906120588

120583120574

120588= 120575

120574

120588minus 119911119887

120574

120588

119880120572(119909

1 119909

2 119911) = 119906

120572(119909

1 119909

2) minus 119911120579

120572(119909

1 119909

2)

+ 1199112120595120572(119909

1 119909

2)

(5)

where (119906120572) are local membrane displacement components

and 119908 = 1199063the transvers displacement

120579120572= (2119887

120591

120572119906120591+ nabla

120572119908)

120595120572= 119887

120574

120572119887120591

120574119906120591+ 119887

120574

120572nabla120574119908

(6)

The general shell deformation tensor derived from the kine-matics defined above is

120576120572120573(119880) = 119890

120572120573(119906) minus 119911119896

120572120573(119906) + 119911

2119876120572120573(119906) (7)

1205761198943(119880) = 0 (8)

where 119890120572120573

is the membrane strain tensor 119896120572120573

is the bendingstrain tensor and 119876

120572120573is the tensor of Gauss curvature

Mathematical Problems in Engineering 3

1

2

3

a1 a3

a3

a2a1

a2

X

C

OR B Y

L

A

Z

M

D

120579

Figure 1 Triangular shell element

The strain tensor contains the change of the third funda-mental form

119876120572120573=1

2(119888120572120573minus 119862

120572120573)

119888 = 119888120572120573= 119887

120583

120572119887120583120573= 119887

120572120583119887120583

120573

119862120572120573= 119861

120583

120572119861120583120573= 119861

120572120583119861120583

120573

(9)

In Kirchhoff-Love thin shells theory the shell is assumedto be thin Following that assumption the three-dimensionaldisplacements field vector 119880

120572(119909

1 119909

2 119911) is taken equal to the

midsurface displacements field

119880120572(119909

1 119909

2 119911) = 119906

120572(119909

1 119909

2) minus 119911nabla

120572119908(119909

1 119909

2)

119882 (1199091 119909

2 119911) = 119908 (119909

1 119909

2) = 119906

3

(10)

It should be noted that the terms120595120572= (119887

120591

120572119887120588

120591119906120588+119887

120591

120572nabla120591119908) in the

current model of thick shells disappear in the classical theoryof thin shells because it is proportional to 1205942 and consideredsmall The thin shell theory with the above assumptioncannot perform good shells behavior because the thicknesshas a considerable influence on the shells behavior The thickshell model used in this survey appears to be suitable tohandle both thin and thick shells because of its completeness

22 Curved Shell Elements Curved shell elements are suitableto model the midsurface geometry more accurately In thecase of certain surfaces such as cylindrical shells it means theexact description of the original surface For more compli-cated cases similar to the displacement field the curvaturesof the surface are approximated by interpolation functions Inthis respect such elements belong to the parametric elementtypes [25]

23 Triangular Curve Shell Element The triangular shellelement is described in Figure 1 it can be double-curved (egspherical structures) or single-curved element (cylindrical

structures) In accordance with the N-Trsquos equations of thetheory of elastic thick shells the geometrical properties of thecylindrical shell are the following

997888997888997888rarr119874119872 =

119883 = 119877 cos120593

119884 = 119877 sin120593

119885 = 119909

(11)

where 119909 and 120593 are curvilinear coordinates 0 le 119909 le 119871 and 0 le120593 le 120579119877 equiv cylinder radius and119883119884 119885 are global coordinates

24 Displacement Field Let 119880120572be the global displacement

field vector defined as follows

119880120572= 119906

120572minus 119911120579

120572+ 119911

2120595120572

119882 = 119908

(12)

With respect to the cylinderrsquos geometry the angles of rota-tions and the Gaussian curvatures are developed in functionof membrane displacements 119906

1and 119906

2and the derivatives

of the transverse displacement 119908120593 119908

119909including curvature

components 119887120591120572

1205791=2119906

1

1198771

minus119908119909

1198601

1205792=2119906

2

1198772

minus119908120593

1198602

120595120572= (119887

120591

120572119887120588

120591119906120588+ 119887

120591

120572nabla120591119908)

(13)

which leads to

1205951= minus

1

1198771

(1199061

1198771

minus1

1198601

119908119909) =

1

1198771

(119887120591

1119906120591+ 120579

1)

1205952= minus

1

1198772

(1199062

1198772

minus1

1198602

119908120593) =

1

1198772

(119887120591

2119906120591+ 120579

2)

(14)


where 1198601 119860

2 119877

1 and 119877

2are geometrical properties of the

midsurface of a shell within the respective coordinate-lines 119909and 120593

In the case of a cylindrical shell they are defined as follows[25]

1198601=10038161003816100381610038161003816

997888rarr1198861

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

997888997888997888rarr120597119872

120597119909

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 1 1198771= infin

1198602=10038161003816100381610038161003816

997888rarr1198862

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

997888997888997888rarr120597119872

120597120593

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119877 1198772= 119877

(15)

The third component of rotation is added for a smoothtransfer of element stiffness matrix from one system ofcoordinates to another

25 Compatibility We recall the strain components in thickshell theory found in (3) as follows

119890120572120573=1

2(nabla

120572V + nabla

120573119906) minus 119887

120572120573119908

119896120572120573= nabla

120572119887120588

120573119906120588+ 119887

120588

120572nabla120573119906120588+ 119887

120588

120573nabla120572119906120588+ nabla

120572nabla120573119908

minus 119887120588

120572119887120588120573119908

119876120572120573= 05 (119887

120583

120572nabla120573119887120588

120583119906120588+ 119887

120583

120572119887120588

120583nabla120573119906120588+ 119887

120583

120573119887120588

120583nabla120572119906120588

+ 119887120583

120573nabla120572119887120588

120583119906120588+ 119887

120583

120572nabla120573nabla120583119908 + 119887

120583

120573nabla120572nabla120583119908)

(16)

The membrane deformation tensor 119890120572120573

and bendingtensor 119896

120572120573are widely used in thin shell theory But the

tensor 119876120572120573

is neglected [24] In classical thin shell andthick shell computation analysis the contribution of thistensor (Gauss curvature tensor) to the deformation energydisappears in the stiffness matrix As the shell becomesthicker the contribution of Gauss curvature tensor in termsof energy is nomore negligible as compared to that of the firsttwo tensors used in classical thin shells So this model is ableto handle thick shells as well as the classical R-M models

We express the strain components using the parametersof the cylindrical shellrsquos midsurface In detail for the case ofa cylindrical shell we have

119890119909= 119906

1119909

119890120593=1

119877(119906

2120593+ 119908)

2119890119909120593=1

1198771199061120593+ 119906

2119909

119896119909= 119908

119909119909

119896120593= minus

1

1198772(2119906

2120593minus 119908

120593120593+ 119908)

2119896119909120593= minus

2

119877(minus119908

119909120593+ 119906

2119909)

119876119909= 0

119876120593= minus

1

1198773(119906

2120593minus 119908

120593120593)

2119876119909120593= minus

1

1198772(119906

2119909minus 119908

119909120593)

(17)

26 Interpolation Theapproachwe are using here is based onstrain interpolationThis method consists in solving a systemof differential equations from strain assumption We have tofind the displacements functions which perfectly capture therigid-like motion then particular displacements are calcu-lated from approximation of the deformations componentsThese components are approximated such that membraneshearing and bending behavior are decoupled So a purebending state or a pure shearing state can be well representedIn this section we consider 119906

1= 119906 and 119906

2= V

The rigid-body-like motion is defined for 120576120572120573

= 0 Itmeans that 119890

120572120573= 0 119896

120572120573= 0 and 119876

120572120573= 0

119906 (119909 120593) = 119906 (120593) (18)

Let119892(120593) be an arbitrary function that can satisfy the followingequations

V (119909 120593) = minus1

1198771199061015840(120593) sdot 119909 + 119892 (120593)

119908 (119909 120593) = +1

11987711990610158401015840(120593) sdot 119909 minus 119892

1015840(120593)

(19)

119906(3)(120593) + 119906

1015840(120593) = 0 (20)

119906(4)(120593) + 119906

(2)(120593) = 0 (21)

119892(3)(120593) + 119892

(1)(120593) = 0 (22)

Since our displacement field is of six components threerotations and three translations we need six independentparameters to define our rigid body-like motion [18 25]

119906 (120593) = 1198861+ 119886

2119877 (cos120593 minus cos 120579) minus 119886

3119877 sin120593 (23)

119892 (120593) = 1198864119877 (cos120593 cos 120579 minus 1) minus 119886

5sin120593 + 119886

6cos120593 (24)

When (23) and (24) are taken into (18) and (19) it comes thatthe rigid body-like motion is well handled by the followingdisplacement components

1199060= 119886

1+ 119886


3119877 sin120593

V0= 119886

2119909 sin120593 + 119886

3119909 cos120593 + 119886

4119877 (cos120593 cos 120579 minus 1)

minus 1198865sin120593 + 119886

6cos120593

1199080= minus119886

2119909 cos120593 + 119886

3119909 sin120593 + 119886

4119877 sin120593 cos 120579

+ 1198865cos120593 + 119886

6sin120593

(25)


The displacements found in (25) are expressed in matrixform

1198800= 119875

01198600 (26)

where 119880119905

0= [1199060 V

01199080]

119905

sdot1198600= [1198861 1198862 1198863 1198864 1198865 1198866] (27)

1198600is the vector of unknown coefficient which captures the

rigid body-like motion

1198750= (

1 119877 (cos120593 minus cos 120579) minus119877 sin120593 0 0 0

0 119909 sin120593 119909 cos120593 119877 (cos120593 cos120593 minus 1) minus sin120593 cos1205930 minus119909 cos120593 119909 sin120593 119877 sin120593 cos 120579 cos120593 sin120593

) (28)

1198750is the matrix of interpolation functionsThe general solution displacements field 119880 is the sum of

the displacements field of rigid body-like motion and that ofa particular solution of the deformation 119880

119889

119880 = 1198800+ 119880

119889=

119905

sdot[119906 V 119908] (29)

where 119905

sdot119880119889= [119906

119889V119889119908119889]

The strain components are interpolated in such a waythat interference between shear strains and bending is notpossible The following displacement components satisfystrains defined above

119906119889= 119886

7119877119909 + 119886

8119877120593 + 119886

9119877119909120593

V119889= 119886

10120593 + 119886

11119909120593

119908119889= 119886

121199092+ 119886

13119909120593 + 119886

141205932+ 119886

151199093+ 119886

161199092120593

+ 11988617119909120593

2+ 119886

181205933

119880119889= 119875

119889119860119889

(30)

with119905

sdot119860119889= [1198867 1198868 1198869 11988610 11988611 11988612 11988613 11988614 11988615 11988616 11988617 11988618]

119875119889

= (

119877119909 119877120593 119877119909120593 0 0 0 0 0 0 0 0 0

0 0 0 120593 119909120593 0 0 0 0 0 0 0

0 0 0 0 0 1199092119909120593 120593

211990931199092120593 119909120593

21205933

)

(31)

119875119889is the interpolation functions matrix related to the partic-

ular solution of displacements fieldFrom the total displacement vector field expression

above we have 18 unknowns It then requires 18 displacementparameters to find all the above unknowns (see Figure 2)

So the total displacement vector over the triangularelement is

119905

sdot119890= [

1V111199091120593112V231199092120593223V33119909312059333] (32)

In the local (curvilinear) systemof coordinates each node hassix (06) components of displacement

27 Stiffness Matrix In order to calculate the stiffness matrixwe have to formulate the variational problem over a domainLet 119878 be the border of the domain then let 120597119878 = 120574

0cup 120574

1be

partitioned in two and the border of the shell 120597Ω = Γ0cup Γ

1

with Γ0= 120574

0timesminusℎ2 ℎ2 and Γ

1= 120574

1timesminusℎ2 ℎ2cupΓ

minuscupΓ

+

We define Γminus= 119878 times minusℎ2 and Γ

+= 119878 times ℎ2 We suppose

here that the shell is clamped on Γ0and loaded by volume

and surface forces as stated above the three-dimensionalvariational equation related to the equilibrium is

find 119880 isin 1198681198671

Γ0

int

sdot

Ω

119879

(119880) (119881) 119889Ω = int

sdot

Ω

119891 sdot 119881119889Ω + int

sdot

120597Ω

119892 sdot 119881 119889D

= 119871 (V) for 119881 isin 1198681198671

Γ0

(33)

119891 is volume forces in the domainΩ and 119892 is surface forces inthe domain 120597Ω

The constitutive law of the linear elastic homogenousmaterial is

= 119888 (34)

where 119888 = 119862119894119895119896119897 = 120582119892119894119895119892119896119897 + 120583(119892119894119896119892119895119897 + 119892119894119897119892119895119896) 120582 and 120583 areLame coefficients which depend on intrinsic properties ofmaterials

Replacing (34) in (33) the problem is stated as follows

find 119906 isin 1198681198671

Γ0

int

sdot

Ω

[120582119892119894119895119892119896119897+ 120583 (119892

119894119896119892119895119897+ 119892

119894119897119892119895119896)] (119906) (V) 119889Ω

= int

sdot

Ω

119891 sdot V 119889Ω + intsdot

120597Ω

119892 sdot V 119889D = 119871 (V)

for V isin 1198681198671

Γ0

(35)


1

2

3

u

w

1205791

1205792

120574

Figure 2 Triangular single-curved element

The strain tensor is defined at (7) if substitution is done inthe left side of the above equation it can now be written

119860 (119906 V) = intsdot

Ω

[120582119892120572120573119892120574120588+ 120583 (119892

120572120574119892120573120588+ 119892

120572120588119892120573120574)]

sdot (119890120572120573(119906) minus 119911119896

120572120573(119906) + 119911

2119876120572120573(119906))

(119890120574120588(V) minus 119911119896

120574120588(V) + 1199112119876

120574120588(V)) 119889Ω = 119871 (V)

(36)

Naturally 1205761198943(119880) = 0 at the limit analysis Then at the midsur-

face 119892120572120573 = (120583minus1)120572120588(120583

minus1)120573

120582119886120588120582 By using Taylorrsquos expansion on

(120583minus1)120572

120588

(120583minus1)120572

120588= 120575

120572

120588+ 119911119887

120572

120588+

infin

sum

119899ge2

119911119899(119887

119899)120572

120588(37)

and truncating at 119899 = 1 we obtain the best first-order two-dimensional

1198601(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573]119876

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

5

80 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573]119876

120572120573(V) 119889119878 = 119871 (V)

(38)

Remark 1 For Taylorrsquos expansion used above if we truncate(37) for 119899 = 0 the bilinear continuous form 119860(119906 V) will takethe value

1198600(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

(39)

which is found in the literature to be K-L shell deformationenergy It is included in 119860

1(119906 V) which is the N-T shell

deformation energy In what follows we compute both 119870N-T119866

and 119870R-M119866

which are respectively stiffness matrix related to1198601(119906 V) and 119860

0(119906 V)

For the computing aim 1198601(119906 V) is transformed

1198601(119906 V) = int

119878

119864119879

V119862119866119864119906119889119878 = int119878

119864119879

V119862119866119864119906 |119869| 119889120585 119889120578 (40)

For this purpose components of strain tensor are expressedas a product of thematrix of interpolation functionsrsquo gradientand that of the total displacement vector over the domainFrom (39) the stress tensor components are deduced infunction of the product above

120576119898= 119890

120572120573= 119863

119898119860 = 119863

119898119875minus1 = 119861

119898 119861

119898= 119863

119898119875minus1

120590119898= 119888120576

119898= 119888119861

119898

(41)

Also the bending stress and strain are expressed

120576119887= 119896

120572120573= 119863

119887119860 = 119863

119887119875minus1 = 119861

119887 119861

119887= 119863

119887119875minus1

120590119887= 119888120576

119887= 119888119861

119887

(42)

Finally as previous the Gauss curvature tensor is expressedas

120576119892= 119876

120572120573= 119863

119892119860 = 119863

119892119875minus1 = 119861

119892 119861

119892= 119863

119892119875minus1

120590119892= 119888120576

119892= 119888119861

119892

(43)


Total displacement is stated as the sum of particular solutionand homogeneous solution of our system of strain differentialequations

119880 = 1198800+ 119880

119889= 119875

01198600+ 119875

119889119860119889= [1198750 119875119889] [

1198600

119860119889

]

= 119873 sdot 119860

(44)

where 119873 = [1198750 119875119889] is the matrix of interpolation functionsand

119880 = 119873 sdot 119860 = 119873 sdot 119875minus1 (45)

The left-hand side of the variational problem reads

1198601(119906 V) =

1

2intΩ

(120576119905

119898119888119905120576119898minus 119911120576

119905

119898119888119905120576119887+ 119911

2120576119905

119898119888119905120576119892

minus 119911120576119905

119887119888119905120576119898+ 119911

2120576119905

119887119888119905120576119887minus 119911

3120576119905

119887119888119905120576119892+ 119911

2120576119905

119892119888119905120576119898

minus 1199113120576119905

119892119888119905120576119887+ 119911

4120576119905

119892119888119905120576119892) 119889Ω = 119871 (V)

(46)

This is written as follows

1198601(119906 V) =

1

2119905

int119878

ℎ119861119905

119898119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119898119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119898119862119905119861119892119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119887119862119905119861119898119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119887119862119905119861119887119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119887119862119905119861119892119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119892119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119892119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ5

80119861119905

119892119862119905119861119892119877119889120593119889119909

= 119871 (V)

(47)

One can deduct from (47) what follows

1198601(119906 V) =

119905

[119870N-T119866]

1198600(119906 V) =

119905

[119870R-M119866]

(48)

where 119870(sdot)

119866is the global stiffness matrix The midsurface is

symmetric in shell thickness that is themidsurface is at+ℎ2from the top surface and at minusℎ2 from the bottom one

119870N-T119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887+ℎ3

12119861119905

119898119862119905119861119892

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

119870R-M119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887)119877119889120593119889119909

(49)

The right-hand side of the best first-order two-dimensionalvariational equation reads

119871 (V) = 119905

intΩ

119905

sdot119875minus1119873

119905119891119889Ω +

119905

int120597Ω

119905


119905119892119889D

= 119905

(119865119881+119872) = 0

(50)

where 119891 and 119892 are defined as in [24] and 119865 is the force vectorfrom distributed load 119865

119881+119872

119865119881= int

sdot

119878

119873119905119891119890119877119889120593119889119909

119872 = int

sdot

119878

119873119905119892119877119889120593119889119909

(51)

By equating1198601(119906 V) to 119871(V) we obtain the structural equilib-

rium equation over the whole area as follows

119870119866 = 119865 (52)

Over a single-curved triangular element of area 119904119890the

elementary stiffness matrix 119870119890and elementary force vectors

119865119890can be locally implemented

119870119890= int

119904119890

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119898119862119905119861119892+ℎ3

12119861119905

119887119862119905119861119887

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

(53)

By regular assembling process of elementary stiffness matrix119870119890and elementary force vectors 119865

119890 we then expressed the

global stiffness matrix and force vector over the whole areaas follows

119870119866=

119873

sum

119890=1

119870119890

119865 =

119873

sum

119890=1

119865119890

(54)

119873 is total number of elements


Table 1 Comparison of transverse displacements at 119861 and 119862

Mesh step 119882119862(cm) reference 0541 119882

119861(cm) reference minus361

DKT18 DKT12 SFE3 CSFE3 DKT18 DKT12 SFE3 CSFE3119873 = 2 0269 0175 0072 0210 minus2893 minus3773 minus2407 minus0470119873 = 4 0330 0330 0292 0060 minus2436 minus2499 minus2541 minus1640119873 = 6 0409 0420 0393 0210 minus2848 minus2882 minus2951 minus2460119873 = 8 0455 0465 0445 0360 minus3106 minus3129 minus3206 minus2940119873 = 10 0482 0489 0475 0460 minus3259 minus3275 minus3363 minus3300119873 = 12 0498 0504 0493 0510 minus3354 minus3365 minus3462 minus3480119873 = 14 0512 0516 0510 0526 minus3438 minus3446 minus3555 minus3580

R

BC

D A

L

120579

diaphragmsRigid

Figure 3 Definition of Scordelis-Lo problem Cylindrical roofsubjected to its self-weight

3 Numerical Study

A numerical study which aims to quantitatively investigatethe accuracy of different thick shell theories for the linearelastic analysis of cylindrical structures and to provide a basisfor the long-term analysis is presented

One of the problems frequently used to evaluate theperformances of a shell element is that of the cylindrical roof(of Scordelis-lo) [4 26] subjected to its self-weight describedin Figure 3 The flat rims are free and the curved edgesrest on rigid diaphragms in their plans The geometrical andmechanical characteristics are indicated for ℎ119877 = 001 119871ℎ= 200 The length of the cylinder 119871 = 600 cm its radius is 119877= 300 cm the angle subtended by the roof is 2120579 = 80 deg thethickness is ℎ = 3 cm Youngrsquos modulus is 119864 = 310119890 + 10Paand Poissonrsquos ratio is ] = 0

31 Convergence The quarter of the roof is discretizedby considering triangular single-curved elements with

3

2

1

0

0 02 04 06 08 1 12214 16 18

2

24

22

18

26

28

3

32

Initial surfaceDeformed surface

B

C

B

C

Figure 4 The undeformed and deformed configurations of one-quarter of the roof

119873 = 4 6 elements on edges 119860119861 and 119860119863 The transversedisplacements at119861 and119862 are plotted (Figure 4) and comparedwith other models of finite elements see Figure 5

The computed deformed limit surface is shown in Fig-ure 4 Here displacement-convergence results are shown inFigure 5 and Table 1 In the diaphragm case we monitorthe displacements under the self-weight at points 119861 and 119862The convergence properties of the method are evident fromFigure 5 and Table 1 As may be seen the CSFE3 convergesas well as both the semifinite elements (SFE) and the DKTelements for the displacements under the loads

32 Scaling and Variation of Displacements A successivescaling is carried out on the thickness ratio ℎ119877 over theScordelis-lo roof We have used the following range of ratios01 02 03 0325 04 05 because it certainly belongs to boththin shells and thick shells ranges of thickness The radius 119877is constant while the thickness ℎ varies with the ratio Thescaled shell obtained here has been implemented using both


0

01

02

03

04

05

06

07

4 6 8 10 12 14 162Number of elements per half side

DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)

Figure 5 Displacement-convergence plot for Scordelis-Lo Roof (a) Convergence of transverse displacements at point 119862 (119882119862) (b)

Convergence of transverse displacements at point 119861 (119882119861)

K-L theory and that proposed by N-T The results are plottedfor each ratio as shown in Figure 6

We observe from the diagrams shown in Figure 6 that forthe thickness ratios ℎ119877 = 01 02 and 03 the transversedisplacements at point 119861 are the same for single-curvedtriangular computation of both K-L and N-T models (seeFigures 6(a) 6(b) and 6(c)) One can verify that at anyother points of the roof structure these same observationsare satisfied For thickness ratios ℎ119877 greater or equal to 0325we observe that transverse displacements computed from theshells equations of K-L and N-T are not the same (see Figures6(d) 6(e) and 6(f)) We can still investigate the exact valueℎ119877 in ]310 1340[ fromwhich displacements results issuedfrom these two approaches of shells theories diverge in acylindrical wall and probably other shells

33 Deviations N-Trsquos equations for elastic thick shell aremore general [24] and with regard to that we assumethe displacements issued from this model of thick shell tobe more accurate We can then investigate the deviationsencountered when using K-L model to evaluate a shell Theinvestigation consists in calculating the deviations (differencebetween K-L displacement and N-T displacement) for eachmesh-step (number of elements per half side) with respect tothe series of thickness ratios ℎ119877 = 01 02 03 0325 04 and05 at the same point This investigation was carried out ontwo different points119862 and 119861 (see Figure 3) and the results areplotted in Figures 7(a) and 7(b)

From diagrams shown in Figures 7(a) and 7(b) weobserve that (i) at points 119861 and 119862 of our cylindrical roofdeviations are encountered from thickness ratios ℎ119877 above03 (ii) deviations increase with mesh-steps at point 119862 anddecrease at point 119861

4 Discussion

We have investigated in this framework the influence ofthe third fundamental form proportional to 1205942 (with 1205942 =(ℎ2119877)

2) included in N-Trsquos model of elastic thick shells Thismodel was implemented using cylindrical shell triangularfinite elements (CSFE3) Based on the results obtained thefollowing comments can be made

The convergence studies of CSFE3 show that this curvedfinite element of three nodes only converges as well as SFEDKT12 and DKT18 which are robust and greedy (memory)[27] The last three finite elements models are based on theo-retical approach of Kirchhoff-Love (thin shells) or Reissner-Mindlin (Thick shells) which neglect the third fundamentalform in their shell kinematic equations In addition to thatthese classical shell theories are based on two powerfulassumptions of K-L or R-M [28 29]

The theoretical equations of thick and thin shells issuedfrom these assumptions can be questioned because thetransverse fiber (normal to the reference surface) has beenforced to follow a specific behavior without any scientific ortechnical background that justifies their kinematic equations[20 24 26] Consequently some quantities without evidentmechanical significations are added in the formulation offinite elementsmodels (likeDKT SFE ) in order to improvecomputational accuracy [7 26]

In the contrary our finite element (CSFE3) is based onN-Trsquos theoretical model of thick shells which is deduced from1205761205723(119906) = 0 obtained at limits analysis of 3-dimensional shell

equations [24] Resultant kinematic equations of shell hereare more general since they contain additional terms to thosefound in classical equations of shellsThey aremathematicallyand technically justified without ad hoc assumption on


4 6 8 10 12 142Number of elements per half side

minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000

Thick shell R-MThick shell Nzengwa

WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)

Figure 6 (a) Variation of transverse displacements at 119861 for ℎ119877 = 01 (b) Variation of transverse displacements at 119861 for ℎ119877 = 02 (c)Variation of transverse displacements at 119861 for ℎ119877 = 03 (d) Variation of transverse displacements at 119861 for ℎ119877 = 0325 (e) Variation oftransverse displacements at 119861 for ℎ119877 = 04 (f) Variation of transverse displacements at 119861 for ℎ119877 = 05


Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)

Figure 7 (a) Evolution of displacements deviations at point 119862 (b) Evolution of displacements deviations at point 119861

transverse fiberrsquos behavior This can explain why a simple3-node curved triangular finite element obtained efficientconvergent result The investigation with a 6-node curvedtriangle will surely improve the convergence rate [7 26]

The successive scaling over the benchmark structurehas presented divergence of displacements results as ℎ119877increases for K-L kinematic model of shells and that ofN-T This is due to the influence of the additional termsfound in N-Trsquos shell theory in the deformation energy of thestructure For homogeneous and isotropic elastic shells witha midsurface between minusℎ2 and +ℎ2 the total deformationenergy containsmembrane stiffness119870

119898 bending stiffness119870

119887

coupled stiffnesses 119870119898119892

and 119870119892119898

(membrane and Gaussianbending) and finally Gaussian stiffness119870

119892

For Reissner-Mindlin and K-L classical model of shellsthe global stiffness matrix 119870

R-M119866

contains only 119870119898

and119870119887while in N-Trsquos model the global stiffness matrix 119870N-T

119866

contains additional terms (119870119898119892

+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866

The proportion of energy contributed byeach stiffness matrix to the global deformation energy is asfollows 119870

119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)

4 Wemonitored that for thickness ratio ℎ119877 less or equal to03 that is (2120594)2 lt 110 the couple stiffness 119870

119898119892and

Gaussian stiffness 119870119892disappear in the global deformation

energy because they are inversely proportional respectivelyto 1000 and 10000 We then obtain 119870N-T

119866asymp 119870

R-M119866

Thisexplains why deviations are zero for thickness ratio lessthan 03 (see Figure 7) For thickness ratio greater or equalto radic110 displacements results encountered for the twomodels are different because the coupled stiffness119870

119898119892and the

Gaussian stiffness 119870119892which are now inversely proportional

respectively to 100 and 1000 cannot disappear anymore inthe global deformation energy of the structure This additiveproportion of energy into the global deformation energy

found in this model improves total stiffness and increases therigidity of the structure

5 Conclusion

A finite element model for predicting the elastic thick shellbehavior of cylindrical structure has been developed in thiswork It has been quantitatively shown that the thickness ratio(thickness on radius) plays important role in their elasticbehavior and structural safety Hence a reliable control ofthis parameter is required N-Trsquos theoretical approach forthe modelling of the displacement and stains in cylindricalstructures have been examined and compared with that of K-L for the case of self-weight loading cylindrical roof

The comparison has revealed that N-Trsquos shell theory issuitable for the analysis of cylindrical thin and thick struc-tures from the perspectives of both accuracy and simplicity A3-node cylindrical shell finite element has been developed forthe simulation of static behaviour of shells in general whichrequires the ability of well handling thickness ratios from thevicinity of zero to one

The ability of the CSFE3 model to deal with a widerange of ratios has been demonstrated through numericalexamples

It has been found that the use of K-L theoretical approachbased on those assumptions leads to inaccurate results as thethickness becomes greater than the square root of 01

It has also been found that above the value square root01 the deviations linearly increase with the increase of thethickness ratio

The issues addressed in this work highlight the needfor N-Trsquos approach to be implemented for the elastic shellsanalysis of cylindrical structures

The analytical model and the numerical results presentedhere contribute to the establishment of a foundation of


theoretical knowledge required for reliable analysis effectivedesign and safe use of cylindrical walled structures

Finally the paper provides a platform for double-curvedfinite elements model to be developed for studying theinfluence of thickness ratio on all shell geometries

Competing Interests

The authors declare that they have no competing interests

References

[1] H Stolarski andT Belytschko ldquoMembrane locking and reducedintegration for curved elementsrdquo Journal of Applied Mechanicsvol 49 no 1 pp 172ndash176 1982

[2] J F DoyleNonlinear Analysis of Thin-Walled Structures StaticsDynamics and Stability Springer Science amp Business Media2013

[3] J R Vinson The Behavior of Thin Walled Structures BeamsPlates and Shells Springer Berlin Germany 2012

[4] J HoefakkerTheory Review for Cylindrical Shells and Paramet-ric Study of Chimneys and Tanks Eburon Uitgeverij BV 2010

[5] W T Koiter and J G Simmonds ldquoFoundations of shelltheoryrdquo inTheoretical and Applied Mechanics E Becker and GMikhailov Eds pp 150ndash176 Springer Berlin Germany 1973

[6] D Chapelle and K J Bathe ldquoFundamental considerations forthe finite element analysis of shell structuresrdquo Computers ampStructures vol 66 no 1 pp 19ndash36 1998

[7] D-N Kim and K-J Bathe ldquoA triangular six-node shell ele-mentrdquoComputers amp Structures vol 87 no 23-24 pp 1451ndash14602009

[8] M Jawad Theory and Design of Plate and Shell StructuresSpringer Berlin Germany 2012

[9] M Z Nejad M Jabbari and M Ghannad ldquoElastic analysis ofaxially functionally graded rotating thick cylinder with variablethickness under non-uniform arbitrarily pressure loadingrdquoInternational Journal of Engineering Science vol 89 pp 86ndash992015

[10] H Zeighampour andY Tadi Beni ldquoCylindrical thin-shellmodelbased onmodified strain gradient theoryrdquo International Journalof Engineering Science vol 78 pp 27ndash47 2014

[11] F Tornabene A Liverani and G Caligiana ldquoGeneral aniso-tropic doubly-curved shell theory a differential quadraturesolution for free vibrations of shells and panels of revolutionwith a free-formmeridianrdquo Journal of Sound and Vibration vol331 no 22 pp 4848ndash4869 2012

[12] F Tornabene E Viola and D J Inman ldquo2-D differential quad-rature solution for vibration analysis of functionally gradedconical cylindrical shell and annul plate structuresrdquo Journal ofSound and Vibration vol 328 no 3 pp 259ndash290 2009

[13] E Viola F Tornabene and N Fantuzzi ldquoGeneralized differ-ential quadrature finite element method for cracked compositestructures of arbitrary shaperdquoComposite Structures vol 106 pp815ndash834 2013

[14] E Pindza and E Mare ldquoDiscrete singular convolution methodfor numerical solutions of fifth order korteweg-de vries equa-tionsrdquo Journal of Applied Mathematics and Physics vol 1 no 7pp 5ndash15 2013

[15] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer Berlin Germany 2012

[16] T Nguyen-Thoi P Phung-Van CThai-Hoang andH Nguyen-Xuan ldquoA cell-based smoothed discrete shear gap method (CS-DSG3) using triangular elements for static and free vibrationanalyses of shell structuresrdquo International Journal of MechanicalSciences vol 74 pp 32ndash45 2013

[17] R Echter B Oesterle and M Bischoff ldquoA hierarchic familyof isogeometric shell finite elementsrdquo Computer Methods inApplied Mechanics and Engineering vol 254 pp 170ndash180 2013

[18] H Nguyen-Xuan L V Tran C H Thai and T Nguyen-ThoildquoAnalysis of functionally graded plates by an efficient finiteelement method with node-based strain smoothingrdquo Thin-Walled Structures vol 54 pp 1ndash18 2012

[19] F Tornabene A Liverani and G Caligiana ldquoStatic analysisof laminated composite curved shells and panels of revolutionwith a posteriori shear and normal stress recovery using gener-alized differential quadrature methodrdquo International Journal ofMechanical Sciences vol 61 no 1 pp 71ndash87 2012

[20] Y-G Fang J-H Pan and W-X Chen ldquoTheory of thick-walled shells and its application in cylindrical shellrdquo AppliedMathematics and Mechanics vol 13 no 11 pp 1055ndash1065 1992

[21] D Chapelle and K-J Bathe The Finite Element Analysis ofShells-Fundamentals Springer Berlin Germany 2010

[22] M Bernadou P G Ciarlet and B Miara ldquoExistence theoremsfor two-dimensional linear shell theoriesrdquo Journal of Elasticityvol 34 no 2 pp 111ndash138 1994

[23] S Saeligvik ldquoTheoretical and experimental studies of stresses inflexible pipesrdquo Computers amp Structures vol 89 no 23-24 pp2273ndash2291 2011

[24] R Nzengwa and B H T Simo ldquoA two-dimensional model forlinear elastic thick shellsrdquo International Journal of Solids andStructures vol 36 no 34 pp 5141ndash5176 1999

[25] IMoharos I Oldal andA Szekrenyes Finite ElementMethodeTypotex Publishing House Budapest Hungary 2012

[26] M Bernadou PM Eiroa andP Trouve ldquoOn the approximationof general linear thin shell problems BYD KT methodsrdquoComputation and Applied Mathematics vol 10 article 103 1991

[27] L Della Croce and P Venini ldquoFinite elements for functionallygraded Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 193 no 9ndash11 pp 705ndash725 2004

[28] E Carrera G Giunta P Nali and M Petrolo ldquoRefined beamelements with arbitrary cross-section geometriesrdquoComputers ampStructures vol 88 no 5-6 pp 283ndash293 2010

[29] E Carrera S Brischetto M Cinefra and M Soave ldquoEffects ofthickness stretching in functionally graded plates and shellsrdquoComposites Part B Engineering vol 42 no 2 pp 123ndash133 2011

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Stochastic AnalysisInternational Journal of


convenience and simplicity [4 11] Hence it is not surprisingthat the development of the numerical formulations sincethe 1950s has led to a gradual cessation of attempts to findclosed-form solutions to rigorous formulations [4] But withtodayrsquos availability of greatly increased computing power(also since the mid-twentieth century) completeness ratherthan simplicity is given more emphasis

In computationalmechanics many numerical techniquesare developed to solve governing equations of shells We canidentify for this purpose the generalized differential quadra-ture finite elements (DGQFEM) [11ndash13] discrete singularconvolution method (DSC) [14] the finite difference (FD)the discrete volume finite element (DVFEM) and the finiteelements method (FEM) under which solid-shells finite ele-ments (SSFEM) and mixt finite element (MFEM) are found[15ndash18] Our approach is found under the MFEM category inwhich three-dimensional shell parameters are reduced to themidsurface of the shell of thickness ℎ Some finite elements inthis category describewell the behavior of shells under severalloads All these models of finite elements are link with eitherKirchhoff-Love (K-L) or Reissner-Mindlin (R-M) hypothesiswhich neglected the effect of curvature in elastic stiffness[11 19] This is one of the main reasons why K-L and R-Mtheories of shells are incomplete and unsuitable to handlevarious types of shells [20] Structure engineers emphasismore on simplifying assumptions than limit analysis Becauseof the third fundamental form that has been neglected intheir theoretical approach of shells governing equations [21]and severe assumption on shell thickness somemathematicalterms without evident mechanic signification are introducedin the stiffness matrix in order to improve their efficiency[6 22]

After a brief presentation of Nzengwa and Tagne (N-T) kinematic equations of elastic thick shells we develop acurved cylindrical 3-node finite element based on both K-L and N-Trsquos shell models using strain tensor interpolationassumption [23] Next through numerical implementationon classical benchmark convergence of N-Trsquos shell modelis investigated The deviation and the limit thickness ratiobetween the two theoretical approaches of thin shells andthick shells are established and discussed by applying succes-sive scaling of a sample of thin shell

2 Materials and Methods

21 N-Trsquos Two-Dimensional ElasticThick ShellsModel TheN-Trsquos two-dimensional model for linear elastic thick shells hasbeen deduced from the three-dimensional problem withoutany ad hoc assumption whether of geometrical or mechan-ical nature The two-dimensional equations are deducedby applying asymptotic analysis on a family of variationalequations obtained from an abstract scaled shell throughmultiple scaling of the initial three-dimensional equationsThe theoretical elastic thick shells governing equations andlimits displacements obtained from this approach (at the limitanalysis) are more general because they contain additionalterms to those found in Kirchhoff-Love model The modelalso completes the thick shells theory of Reissner-Mindlin[24]

For a three-dimensional shell the following bases (1198921 11989221198923) (119892

1 119892

2 119892

3) are dual bases and (119886

1 119886

2 119886

3) (119886

1 119886

2 119886

3)

defined on the midsurface also constitute dual bases ofshell if 120594 lt 1 Let 119880

119894(119909

1 119909

2 119911) be the three-dimensional

displacement field vector it is defined as follows in the localcoordinate system

119880 = 119880120572119892120572+ 119880

31198923= 119906

120572119886120572+ 119906

31198863

119906120588= 119906

120588(119909

1 119909

2 119911)

1199063= 119906

3(119909

1 119909

2 119911)

(1)

The strain tensor for a three-dimensional shell reads

120576120572120573(119880) =

1

2(119880

120572120573+ 119880

120573120572) = 120576

120572120573(119906)

=[120583

]120572(nabla

120573119906] minus 1198871205721205731199063) + 120583

]120573(nabla

120572119906] minus 119887]1205731199063)]

2

(2)

1205761205723(119880) =

1

2(119880

1205723+ 119880

3120572) = 120576

1205723(119906)

=[120583

]120572119906]3 + (1199063120572 + 119887

]120572119906])]

2

(3)

12057633(119880) = 119880

33= 119880

33= 119906

33= 120576

33(119906) (4)

From the limit analysis N-T demonstrated that the displace-ments field satisfies the equation 120576

1198943(119880) = 0 and the unique

solution is

119906120588(119909

1 119909

2 119911) = 120583

120574

120588119906120574(119909

1 119909

2) minus 119911120597

120588119908

119882 = 119908 = 1199063

119880120572(119909

1 119909

2 119911) = 120583

120588

120572119906120588

120583120574

120588= 120575

120574

120588minus 119911119887

120574

120588

119880120572(119909

1 119909

2 119911) = 119906

120572(119909

1 119909

2) minus 119911120579

120572(119909

1 119909

2)

+ 1199112120595120572(119909

1 119909

2)

(5)

where (119906120572) are local membrane displacement components

and 119908 = 1199063the transvers displacement

120579120572= (2119887

120591

120572119906120591+ nabla

120572119908)

120595120572= 119887

120574

120572119887120591

120574119906120591+ 119887

120574

120572nabla120574119908

(6)

The general shell deformation tensor derived from the kine-matics defined above is

120576120572120573(119880) = 119890

120572120573(119906) minus 119911119896

120572120573(119906) + 119911

2119876120572120573(119906) (7)

1205761198943(119880) = 0 (8)

where 119890120572120573

is the membrane strain tensor 119896120572120573

is the bendingstrain tensor and 119876

120572120573is the tensor of Gauss curvature


1

2

3

a1 a3

a3

a2a1

a2

X

C

OR B Y

L

A

Z

M

D

120579



119876120572120573=1

2(119888120572120573minus 119862

120572120573)

119888 = 119888120572120573= 119887

120583

120572119887120583120573= 119887

120572120583119887120583

120573

119862120572120573= 119861

120583

120572119861120583120573= 119861

120572120583119861120583

120573

(9)


120572(119909

1 119909



119880120572(119909

1 119909

2 119911) = 119906

120572(119909

1 119909


120572119908(119909

1 119909

2)

119882 (1199091 119909

2 119911) = 119908 (119909

1 119909

2) = 119906

3

(10)


120591

120572119887120588

120591119906120588+119887

120591

120572nabla120591119908) in the





997888997888997888rarr119874119872 =

119883 = 119877 cos120593

119884 = 119877 sin120593

119885 = 119909

(11)




119880120572= 119906

120572minus 119911120579

120572+ 119911

2120595120572

119882 = 119908

(12)


1and 119906




components 119887120591120572

1205791=2119906

1

1198771

minus119908119909

1198601

1205792=2119906

2

1198772

minus119908120593

1198602

120595120572= (119887

120591

120572119887120588

120591119906120588+ 119887

120591

120572nabla120591119908)

(13)

which leads to

1205951= minus

1

1198771

(1199061

1198771

minus1

1198601

119908119909) =

1

1198771

(119887120591

1119906120591+ 120579

1)

1205952= minus

1

1198772

(1199062

1198772

minus1

1198602

119908120593) =

1

1198772

(119887120591

2119906120591+ 120579

2)

(14)


where 1198601 119860

2 119877

1 and 119877




1198601=10038161003816100381610038161003816

997888rarr1198861

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

997888997888997888rarr120597119872

120597119909

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 1 1198771= infin

1198602=10038161003816100381610038161003816

997888rarr1198862

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

997888997888997888rarr120597119872

120597120593

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119877 1198772= 119877

(15)



119890120572120573=1

2(nabla

120572V + nabla

120573119906) minus 119887

120572120573119908

119896120572120573= nabla

120572119887120588

120573119906120588+ 119887

120588

120572nabla120573119906120588+ 119887

120588

120573nabla120572119906120588+ nabla

120572nabla120573119908

minus 119887120588

120572119887120588120573119908

119876120572120573= 05 (119887

120583

120572nabla120573119887120588

120583119906120588+ 119887

120583

120572119887120588

120583nabla120573119906120588+ 119887

120583

120573119887120588

120583nabla120572119906120588

+ 119887120583

120573nabla120572119887120588

120583119906120588+ 119887

120583

120572nabla120573nabla120583119908 + 119887

120583

120573nabla120572nabla120583119908)

(16)




tensor 119876120572120573



119890119909= 119906

1119909

119890120593=1

119877(119906

2120593+ 119908)

2119890119909120593=1

1198771199061120593+ 119906

2119909

119896119909= 119908

119909119909

119896120593= minus

1

1198772(2119906

2120593minus 119908

120593120593+ 119908)

2119896119909120593= minus

2

119877(minus119908

119909120593+ 119906

2119909)

119876119909= 0

119876120593= minus

1

1198773(119906

2120593minus 119908

120593120593)

2119876119909120593= minus

1

1198772(119906

2119909minus 119908

119909120593)

(17)


1= 119906 and 119906

2= V



120572120573= 0 119896

120572120573= 0 and 119876

120572120573= 0

119906 (119909 120593) = 119906 (120593) (18)


V (119909 120593) = minus1

1198771199061015840(120593) sdot 119909 + 119892 (120593)

119908 (119909 120593) = +1

11987711990610158401015840(120593) sdot 119909 minus 119892

1015840(120593)

(19)

119906(3)(120593) + 119906

1015840(120593) = 0 (20)

119906(4)(120593) + 119906

(2)(120593) = 0 (21)

119892(3)(120593) + 119892

(1)(120593) = 0 (22)


119906 (120593) = 1198861+ 119886


3119877 sin120593 (23)


5sin120593 + 119886

6cos120593 (24)


1199060= 119886

1+ 119886


3119877 sin120593

V0= 119886

2119909 sin120593 + 119886

3119909 cos120593 + 119886

4119877 (cos120593 cos 120579 minus 1)

minus 1198865sin120593 + 119886

6cos120593

1199080= minus119886

2119909 cos120593 + 119886

3119909 sin120593 + 119886

4119877 sin120593 cos 120579

+ 1198865cos120593 + 119886

6sin120593

(25)



1198800= 119875

01198600 (26)

where 119880119905

0= [1199060 V

01199080]

119905

sdot1198600= [1198861 1198862 1198863 1198864 1198865 1198866] (27)



1198750= (



) (28)



119889

119880 = 1198800+ 119880

119889=

119905

sdot[119906 V 119908] (29)

where 119905

sdot119880119889= [119906

119889V119889119908119889]


119906119889= 119886

7119877119909 + 119886

8119877120593 + 119886

9119877119909120593

V119889= 119886

10120593 + 119886

11119909120593

119908119889= 119886

121199092+ 119886

13119909120593 + 119886

141205932+ 119886

151199093+ 119886

161199092120593

+ 11988617119909120593

2+ 119886

181205933

119880119889= 119875

119889119860119889

(30)

with119905

sdot119860119889= [1198867 1198868 1198869 11988610 11988611 11988612 11988613 11988614 11988615 11988616 11988617 11988618]

119875119889

= (

119877119909 119877120593 119877119909120593 0 0 0 0 0 0 0 0 0

0 0 0 120593 119909120593 0 0 0 0 0 0 0

0 0 0 0 0 1199092119909120593 120593

211990931199092120593 119909120593

21205933

)

(31)





119905

sdot119890= [

1V111199091120593112V231199092120593223V33119909312059333] (32)



0cup 120574

1be


1

with Γ0= 120574


1= 120574


minuscupΓ

+





find 119880 isin 1198681198671

Γ0

int

sdot

Ω

119879

(119880) (119881) 119889Ω = int

sdot

Ω

119891 sdot 119881119889Ω + int

sdot

120597Ω

119892 sdot 119881 119889D

= 119871 (V) for 119881 isin 1198681198671

Γ0

(33)



= 119888 (34)



find 119906 isin 1198681198671

Γ0

int

sdot

Ω

[120582119892119894119895119892119896119897+ 120583 (119892

119894119896119892119895119897+ 119892

119894119897119892119895119896)] (119906) (V) 119889Ω

= int

sdot

Ω


120597Ω

119892 sdot V 119889D = 119871 (V)

for V isin 1198681198671

Γ0

(35)


1

2

3

u

w

1205791

1205792

120574



119860 (119906 V) = intsdot

Ω

[120582119892120572120573119892120574120588+ 120583 (119892

120572120574119892120573120588+ 119892

120572120588119892120573120574)]

sdot (119890120572120573(119906) minus 119911119896

120572120573(119906) + 119911

2119876120572120573(119906))

(119890120574120588(V) minus 119911119896

120574120588(V) + 1199112119876

120574120588(V)) 119889Ω = 119871 (V)

(36)


face 119892120572120573 = (120583minus1)120572120588(120583

minus1)120573


(120583minus1)120572

120588

(120583minus1)120572

120588= 120575

120572

120588+ 119911119887

120572

120588+

infin

sum

119899ge2

119911119899(119887

119899)120572

120588(37)


1198601(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573]119876

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

5

80 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573]119876

120572120573(V) 119889119878 = 119871 (V)

(38)


1198600(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

(39)




and 119870R-M119866


0(119906 V)


1198601(119906 V) = int

119878

119864119879

V119862119866119864119906119889119878 = int119878

119864119879

V119862119866119864119906 |119869| 119889120585 119889120578 (40)


120576119898= 119890

120572120573= 119863

119898119860 = 119863

119898119875minus1 = 119861

119898 119861

119898= 119863

119898119875minus1

120590119898= 119888120576

119898= 119888119861

119898

(41)


120576119887= 119896

120572120573= 119863

119887119860 = 119863

119887119875minus1 = 119861

119887 119861

119887= 119863

119887119875minus1

120590119887= 119888120576

119887= 119888119861

119887

(42)


120576119892= 119876

120572120573= 119863

119892119860 = 119863

119892119875minus1 = 119861

119892 119861

119892= 119863

119892119875minus1

120590119892= 119888120576

119892= 119888119861

119892

(43)



119880 = 1198800+ 119880

119889= 119875

01198600+ 119875

119889119860119889= [1198750 119875119889] [

1198600

119860119889

]

= 119873 sdot 119860

(44)


119880 = 119873 sdot 119860 = 119873 sdot 119875minus1 (45)


1198601(119906 V) =

1

2intΩ

(120576119905

119898119888119905120576119898minus 119911120576

119905

119898119888119905120576119887+ 119911

2120576119905

119898119888119905120576119892

minus 119911120576119905

119887119888119905120576119898+ 119911

2120576119905

119887119888119905120576119887minus 119911

3120576119905

119887119888119905120576119892+ 119911

2120576119905

119892119888119905120576119898

minus 1199113120576119905

119892119888119905120576119887+ 119911

4120576119905

119892119888119905120576119892) 119889Ω = 119871 (V)

(46)


1198601(119906 V) =

1

2119905

int119878

ℎ119861119905

119898119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119898119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119898119862119905119861119892119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119887119862119905119861119898119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119887119862119905119861119887119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119887119862119905119861119892119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119892119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119892119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ5

80119861119905

119892119862119905119861119892119877119889120593119889119909

= 119871 (V)

(47)


1198601(119906 V) =

119905

[119870N-T119866]

1198600(119906 V) =

119905

[119870R-M119866]

(48)

where 119870(sdot)



119870N-T119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887+ℎ3

12119861119905

119898119862119905119861119892

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

119870R-M119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887)119877119889120593119889119909

(49)


119871 (V) = 119905

intΩ

119905


119905119891119889Ω +

119905

int120597Ω

119905


119905119892119889D

= 119905

(119865119881+119872) = 0

(50)


119881+119872

119865119881= int

sdot

119878

119873119905119891119890119877119889120593119889119909

119872 = int

sdot

119878

119873119905119892119877119889120593119889119909

(51)



119870119866 = 119865 (52)




119870119890= int

119904119890

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119898119862119905119861119892+ℎ3

12119861119905

119887119862119905119861119887

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

(53)




119870119866=

119873

sum

119890=1

119870119890

119865 =

119873

sum

119890=1

119865119890

(54)







R

BC

D A

L

120579

diaphragmsRigid


3 Numerical Study




3

2

1

0

0 02 04 06 08 1 12214 16 18

2

24

22

18

26

28

3

32


B

C

B

C






0

01

02

03

04

05

06

07


DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)







4 Discussion









minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

a1 a3

a3

a2a1

a2

X

C

OR B Y

L

A

Z

M

D

120579



119876120572120573=1

2(119888120572120573minus 119862

120572120573)

119888 = 119888120572120573= 119887

120583

120572119887120583120573= 119887

120572120583119887120583

120573

119862120572120573= 119861

120583

120572119861120583120573= 119861

120572120583119861120583

120573

(9)


120572(119909

1 119909



119880120572(119909

1 119909

2 119911) = 119906

120572(119909

1 119909


120572119908(119909

1 119909

2)

119882 (1199091 119909

2 119911) = 119908 (119909

1 119909

2) = 119906

3

(10)


120591

120572119887120588

120591119906120588+119887

120591

120572nabla120591119908) in the





997888997888997888rarr119874119872 =

119883 = 119877 cos120593

119884 = 119877 sin120593

119885 = 119909

(11)




119880120572= 119906

120572minus 119911120579

120572+ 119911

2120595120572

119882 = 119908

(12)


1and 119906




components 119887120591120572

1205791=2119906

1

1198771

minus119908119909

1198601

1205792=2119906

2

1198772

minus119908120593

1198602

120595120572= (119887

120591

120572119887120588

120591119906120588+ 119887

120591

120572nabla120591119908)

(13)

which leads to

1205951= minus

1

1198771

(1199061

1198771

minus1

1198601

119908119909) =

1

1198771

(119887120591

1119906120591+ 120579

1)

1205952= minus

1

1198772

(1199062

1198772

minus1

1198602

119908120593) =

1

1198772

(119887120591

2119906120591+ 120579

2)

(14)


where 1198601 119860

2 119877

1 and 119877




1198601=10038161003816100381610038161003816

997888rarr1198861

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

997888997888997888rarr120597119872

120597119909

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 1 1198771= infin

1198602=10038161003816100381610038161003816

997888rarr1198862

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

997888997888997888rarr120597119872

120597120593

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119877 1198772= 119877

(15)



119890120572120573=1

2(nabla

120572V + nabla

120573119906) minus 119887

120572120573119908

119896120572120573= nabla

120572119887120588

120573119906120588+ 119887

120588

120572nabla120573119906120588+ 119887

120588

120573nabla120572119906120588+ nabla

120572nabla120573119908

minus 119887120588

120572119887120588120573119908

119876120572120573= 05 (119887

120583

120572nabla120573119887120588

120583119906120588+ 119887

120583

120572119887120588

120583nabla120573119906120588+ 119887

120583

120573119887120588

120583nabla120572119906120588

+ 119887120583

120573nabla120572119887120588

120583119906120588+ 119887

120583

120572nabla120573nabla120583119908 + 119887

120583

120573nabla120572nabla120583119908)

(16)




tensor 119876120572120573



119890119909= 119906

1119909

119890120593=1

119877(119906

2120593+ 119908)

2119890119909120593=1

1198771199061120593+ 119906

2119909

119896119909= 119908

119909119909

119896120593= minus

1

1198772(2119906

2120593minus 119908

120593120593+ 119908)

2119896119909120593= minus

2

119877(minus119908

119909120593+ 119906

2119909)

119876119909= 0

119876120593= minus

1

1198773(119906

2120593minus 119908

120593120593)

2119876119909120593= minus

1

1198772(119906

2119909minus 119908

119909120593)

(17)


1= 119906 and 119906

2= V



120572120573= 0 119896

120572120573= 0 and 119876

120572120573= 0

119906 (119909 120593) = 119906 (120593) (18)


V (119909 120593) = minus1

1198771199061015840(120593) sdot 119909 + 119892 (120593)

119908 (119909 120593) = +1

11987711990610158401015840(120593) sdot 119909 minus 119892

1015840(120593)

(19)

119906(3)(120593) + 119906

1015840(120593) = 0 (20)

119906(4)(120593) + 119906

(2)(120593) = 0 (21)

119892(3)(120593) + 119892

(1)(120593) = 0 (22)


119906 (120593) = 1198861+ 119886


3119877 sin120593 (23)


5sin120593 + 119886

6cos120593 (24)


1199060= 119886

1+ 119886


3119877 sin120593

V0= 119886

2119909 sin120593 + 119886

3119909 cos120593 + 119886

4119877 (cos120593 cos 120579 minus 1)

minus 1198865sin120593 + 119886

6cos120593

1199080= minus119886

2119909 cos120593 + 119886

3119909 sin120593 + 119886

4119877 sin120593 cos 120579

+ 1198865cos120593 + 119886

6sin120593

(25)



1198800= 119875

01198600 (26)

where 119880119905

0= [1199060 V

01199080]

119905

sdot1198600= [1198861 1198862 1198863 1198864 1198865 1198866] (27)



1198750= (



) (28)



119889

119880 = 1198800+ 119880

119889=

119905

sdot[119906 V 119908] (29)

where 119905

sdot119880119889= [119906

119889V119889119908119889]


119906119889= 119886

7119877119909 + 119886

8119877120593 + 119886

9119877119909120593

V119889= 119886

10120593 + 119886

11119909120593

119908119889= 119886

121199092+ 119886

13119909120593 + 119886

141205932+ 119886

151199093+ 119886

161199092120593

+ 11988617119909120593

2+ 119886

181205933

119880119889= 119875

119889119860119889

(30)

with119905

sdot119860119889= [1198867 1198868 1198869 11988610 11988611 11988612 11988613 11988614 11988615 11988616 11988617 11988618]

119875119889

= (

119877119909 119877120593 119877119909120593 0 0 0 0 0 0 0 0 0

0 0 0 120593 119909120593 0 0 0 0 0 0 0

0 0 0 0 0 1199092119909120593 120593

211990931199092120593 119909120593

21205933

)

(31)





119905

sdot119890= [

1V111199091120593112V231199092120593223V33119909312059333] (32)



0cup 120574

1be


1

with Γ0= 120574


1= 120574


minuscupΓ

+





find 119880 isin 1198681198671

Γ0

int

sdot

Ω

119879

(119880) (119881) 119889Ω = int

sdot

Ω

119891 sdot 119881119889Ω + int

sdot

120597Ω

119892 sdot 119881 119889D

= 119871 (V) for 119881 isin 1198681198671

Γ0

(33)



= 119888 (34)



find 119906 isin 1198681198671

Γ0

int

sdot

Ω

[120582119892119894119895119892119896119897+ 120583 (119892

119894119896119892119895119897+ 119892

119894119897119892119895119896)] (119906) (V) 119889Ω

= int

sdot

Ω


120597Ω

119892 sdot V 119889D = 119871 (V)

for V isin 1198681198671

Γ0

(35)


1

2

3

u

w

1205791

1205792

120574



119860 (119906 V) = intsdot

Ω

[120582119892120572120573119892120574120588+ 120583 (119892

120572120574119892120573120588+ 119892

120572120588119892120573120574)]

sdot (119890120572120573(119906) minus 119911119896

120572120573(119906) + 119911

2119876120572120573(119906))

(119890120574120588(V) minus 119911119896

120574120588(V) + 1199112119876

120574120588(V)) 119889Ω = 119871 (V)

(36)


face 119892120572120573 = (120583minus1)120572120588(120583

minus1)120573


(120583minus1)120572

120588

(120583minus1)120572

120588= 120575

120572

120588+ 119911119887

120572

120588+

infin

sum

119899ge2

119911119899(119887

119899)120572

120588(37)


1198601(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573]119876

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

5

80 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573]119876

120572120573(V) 119889119878 = 119871 (V)

(38)


1198600(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

(39)




and 119870R-M119866


0(119906 V)


1198601(119906 V) = int

119878

119864119879

V119862119866119864119906119889119878 = int119878

119864119879

V119862119866119864119906 |119869| 119889120585 119889120578 (40)


120576119898= 119890

120572120573= 119863

119898119860 = 119863

119898119875minus1 = 119861

119898 119861

119898= 119863

119898119875minus1

120590119898= 119888120576

119898= 119888119861

119898

(41)


120576119887= 119896

120572120573= 119863

119887119860 = 119863

119887119875minus1 = 119861

119887 119861

119887= 119863

119887119875minus1

120590119887= 119888120576

119887= 119888119861

119887

(42)


120576119892= 119876

120572120573= 119863

119892119860 = 119863

119892119875minus1 = 119861

119892 119861

119892= 119863

119892119875minus1

120590119892= 119888120576

119892= 119888119861

119892

(43)



119880 = 1198800+ 119880

119889= 119875

01198600+ 119875

119889119860119889= [1198750 119875119889] [

1198600

119860119889

]

= 119873 sdot 119860

(44)


119880 = 119873 sdot 119860 = 119873 sdot 119875minus1 (45)


1198601(119906 V) =

1

2intΩ

(120576119905

119898119888119905120576119898minus 119911120576

119905

119898119888119905120576119887+ 119911

2120576119905

119898119888119905120576119892

minus 119911120576119905

119887119888119905120576119898+ 119911

2120576119905

119887119888119905120576119887minus 119911

3120576119905

119887119888119905120576119892+ 119911

2120576119905

119892119888119905120576119898

minus 1199113120576119905

119892119888119905120576119887+ 119911

4120576119905

119892119888119905120576119892) 119889Ω = 119871 (V)

(46)


1198601(119906 V) =

1

2119905

int119878

ℎ119861119905

119898119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119898119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119898119862119905119861119892119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119887119862119905119861119898119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119887119862119905119861119887119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119887119862119905119861119892119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119892119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119892119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ5

80119861119905

119892119862119905119861119892119877119889120593119889119909

= 119871 (V)

(47)


1198601(119906 V) =

119905

[119870N-T119866]

1198600(119906 V) =

119905

[119870R-M119866]

(48)

where 119870(sdot)



119870N-T119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887+ℎ3

12119861119905

119898119862119905119861119892

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

119870R-M119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887)119877119889120593119889119909

(49)


119871 (V) = 119905

intΩ

119905


119905119891119889Ω +

119905

int120597Ω

119905


119905119892119889D

= 119905

(119865119881+119872) = 0

(50)


119881+119872

119865119881= int

sdot

119878

119873119905119891119890119877119889120593119889119909

119872 = int

sdot

119878

119873119905119892119877119889120593119889119909

(51)



119870119866 = 119865 (52)




119870119890= int

119904119890

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119898119862119905119861119892+ℎ3

12119861119905

119887119862119905119861119887

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

(53)




119870119866=

119873

sum

119890=1

119870119890

119865 =

119873

sum

119890=1

119865119890

(54)







R

BC

D A

L

120579

diaphragmsRigid


3 Numerical Study




3

2

1

0

0 02 04 06 08 1 12214 16 18

2

24

22

18

26

28

3

32


B

C

B

C






0

01

02

03

04

05

06

07


DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)







4 Discussion









minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 1198601 119860

2 119877

1 and 119877




1198601=10038161003816100381610038161003816

997888rarr1198861

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

997888997888997888rarr120597119872

120597119909

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 1 1198771= infin

1198602=10038161003816100381610038161003816

997888rarr1198862

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

997888997888997888rarr120597119872

120597120593

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119877 1198772= 119877

(15)



119890120572120573=1

2(nabla

120572V + nabla

120573119906) minus 119887

120572120573119908

119896120572120573= nabla

120572119887120588

120573119906120588+ 119887

120588

120572nabla120573119906120588+ 119887

120588

120573nabla120572119906120588+ nabla

120572nabla120573119908

minus 119887120588

120572119887120588120573119908

119876120572120573= 05 (119887

120583

120572nabla120573119887120588

120583119906120588+ 119887

120583

120572119887120588

120583nabla120573119906120588+ 119887

120583

120573119887120588

120583nabla120572119906120588

+ 119887120583

120573nabla120572119887120588

120583119906120588+ 119887

120583

120572nabla120573nabla120583119908 + 119887

120583

120573nabla120572nabla120583119908)

(16)




tensor 119876120572120573



119890119909= 119906

1119909

119890120593=1

119877(119906

2120593+ 119908)

2119890119909120593=1

1198771199061120593+ 119906

2119909

119896119909= 119908

119909119909

119896120593= minus

1

1198772(2119906

2120593minus 119908

120593120593+ 119908)

2119896119909120593= minus

2

119877(minus119908

119909120593+ 119906

2119909)

119876119909= 0

119876120593= minus

1

1198773(119906

2120593minus 119908

120593120593)

2119876119909120593= minus

1

1198772(119906

2119909minus 119908

119909120593)

(17)


1= 119906 and 119906

2= V



120572120573= 0 119896

120572120573= 0 and 119876

120572120573= 0

119906 (119909 120593) = 119906 (120593) (18)


V (119909 120593) = minus1

1198771199061015840(120593) sdot 119909 + 119892 (120593)

119908 (119909 120593) = +1

11987711990610158401015840(120593) sdot 119909 minus 119892

1015840(120593)

(19)

119906(3)(120593) + 119906

1015840(120593) = 0 (20)

119906(4)(120593) + 119906

(2)(120593) = 0 (21)

119892(3)(120593) + 119892

(1)(120593) = 0 (22)


119906 (120593) = 1198861+ 119886


3119877 sin120593 (23)


5sin120593 + 119886

6cos120593 (24)


1199060= 119886

1+ 119886


3119877 sin120593

V0= 119886

2119909 sin120593 + 119886

3119909 cos120593 + 119886

4119877 (cos120593 cos 120579 minus 1)

minus 1198865sin120593 + 119886

6cos120593

1199080= minus119886

2119909 cos120593 + 119886

3119909 sin120593 + 119886

4119877 sin120593 cos 120579

+ 1198865cos120593 + 119886

6sin120593

(25)



1198800= 119875

01198600 (26)

where 119880119905

0= [1199060 V

01199080]

119905

sdot1198600= [1198861 1198862 1198863 1198864 1198865 1198866] (27)



1198750= (



) (28)



119889

119880 = 1198800+ 119880

119889=

119905

sdot[119906 V 119908] (29)

where 119905

sdot119880119889= [119906

119889V119889119908119889]


119906119889= 119886

7119877119909 + 119886

8119877120593 + 119886

9119877119909120593

V119889= 119886

10120593 + 119886

11119909120593

119908119889= 119886

121199092+ 119886

13119909120593 + 119886

141205932+ 119886

151199093+ 119886

161199092120593

+ 11988617119909120593

2+ 119886

181205933

119880119889= 119875

119889119860119889

(30)

with119905

sdot119860119889= [1198867 1198868 1198869 11988610 11988611 11988612 11988613 11988614 11988615 11988616 11988617 11988618]

119875119889

= (

119877119909 119877120593 119877119909120593 0 0 0 0 0 0 0 0 0

0 0 0 120593 119909120593 0 0 0 0 0 0 0

0 0 0 0 0 1199092119909120593 120593

211990931199092120593 119909120593

21205933

)

(31)





119905

sdot119890= [

1V111199091120593112V231199092120593223V33119909312059333] (32)



0cup 120574

1be


1

with Γ0= 120574


1= 120574


minuscupΓ

+





find 119880 isin 1198681198671

Γ0

int

sdot

Ω

119879

(119880) (119881) 119889Ω = int

sdot

Ω

119891 sdot 119881119889Ω + int

sdot

120597Ω

119892 sdot 119881 119889D

= 119871 (V) for 119881 isin 1198681198671

Γ0

(33)



= 119888 (34)



find 119906 isin 1198681198671

Γ0

int

sdot

Ω

[120582119892119894119895119892119896119897+ 120583 (119892

119894119896119892119895119897+ 119892

119894119897119892119895119896)] (119906) (V) 119889Ω

= int

sdot

Ω


120597Ω

119892 sdot V 119889D = 119871 (V)

for V isin 1198681198671

Γ0

(35)


1

2

3

u

w

1205791

1205792

120574



119860 (119906 V) = intsdot

Ω

[120582119892120572120573119892120574120588+ 120583 (119892

120572120574119892120573120588+ 119892

120572120588119892120573120574)]

sdot (119890120572120573(119906) minus 119911119896

120572120573(119906) + 119911

2119876120572120573(119906))

(119890120574120588(V) minus 119911119896

120574120588(V) + 1199112119876

120574120588(V)) 119889Ω = 119871 (V)

(36)


face 119892120572120573 = (120583minus1)120572120588(120583

minus1)120573


(120583minus1)120572

120588

(120583minus1)120572

120588= 120575

120572

120588+ 119911119887

120572

120588+

infin

sum

119899ge2

119911119899(119887

119899)120572

120588(37)


1198601(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573]119876

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

5

80 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573]119876

120572120573(V) 119889119878 = 119871 (V)

(38)


1198600(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

(39)




and 119870R-M119866


0(119906 V)


1198601(119906 V) = int

119878

119864119879

V119862119866119864119906119889119878 = int119878

119864119879

V119862119866119864119906 |119869| 119889120585 119889120578 (40)


120576119898= 119890

120572120573= 119863

119898119860 = 119863

119898119875minus1 = 119861

119898 119861

119898= 119863

119898119875minus1

120590119898= 119888120576

119898= 119888119861

119898

(41)


120576119887= 119896

120572120573= 119863

119887119860 = 119863

119887119875minus1 = 119861

119887 119861

119887= 119863

119887119875minus1

120590119887= 119888120576

119887= 119888119861

119887

(42)


120576119892= 119876

120572120573= 119863

119892119860 = 119863

119892119875minus1 = 119861

119892 119861

119892= 119863

119892119875minus1

120590119892= 119888120576

119892= 119888119861

119892

(43)



119880 = 1198800+ 119880

119889= 119875

01198600+ 119875

119889119860119889= [1198750 119875119889] [

1198600

119860119889

]

= 119873 sdot 119860

(44)


119880 = 119873 sdot 119860 = 119873 sdot 119875minus1 (45)


1198601(119906 V) =

1

2intΩ

(120576119905

119898119888119905120576119898minus 119911120576

119905

119898119888119905120576119887+ 119911

2120576119905

119898119888119905120576119892

minus 119911120576119905

119887119888119905120576119898+ 119911

2120576119905

119887119888119905120576119887minus 119911

3120576119905

119887119888119905120576119892+ 119911

2120576119905

119892119888119905120576119898

minus 1199113120576119905

119892119888119905120576119887+ 119911

4120576119905

119892119888119905120576119892) 119889Ω = 119871 (V)

(46)


1198601(119906 V) =

1

2119905

int119878

ℎ119861119905

119898119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119898119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119898119862119905119861119892119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119887119862119905119861119898119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119887119862119905119861119887119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119887119862119905119861119892119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119892119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119892119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ5

80119861119905

119892119862119905119861119892119877119889120593119889119909

= 119871 (V)

(47)


1198601(119906 V) =

119905

[119870N-T119866]

1198600(119906 V) =

119905

[119870R-M119866]

(48)

where 119870(sdot)



119870N-T119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887+ℎ3

12119861119905

119898119862119905119861119892

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

119870R-M119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887)119877119889120593119889119909

(49)


119871 (V) = 119905

intΩ

119905


119905119891119889Ω +

119905

int120597Ω

119905


119905119892119889D

= 119905

(119865119881+119872) = 0

(50)


119881+119872

119865119881= int

sdot

119878

119873119905119891119890119877119889120593119889119909

119872 = int

sdot

119878

119873119905119892119877119889120593119889119909

(51)



119870119866 = 119865 (52)




119870119890= int

119904119890

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119898119862119905119861119892+ℎ3

12119861119905

119887119862119905119861119887

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

(53)




119870119866=

119873

sum

119890=1

119870119890

119865 =

119873

sum

119890=1

119865119890

(54)







R

BC

D A

L

120579

diaphragmsRigid


3 Numerical Study




3

2

1

0

0 02 04 06 08 1 12214 16 18

2

24

22

18

26

28

3

32


B

C

B

C






0

01

02

03

04

05

06

07


DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)







4 Discussion









minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198800= 119875

01198600 (26)

where 119880119905

0= [1199060 V

01199080]

119905

sdot1198600= [1198861 1198862 1198863 1198864 1198865 1198866] (27)



1198750= (



) (28)



119889

119880 = 1198800+ 119880

119889=

119905

sdot[119906 V 119908] (29)

where 119905

sdot119880119889= [119906

119889V119889119908119889]


119906119889= 119886

7119877119909 + 119886

8119877120593 + 119886

9119877119909120593

V119889= 119886

10120593 + 119886

11119909120593

119908119889= 119886

121199092+ 119886

13119909120593 + 119886

141205932+ 119886

151199093+ 119886

161199092120593

+ 11988617119909120593

2+ 119886

181205933

119880119889= 119875

119889119860119889

(30)

with119905

sdot119860119889= [1198867 1198868 1198869 11988610 11988611 11988612 11988613 11988614 11988615 11988616 11988617 11988618]

119875119889

= (

119877119909 119877120593 119877119909120593 0 0 0 0 0 0 0 0 0

0 0 0 120593 119909120593 0 0 0 0 0 0 0

0 0 0 0 0 1199092119909120593 120593

211990931199092120593 119909120593

21205933

)

(31)





119905

sdot119890= [

1V111199091120593112V231199092120593223V33119909312059333] (32)



0cup 120574

1be


1

with Γ0= 120574


1= 120574


minuscupΓ

+





find 119880 isin 1198681198671

Γ0

int

sdot

Ω

119879

(119880) (119881) 119889Ω = int

sdot

Ω

119891 sdot 119881119889Ω + int

sdot

120597Ω

119892 sdot 119881 119889D

= 119871 (V) for 119881 isin 1198681198671

Γ0

(33)



= 119888 (34)



find 119906 isin 1198681198671

Γ0

int

sdot

Ω

[120582119892119894119895119892119896119897+ 120583 (119892

119894119896119892119895119897+ 119892

119894119897119892119895119896)] (119906) (V) 119889Ω

= int

sdot

Ω


120597Ω

119892 sdot V 119889D = 119871 (V)

for V isin 1198681198671

Γ0

(35)


1

2

3

u

w

1205791

1205792

120574



119860 (119906 V) = intsdot

Ω

[120582119892120572120573119892120574120588+ 120583 (119892

120572120574119892120573120588+ 119892

120572120588119892120573120574)]

sdot (119890120572120573(119906) minus 119911119896

120572120573(119906) + 119911

2119876120572120573(119906))

(119890120574120588(V) minus 119911119896

120574120588(V) + 1199112119876

120574120588(V)) 119889Ω = 119871 (V)

(36)


face 119892120572120573 = (120583minus1)120572120588(120583

minus1)120573


(120583minus1)120572

120588

(120583minus1)120572

120588= 120575

120572

120588+ 119911119887

120572

120588+

infin

sum

119899ge2

119911119899(119887

119899)120572

120588(37)


1198601(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573]119876

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

5

80 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573]119876

120572120573(V) 119889119878 = 119871 (V)

(38)


1198600(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

(39)




and 119870R-M119866


0(119906 V)


1198601(119906 V) = int

119878

119864119879

V119862119866119864119906119889119878 = int119878

119864119879

V119862119866119864119906 |119869| 119889120585 119889120578 (40)


120576119898= 119890

120572120573= 119863

119898119860 = 119863

119898119875minus1 = 119861

119898 119861

119898= 119863

119898119875minus1

120590119898= 119888120576

119898= 119888119861

119898

(41)


120576119887= 119896

120572120573= 119863

119887119860 = 119863

119887119875minus1 = 119861

119887 119861

119887= 119863

119887119875minus1

120590119887= 119888120576

119887= 119888119861

119887

(42)


120576119892= 119876

120572120573= 119863

119892119860 = 119863

119892119875minus1 = 119861

119892 119861

119892= 119863

119892119875minus1

120590119892= 119888120576

119892= 119888119861

119892

(43)



119880 = 1198800+ 119880

119889= 119875

01198600+ 119875

119889119860119889= [1198750 119875119889] [

1198600

119860119889

]

= 119873 sdot 119860

(44)


119880 = 119873 sdot 119860 = 119873 sdot 119875minus1 (45)


1198601(119906 V) =

1

2intΩ

(120576119905

119898119888119905120576119898minus 119911120576

119905

119898119888119905120576119887+ 119911

2120576119905

119898119888119905120576119892

minus 119911120576119905

119887119888119905120576119898+ 119911

2120576119905

119887119888119905120576119887minus 119911

3120576119905

119887119888119905120576119892+ 119911

2120576119905

119892119888119905120576119898

minus 1199113120576119905

119892119888119905120576119887+ 119911

4120576119905

119892119888119905120576119892) 119889Ω = 119871 (V)

(46)


1198601(119906 V) =

1

2119905

int119878

ℎ119861119905

119898119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119898119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119898119862119905119861119892119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119887119862119905119861119898119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119887119862119905119861119887119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119887119862119905119861119892119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119892119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119892119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ5

80119861119905

119892119862119905119861119892119877119889120593119889119909

= 119871 (V)

(47)


1198601(119906 V) =

119905

[119870N-T119866]

1198600(119906 V) =

119905

[119870R-M119866]

(48)

where 119870(sdot)



119870N-T119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887+ℎ3

12119861119905

119898119862119905119861119892

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

119870R-M119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887)119877119889120593119889119909

(49)


119871 (V) = 119905

intΩ

119905


119905119891119889Ω +

119905

int120597Ω

119905


119905119892119889D

= 119905

(119865119881+119872) = 0

(50)


119881+119872

119865119881= int

sdot

119878

119873119905119891119890119877119889120593119889119909

119872 = int

sdot

119878

119873119905119892119877119889120593119889119909

(51)



119870119866 = 119865 (52)




119870119890= int

119904119890

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119898119862119905119861119892+ℎ3

12119861119905

119887119862119905119861119887

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

(53)




119870119866=

119873

sum

119890=1

119870119890

119865 =

119873

sum

119890=1

119865119890

(54)







R

BC

D A

L

120579

diaphragmsRigid


3 Numerical Study




3

2

1

0

0 02 04 06 08 1 12214 16 18

2

24

22

18

26

28

3

32


B

C

B

C






0

01

02

03

04

05

06

07


DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)







4 Discussion









minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

u

w

1205791

1205792

120574



119860 (119906 V) = intsdot

Ω

[120582119892120572120573119892120574120588+ 120583 (119892

120572120574119892120573120588+ 119892

120572120588119892120573120574)]

sdot (119890120572120573(119906) minus 119911119896

120572120573(119906) + 119911

2119876120572120573(119906))

(119890120574120588(V) minus 119911119896

120574120588(V) + 1199112119876

120574120588(V)) 119889Ω = 119871 (V)

(36)


face 119892120572120573 = (120583minus1)120572120588(120583

minus1)120573


(120583minus1)120572

120588

(120583minus1)120572

120588= 120575

120572

120588+ 119911119887

120572

120588+

infin

sum

119899ge2

119911119899(119887

119899)120572

120588(37)


1198601(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573]119876

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

5

80 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119876120572120573(119906) + V119876120582

120582119886120572120573]119876

120572120573(V) 119889119878 = 119871 (V)

(38)


1198600(119906 V) =

119864ℎ

1 minus V2

sdot int

sdot

119878

[(1 minus V) 119890120572120573 (119906) + V119890120582120582119886120572120573] 119890

120572120573(V) 119889119878

+119864ℎ

3

12 (1 minus V2)

sdot int

sdot

119878

[(1 minus V) 119896120572120573 (119906) + V119896120582120582119886120572120573] 119896

120572120573(V) 119889119878

(39)




and 119870R-M119866


0(119906 V)


1198601(119906 V) = int

119878

119864119879

V119862119866119864119906119889119878 = int119878

119864119879

V119862119866119864119906 |119869| 119889120585 119889120578 (40)


120576119898= 119890

120572120573= 119863

119898119860 = 119863

119898119875minus1 = 119861

119898 119861

119898= 119863

119898119875minus1

120590119898= 119888120576

119898= 119888119861

119898

(41)


120576119887= 119896

120572120573= 119863

119887119860 = 119863

119887119875minus1 = 119861

119887 119861

119887= 119863

119887119875minus1

120590119887= 119888120576

119887= 119888119861

119887

(42)


120576119892= 119876

120572120573= 119863

119892119860 = 119863

119892119875minus1 = 119861

119892 119861

119892= 119863

119892119875minus1

120590119892= 119888120576

119892= 119888119861

119892

(43)



119880 = 1198800+ 119880

119889= 119875

01198600+ 119875

119889119860119889= [1198750 119875119889] [

1198600

119860119889

]

= 119873 sdot 119860

(44)


119880 = 119873 sdot 119860 = 119873 sdot 119875minus1 (45)


1198601(119906 V) =

1

2intΩ

(120576119905

119898119888119905120576119898minus 119911120576

119905

119898119888119905120576119887+ 119911

2120576119905

119898119888119905120576119892

minus 119911120576119905

119887119888119905120576119898+ 119911

2120576119905

119887119888119905120576119887minus 119911

3120576119905

119887119888119905120576119892+ 119911

2120576119905

119892119888119905120576119898

minus 1199113120576119905

119892119888119905120576119887+ 119911

4120576119905

119892119888119905120576119892) 119889Ω = 119871 (V)

(46)


1198601(119906 V) =

1

2119905

int119878

ℎ119861119905

119898119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119898119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119898119862119905119861119892119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119887119862119905119861119898119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119887119862119905119861119887119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119887119862119905119861119892119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119892119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119892119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ5

80119861119905

119892119862119905119861119892119877119889120593119889119909

= 119871 (V)

(47)


1198601(119906 V) =

119905

[119870N-T119866]

1198600(119906 V) =

119905

[119870R-M119866]

(48)

where 119870(sdot)



119870N-T119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887+ℎ3

12119861119905

119898119862119905119861119892

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

119870R-M119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887)119877119889120593119889119909

(49)


119871 (V) = 119905

intΩ

119905


119905119891119889Ω +

119905

int120597Ω

119905


119905119892119889D

= 119905

(119865119881+119872) = 0

(50)


119881+119872

119865119881= int

sdot

119878

119873119905119891119890119877119889120593119889119909

119872 = int

sdot

119878

119873119905119892119877119889120593119889119909

(51)



119870119866 = 119865 (52)




119870119890= int

119904119890

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119898119862119905119861119892+ℎ3

12119861119905

119887119862119905119861119887

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

(53)




119870119866=

119873

sum

119890=1

119870119890

119865 =

119873

sum

119890=1

119865119890

(54)







R

BC

D A

L

120579

diaphragmsRigid


3 Numerical Study




3

2

1

0

0 02 04 06 08 1 12214 16 18

2

24

22

18

26

28

3

32


B

C

B

C






0

01

02

03

04

05

06

07


DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)







4 Discussion









minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 = 1198800+ 119880

119889= 119875

01198600+ 119875

119889119860119889= [1198750 119875119889] [

1198600

119860119889

]

= 119873 sdot 119860

(44)


119880 = 119873 sdot 119860 = 119873 sdot 119875minus1 (45)


1198601(119906 V) =

1

2intΩ

(120576119905

119898119888119905120576119898minus 119911120576

119905

119898119888119905120576119887+ 119911

2120576119905

119898119888119905120576119892

minus 119911120576119905

119887119888119905120576119898+ 119911

2120576119905

119887119888119905120576119887minus 119911

3120576119905

119887119888119905120576119892+ 119911

2120576119905

119892119888119905120576119898

minus 1199113120576119905

119892119888119905120576119887+ 119911

4120576119905

119892119888119905120576119892) 119889Ω = 119871 (V)

(46)


1198601(119906 V) =

1

2119905

int119878

ℎ119861119905

119898119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119898119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119898119862119905119861119892119877119889120593119889119909

minus1

2119905

int119878

nablaℎ2

8119861119905

119887119862119905119861119898119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119887119862119905119861119887119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119887119862119905119861119892119877119889120593119889119909

+1

2119905

int119878

ℎ3

12119861119905

119892119862119905119861119898119877119889120593119889119909

minus1

2119905

int119878

nablaℎ4

32119861119905

119892119862119905119861119887119877119889120593119889119909

+1

2119905

int119878

ℎ5

80119861119905

119892119862119905119861119892119877119889120593119889119909

= 119871 (V)

(47)


1198601(119906 V) =

119905

[119870N-T119866]

1198600(119906 V) =

119905

[119870R-M119866]

(48)

where 119870(sdot)



119870N-T119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887+ℎ3

12119861119905

119898119862119905119861119892

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

119870R-M119866

= int119878

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119887119862119905119861119887)119877119889120593119889119909

(49)


119871 (V) = 119905

intΩ

119905


119905119891119889Ω +

119905

int120597Ω

119905


119905119892119889D

= 119905

(119865119881+119872) = 0

(50)


119881+119872

119865119881= int

sdot

119878

119873119905119891119890119877119889120593119889119909

119872 = int

sdot

119878

119873119905119892119877119889120593119889119909

(51)



119870119866 = 119865 (52)




119870119890= int

119904119890

(ℎ119861119905

119898119862119905119861119898+ℎ3

12119861119905

119898119862119905119861119892+ℎ3

12119861119905

119887119862119905119861119887

+ℎ3

12119861119905

119892119862119905119861119898+ℎ5

80119861119905

119892119862119905119861119892)119877119889120593119889119909

(53)




119870119866=

119873

sum

119890=1

119870119890

119865 =

119873

sum

119890=1

119865119890

(54)







R

BC

D A

L

120579

diaphragmsRigid


3 Numerical Study




3

2

1

0

0 02 04 06 08 1 12214 16 18

2

24

22

18

26

28

3

32


B

C

B

C






0

01

02

03

04

05

06

07


DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)







4 Discussion









minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














R

BC

D A

L

120579

diaphragmsRigid


3 Numerical Study




3

2

1

0

0 02 04 06 08 1 12214 16 18

2

24

22

18

26

28

3

32


B

C

B

C






0

01

02

03

04

05

06

07


DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)







4 Discussion









minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

01

02

03

04

05

06

07


DKT18DKT12

SFE3CSFE3

Tran

sver

se d

ispla

cem

entW

C(c

m)

WCref

(a)

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0


DKT18DKT12

SFE3CSFE3

WBref

Tran

sver

se d

ispla

cem

entW

B(c

m)

(b)







4 Discussion









minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus2300

minus2250

minus2200

minus2150

minus2100

minus2050

minus2000


WBtimesE+6

(m)

(a)



minus1300

minus1290

minus1280

minus1270

minus1260

minus1250

minus1240

minus1230

minus1220

minus1210

minus1200

WBtimesE+6

(m)

(b)



minus900

minus880

minus860

minus840

minus820

minus800

minus780

minus760

WBtimesE+6

(m)

(c)



minus820

minus800

minus780

minus760

minus740

minus720

minus700

WBtimesE+6

(m)

(d)



minus640

minus630

minus620

minus610

minus600

minus590

minus580

minus570

minus560

WBtimesE+6

(m)

(e)



minus470

minus465

minus460

minus455

minus450

minus445

minus440

minus435

minus430

minus425

WBtimesE+6

(m)

(f)



Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ratio hR

Dev

iatio

ns at

Ctimes10minus6

minus15

minus1

minus05

0

05

1

15

2

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 24 times 4

(a)

Ratio hR

Dev

iatio

ns at

B

times10minus6

05

01

012

50

150

175

02

022

50

250

275

03

032

50

350

375

04

042

50

450

475

14 times 14

12 times 12

10 times 10

8 times 8

6 times 6

2 times 2

4 times 4

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

(b)





119887


and 119870119892119898


119892


R-M119866



119866


+ 119870119892119898

+ 119870119892) to those

found in 119870R-M119866


119898= 120572

119898119864(ℎ119877) 119870

119887= 120572

119887119864(ℎ119877)(ℎ119877)

2 119870119898119892

=

120572119898119892119864(ℎ119877)10

minus1(ℎ119877)

2 and 119870119892= 120572

119892119864(ℎ119877)10

minus1(ℎ119877)


119898119892and



119866asymp 119870

R-M119866


119898119892and the




5 Conclusion











Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Competing Interests


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Approximation of Linear Elastic Shells by ... · today s availability of greatly...

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Transcript of Research Article Approximation of Linear Elastic Shells by ... · today s availability of greatly...