Research ArticleRobustness of Hierarchical Laminated Shell Element Based onEquivalent Single-Layer Theory

Jae S Ahn1 Seung H Yang2 and Kwang S Woo2

1School of General Education Yeungnam University 280 Daehak-ro Gyeongsan Gyeongbuk 712-749 Republic of Korea2Department of Civil Engineering Yeungnam University 280 Daehak-ro Gyeongsan Gyeongbuk 712-749 Republic of Korea

Correspondence should be addressed to Kwang S Woo kswooyuackr

Received 15 September 2014 Accepted 26 February 2015

Academic Editor Sellakkutti Rajendran

Copyright copy 2015 Jae S Ahn et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the hierarchical laminated shell elements with nonsensitivity to adverse conditions for linear static analysis ofcylindrical problems Displacement approximation of the elements is established by high-order shape functions using the integralsof Legendre polynomials to ensure1198620 continuity at the interface between adjacent elements For exact linearmapping of cylindricalshell problems cylindrical coordinate is adopted To find global response of laminated composite shells equivalent single-layertheory is also considered Thus the proposed elements are formulated by the dimensional reduction from three-dimensional solidto two-dimensional plane which allows the first-order shear deformation and considers anisotropy due to fiber orientation Thesensitivity tests are implemented to show robustness of the present elements with respect to severe element distortions very highaspect ratios of elements and very large radius-to-thickness ratios of shells In addition this element has investigated whethermaterial conditions such as isotropic and orthotropic properties may affect the accuracy as the element distortion ratio is increasedThe robustness of present element has been compared with that of several shell elements available in ANSYS program

1 Introduction

Finite element methods generally involve finding approxi-mate solutions in a space of piecewise polynomials of degreewhich is often designated by the letter 119901 on a grid of meshsize ℎ In general conventional finite elements based onmeshrefinement have given reliable solutions when discretizingmesh is regular while poor performance would be obtainedwhen the element geometry is distorted Also it is knownwell that the interpolation precision of quadrilateral finiteelements deteriorates if the element geometry considerablydifferentiated itself with a square In this sense loss of elementperformance is commonly associated with a gradual increaseof the stiffness leading to sort of locking the elementrsquosresponse It brings about the inability of the element tofind a good approximation of the solution Moreover thingswould turn for the worse when conventional low-order finiteelement formulation is used in thin plates or shells sinceshear or membrane locking phenomena arise Various robustschemes have been suggested for the problems involvinglocking One possibility is to use higher-order elements It is

known that the locking completely eliminates certain typesof locking [1]The high-order finite element implementationshave also been advocated in recent years as a means ofeliminating the locking phenomena completely The relevantworks have been implemented by some researchers [2ndash5]Theissue of locking using lower-order elements has been mostprominently addressed through the use of low-order finitetechnology using some mixed variational principles [6ndash8]However these depend upon reformulating the problems inspecial ways which have not been required in higher-orderelements In this paper wewill address only the finite elementformulation for laminated shell behavior using the 119901-versionapproach The first successful 119901-version formulation relatedto shells was reported byWoo andBasu [9] who presented thehierarchical 1198620-shell element formulation in the cylindricalcoordinates associated with a suitable transfinite mappingfunction to represent the curved geometry This elementshowed not only a constant strain and a rigid body motionfrom patch and eigenvalue tests but also a strong robustnesswith regard to very large aspect ratio and severely distorted

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 301054 9 pageshttpdxdoiorg1011552015301054

2 Mathematical Problems in Engineering

mesh The proposed hierarchical 1198620-shell element can alsobe successfully extended to singularity problems includinga crack and a cut-out Surana and Sorem [10] developed athree-dimensional curved shell element with a higher-orderhierarchical displacement approximation in the thicknessdirection of the shell The extension of this idea to laminatedplates and shells was accomplished by the same authors [11]Laminated plates and shells have usually been analyzed bythe use of ESL (equivalent single-layer) theories based oneither the classical Kirchhoff-Love hypothesis or first-ordershear deformation theories A drawback of ESL theoriesin modeling composite laminated plates and shells is thatthe transversal strain components are continuous acrossinterfaces between dissimilar materials for the perfectlybonded layers Therefore the transverse stress componentsare discontinuous at the layer interfaces On the other handlayerwise theories are considerably more accurate than thepreceding theories Thus Ahn andWoo [12 13] extended the119901-version approach to efficient ldquomixed model analysisrdquo oftencalled ldquoglobal-local approachrdquo to obtain interlaminar stressesat free edges in the laminated composite plate under exten-sion and flexure on the basis of layerwise theoriesThe aim ofthis study is to present a simple high-order hierarchical 1198620-element for laminated composite shells using ESL conceptsprior to layerwise theories in the cylindrical coordinatesThisapproach is computationally less expensive as compared tothose obtained by three-dimensional elasticity solutions andlayerwise finite element models

In general the most important symptoms of accuracyfailure in modern finite elements are spurious mechanismlocking elementary defects like violation of rigid body prop-erty and invariance to node numbering Also parameterswhich affect accuracy are loading element geometry prob-lem geometry material properties material anisotropy andso on Because of these reasons governmental concern in theUSA for the accuracy of finite element analysis is evidencedby NRC (Nuclear Regulatory Commission) requirement forstructural analysis computer program validation and theformation of NAFEM (National Agency for Finite ElementMethods and Standards) in the United Kingdom

In this study the sensitivity test has been carried outto verify the robustness of proposed element in relation todistortion effect ofmesh very high aspect ratio and very largeradius-to-thickness ratio In addition to these the orthotropiccylindrical shell stacking with different fiber orientations istested to check whether the anisotropy ofmaterialsmay affectthe accuracy as the element distortion ratio is increasedThe numerical solutions obtained by present element arecompared with several shell elements available in ANSYSprogram [14]

2 Formulation of HierarchicalLaminated Shell Element

21 Hierarchical Shape Functions for Quadrilaterals A stan-dard quadrilateral element is shown in Figure 1 Two-dimensional shape functions in the element can be classified

120585

120578

Node 1 Node 2

Node 3Node 4

Edge 1

Edge 2

Edge 3

Edge 40

(1 1)

(1 minus1)(minus1 minus1)

(minus1 1)

Figure 1 Standard quadrilateral element

into three groups such as nodal edge and internal shapefunctions The nodal shape functions can be defined as

119878119873119894

11(120585 120578) = 025 (1 + 120585

119894120585) (1 + 120578

119894120578) 119894 = 1 2 3 4 (1)

120585119894and 120578

119894denote the local coordinate of the 119894th node in the

Figure 1 Also the edge shape functions with any order 119901which vanish in all other edges are defined separately for eachindividual edge Consider

119878119864119898

1198941(120585 120578) = 05 (1 + 120575

120585119898120585119899120585119898120585 + 120575120578119898120578119899

120578119898120578)

sdot Φ119894(

10038161003816100381610038161003816100381610038161003816

120585119898minus 120585119899

2

10038161003816100381610038161003816100381610038161003816

120585 +

1003816100381610038161003816100381610038161003816

120578119898minus 120578119899

2

1003816100381610038161003816100381610038161003816120578)

119894 = 2 3 119901 119898 = 1 2 3 4 119899 =

119898 + 1 119898 = 4

1 119898 = 4

(2)

The quantity 120575 refers to the Kronecker tensor 120585119898

and 120578119899

denote the local coordinate of both nodes in the 119898th edgeshown in the Figure 1 The functions Φ

119894which have one

independent variable of the two variables 120585 and 120578 are definedas follows

Φ119894(120585) = radic

2119894 minus 1

2int

120585

minus1

119871119894minus1(119909) 119889119909 119894 = 2 3 119901 (3)

where the functions 119871 are the Legendre polynomials Thesefunctions with integrals of Legendre polynomials are wellsuited for computer implementation and have very favorableproperties from the point of view of numerical stability [15]Lastly there are 05(119901 minus 2)(119901 minus 3) internal shape functions in119901 ge 4 These can be written as

119878int119894119895(120585 120578) = Φ

119894 (120585)Φ119895 (120578) 119894 119895 = 2 3 119894 + 119895 = 119901 (4)

22 Geometry Fields One element with a curved surface ison a cylindrical shell shown in Figure 2(a) Linear mappingon a general Cartesian coordinate is not accurate enough for

Mathematical Problems in Engineering 3

120585

120578

Node 1 Node 2

Node 3Node 4

Edge 1

Edge 2

Edge 3

Edge 40

Node 2

Node 1

Node 3

Node 4

Edge 1

Edge 2Edge 3

Edge 4

120579

R

r

x

(a) One plane element with cylindrical shape (b) Standard quadrilateral element

(x1 1205791)

(x2 1205792)

(x3 1205793)

(x4 1205794) (1 1)

(1 minus1)(minus1 minus1)

(minus1 1)

Figure 2 Mapping concept for planes

the element In the case of degenerated shell concept basedon ℎ-FEMwhich is normally adopted in commercial codes anodal coordinate system defined at each nodal point with anyreference surface is considered [16] However the mappingtechnique explains the curved surface approximately notexactly in which the size of the elements is made smallenough to follow the curved surface as closely as desired

In 119901-FEM this option is not acceptable and to avoid geo-metrical sources of error exactmapping of a curved surface isnecessary because one of the sources of efficiency of119901-FEM isthe use of as few elements as possible Woo and Basu [9] pro-posed the transfinitemappingwith trigonometric polynomialinterpolation projects based on blending function methodof 119901-FEM The mapping technique ought to be consideredfor circular or oval hole to highlight modeling simplicity of119901-FEM However as surfaces on cylindrical shells are onlyconsidered in the present study the simple linear mappingon cylindrical coordinate shown in Figure 2(a) is enoughFigures 2(a) and 2(b) present the mapping concept of thecomputational domainwith the shape functions of four nodesin (1) aforementioned For the mapping process of the curvedsurfaces on cylindrical shells with constant curvature radius119877 along an axial direction the equation can be written as

∬119860

() 119877 119889120579 119889119909 = ∬

1

minus1

()120597 (119909 120579)

120597 (120585 120578)119877 119889120578 119889120585 (5)

23 Displacement Fields For modeling of composite struc-tures ESL elements (Figure 3) derived from 119862

1

119903function the-

ories [17] in a laminated system are obtained by dimensional

x

r

NodesSide modesInternal modes

Layer 1Layer 2

Layer 3

O Constant curvature 120581

Figure 3 Modeling scheme of the hierarchical laminated shellelement

reduction from three-dimensional solid to two-dimensionalrepresentative surface satisfying plane stress theory formem-brane behavior and first-order shear deformation theory forbending behavior

Thus in the case of a quadrilateral subparametric1198620119909120579ele-

ment there are three translational displacement componentsand two rotational components corresponding to each vertexside and internal mode For ESL model of an element with

4 Mathematical Problems in Engineering

any 119901-level the two in-plane displacement components canbe defined as

119906119888= 119878119873119894

11(120585 120578) (119860

119888

119894+ 120577119861119888

119894) + 119878119864119898

1198941(120585 120578) (119862

119888

119894+ 120577119863119888

119894)

+ 119878int119894119895(120585 120578) (119864

119888

119894119895+ 120577119865119888

119894119895) in 119888 = 1 2

(6)

Here the shape function functions 119878 are defined in (1) (2)and (4) Also 119860 and 119861 are the nodal variables for translationand rotation corresponding to the four nodes Likewise thevalues (119862 119864) and (119863 119865) denote translational and rotationalnodal variables corresponding to side and internal modesrespectively The variable 120577 denotes the distance from thereference surface in the thickness direction of a point ofinterest on standard coordinateThe transverse displacementfield of a point in the ESL element with any 119901-level is definedas

119908 = 119878119873119894

11(120585 120578) 119866

119894+ 119878119864119898

1198941(120585 120578)119867

119894+ 119878

int119894119895(120585 120578) 119869

119894119895 (7)

In this equation 119866 refers to the nodal variable for the fournodes and119867 and 119869 correspond to the nodal variables for sideand internal modes respectively Equation (7) connotes thatthe lateral displacement is independent of thickness coordi-nate which is contrary to in-plane displacement componentsThis choice signifies that the transverse normal stress is zeroThe constitutive relationship for the ESL element with respectto the local element coordinate system (119909 119910 119911) in referencesurface can be defined as

⟨119909120579119903

⟩T1times8

= [119871]8times8 ⟨120576119909120579119903⟩T1times8

(8)

where [119871]8times8

refers to the full elasticity matrix of a 119899-layeredESL element and is composed of submatrices as shown belowConsider

[119871]8times8 =

119899

sum

119894=12

[[[

[

[119864]119894

3times3[119862]119894

3times3[0]2times2

[119862]119894

3times3[119861]119894

3times3[0]2times2

[0]2times2 [0]2times2 [119876]119894

2times2

]]]

]8times8

(9)

Here [119864] is the extensional elasticity matrix [119861] is the bend-ing elasticity matrix [119862] is the coupling between bendingand extensional elasticitymatrices [119876] is the transverse-shearelasticity matrix and [0] denotes a null matrix The sum-mation accounts for the contributions from all the 119899 layersFor a typical layer 119897 the submatrices ([119864]119897

3times3 [119862]119897

3times3 [119876]119897

2times2)

are calculated using its elasticity matrix [119863]119897

5times5allowing

for anisotropy with three mutually orthogonal planes ofsymmetry The elasticity matrix [119863]

119897

5times5with reference to

the elemental coordinate system is obtained from the elas-ticity matrix [119863]119897

5times5referred to the material axes by using

the coordinate transformation matrix [119867]5times5

The resultingtransformation relationship takes the following triple productstandard form

[119863]119897

5times5= [119867]

T5times5

[119863]119897

5times5[119867]5times5 (10)

In (10) the elasticity matrix [119863]1198975times5

includes shear correctionfactors in order to allow for the error resulting from the use of

transverse-shear strain energy on an average basis In the caseof heterogeneous plates this factor is based on equilibriumequations and strain energy components used in [18] Theindividual strain components with respect to 119909 120579 and 119903 axesshown in (8) can be identified as

⟨120576119909120579119903

⟩T1times8

= ⟨120576119898

119909119909120576119898

120579120579120576119898

119909120579120576119887

119909119909120576119887

120579120579120576119887

119909120579120576119905

119909119903120576119905

120579119903⟩T

1times8

(11)

where superscripts 119898 119887 and 119905 refer to membrane bendingand transverse strain components respectively The totalstrain vector for an arbitrary point will have five componentsas

⟨120576119909120579119903

⟩T1times5

= ⟨120576119909119909 120576120579120579 120576119909120579 120576119909119903 120576120579119903⟩T (12)

For a point at a distance 119911 from the reference surfacethe flexural components of planar strains can be separated asfollows

⟨120576119909120579119903

⟩T1times5

= ⟨120576119898

119909119909120576119898

120579120579120576119898

119909120579120576119905

119909119903120576119905

120579119903⟩T

+ 119911 ⟨120576119887

119909119909120576119887

120579120579120576119887

1199091205790 0⟩

T

(13)

Based on the strain vector in (8) the general strain vector foran ESL element can be expressed as

⟨120576119909120579119903

⟩T1times8

= [119861]8times119902 ⟨120575⟩T1times119902

(14)

where [119861] is the strain matrix composed of derivatives ofhierarchical shape functions and ⟨120575⟩ is the vector of variablesof the ESL element with any number 119902 of degrees of freedomrelated to 119901 order The corresponding stress resultants in anylayer 119894 can now be expressed as

1198733times1

=

119899

sum

119894=1

int

119903top119894

119903bot119894

[[[

[

11986311

11986312

11986313

11986321

11986322

11986323

11986331

11986332

11986333

]]]

]

119894

120576119898

119909119909+ 119911120576119887

119909119909

120576119898

120579120579+ 119911120576119887

120579120579

120576119898

119909120579+ 119911120576119887

119909120579

119889119903

1198723times1

=

119899

sum

119894=1

int

119903119911top119894

119903bot119894

[[[

[

11986311

11986312

11986313

11986321

11986322

11986323

11986331

11986332

11986333

]]]

]

119894

119911120576119898

119909119909+ 1199112120576119887

119909119909

119911120576119898

120579120579+ 1199112120576119887

120579120579

119911120576119898

119909120579+ 1199112120576119887

119909120579

119889119903

1198772times1 =

119899

sum

119894=1

int

119903top119894

119903bot119894

[11986344

11986345

11986354

11986355

]

119894

120574119905

119909119903

120574119905

120579119903

119889119903

(15)

where 119873 119872 and 119877 are membrane force bending momentand transversal shear resultants respectively 119903bot and 119903top arethe 119903-axes of bottom and top surfaces of a typical layer withrespect to the reference surface

Mathematical Problems in Engineering 5

z

P

P

y

x

Computational region

L2 L2

(a) Pinched cylinder

P4

Δ2

Δ1

L1

L2x

y

z

Free

Symmetry

Symmetry

Symmetry

R

t

L4

L4

(b) Finite element modeling in computationaldomain

Figure 4 Geometric configuration and mesh refinement of pinched cylinder problem

3 Sensitivity to Geometric Parameters

In this study the sensitivity test has been carried out to verifythe robustness of present elementwith respect to three criticalconditions including severe element distortion very highaspect ratio and very large thickness ratio The cylindricalshell with length radius and thickness dimensions specifiedby 119871 119877 and 119905 respectively is subjected to a pinch load119875 as shown in Figure 4 Due to the symmetry conditionone octant of the shell is modeled by 2 times 2 mesh designThe sensitivity to input parameters by the present elementwith different 119901-levels has been compared with several shellelements available in ANSYS program which are two ℎ-convergent elements of SHELL181 (4-node) and SHELL281(8-node) and one 119901-convergent element of SHELL150 In thecase of SHELL150 element the shape functions are formu-lated by Lagrangian polynomials and curvilinear mappingtechnique is adopted in Cartesian coordinate On the otherhand linear exact mapping in cylindrical coordinate is usedin the present 119901-FEM For three cases of sensitivity teststwo material properties are considered in (16) and (17) asfollows

Case 1 Isotropy is as follows

119864 = 105 times 106

] = 03125

119871 = 1035

119877 = 4953

119905 = 0094

119875 = 100

(16)

Case 2 Orthotropy is as follows

1198641= 20 times 10

7

1198642= 10 times 10

6

11986612= 11986613= 50 times 10

5

11986623= 20 times 10

5

]12= 025

(17)

Also the numerical results are normalized using the classicalsolution or the estimated exact solutionThus the normalizedcentral deflection is used in this study which is defined in

119908 =119908max119908ref

(18)

Here 119908max is the maximum deflection in the center of shellsthat is obtained by numerical analysis and119908ref represents theestimated exact solution

31 Effect of ElementDistortion In this section the sensitivityof the present element has been tested with respect tosevere element distortion for both isotropic and orthotropiccases The material conditions are assumed to be isotropic ifthere is no mention about material property To check theelement accuracy for element distortion effects we employthe element distortion ratios denoted by 120572 to the centralnode of the 2 times 2 mesh when 119877119905 is fixed as 53 Thus theelement distortion ratios 120572 are defined by (19) In the casesof SHELL181 and SHELL281 two mesh types (2 times 2 and 4 times4) are investigated to model one octant of the shell exploitingsymmetry

120572def=Δ1

1198711

times 100 () def=Δ1

1198712

times 100 () (19)

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

1 2 3 4 5 6 7

ANSYS [

14

Nor

mal

ized

max

imum

defl

ectio

nw

Present [2 times 2 mesh]2 times 2 mesh]

p-level

Takemoto and Cook [1973]

SHELL150

Figure 5 119901-convergence of normalized maximum deflection when120572 = 0 and 119877119905 = 53

The convergence characteristics of the present elementswith respect to different 119901-levels are plotted in Figure 5when 120572 = 0 and 119877119905 = 53 From this figure it isseen that the convergence begins from 119901 = 5 by usingthe hierarchical shell elements as well as the SHELL150elements based on 119901-FEM Those results are compared withthe analytical solution by Takemoto and Cook [19] Thepresent solution virtually converges to the normalized exactsolution denoted by 10 However the numerical solutionby SHELL150 elements is converged to upper bound of12 This tendency is mainly due to the use of differentshape functions and mapping technique for curved bound-ary Linear mapping technique on rectangular coordinate isconsidered as interpolation functions of geometry fields forSHELL150 elements Therefore SHELL150 elements requiremesh refinement for curved shapes to improve accuracyalthough they are 119901-FEM In Table 1 the numerical resultsobtained by the present elements are tabulated for ] = 03125with different element distortion ratios and compared withthe references It is noted that the hierarchical shell elementsbased on 2 times 2 mesh with 119901 = 5 show an excellent behavioreven for extremely distorted mesh of 120572 = 40 All resultsare very close to the normalized displacement 10 whichrepresents the approximate solutions equal to exact solutionHowever other solutions by SHELL181 and SHELL281 showpoor accuracy and behave relatively stiff and converge veryslowly even though the meshes are refined up to 16 elements(4 times 4 mesh) Thus it is concluded that numerical solutionsin references by ANSYS program degrade significantly withincreasing element distortion These remarks can be recon-firmed by the graphical illustration as shown in Figure 6From this figure it is advised that the element distortion ratioshould not exceed roughly 10 for 4-node and 20 for 8-node ℎ-convergent shell elements to get good displacementresults On the contrary the effect of element distortion onthe numerical results by the present element is plotted inFigure 7 as the element distortion ratios 120572 vary from 00

Table 1 Comparison of normalized maximum deflection withrespect to element distortion ratio

120572 () 0 10 20 30 40

ℎ-FEM

SHELL181 (2 times 2) 0617 0363 0158 0026 0052SHELL181 (4 times 4) 0880 0807 0278 0194 0030SHELL281 (2 times 2) 0975 0937 0855 0740 0616SHELL281 (4 times 4) 0996 0975 0908 0792 0646

p-FEM

SHELL150 (2 times 2)119901 = 4

1161 1159 1150 1120 1055

SHELL150 (2 times 2)119901 = 5

1168 1167 1166 1163 1159

Present (2 times 2)119901 = 4

0979 0979 0974 0954 0909

Present (2 times 2)119901 = 5

0988 0988 0987 0985 0980

0

02

04

06

08

1

12

14

0 10 20 30 40

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 5]p = 5]

ANSYS [4 times 4 mesh]ANSYS [4 times 4 mesh]

ANSYS [2 times 2 mesh

Takemoto and Cook [1973]

SHELL150SHELL281SHELL181

Figure 6 Variation of normalized maximum deflection withincreasing distortion parameter 120572

to 90 on 2 times 2 mesh design It is observed that the presentelement shows strong robustness to severe element distortioneven for extreme case of 120572 = 90 especially for 119901 = 6

Next example is another pinched cylindrical problemcomposed of orthotropic materials defined in (17) Theorthotropic pinched cylindrical shell stacking with differentfiber orientations is tested in order to check whether materialconditions may affect the accuracy as the element distortionratio is increased For this purpose the laminated cylindricalshell with a pinch load is modeled by two layers with cross-ply (0∘90∘) Similar to the isotropic problem previously thepresent elements may endure severe element distortion up to120572 = 90 from 119901 = 6 The details of numerical solutionsare presented in Table 2 as the 119901-level increases from 1 to10 It is noticed that the present shell elements based on 119901-FEM exhibit strong robustness with respect to severe elementdistortion regardless of the anisotropy of materials

Mathematical Problems in Engineering 7

Table 2 p-convergence of normalized maximum deflection with increasing distortion ratios for orthotropic materials (0∘90∘)

120572 () 0 10 20 30 40 50 60 70 80 90119901 = 1 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002119901 = 2 0029 0029 0029 0029 0029 0028 0027 0029 0015 0026119901 = 3 0681 0640 0548 0414 0280 0176 0134 0081 0074 0069119901 = 4 0966 0966 0964 0952 0918 0846 0733 0549 0462 0367119901 = 5 0979 0979 0978 0975 0968 0952 0928 0891 0874 0813119901 = 6 0985 0985 0985 0985 0985 0985 0984 0982 0979 0972119901 = 7 0989 0989 0989 0989 0989 0989 0989 0989 0989 0989119901 = 8 0991 0991 0991 0991 0992 0992 0992 0992 0992 0992119901 = 9 0993 0993 0993 0993 0993 0993 0993 0993 0993 0993119901 = 10 0994 0994 0994 0994 0994 0994 0994 0994 0994 0994

0

02

04

06

08

1

12

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 4]Present [2 times 2 mesh p = 5]Present [2 times 2 mesh p = 6]

Takemoto and Cook [1973]

Figure 7 Robustness of hierarchical shell element with respect todistortion parameter 120572

32 Effect of Aspect Ratios of Elements In general Robinson[20] andMacneal and Harder [21] suggested a single elementtest in which response is examined as the element aspect ratiois changed It is noticed that element aspect ratios shouldnot exceed roughly seven for good displacement resultsand roughly three for good stress results Of course thisconclusion can be changed according to types of problemsHowever since large aspect ratio is one of the sources causingnumerical errors it is desirable to make the aspect ratio oneespecially in the vicinity of a singular point The aspect ratio120573 is defined by (20) and 119871

3and 119871

4are denoted in Figure 8

Consider

120573 =1198714

1198713

(20)

The finite element results obtained by present elements with119901 = 5 are presented in Table 3 with respect to aspect ratio 120573and comparedwith reference values by SHELL181 SHELL281and SHELL150 (119901 = 5) All results are based on 2 times 2mesh design When the aspect ratio 120573 is 10 the normalizedtransverse deflections at the center by SHELL281 SHELL

P4

L4

L3

x

y

z

R

L4

L4

Figure 8 Mesh design associated with aspect ratio

150 and present elements are close to an exact solutionHowever the results by SHELL181 elements fail to convergePrecisely speaking the relative errors of present elements toan exact solution are below 3 until 120573 = 1000 On the otherhand the relative errors of SHELL281 and SHELL150 showapproximately more than 10 from the aspect ratio 120573 = 20The graphical solution is also presented in Figure 9 to easilyunderstand the deviation of numerical errors in referenceto the exact solution The tolerance of present elements isrepresented in Figure 10 as the aspect ratios 120573 are increasedfrom 1 to 1000 It is seen that the hierarchical shell elementtolerates the large aspect ratios up to 1000 under 3 accuracyof relative error

33 Effect of Thickness Ratios In order to investigate themembrane and shear locking phenomena of hierarchicalshell elements the central deflections have been calculatedunder very large radius-to-thickness ratios (119877119905) Thus thetolerance of numerical solutions has been exhibited in Table 4in regard to 119877119905 ratios ranging from 10 (thick) to 1000(extremely thin) Present solutions show good agreementwith those by 8-node shell element denoted by SHELL281available in ANSYS program It is noted that both elementsavoid the shear locking regardless of extremely thin case

8 Mathematical Problems in Engineering

Table 3 Change of the normalized maximum deflection with increasing aspect ratio

120573 1 2 4 6 8 10 20 50 100 500 1000

h-FEM (2 times 2) SHELL 181 0616 0609 0588 0551 0499 0431 0347 0248 0133 0133 0133SHELL 281 0974 0971 0965 0956 0944 0930 0913 0894 0872 0872 0872

p-FEM (2 times 2)SHELL 150119901 = 5

1170 1159 1147 1134 1123 1118 1121 1133 1156 1159 1161

Present119901 = 5

0990 0986 0982 0980 0978 0976 0974 0972 0971 0970 0970

Table 4 Comparison of central deflection with increasing 119877119905

ratios

119877119905 10 100 500 1000SHELL281(4 times 4) 000088 07615 931720 7430300

Present(2 times 2 119901 = 5) 000087 07558 928663 7416710

0

02

04

06

08

1

12

14

1 2 3 4 5 6 7 8 9 10 11

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 5][p = 5]ANSYS

ANSYS ANSYS

Takemoto and Cook [1973]

SHELL150

SHELL281SHELL181

Figure 9 Variation of normalized maximum deflections withincreasing aspect ratios 120573

4 Conclusions

In the mesh design of the finite element analysis of shellsthe sources of numerical errors such as severe elementdistortions very high aspect ratios and very large radius-to-thickness ratios cause the numerical instability of a stiffnessmatrixThe proposed hierarchical shell elements based on 119901-FEM have been tested from the point of view of the elementrobustnessThe obtained results and subsequent future workscan be summarized as follows

(1) It is observed that the present elements show strongrobustness to severe element distortion even forextreme case of 120572 = 90 especially for 119901 = 6

(2) It is noticed that the hierarchical shell elements basedon 119901-FEM exhibit a strong robustness with respect

0

02

04

06

08

1

12

1 2 4 6 8 10 20 50 100 500 1000

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 4]Present [p = 5]Present [p = 6]

Takemoto and Cook [1973]

Figure 10 Robustness of hierarchical shell elements with respect toaspect ratios 120573

to severe element distortion regardless of cross-plylamination scheme of materials

(3) It is seen that the proposed elements tolerate the largeaspect ratio up to 1000 under 3 accuracy of relativeerror

(4) It is noted that the present elements avoid the shearand membrane locking regardless of the thin casesextremely denoted by 119877119905 = 1000

(5) In this paper performance of the present elementsis only limited to cylindrical shells based on linearanalysis From these results in future it is necessary toinvestigate geometrical nonlinearity and critical shellshapes such as spherical hyperbolic shells whichwould differ with cylindrical shells

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

This work was supported by the 2014 Yeungnam UniversityResearch Grant

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

mesh The proposed hierarchical 1198620-shell element can alsobe successfully extended to singularity problems includinga crack and a cut-out Surana and Sorem [10] developed athree-dimensional curved shell element with a higher-orderhierarchical displacement approximation in the thicknessdirection of the shell The extension of this idea to laminatedplates and shells was accomplished by the same authors [11]Laminated plates and shells have usually been analyzed bythe use of ESL (equivalent single-layer) theories based oneither the classical Kirchhoff-Love hypothesis or first-ordershear deformation theories A drawback of ESL theoriesin modeling composite laminated plates and shells is thatthe transversal strain components are continuous acrossinterfaces between dissimilar materials for the perfectlybonded layers Therefore the transverse stress componentsare discontinuous at the layer interfaces On the other handlayerwise theories are considerably more accurate than thepreceding theories Thus Ahn andWoo [12 13] extended the119901-version approach to efficient ldquomixed model analysisrdquo oftencalled ldquoglobal-local approachrdquo to obtain interlaminar stressesat free edges in the laminated composite plate under exten-sion and flexure on the basis of layerwise theoriesThe aim ofthis study is to present a simple high-order hierarchical 1198620-element for laminated composite shells using ESL conceptsprior to layerwise theories in the cylindrical coordinatesThisapproach is computationally less expensive as compared tothose obtained by three-dimensional elasticity solutions andlayerwise finite element models

In general the most important symptoms of accuracyfailure in modern finite elements are spurious mechanismlocking elementary defects like violation of rigid body prop-erty and invariance to node numbering Also parameterswhich affect accuracy are loading element geometry prob-lem geometry material properties material anisotropy andso on Because of these reasons governmental concern in theUSA for the accuracy of finite element analysis is evidencedby NRC (Nuclear Regulatory Commission) requirement forstructural analysis computer program validation and theformation of NAFEM (National Agency for Finite ElementMethods and Standards) in the United Kingdom

In this study the sensitivity test has been carried outto verify the robustness of proposed element in relation todistortion effect ofmesh very high aspect ratio and very largeradius-to-thickness ratio In addition to these the orthotropiccylindrical shell stacking with different fiber orientations istested to check whether the anisotropy ofmaterialsmay affectthe accuracy as the element distortion ratio is increasedThe numerical solutions obtained by present element arecompared with several shell elements available in ANSYSprogram [14]

2 Formulation of HierarchicalLaminated Shell Element

21 Hierarchical Shape Functions for Quadrilaterals A stan-dard quadrilateral element is shown in Figure 1 Two-dimensional shape functions in the element can be classified

120585

120578

Node 1 Node 2

Node 3Node 4

Edge 1

Edge 2

Edge 3

Edge 40

(1 1)

(1 minus1)(minus1 minus1)

(minus1 1)

Figure 1 Standard quadrilateral element

into three groups such as nodal edge and internal shapefunctions The nodal shape functions can be defined as

119878119873119894

11(120585 120578) = 025 (1 + 120585

119894120585) (1 + 120578

119894120578) 119894 = 1 2 3 4 (1)

120585119894and 120578

119894denote the local coordinate of the 119894th node in the

Figure 1 Also the edge shape functions with any order 119901which vanish in all other edges are defined separately for eachindividual edge Consider

119878119864119898

1198941(120585 120578) = 05 (1 + 120575

120585119898120585119899120585119898120585 + 120575120578119898120578119899

120578119898120578)

sdot Φ119894(

10038161003816100381610038161003816100381610038161003816

120585119898minus 120585119899

2

10038161003816100381610038161003816100381610038161003816

120585 +

1003816100381610038161003816100381610038161003816

120578119898minus 120578119899

2

1003816100381610038161003816100381610038161003816120578)

119894 = 2 3 119901 119898 = 1 2 3 4 119899 =

119898 + 1 119898 = 4

1 119898 = 4

(2)

The quantity 120575 refers to the Kronecker tensor 120585119898

and 120578119899

denote the local coordinate of both nodes in the 119898th edgeshown in the Figure 1 The functions Φ

119894which have one

independent variable of the two variables 120585 and 120578 are definedas follows

Φ119894(120585) = radic

2119894 minus 1

2int

120585

minus1

119871119894minus1(119909) 119889119909 119894 = 2 3 119901 (3)

where the functions 119871 are the Legendre polynomials Thesefunctions with integrals of Legendre polynomials are wellsuited for computer implementation and have very favorableproperties from the point of view of numerical stability [15]Lastly there are 05(119901 minus 2)(119901 minus 3) internal shape functions in119901 ge 4 These can be written as

119878int119894119895(120585 120578) = Φ

119894 (120585)Φ119895 (120578) 119894 119895 = 2 3 119894 + 119895 = 119901 (4)

22 Geometry Fields One element with a curved surface ison a cylindrical shell shown in Figure 2(a) Linear mappingon a general Cartesian coordinate is not accurate enough for

Mathematical Problems in Engineering 3

120585

120578

Node 1 Node 2

Node 3Node 4

Edge 1

Edge 2

Edge 3

Edge 40

Node 2

Node 1

Node 3

Node 4

Edge 1

Edge 2Edge 3

Edge 4

120579

R

r

x

(a) One plane element with cylindrical shape (b) Standard quadrilateral element

(x1 1205791)

(x2 1205792)

(x3 1205793)

(x4 1205794) (1 1)

(1 minus1)(minus1 minus1)

(minus1 1)

Figure 2 Mapping concept for planes

the element In the case of degenerated shell concept basedon ℎ-FEMwhich is normally adopted in commercial codes anodal coordinate system defined at each nodal point with anyreference surface is considered [16] However the mappingtechnique explains the curved surface approximately notexactly in which the size of the elements is made smallenough to follow the curved surface as closely as desired

In 119901-FEM this option is not acceptable and to avoid geo-metrical sources of error exactmapping of a curved surface isnecessary because one of the sources of efficiency of119901-FEM isthe use of as few elements as possible Woo and Basu [9] pro-posed the transfinitemappingwith trigonometric polynomialinterpolation projects based on blending function methodof 119901-FEM The mapping technique ought to be consideredfor circular or oval hole to highlight modeling simplicity of119901-FEM However as surfaces on cylindrical shells are onlyconsidered in the present study the simple linear mappingon cylindrical coordinate shown in Figure 2(a) is enoughFigures 2(a) and 2(b) present the mapping concept of thecomputational domainwith the shape functions of four nodesin (1) aforementioned For the mapping process of the curvedsurfaces on cylindrical shells with constant curvature radius119877 along an axial direction the equation can be written as

∬119860

() 119877 119889120579 119889119909 = ∬

1

minus1

()120597 (119909 120579)

120597 (120585 120578)119877 119889120578 119889120585 (5)

23 Displacement Fields For modeling of composite struc-tures ESL elements (Figure 3) derived from 119862

1

119903function the-

ories [17] in a laminated system are obtained by dimensional

x

r

NodesSide modesInternal modes

Layer 1Layer 2

Layer 3

O Constant curvature 120581

Figure 3 Modeling scheme of the hierarchical laminated shellelement

reduction from three-dimensional solid to two-dimensionalrepresentative surface satisfying plane stress theory formem-brane behavior and first-order shear deformation theory forbending behavior

Thus in the case of a quadrilateral subparametric1198620119909120579ele-

ment there are three translational displacement componentsand two rotational components corresponding to each vertexside and internal mode For ESL model of an element with

4 Mathematical Problems in Engineering

any 119901-level the two in-plane displacement components canbe defined as

119906119888= 119878119873119894

11(120585 120578) (119860

119888

119894+ 120577119861119888

119894) + 119878119864119898

1198941(120585 120578) (119862

119888

119894+ 120577119863119888

119894)

+ 119878int119894119895(120585 120578) (119864

119888

119894119895+ 120577119865119888

119894119895) in 119888 = 1 2

(6)

Here the shape function functions 119878 are defined in (1) (2)and (4) Also 119860 and 119861 are the nodal variables for translationand rotation corresponding to the four nodes Likewise thevalues (119862 119864) and (119863 119865) denote translational and rotationalnodal variables corresponding to side and internal modesrespectively The variable 120577 denotes the distance from thereference surface in the thickness direction of a point ofinterest on standard coordinateThe transverse displacementfield of a point in the ESL element with any 119901-level is definedas

119908 = 119878119873119894

11(120585 120578) 119866

119894+ 119878119864119898

1198941(120585 120578)119867

119894+ 119878

int119894119895(120585 120578) 119869

119894119895 (7)

In this equation 119866 refers to the nodal variable for the fournodes and119867 and 119869 correspond to the nodal variables for sideand internal modes respectively Equation (7) connotes thatthe lateral displacement is independent of thickness coordi-nate which is contrary to in-plane displacement componentsThis choice signifies that the transverse normal stress is zeroThe constitutive relationship for the ESL element with respectto the local element coordinate system (119909 119910 119911) in referencesurface can be defined as

⟨119909120579119903

⟩T1times8

= [119871]8times8 ⟨120576119909120579119903⟩T1times8

(8)

where [119871]8times8

refers to the full elasticity matrix of a 119899-layeredESL element and is composed of submatrices as shown belowConsider

[119871]8times8 =

119899

sum

119894=12

[[[

[

[119864]119894

3times3[119862]119894

3times3[0]2times2

[119862]119894

3times3[119861]119894

3times3[0]2times2

[0]2times2 [0]2times2 [119876]119894

2times2

]]]

]8times8

(9)

Here [119864] is the extensional elasticity matrix [119861] is the bend-ing elasticity matrix [119862] is the coupling between bendingand extensional elasticitymatrices [119876] is the transverse-shearelasticity matrix and [0] denotes a null matrix The sum-mation accounts for the contributions from all the 119899 layersFor a typical layer 119897 the submatrices ([119864]119897

3times3 [119862]119897

3times3 [119876]119897

2times2)

are calculated using its elasticity matrix [119863]119897

5times5allowing

for anisotropy with three mutually orthogonal planes ofsymmetry The elasticity matrix [119863]

119897

5times5with reference to

the elemental coordinate system is obtained from the elas-ticity matrix [119863]119897

5times5referred to the material axes by using

the coordinate transformation matrix [119867]5times5

The resultingtransformation relationship takes the following triple productstandard form

[119863]119897

5times5= [119867]

T5times5

[119863]119897

5times5[119867]5times5 (10)

In (10) the elasticity matrix [119863]1198975times5

includes shear correctionfactors in order to allow for the error resulting from the use of

transverse-shear strain energy on an average basis In the caseof heterogeneous plates this factor is based on equilibriumequations and strain energy components used in [18] Theindividual strain components with respect to 119909 120579 and 119903 axesshown in (8) can be identified as

⟨120576119909120579119903

⟩T1times8

= ⟨120576119898

119909119909120576119898

120579120579120576119898

119909120579120576119887

119909119909120576119887

120579120579120576119887

119909120579120576119905

119909119903120576119905

120579119903⟩T

1times8

(11)

where superscripts 119898 119887 and 119905 refer to membrane bendingand transverse strain components respectively The totalstrain vector for an arbitrary point will have five componentsas

⟨120576119909120579119903

⟩T1times5

= ⟨120576119909119909 120576120579120579 120576119909120579 120576119909119903 120576120579119903⟩T (12)

For a point at a distance 119911 from the reference surfacethe flexural components of planar strains can be separated asfollows

⟨120576119909120579119903

⟩T1times5

= ⟨120576119898

119909119909120576119898

120579120579120576119898

119909120579120576119905

119909119903120576119905

120579119903⟩T

+ 119911 ⟨120576119887

119909119909120576119887

120579120579120576119887

1199091205790 0⟩

T

(13)

Based on the strain vector in (8) the general strain vector foran ESL element can be expressed as

⟨120576119909120579119903

⟩T1times8

= [119861]8times119902 ⟨120575⟩T1times119902

(14)

where [119861] is the strain matrix composed of derivatives ofhierarchical shape functions and ⟨120575⟩ is the vector of variablesof the ESL element with any number 119902 of degrees of freedomrelated to 119901 order The corresponding stress resultants in anylayer 119894 can now be expressed as

1198733times1

=

119899

sum

119894=1

int

119903top119894

119903bot119894

[[[

[

11986311

11986312

11986313

11986321

11986322

11986323

11986331

11986332

11986333

]]]

]

119894

120576119898

119909119909+ 119911120576119887

119909119909

120576119898

120579120579+ 119911120576119887

120579120579

120576119898

119909120579+ 119911120576119887

119909120579

119889119903

1198723times1

=

119899

sum

119894=1

int

119903119911top119894

119903bot119894

[[[

[

11986311

11986312

11986313

11986321

11986322

11986323

11986331

11986332

11986333

]]]

]

119894

119911120576119898

119909119909+ 1199112120576119887

119909119909

119911120576119898

120579120579+ 1199112120576119887

120579120579

119911120576119898

119909120579+ 1199112120576119887

119909120579

119889119903

1198772times1 =

119899

sum

119894=1

int

119903top119894

119903bot119894

[11986344

11986345

11986354

11986355

]

119894

120574119905

119909119903

120574119905

120579119903

119889119903

(15)

where 119873 119872 and 119877 are membrane force bending momentand transversal shear resultants respectively 119903bot and 119903top arethe 119903-axes of bottom and top surfaces of a typical layer withrespect to the reference surface

Mathematical Problems in Engineering 5

z

P

P

y

x

Computational region

L2 L2

(a) Pinched cylinder

P4

Δ2

Δ1

L1

L2x

y

z

Free

Symmetry

Symmetry

Symmetry

R

t

L4

L4

(b) Finite element modeling in computationaldomain

Figure 4 Geometric configuration and mesh refinement of pinched cylinder problem

3 Sensitivity to Geometric Parameters

In this study the sensitivity test has been carried out to verifythe robustness of present elementwith respect to three criticalconditions including severe element distortion very highaspect ratio and very large thickness ratio The cylindricalshell with length radius and thickness dimensions specifiedby 119871 119877 and 119905 respectively is subjected to a pinch load119875 as shown in Figure 4 Due to the symmetry conditionone octant of the shell is modeled by 2 times 2 mesh designThe sensitivity to input parameters by the present elementwith different 119901-levels has been compared with several shellelements available in ANSYS program which are two ℎ-convergent elements of SHELL181 (4-node) and SHELL281(8-node) and one 119901-convergent element of SHELL150 In thecase of SHELL150 element the shape functions are formu-lated by Lagrangian polynomials and curvilinear mappingtechnique is adopted in Cartesian coordinate On the otherhand linear exact mapping in cylindrical coordinate is usedin the present 119901-FEM For three cases of sensitivity teststwo material properties are considered in (16) and (17) asfollows

Case 1 Isotropy is as follows

119864 = 105 times 106

] = 03125

119871 = 1035

119877 = 4953

119905 = 0094

119875 = 100

(16)

Case 2 Orthotropy is as follows

1198641= 20 times 10

7

1198642= 10 times 10

6

11986612= 11986613= 50 times 10

5

11986623= 20 times 10

5

]12= 025

(17)

Also the numerical results are normalized using the classicalsolution or the estimated exact solutionThus the normalizedcentral deflection is used in this study which is defined in

119908 =119908max119908ref

(18)

Here 119908max is the maximum deflection in the center of shellsthat is obtained by numerical analysis and119908ref represents theestimated exact solution

31 Effect of ElementDistortion In this section the sensitivityof the present element has been tested with respect tosevere element distortion for both isotropic and orthotropiccases The material conditions are assumed to be isotropic ifthere is no mention about material property To check theelement accuracy for element distortion effects we employthe element distortion ratios denoted by 120572 to the centralnode of the 2 times 2 mesh when 119877119905 is fixed as 53 Thus theelement distortion ratios 120572 are defined by (19) In the casesof SHELL181 and SHELL281 two mesh types (2 times 2 and 4 times4) are investigated to model one octant of the shell exploitingsymmetry

120572def=Δ1

1198711

times 100 () def=Δ1

1198712

times 100 () (19)

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

1 2 3 4 5 6 7

ANSYS [

14

Nor

mal

ized

max

imum

defl

ectio

nw

Present [2 times 2 mesh]2 times 2 mesh]

p-level

Takemoto and Cook [1973]

SHELL150

Figure 5 119901-convergence of normalized maximum deflection when120572 = 0 and 119877119905 = 53

The convergence characteristics of the present elementswith respect to different 119901-levels are plotted in Figure 5when 120572 = 0 and 119877119905 = 53 From this figure it isseen that the convergence begins from 119901 = 5 by usingthe hierarchical shell elements as well as the SHELL150elements based on 119901-FEM Those results are compared withthe analytical solution by Takemoto and Cook [19] Thepresent solution virtually converges to the normalized exactsolution denoted by 10 However the numerical solutionby SHELL150 elements is converged to upper bound of12 This tendency is mainly due to the use of differentshape functions and mapping technique for curved bound-ary Linear mapping technique on rectangular coordinate isconsidered as interpolation functions of geometry fields forSHELL150 elements Therefore SHELL150 elements requiremesh refinement for curved shapes to improve accuracyalthough they are 119901-FEM In Table 1 the numerical resultsobtained by the present elements are tabulated for ] = 03125with different element distortion ratios and compared withthe references It is noted that the hierarchical shell elementsbased on 2 times 2 mesh with 119901 = 5 show an excellent behavioreven for extremely distorted mesh of 120572 = 40 All resultsare very close to the normalized displacement 10 whichrepresents the approximate solutions equal to exact solutionHowever other solutions by SHELL181 and SHELL281 showpoor accuracy and behave relatively stiff and converge veryslowly even though the meshes are refined up to 16 elements(4 times 4 mesh) Thus it is concluded that numerical solutionsin references by ANSYS program degrade significantly withincreasing element distortion These remarks can be recon-firmed by the graphical illustration as shown in Figure 6From this figure it is advised that the element distortion ratioshould not exceed roughly 10 for 4-node and 20 for 8-node ℎ-convergent shell elements to get good displacementresults On the contrary the effect of element distortion onthe numerical results by the present element is plotted inFigure 7 as the element distortion ratios 120572 vary from 00

Table 1 Comparison of normalized maximum deflection withrespect to element distortion ratio

120572 () 0 10 20 30 40

ℎ-FEM

SHELL181 (2 times 2) 0617 0363 0158 0026 0052SHELL181 (4 times 4) 0880 0807 0278 0194 0030SHELL281 (2 times 2) 0975 0937 0855 0740 0616SHELL281 (4 times 4) 0996 0975 0908 0792 0646

p-FEM

SHELL150 (2 times 2)119901 = 4

1161 1159 1150 1120 1055

SHELL150 (2 times 2)119901 = 5

1168 1167 1166 1163 1159

Present (2 times 2)119901 = 4

0979 0979 0974 0954 0909

Present (2 times 2)119901 = 5

0988 0988 0987 0985 0980

0

02

04

06

08

1

12

14

0 10 20 30 40

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 5]p = 5]

ANSYS [4 times 4 mesh]ANSYS [4 times 4 mesh]

ANSYS [2 times 2 mesh

Takemoto and Cook [1973]

SHELL150SHELL281SHELL181

Figure 6 Variation of normalized maximum deflection withincreasing distortion parameter 120572

to 90 on 2 times 2 mesh design It is observed that the presentelement shows strong robustness to severe element distortioneven for extreme case of 120572 = 90 especially for 119901 = 6

Next example is another pinched cylindrical problemcomposed of orthotropic materials defined in (17) Theorthotropic pinched cylindrical shell stacking with differentfiber orientations is tested in order to check whether materialconditions may affect the accuracy as the element distortionratio is increased For this purpose the laminated cylindricalshell with a pinch load is modeled by two layers with cross-ply (0∘90∘) Similar to the isotropic problem previously thepresent elements may endure severe element distortion up to120572 = 90 from 119901 = 6 The details of numerical solutionsare presented in Table 2 as the 119901-level increases from 1 to10 It is noticed that the present shell elements based on 119901-FEM exhibit strong robustness with respect to severe elementdistortion regardless of the anisotropy of materials

Mathematical Problems in Engineering 7

Table 2 p-convergence of normalized maximum deflection with increasing distortion ratios for orthotropic materials (0∘90∘)

120572 () 0 10 20 30 40 50 60 70 80 90119901 = 1 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002119901 = 2 0029 0029 0029 0029 0029 0028 0027 0029 0015 0026119901 = 3 0681 0640 0548 0414 0280 0176 0134 0081 0074 0069119901 = 4 0966 0966 0964 0952 0918 0846 0733 0549 0462 0367119901 = 5 0979 0979 0978 0975 0968 0952 0928 0891 0874 0813119901 = 6 0985 0985 0985 0985 0985 0985 0984 0982 0979 0972119901 = 7 0989 0989 0989 0989 0989 0989 0989 0989 0989 0989119901 = 8 0991 0991 0991 0991 0992 0992 0992 0992 0992 0992119901 = 9 0993 0993 0993 0993 0993 0993 0993 0993 0993 0993119901 = 10 0994 0994 0994 0994 0994 0994 0994 0994 0994 0994

0

02

04

06

08

1

12

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 4]Present [2 times 2 mesh p = 5]Present [2 times 2 mesh p = 6]

Takemoto and Cook [1973]

Figure 7 Robustness of hierarchical shell element with respect todistortion parameter 120572

32 Effect of Aspect Ratios of Elements In general Robinson[20] andMacneal and Harder [21] suggested a single elementtest in which response is examined as the element aspect ratiois changed It is noticed that element aspect ratios shouldnot exceed roughly seven for good displacement resultsand roughly three for good stress results Of course thisconclusion can be changed according to types of problemsHowever since large aspect ratio is one of the sources causingnumerical errors it is desirable to make the aspect ratio oneespecially in the vicinity of a singular point The aspect ratio120573 is defined by (20) and 119871

3and 119871

4are denoted in Figure 8

Consider

120573 =1198714

1198713

(20)

The finite element results obtained by present elements with119901 = 5 are presented in Table 3 with respect to aspect ratio 120573and comparedwith reference values by SHELL181 SHELL281and SHELL150 (119901 = 5) All results are based on 2 times 2mesh design When the aspect ratio 120573 is 10 the normalizedtransverse deflections at the center by SHELL281 SHELL

P4

L4

L3

x

y

z

R

L4

L4

Figure 8 Mesh design associated with aspect ratio

150 and present elements are close to an exact solutionHowever the results by SHELL181 elements fail to convergePrecisely speaking the relative errors of present elements toan exact solution are below 3 until 120573 = 1000 On the otherhand the relative errors of SHELL281 and SHELL150 showapproximately more than 10 from the aspect ratio 120573 = 20The graphical solution is also presented in Figure 9 to easilyunderstand the deviation of numerical errors in referenceto the exact solution The tolerance of present elements isrepresented in Figure 10 as the aspect ratios 120573 are increasedfrom 1 to 1000 It is seen that the hierarchical shell elementtolerates the large aspect ratios up to 1000 under 3 accuracyof relative error

33 Effect of Thickness Ratios In order to investigate themembrane and shear locking phenomena of hierarchicalshell elements the central deflections have been calculatedunder very large radius-to-thickness ratios (119877119905) Thus thetolerance of numerical solutions has been exhibited in Table 4in regard to 119877119905 ratios ranging from 10 (thick) to 1000(extremely thin) Present solutions show good agreementwith those by 8-node shell element denoted by SHELL281available in ANSYS program It is noted that both elementsavoid the shear locking regardless of extremely thin case

8 Mathematical Problems in Engineering

Table 3 Change of the normalized maximum deflection with increasing aspect ratio

120573 1 2 4 6 8 10 20 50 100 500 1000

h-FEM (2 times 2) SHELL 181 0616 0609 0588 0551 0499 0431 0347 0248 0133 0133 0133SHELL 281 0974 0971 0965 0956 0944 0930 0913 0894 0872 0872 0872

p-FEM (2 times 2)SHELL 150119901 = 5

1170 1159 1147 1134 1123 1118 1121 1133 1156 1159 1161

Present119901 = 5

0990 0986 0982 0980 0978 0976 0974 0972 0971 0970 0970

Table 4 Comparison of central deflection with increasing 119877119905

ratios

119877119905 10 100 500 1000SHELL281(4 times 4) 000088 07615 931720 7430300

Present(2 times 2 119901 = 5) 000087 07558 928663 7416710

0

02

04

06

08

1

12

14

1 2 3 4 5 6 7 8 9 10 11

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 5][p = 5]ANSYS

ANSYS ANSYS

Takemoto and Cook [1973]

SHELL150

SHELL281SHELL181

Figure 9 Variation of normalized maximum deflections withincreasing aspect ratios 120573

4 Conclusions

In the mesh design of the finite element analysis of shellsthe sources of numerical errors such as severe elementdistortions very high aspect ratios and very large radius-to-thickness ratios cause the numerical instability of a stiffnessmatrixThe proposed hierarchical shell elements based on 119901-FEM have been tested from the point of view of the elementrobustnessThe obtained results and subsequent future workscan be summarized as follows

(1) It is observed that the present elements show strongrobustness to severe element distortion even forextreme case of 120572 = 90 especially for 119901 = 6

(2) It is noticed that the hierarchical shell elements basedon 119901-FEM exhibit a strong robustness with respect

0

02

04

06

08

1

12

1 2 4 6 8 10 20 50 100 500 1000

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 4]Present [p = 5]Present [p = 6]

Takemoto and Cook [1973]

Figure 10 Robustness of hierarchical shell elements with respect toaspect ratios 120573

to severe element distortion regardless of cross-plylamination scheme of materials

(3) It is seen that the proposed elements tolerate the largeaspect ratio up to 1000 under 3 accuracy of relativeerror

(4) It is noted that the present elements avoid the shearand membrane locking regardless of the thin casesextremely denoted by 119877119905 = 1000

(5) In this paper performance of the present elementsis only limited to cylindrical shells based on linearanalysis From these results in future it is necessary toinvestigate geometrical nonlinearity and critical shellshapes such as spherical hyperbolic shells whichwould differ with cylindrical shells

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

This work was supported by the 2014 Yeungnam UniversityResearch Grant

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

120585

120578

Node 1 Node 2

Node 3Node 4

Edge 1

Edge 2

Edge 3

Edge 40

Node 2

Node 1

Node 3

Node 4

Edge 1

Edge 2Edge 3

Edge 4

120579

R

r

x

(a) One plane element with cylindrical shape (b) Standard quadrilateral element

(x1 1205791)

(x2 1205792)

(x3 1205793)

(x4 1205794) (1 1)

(1 minus1)(minus1 minus1)

(minus1 1)

Figure 2 Mapping concept for planes

the element In the case of degenerated shell concept basedon ℎ-FEMwhich is normally adopted in commercial codes anodal coordinate system defined at each nodal point with anyreference surface is considered [16] However the mappingtechnique explains the curved surface approximately notexactly in which the size of the elements is made smallenough to follow the curved surface as closely as desired

In 119901-FEM this option is not acceptable and to avoid geo-metrical sources of error exactmapping of a curved surface isnecessary because one of the sources of efficiency of119901-FEM isthe use of as few elements as possible Woo and Basu [9] pro-posed the transfinitemappingwith trigonometric polynomialinterpolation projects based on blending function methodof 119901-FEM The mapping technique ought to be consideredfor circular or oval hole to highlight modeling simplicity of119901-FEM However as surfaces on cylindrical shells are onlyconsidered in the present study the simple linear mappingon cylindrical coordinate shown in Figure 2(a) is enoughFigures 2(a) and 2(b) present the mapping concept of thecomputational domainwith the shape functions of four nodesin (1) aforementioned For the mapping process of the curvedsurfaces on cylindrical shells with constant curvature radius119877 along an axial direction the equation can be written as

∬119860

() 119877 119889120579 119889119909 = ∬

1

minus1

()120597 (119909 120579)

120597 (120585 120578)119877 119889120578 119889120585 (5)

23 Displacement Fields For modeling of composite struc-tures ESL elements (Figure 3) derived from 119862

1

119903function the-

ories [17] in a laminated system are obtained by dimensional

x

r

NodesSide modesInternal modes

Layer 1Layer 2

Layer 3

O Constant curvature 120581

Figure 3 Modeling scheme of the hierarchical laminated shellelement

reduction from three-dimensional solid to two-dimensionalrepresentative surface satisfying plane stress theory formem-brane behavior and first-order shear deformation theory forbending behavior

Thus in the case of a quadrilateral subparametric1198620119909120579ele-

ment there are three translational displacement componentsand two rotational components corresponding to each vertexside and internal mode For ESL model of an element with

4 Mathematical Problems in Engineering

any 119901-level the two in-plane displacement components canbe defined as

119906119888= 119878119873119894

11(120585 120578) (119860

119888

119894+ 120577119861119888

119894) + 119878119864119898

1198941(120585 120578) (119862

119888

119894+ 120577119863119888

119894)

+ 119878int119894119895(120585 120578) (119864

119888

119894119895+ 120577119865119888

119894119895) in 119888 = 1 2

(6)

Here the shape function functions 119878 are defined in (1) (2)and (4) Also 119860 and 119861 are the nodal variables for translationand rotation corresponding to the four nodes Likewise thevalues (119862 119864) and (119863 119865) denote translational and rotationalnodal variables corresponding to side and internal modesrespectively The variable 120577 denotes the distance from thereference surface in the thickness direction of a point ofinterest on standard coordinateThe transverse displacementfield of a point in the ESL element with any 119901-level is definedas

119908 = 119878119873119894

11(120585 120578) 119866

119894+ 119878119864119898

1198941(120585 120578)119867

119894+ 119878

int119894119895(120585 120578) 119869

119894119895 (7)

In this equation 119866 refers to the nodal variable for the fournodes and119867 and 119869 correspond to the nodal variables for sideand internal modes respectively Equation (7) connotes thatthe lateral displacement is independent of thickness coordi-nate which is contrary to in-plane displacement componentsThis choice signifies that the transverse normal stress is zeroThe constitutive relationship for the ESL element with respectto the local element coordinate system (119909 119910 119911) in referencesurface can be defined as

⟨119909120579119903

⟩T1times8

= [119871]8times8 ⟨120576119909120579119903⟩T1times8

(8)

where [119871]8times8

refers to the full elasticity matrix of a 119899-layeredESL element and is composed of submatrices as shown belowConsider

[119871]8times8 =

119899

sum

119894=12

[[[

[

[119864]119894

3times3[119862]119894

3times3[0]2times2

[119862]119894

3times3[119861]119894

3times3[0]2times2

[0]2times2 [0]2times2 [119876]119894

2times2

]]]

]8times8

(9)

Here [119864] is the extensional elasticity matrix [119861] is the bend-ing elasticity matrix [119862] is the coupling between bendingand extensional elasticitymatrices [119876] is the transverse-shearelasticity matrix and [0] denotes a null matrix The sum-mation accounts for the contributions from all the 119899 layersFor a typical layer 119897 the submatrices ([119864]119897

3times3 [119862]119897

3times3 [119876]119897

2times2)

are calculated using its elasticity matrix [119863]119897

5times5allowing

for anisotropy with three mutually orthogonal planes ofsymmetry The elasticity matrix [119863]

119897

5times5with reference to

the elemental coordinate system is obtained from the elas-ticity matrix [119863]119897

5times5referred to the material axes by using

the coordinate transformation matrix [119867]5times5

The resultingtransformation relationship takes the following triple productstandard form

[119863]119897

5times5= [119867]

T5times5

[119863]119897

5times5[119867]5times5 (10)

In (10) the elasticity matrix [119863]1198975times5

includes shear correctionfactors in order to allow for the error resulting from the use of

transverse-shear strain energy on an average basis In the caseof heterogeneous plates this factor is based on equilibriumequations and strain energy components used in [18] Theindividual strain components with respect to 119909 120579 and 119903 axesshown in (8) can be identified as

⟨120576119909120579119903

⟩T1times8

= ⟨120576119898

119909119909120576119898

120579120579120576119898

119909120579120576119887

119909119909120576119887

120579120579120576119887

119909120579120576119905

119909119903120576119905

120579119903⟩T

1times8

(11)

where superscripts 119898 119887 and 119905 refer to membrane bendingand transverse strain components respectively The totalstrain vector for an arbitrary point will have five componentsas

⟨120576119909120579119903

⟩T1times5

= ⟨120576119909119909 120576120579120579 120576119909120579 120576119909119903 120576120579119903⟩T (12)

For a point at a distance 119911 from the reference surfacethe flexural components of planar strains can be separated asfollows

⟨120576119909120579119903

⟩T1times5

= ⟨120576119898

119909119909120576119898

120579120579120576119898

119909120579120576119905

119909119903120576119905

120579119903⟩T

+ 119911 ⟨120576119887

119909119909120576119887

120579120579120576119887

1199091205790 0⟩

T

(13)

Based on the strain vector in (8) the general strain vector foran ESL element can be expressed as

⟨120576119909120579119903

⟩T1times8

= [119861]8times119902 ⟨120575⟩T1times119902

(14)

where [119861] is the strain matrix composed of derivatives ofhierarchical shape functions and ⟨120575⟩ is the vector of variablesof the ESL element with any number 119902 of degrees of freedomrelated to 119901 order The corresponding stress resultants in anylayer 119894 can now be expressed as

1198733times1

=

119899

sum

119894=1

int

119903top119894

119903bot119894

[[[

[

11986311

11986312

11986313

11986321

11986322

11986323

11986331

11986332

11986333

]]]

]

119894

120576119898

119909119909+ 119911120576119887

119909119909

120576119898

120579120579+ 119911120576119887

120579120579

120576119898

119909120579+ 119911120576119887

119909120579

119889119903

1198723times1

=

119899

sum

119894=1

int

119903119911top119894

119903bot119894

[[[

[

11986311

11986312

11986313

11986321

11986322

11986323

11986331

11986332

11986333

]]]

]

119894

119911120576119898

119909119909+ 1199112120576119887

119909119909

119911120576119898

120579120579+ 1199112120576119887

120579120579

119911120576119898

119909120579+ 1199112120576119887

119909120579

119889119903

1198772times1 =

119899

sum

119894=1

int

119903top119894

119903bot119894

[11986344

11986345

11986354

11986355

]

119894

120574119905

119909119903

120574119905

120579119903

119889119903

(15)

where 119873 119872 and 119877 are membrane force bending momentand transversal shear resultants respectively 119903bot and 119903top arethe 119903-axes of bottom and top surfaces of a typical layer withrespect to the reference surface

Mathematical Problems in Engineering 5

z

P

P

y

x

Computational region

L2 L2

(a) Pinched cylinder

P4

Δ2

Δ1

L1

L2x

y

z

Free

Symmetry

Symmetry

Symmetry

R

t

L4

L4

(b) Finite element modeling in computationaldomain

Figure 4 Geometric configuration and mesh refinement of pinched cylinder problem

3 Sensitivity to Geometric Parameters

In this study the sensitivity test has been carried out to verifythe robustness of present elementwith respect to three criticalconditions including severe element distortion very highaspect ratio and very large thickness ratio The cylindricalshell with length radius and thickness dimensions specifiedby 119871 119877 and 119905 respectively is subjected to a pinch load119875 as shown in Figure 4 Due to the symmetry conditionone octant of the shell is modeled by 2 times 2 mesh designThe sensitivity to input parameters by the present elementwith different 119901-levels has been compared with several shellelements available in ANSYS program which are two ℎ-convergent elements of SHELL181 (4-node) and SHELL281(8-node) and one 119901-convergent element of SHELL150 In thecase of SHELL150 element the shape functions are formu-lated by Lagrangian polynomials and curvilinear mappingtechnique is adopted in Cartesian coordinate On the otherhand linear exact mapping in cylindrical coordinate is usedin the present 119901-FEM For three cases of sensitivity teststwo material properties are considered in (16) and (17) asfollows

Case 1 Isotropy is as follows

119864 = 105 times 106

] = 03125

119871 = 1035

119877 = 4953

119905 = 0094

119875 = 100

(16)

Case 2 Orthotropy is as follows

1198641= 20 times 10

7

1198642= 10 times 10

6

11986612= 11986613= 50 times 10

5

11986623= 20 times 10

5

]12= 025

(17)

Also the numerical results are normalized using the classicalsolution or the estimated exact solutionThus the normalizedcentral deflection is used in this study which is defined in

119908 =119908max119908ref

(18)

Here 119908max is the maximum deflection in the center of shellsthat is obtained by numerical analysis and119908ref represents theestimated exact solution

31 Effect of ElementDistortion In this section the sensitivityof the present element has been tested with respect tosevere element distortion for both isotropic and orthotropiccases The material conditions are assumed to be isotropic ifthere is no mention about material property To check theelement accuracy for element distortion effects we employthe element distortion ratios denoted by 120572 to the centralnode of the 2 times 2 mesh when 119877119905 is fixed as 53 Thus theelement distortion ratios 120572 are defined by (19) In the casesof SHELL181 and SHELL281 two mesh types (2 times 2 and 4 times4) are investigated to model one octant of the shell exploitingsymmetry

120572def=Δ1

1198711

times 100 () def=Δ1

1198712

times 100 () (19)

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

1 2 3 4 5 6 7

ANSYS [

14

Nor

mal

ized

max

imum

defl

ectio

nw

Present [2 times 2 mesh]2 times 2 mesh]

p-level

Takemoto and Cook [1973]

SHELL150

Figure 5 119901-convergence of normalized maximum deflection when120572 = 0 and 119877119905 = 53

The convergence characteristics of the present elementswith respect to different 119901-levels are plotted in Figure 5when 120572 = 0 and 119877119905 = 53 From this figure it isseen that the convergence begins from 119901 = 5 by usingthe hierarchical shell elements as well as the SHELL150elements based on 119901-FEM Those results are compared withthe analytical solution by Takemoto and Cook [19] Thepresent solution virtually converges to the normalized exactsolution denoted by 10 However the numerical solutionby SHELL150 elements is converged to upper bound of12 This tendency is mainly due to the use of differentshape functions and mapping technique for curved bound-ary Linear mapping technique on rectangular coordinate isconsidered as interpolation functions of geometry fields forSHELL150 elements Therefore SHELL150 elements requiremesh refinement for curved shapes to improve accuracyalthough they are 119901-FEM In Table 1 the numerical resultsobtained by the present elements are tabulated for ] = 03125with different element distortion ratios and compared withthe references It is noted that the hierarchical shell elementsbased on 2 times 2 mesh with 119901 = 5 show an excellent behavioreven for extremely distorted mesh of 120572 = 40 All resultsare very close to the normalized displacement 10 whichrepresents the approximate solutions equal to exact solutionHowever other solutions by SHELL181 and SHELL281 showpoor accuracy and behave relatively stiff and converge veryslowly even though the meshes are refined up to 16 elements(4 times 4 mesh) Thus it is concluded that numerical solutionsin references by ANSYS program degrade significantly withincreasing element distortion These remarks can be recon-firmed by the graphical illustration as shown in Figure 6From this figure it is advised that the element distortion ratioshould not exceed roughly 10 for 4-node and 20 for 8-node ℎ-convergent shell elements to get good displacementresults On the contrary the effect of element distortion onthe numerical results by the present element is plotted inFigure 7 as the element distortion ratios 120572 vary from 00

Table 1 Comparison of normalized maximum deflection withrespect to element distortion ratio

120572 () 0 10 20 30 40

ℎ-FEM

SHELL181 (2 times 2) 0617 0363 0158 0026 0052SHELL181 (4 times 4) 0880 0807 0278 0194 0030SHELL281 (2 times 2) 0975 0937 0855 0740 0616SHELL281 (4 times 4) 0996 0975 0908 0792 0646

p-FEM

SHELL150 (2 times 2)119901 = 4

1161 1159 1150 1120 1055

SHELL150 (2 times 2)119901 = 5

1168 1167 1166 1163 1159

Present (2 times 2)119901 = 4

0979 0979 0974 0954 0909

Present (2 times 2)119901 = 5

0988 0988 0987 0985 0980

0

02

04

06

08

1

12

14

0 10 20 30 40

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 5]p = 5]

ANSYS [4 times 4 mesh]ANSYS [4 times 4 mesh]

ANSYS [2 times 2 mesh

Takemoto and Cook [1973]

SHELL150SHELL281SHELL181

Figure 6 Variation of normalized maximum deflection withincreasing distortion parameter 120572

to 90 on 2 times 2 mesh design It is observed that the presentelement shows strong robustness to severe element distortioneven for extreme case of 120572 = 90 especially for 119901 = 6

Next example is another pinched cylindrical problemcomposed of orthotropic materials defined in (17) Theorthotropic pinched cylindrical shell stacking with differentfiber orientations is tested in order to check whether materialconditions may affect the accuracy as the element distortionratio is increased For this purpose the laminated cylindricalshell with a pinch load is modeled by two layers with cross-ply (0∘90∘) Similar to the isotropic problem previously thepresent elements may endure severe element distortion up to120572 = 90 from 119901 = 6 The details of numerical solutionsare presented in Table 2 as the 119901-level increases from 1 to10 It is noticed that the present shell elements based on 119901-FEM exhibit strong robustness with respect to severe elementdistortion regardless of the anisotropy of materials

Mathematical Problems in Engineering 7

Table 2 p-convergence of normalized maximum deflection with increasing distortion ratios for orthotropic materials (0∘90∘)

120572 () 0 10 20 30 40 50 60 70 80 90119901 = 1 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002119901 = 2 0029 0029 0029 0029 0029 0028 0027 0029 0015 0026119901 = 3 0681 0640 0548 0414 0280 0176 0134 0081 0074 0069119901 = 4 0966 0966 0964 0952 0918 0846 0733 0549 0462 0367119901 = 5 0979 0979 0978 0975 0968 0952 0928 0891 0874 0813119901 = 6 0985 0985 0985 0985 0985 0985 0984 0982 0979 0972119901 = 7 0989 0989 0989 0989 0989 0989 0989 0989 0989 0989119901 = 8 0991 0991 0991 0991 0992 0992 0992 0992 0992 0992119901 = 9 0993 0993 0993 0993 0993 0993 0993 0993 0993 0993119901 = 10 0994 0994 0994 0994 0994 0994 0994 0994 0994 0994

0

02

04

06

08

1

12

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 4]Present [2 times 2 mesh p = 5]Present [2 times 2 mesh p = 6]

Takemoto and Cook [1973]

Figure 7 Robustness of hierarchical shell element with respect todistortion parameter 120572

32 Effect of Aspect Ratios of Elements In general Robinson[20] andMacneal and Harder [21] suggested a single elementtest in which response is examined as the element aspect ratiois changed It is noticed that element aspect ratios shouldnot exceed roughly seven for good displacement resultsand roughly three for good stress results Of course thisconclusion can be changed according to types of problemsHowever since large aspect ratio is one of the sources causingnumerical errors it is desirable to make the aspect ratio oneespecially in the vicinity of a singular point The aspect ratio120573 is defined by (20) and 119871

3and 119871

4are denoted in Figure 8

Consider

120573 =1198714

1198713

(20)

The finite element results obtained by present elements with119901 = 5 are presented in Table 3 with respect to aspect ratio 120573and comparedwith reference values by SHELL181 SHELL281and SHELL150 (119901 = 5) All results are based on 2 times 2mesh design When the aspect ratio 120573 is 10 the normalizedtransverse deflections at the center by SHELL281 SHELL

P4

L4

L3

x

y

z

R

L4

L4

Figure 8 Mesh design associated with aspect ratio

150 and present elements are close to an exact solutionHowever the results by SHELL181 elements fail to convergePrecisely speaking the relative errors of present elements toan exact solution are below 3 until 120573 = 1000 On the otherhand the relative errors of SHELL281 and SHELL150 showapproximately more than 10 from the aspect ratio 120573 = 20The graphical solution is also presented in Figure 9 to easilyunderstand the deviation of numerical errors in referenceto the exact solution The tolerance of present elements isrepresented in Figure 10 as the aspect ratios 120573 are increasedfrom 1 to 1000 It is seen that the hierarchical shell elementtolerates the large aspect ratios up to 1000 under 3 accuracyof relative error

33 Effect of Thickness Ratios In order to investigate themembrane and shear locking phenomena of hierarchicalshell elements the central deflections have been calculatedunder very large radius-to-thickness ratios (119877119905) Thus thetolerance of numerical solutions has been exhibited in Table 4in regard to 119877119905 ratios ranging from 10 (thick) to 1000(extremely thin) Present solutions show good agreementwith those by 8-node shell element denoted by SHELL281available in ANSYS program It is noted that both elementsavoid the shear locking regardless of extremely thin case

8 Mathematical Problems in Engineering

Table 3 Change of the normalized maximum deflection with increasing aspect ratio

120573 1 2 4 6 8 10 20 50 100 500 1000

h-FEM (2 times 2) SHELL 181 0616 0609 0588 0551 0499 0431 0347 0248 0133 0133 0133SHELL 281 0974 0971 0965 0956 0944 0930 0913 0894 0872 0872 0872

p-FEM (2 times 2)SHELL 150119901 = 5

1170 1159 1147 1134 1123 1118 1121 1133 1156 1159 1161

Present119901 = 5

0990 0986 0982 0980 0978 0976 0974 0972 0971 0970 0970

Table 4 Comparison of central deflection with increasing 119877119905

ratios

119877119905 10 100 500 1000SHELL281(4 times 4) 000088 07615 931720 7430300

Present(2 times 2 119901 = 5) 000087 07558 928663 7416710

0

02

04

06

08

1

12

14

1 2 3 4 5 6 7 8 9 10 11

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 5][p = 5]ANSYS

ANSYS ANSYS

Takemoto and Cook [1973]

SHELL150

SHELL281SHELL181

Figure 9 Variation of normalized maximum deflections withincreasing aspect ratios 120573

4 Conclusions

In the mesh design of the finite element analysis of shellsthe sources of numerical errors such as severe elementdistortions very high aspect ratios and very large radius-to-thickness ratios cause the numerical instability of a stiffnessmatrixThe proposed hierarchical shell elements based on 119901-FEM have been tested from the point of view of the elementrobustnessThe obtained results and subsequent future workscan be summarized as follows

(1) It is observed that the present elements show strongrobustness to severe element distortion even forextreme case of 120572 = 90 especially for 119901 = 6

(2) It is noticed that the hierarchical shell elements basedon 119901-FEM exhibit a strong robustness with respect

0

02

04

06

08

1

12

1 2 4 6 8 10 20 50 100 500 1000

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 4]Present [p = 5]Present [p = 6]

Takemoto and Cook [1973]

Figure 10 Robustness of hierarchical shell elements with respect toaspect ratios 120573

to severe element distortion regardless of cross-plylamination scheme of materials

(3) It is seen that the proposed elements tolerate the largeaspect ratio up to 1000 under 3 accuracy of relativeerror

(4) It is noted that the present elements avoid the shearand membrane locking regardless of the thin casesextremely denoted by 119877119905 = 1000

(5) In this paper performance of the present elementsis only limited to cylindrical shells based on linearanalysis From these results in future it is necessary toinvestigate geometrical nonlinearity and critical shellshapes such as spherical hyperbolic shells whichwould differ with cylindrical shells

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

This work was supported by the 2014 Yeungnam UniversityResearch Grant

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

any 119901-level the two in-plane displacement components canbe defined as

119906119888= 119878119873119894

11(120585 120578) (119860

119888

119894+ 120577119861119888

119894) + 119878119864119898

1198941(120585 120578) (119862

119888

119894+ 120577119863119888

119894)

+ 119878int119894119895(120585 120578) (119864

119888

119894119895+ 120577119865119888

119894119895) in 119888 = 1 2

(6)

Here the shape function functions 119878 are defined in (1) (2)and (4) Also 119860 and 119861 are the nodal variables for translationand rotation corresponding to the four nodes Likewise thevalues (119862 119864) and (119863 119865) denote translational and rotationalnodal variables corresponding to side and internal modesrespectively The variable 120577 denotes the distance from thereference surface in the thickness direction of a point ofinterest on standard coordinateThe transverse displacementfield of a point in the ESL element with any 119901-level is definedas

119908 = 119878119873119894

11(120585 120578) 119866

119894+ 119878119864119898

1198941(120585 120578)119867

119894+ 119878

int119894119895(120585 120578) 119869

119894119895 (7)

In this equation 119866 refers to the nodal variable for the fournodes and119867 and 119869 correspond to the nodal variables for sideand internal modes respectively Equation (7) connotes thatthe lateral displacement is independent of thickness coordi-nate which is contrary to in-plane displacement componentsThis choice signifies that the transverse normal stress is zeroThe constitutive relationship for the ESL element with respectto the local element coordinate system (119909 119910 119911) in referencesurface can be defined as

⟨119909120579119903

⟩T1times8

= [119871]8times8 ⟨120576119909120579119903⟩T1times8

(8)

where [119871]8times8

refers to the full elasticity matrix of a 119899-layeredESL element and is composed of submatrices as shown belowConsider

[119871]8times8 =

119899

sum

119894=12

[[[

[

[119864]119894

3times3[119862]119894

3times3[0]2times2

[119862]119894

3times3[119861]119894

3times3[0]2times2

[0]2times2 [0]2times2 [119876]119894

2times2

]]]

]8times8

(9)

Here [119864] is the extensional elasticity matrix [119861] is the bend-ing elasticity matrix [119862] is the coupling between bendingand extensional elasticitymatrices [119876] is the transverse-shearelasticity matrix and [0] denotes a null matrix The sum-mation accounts for the contributions from all the 119899 layersFor a typical layer 119897 the submatrices ([119864]119897

3times3 [119862]119897

3times3 [119876]119897

2times2)

are calculated using its elasticity matrix [119863]119897

5times5allowing

for anisotropy with three mutually orthogonal planes ofsymmetry The elasticity matrix [119863]

119897

5times5with reference to

the elemental coordinate system is obtained from the elas-ticity matrix [119863]119897

5times5referred to the material axes by using

the coordinate transformation matrix [119867]5times5

The resultingtransformation relationship takes the following triple productstandard form

[119863]119897

5times5= [119867]

T5times5

[119863]119897

5times5[119867]5times5 (10)

In (10) the elasticity matrix [119863]1198975times5

includes shear correctionfactors in order to allow for the error resulting from the use of

transverse-shear strain energy on an average basis In the caseof heterogeneous plates this factor is based on equilibriumequations and strain energy components used in [18] Theindividual strain components with respect to 119909 120579 and 119903 axesshown in (8) can be identified as

⟨120576119909120579119903

⟩T1times8

= ⟨120576119898

119909119909120576119898

120579120579120576119898

119909120579120576119887

119909119909120576119887

120579120579120576119887

119909120579120576119905

119909119903120576119905

120579119903⟩T

1times8

(11)

where superscripts 119898 119887 and 119905 refer to membrane bendingand transverse strain components respectively The totalstrain vector for an arbitrary point will have five componentsas

⟨120576119909120579119903

⟩T1times5

= ⟨120576119909119909 120576120579120579 120576119909120579 120576119909119903 120576120579119903⟩T (12)

For a point at a distance 119911 from the reference surfacethe flexural components of planar strains can be separated asfollows

⟨120576119909120579119903

⟩T1times5

= ⟨120576119898

119909119909120576119898

120579120579120576119898

119909120579120576119905

119909119903120576119905

120579119903⟩T

+ 119911 ⟨120576119887

119909119909120576119887

120579120579120576119887

1199091205790 0⟩

T

(13)

Based on the strain vector in (8) the general strain vector foran ESL element can be expressed as

⟨120576119909120579119903

⟩T1times8

= [119861]8times119902 ⟨120575⟩T1times119902

(14)

where [119861] is the strain matrix composed of derivatives ofhierarchical shape functions and ⟨120575⟩ is the vector of variablesof the ESL element with any number 119902 of degrees of freedomrelated to 119901 order The corresponding stress resultants in anylayer 119894 can now be expressed as

1198733times1

=

119899

sum

119894=1

int

119903top119894

119903bot119894

[[[

[

11986311

11986312

11986313

11986321

11986322

11986323

11986331

11986332

11986333

]]]

]

119894

120576119898

119909119909+ 119911120576119887

119909119909

120576119898

120579120579+ 119911120576119887

120579120579

120576119898

119909120579+ 119911120576119887

119909120579

119889119903

1198723times1

=

119899

sum

119894=1

int

119903119911top119894

119903bot119894

[[[

[

11986311

11986312

11986313

11986321

11986322

11986323

11986331

11986332

11986333

]]]

]

119894

119911120576119898

119909119909+ 1199112120576119887

119909119909

119911120576119898

120579120579+ 1199112120576119887

120579120579

119911120576119898

119909120579+ 1199112120576119887

119909120579

119889119903

1198772times1 =

119899

sum

119894=1

int

119903top119894

119903bot119894

[11986344

11986345

11986354

11986355

]

119894

120574119905

119909119903

120574119905

120579119903

119889119903

(15)

where 119873 119872 and 119877 are membrane force bending momentand transversal shear resultants respectively 119903bot and 119903top arethe 119903-axes of bottom and top surfaces of a typical layer withrespect to the reference surface

Mathematical Problems in Engineering 5

z

P

P

y

x

Computational region

L2 L2

(a) Pinched cylinder

P4

Δ2

Δ1

L1

L2x

y

z

Free

Symmetry

Symmetry

Symmetry

R

t

L4

L4

(b) Finite element modeling in computationaldomain

Figure 4 Geometric configuration and mesh refinement of pinched cylinder problem

3 Sensitivity to Geometric Parameters

In this study the sensitivity test has been carried out to verifythe robustness of present elementwith respect to three criticalconditions including severe element distortion very highaspect ratio and very large thickness ratio The cylindricalshell with length radius and thickness dimensions specifiedby 119871 119877 and 119905 respectively is subjected to a pinch load119875 as shown in Figure 4 Due to the symmetry conditionone octant of the shell is modeled by 2 times 2 mesh designThe sensitivity to input parameters by the present elementwith different 119901-levels has been compared with several shellelements available in ANSYS program which are two ℎ-convergent elements of SHELL181 (4-node) and SHELL281(8-node) and one 119901-convergent element of SHELL150 In thecase of SHELL150 element the shape functions are formu-lated by Lagrangian polynomials and curvilinear mappingtechnique is adopted in Cartesian coordinate On the otherhand linear exact mapping in cylindrical coordinate is usedin the present 119901-FEM For three cases of sensitivity teststwo material properties are considered in (16) and (17) asfollows

Case 1 Isotropy is as follows

119864 = 105 times 106

] = 03125

119871 = 1035

119877 = 4953

119905 = 0094

119875 = 100

(16)

Case 2 Orthotropy is as follows

1198641= 20 times 10

7

1198642= 10 times 10

6

11986612= 11986613= 50 times 10

5

11986623= 20 times 10

5

]12= 025

(17)

Also the numerical results are normalized using the classicalsolution or the estimated exact solutionThus the normalizedcentral deflection is used in this study which is defined in

119908 =119908max119908ref

(18)

Here 119908max is the maximum deflection in the center of shellsthat is obtained by numerical analysis and119908ref represents theestimated exact solution

31 Effect of ElementDistortion In this section the sensitivityof the present element has been tested with respect tosevere element distortion for both isotropic and orthotropiccases The material conditions are assumed to be isotropic ifthere is no mention about material property To check theelement accuracy for element distortion effects we employthe element distortion ratios denoted by 120572 to the centralnode of the 2 times 2 mesh when 119877119905 is fixed as 53 Thus theelement distortion ratios 120572 are defined by (19) In the casesof SHELL181 and SHELL281 two mesh types (2 times 2 and 4 times4) are investigated to model one octant of the shell exploitingsymmetry

120572def=Δ1

1198711

times 100 () def=Δ1

1198712

times 100 () (19)

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

1 2 3 4 5 6 7

ANSYS [

14

Nor

mal

ized

max

imum

defl

ectio

nw

Present [2 times 2 mesh]2 times 2 mesh]

p-level

Takemoto and Cook [1973]

SHELL150

Figure 5 119901-convergence of normalized maximum deflection when120572 = 0 and 119877119905 = 53

The convergence characteristics of the present elementswith respect to different 119901-levels are plotted in Figure 5when 120572 = 0 and 119877119905 = 53 From this figure it isseen that the convergence begins from 119901 = 5 by usingthe hierarchical shell elements as well as the SHELL150elements based on 119901-FEM Those results are compared withthe analytical solution by Takemoto and Cook [19] Thepresent solution virtually converges to the normalized exactsolution denoted by 10 However the numerical solutionby SHELL150 elements is converged to upper bound of12 This tendency is mainly due to the use of differentshape functions and mapping technique for curved bound-ary Linear mapping technique on rectangular coordinate isconsidered as interpolation functions of geometry fields forSHELL150 elements Therefore SHELL150 elements requiremesh refinement for curved shapes to improve accuracyalthough they are 119901-FEM In Table 1 the numerical resultsobtained by the present elements are tabulated for ] = 03125with different element distortion ratios and compared withthe references It is noted that the hierarchical shell elementsbased on 2 times 2 mesh with 119901 = 5 show an excellent behavioreven for extremely distorted mesh of 120572 = 40 All resultsare very close to the normalized displacement 10 whichrepresents the approximate solutions equal to exact solutionHowever other solutions by SHELL181 and SHELL281 showpoor accuracy and behave relatively stiff and converge veryslowly even though the meshes are refined up to 16 elements(4 times 4 mesh) Thus it is concluded that numerical solutionsin references by ANSYS program degrade significantly withincreasing element distortion These remarks can be recon-firmed by the graphical illustration as shown in Figure 6From this figure it is advised that the element distortion ratioshould not exceed roughly 10 for 4-node and 20 for 8-node ℎ-convergent shell elements to get good displacementresults On the contrary the effect of element distortion onthe numerical results by the present element is plotted inFigure 7 as the element distortion ratios 120572 vary from 00

Table 1 Comparison of normalized maximum deflection withrespect to element distortion ratio

120572 () 0 10 20 30 40

ℎ-FEM

SHELL181 (2 times 2) 0617 0363 0158 0026 0052SHELL181 (4 times 4) 0880 0807 0278 0194 0030SHELL281 (2 times 2) 0975 0937 0855 0740 0616SHELL281 (4 times 4) 0996 0975 0908 0792 0646

p-FEM

SHELL150 (2 times 2)119901 = 4

1161 1159 1150 1120 1055

SHELL150 (2 times 2)119901 = 5

1168 1167 1166 1163 1159

Present (2 times 2)119901 = 4

0979 0979 0974 0954 0909

Present (2 times 2)119901 = 5

0988 0988 0987 0985 0980

0

02

04

06

08

1

12

14

0 10 20 30 40

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 5]p = 5]

ANSYS [4 times 4 mesh]ANSYS [4 times 4 mesh]

ANSYS [2 times 2 mesh

Takemoto and Cook [1973]

SHELL150SHELL281SHELL181

Figure 6 Variation of normalized maximum deflection withincreasing distortion parameter 120572

to 90 on 2 times 2 mesh design It is observed that the presentelement shows strong robustness to severe element distortioneven for extreme case of 120572 = 90 especially for 119901 = 6

Next example is another pinched cylindrical problemcomposed of orthotropic materials defined in (17) Theorthotropic pinched cylindrical shell stacking with differentfiber orientations is tested in order to check whether materialconditions may affect the accuracy as the element distortionratio is increased For this purpose the laminated cylindricalshell with a pinch load is modeled by two layers with cross-ply (0∘90∘) Similar to the isotropic problem previously thepresent elements may endure severe element distortion up to120572 = 90 from 119901 = 6 The details of numerical solutionsare presented in Table 2 as the 119901-level increases from 1 to10 It is noticed that the present shell elements based on 119901-FEM exhibit strong robustness with respect to severe elementdistortion regardless of the anisotropy of materials

Mathematical Problems in Engineering 7

Table 2 p-convergence of normalized maximum deflection with increasing distortion ratios for orthotropic materials (0∘90∘)

120572 () 0 10 20 30 40 50 60 70 80 90119901 = 1 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002119901 = 2 0029 0029 0029 0029 0029 0028 0027 0029 0015 0026119901 = 3 0681 0640 0548 0414 0280 0176 0134 0081 0074 0069119901 = 4 0966 0966 0964 0952 0918 0846 0733 0549 0462 0367119901 = 5 0979 0979 0978 0975 0968 0952 0928 0891 0874 0813119901 = 6 0985 0985 0985 0985 0985 0985 0984 0982 0979 0972119901 = 7 0989 0989 0989 0989 0989 0989 0989 0989 0989 0989119901 = 8 0991 0991 0991 0991 0992 0992 0992 0992 0992 0992119901 = 9 0993 0993 0993 0993 0993 0993 0993 0993 0993 0993119901 = 10 0994 0994 0994 0994 0994 0994 0994 0994 0994 0994

0

02

04

06

08

1

12

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 4]Present [2 times 2 mesh p = 5]Present [2 times 2 mesh p = 6]

Takemoto and Cook [1973]

Figure 7 Robustness of hierarchical shell element with respect todistortion parameter 120572

32 Effect of Aspect Ratios of Elements In general Robinson[20] andMacneal and Harder [21] suggested a single elementtest in which response is examined as the element aspect ratiois changed It is noticed that element aspect ratios shouldnot exceed roughly seven for good displacement resultsand roughly three for good stress results Of course thisconclusion can be changed according to types of problemsHowever since large aspect ratio is one of the sources causingnumerical errors it is desirable to make the aspect ratio oneespecially in the vicinity of a singular point The aspect ratio120573 is defined by (20) and 119871

3and 119871

4are denoted in Figure 8

Consider

120573 =1198714

1198713

(20)

The finite element results obtained by present elements with119901 = 5 are presented in Table 3 with respect to aspect ratio 120573and comparedwith reference values by SHELL181 SHELL281and SHELL150 (119901 = 5) All results are based on 2 times 2mesh design When the aspect ratio 120573 is 10 the normalizedtransverse deflections at the center by SHELL281 SHELL

P4

L4

L3

x

y

z

R

L4

L4

Figure 8 Mesh design associated with aspect ratio

150 and present elements are close to an exact solutionHowever the results by SHELL181 elements fail to convergePrecisely speaking the relative errors of present elements toan exact solution are below 3 until 120573 = 1000 On the otherhand the relative errors of SHELL281 and SHELL150 showapproximately more than 10 from the aspect ratio 120573 = 20The graphical solution is also presented in Figure 9 to easilyunderstand the deviation of numerical errors in referenceto the exact solution The tolerance of present elements isrepresented in Figure 10 as the aspect ratios 120573 are increasedfrom 1 to 1000 It is seen that the hierarchical shell elementtolerates the large aspect ratios up to 1000 under 3 accuracyof relative error

33 Effect of Thickness Ratios In order to investigate themembrane and shear locking phenomena of hierarchicalshell elements the central deflections have been calculatedunder very large radius-to-thickness ratios (119877119905) Thus thetolerance of numerical solutions has been exhibited in Table 4in regard to 119877119905 ratios ranging from 10 (thick) to 1000(extremely thin) Present solutions show good agreementwith those by 8-node shell element denoted by SHELL281available in ANSYS program It is noted that both elementsavoid the shear locking regardless of extremely thin case

8 Mathematical Problems in Engineering

Table 3 Change of the normalized maximum deflection with increasing aspect ratio

120573 1 2 4 6 8 10 20 50 100 500 1000

h-FEM (2 times 2) SHELL 181 0616 0609 0588 0551 0499 0431 0347 0248 0133 0133 0133SHELL 281 0974 0971 0965 0956 0944 0930 0913 0894 0872 0872 0872

p-FEM (2 times 2)SHELL 150119901 = 5

1170 1159 1147 1134 1123 1118 1121 1133 1156 1159 1161

Present119901 = 5

0990 0986 0982 0980 0978 0976 0974 0972 0971 0970 0970

Table 4 Comparison of central deflection with increasing 119877119905

ratios

119877119905 10 100 500 1000SHELL281(4 times 4) 000088 07615 931720 7430300

Present(2 times 2 119901 = 5) 000087 07558 928663 7416710

0

02

04

06

08

1

12

14

1 2 3 4 5 6 7 8 9 10 11

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 5][p = 5]ANSYS

ANSYS ANSYS

Takemoto and Cook [1973]

SHELL150

SHELL281SHELL181

Figure 9 Variation of normalized maximum deflections withincreasing aspect ratios 120573

4 Conclusions

In the mesh design of the finite element analysis of shellsthe sources of numerical errors such as severe elementdistortions very high aspect ratios and very large radius-to-thickness ratios cause the numerical instability of a stiffnessmatrixThe proposed hierarchical shell elements based on 119901-FEM have been tested from the point of view of the elementrobustnessThe obtained results and subsequent future workscan be summarized as follows

(1) It is observed that the present elements show strongrobustness to severe element distortion even forextreme case of 120572 = 90 especially for 119901 = 6

(2) It is noticed that the hierarchical shell elements basedon 119901-FEM exhibit a strong robustness with respect

0

02

04

06

08

1

12

1 2 4 6 8 10 20 50 100 500 1000

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 4]Present [p = 5]Present [p = 6]

Takemoto and Cook [1973]

Figure 10 Robustness of hierarchical shell elements with respect toaspect ratios 120573

to severe element distortion regardless of cross-plylamination scheme of materials

(3) It is seen that the proposed elements tolerate the largeaspect ratio up to 1000 under 3 accuracy of relativeerror

(4) It is noted that the present elements avoid the shearand membrane locking regardless of the thin casesextremely denoted by 119877119905 = 1000

(5) In this paper performance of the present elementsis only limited to cylindrical shells based on linearanalysis From these results in future it is necessary toinvestigate geometrical nonlinearity and critical shellshapes such as spherical hyperbolic shells whichwould differ with cylindrical shells

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

This work was supported by the 2014 Yeungnam UniversityResearch Grant

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

z

P

P

y

x

Computational region

L2 L2

(a) Pinched cylinder

P4

Δ2

Δ1

L1

L2x

y

z

Free

Symmetry

Symmetry

Symmetry

R

t

L4

L4

(b) Finite element modeling in computationaldomain

Figure 4 Geometric configuration and mesh refinement of pinched cylinder problem

3 Sensitivity to Geometric Parameters

In this study the sensitivity test has been carried out to verifythe robustness of present elementwith respect to three criticalconditions including severe element distortion very highaspect ratio and very large thickness ratio The cylindricalshell with length radius and thickness dimensions specifiedby 119871 119877 and 119905 respectively is subjected to a pinch load119875 as shown in Figure 4 Due to the symmetry conditionone octant of the shell is modeled by 2 times 2 mesh designThe sensitivity to input parameters by the present elementwith different 119901-levels has been compared with several shellelements available in ANSYS program which are two ℎ-convergent elements of SHELL181 (4-node) and SHELL281(8-node) and one 119901-convergent element of SHELL150 In thecase of SHELL150 element the shape functions are formu-lated by Lagrangian polynomials and curvilinear mappingtechnique is adopted in Cartesian coordinate On the otherhand linear exact mapping in cylindrical coordinate is usedin the present 119901-FEM For three cases of sensitivity teststwo material properties are considered in (16) and (17) asfollows

Case 1 Isotropy is as follows

119864 = 105 times 106

] = 03125

119871 = 1035

119877 = 4953

119905 = 0094

119875 = 100

(16)

Case 2 Orthotropy is as follows

1198641= 20 times 10

7

1198642= 10 times 10

6

11986612= 11986613= 50 times 10

5

11986623= 20 times 10

5

]12= 025

(17)

Also the numerical results are normalized using the classicalsolution or the estimated exact solutionThus the normalizedcentral deflection is used in this study which is defined in

119908 =119908max119908ref

(18)

Here 119908max is the maximum deflection in the center of shellsthat is obtained by numerical analysis and119908ref represents theestimated exact solution

31 Effect of ElementDistortion In this section the sensitivityof the present element has been tested with respect tosevere element distortion for both isotropic and orthotropiccases The material conditions are assumed to be isotropic ifthere is no mention about material property To check theelement accuracy for element distortion effects we employthe element distortion ratios denoted by 120572 to the centralnode of the 2 times 2 mesh when 119877119905 is fixed as 53 Thus theelement distortion ratios 120572 are defined by (19) In the casesof SHELL181 and SHELL281 two mesh types (2 times 2 and 4 times4) are investigated to model one octant of the shell exploitingsymmetry

120572def=Δ1

1198711

times 100 () def=Δ1

1198712

times 100 () (19)

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

1 2 3 4 5 6 7

ANSYS [

14

Nor

mal

ized

max

imum

defl

ectio

nw

Present [2 times 2 mesh]2 times 2 mesh]

p-level

Takemoto and Cook [1973]

SHELL150

Figure 5 119901-convergence of normalized maximum deflection when120572 = 0 and 119877119905 = 53

The convergence characteristics of the present elementswith respect to different 119901-levels are plotted in Figure 5when 120572 = 0 and 119877119905 = 53 From this figure it isseen that the convergence begins from 119901 = 5 by usingthe hierarchical shell elements as well as the SHELL150elements based on 119901-FEM Those results are compared withthe analytical solution by Takemoto and Cook [19] Thepresent solution virtually converges to the normalized exactsolution denoted by 10 However the numerical solutionby SHELL150 elements is converged to upper bound of12 This tendency is mainly due to the use of differentshape functions and mapping technique for curved bound-ary Linear mapping technique on rectangular coordinate isconsidered as interpolation functions of geometry fields forSHELL150 elements Therefore SHELL150 elements requiremesh refinement for curved shapes to improve accuracyalthough they are 119901-FEM In Table 1 the numerical resultsobtained by the present elements are tabulated for ] = 03125with different element distortion ratios and compared withthe references It is noted that the hierarchical shell elementsbased on 2 times 2 mesh with 119901 = 5 show an excellent behavioreven for extremely distorted mesh of 120572 = 40 All resultsare very close to the normalized displacement 10 whichrepresents the approximate solutions equal to exact solutionHowever other solutions by SHELL181 and SHELL281 showpoor accuracy and behave relatively stiff and converge veryslowly even though the meshes are refined up to 16 elements(4 times 4 mesh) Thus it is concluded that numerical solutionsin references by ANSYS program degrade significantly withincreasing element distortion These remarks can be recon-firmed by the graphical illustration as shown in Figure 6From this figure it is advised that the element distortion ratioshould not exceed roughly 10 for 4-node and 20 for 8-node ℎ-convergent shell elements to get good displacementresults On the contrary the effect of element distortion onthe numerical results by the present element is plotted inFigure 7 as the element distortion ratios 120572 vary from 00

Table 1 Comparison of normalized maximum deflection withrespect to element distortion ratio

120572 () 0 10 20 30 40

ℎ-FEM

SHELL181 (2 times 2) 0617 0363 0158 0026 0052SHELL181 (4 times 4) 0880 0807 0278 0194 0030SHELL281 (2 times 2) 0975 0937 0855 0740 0616SHELL281 (4 times 4) 0996 0975 0908 0792 0646

p-FEM

SHELL150 (2 times 2)119901 = 4

1161 1159 1150 1120 1055

SHELL150 (2 times 2)119901 = 5

1168 1167 1166 1163 1159

Present (2 times 2)119901 = 4

0979 0979 0974 0954 0909

Present (2 times 2)119901 = 5

0988 0988 0987 0985 0980

0

02

04

06

08

1

12

14

0 10 20 30 40

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 5]p = 5]

ANSYS [4 times 4 mesh]ANSYS [4 times 4 mesh]

ANSYS [2 times 2 mesh

Takemoto and Cook [1973]

SHELL150SHELL281SHELL181

Figure 6 Variation of normalized maximum deflection withincreasing distortion parameter 120572

to 90 on 2 times 2 mesh design It is observed that the presentelement shows strong robustness to severe element distortioneven for extreme case of 120572 = 90 especially for 119901 = 6

Next example is another pinched cylindrical problemcomposed of orthotropic materials defined in (17) Theorthotropic pinched cylindrical shell stacking with differentfiber orientations is tested in order to check whether materialconditions may affect the accuracy as the element distortionratio is increased For this purpose the laminated cylindricalshell with a pinch load is modeled by two layers with cross-ply (0∘90∘) Similar to the isotropic problem previously thepresent elements may endure severe element distortion up to120572 = 90 from 119901 = 6 The details of numerical solutionsare presented in Table 2 as the 119901-level increases from 1 to10 It is noticed that the present shell elements based on 119901-FEM exhibit strong robustness with respect to severe elementdistortion regardless of the anisotropy of materials

Mathematical Problems in Engineering 7

Table 2 p-convergence of normalized maximum deflection with increasing distortion ratios for orthotropic materials (0∘90∘)

120572 () 0 10 20 30 40 50 60 70 80 90119901 = 1 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002119901 = 2 0029 0029 0029 0029 0029 0028 0027 0029 0015 0026119901 = 3 0681 0640 0548 0414 0280 0176 0134 0081 0074 0069119901 = 4 0966 0966 0964 0952 0918 0846 0733 0549 0462 0367119901 = 5 0979 0979 0978 0975 0968 0952 0928 0891 0874 0813119901 = 6 0985 0985 0985 0985 0985 0985 0984 0982 0979 0972119901 = 7 0989 0989 0989 0989 0989 0989 0989 0989 0989 0989119901 = 8 0991 0991 0991 0991 0992 0992 0992 0992 0992 0992119901 = 9 0993 0993 0993 0993 0993 0993 0993 0993 0993 0993119901 = 10 0994 0994 0994 0994 0994 0994 0994 0994 0994 0994

0

02

04

06

08

1

12

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 4]Present [2 times 2 mesh p = 5]Present [2 times 2 mesh p = 6]

Takemoto and Cook [1973]

Figure 7 Robustness of hierarchical shell element with respect todistortion parameter 120572

32 Effect of Aspect Ratios of Elements In general Robinson[20] andMacneal and Harder [21] suggested a single elementtest in which response is examined as the element aspect ratiois changed It is noticed that element aspect ratios shouldnot exceed roughly seven for good displacement resultsand roughly three for good stress results Of course thisconclusion can be changed according to types of problemsHowever since large aspect ratio is one of the sources causingnumerical errors it is desirable to make the aspect ratio oneespecially in the vicinity of a singular point The aspect ratio120573 is defined by (20) and 119871

3and 119871

4are denoted in Figure 8

Consider

120573 =1198714

1198713

(20)

The finite element results obtained by present elements with119901 = 5 are presented in Table 3 with respect to aspect ratio 120573and comparedwith reference values by SHELL181 SHELL281and SHELL150 (119901 = 5) All results are based on 2 times 2mesh design When the aspect ratio 120573 is 10 the normalizedtransverse deflections at the center by SHELL281 SHELL

P4

L4

L3

x

y

z

R

L4

L4

Figure 8 Mesh design associated with aspect ratio

150 and present elements are close to an exact solutionHowever the results by SHELL181 elements fail to convergePrecisely speaking the relative errors of present elements toan exact solution are below 3 until 120573 = 1000 On the otherhand the relative errors of SHELL281 and SHELL150 showapproximately more than 10 from the aspect ratio 120573 = 20The graphical solution is also presented in Figure 9 to easilyunderstand the deviation of numerical errors in referenceto the exact solution The tolerance of present elements isrepresented in Figure 10 as the aspect ratios 120573 are increasedfrom 1 to 1000 It is seen that the hierarchical shell elementtolerates the large aspect ratios up to 1000 under 3 accuracyof relative error

33 Effect of Thickness Ratios In order to investigate themembrane and shear locking phenomena of hierarchicalshell elements the central deflections have been calculatedunder very large radius-to-thickness ratios (119877119905) Thus thetolerance of numerical solutions has been exhibited in Table 4in regard to 119877119905 ratios ranging from 10 (thick) to 1000(extremely thin) Present solutions show good agreementwith those by 8-node shell element denoted by SHELL281available in ANSYS program It is noted that both elementsavoid the shear locking regardless of extremely thin case

8 Mathematical Problems in Engineering

Table 3 Change of the normalized maximum deflection with increasing aspect ratio

120573 1 2 4 6 8 10 20 50 100 500 1000

h-FEM (2 times 2) SHELL 181 0616 0609 0588 0551 0499 0431 0347 0248 0133 0133 0133SHELL 281 0974 0971 0965 0956 0944 0930 0913 0894 0872 0872 0872

p-FEM (2 times 2)SHELL 150119901 = 5

1170 1159 1147 1134 1123 1118 1121 1133 1156 1159 1161

Present119901 = 5

0990 0986 0982 0980 0978 0976 0974 0972 0971 0970 0970

Table 4 Comparison of central deflection with increasing 119877119905

ratios

119877119905 10 100 500 1000SHELL281(4 times 4) 000088 07615 931720 7430300

Present(2 times 2 119901 = 5) 000087 07558 928663 7416710

0

02

04

06

08

1

12

14

1 2 3 4 5 6 7 8 9 10 11

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 5][p = 5]ANSYS

ANSYS ANSYS

Takemoto and Cook [1973]

SHELL150

SHELL281SHELL181

Figure 9 Variation of normalized maximum deflections withincreasing aspect ratios 120573

4 Conclusions

In the mesh design of the finite element analysis of shellsthe sources of numerical errors such as severe elementdistortions very high aspect ratios and very large radius-to-thickness ratios cause the numerical instability of a stiffnessmatrixThe proposed hierarchical shell elements based on 119901-FEM have been tested from the point of view of the elementrobustnessThe obtained results and subsequent future workscan be summarized as follows

(1) It is observed that the present elements show strongrobustness to severe element distortion even forextreme case of 120572 = 90 especially for 119901 = 6

(2) It is noticed that the hierarchical shell elements basedon 119901-FEM exhibit a strong robustness with respect

0

02

04

06

08

1

12

1 2 4 6 8 10 20 50 100 500 1000

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 4]Present [p = 5]Present [p = 6]

Takemoto and Cook [1973]

Figure 10 Robustness of hierarchical shell elements with respect toaspect ratios 120573

to severe element distortion regardless of cross-plylamination scheme of materials

(3) It is seen that the proposed elements tolerate the largeaspect ratio up to 1000 under 3 accuracy of relativeerror

(4) It is noted that the present elements avoid the shearand membrane locking regardless of the thin casesextremely denoted by 119877119905 = 1000

(5) In this paper performance of the present elementsis only limited to cylindrical shells based on linearanalysis From these results in future it is necessary toinvestigate geometrical nonlinearity and critical shellshapes such as spherical hyperbolic shells whichwould differ with cylindrical shells

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

This work was supported by the 2014 Yeungnam UniversityResearch Grant

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

1 2 3 4 5 6 7

ANSYS [

14

Nor

mal

ized

max

imum

defl

ectio

nw

Present [2 times 2 mesh]2 times 2 mesh]

p-level

Takemoto and Cook [1973]

SHELL150

Figure 5 119901-convergence of normalized maximum deflection when120572 = 0 and 119877119905 = 53

The convergence characteristics of the present elementswith respect to different 119901-levels are plotted in Figure 5when 120572 = 0 and 119877119905 = 53 From this figure it isseen that the convergence begins from 119901 = 5 by usingthe hierarchical shell elements as well as the SHELL150elements based on 119901-FEM Those results are compared withthe analytical solution by Takemoto and Cook [19] Thepresent solution virtually converges to the normalized exactsolution denoted by 10 However the numerical solutionby SHELL150 elements is converged to upper bound of12 This tendency is mainly due to the use of differentshape functions and mapping technique for curved bound-ary Linear mapping technique on rectangular coordinate isconsidered as interpolation functions of geometry fields forSHELL150 elements Therefore SHELL150 elements requiremesh refinement for curved shapes to improve accuracyalthough they are 119901-FEM In Table 1 the numerical resultsobtained by the present elements are tabulated for ] = 03125with different element distortion ratios and compared withthe references It is noted that the hierarchical shell elementsbased on 2 times 2 mesh with 119901 = 5 show an excellent behavioreven for extremely distorted mesh of 120572 = 40 All resultsare very close to the normalized displacement 10 whichrepresents the approximate solutions equal to exact solutionHowever other solutions by SHELL181 and SHELL281 showpoor accuracy and behave relatively stiff and converge veryslowly even though the meshes are refined up to 16 elements(4 times 4 mesh) Thus it is concluded that numerical solutionsin references by ANSYS program degrade significantly withincreasing element distortion These remarks can be recon-firmed by the graphical illustration as shown in Figure 6From this figure it is advised that the element distortion ratioshould not exceed roughly 10 for 4-node and 20 for 8-node ℎ-convergent shell elements to get good displacementresults On the contrary the effect of element distortion onthe numerical results by the present element is plotted inFigure 7 as the element distortion ratios 120572 vary from 00

Table 1 Comparison of normalized maximum deflection withrespect to element distortion ratio

120572 () 0 10 20 30 40

ℎ-FEM

SHELL181 (2 times 2) 0617 0363 0158 0026 0052SHELL181 (4 times 4) 0880 0807 0278 0194 0030SHELL281 (2 times 2) 0975 0937 0855 0740 0616SHELL281 (4 times 4) 0996 0975 0908 0792 0646

p-FEM

SHELL150 (2 times 2)119901 = 4

1161 1159 1150 1120 1055

SHELL150 (2 times 2)119901 = 5

1168 1167 1166 1163 1159

Present (2 times 2)119901 = 4

0979 0979 0974 0954 0909

Present (2 times 2)119901 = 5

0988 0988 0987 0985 0980

0

02

04

06

08

1

12

14

0 10 20 30 40

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 5]p = 5]

ANSYS [4 times 4 mesh]ANSYS [4 times 4 mesh]

ANSYS [2 times 2 mesh

Takemoto and Cook [1973]

SHELL150SHELL281SHELL181

Figure 6 Variation of normalized maximum deflection withincreasing distortion parameter 120572

to 90 on 2 times 2 mesh design It is observed that the presentelement shows strong robustness to severe element distortioneven for extreme case of 120572 = 90 especially for 119901 = 6

Next example is another pinched cylindrical problemcomposed of orthotropic materials defined in (17) Theorthotropic pinched cylindrical shell stacking with differentfiber orientations is tested in order to check whether materialconditions may affect the accuracy as the element distortionratio is increased For this purpose the laminated cylindricalshell with a pinch load is modeled by two layers with cross-ply (0∘90∘) Similar to the isotropic problem previously thepresent elements may endure severe element distortion up to120572 = 90 from 119901 = 6 The details of numerical solutionsare presented in Table 2 as the 119901-level increases from 1 to10 It is noticed that the present shell elements based on 119901-FEM exhibit strong robustness with respect to severe elementdistortion regardless of the anisotropy of materials

Mathematical Problems in Engineering 7

Table 2 p-convergence of normalized maximum deflection with increasing distortion ratios for orthotropic materials (0∘90∘)

120572 () 0 10 20 30 40 50 60 70 80 90119901 = 1 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002119901 = 2 0029 0029 0029 0029 0029 0028 0027 0029 0015 0026119901 = 3 0681 0640 0548 0414 0280 0176 0134 0081 0074 0069119901 = 4 0966 0966 0964 0952 0918 0846 0733 0549 0462 0367119901 = 5 0979 0979 0978 0975 0968 0952 0928 0891 0874 0813119901 = 6 0985 0985 0985 0985 0985 0985 0984 0982 0979 0972119901 = 7 0989 0989 0989 0989 0989 0989 0989 0989 0989 0989119901 = 8 0991 0991 0991 0991 0992 0992 0992 0992 0992 0992119901 = 9 0993 0993 0993 0993 0993 0993 0993 0993 0993 0993119901 = 10 0994 0994 0994 0994 0994 0994 0994 0994 0994 0994

0

02

04

06

08

1

12

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 4]Present [2 times 2 mesh p = 5]Present [2 times 2 mesh p = 6]

Takemoto and Cook [1973]

Figure 7 Robustness of hierarchical shell element with respect todistortion parameter 120572

32 Effect of Aspect Ratios of Elements In general Robinson[20] andMacneal and Harder [21] suggested a single elementtest in which response is examined as the element aspect ratiois changed It is noticed that element aspect ratios shouldnot exceed roughly seven for good displacement resultsand roughly three for good stress results Of course thisconclusion can be changed according to types of problemsHowever since large aspect ratio is one of the sources causingnumerical errors it is desirable to make the aspect ratio oneespecially in the vicinity of a singular point The aspect ratio120573 is defined by (20) and 119871

3and 119871

4are denoted in Figure 8

Consider

120573 =1198714

1198713

(20)

The finite element results obtained by present elements with119901 = 5 are presented in Table 3 with respect to aspect ratio 120573and comparedwith reference values by SHELL181 SHELL281and SHELL150 (119901 = 5) All results are based on 2 times 2mesh design When the aspect ratio 120573 is 10 the normalizedtransverse deflections at the center by SHELL281 SHELL

P4

L4

L3

x

y

z

R

L4

L4

Figure 8 Mesh design associated with aspect ratio

150 and present elements are close to an exact solutionHowever the results by SHELL181 elements fail to convergePrecisely speaking the relative errors of present elements toan exact solution are below 3 until 120573 = 1000 On the otherhand the relative errors of SHELL281 and SHELL150 showapproximately more than 10 from the aspect ratio 120573 = 20The graphical solution is also presented in Figure 9 to easilyunderstand the deviation of numerical errors in referenceto the exact solution The tolerance of present elements isrepresented in Figure 10 as the aspect ratios 120573 are increasedfrom 1 to 1000 It is seen that the hierarchical shell elementtolerates the large aspect ratios up to 1000 under 3 accuracyof relative error

33 Effect of Thickness Ratios In order to investigate themembrane and shear locking phenomena of hierarchicalshell elements the central deflections have been calculatedunder very large radius-to-thickness ratios (119877119905) Thus thetolerance of numerical solutions has been exhibited in Table 4in regard to 119877119905 ratios ranging from 10 (thick) to 1000(extremely thin) Present solutions show good agreementwith those by 8-node shell element denoted by SHELL281available in ANSYS program It is noted that both elementsavoid the shear locking regardless of extremely thin case

8 Mathematical Problems in Engineering

Table 3 Change of the normalized maximum deflection with increasing aspect ratio

120573 1 2 4 6 8 10 20 50 100 500 1000

h-FEM (2 times 2) SHELL 181 0616 0609 0588 0551 0499 0431 0347 0248 0133 0133 0133SHELL 281 0974 0971 0965 0956 0944 0930 0913 0894 0872 0872 0872

p-FEM (2 times 2)SHELL 150119901 = 5

1170 1159 1147 1134 1123 1118 1121 1133 1156 1159 1161

Present119901 = 5

0990 0986 0982 0980 0978 0976 0974 0972 0971 0970 0970

Table 4 Comparison of central deflection with increasing 119877119905

ratios

119877119905 10 100 500 1000SHELL281(4 times 4) 000088 07615 931720 7430300

Present(2 times 2 119901 = 5) 000087 07558 928663 7416710

0

02

04

06

08

1

12

14

1 2 3 4 5 6 7 8 9 10 11

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 5][p = 5]ANSYS

ANSYS ANSYS

Takemoto and Cook [1973]

SHELL150

SHELL281SHELL181

Figure 9 Variation of normalized maximum deflections withincreasing aspect ratios 120573

4 Conclusions

In the mesh design of the finite element analysis of shellsthe sources of numerical errors such as severe elementdistortions very high aspect ratios and very large radius-to-thickness ratios cause the numerical instability of a stiffnessmatrixThe proposed hierarchical shell elements based on 119901-FEM have been tested from the point of view of the elementrobustnessThe obtained results and subsequent future workscan be summarized as follows

(1) It is observed that the present elements show strongrobustness to severe element distortion even forextreme case of 120572 = 90 especially for 119901 = 6

(2) It is noticed that the hierarchical shell elements basedon 119901-FEM exhibit a strong robustness with respect

0

02

04

06

08

1

12

1 2 4 6 8 10 20 50 100 500 1000

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 4]Present [p = 5]Present [p = 6]

Takemoto and Cook [1973]

Figure 10 Robustness of hierarchical shell elements with respect toaspect ratios 120573

to severe element distortion regardless of cross-plylamination scheme of materials

(3) It is seen that the proposed elements tolerate the largeaspect ratio up to 1000 under 3 accuracy of relativeerror

(4) It is noted that the present elements avoid the shearand membrane locking regardless of the thin casesextremely denoted by 119877119905 = 1000

(5) In this paper performance of the present elementsis only limited to cylindrical shells based on linearanalysis From these results in future it is necessary toinvestigate geometrical nonlinearity and critical shellshapes such as spherical hyperbolic shells whichwould differ with cylindrical shells

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

This work was supported by the 2014 Yeungnam UniversityResearch Grant

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 p-convergence of normalized maximum deflection with increasing distortion ratios for orthotropic materials (0∘90∘)

120572 () 0 10 20 30 40 50 60 70 80 90119901 = 1 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002119901 = 2 0029 0029 0029 0029 0029 0028 0027 0029 0015 0026119901 = 3 0681 0640 0548 0414 0280 0176 0134 0081 0074 0069119901 = 4 0966 0966 0964 0952 0918 0846 0733 0549 0462 0367119901 = 5 0979 0979 0978 0975 0968 0952 0928 0891 0874 0813119901 = 6 0985 0985 0985 0985 0985 0985 0984 0982 0979 0972119901 = 7 0989 0989 0989 0989 0989 0989 0989 0989 0989 0989119901 = 8 0991 0991 0991 0991 0992 0992 0992 0992 0992 0992119901 = 9 0993 0993 0993 0993 0993 0993 0993 0993 0993 0993119901 = 10 0994 0994 0994 0994 0994 0994 0994 0994 0994 0994

0

02

04

06

08

1

12

0 10 20 30 40 50 60 70 80 90

Nor

mal

ized

max

imum

defl

ectio

nw

Mesh distortion ratio 120572

Present [2 times 2 mesh p = 4]Present [2 times 2 mesh p = 5]Present [2 times 2 mesh p = 6]

Takemoto and Cook [1973]

Figure 7 Robustness of hierarchical shell element with respect todistortion parameter 120572

32 Effect of Aspect Ratios of Elements In general Robinson[20] andMacneal and Harder [21] suggested a single elementtest in which response is examined as the element aspect ratiois changed It is noticed that element aspect ratios shouldnot exceed roughly seven for good displacement resultsand roughly three for good stress results Of course thisconclusion can be changed according to types of problemsHowever since large aspect ratio is one of the sources causingnumerical errors it is desirable to make the aspect ratio oneespecially in the vicinity of a singular point The aspect ratio120573 is defined by (20) and 119871

3and 119871

4are denoted in Figure 8

Consider

120573 =1198714

1198713

(20)

The finite element results obtained by present elements with119901 = 5 are presented in Table 3 with respect to aspect ratio 120573and comparedwith reference values by SHELL181 SHELL281and SHELL150 (119901 = 5) All results are based on 2 times 2mesh design When the aspect ratio 120573 is 10 the normalizedtransverse deflections at the center by SHELL281 SHELL

P4

L4

L3

x

y

z

R

L4

L4

Figure 8 Mesh design associated with aspect ratio

150 and present elements are close to an exact solutionHowever the results by SHELL181 elements fail to convergePrecisely speaking the relative errors of present elements toan exact solution are below 3 until 120573 = 1000 On the otherhand the relative errors of SHELL281 and SHELL150 showapproximately more than 10 from the aspect ratio 120573 = 20The graphical solution is also presented in Figure 9 to easilyunderstand the deviation of numerical errors in referenceto the exact solution The tolerance of present elements isrepresented in Figure 10 as the aspect ratios 120573 are increasedfrom 1 to 1000 It is seen that the hierarchical shell elementtolerates the large aspect ratios up to 1000 under 3 accuracyof relative error

33 Effect of Thickness Ratios In order to investigate themembrane and shear locking phenomena of hierarchicalshell elements the central deflections have been calculatedunder very large radius-to-thickness ratios (119877119905) Thus thetolerance of numerical solutions has been exhibited in Table 4in regard to 119877119905 ratios ranging from 10 (thick) to 1000(extremely thin) Present solutions show good agreementwith those by 8-node shell element denoted by SHELL281available in ANSYS program It is noted that both elementsavoid the shear locking regardless of extremely thin case

8 Mathematical Problems in Engineering

Table 3 Change of the normalized maximum deflection with increasing aspect ratio

120573 1 2 4 6 8 10 20 50 100 500 1000

h-FEM (2 times 2) SHELL 181 0616 0609 0588 0551 0499 0431 0347 0248 0133 0133 0133SHELL 281 0974 0971 0965 0956 0944 0930 0913 0894 0872 0872 0872

p-FEM (2 times 2)SHELL 150119901 = 5

1170 1159 1147 1134 1123 1118 1121 1133 1156 1159 1161

Present119901 = 5

0990 0986 0982 0980 0978 0976 0974 0972 0971 0970 0970

Table 4 Comparison of central deflection with increasing 119877119905

ratios

119877119905 10 100 500 1000SHELL281(4 times 4) 000088 07615 931720 7430300

Present(2 times 2 119901 = 5) 000087 07558 928663 7416710

0

02

04

06

08

1

12

14

1 2 3 4 5 6 7 8 9 10 11

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 5][p = 5]ANSYS

ANSYS ANSYS

Takemoto and Cook [1973]

SHELL150

SHELL281SHELL181

Figure 9 Variation of normalized maximum deflections withincreasing aspect ratios 120573

4 Conclusions

In the mesh design of the finite element analysis of shellsthe sources of numerical errors such as severe elementdistortions very high aspect ratios and very large radius-to-thickness ratios cause the numerical instability of a stiffnessmatrixThe proposed hierarchical shell elements based on 119901-FEM have been tested from the point of view of the elementrobustnessThe obtained results and subsequent future workscan be summarized as follows

(1) It is observed that the present elements show strongrobustness to severe element distortion even forextreme case of 120572 = 90 especially for 119901 = 6

(2) It is noticed that the hierarchical shell elements basedon 119901-FEM exhibit a strong robustness with respect

0

02

04

06

08

1

12

1 2 4 6 8 10 20 50 100 500 1000

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 4]Present [p = 5]Present [p = 6]

Takemoto and Cook [1973]

Figure 10 Robustness of hierarchical shell elements with respect toaspect ratios 120573

to severe element distortion regardless of cross-plylamination scheme of materials

(3) It is seen that the proposed elements tolerate the largeaspect ratio up to 1000 under 3 accuracy of relativeerror

(4) It is noted that the present elements avoid the shearand membrane locking regardless of the thin casesextremely denoted by 119877119905 = 1000

(5) In this paper performance of the present elementsis only limited to cylindrical shells based on linearanalysis From these results in future it is necessary toinvestigate geometrical nonlinearity and critical shellshapes such as spherical hyperbolic shells whichwould differ with cylindrical shells

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

This work was supported by the 2014 Yeungnam UniversityResearch Grant

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 3 Change of the normalized maximum deflection with increasing aspect ratio

120573 1 2 4 6 8 10 20 50 100 500 1000

h-FEM (2 times 2) SHELL 181 0616 0609 0588 0551 0499 0431 0347 0248 0133 0133 0133SHELL 281 0974 0971 0965 0956 0944 0930 0913 0894 0872 0872 0872

p-FEM (2 times 2)SHELL 150119901 = 5

1170 1159 1147 1134 1123 1118 1121 1133 1156 1159 1161

Present119901 = 5

0990 0986 0982 0980 0978 0976 0974 0972 0971 0970 0970

Table 4 Comparison of central deflection with increasing 119877119905

ratios

119877119905 10 100 500 1000SHELL281(4 times 4) 000088 07615 931720 7430300

Present(2 times 2 119901 = 5) 000087 07558 928663 7416710

0

02

04

06

08

1

12

14

1 2 3 4 5 6 7 8 9 10 11

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 5][p = 5]ANSYS

ANSYS ANSYS

Takemoto and Cook [1973]

SHELL150

SHELL281SHELL181

Figure 9 Variation of normalized maximum deflections withincreasing aspect ratios 120573

4 Conclusions

In the mesh design of the finite element analysis of shellsthe sources of numerical errors such as severe elementdistortions very high aspect ratios and very large radius-to-thickness ratios cause the numerical instability of a stiffnessmatrixThe proposed hierarchical shell elements based on 119901-FEM have been tested from the point of view of the elementrobustnessThe obtained results and subsequent future workscan be summarized as follows

(1) It is observed that the present elements show strongrobustness to severe element distortion even forextreme case of 120572 = 90 especially for 119901 = 6

(2) It is noticed that the hierarchical shell elements basedon 119901-FEM exhibit a strong robustness with respect

0

02

04

06

08

1

12

1 2 4 6 8 10 20 50 100 500 1000

Nor

mal

ized

max

imum

defl

ectio

nw

Aspect ratio 120573

Present [p = 4]Present [p = 5]Present [p = 6]

Takemoto and Cook [1973]

Figure 10 Robustness of hierarchical shell elements with respect toaspect ratios 120573

to severe element distortion regardless of cross-plylamination scheme of materials

(3) It is seen that the proposed elements tolerate the largeaspect ratio up to 1000 under 3 accuracy of relativeerror

(4) It is noted that the present elements avoid the shearand membrane locking regardless of the thin casesextremely denoted by 119877119905 = 1000

(5) In this paper performance of the present elementsis only limited to cylindrical shells based on linearanalysis From these results in future it is necessary toinvestigate geometrical nonlinearity and critical shellshapes such as spherical hyperbolic shells whichwould differ with cylindrical shells

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

This work was supported by the 2014 Yeungnam UniversityResearch Grant

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

References

[1] M Suri ldquoOn the robustness of the h- and p-versions of thefinite-element methodrdquo Journal of Computational and AppliedMathematics vol 35 no 1ndash3 pp 303ndash310 1991

[2] J P Pontaza and J N Reddy ldquoMixed plate bending elementsbased on least-squares formulationrdquo International Journal forNumerical Methods in Engineering vol 60 no 5 pp 891ndash9222004

[3] J P Pontaza and J N Reddy ldquoLeast-squares finite elementformulation for shear-deformable shellsrdquo Computer Methodsin Applied Mechanics and Engineering vol 194 no 21ndash24 pp2464ndash2493 2005

[4] R A Arciniega and J N Reddy ldquoTensor-based finite elementformulation for geometrically nonlinear analysis of shell struc-turesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 4ndash6 pp 1048ndash1073 2007

[5] R A Arciniega and J N Reddy ldquoLarge deformation analysis offunctionally graded shellsrdquo International Journal of Solids andStructures vol 44 no 6 pp 2036ndash2052 2007

[6] E N Dvorkin and K-J Bathe ldquoA continuum mechanicsbased four-node shell element for general non-linear analysisrdquoEngineering Computations vol 1 no 1 pp 77ndash88 1984

[7] E Hinton and H C Huang ldquoA family of quadrilateral Mindlinplate elements with substitute shear strain fieldsrdquo Computers ampStructures vol 23 no 3 pp 409ndash431 1986

[8] J C Simo and M S Rifai ldquoA class of mixed assumed strainmethods and themethod of incompatible modesrdquo InternationalJournal for Numerical Methods in Engineering vol 29 no 8 pp1595ndash1638 1990

[9] K S Woo and P K Basu ldquoAnalysis of singular cylindricalshells by p-version of FEMrdquo International Journal of Solids andStructures vol 25 no 2 pp 151ndash165 1989

[10] K S Surana and R M Sorem ldquoCurved shell elements for elas-tostatics with p-version in the thickness directionrdquo Computersamp Structures vol 36 no 4 pp 701ndash719 1990

[11] K S Surana and R M Sorem ldquoCompletely hierarchical p-version curved shell element for laminated composite plates andshellsrdquoComputational Mechanics vol 7 no 4 pp 237ndash251 1991

[12] J-S Ahn Y-W Kim and K-S Woo ldquoAnalysis of circular freeedge effect in composite laminates by p-convergent global-localmodelrdquo International Journal ofMechanical Sciences vol 66 pp149ndash155 2013

[13] J-S Ahn and K-S Woo ldquoInterlaminar stress distribution oflaminated composites using the mixed-dimensional transitionelementrdquo Journal of CompositeMaterials vol 48 no 1 pp 3ndash202014

[14] ANSYSTheoryReference for theMechanical APDLandMechan-ical Applications Release 120 ANSYS Inc Canonsburg PaUSA 2008

[15] B Szabo and I Babuska Finite Element Analysis John Wiley ampSons New York NY USA 1991

[16] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquoInternational Journal for Numerical Methods in Engineering vol2 no 3 pp 419ndash451 1970

[17] K Rohwer S Friedrichs and C Wehmeyer ldquoAnalyzing lami-nated structures fromfiber-reinforced compositematerialsmdashanassessmentrdquo Technische Mechanik vol 25 pp 59ndash79 2005

[18] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press 1984

[19] H Takemoto and R D Cook ldquoSome modifications of anisoparametric shell elementrdquo International Journal for Numer-ical Methods in Engineering vol 7 no 3 pp 401ndash405 1973

[20] J Robinson ldquoA single element testrdquo Computer Methods inApplied Mechanics and Engineering vol 7 no 2 pp 191ndash2001976

[21] R H Macneal and R L Harder ldquoA proposed standard set ofproblems to test finite element accuracyrdquo Finite Elements inAnalysis and Design vol 1 no 1 pp 3ndash20 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of