Engineering ComputationsA 4-node quadrilateral flat shell element formulated by the shape-free HDF plateand HSF membrane elementsYan Shang Song Cen Chen-Feng Li

Article information:To cite this document:Yan Shang Song Cen Chen-Feng Li , (2016)," A 4-node quadrilateral flat shell element formulated bythe shape-free HDF plate and HSF membrane elements ", Engineering Computations, Vol. 33 Iss 3pp. 713 - 741Permanent link to this document:http://dx.doi.org/10.1108/EC-04-2015-0102

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A 4-node quadrilateral flat shellelement formulated by the

shape-free HDF plate and HSFmembrane elements

Yan ShangDepartment of Engineering Mechanics, School of Aerospace Engineering,

Tsinghua University, Beijing, ChinaSong Cen

Department of Engineering Mechanics andKey Laboratory of Applied Mechanics, School of Aerospace Engineering,

Tsinghua University, Beijing, China, andChen-Feng Li

Zienkiewicz Centre for Computational Engineering andEnergy Safety Research Institute, College of Engineering,

Swansea University, Swansea, UK

AbstractPurpose – The purpose of this paper is to propose an efficient low-order quadrilateral flat shellelement that possesses all outstanding advantages of novel shape-free plate bending and planemembrane elements proposed recently.Design/methodology/approach – By assembling a shape-free quadrilateral hybrid displacement-function (HDF) plate bending element HDF-P4-11ß (Cen et al., 2014) and a shape-free quadrilateralhybrid stress-function (HSF) plane membrane element HSF-Q4θ-7ß (Cen et al., 2011b) with drillingdegrees of freedom (DOF), a new 4-node, 24-DOF (six DOFs per node) quadrilateral flat shell element issuccessfully constructed. The trial functions for resultant/stress fields within the element are derivedfrom the analytical solutions of displacement and stress functions for plate bending and planeproblems, respectively, so that they can a priori satisfy the related governing equations. Furthermore,in order to take the influences of moderately warping geometry into consideration, the rigid linkcorrection strategy (Taylor, 1987) is also employed.Findings – The element stiffness matrix of a new simple 4-node, 24-DOF quadrilateral flat shellelement is obtained. The resulting models, denoted as HDF-SH4, not only possesses all advantages oforiginal HDF plate and HSF plane elements when analyzing plate and plane structures, but alsoexhibits good performances for analyses of complicated spatial shell structures. Especially, it is quiteinsensitive to mesh distortions.Originality/value – This work presents a new scheme, which can take the advantages of bothanalytical and discrete methods, to develop low-order mesh-distortion resistant flat shell elements.Keywords Drilling degrees of freedom, Flat shell element, Hybrid displacement-function (HDF),Hybrid stress-function (HSF), Mesh distortionPaper type Research paper

Engineering Computations:International Journal for Computer-Aided Engineering and Software

Vol. 33 No. 3, 2016pp. 713-741

©Emerald Group Publishing Limited0264-4401

DOI 10.1108/EC-04-2015-0102

Received 3 May 2015Revised 2 August 2015

Accepted 21 August 2015

The current issue and full text archive of this journal is available on Emerald Insight at:www.emeraldinsight.com/0264-4401.htm

This work is financially supported by the National Natural Science Foundation of China(11272181), the Specialized Research Fund for the Doctoral Program of Higher Education ofChina (20120002110080) and the Tsinghua University Initiative Scientific Research Program(2014z09099).

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1. IntroductionThe finite element method (FEM) is an efficient tool for analyses of shell/platestructures. During the past decades, numerous efforts have been made for developingvarious shell element models (Long et al., 2009). At present, most existing elements canbe classified into three categories (Yang et al., 2000): first, the curved shell elementsbased on the classical shell theory; second, the degenerated shell elements derived fromthe 3D solid theory; and third, the flat shell elements obtained by combining the platebending and the membrane elements. In general, it is difficult to identify which kind isthe best choice. But the flat shell element is often treated as the simplest one since it canbe easily constructed and efficiently applied without complicated treatments inmathematical derivation. Furthermore, it is more convenient when handling complexboundary conditions and analyzing folded plate structures.

Since 1970s, many flat shell elements have been successfully proposed, in which thetriangular flat elements are generally regarded as more reasonable models becausethey can be used to discretize an arbitrary geometry without influence caused byelement warping (Gal and Levy, 2006). However, since the quadrilateral elementsusually have a much better predictive capability (Lee and Bathe, 2004) than triangularones, they still attract the interests from many FEM researchers. For examples,Ibrahimbegovic and Wilson (1991) proposed a 4-node quadrilateral element which canallow two nodes to be coalesced with a trivial modification; Kim et al. (2003) presentedtwo 4-node quasi-conforming shell elements based on the assumed strain method, inwhich the explicit stiffness matrices were obtained; Wang et al. (2012) also developedmodels within the quasi-conforming framework by employing the formulae ofTimoshenko’s beam as the string net functions along element boundaries; Moreira andRodrigues (2011) constructed an element in which the rotation components areincompatible and independently interpolated; Choi et al. (1999) formulated anonconforming model by using a modified integration scheme to overcome shearlocking; and so on.

Although many quadrilateral flat shell elements can be found in various literatures,there is still demand for further enhancement of element performance, especially for thebehaviors in distorted meshes. Actually, how to develop lower-order distortion-resistant models is still a great challenge nowadays. On this topic, some works areworthy of mention. The smoothed finite element method (SFEM) (Liu et al., 2007), whichintegrates the strain smoothing technique (Chen et al., 2001) into the conventional FEM,has been successfully applied in different problems (Nguyen-Xuan et al., 2008; Cui et al.,2010; Nguyen-Thanh et al., 2008; Wu and Wang, 2013). Nguyen-Van et al. (2009)developing a quadrilateral flat shell element by introducing the smoothed techniqueindependently into plate bending part and membrane part. With the aid of smoothingoperation, the resulting element can work well in distorted meshes. In addition to theSFEM, the present authors proposed a hybrid displacement-function (HDF) method(Cen et al., 2014; Shang et al., 2015b) and a hybrid stress-function (HSF) method (Cenet al., 2011a, b, 2013; Fu et al., 2010) for developing Mindlin-Reissner plate bending andplane membrane elements, respectively. Both the HDF and the HSF methods are basedon the principle of minimum complementary energy. In the HDF method, the trialfunctions for the resultants within the element are derived from the analytical solutionsof the displacement function (Hu, 1984; Shang et al., 2015a; Cen and Shang, 2015). Thus,they can a priori satisfy all the related governing equations of the Mindlin-Reissnerplate. Furthermore, the formulae of the locking-free Timoshenko’s beam are utilized todetermine the element boundary displacement modes. Therefore, the resulting element

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HDF-P4-11β (Cen et al., 2014) is free of shear locking and can provide excellent results,even in severely distorted meshes. Similarly, in the HSF method, the stress trialfunctions are derived from the Airy stress function, and can satisfy the relatedgoverning equations of plane elasticity. The resulting 4-node membrane element HSF-Q4θ-7β (Cen et al., 2011b) with Allman’s drilling degrees of freedom (DOF) (Allman,1984) is also less sensitive to severe mesh distortions.

In this work, by combination of the HDF plate bending element HDF-P4-11β (Cenet al., 2014) and the HSF plane membrane element HSF-Q4θ-7β (Cen et al., 2011b), a new4-node quadrilateral flat shell element is constructed. This new element, named byHDF-SH4, has definite six DOFs per node, so that the singularity problem of the globalstiffness matrix on the condition that all adjacent elements are coplanar can be avoidednaturally. Furthermore, in order to take the influences of moderately warped geometryinto account, the rigid link correction strategy proposed by Taylor (1987) is employed.Since the plate and membrane components are separately accurate and robust even indistorted meshes, the performance of the new shell element is desirable.

Several classical benchmarks are employed for assessing the new 4-node, 24-DOFquadrilateral flat shell element. Numerical results show that the new model inherits theadvantages of its parent components and is efficient for analysis of shell structureusing both regular and distorted meshes.

2. The formulations of new flat shell element HDF-SH42.1 The geometry and DOFs of the flat shell elementAs shown in Figure 1, the geometry of a 4-node quadrilateral flat shell element isrepresented by its mid-surface 1234, in which points 1~4 are the element nodes. Points5~8 are the mid-side points of each edge, so that they must be coplanar. A localCartesian coordinate system (x′, y′, z′) is established according to the position of the

x

y

u

v

w

z

�x

�z �y

� ′y � ′x

� ′zu ′

w ′x ′y ′

z ′

v ′

4 (x4, y4, z4)

3 (x3, y3, z3)

1 (x1, y1, z1)

2 (x2, y2, z2)

1

2

3

4

V1

V3

V2

x ′

z ′y ′

Figure 1.The geometry of the

quadrilateral flatshell element HDF-

SH4 and twoCartesian coordinate

systems

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planar face 5678, in which x′- and y′- axes are parallel to 5678 while z′-axis isperpendicular to it, and x′-axis is also parallel to line 57.

In the global Cartesian coordinate system (x, y, z), the element nodal DOF vector ae isdefined by (see Figure 1):

ae ¼

ae1ae2ae3ae4

8>>>><>>>>:

9>>>>=>>>>;; with aei ¼ ui vi wi yxi yyi yzi

h iT; i ¼ 1� 4ð Þ: (1)

And the local element nodal DOF vector a′e is defined in the local Cartesian coordinatesystem (x′, y′, z′) (see Figure 1), and is given by:

a0e ¼

a0e1a0e2a0e3a0e4

8>>>><>>>>:

9>>>>=>>>>;; with a0ei ¼ u0i v0i w0

i y0xi y0yi y0zih iT

; i ¼ 1� 4ð Þ: (2)

The transformation between two coordinate systems is:

x0

y0

z0

8><>:

9>=>; ¼

lx0x lx0y lx0zly0x ly0y ly0zlz0x lz0y lz0z

264

375

x

y

z

8><>:

9>=>; ¼ k

x

y

z

8><>:

9>=>;; with k ¼

lx0x lx0y lx0zly0x ly0y ly0zlz0x lz0y lz0z

264

375; (3)

in which:

lx0xlx0ylx0z

8><>:

9>=>; ¼ V1

:V1:2; V1 ¼

x7�x5y7�y5z7�z5

8><>:

9>=>;; (4)

lz0xlz0ylz0z

8><>:

9>=>; ¼ V3

:V3:2; V3 ¼ V1 � V2;V2 ¼

x8�x6y8�y6z8�z6

8><>:

9>=>;; (5)

l0y0x

ly0yly0z

8><>:

9>=>; ¼ V3 � V1

:V3 � V1:2: (6)

Here, for a vector b ¼ ½ b1 b2 b3 �T, the expression of :b:2 is:

:b:2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21þb22þb23

q: (7)

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Thus, the relationship between ae and a′e can be obtained:

a0e ¼ Tae; (8)

where:

T ¼

k

k

k

k

k

k

k

k

266666666666664

37777777777777524�24

: (9)

Accordingly, the final element stiffness matrix ðKeglobalÞ24�24 in the global coordinate

system can be transformed from its local form ðKelocal Þ24�24:

Keglobal ¼ TTKe

localT: (10)

2.2 The element stiffness matrix of flat shell element in local coordinate systemFor a 4-node quadrilateral flat shell element, the element stiffness matrix in the localCartesian coordinate system can be obtained by assembling the element stiffnessmatrices of the plate bending and the membrane elements, as shown in Figure 2.

Plate part Membrane part

Flat shell element

1

2

3

4

x ′

y ′z ′

�′z4

�′z1

�′z3

�′z2

v ′y4

v ′y3

v ′y2

v ′y1

u ′x4

u ′x1 u ′x3

u ′x2

w4′

w1′w3′

w2′

�′y4�′x4

�′y3

�′x3

�′x2�′y2

�′y1

�′x1

(a)

(b) (c)

Figure 2.Flat shell element inthe local Cartesiancoordinate systems

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In Equation (2), the DOFs of a shell element related to the plate part are:

a0eP ¼

a0eP1a0eP2a0eP3a0eP4

8>>>><>>>>:

9>>>>=>>>>;; with a0ePi ¼ w0

i y0xi y0yih iT

; i ¼ 1� 4ð Þ; (11)

and the DOFs related to the membrane part are:

a0em ¼

a0em1

a0em2

a0em3

a0em4

8>>>><>>>>:

9>>>>=>>>>;; with a0emi ¼ u0i v0i y0zi

h iT; i ¼ 1� 4ð Þ: (12)

2.2.1 The plate part of HDF-SH4. In the formulations of the new flat shell elementHDF-SH4, the plate part comes from the 4-node Mindlin-Reissner plate elementHDF-P4-11β (Cen et al., 2014) proposed recently. This element was developed based ona so-called HDF method, in which the trial functions for the resultants within theelement are derived from the analytical solutions of the displacement functionF, instead of being directly assumed. This element is free of shear locking, and can stillperform well in severely distorted meshes, even when the element shape degeneratesinto a concave quadrilateral. Hence, it can be regarded as a kind of shape-free Mindlin-Reissner plate element.

For the Mindlin-Reissner plate element HDF-P4-11β, the complementary energyfunctional is expressed by (Cen et al., 2014):

PCP ¼ 1

2

ZZAe

SPbPþRn

P

� �TCP SPbPþRn

P

� �dx0dy0

þZSe

SPbPþRn

P

� �TLP

TN��P q

eds; (13)

in which SP is the resultant/stress trial function matrix:

Sp ¼

M 0x01 M 0

x02 M 0x03 � � � M 0

x011

M 0y01 M 0

y02 M 0y03 � � � M 0

y011

M 0x0y01 M 0

x0y02 M 0x0y03 � � � M 0

x0y011

T0x01 T0

x02 T0x03 � � � T0

x011

T0y01 T0

y02 T0y03 � � � T0

y011

2666666664

3777777775; (14)

and its components are all analytical solutions for plate resultants and are given inTable I; bP is the corresponding coefficient vector containing eleven unknown

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coefficients; Cp is the elasticity matrix of compliance for Mindlin-Reissner plate:

Cp ¼

1D 1�m2ð Þ

�mD 1�m2ð Þ 0 0 0

�mD 1�m2ð Þ

1D 1�m2ð Þ 0 0 0

0 0 2D 1�mð Þ 0 0

0 0 0 1C 0

0 0 0 0 1C

26666666664

37777777775; (15)

with:

D ¼ Eh3=12 1�mð Þ; C ¼ 5=6 Gh; (16)

where μ is Poisson’s ratio, E is Young’s modulus, G¼E/[2(1+μ)] is shear modulus, andh is the element thickness; Rn

P is the particular solution part related to a distributed load(Cen et al., 2014); LP is the direction cosine matrix along the element’s edges:

LP ¼l2 m2 2lm 0 0

�lm lm l2�m2 0 0

0 0 0 �l �m

264

375; (17)

where l and m denote the direction cosines of the element boundaries’ outer normal n;N��P is the interpolation function matrix for boundary displacements determined by

using the locking-free formulae of Timoshenko’s beam (Hu, 1984; Soh et al., 1999, 2001),and its detailed expressions are given in Appendix 1; qe is the DOF vector used for theplate element HDF-P4-11β:

qe ¼

qe1

qe2

qe3

qe4

8>>>><>>>>:

9>>>>=>>>>;; with qe

i ¼ w0i Ψ 0

xi Ψ 0yi

h iT; i ¼ 1� 4ð Þ: (18)

As shown in Figure 3, it should be noted that qe is different from a0eP in Equation (11)defined for the plate part of the shell element, and their transform relationships areas follows:

qe ¼ TPa0eP ; (19)

i 1 2 3 4 5 6 7 8 9 10 11

M0x0 i 2 0 2μ 6x′ 2y′ 2μx′ 6μy′ 6x′y′ 6μx′y′ 12(x′2−μy′2) 12(1−μ)(y′2−x′2)

M0y0 i

2μ 0 2 6μx′ 2μy′ 2x′ 6y′ 6μx′y′ 6x′y′ −12(y′2−μx′2) 12(1−μ)(x′2−y′2)M0

x0y0 i0 1−μ 0 0 2(1−μ)x′ 2(1−μ)y′ 0 3(1−μ)x′2 3(1−μ)y′2 0 24(1−μ)x′y′

T0x0 i 0 0 0 6 0 2 0 6y′ 6y′ 24x′ 0

T0y0 i

0 0 0 0 2 0 6 6x′ 6x′ −24y′ 0

Table I.Eleven resultant/

stress trial functionsof the plate element

HDP-P4-11β

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with:

TP ¼

tptp

tptp

266664

377775; tp ¼

1 0 0

0 0 �1

0 1 0

264

375: (20)

Then, according to Equation (13) and the principle of minimum complementary energy,the element stiffness matrix Ke

P

� �12�12 and the equivalent nodal load vector Pe

P

� �12�1

caused by a distributed load can be obtained:

KeP ¼ HT

P�1P HP ; (21)

PeP ¼ VT

P�HTPΜ

�1P Μn�1

P ; (22)in which:

ΜP ¼ZZ

AeSP

TCP SPdx0dy0; HP ¼

ZSeSP

TLP

TN��Pds; (23)

Μn

P ¼ZZ

AeSP

TCPR

n

Pdx0dy0; VP ¼

ZSeRnTP LT

PN��Pds: (24)

Finally, according to Equation (19), KeP and Pe

P should be transformed into KeSP and

PeSP which are related to a0eP given by Equation (11):

KeSP ¼ TT

PKePTP ; (25)

PeSP ¼ TT

PPeP : (26)

2.2.2 The membrane part of HDF-SH4. The membrane part of the elementHDF-SH4 comes from the 4-node quadrilateral membrane element HSF-Q4θ-7β(Cen et al., 2011b). This element was developed based on a so-called HSF method, inwhich the trial functions for stresses within the element are derived from the analyticalsolutions of Airy stress function. This element is also less sensitive to severe meshdistortions. Since the Allman’s drilling DOF (Allman, 1984) are employed, the resulting

x ′

y ′

z ′

x ′

y ′

z ′

w ′ w ′�y′ �x′

�y′�x′

(a) (b)

Figure 3.The DOFs related toplate part used for(a) shell elementHDF-SH4 and(b) plate elementHDF-P4-11β

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shell element will possess six DOFs per node. Hence, the singularity problem of theglobal stiffness matrix, which will arise when all adjacent elements are coplanar, can beavoided naturally.

For the membrane element HSF-Q4θ-7β, the complementary energy functional canbe expressed as (Cen et al., 2011b):

PCm ¼ 1

2

ZZAe

SmbmþRn

m

� �TCm SmbmþRn

m

� �dx0dy0

þZSe

SmbmþRn

m

� �TLm

TN��ma

0meds; (27)

where Sm is the stress trial function matrix of the membrane element HSF-Q4θ-7β:

Sm ¼sx01 sx02 sx03 � � � sx07sy01 sy02 sy03 � � � sy07tx0y01 tx0y02 tx0y03 � � � tx0y07

264

375; (28)

and its components are all analytical solutions for stresses of plane problem and aregiven in Table II; bm is the corresponding coefficient vector containing seven unknowncoefficients; Rn

m is the particular solution part related to a distributed load (Cen et al.,2011b); Cm is the elasticity compliance matrix for plane stress problem:

Cm ¼ 1E

1 �m 0

�m 1 0

0 0 2 1þmð Þ

264

375; (29)

Lm is the direction cosine matrix along the element’s edges:

Lm ¼ l 0 m

0 m l

� ; (30)

N��m is the interpolation function matrix for boundary displacements derived from the

displacement fields of Allman (1984), and the detailed expressions of its componentsare given in Appendix 2.

Then, by applying the principle of minimum complementary energy, the elementstiffness matrix Ke

m

� �12�12 and the equivalent nodal load vector Pe

m

� �12�1 related to the

distributed load can be obtained:

Kem ¼ HT

m�1m Hm; (31)

i 1 2 3 4 5 6 7

σ x′i 0 0 2 0 0 2x′ 6y′σ y′i 2 0 0 6x′ 2y′ 0 0τx′y′i 0 –1 0 0 –2x′ –2y′ 0

Table II.Seven stress trialfunctions of the

membrane elementHSF-Q4θ-7β

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Pem ¼ VT

m�HTmΜ

�1m Μn�1

m ; (32)with:

Μm ¼ZZ

AeSm

TCmSmtdx0dy0; Hm ¼

ZSeSm

TLm

TN��mtds; (33)

Μn

m ¼ZZ

AeSm

TCmR

n

mtdx0dy0; Vm ¼

ZSeRnTm LT

mN��mtds: (34)

2.3 The correction for the element’s warped geometryBy assembling Equations (25) and (31) according to the DOFs’ sequence in Equation (2),the element stiffness matrix ðKe

f latÞ24�24, which corresponds to a flat shell, can beobtained. When all nodes of the flat shell element are coplanar, the element stiffnessmatrix in the local Cartesian coordinate system can be directly obtained:

Kelocal ¼ Ke

f lat : (35)

Then, the final element stiffness matrix in the global Cartesian coordinate system canbe derived by substituting Equation (35) into Equation (10). Similarly, by assemblingEquations (26) and (32), the final equivalent nodal load vector ðPe

f latÞ24�1 of the flat shellelement in the local Cartesian coordinate system, which is caused by a distributed load,can also be obtained. Next, it will be transformed into the form in the global Cartesiancoordinate system. In fact, when incorporated into commercial FEM software, the loadcan be directly applied to structures by using the existing method in FEM software forconvenience.

However, if the element is not so “flat,” some inaccuracies will be introduced. Inorder to take such influences brought by warping geometry into consideration, therigid link correction strategy proposed by Taylor (1987) will be employed here. For awarping element, as shown in Figure 4, the following equation is used to transform theDOFs of a real warping geometry 1234 into the projected flat plane 1′2′3′4′:

ufivfiwfi

yfxiyfyi

yfzi

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

¼

1

0 1

0 0 1

0 hi 0 1

�hi 0 0 0 1

0 0 0 0 0 1

2666666664

3777777775

urivriwri

yrxiyryiyrzi

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼ Wi

urivriwri

yrxiyryiyrzi

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;; i ¼ 1� 4ð Þ; (36)

in which, the superscript f means the flat plane, and r means the real warpinggeometry; hi denotes the warping offset of the node i. Therefore, the stiffness matrixKe

local is rewritten as:

Kelocal ¼ WKK

ef latW

TK ; (37)

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with:

WK ¼

W1

W2

W3

W4

26664

37775: (38)

Then, substitution of Equation (37) into Equation (10) yields the final element stiffnessmatrix in the global Cartesian coordinate system.

Since the plate and the membrane parent parts of the shell element are bothinsensitive to mesh distortions, the performance of the new flat shell element HDF-SH4in distorted meshes can be desirable.

3. Numerical testsTo assess the performance of the element HDF-SH4, several classical benchmarks areinvestigated. These benchmarks are operated through integrating the presentelement’s scheme into the Abaqus/Standard UEL. For comparison, the resultscalculated by following existing shell elements are also provided for comparison:

S4 4-node shell element in the software Abaqus/Standard (Abaqus, 2004).QC5D-SA 4-node flat shell element with the use of 5-point integration scheme

(Groenwold and Stander, 1995).MITC4 4-node general shell element based on the mixed interpolation of tensorial

component method (Bathe and Dvorkin, 1986).MIST1 4-node flat shell element based on mixed interpolation with smoothed

bending and membrane strain (Nguyen-Thanh et al., 2008).QPH 4-node shell element with physical hourglass control (Belytschko and

Leviathan, 1994).XSHELL414-node assumed-strain quasi-conforming flat shell element (Kim et al., 2003).QCS1 4-node assumed-displacement quasi-conforming flat shell element

(Wang et al., 2012).MISQ24 4-node smoothed displacement-based flat shell element (Nguyen-Van et al.,

2009).

4′

1′

2′

3′

1 2

3

4

56

78

Figure 4.The projection of awarping element

1234 into a flat plane1′2′3′4′

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CYSE 4-node one point quadrature general shell element with physicalstabilization scheme (Cardoso et al., 2002).

Simo 4-node shell element based on the classical linear stress-resultant shelltheory (Simo et al., 1989).

Schwarze Reduced integration solid-shell element based on the EAS and ANS concept(Schwarze and Reese, 2009).

SRI bilinear degenerated shell element with selective reduced integration(Hughes and Liu, 1981).

Q4DRL 4-node non-conforming flat shell element (Moreira and Rodrigues, 2011).RQS20 4-node refined discrete degenerated shell element with 20 DOFs (Chen and

Cheung, 2005).QS20-R 4-node selective reduced integration degenerated shell element with 20

DOFs (Chen and Cheung, 2005).Shin 3-node triangular flat shell element based on the assumed natural deviatoric

strain (ANDES) formulation (Shin and Lee, 2014).MISC2 4-node smoothed Mindin-Reissner plate element (Nguyen-Xuan et al., 2008).DKMQ 4-node discrete Kirchoff-Mindlin plate element (Katili, 1993).ARS-Q12 4-node Mindlin-Reissner plate element based on the use of locking-free

Timoshenko’s beam (Soh et al., 2001).AC-MQ4 4-node Mindlin-Reissner plate element based on the quadrilateral area

coordinate method (Cen et al., 2006).

3.1 The patch tests for plate bending and plane membrane elements

Since the parent plate and membrane components can strictly pass all patch tests forplate and membrane elements, respectively, the resulting flat shell element HDF-SH4can also pass these tests.

3.2 Cook’s trapezoidal skew beamAs shown in Figure 5, a clamped trapezoidal skew beam (Cook et al., 2001) is subjectedto a uniformly distributed shear load along its free edge. The material parameters andthe typical mesh are also given in Figure 5. In this problem, the in-plane sheardeformation is predominant. Thus, it is useful to validate the element’s ability inhandling membrane deformation situation. For comparison, the normalized results ofthe y-direction displacements at the point C, calculated from different elements, areplotted in Figure 6. Since there is no analytical solution for this problem, an overkillsolution 23.96 (Long and Xu, 1994) is used as the reference solution. It can be seen that,the present shell element HDF-SH4 exhibits much better performance than othermodels in this test.

3.3 Razzaque’ s skew plateThe Razzaque 60° rhombic plate (Razzaque, 1973) is subjected to a uniformlydistributed load fz, as shown in Figure 7. This benchmark is utilized to assess theelement’s ability to model the plate bending problem in skew meshes. The normalizedcentral transverse deflections, obtained by different elements, are given in Table III,and the corresponding convergence plots are shown in Figure 8. This test shows thatthe new element exhibits the best performance among all models.

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3.4 Pinched cylinderFigure 9 depicts a pinched cylinder with end diaphragms subjected to a pair of oppositeconcentrated forces in the mid-span. Due to the symmetry, only one-eighth structure isanalyzed. As shown in Figure 10, two different meshes are considered: regular anddistorted meshes. The dsitroted mesh is obtained by following the strategy proposedby Nguyen-Thanh et al. (2008): the coordinates of interior nodes in the initial regularmesh are replaced by:

zdis ¼ zreþ0:5arDzre; (39)

44

P

A

BC

x

y

44

16

48

t=1, E=1, �=1/3, P=1

z

Figure 5.Cook’s trapezoidal

skew beam

1.00

Nor

mal

ized

def

lect

ion

0.95

0.90

0.85

0.80

2 4 8

Mesh N×N (log2scale)

16

ReferenceS4QC5D-SAMITC4XSHELL41CYSESimoQ4DRLHDF-SH4

Figure 6.The convergence

plot of Cook’s beam

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A B

CD

x

y

z

O

SS1

SS1

FreeFree

60°

L

Notes: L= 100, t= 0.1, E= 10.92, � = 0.3, Uniformdistributed load fz= –1

Figure 7.Razzaque’sskew plate

Mesh 2× 2 4× 4 6× 6 8× 8 12× 12

HDF-SH4 0.9624 0.9931 0.9947 0.9953 0.9956S4 0.4854 0.8462 0.9261 0.9556 0.9774MITC4 0.5004 0.8480 0.9278 0.9578 0.9799MISC2 0.4709 0.8464 0.9285 0.9578 0.9787DKMQ 0.8390 0.9685 0.9854 0.9913 0.9953ARS-Q12 0.8390 0.9680 0.9854 0.9913 0.9955AC-MQ4 0.9539 0.9950 0.9974 0.9981 0.9985QCS1 0.8688 0.9508 0.9782 0.9849 0.9909Note: The reference solution is 0.7945

Table III.The normalizedcentral deflectionsof the Razzaque’sskew plate(Figures 7 and 8)

1.0

Nor

mal

ized

def

lect

ion 0.9

0.8

0.7

0.6

0.5

0.42 4 6 8 10 12

Mesh N×N

ReferenceS4

QCS1

MITC4MITC2DKMQARS-Q12RDKQM

HDF-SH4

Figure 8.The convergenceplot of the centraldeflection ofRazzaque’sskew plate

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in which αr is a random number between −1.0 and 1.0; and Δzre is the initial regularelement size along the z-direction.

This is a severe test for both inextensional bending and complex membrane states.The reference vertical defelction at the load point is 1.8248×10−5 (Belytschko andLeviathan, 1994). Table IV gives the normalized results calculated by these two meshes.Furthermore, the results of other elements in the regular mesh are also listed inTable IV. Their convergence plots are shown in Figure 11. It can be observed that thepresent shell element HDF-SH4 is rather good when comparing with other element,even in the distorted mesh, see Table IV and Figure 11.

3.5 Hemispherical shell3.5.1 Hemispherical shell with 18° open hole. A hemispherical shell with 18° open holeat its top is subjected to concentrated loads, as shown in Figure 12(a). Owing to thesymmetry, only a quarter is analyzed. This problem, proposed by Macneal and Harder(1985), is investigated to test the element’s ability in modeling inextensional bendingdeformations and rigid body rotations about the normal to the shell surface. Themembrane locking and geometry warping may result in deteriorations of element’s

Regular mesh Distorted mesh

Y

XZ

Y

XZ

(a) (b)

Figure 10.The contour plots of

y-directiondisplacement of thepinched cylinder on

different meshes

P

sym

sym

P

R

Diaphrag Diaphragm

L /2 L /2

x

y

z

Notes: R= 300, L= 600, t= 3, E= 3×107, � = 0.3, P =1

Figure 9.The pinchedcylinder with

diaphragms ends

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perfomance. Table V and Figure 13 present the radial deflections at the point A and thecorresponding covergence plots, respectively. The results have been normalized by thereference solution 0.094 (Macneal and Harder, 1985), although a slightly lower one 0.093is proposed latter (Simo et al., 1989). It can be seen that, as the discretization errorsdecrease, the results of element HDF-SH4 converge rapidly to the reference solution.

3.5.2 Full hemispherical shell. The discretization of a full hemispherical shell isshown in Figure 12(b). This shell has the same geometry and material parameters withthe previous one. In this problem, the elements present more pronounced warpingconfigurations. Thus it can help to study the effect of the warping element geometry.The radial deflections along the load direction at point A are normalized by thereference solution 0.0924 (Parisch, 1991), as listed in Table VI. And the convergenceplots are given in Figure 14. This test proves once again that the element HDF-SH4 canrepresent the inextensional bending modes and rigid body rotations well.

3.6 Scordelis-Lo roofFigure 15 illustrates that the Scordelis-Lo’s cylindrical roof structure with end rigiddiaphragms is subjected to a self-weight load. Taking advantage of its symmetry, only

Mesh N×N 4× 4 8× 8 16× 16 24× 24

HDF-SH4 (mesh a) 0.6292 0.9199 0.9919 1.0024HDF-SH4 (mesh b) 0.6392 0.9264 0.9933 0.9985S4 0.3882 0.7543 0.9328 0.9743QC5D-SA 0.3759 0.7464 0.9300 0.9919MIST1 0.4705 0.8016 0.9482 0.9794MITC4 0.3677 0.7363 0.9203 0.9644QPH 0.37 0.74 0.93 –QCS1 0.6089 0.9255 0.9818 –XSHELL41 0.625 0.926 0.995 –MISQ24 0.6416 0.9411 1.0018 –Q4DRL 0.348 0.732 0.926 –

Table IV.The normalizedvertical defelction atthe load point of thepinched cylinder(Figures 9-11)

1.0

Nor

mal

ized

def

lect

ion

0.9

0.8

0.7

0.6

0.5

0.4

0.34 8 12 16 20 24

Mesh N×N

ReferenceS4

QCS1

MITC4

QC5D-SA

QPH

Q4DRL

XSHELL41

MIST1

MISQ24

HDF-SH4 (mesh a)HDF-SH4 (mesh b)

Figure 11.The convergenceplot of the verticaldeflection at loadpoint of thepinched cylinder

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F

x

sym sym

Free

Free

F

z

y

A

(a)

FF

x

sym

z

y

sym

Free

A

(b)

Notes: R= 10, � = 0.3, t= 0.04, E= 6.825×107, F =1

Figure 12.(a) Hemispherical

shell with 18°hole; (b) full

hemispherical shell

Mesh 4× 4 8× 8 16× 16

HDF-SH4 0.5220 0.9614 0.9927S4 0.9671 0.9829 0.9892QC5D-SA 0.3863 0.9505 0.9906XSHELL41a 1.038 1.012 1.001QCS1 0.4851 0.9564 0.9926Q4DRL 0.990 0.986 0.988MISQ24 0.7670 0.9798 0.9960Note: aData which are normalized by 0.093 (Kim et al., 2003)

Table V.The normalized

radial defections ofthe hemispherical

shell with 18°open hole (Figures

12(a) and 13)

1.0

Nor

mal

ized

def

lect

ion

0.8

0.6

0.4

4 8 12 16

Mesh N×N

Reference

S4

QCS1

MISQ24

QC5D-SA

Q4DRL

XSHELL41

HDF-SH4

Figure 13.The convergenceplot of the radialdeflections of the

hemispherical shellwith 18° open hole

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one quarter of the roof is considered. As shown in Figure 16, two different typicalmeshes are considered: the regular mesh and the distorted mesh. The distorted mesh isgenerated by the same strategy given in Equation (39). The vertical deflections at thepoint A are listed in Table VII, which have been normalized by the reference solution0.3024. And the convergence plots are given in Figure 17. The results show that theelement HDF-SH4 performs better than most elements. Furthermore, it can providesatisfactory results even in distorted mesh.

3.7 Twisted beamAs shown in Figure 18, the 90° pre-twisted beam is subjected to a load at its tip. Twoload cases are considered: in-plane case and out-plane case. This problem wasoriginally proposed by Macneal and Harder (1985) and then modified by Belytschkoet al. (1989) through reducing the shell thickness. The latter case is considered in ourwork and is a more challenging test due to the warping geometry and membranelocking. The deflections along the load directions at the tip mid-point and theirconvergence plots are given in Table VIII and Figure 19. Results of two different caseshave been normalized by the reference deflections 5,256 and 1,294 (Cook, 1993),

N elements per edge 4 8 16

HDF-SH4 0.4865 0.9560 0.9971S4 0.5157 0.9594 0.9916QC5D-SA 0.3101 0.9310 0.9926XSHELL41 1.007 0.998 –QCS1 0.3571 0.9188 0.9913QPH 0.28 0.86 0.99CYSE 0.68 0.98 0.99Schwarze 0.418 0.956 0.996

Table VI.The normalizedradial defections ofthe full hemisphericalshell (Figures12(b) and 14)

1.0

Nor

mal

ized

def

lect

ion

0.8

0.6

0.4

0.24 8 12 16

N

Reference

S4

QCS1

CYSE

Schwarze

QC5D-SA

QPH

XSHELL41

HDF-SH4Figure 14.The convergenceplot of the radialdeflections of the fullhemispherical shell

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Diaphragm

sym

sym

Free

Free

40°

R

L

x

y

z

A

Notes: R= 25, L= 50, t= 0.25, E= 4.32×108, � = 0,self-weight 90/area

Figure 15.Scordelis-Lo roof

Regular mesh Distorted mesh

Y

XZ

Y

XZ

(a) (b)

Figure 16.The contour plots ofy-direction deflectionof Scordelis-Lo roofon different meshes

Mesh 4× 4 8× 8 16× 16 32× 32

HDF-SH4 (mesh a) 1.0374 1.0029 0.9961 0.9951HDF-SH4 (mesh b) 1.0387 1.0014 0.9956 0.9951S4 1.0356 1.0019 0.9966 0.9961MIST1 1.168 1.028 1.008 –MITC4 0.9284 0.9609 0.9908 –XSHELL41 1.035 1.002 – –QCS1 0.7712 0.8307 0.9431 –MISQ24 1.1912 1.0420 1.0063 –Q4DRL 0.775 0.913 – –SRI 0.964 0.984 0.999 –CYSE 0.942 1.008 1.005 1.006

Table VII.The normalized

vertical deflectionat point A of theScordelis-Lo roof(Figures 15-17)

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1.0

1.2

Nor

mal

ized

def

lect

ion

0.8

0.6

4 8 12 16

Mesh N×N

ReferenceS4MIST1MITC4

QCS1

SRICYSE

MISQ24

XSHELL41

HDF-SH4 (mesh a)HDF-SH4 (mesh b)

Figure 17.The convergenceplot of verticaldeflections atpoint a of theScordelis-Lo roof

L

z

y

x

P2

P1

w

Notes: L= 11, w= 1.1, t= 0.0032, E= 2.9×107, � = 0.22, P1=1,P2=1

Figure 18.Twisted beam

Mesh 2× 4 2× 8 2× 16

In-plane caseHDF-SH4 0.9966 1.0034 1.0051S4 1.0031 0.9837 0.9929RQS20 0.961 0.977 0.990QS20-R 0.965 0.974 0.987MITC4 0.905 0.958 0.983Shin 0.8606 0.9732 0.9983

Out-plane caseHDF-SH4 1.0228 1.0086 1.0043S4 0.8731 0.9622 0.9927RQS20 0.864 0.957 0.988QS20-R 0.835 0.949 0.987MITC4 0.789 0.935 0.983Shin 0.8658 0.9703 0.99435

Table VIII.The normalizeddeflections alongthe load direction ofthe twisted beam(Figures 18 and 19)

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respectively. Even though the shell is extremely thin, the present element HDF-SH4experiences no locking problems and exhibits an excellent performance.

3.8 Hyperbolic paraboloid (hypa) shellFigure 20 shows a hyperbolic paraboloid (hypa) shell subjected to a self-weight load(Gruttmann and Wagner, 2005). The geometric parameters are: z¼ xy/8L, L¼ 20 and x,y⊆[−10, 10]. At the shell’s boundary, the deflections along z-direction are restrained.And the following boundary conditions are also imposed:

uE ¼ uG ¼ 0; vF ¼ vH ¼ 0: (40)

This problem is also used to assess the element performance in dealing with membranelocking under a warping geometry. Table IX and Figure 21, respectively give thevertical deflections at the central point and the corresponding convergence plots.All the results have been normalized by an analytical Kirchhoff solution 0.046. Notethat the present element HDF-SH4 is only a simple flat shell element, it performsreasonably well.

In-plane case Out-plane case

1.00

0.95

Nor

mal

ized

def

lect

ion

Nor

mal

ized

def

lect

ion

0.90

0.85

0.80

1.00

1.05

0.95

0.90

0.85

0.80

0.75

4 8 12

Reference

S4S4

RQS20

QS20-R

MITC4

Shin

HDF-SH4

Reference

RQS20QS20-R

MITC4

Shin

HDF-SH4

N N

16 4 8 12 16

(a) (b)

Figure 19.The convergence

plot of the twistedbeam

L

L

A

D

C

B

y

z

x

O

E

F

G

H

Notes: L= 20, t= 0.2, z= xy/8L, E= 108, � = 0, fz= –5.0

Figure 20.A hyperbolic

paraboloid shell

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3.9 Partly clamped hyperbolic paraboloid shellFigure 22 illustrates a partly clamped hyperbolic paraboloid shell. It is clamped at oneside and subjected to a self-weight load. This problem is suggested as a challenging testfor modeling pure bending dominated behavior (Bathe et al., 2000). Due to thesymmetry, only a half shell is modeled by using the mesh of N×N/2. Table X presentsthe vertical deflections at the tip point A. And the solution obtained by using the high-order element MITC16 (Bathe et al., 2000) in a refined mesh is given for comparison.The results show that the element HDF-SH4 has a satisfactory performance.

3.10 Raasch’s hookThe geometry and typical mesh of the Raash’s hook is illustrated in Figure 23. Thehook contains two different curved segments which are tangent at the intersection.Both segments have the same thickness and widths. One end of the hook is clampedand the other one is subjected to a unit shear load. This challenge problem is firstreported by Ingo Raash of BMW Corp (Harder, 1991). When he used the refined mesh toobtain better results, he found that the results diverge unexpectedly. The reason ismainly related to the incorrect transfer of twisting moments between adjacent flatelements which are not in the same plane (MacNeal et al., 1998). Knight (1997) examinedhow the element’s feathers affect the solutions, and reported that this divergenceproblem can be suppressed by omitting the shear deformations of shell elements.

Mesh 8× 8 16× 16 32× 32 64× 64

HDF-SH4 0.9688 0.9873 0.9926 0.9969S4 0.9794 0.9928 0.9989 1.0027MISQ24 0.994 0.998 0.999 1.000Taylor 0.980 0.989 0.991 0.993MITC4 0.811 0.934 0.972 0.984MIST1 0.979 0.982 0.984 0.987

Table IX.The normalizedcentral deflectionsof the hypa shell(Figures 20 and 21)

1.00

1.05

Nor

mal

ized

def

lect

ion

0.95

0.90

0.85

0.80

0.758 16 32 64

Mesh N×N (log2scale)

Reference

S4

MISQ24

MITC4

Taylor

MIST1

HDF-SH4Figure 21.The convergenceplot of the centraldeflection of thehypa shell

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By following his suggestion, a modification scheme for overcoming this problem isintroduced here.

In the original form of locking-free Timoshenko’s beam, which is employed todetermine the boundary displacement model of the plate bending part (Cen et al., 2014),the shear strain γij of the edge ij is assumed as:

gij ¼ dijG; (41)

with:

G ¼ 2l ij

�wiþwj� ��csi�csj; dij ¼

6lij1þ12lij

; lij ¼D

Cl2ij: (42)

Here, a modification factor is introduced into Equation (41):

gij ¼ amdijG; (43)

in which αm is suggested to vary from 0.25 to 3. Through this modification, the effect ofthe plate’s shear deformation is reduced. Figure 24 presents the normalized deflectionsalong the load direction at the free edge, in which the number of elements along thelongitudinal direction is used as the y-axis. Here, the reference solution is set as 5.027(Kemp et al., 1998), although different solutions have been proposed. For comparison,

clampedsym

A

Notes: t= 0.001, E= 2×1011, � = 0.3, self-weight 8.0/areaz= x2–y2, x (–0.5, 0.5), y (–0.5, 0)

Figure 22.Partly clamped

hyperbolicparaboloid shell

Mesh 8× 4 16× 8 32× 16 48× 24

HDF-SH4 0.004105 0.006148 0.006324 0.006365S4 0.006129 0.006187 0.006297 0.006359MISQ24 0.007121 0.006713 0.006468 0.006426MITC4 0.004758 0.005808 0.006190 0.006294MIST1 0.005586 0.006190 0.006347 0.006383MITC16 0.006394

Table X.The vertical

deflection at point aof the partly

clamped hyperbolicparaboloid shell

(Figure 22)

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the results of some other elements (Knight, 1997) are also presented. It can be seen that,the models with modification can provide reasonably convergent results, while resultsof the original element diverge, as well as many other flat shell elements.

4. ConclusionsRecently, the present authors have developed the HDF method (Cen et al., 2014; Shanget al., 2015b) and the HSF method (Cen et al., 2011a, b) for constructing shape-free

R1

R2

P

30°x

y

z

P

w

Notes: t= 2, E= 3,300, � = 0.35, w= 20, R1 = 14, R2 = 46, P = 1

Figure 23.The model and meshof Raasch’s hook

3

Reference

S44-STG

4-HYB

4-ANS

AQ4

HDF-SH4

HDF-SH4 α=0.3

HDF-SH4 α=0.25

Nor

mal

ized

def

lect

ion

2

1

17 34 51 68

N

85 102 119 136

Figure 24.Normalized results ofthe Raasch’s hook

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Mindlin-Reissner plate bending and plane membrane elements, respectively. In thesetwo methods, the resultant/stress trial functions within the element are derived fromthe analytical solutions of the displacement function/Airy stress function, instead ofbeing directly assumed. Thus, these trial functions can a priori satisfy all the relatedgoverning equations for plate bending and plane problems, respectively. The derivedplate and membrane element have an excellent performance and are insensitive tovarious mesh distortions. In this work, by combination of the HDF method andHSF method, a 4-node quadrilateral flat shell element HDF-SH4 with six DOFs per nodeis proposed.

In order to investigate the performance of the present shell element, several classicaland severe benchmarks are employed. Numerical results show that the elementHDF-SH4 is efficient for analysis of shell structures and inherits the advantages ofits parent components, i.e. free of shear/membrane locking and less insensitive tomesh distortions.

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Appendix 1. The non-zero components of the matrix N��P in Equation (13)

For the edge ij i ¼ 1; 2; 3; 4!

; j ¼ 2; 3; 4; 1!� �

, the following labels are defined:

Lab1 ¼ 3� i�1ð Þþ1; Lab2 ¼ 3� j�1ð Þþ1: (A1)

Then the non-zero components of N��P are:

N��P 1; lab1þ1ð Þ ¼ �yij

l ij1�sð Þ; N

��P 1; lab1þ2ð Þ ¼ xij

l ij1�sð Þ;

N��P 1; lab2þ1ð Þ ¼ �yij

l ijs; N

��P 1; lab2þ2ð Þ ¼ xij

l ijs;

N��P 2; lab1ð Þ ¼ �6

l ij1�2dij� �

Z 2; N��P 2; lab1þ1ð Þ ¼ �xij

l ij1�s�3 1�2dij

� �Z 2

� �;

N��P 2; lab1þ2ð Þ ¼ �yij

l ij1�s�3 1�2dij

� �Z 2

� �; N

��P 2; lab2ð Þ ¼ 6

l ij1�2dij� �

Z 2;

N��P 2; lab2þ1ð Þ ¼ �xij

l ijs�3 1�2dij

� �Z 2

� �; N

��P 2; lab2þ2ð Þ ¼ �yij

l ijs�3 1�2dij

� �Z 2

� �;

N��P 3; lab1ð Þ ¼ 1�sþ 1�2dij

� �Z 3; N

��P 3; lab1þ1ð Þ ¼ �xij

2Z 2þ 1�2dij

� �Z 3

� �;

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N��P 3; lab1þ2ð Þ ¼ �yij

2Z 2þ 1�2dij

� �Z 3

� �; N

��P 3; lab2ð Þ ¼ s� 1�2dij

� �Z 3;

N��P 3; lab2þ1ð Þ ¼ xij

2Z 2� 1�2dij

� �Z 3

� �; N

��P 3; lab2þ2ð Þ ¼ yij

2Z 2� 1�2dij

� �Z 3

� �; (A2)

with:

Z 2 ¼ s 1�sð Þ; Z 3 ¼ s 1�sð Þ 1�2sð Þ; dij ¼ 6lij= 1þ12lij� �

; lij ¼ D=Cl2ij; (A3)

where D and C are given in Equation (16); s ¼ 1þx2 ; x �1pxp1ð Þ is the one-dimension

isoparametric coordinate along the edge ij; l2ij ¼ x2ijþy2ij with xij¼ xi−xj, yij¼ yi−yj.

Appendix 2. The non-zero components of the matrix N��m in Equation (27)

For the edge ij i ¼ 1; 2; 3; 4!

; j ¼ 2; 3; 4; 1!� �

, the following labels are defined:

Lab1 ¼ 3� i�1ð Þþ1; Lab2 ¼ 3� j�1ð Þþ1: (A4)

Then the non-zero components of N��m are:

N��m 1; lab1ð Þ ¼ 1

21�xð Þ; N

��m 1; lab1þ2ð Þ ¼ yij

81�x2� �

;

N��m 1; lab2ð Þ ¼ 1

21þxð Þ; N

��m 1; lab2þ2ð Þ ¼ �yij

81�x2� �

;

N��m 2; lab1þ1ð Þ ¼ 1

21�xð Þ; N

��m 2; lab1þ2ð Þ ¼ �xij

81�x2� �

;

N��m 2; lab2þ1ð Þ ¼ 1

21þxð Þ; N

��m 2; lab2þ2ð Þ ¼ xij

81�x2� �

; (A5)

where xij¼ xi−xj, yij¼ yi−yj; ξ (−1 ⩽ ξ ⩽ 1) is the one-dimension isoparametric coordinate alongthe edge ij.

Corresponding authorSong Cen can be contacted at: censong@mail.tsinghua.edu.cn

For instructions on how to order reprints of this article, please visit our website:www.emeraldgrouppublishing.com/licensing/reprints.htmOr contact us for further details: permissions@emeraldinsight.com

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