Computers and Structures 185 (2017) 1–14

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Computers and Structures

journal homepage: www.elsevier .com/locate /compstruc

The MITC4+ shell element in geometric nonlinear analysis

http://dx.doi.org/10.1016/j.compstruc.2017.01.0150045-7949/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: phillseung@kaist.edu (P.-S. Lee).

Yeongbin Ko a, Phill-Seung Lee a,⇑, Klaus-Jürgen Bathe b

aDepartment of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of KoreabDepartment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

a r t i c l e i n f o

Article history:Received 27 December 2016Accepted 27 January 2017

Keywords:Shell structures4-node shell finite elementsMITC methodLarge displacements and rotationsShear and membrane lockingReliability in geometric nonlinear analysis

a b s t r a c t

We present the large displacement and rotation formulation of the new MITC4+ shell finite elementrecently proposed by Ko, Lee and Bathe for linear analysis (Ko et al., 2017) and demonstrate the perfor-mance in geometric nonlinear analysis. The element shows in linear analysis an almost ideal convergencebehavior since shear and membrane locking is alleviated using the MITC approach. We show now thatusing the total Lagrangian formulation for large displacements and large rotations, the element is alsorobust and efficient in nonlinear analysis. We demonstrate the element performance through the solu-tions of various benchmark problems and reach the important conclusion that the MITC4+ shell elementperforms reliably and well even when the mesh undergoes large displacements and significant distor-tions during the response.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

For the analysis of shell structures, developing ‘‘ideal” shellfinite elements that satisfy the ellipticity, consistency and inf-supconditions has been of great interest [1–10]. Such shell elementsshould pass the basic tests (the isotropy, zero energy mode andpatch tests), show uniformly optimal convergence behavior inany shell problem irrespective of the shell geometry, loading andboundary conditions, and do so when regular and even distortedmeshes are used [7–10]. Also, the shell elements need to performequally well in geometric nonlinear analysis where an effectivebehavior in the nonlinear response predictions is important [4].

In geometric nonlinear analysis of shell structures, significantmesh distortions can occur as the geometry of the elementschanges during the response [11–15]. These element geometricchanges can lead to locking in bending-dominated shell problems[4,8], that is, an overly stiff behavior of the shell discretization isseen, which can be particularly severe when 4-node shell elementsare used to model thin shell structures [1,2,4,8]. Hence, in geomet-ric nonlinear analysis, locking due to the discretization undergoinglarge displacements can lead to erroneous predictions of load-displacement trajectories and critical loads.

The MITC (Mixed Interpolation of Tensorial Components)method [1–4,16–25] has been used effectively to remedy shearand membrane locking. The classical 4-node MITC shell element(labeled as MITC4 element) has been widely used in practice for

both linear and nonlinear analyses. However, in the original formu-lation of the MITC4 shell element, membrane locking was not trea-ted, and thus solution accuracy can deteriorate when curvedgeometries are modeled with distorted meshes [1,2]. Followingvarious attempts to alleviate membrane locking of 4-node shellelements [26–28], we recently presented the new MITC4+ shellelement for general linear analysis [1]. This element satisfies allthe basic element tests, contains no parameter to adjust, andshows an almost optimal convergence behavior in the solutionsof a ‘behavior-encompassing’ set of benchmark problems usingregular and distorted meshes. The fact that the element perfor-mance is also very good in highly distorted meshes is particularlynoteworthy and makes this element an excellent candidate forgeneral large displacement and rotation nonlinear analysis.

This expectation is reasonable because our experience is that ifa well-formulated MITC element has been established for linearanalysis, then this element formulation can directly be extendedto geometric nonlinear analysis without introducing instabilitiesin the element, like seen in formulations based on incompatiblemodes and enhanced assumed strains [4,29,30]. In addition, theincompatible modes and enhanced assumed strain elements arecomputationally more expensive.

In this paper, we present the geometric nonlinear formulationof the newMITC4+ shell. We develop the assumed shear and mem-brane fields of the element for the total Lagrangian formulationusing the Green-Lagrange strains and incremental Green-Lagrange strains to obtain the tangent stiffness matrix and internalforce vector [4]. We demonstrate the performance of the elementin geometric nonlinear solutions by solving various shell problems

2 Y. Ko et al. / Computers and Structures 185 (2017) 1–14

with uniform and distorted meshes. To assess the accuracy of thesolutions, we compare the predicted response with analytical dataand finite element solutions obtained using the MITC4 and MITC9shell elements. An important conclusion is that the new MITC4+shell element performs well even when a mesh undergoes signifi-cant displacements that could induce some locking, like might beseen in solutions using the classical MITC4 shell element.

2. Geometric nonlinear formulation

In this section, we present the geometric nonlinear formulationof the MITC4+ shell element. In the total Lagrangian formulation,the left superscript t denotes ‘‘time” for a general analysis (in staticsolutions ‘‘time” simply denotes the load step and configuration)and the left subscript 0 is used to denote the initial (reference) con-figuration [4].

2.1. Geometry and displacement interpolations

The geometry of the MITC4+ shell element in the configurationat time t shown in Fig. 1 is interpolated using [1,2]

txðr; s; fÞ ¼ txm þ ftxb with txm ¼X4i¼1

hiðr; sÞtxi and

txb ¼ 12

X4i¼1

aihiðr; sÞtVin; ð1Þ

where hiðr; sÞ is the two-dimensional interpolation function of thestandard isoparametric procedure corresponding to node i, txi is

the position vector of node i, and ai and tVin denote the shell thick-

ness and the director vector at the node, see Fig. 1.The following representation of interpolation function hiðr; sÞ is

useful in the element formulation:

hiðr; sÞ ¼ 14ð1þ nirÞð1þ gisÞ with i ¼ 1;2;3;4;

n1 n2 n3 n4½ � ¼ 1 �1 �1 1½ �;g1 g2 g3 g4½ � ¼ 1 1 �1 �1½ �; ð2Þ

in which ni and gi are permuted together.The incremental displacement vector u from the configuration

at time t to the configuration at time t þ Dt is

uðr; s; fÞ ¼ tþDtxðr; s; fÞ � txðr; s; fÞ: ð3Þ

ζ

2 1

4

3

rs

31V

t

32V

t

3n

tV

xi

yizi

z

x

y

Fig. 1. A standard 4-node quadrilateral continuum mechanics based shell finiteelement in the configuration at time t.

Using Eq. (1) in Eq. (3), we obtain

u ¼X4i¼1

hiðr; sÞðtþDtxi � txiÞ þ f2

X4i¼1

aihiðr; sÞðtþDtVin � tVi

nÞ; ð4aÞ

with

tþDtxi � txi ¼ uiix þ v iiy þwiiz; ð4bÞand to quadratic order

tþDtVin� tVi

n ¼ hi� tVinþ

12hi�ðhi� tVi

nÞ; hi ¼ tVi1aiþ tVi

2bi; ð4cÞ

in which ix, iy and iz are the base vectors of the global Cartesiancoordinate system, and at node i, ui, v i and wi are the corresponding

displacement components, tVi1 and tVi

2 are unit vectors orthogonal

to the director vector (tVin) and to each other, and ai and bi denote

the rotations of the director vector about tVi1 and tVi

2, respectively[4].

Substituting from Eqs. (4b) and (4c) into Eq. (4a), the incremen-tal displacement is obtained as

uðr; s; fÞ ¼ um þ fðub1 þ ub2Þ; ð5aÞwhere

um ¼X4i¼1

hiðr; sÞui; ð5bÞ

ub1 ¼12

X4i¼1

aihiðr;sÞð�lVi2aiþ lVi

1biÞ;ub2 ¼�14

X4i¼1

aihiðr;sÞða2i þb2

i ÞlVin:

ð5cÞWe next group the displacement terms in Eq. (5) as

u1 ¼ um þ fub1; u2 ¼ fub2; ð6Þwhere u1 and u2 contain the linear and quadratic terms of unknowndisplacements and rotations, respectively.

2.2. Green-Lagrange strains

The covariant Green-Lagrange strain components in the config-uration at time t with respect to the reference configuration at time0 are defined by

t0e ijðr; s; fÞ ¼

12ðtgi � tgj � 0gi � 0gjÞ; ð7Þ

in which tgi ¼ @tx@ri

are covariant base vectors with r1 ¼ r, r2 ¼ s,

r3 ¼ f.Using Eq. (3) in Eq. (7) applied at time t and t þ Dt, the incre-

mental covariant strain components are

0eijðr;s;fÞ¼ tþDt0 eijðr;s;fÞ� t

0e ijðr;s;fÞ¼12

tgi �u;jþu;i � tgjþu;i �u;j� �

;

ð8Þwith u;i ¼ @u

@ri.

Substituting from Eq. (6) into Eq. (8) and retaining only thestrain terms up to second order of unknowns, the incrementalstrain components can be written as

0eijðr; s; fÞ ¼ 0eijðr; s; fÞ þ 0gijðr; s; fÞ; ð9Þwith

0eijðr; s; fÞ ¼ 12

tgi � u1;j þ u1;i � tgj

� �;

1

1

1r

s

1

.0 constε ζst

.0 constε ζrt

)(0

Aζr

t ε

)(0

Bζr

t ε

)(0

Cζs

t ε)(0

Dζs

t ε

.0 constε ζrt

.0 constε ζst

Fig. 2. Tying positions (A), (B), (C) and (D) for the assumed transverse shear strainfield of the MITC4 shell element. The constant transverse shear strain conditions areimposed along its edges.

1

1

1r

s

1

)(0

Amrr

t ε

)(0

Bmrr

t ε

)(0

Cmss

t ε)(0

Dmss

t ε)(

0Em

rst ε

Y. Ko et al. / Computers and Structures 185 (2017) 1–14 3

0gijðr; s; fÞ ¼12

u1;i � u1;j þ tgi � u2;j þ u2;i � tgj

� �with

u1;i ¼ @u1

@ri; u2;i ¼ @u2

@ri;

where 0eij and 0gij are the linear and nonlinear parts, respectively[4,31].

2.3. Assumed Green-Lagrange strains

To alleviate shear locking, the assumed transverse shear strainsof the classical MITC4 shell element are used [2]

t0~erf ¼

12ð1þ sÞt0e

ðAÞrf þ 1

2ð1� sÞt0e

ðBÞrf ;

t0~esf ¼

12ð1þ rÞt0e

ðCÞsf þ 1

2ð1� rÞt0e

ðDÞsf ; ð10Þ

where the tying positions ðAÞ, ðBÞ, ðCÞ and ðDÞ are shown in Fig. 2[1,2,8].

In order to alleviate membrane locking, we separate the corre-sponding membrane strains from the in-plane strains. The covari-ant in-plane strains in Eq. (7) are expressed as

t0e ij ¼ t

0emij þ f t

0eb1ij þ f2t0e

b2ij with i; j ¼ 1;2; ð11aÞ

3

1

4

2r

s

P rt xs

t xnt

Fig. 3. Characteristic vectors for the element geometry at time t. (a) Two in-plane vec

in which

t0e

mij ¼ 1

2txm;i � txm;j þ txm;j � txm;i� �� 1

20xm;i � 0xm;j þ 0xm;j � 0xm;i� �

;

ð11bÞ

t0e

b1ij ¼ txm;i � txb;j þ txm;j � txb;i

� �� 0xm;i � 0xb;j þ 0xm;j � 0xb;i

� �; ð11cÞ

t0e

b2ij ¼ 1

2txb;i � txb;j þ txb;j � txb;i

� �� 12

0xb;i � 0xb;j þ 0xb;j � 0xb;i

� �;

ð11dÞwith

txm;i ¼ @txm

@ri; txb;i ¼ @txb

@ri:

In Eq. (11a), the term t0e

mij is the covariant in-plane strain at the

shell mid-surface (f ¼ 0), which in general can induce membranelocking.

In the geometric nonlinear formulation of the MITC4+ shell ele-ment, the assumed membrane strain fields are applied based onthe current configuration. Hence the covariant membrane strainin Eq. (11b) is considered

t0e

mij ¼ 1

2tgm

ij �12

0gmij with i; j ¼ 1;2; ð12aÞ

withtgm

ij ¼ txm;i � txm;j þ txm;j � txm;i; ð12bÞ

3

1

4

2

dt x2

tors txr and txs , and the plane P with normal vector tn. (b) Distortion vector txd .

Fig. 4. Tying positions (A), (B), (C), (D) and (E) for the assumed membrane strainfield.

(b)

(a)

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

reference16x1 MITC416x1 MITC4+

max/ PPtipwLoad

( )

tipu

4 Y. Ko et al. / Computers and Structures 185 (2017) 1–14

0gmij ¼ 0xm;i � 0xm;j þ 0xm;j � 0xm;i: ð12cÞWe next define the three characteristic vectors in the configura-

tion at time t [1]

txr ¼ 14

X4i¼1

nitxi;

txs ¼ 14

X4i¼1

gitxi;

txd ¼ 14

X4i¼1

nigitxi; ð13Þ

in which ni and gi are given in Eq. (2). The geometric representa-tions of the three vectors at time t are shown in Fig. 3.

As shown in Fig. 3(a), the two vectors txr and txs form the planeP with the normal vector tn

tn ¼txr � txs

ktxr � txsk ; ð14Þ

and the dual base vectors tmr and tms on the plane

tmri � txrj ¼ dij;tmri � tn ¼ 0 with i; j ¼ 1;2:

For the MITC4+ shell element, the following assumed field isused for the covariant membrane strain [1]

t0~emij ¼ 1

2t~gm

ij �12

0~gmij with i; j ¼ 1;2 ð15aÞ

where

t~gmrr ¼

12

1� 2taA þ sþ 2taA � s2� �

tgmðAÞrr

þ 12

1� 2taB � sþ 2taB � s2� �

tgmðBÞrr

þ taC �1þ s2� �

tgmðCÞss þ taD �1þ s2

� �tgmðDÞ

ss

þ taE �1þ s2� �

tgmðEÞrs ; ð15bÞ

t~gmss ¼ taA �1þ r2

� �tgmðAÞ

rr þ taB �1þ r2� �

tgmðBÞrr

þ 12

1� 2taC þ r þ 2taC � r2� �

tgmðCÞss

þ 12

1� 2taD � r þ 2taD � r2� �tgmðDÞ

ss

þ taE �1þ r2� �

tgmðEÞrs ; ð15cÞ

t~gmrs ¼

14

r þ 4taA � rs� �

tgmðAÞrr þ 1

4�r þ 4taB � rs� �

tgmðBÞrr

þ 14

sþ 4taC � rs� �

tgmðCÞss þ 1

4�sþ 4taD � rs� �

tgmðDÞss

þ 1þ taE � rs� �

tgmðEÞrs ; ð15dÞ

in which the tying positions ðAÞ, ðBÞ, ðCÞ, ðDÞ and ðEÞ for the corre-sponding strain components are shown in Fig. 4, and the geometriccoefficients are

z

M

b

L

x y

P (b)(a)

Fig. 5. Cantilever problems (16 � 1 mesh, width b ¼ 1:0, thickness a ¼ 0:1,E ¼ 1:2� 106 and m ¼ 0:0). (a) Case of the tip shearing force (L ¼ 10:0). (b) Case ofthe tip moment (L ¼ 12:0).

taA ¼tcr tcr � 1ð Þ

2td; taB ¼

tcrðtcr þ 1Þ2td

;

taC ¼tcsðtcs � 1Þ

2td; taD ¼

tcsðtcs þ 1Þ2td

;

taE ¼ 2tcrtcstd

; td¼ tc2r þ tc2s � 1; tcr ¼ tmr � txd;tcs ¼ tms � txd:

ð15eÞUsing the assumed membrane strain fields, the in-plane strain

components are constructed

t0~e ij ¼ t

0~emij þ f t

0eb1ij þ f2t0e

b2ij with i; j ¼ 1;2: ð16Þ

The shell-aligned local Cartesian coordinate system in the con-figuration at time 0 is defined using the unit vectors 0Li, i ¼ 1;2;3,

0L3 ¼0g3

k0g3k; 0L1 ¼

0g2 � 0L3

k0g2 � 0L3k ;0L2 ¼ 0L3 � 0L1; ð17Þ

and the corresponding local strain components are

t0�e ij ¼ t

0~eklð0Li � 0gkÞð0Lj � 0glÞ with 0gi � 0g j ¼ d j

i : ð18Þ

Displacements

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

analytic solution16x1 MITC416x1 MITC4+

max/MMLoad

( )

tipw tipu

Fig. 6. Load-displacement curves for the cantilever. (a) Case of the tip shearingforce. (b) Case of the tip moment.

x y

z

maxPP

Initial m

esh

max25.0 PP

P

Initial m

esh

z

maxMM

max5.0 MM

max25.0 MM

M

x y

Fig. 7. Deformed shapes of the cantilever. (a) Case of the tip shearing force. (b) Case of the tip moment.

A

B

P

oR iR

x y

z

Fig. 8. Slit annular plate problem (5 � 40 mesh, Ro ¼ 10:0, Ri ¼ 6:0, thicknessa ¼ 0:03, E ¼ 2:1� 107 and m ¼ 0:0).

Y. Ko et al. / Computers and Structures 185 (2017) 1–14 5

2.4. Assumed incremental Green-Lagrange strains

From the assumed Green-Lagrange strains in Section 2.3, weproceed to obtain the corresponding assumed incrementalGreen-Lagrange strains in a consistent manner.

For the transverse shear we use the following assumed incre-mental shear strains

0~erf¼12ð1þsÞ0eðAÞrf þ1

2ð1�sÞ0eðBÞrf ; 0~esf¼1

2ð1þrÞ0eðCÞsf þ1

2ð1�rÞ0eðDÞsf ;

0~grf¼12ð1þsÞ0gðAÞ

rf þ12ð1�sÞ0gðBÞ

rf ; 0~gsf¼12ð1þrÞ0gðCÞ

sf þ12ð1�rÞ0gðDÞ

sf ;

ð19Þwhere the tying positions ðAÞ, ðBÞ, ðCÞ and ðDÞ are shown in Fig. 2[1,2,4,8].

The linear and nonlinear parts of the incremental covariant in-plane strains in Eq. (9) are expressed as

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(b)

Displacements

BwAw

BwAw

)/( io RRPLoad

( )

)/( io RRPLoad

( )

(a)

reference3x24 MITC43x24 MITC4+

reference5x40 MITC45x40 MITC4+

Fig. 9. Load-displacement curves for the slit annular plate problem. (a) 3 � 24mesh. (b) 5 � 40 mesh.

Initial shape

z

Defor

med

mes

h

maxPP

x y

Fig. 10. Deformed shape for the slit annular plate problem.

x

A

B

Free

edge

C

D

P

P

Free

edge

R

y

z

L

β

Fig. 11. Pull-out of a free cylindrical shell structure (12 � 12 mesh, R ¼ 4:953,L ¼ 10:35, thickness a ¼ 0:094, E ¼ 1:05� 107 and m ¼ 0:3125).

6 Y. Ko et al. / Computers and Structures 185 (2017) 1–14

0eij ¼ 0emij þ f 0eb1ij þ f20eb2ij ;

0gij ¼ 0gmij þ f0gb1

ij þ f20gb2ij with i; j ¼ 1;2; ð20aÞ

in which

0emij ¼ 12ðtxm;i � um;j þ txm;j � um;iÞ; ð20bÞ

0eb1ij ¼ 12ðtxm;i � ub1;j þ txm;j � ub1;i þ txb;i � um;j þ txb;j � um;iÞ; ð20cÞ

0eb2ij ¼ 12ðtxb;i � ub1;j þ txb;j � ub1;iÞ; ð20dÞ

0gmij ¼ 1

2um;i � um;j; ð20eÞ

0gb1ij ¼ 1

2ðum;i � ub1;j þ um;j � ub1;i þ txm;i � ub2;j þ txm;j � ub2;iÞ; ð20fÞ

0gb2ij ¼ 1

2ðub1;i � ub1;j þ txb;i � ub2;j þ txb;j � ub2;iÞ; ð20gÞ

with

txm;i ¼ @txm

@ri; txb;i ¼ @txb

@ri; um;i ¼ @um

@ri;

ub1;i ¼ @ub1

@ri; ub2;i ¼ @ub2

@ri:

maxPP

Y. Ko et al. / Computers and Structures 185 (2017) 1–14 7

In Eq. (20a), the 0emij and 0gmij terms are the linear and nonlinear

incremental covariant in-plane strains at the shell mid-surface(f ¼ 0); these in-plane membrane strains can in general inducemembrane locking.

From the assumed Green-Lagrange membrane strains in Eq.(15), we derive the assumed incremental linear membrane strain0emij in Eq. (20b)

0~emrr ¼12ð1� 2taA þ sþ 2taA � s2Þ0emðAÞ

rr

þ 12ð1� 2taB � sþ 2taB � s2Þ0emðBÞ

rr

þ taCð�1þ s2Þ0emðCÞss þ taDð�1þ s2Þ0emðDÞ

ss

þ taEð�1þ s2Þ0emðEÞrs ; ð21aÞ

0~emss ¼ taAð�1þ r2Þ0emðAÞrr þ taBð�1þ r2Þ0emðBÞ

rr

þ 12ð1� 2taC þ r þ 2taC � r2Þ0emðCÞ

ss

þ 12ð1� 2taD � r þ 2taD � r2Þ0emðDÞ

ss þ taEð�1þ r2Þ0emðEÞrs ; ð21bÞ

0~emrs ¼14ðr þ 4taA � rsÞ0emðAÞ

rr þ 14ð�r þ 4taB � rsÞ0emðBÞ

rr

þ 14ðsþ 4taC � rsÞ0emðCÞ

ss þ 14ð�sþ 4taD � rsÞ0emðDÞ

ss

þ ð1þ taE � rsÞ0emðEÞrs ; ð21cÞ

with the same geometric coefficients as in Eq. (15e) and the tyingpositions ðAÞ, ðBÞ, ðCÞ, ðDÞ and ðEÞ in Fig. 4.

We employ the same assumed strain field for the incrementalnonlinear membrane strain 0gm

ij , and hence the incremental in-plane strain components are

0~eij ¼ 0~emij þ f 0eb1ij þ f20eb2ij and

0~gij ¼ 0~gmij þ f0gb1

ij þ f20gb2ij with i; j ¼ 1;2: ð22Þ

x104

0

0.5

1

1.5

2

2.5

3

3.5

4

Displacements

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

PLoad( ) AwDw

Bu

Bu Cu

Cu

reference12x12 MITC412x12 MITC4+

Fig. 12. Load-displacement curves for the pull-out of the free cylindrical shellstructure.

Using the shell-aligned local Cartesian coordinate systemdefined in Eq. (17), the linear and nonlinear parts of the incremen-tal local strains are calculated using the following transformations

0�eij ¼ 0~eklð0Li � 0gkÞð0Lj � 0glÞ;0�gij ¼ 0~gklð0Li � 0gkÞð0Lj � 0glÞ with 0gi � 0g j ¼ d j

i : ð23Þ

2.5. Stiffness matrix and internal force vector

Using the standard total Lagrangian formulation [4,18,31], thetangent stiffness matrix (tKe) and internal force vector (t0Fe) ofthe MITC4+ shell element are obtained

tKe ¼Z

0VBTij�CijklBkld

0V þZ

0V

t0�SijNijd

0V ; ð24aÞ

t0Fe ¼

Z0V

BTijt0�Sijd

0V ; ð24bÞ

in which 0V is the volume of the shell element at time 0, and �Cijkl

and t0�Sij denote, respectively, the material law tensor and the second

Piola-Kirchhoff stress measured in the local Cartesian coordinatesystem.

In Eq. (24), the strain-displacement matrices, Bij and Nij, aredefined by

0�eij ¼ BijUe; d0�gij ¼ dUTeNijUe; ð25Þ

where Ue is the incremental nodal displacement vector

Ue ¼ UT1 UT

2 UT3 UT

4

� �T with Ui ¼ ui v i wi ai bi½ �T :

x y

maxPP

z

Fig. 13. Deformed shape for the pull-out of the free cylindrical shell structure.

(b)

B D

Freeed

ge

R

Lβ

x y

z

θ

egdeeerFC

E

M

A

(c)

(a)

1L2L

3L

4L

1L

2L

3L

4L

CB A

E D

Fig. 14. Bending of a cylindrical shell structure (R ¼ 10:0, h ¼ 30� , L ¼ 20:0, thickness a, E ¼ 2:1� 106 and m ¼ 0:0). (a) Problem description (12 � 12 uniform mesh). (b)Distorted mesh pattern (4 � 4 mesh). (c) Distorted mesh pattern used (12 � 12 mesh).

8 Y. Ko et al. / Computers and Structures 185 (2017) 1–14

In the nonlinear solution procedure, the nodal geometry is

updated using Eq. (4b), and the vectors tVi1,

tVi2 and tVi

n at node iare updated using the following equations:

tþDtVin ¼ Q tVi

n;tþDtVi

1 ¼ Q tVi1;

tþDtVi2 ¼ Q tVi

2; ð26Þwith

Q ¼ 2

q20 þ q2

1 � 12 q1q2 � q0q3 q3q1 þ q0q2

q1q2 þ q0q3 q20 þ q2

2 � 12 q2q3 � q0q1

q3q1 � q0q2 q2q3 þ q0q1 q20 þ q2

3 � 12

0BBB@

1CCCA;

q0 ¼ coshi2

� �; q1 q2 q3½ �T ¼ hi

hisin

hi2

� �; hi ¼ khik;

in which a quaternion representation of large rotations is utilized[32].

In the finite element solutions, we use 2� 2� 2 Gauss integra-tion over the element volume for the 4-node shell elementsconsidered. The computational cost of the MITC4+ shell element

is only slightly higher than the cost of the classical MITC4 shellelement due to the use of the assumed covariant membranestrains.

3. Numerical examples

In this section, several numerical examples are solved todemonstrate the performance of the MITC4+ shell element in geo-metric nonlinear analysis. The results are compared with thoseobtained using of the classical MITC4 shell element. The referencesolutions are analytical data or are obtained using a fine uniformmesh of the MITC9 shell element, which is known to satisfy theellipticity and consistency conditions and to show goodconvergence behavior in both linear and nonlinear analyses[4–8,17–20].

We show in the example solutions that both the MITC4 andMITC4+ elements work well when uniform meshes are used anddue to the specific physical problem the large displacements ofthe meshes do not induce locking, but in contrast to the MITC4

(c)

(b)

(a)

Load( )0M

Aw

Displacements

Aw

Aw

Load( )0M

Load( )0M

Au

Au

Au

x104

x104

x104

0 1 2 3 4 5 6 7 8 90

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

0 1 2 3 4 5 6 7 8 90

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

0 1 2 3 4 5 6 7 8 90

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

reference12x12 MITC412x12 MITC4+

reference12x12 MITC412x12 MITC4+

reference12x12 MITC412x12 MITC4+

Fig. 15. Load-displacement curves for the bending of the cylindrical shell structurewith the uniform mesh. (a) a=R ¼ 1=100. (b) a=R ¼ 1=1000. (c) a=R ¼ 1=10;000.

(c)

(b)

(a)

Displacements

Aw

Aw

Load( )0M

Load( )0M

Load( )0M

Au

Au

Au

x104

0 1 2 3 4 5 6 7 8 9

Aw

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

x104

x1040 1 2 3 4 5 6 7 8 9

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

0 1 2 3 4 5 6 7 8 90

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

reference12x12 MITC412x12 MITC4+

reference12x12 MITC412x12 MITC4+

reference12x12 MITC412x12 MITC4+

Fig. 16. Load-displacement curves for the bending of the cylindrical shell structurewith the distorted mesh. (a) a=R ¼ 1=100. (b) a=R ¼ 1=1000. (c) a=R ¼ 1=10;000.

Y. Ko et al. / Computers and Structures 185 (2017) 1–14 9

A

D C

B

PP

P PA B

R

x y

z

0φ

CD

β

(b) (c)

(a)

1L2L

3L

4L

1L

2L

3L

4L

Fig. 17. Hemispherical shell problem (R ¼ 10:0, u0 ¼ 18� , thickness a ¼ 0:04, E ¼ 6:825� 107 and m ¼ 0:3). (a) Problem description (12 � 12 uniform mesh). (b) Distortedmesh pattern (4 � 4 mesh). (c) Distorted mesh pattern applied (12 � 12 mesh).

10 Y. Ko et al. / Computers and Structures 185 (2017) 1–14

shell element, the MITC4+ element also works well when an ini-tially distorted mesh is used for such problem solutions, see Sec-tions 3.1–3.4.

We also show the important point that the MITC4+ element ismore effective than the MITC4 element in response solutions whendue to the physical nature of the shell problem an initially uniformmesh in the large displacement response can induce locking. In thiscase, the MITC4 element locks in membrane actions whereas theMITC4+ element continues to work well, see Sections 3.5–3.7.

3.1. Cantilever problem

We consider the cantilever bending problem in Fig. 5 [11–13,33]. The cantilever fully clamped at one end is subjected toeither a shearing force P or bending moment M at the free tip.The cantilever is modeled with a 16 � 1 mesh for the MITC4 andMITC4 + shell elements.

For the shearing load case, the reference solution is obtainedusing a 32 � 1 mesh of MITC9 shell elements. We consider the

maximum load of Pmax ¼ 4P0 with P0 ¼ EI=L and I ¼ ba3=12 .

For the moment load case, the cantilever develops to form a cir-

cular arc of radius R ¼ EI=M with I ¼ ba3=12. Using this formula,

the following analytical tip displacements are obtained [12,13]

utip

L¼ M0

Msin

MM0

� 1;wtip

L¼ M0

M1� cos

MM0

� �; M0 ¼ EI

L: ð27Þ

The cantilever beam should bend into a complete circle at theapplied tip moment Mmax ¼ 2pM0.

Fig. 6 shows the load-displacement curves of the MITC4 andMITC4+ shell elements. The solutions using both elements agreewell with the reference and analytic solutions. Fig. 7 shows thedeformed shapes at successive load levels P=Pmax ¼ 0:25 and 1.0for the shearing load case and M=Mmax ¼ 0:25, 0.5 and 1.0 for themoment load case.

3.2. Slit annular plate problem

We next consider the slit annular plate problem shown in Fig. 8[12,13,15,34]. The shearing force P is applied at one end of the slitwhile the other end is fully clamped. We use 3 � 24 and 5 � 40meshes for the MITC4 and MITC4 + shell elements. The maximumload per unit length Pmax=ðRo � RiÞ ¼ 0:8 is considered. The refer-ence solution is obtained using a 10 � 80 mesh of MITC9 shell ele-ments. Fig. 9 shows the load-displacement curves. The solutionobtained using the MITC4+ shell element is slightly better thanthe solution using the MITC4 shell element. The final deformedshape of the structure calculated using the MITC4+ shell elementis presented in Fig. 10.

3.3. Pull-out of a free cylindrical shell

We consider a pull-out of the free cylindrical shell structureshown in Fig. 11 [12,13,34]. The shell structure is subjected to a

Fig. 18. Load-displacement curves for the hemispherical shell problem with theuniform mesh shown in Fig. 8(a). (a) 8 � 8 mesh. (b) 12 � 12 mesh.

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 90

50

100

150

200

250

300

350

400

Displacements

PLoad( )

PLoad( )

(b)

(a)

Au

Au

Bv

Bv

reference8x8 MITC48x8 MITC4+

reference12x12 MITC412x12 MITC4+

Fig. 19. Load-displacement curves for the hemispherical shell problem with thedistorted mesh shown in Fig. 8(c). (a) 8 � 8 mesh. (b) 12 � 12 mesh.

maxPP

maxPP

Y. Ko et al. / Computers and Structures 185 (2017) 1–14 11

pair of pull-out loads (P) at its center. Due to symmetry, only one-eighth of the structure corresponding to the shaded region ABCD inFig. 11 is modeled using a 12 � 12 mesh of the 4-node shell ele-ments. We use the following boundary conditions:w ¼ b ¼ 0 alongBC, u ¼ b ¼ 0 along AD, and v ¼ a ¼ 0 along AB. The analysis is per-formed up to Pmax ¼ 4� 104. The reference solutions are obtainedusing a 32 � 32 mesh of MITC9 shell elements. Fig. 12 shows theresulting load-displacement curves. The MITC4 and MITC4+ shellelements perform very well. Fig. 13 presents the final deformedshape of the structure obtained using the MITC4+ shell element.

x y

z maxPP

maxPP

Fig. 20. Deformed shape of the hemispherical shell (12 � 12 uniform mesh used).

3.4. Bending of a cylindrical shell structure

We solve the bending problem of a cylindrical shell structureshown in Fig. 14(a) [11,16–18]. The structure is subjected to uni-form bending momentM along BC. Three thickness to radius ratios,a=R ¼ 1=100, a=R ¼ 1=1000 and a=R ¼ 1=10;000, are considered.The applied moment varies with the thickness a considered

PIn-plane

POut-of-plane

zb

L

x y

A

Fig. 21. Twisted cantilever beam problems (4 � 24 mesh, L ¼ 12:0, b ¼ 1:1,thickness a ¼ 0:0032, E ¼ 2:9� 107 and m ¼ 0:22).

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7 8 9 10

8 9

11 120

0.5

1

1.5

2

2.5

3

3.5

4

(b)

Displacements

(a)x10-2

x10-2

PLoad( )

PLoad( )

Au

Au

Aw

Aw

reference4x24 MITC44x24 MITC4+

reference4x24 MITC44x24 MITC4+

Fig. 22. Load-displacement curves for the twisted cantilever beam problem. (a)Case of the in-plane load. (b) Case of the out-of-plane load.

12 Y. Ko et al. / Computers and Structures 185 (2017) 1–14

according to M ¼ M0a3. A fully clamped boundary condition isapplied, u ¼ v ¼ w ¼ a ¼ b ¼ 0 along DE. For each thickness, weconsider the load level up to ðM0Þmax ¼ 4:0� 104.

In addition to the uniform mesh in Fig. 14(a), we also considerthe distorted mesh pattern shown in Fig. 14(b). For an N � N ele-ment mesh, a pair of edges are discretized in the following ratio:L1: L2:L3: . . . LN = 1: 2: 3: . . . N. The distorted mesh is shown inFig. 14(c). For the 4-node shell elements a 12 � 12 mesh is used.A 32 � 32 mesh of MITC9 shell elements is employed to obtainthe reference solution.

Fig. 15 shows the load-displacement curves for the uniformmesh, where the solutions obtained using the MITC4 and MITC4+shell elements agree well with the reference solution. Fig. 16shows the load-displacement curves when the distorted mesh isused. As the shell thickness decreases, the solutions obtained usingthe MITC4 shell element depart from the reference solution. How-ever, using the MITC4+ shell element good response predictionsare always obtained.

3.5. Hemispherical shell problem

We solve the hemispherical shell problem shown in Fig. 17(a)[13–15,17,18,34]. The hemispherical shell with an 18� cutout atits pole is subjected to alternating radial point forces (P) at itsequator. In this bending problem, the shell structure undergoesalmost inextensional deformations and thus we test whethermembrane locking occurs. Due to symmetry, only one quarter ofthe structure corresponding to the shaded region ABCD in Fig. 17(a) is modeled using uniform meshes of 8 � 8 and 12 � 124-node shell elements. We use the following boundary conditions:u ¼ b ¼ 0 along BC, v ¼ b ¼ 0 along AD, and w ¼ 0 at A. The max-imum load of Pmax ¼ 400 is considered.

In addition to the uniform mesh in Fig. 17(a), we consider thedistorted mesh pattern shown in Fig. 17(b) in which we use anN � N element mesh, each edge is discretized in the followingratio: L1: L2: L3: . . . LN = 1: 2: 3: . . . N. The distorted mesh in onequarter of the hemisphere is shown in Fig. 17(c). To obtain the ref-erence solution, a 32 � 32 uniformmesh of MITC9 shell elements isemployed.

Figs. 18 and 19 present the load-displacement curves for theuniform and distorted meshes, respectively. As the mesh is refined,the solutions obtained using the MITC4+ shell element converge tothe reference solution more rapidly than those obtained with theMITC4 shell element. For the distorted mesh cases, the MITC4 shellelement gives a response with a large error from the referencesolution. However, the MITC4+ shell element still shows a goodbehavior. Fig. 20 gives the final deformed shape of the hemispher-ical shell calculated using the MITC4+ shell element.

3.6. Twisted cantilever beam problems

We consider the twisted cantilever beam problems shown inFig. 21 [26,34,35]. The initially twisted beam is fully clamped atone end and is loaded by a point load P at the center of the freetip. Two load cases are considered: an in-plane and anout-of-plane load as shown in Fig. 21. We use a 4 � 24 mesh ofthe 4-node elements while an 8 � 48 mesh of MITC9 shell ele-ments is used for the reference solutions. The maximum load levelis Pmax ¼ 4� 10�2 for both load cases.

Fig. 22 gives the load-displacement curves for both the in-planeand out-of-plane loads. No severe locking is present for the in-plane load case, where both 4-node shell elements perform well.When the out-of-plane load is applied, the response predictedusing the MITC4 shell element deviates significantly from the ref-erence solution due to membrane locking. However, the MITC4+

Fig. 23. Deformed shapes of the cantilever beam. (a) Case of the in-plane load. (b)Case of the out-of-plane load.

A

xy1R

2R

1θ

2θ

b

PA

x y

z8 elements

12 elements

Fig. 24. Hook problem (4 � 20 mesh, R1 ¼ 14, h1 ¼ 60� , R2 ¼ 46, h2 ¼ 150� , b ¼ 20,thickness a ¼ 0:02, E ¼ 3:3� 103 and m ¼ 0:3).

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x10-4

PLoad( )

reference4x20 MITC44x20 MITC4+

Au Aw

Y. Ko et al. / Computers and Structures 185 (2017) 1–14 13

shell element shows a good predictive capability. The finaldeformed shapes of the cantilever beam obtained using theMITC4+ shell element are presented in Fig. 23.

0 10 20 30 40 50 60 70 800

0.1

Displacements

Fig. 25. Load-displacement curves for the hook problem.

3.7. Hook problem

Finally, we consider the hook problem shown in Fig. 24, referredto in linear analysis as the Raasch challenge, see Ref. [36]. Thestructure is fully clamped at one end and is loaded by a shear loadP applied as a uniformly distributed traction at the free tip. For thesolution, we use a 4 � 20 mesh with the MITC4 and MITC4+ shellelements and an 8 � 40 mesh of MITC9 shell elements to obtainthe reference solution. The load is applied up toPmax ¼ 1:0� 10�4. Fig. 25 shows the resulting load-displacementcurves. Using the MITC4+ shell element produces a significantly

more accurate solution than using the MITC4 shell element. Thefinal deformed shape of the hook obtained using the MITC4+ shellelement is shown in Fig. 26.

x y

z

maxPP

Initial shape

Deformed mesh

Fig. 26. Deformed shape of the hook.

14 Y. Ko et al. / Computers and Structures 185 (2017) 1–14

4. Concluding remarks

We presented the geometric nonlinear formulation of theMITC4+ continuum mechanics-based shell element which is for-mulated using the MITC approach to alleviate shear and membranelocking. The assumed shear and membrane strain fields used in lin-ear analysis are developed for geometric nonlinear analysis in aconsistent manner. The nonlinear performance of the MITC4+ shellelement is numerically tested through the solutions of variousexamples. The computational cost of the element is only slightlyhigher than the cost of the classical MITC4 shell element.

We can conclude that the MITC4+ shell element provides reli-able and efficient solutions in large displacement problems. Com-pared to the original MITC4 element, the MITC4+ shell elementshows improved performance when distorted meshes in the initialconfiguration are used. Moreover, the MITC4+ element performsmuch better than the MITC4 shell element when due to the natureof the shell problem, the large displacements of the mesh caninduce locking. Hence, we can conclude that the MITC4+ shell ele-ment shows excellent behavior in both linear and nonlinearanalyses.

Acknowledgments

This work was supported by the Basic Science Research Pro-gram through the National Research Foundation of Korea fundedby the Ministry of Science, ICT, and Future Planning (No.2014R1A1A1A05007219), and a grant (MPSS-CG-2016-04) throughthe Disaster and Safety Management Institute funded by Ministryof Public Safety and Security of Korean government.

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