Finite Elements in Analysis and Design 46 (2010) 1031–1036

Contents lists available at ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

n Corr

E-m

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

An effective planar triangular element with drilling rotation

Ming Huang n, Zhenfeng Zhao, Changyu Shen

APPT NERC, Engineering College, Zhengzhou University, 97 Wenhua Road, Campus Box 148, Zhengzhou 450002, PR China

a r t i c l e i n f o

Article history:

Received 12 December 2008

Received in revised form

18 June 2010

Accepted 22 July 2010

Keywords:

Finite element

Drilling freedom

Triangle

Membrane

Shell

4X/$ - see front matter & 2010 Elsevier B.V. A

016/j.finel.2010.07.019

esponding author. Tel.: +86 371 63888582; f

ail address: huangming@zzu.edu.cn (M. Huan

a b s t r a c t

A modified Allman’s triangular planar element with drilling degree of freedom is formulated. The basic

formulation is similar to the original Allman’s element, but a constraint is introduced which ensures

that the drilling degree of freedom is a true rotation in elasticity. The new element keeps all the

advantages of Allman’s element, but has no spurious energy mode. It also shows that the drilling

parameter of the present element can correctly carry over the twist moment to the other kinds of

element at nodes. A few examples are given for illustration.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

The combination of a membrane element and a plate-bendingelement provides a versatile tool for analysis of shell or folded-plate structures. However, the general membrane element hasonly two degrees of freedom at each node, which leads to therelative shell element having only five independent degrees offreedom—one less than that required in engineering applications.Though we can deal with this problem using some special skills,the best solution is to develop the membrane element with adrilling degree of freedom. To realize this, some early attemptswere made. In the mid 1980s, two independent investigations byAllman [1] and Bergan and Felippa [2] tried new approaches tothe problem. Their research, especially the interpolation scheme,made great contribution to developing membrane element withdrilling degree of freedom. The original Allman’s element hadspurious zero energy mode, but the problem was solved by thelater works [3,4]. Since then, a lot of papers about such elementhave been published [5–10]. Even in recent years, we still couldfind some studies [11–19]. Some of them were directly based onthe Allman’s element, while others employed hybrid methods orother techniques.

Generally speaking, we have two purposes to developthe planar element with drilling degree of freedom. The first isto improve the elemental accuracy, and the other is to providean ideal membrane element to form shell element or connectwith other kinds of elements with rotational degree of freedom

ll rights reserved.

ax: +86 371 63887570.

g).

such as beam element to do some complicated engineeringanalysis. To a certain degree, the later is more important. It istrue that all planar elements with drilling degree of freedomcan nominally do this job, but it does not mean that all thoseelements can correctly do the job. To ensure the correctness of theelement, we need to clarify two things: what is the physicalmeaning of the drilling degree of freedom and how to evaluate itscorrectness. However, in the most papers that studied drillingdegree of freedom, only accuracy was investigated, and thecorrectness of the drilling degree of freedom was seldomconcerned.

This paper presents a modified Allman’s triangular elementand tries to give the clear definition of the drilling degree offreedom. To begin with, it is proved that the drilling degree offreedom should be the rotation component of the elasticityby use of the theory of work of external force. Next, theinterpolation scheme of the original Allman’s element is deeplystudied. It shows that the drilling parameter in Allman’selement is not the rotation of elasticity. However, it can becomethe true rotation of elasticity by introducing a constraint. Inaddition, the constraint also eliminates the spurious zero mode. Inthis way, a new element with drilling degree of freedomis derived. The new element needs only slight change to theoriginal Allman’s element. It still keeps the advantages ofsimplicity, accuracy and compatibility of Allman’s element,but has no spurious zero energy modes. More importantly,the drilling parameter of the present element can correctlyundertake the external in-plane moments and carry over thetorque to other elements such as 1D beam elements, which is asignificant feature that many other similar elements do nothave.

δy

δx

τϕ

x

y

M

r

o

us

Fig. 2. Approximate expression of moment in 2D space.

M. Huang et al. / Finite Elements in Analysis and Design 46 (2010) 1031–10361032

2. Physical meaning of drilling degree of freedom

Generally, there are two types of elements. The first is directlyderived from elasticity theory, such as 2D membrane element and3D cubic element whose all nodal parameters are translationaldisplacements u, v and w. The second includes those that also takerotations as nodal parameters, like beam and plate bendingelements. In the second kind of elements, the nodal rotation musthave the functionality: it can support the related foreign momentand their product represents the work done by the moment. Onlyunder this condition, the internal moment in an element can becarried over to other kind of elements through the rotation degreeof freedom, and the different kind of elements can be assembledtogether to solve complicated engineering problem. To checkwhether the product of the moment and the rotation is the workof the foreign force, the elasticity theory can be employed. Forexample, we take the beam bending problem into consideration,as shown in Fig. 1. According to Kirchhoff hypothesis, thedisplacement field can be expressed as

u¼ zyy, v¼ 0 ð1Þ

where yy is the rotation of the beam around y-axis. Assuming thatthe height of the beam is h, the moment is defined as

M¼

Z h=2

�h=2szdz ð2Þ

The work done by the external force is

w¼

Z h=2

�h=2usdz ð3Þ

Substitution of Eqs. (1) and (2) into Eq. (3) yields

w¼

Z h=2

�h=2yyszdz¼Myy ð4Þ

It is therefore proved that the product of the moment androtation in beam bending represents the work done by theexternal loading. In same way, we can have similar conclusion forother cases in which the rotation is introduced. In those problems,there is a common point: the rotation of a point is rigid rotation,i.e., the shear strain is zero, so the displacement can be expressedpartly using the rotation. The rotation has a very clear physicalmeaning, such as yx¼qw/qy, yy¼�qw/qx in plate bending.

But in the case of membrane element, the situation becomescompletely different. Except some very special cases, we cannotspecify shear strain to be zero. Thus the question appears: What isthe drilling degree of freedom in membrane element? Does ithave same meaning as the rotation degree of freedom in the plateor beam has? If answer is no, how can we use it as global degree offreedom to assemble different kinds of elements? If yes, how toprove it?

To answer the question, we still can use the concept of thework. Assuming that drilling degree of freedom y is subjected to amoment M, we need to ensure that the work can be written as

w¼My ð5Þ

In fact, not like other forces or stresses, the moment inelasticity has also no unique definition. It can be a couple of forceswith opposite direction, and it can also be the integration of the

x

z

M

�

Fig. 1. Moment and its work in beam element.

distributed forces. But with Saint-Venant Principle, the differentdefinitions can only affect the solution in a very limited area. Toavoid the element coordinate dependent, we define the momentin our problem as uniformly distributed constant shear forcearound a small circle at the center of which the drilling degree offreedom is located, as shown in Fig. 2. The shear force andtangential displacement have the expressions:

t¼ M

2pr2ð6Þ

us ¼ vcosj�usinj ð7Þ

in which v and u can be linearly expanded at the center of thecircle:

v¼ v0þ@v0

@xdxþ

@v0

@ydy

u¼ u0þ@u0

@xdxþ

@u0

@ydy

8>>><>>>:

ð8Þ

with

dx¼ r cosj, dy¼ rsinj ð9Þ

Based on the definition, the work of the moment M can beexpressed as

w¼

Iustds ð10Þ

Substituting Eqs. (6)–(9) into Eq. (10), and also using the belowformulas:H

sinjds¼H

cosjds¼H

sinjcosjds¼ 0Hsin2jds¼

Hcos2jds¼ r

Hsin2ydy¼ rp

(ð11Þ

We finally have

w¼M1

2

@v

@x�@u

@y

� �¼Mo ð12Þ

where o is the rotation component defined in the elasticity.Comparing Eq. (12) with Eq. (5), we can conclude that the drillingdegree of freedom in the membrane element should be therotation component of the elasticity

y¼o ð13Þ

If the purpose of using the membrane element with drillingdegree of freedom is only to improve the accuracy of computa-tion, it seems no problem to use any such kind of element. But ifthe purpose is to form shell element or connect to plate bendingor beam element, Eq. (13) should be checked, otherwise it willlead to a wrong result.

3. Analysis to original Allman’s element

In Ref. [1], Allman derived a well-known triangular element withdrilling degree of freedom, whose displacement interpolation can be

M. Huang et al. / Finite Elements in Analysis and Design 46 (2010) 1031–1036 1033

expressed as

8>>><>>>:

u¼ u0þ1

2ðy1�y2Þb3x1x2þ

1

2ðy2�y3Þb1x2x3þ

1

2ðy3�y1Þb2 x3x1

v¼ v0þ1

2ðy1�y2Þc3x1x2þ

1

2ðy2�y3Þc1x2x3þ

1

2ðy3�y1Þc2x3x1

ð14Þ

where x1, x2, x3 are triangular area coordinates, y1, y2, y3 the vertexrotational degrees of freedom and u0, v0 the displacement interpola-tion of the constant strain element, viz.

u0 ¼ u1x1þu2x2þu3x3

v0 ¼ v1x1þv2x2þv3x3

(ð15Þ

and bi, ci are defined as

bi ¼ yj�yk, ci ¼ xk�xj ð16Þ

in which i¼(1,2,3), j¼mod(i/3)+1 and k¼mod((i+1)/3)+1. Thiselement is compatible, provides good accuracy and is much simplerthan many other similar elements. However, this element has twodisadvantages which limit its application. Firstly, it has a spuriouszero energy mode, i.e.,

y1 ¼ y2 ¼ y3a0

ui and vi : any rigid body movement

(ð17Þ

which results in a singular stiffness matrix even if the general rigidbody movements are suppressed. Although this problem can beovercome if the nodal rotations are known at some nodes, wesometimes have no way to know this information, as shown in theexamples of this paper. Secondly, the drilling degree of freedom yi inthis element has no exact physical meaning. In other words, it cannotpass the test of Eq. (13). For example, not losing the generality, basedon the definition of elastic theory, the rotational component at node 1is expressed as

o1 ¼1

2

@v

@x�@u

@yÞ1

�ð18Þ

Substituting Eq. (14) into Eq. (18), we have

o1 ¼1

2

@v

@x�@u

@y

� �1

¼o0þ1

8Dðy1�y2Þðc3b2�c2b3Þþðy3�y1Þðc2b3�c3b2Þ� �

¼o0þ1

8Dð2y1�y2�y3Þðc3b2�c2b3Þ� �

¼o0þ1

4ð2y1�y2�y3Þ ð19Þ

where D is the area of the element and

o0 ¼1

2

@v0

@x�@u0

@y

� �ð20Þ

is the rotation of the constant stain element. In the derivation ofEq. (19), the below formulas are employed:

@xi

@x¼

bi

2D,

@xi

@y¼

ci

2Dand 2D¼ ðc3b2�c2b3Þ ð21Þ

It can be found that three drilling variables in Eq. (19) are onlysome generalized parameters. They may have a certain contribu-tion to the nodal rotation, but oi is generally not equal to yi.Therefore, this element can be used to improve the elementaccuracy for the planar problem, but it cannot be used to formshell element or assembled with beam or plate bending elements.The present paper tries to solve this problem.

As mentioned above, y1, y2, y3 are only the generalizedparameters, so the coefficients related to them in Eq. (14) arenot necessary to be 1/2. For example, changing 1/2 to 1 will not

affect the solution to the displacement. Of course, the solution toy1, y2, y3 will change, but we are not interested in them becausethey have no exact physical meaning. Thus Eq. (14) can berewritten as8<:

u¼ u0þeðy1�y2Þb3x1x2þeðy2�y3Þb1x2x3þeðy3�y1Þb2 x3x1

v¼ v0þeðy1�y2Þc3x1x2þeðy2�y3Þc1x2x3þeðy3�y1Þc2x3x1

ð22Þ

where e is an undetermined factor. Our purpose here is to find asuitable value for e and let the nodal rotation oi equal to nodalparameter yi. Similar to the derivation of Eq. (19), the nodalrotation based on Eq. (22) can easily be expressed as

oi ¼o0þe

2ð3yi�y1�y2�y3Þ, i¼ 1,2,3 ð23Þ

It is obvious that yi will become true nodal rotation, i.e., yi¼oi,so long as we choose (e¼2/3) and at same time make

o0 ¼y1þy2þy3

3ð24Þ

In the original Allman’s element, 9 equilibrium equations canbe derived for each element, in which only 5 are independent dueto 3 rigid body movement modes and 1 spurious energy mode.This means that the independent equations are 1 less than thatwe require. If Eq. (24) can somehow be added to the system of theelemental equilibrium equations, the problem will be naturallysolved. In this way, we not only make the drilling parametersbecome real nodal rotation, but also remove the spurious energymode.

4. Improvement of Allman’s element

As discussed above, the new element basically still usesAllman’s displacement interpolation, but the coefficient 1/2 willbe changed into 2/3, i.e.,8>>><>>>:

u¼ u0þ2

3ðy1�y2Þb3x1x2þ

2

3ðy2�y3Þb1x2x3þ

2

3ðy3�y1Þb2 x3x1

v¼ v0þ2

3ðy1�y2Þc3x1x2þ

2

3ðy2�y3Þc1x2x3þ

2

3ðy3�y1Þc2x3x1

ð25Þ

As an additional condition, Eq. (24) must also be introduced atsame time.

There are two ways to add Eq. (24) to the elementalequilibrium equation. Assume that the system of elementalequilibrium equation based on Eq. (25) is

Ke0q¼ fe ð26Þ

Substituting Eqs. (15) and (20) into Eq. (24), we have thediscrete form:

Aq¼ 0 ð27Þ

To ensure that the final elemental stiffness matrix keepssymmetric, Eq. (27) is expanded as

ATAq¼ 0 ð28Þ

Adding Eq. (28) to Eq. (26), the final elemental stiffness matrixis obtained in the form

Ke ¼Ke0þtGDATA ð29Þ

where t is thickness and G is the shear elastic modulus whoseintroduction is to let the additional matrix and the originalstiffness matrix have same dimension. The detailed formulationwill be given in the next section.

Eq. (29) can also be derived by the functional variation.Assuming that the functional of minimum potential energyprinciple is P, we can add Eq. (24), which acts as the penalty

M. Huang et al. / Finite Elements in Analysis and Design 46 (2010) 1031–10361034

function, into P and have a modified functional

P� ¼PþtG

2

ZZ1

3ðy1þy2þy3Þ�

1

2

@v0

@x�@u0

@y

� �� �2

dxdy ð30Þ

Using the general procedure of formulating a finiteelement, we readily get the new elemental stiffness matrix, i.e.,Eq. (29).

In above formulation, G seems a subjective factor. However,the numerical test has proved that its value does not affect thefinal solution significantly so long as it is large enough. Forexample, G, 10 or 0.1 G will give very close results.

Fig. 3. Rigid body rotation.

Table 1Nodal displacements and rotations in example 1.

Nodes i j k m n

u 0 �0.1 0 0.1 0

v 0.1 0 �0.1 0 0

h 0.1 0.1 0.1 0.1 0.1

5. Element formulation

5.1. Strain

The components of strain are given by the kinematicalequation:

e¼

ex

ey

gxy

8><>:

9>=>;¼

@

@x0

0@

@y@

@y

@

@x

266666664

377777775

u

v

� ð31Þ

Substitution of Eq. (25) into Eq. (31) yields

e¼ Bq¼ B1 B2 B3� � q1

q2

q3

8><>:

9>=>; ð32Þ

where

Bi ¼1

2D

bi 02

3biðbkxj�bjxkÞ

0 ci2

3ciðckxj�cjxkÞ

ci bi2

3ðbkciþbickÞxj�ðbicjþbjciÞxk

� �

26666664

37777775

ð33Þ

qi ¼ ui vi yi

n oTð34Þ

5.2. Stress

The multiplication of elastic matrix D by strain vector will givestress vector

r¼De ð35Þ

where

r¼ sx sy txy

n oTð36Þ

In the case of planar stress in an isotropic material

D¼E

1�n2

1 n 0

n 1 0

0 01�n

2

2664

3775 ð37Þ

5.3. Elemental stiffness matrix

Following standard procedure of deriving finite element,we readily write the main part of the elemental stiffnessmatrix as

Ke0 ¼

ZZeBTDBt dxdy ð38Þ

where t is the thickness of the element. Substitution of Eqs. (15)and (20) into Eq. (24) yields

1

3ðy1þy2þy3Þ�

1

2

@v0

@x�@u0

@y

� �¼ Aq¼ A1 A2 A3

� � q1

q2

q3

8><>:

9>=>;¼ 0

ð39Þ

where

Ai ¼ci

2

�bi

2

1

3

� �ð40Þ

Thus the additional part of the elemental stiffness matrix canbe written as

Ked ¼

ZZeGATAt dxdy¼ tGDATA ð41Þ

The sum of Ke0 and Ked gives the final elemental stiffnessmatrix.

6. Numerical examples

6.1. Example 1. Rigid body rotation

Consider a square plate in which 4 elements are drawn, as shownin Fig. 3. At point n, let un¼vn¼0. An arbitrary set of reasonableelastic data are given. A certain rigid body rotation is assumed. Thepurpose of this example is to check if the accurate solution can beobtained, and if the spurious energy mode can be eliminated.

Firstly, let vi¼0.1. As it is well known, in the problem of smalldeformation, the rotational angle should be 0.1 for all nodes. Thesolution of the present element is presented in Table 1. It is clearlyshown that the accurate solution, especially rotation, is obtainedat all nodes.

Secondly, release the given displacement vi¼0.1. Instead, welet yn¼0.1. The solution in this case is exactly same as the firstcase.

The example verifies that the drilling degree of freedom of thepresent element is consistent with the rotational variable ofelasticity. More importantly, only 3 constraints are needed tosuppress rigid body movement, which proves that the elementhas no spurious energy mode. Due to this good feature, it is very

24 24

1644

AP

x

y

Fig. 5. Cook cantilever beam.

Table 3Deflection of cook cantilever beam.

Mesh 2�2 4�4 8�8 Ref. [20]

AT3 19.64 22.41 23.70 23.96

Present 18.61 21.86 23.48

P

P

1 2 3

4

5 6 7

8 9

10

Fig. 6. Connected beam and plate.

M. Huang et al. / Finite Elements in Analysis and Design 46 (2010) 1031–1036 1035

convenient for us to deal with some statically terminate problemin which the force is self-balanced and no constraint is given. Inthis case, just one node needs to be selected and fixed completelyto suppress rigid body movement, while 2 nodes for 2D problemor 3 nodes for 3D problem are needed when using some othermembrane elements. Although this example is simple, not allelements with drilling degree of freedom can pass this test. Forinstance, the elements with spurious energy mode, including theoriginal Allman’s element, cannot obtain the results if justconstraining 3 nodal parameters.

6.2. Example 2. Purely bending

The problem is defined in Fig. 4. A rectangular plate is bentunder the moment at its ends. Considering the symmetry, onlythe half is taken for the investigation. The length of the plate is 8,width 5, and thickness 1; the elastic modulus is 1 and Poisson’sratio 0.2. Two ways are employed to load the moment. First, acouple of forces are loaded at points A and B with oppositedirection, where p¼0.2, acting on uA and uB; then the momentM¼0.5 is used, acting on yA and yB. All numbers aredimensionless. According to Saint-Venant’s Principle, thesolutions of both cases should be very close. The deflections ofpoint A and B are given in Table 2.

This example has two purposes: firstly, it can be used toexamine the elemental accuracy; secondly, to check if the drillingdegree of freedom can correctly support a torque. To a certaindegree, the later is more important than the former because theintroduction of the drilling degree of freedom is mainly forproviding a suitable element which can combine with platebending element to form shell element or connect with 1Delement to analyze some complicated structure. If an elementcannot pass this test, its usefulness is limited. Table 2 shows thatthe result of the present element is satisfying, while the originalAllman’s element (AT3) fails to pass the torque test. Becausesimilar test has not been found in references, it is difficult toevaluate other same kinds of elements, but it is very possible thata lot of the elements could not pass the test.

6.3. Example 3. Cook’s problem

As shown in Fig. 5, a cantilever beam is subjected to aconcentrated load. Assuming the thickness of the beam is t, thedeflection of the point A can be written as VA ¼ c P=Et

�. The

A

B

P

M

Fig. 4. Purely bending plate.

Table 2Deflection of purely bending plate.

Loading P M

VA VB VA VB

AT3 �3.20 �3.21 �5.48 �5.73

Present �3.10 �3.10 �3.22 �3.26

Analytical �3.13

solution of coefficient c is presented in Table 3. To compare theaccuracy of the present element, the result of the original Allman’selement (AT3) is also given. The reference result is taken from Ref.[20], and 64�64 mesh is used, which can be considered accurateenough.

The result shows that the accuracy of the present element isstill good. Due to the extra constraint of Eq. (24), the accuracy isslightly lower than the original Allman’s element, but acceptable.

6.4. Example 4. Connection to beam element

A square plate, with side length being 0.4 m and thickness0.01 m, is investigated (Fig. 6). At the center of the plate, itconnects to a rod whose length is 0.2 m and diameter 0.02 m. Theother end of the rod (node 10) is completely fixed, i.e., all its 6degrees of freedom are zero. The elastic data and load areE¼1�109 Pa, u¼0.3, P¼5 N, respectively. The test is used toexamine whether the torque produced by P is equal to the countertorque at the fixed end of the rod, thus we will know if the drillingdegree of freedom can effectively carry over the twist moment.The problem is statically determinate, so it is easy to obtain theaccurate counter torque at node 10:

M10 ¼�5N� 0:4m¼�2Nm

The calculation of present element gives above accuratesolution. Due to the spurious energy mode, the original Allman’s

M

x

y

a b

c

30°

r = 50mm

Fig. 7. Wedge subjected concentrated moment.

Table 4Shear stress of wedge.

Coordinate (mm) (9.47, 0) (20.60, 0) (30.60, 0) (40.60, 0)

Present (Pa) 32 092.34 6782.14 3073.68 1746.02Analytical (Pa) 30 329.16 6640.65 3043.33 1736.11Error percentage 5.49 2.09 0.99 0.57

M. Huang et al. / Finite Elements in Analysis and Design 46 (2010) 1031–10361036

element cannot be used in this test because we have no way to seta rotation for the membrane element. This discloses the commondisadvantage of the elements with drilling degree of freedom thathave the spurious energy mode. Correctly connecting with 1Delements and passing torque should be one of the importantfunction of the element with drilling degree of freedom. For anymembrane element, so long as it has drilling degree of freedom, italways looks like to have the ability to connect with a 1D element,but this does not mean that the connection is correct. The testsimilar to the current example should be one of the standards toevaluate this kind of elements.

6.5. Example 5. Wedge subjected to concentrated moment

As shown in Fig. 7, a wedge is subjected to a concentratedmoment at C, and M¼1 Nm. The meshes are generated with thesize of about 1 mm. The boundary conditions along the ab aregiven according to the analytical solution:

sr ¼24M sin2yð6�p

ffiffiffi3pÞr2

, sy ¼ 0, try ¼12Mð

ffiffiffi3p

=2�cos2yÞð6�p

ffiffiffi3pÞr2

The numerical result of the present element is given in Table 4.It shows that the present element is still good, especially at theplace far away from point c. Around c, the error is little larger, butit is common in finite element analysis when dealing withconcentrated loading if the coarse mesh is used. The exampleagain proves that the present element can well handle momentloading.

7. Conclusion

Without question, Allman’s element plays very important rolein the development of element with drilling degree of freedom.For most similar elements, including the present one, Allman’sidea has been used in their derivation. However, the originalAllman’s element can be used only to improve the accuracy ofmembrane element, while more important function, forming shellelement with plate bending element or connecting to other kinds

of elements with rotational degree of freedom, seems question-able. As disclosed in the examples, its drilling degree of freedom isunreliable. In fact, this seems to be the common disadvantage formost similar elements. Up to date, there have been many studiespublished, in which the spurious energy mode has been removed,and accuracy has also been increased. However, few of themtested the correctness of drilling degree of freedom.

On the basis of the original Allman’s element, only slightchange was made in the present element. It keeps the accuracy ofAllman’s element, but has no disadvantage. It has no spuriousenergy mode, its drilling parameter is real rotation of elasticity,and more importantly, it can pass the correctness test of drillingdegree of freedom. In the family of this kind of elements, thepresent one is also one of the simplest elements. In theengineering practice, the present element would be one of idealchoices when the triangular membrane element with drillingdegree of freedom is needed.

Acknowlegment

This work is supported by the National Natural Science Fund ofChina (10590352).

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