analysis of shear wall structure using optimal membrane triangle element

Finite Elements in Analysis and Design 43 (2007) 861–869www.elsevier.com/locate/finel

Analysis of shear wall structure using optimal membrane triangle element

M. Paknahad∗, J. Noorzaei, M.S. Jaafar, Waleed A. ThanoonCivil Engineering Department, Faculty of Engineering, University Putra Malaysia, 43400 UPM-Serdang, Malaysia

Received 25 January 2006; received in revised form 9 March 2007; accepted 30 May 2007Available online 19 July 2007

Abstract

In this study an alternate formulation, using optimal membrane triangle elements in finite element (FE) programming has been implemented.The formulation showed that more efficient computation was achieved and the accuracy of the FE program was established using some standardbenchmark examples. Numerical studies indicate that the FE idealization, with coarse mesh using this alternative optimal membrane triangleelement, produced good results for the analysis of shear wall structures. The results were found to be satisfactory with a wide range of elementaspect ratio.� 2007 Elsevier B.V. All rights reserved.

Keywords: Finite elements; High performance element; Drilling degrees of freedom; Shear wall structures with/without opening

1. Introduction

The idea of including normal-rotation degrees of freedomat corner points of plane-stress finite elements (FEs) (the so-called drilling freedom) is an old one [1–8]. Many efforts todevelop membrane elements with drilling degrees of freedomwere made during the period 1964–1975, which came out withinconclusive results. The classical FE formulation to developmembrane elements with drilling degrees of freedom was un-successful. These unsatisfactory endeavors caused Irons andAhmad [9] to view it as futile for any further attempt to de-velop membrane elements with drilling degrees of freedom.The main motivations behind inclusion of drilling degrees offreedom were:

(i) To improve the element performance while avoiding theuse of midpoint degrees of freedom. The midpoint nodeshave lower valency respect to corner nodes, demand extraeffort in FE mesh, and can cause modeling difficulties innonlinear and dynamics analysis.

(ii) To solve the “normal rotation problem” of shells analyzedwith FE programs that carry six degrees of freedom pernode.

∗ Corresponding author.E-mail address: [email protected] (M. Paknahad).

0168-874X/$ - see front matter � 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.finel.2007.05.010

(iii) To simplify the modeling of connections between plates,shells and beams.

Numerical techniques based on FEs with drilling degrees offreedom have received attention in recent years. For examples,MacNeal [10–12] reported the processes of formulation, com-putation and validation of a defect free performance of four-node membrane quadrilateral element, with drilling degrees offreedom based on the Isoparametric principles. Sze et al. [13]proposed a mixed quadrilateral plane element with drillingdegrees of freedom using Allman’s interpolation scheme,Hellinger–Reissener functional and assumed stress field. Theelement stiffness matrix is generated based on numerical in-tegration scheme. The performance of the element has beenestablished by analyzing few standard benchmark examples.Piancastelli [14] introduced a plate-type FE with six degreesof freedom for each node to analyze anisotropic compositematerials. A similar study on composite folded anisotropicstructures using plate element with drilling degrees of freedomwas reported by Lee et al. [15].

Pimpinelli [16] studied a four nodes quadrilateral membranewith drilling degrees of freedom. The formulation was mainlybased on minimization of the modified Hu–Washizu functional,in which the enhanced strain and rotation fields were included.Hughes et al. [17] exploited a proper functional to describe,in weak form, the equilibrium problem associated with the

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862 M. Paknahad et al. / Finite Elements in Analysis and Design 43 (2007) 861–869

boundary values in the presence of independent rotation fields.In the two-dimensional case, these lead to membrane elementswith drilling degrees of freedom.

Cook [18] developed a 24 degrees of freedom quadrilateralshell element by including the drilling degrees of freedom. Theauthor concluded that numerical results are good but the el-ement is not the best available four-node shell element in alltest cases. Chinosi [19] combined a membrane element withplate bending element which lead to a shell elements with sixdegrees of freedom. Further more, Chinosi and collaborators[20] constructed a new FE with drilling degrees of freedom forlinear elasticity problems. They showed that the new elementwas more efficient compared with the author’s earlier proposedshell element. Zhu et al. [21] discussed the development of anew quadrilateral shell element with drilling degrees of free-dom. One point quadrature was used for the analysis of non-linear geometrical and material problems.

Ibrahimbegovic [22–25] presented membrane elements withdrilling degrees of freedom based on a variational formulationwhich employs an independent rotation field. Two new mem-brane elements, namely MQ2 and MQ3 were developed. Theseelements exhibited good performance over a set of problems.Furthermore, the author demonstrated the application of theseelements for geometrically nonlinear shell theory. Felippa et al.[26] studied the formulation of 3-node, 9-dof membrane ele-ments with the drilling degrees of freedom within the contextof parameterized variational principles. The investigation hasconstructed an element of this type, using the extended freeformulation (EFF). They constructed this element within thecontext of the assumed natural deviatoric strain (ANDES) for-mulation. The resulting formulation has five free parameters.These parameters are optimized against pure bending by energybalance methods. Furthermore, Felippa [27] compared deriva-tion methods for constructing optimal membrane triangles withcorner drilling freedoms. In this report a comprehensive sum-mary of element formulation approaches and the constructionof an optimal 3-node triangle (OPT) using the ANDES for-mulation were presented. Based on the extensive review of theliterature on elements with degrees of freedom derived on dif-ferent principles, it was seen that for two dimensional planestress problems Felippa and co-workers optimal membrane tri-angle element based on ANDES formulation, was found to bemost efficient and suitable.

In the present work, an attempt has been made to furtherenhance the formulation of Felippa’s work [26–29]. Hence, analternative formulation of triangle optimal element has beenproposed to make the triangle optimal element more attractiveas far as its programming effort and computational efficiencyare concerned. The objectives of the present work are to:

(i) carry out reformulation (or alternate formulation) of OPTelement in an alternative form,

(ii) implement this reformulation and its computational algo-rithm in a computer coding,

(iii) test the FE code against the standard benchmark examples,(iv) apply the proposed code to shear wall structures with and

without openings.

2. FE formulation of optimal membrane triangle element

The ANDES formulation is a combination of the free for-mulation (FF) of Bergan and a variant of the assumed naturalstrain (ANS) method due to Park et al. [1,3]. Extensive formu-lations of ANS and ANDES were published by Felippa et al.[26–29]. The basic steps of the formulations are summarizedin this section.

Assuming that the element to be constructed has nodal dis-placement degrees of freedom collected in vector v, elasticmodulus matrix E, and volume V, stiffness matrix is con-structed by using the fundamental decomposition of stiffnessequations:

KR = (Kb + �Kh)V . (1)

Here Kb is the basic stiffness, which takes care of consistency,and Kh is the higher order stiffness, which takes care of stability(rank sufficiency) and accuracy. This decomposition was foundby Bergan [6] as part of the FF and � is a scaling coefficient(� > 0). The basic stiffness matrix Kb is constructed by thestandard procedure (CST element). The main portion of thestrains is left to be determined variationally from the constantstress assumptions which are used to develop Kh.

2.1. Element description

The membrane triangle shown in Fig. 1 has straight sidesjoining the corners and is defined by the coordinates {xi, yi},i = 1, 2, 3. Coordinate differences are abbreviated as

xij = xi − xj and yij = yi − yj . (2)

The area A is given by

2A = (x2y3 − x3y2) + (x3y1 − x1y3) + (x1y2 − x2y1)

= y21x13 − x21y13. (3)

In addition to the corner nodes 1, 2 and 3, midpoints 4, 5 and6 shall also be used for derivations, although these nodes donot appear in the final equations of the element. Midpoints4, 5, 6 are located at the opposite corners 3, 1 and 2, re-spectively. As shown in Fig. 1, intrinsic coordinate systemsare used over each side and the Lij ’s are the lengths of the

Fig. 1. Triangle geometry.

M. Paknahad et al. / Finite Elements in Analysis and Design 43 (2007) 861–869 863

sides [27]. The triangle will be assumed to have constantthickness h and uniform plane stress constitutive properties.The well-known triangle coordinates showed in Fig. 2, aredenoted by �1, �2 and �3, which satisfy �1 + �2 + �3 = 1. Thedegrees of freedom are collected in the node displacementvector uR as

uR = [ux1uy1�1ux2uy2�2ux3uy3�3]T. (4)

Here uxiand uyi

denote the nodal values of the translationaldisplacements along x and y, respectively, and �i is the “drillingrotations” about z.

2.2. Natural strains

In the derivation of the higher order stiffness by ANDES,natural strains play a key role. Strains along the three sidedirections were used in [28]. The natural strains are collectedin the three vectors. The natural strains are related to Cartesianstrains by the “strain gage rosette” transformation:

[�] = [�21 �32 �13]T, [e] = [exx eyy 2exy]T, � = T −1e e,

(5)

Te = 1

4A2

⎡⎢⎢⎣

y23y13l221 y31y21l

232 y12y32l

213

x23x13l221 x31x21l

232 x12x32l

213

(y23x31 + x32y13)l221 (y31x12 + x13y21)l

232 (y12x23 + x21y32)l

213

⎤⎥⎥⎦ , (6)

Enat = T Te ET e, (7)

where Enat is the natural stress–strain matrix defined which isconstant over the triangle.

2.3. The basic stiffness

An explicit form of the basic stiffness was published byBergan et al. [6]. It can be expressed as

Kb = V −1LELT,

where V = Ah is the element volume and L is a 3 × 9 matrixthat contains a free parameter �b [26–29]:

L = 12h

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

y23 0 x230 x23 y23

16�by23(y13 − y21)

16�bx32(x31 − x12)

13�b(x31y13 − x12y21)

y31 0 x130 x13 y31

16�by31(y21 − y32)

16�bx13(x12 − x23)

13�b(x12y21 − x23y32)

y12 0 x210 x21 y12

16�by12(y32 − y13)

16�bx21(x23 − x31)

13�b(x23y32 − x31y13)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (8)

If �b=0, the basic stiffness reduces to the stiffness matrix of theCST element. In this case, the rows and columns associated withthe drilling rotations vanish. In the direct fabrication approach,the decomposition is explicitly used to construct the stiffnessmatrix in two stages, first Kb and then Kh.

2.4. The higher order stiffness

The ANDES form of higher order stiffness matrix Kh devel-oped in [8], is

Kh = CfacTT�u

K�T�u, (9)

where K� is the 3 × 3 higher order stiffness in terms of thehierarchical rotations and Cfac is a scaling factor. To expressK� compactly, the following matrices are introduced:

Q1 = 2A

3

⎡⎢⎢⎢⎢⎢⎢⎣

�1

l221

�2

l221

�3

l221

�4

l232

�5

l232

�6

l232

�7

l213

�8

l213

�9

l213

⎤⎥⎥⎥⎥⎥⎥⎦

,

Q2 = 2A

3

⎡⎢⎢⎢⎢⎢⎢⎣

�9

l221

�7

l221

�8

l221

�3

l232

�1

l232

�2

l232

�6

l213

�4

l213

�5

l213

⎤⎥⎥⎥⎥⎥⎥⎦

,

Q3 = 2A

3

⎡⎢⎢⎢⎢⎢⎢⎣

�5

l221

�6

l221

�4

l221

�8

l232

�9

l232

�7

l232

�2

l213

�3

l213

�1

l213

⎤⎥⎥⎥⎥⎥⎥⎦

. (10)

Depending on nine free dimensionless parameters; �1–�9, thescaling by 2A/3 is for convenience in correlating with priordevelopments. Matrix Qi are evaluated at the midpoints by thefollowing terms:

Q4 = 12 (Q1 + Q2), Q5 = 1

2 (Q2 + Q3),

Q6 = 12 (Q3 + Q1),

K� = h(QT4 EnatQ4 + QT

5 EnatQ5 + QT6 EnatQ6),

Kh = 34�0T

T�u

K�T�u, (11)


Fig. 2. Natural strains, along side directions.

Table 1Dimensionless parameter of OPT element

�b �0 �1 �2 �3 �4 �5 �6 �7 �8 �9

32

12 1 2 1 0 1 −1 −1 −1 −2

where �0 is an overall scaling coefficient. So finally KR as-sumes a template form with 11 dimensionless parameters: �b,�0, �1, . . . , �9:

KR(�b, �0, �1, . . . , �9) = V −1LELT + 34�T T

�uK�T�u. (12)

The free dimensionless parameters are determined from ahigher order patch test which tunes up the higher order stiff-ness of triangular elements. These parameters are collectedand tabulated in Table 1 [27].

3. Alternative formulation of higher order stiffness matrix

In the present study an alternative formulation of the aboveelement is presented, which is more efficient compared to theoptimal element constructed by Felippa [27]. In the subsequentdiscussion these formulations are presented in detail.

Let us assume LL matrix as

LL =⎡⎣ l2

21 0 00 l2

32 00 0 l2

13

⎤⎦ . (13)

Then the formulation of Qi and Te (Eqs. (6) and (10)) has beenrewritten in the following form:

Qi = LL−1Q∗i , Te = T ∗

e LL and E∗nat = T ∗T

e ET ∗e , (14)

T ∗e = 1

4A2

⎡⎢⎣

y23y13 y31y21 y12y32x23x13 x31x21 x12x32y23x31 y31x12 y12x23+x32y13 +x13y21 +x21y32

⎤⎥⎦ , (15)

where

Q∗1 = 2A

3

[�1 �2 �3�4 �5 �6�7 �8 �9

], Q∗

2 = 2A

3

[�9 �7 �8�3 �1 �2�6 �4 �5

],

Q∗3 = 2A

3

[�5 �6 �4�8 �9 �7�2 �3 �1

](16)

and Eq. (11) can be rewritten as

Q∗4 = LL−1Q4, Q∗

5 = LL−1Q5, Q∗6 = LL−1Q6 (17)

by substituting the new form of Te and Qi in high order stiffnessformulation and expanding it

K� = h((LL−1Q∗4)

T(T ∗e LL)TE(T ∗

e LL)(LL−1Q∗4)

+ (LL−1Q∗5)


e LL)(LL−1Q∗5)

+ (LL−1Q∗6)


e LL)(LL−1Q∗6)), (18)

K� = h((Q∗T4 )(LL−TLLT)(T ∗T

e ET∗e )(LLLL−1)(Q∗

4)

+ (Q∗T5 )(LL−TLLT)(T ∗T


5)

+ (Q∗T6 )(LL−TLLT)(T ∗T


6)) (19)

from definition of matrix LL it can be written:

K� = h(Q∗T4 E∗

natQ∗4 + Q∗T

5 E∗natQ

∗5 + Q∗T

6 E∗natQ

∗6). (20)

Hence the formulation of Kh with new notation is more efficientin view point of computational time and effort compared tothat of Felippa [27]. The same formulation is also adapted forcalculation of stresses.

4. Computational procedure and development of an FEcode

The followings are the major computational steps adopted inimplementing the proposed OPT element:

Step i: Define geometrical parameter of the OPT element tocalculate Eqs. (3) and (15).

Step ii: Evaluate Enat matrix using Eq. (14).Step iii: Determine the basic stiffness matrix Kb.Step iv: Generate Qi for corner and mid-side nodes employ-

ing Eqs. (16) and (17).Step v: Calculate higher order stiffness matrix Kh using

Eqs. (18)–(20).Step vi: Evaluate triangle optimal element stiffness matrix

using Eq. (12).

Hence, based on the above computational steps an existingtwo-dimensional FE analysis program written by Noorzaeiet al. [30] has been extensively modified in view of inclu-sion of triangle optimal element which is based on newformulation presented in this paper [31]. This program ismulti-element, multi-degrees of freedom and has dynamicallydimensioned features. The program was written in FOR-TRAN language and works under FORTRAN power stationenvironment.

5. Testing and verification

In order to validate the formulation, computational algorithmand implementation of new formulation of OPT element, threebenchmark examples which are available in the literature areconsidered [19]. Table 2 shows the notations used for previous


Table 2Identifier of triangle element instances

Name Description

ALL-3I Allman 88 element integrated by 3-point interior ruleALL-3M Allman 88 element integrated by 3-midpoint ruleALL-LS Allman 88 element, least-square strain fitCST Constant strain triangle CST-3/6CLST-Ret Retrofitted LST with �b = 4

3OPT Optimal ANDES template

Fig. 3. Slender cantilever beam under end moment.

Fig. 4. Tip deflection for cantilever beam.

results in literature. In all benchmark examples the used unitswere consistent.

5.1. Example 1—cantilever beam under end moment

The slender cantilever beam of Fig. 3 is subjected to an endmoment M =100. The exact tip deflection is 100. The geomet-ric data, material properties, boundary conditions, loading anddimension of the beam are also presented in this figure. Thebeam has been discretized using regular meshes, ranging from2 × 2 to 32 × 2, with each rectangle mesh unit consisting offour half-thickness overlaid triangles. The element aspect ratiosvary from 1:1 to 16:1.

Fig. 4 shows computed tip deflections for several elementtypes and five aspect ratios (1, 2, 4, 8, and 16), respectively. Itis clear from this plot that the results obtained from this study

Fig. 5. Cantilever under end shear.

Fig. 6. Tip deflection for short cantilever mesh 16 × 4.

are similar to those published by Felippa (using similar OPTelement). The figure also indicates that the OPT element issuperior to other elements as reported by Felippa [26–29].

5.2. Example 2—the shear-loaded short cantilever

The shear-loaded cantilever beam defined in Fig. 5 has beenselected as a test problem for plane stress elements by manyinvestigators since it was originally presented in 1966 [29].The geometrical data, material property, boundary conditionsand loading are shown in Fig. 5. The comparison value is thetip deflection �c at the center of the end-loaded cross sec-tion. An approximate solution derived from two-dimensionalelasticity, based on a polynomial Airy’s stress function, gives�c = 0.35533. Fig. 6 shows computed deflections for rectan-gular mesh units with aspect ratios of 1, 2 and 4, respectively.Mesh units consist of four half-thickness overlaid triangles. Forreporting purposes, the load was scaled, so that the “theoreti-cal solution” becomes 100. The deflection evaluated from thepresent investigation is identical as reported by Felippa [26–29].


5.3. Example 3-cook’s problem

Fig. 8 shows results computed for the plane stress problemdefined in Fig. 7. This problem was proposed by Cook [18] asa test case for nonrectangular quadrilateral elements. There isnot any known analytical solution, but the OPT results for the64 × 64 mesh are used for comparison purposes.

For triangle tests, quadrilaterals were assembled with twotriangles in the shortest-diagonal-cut layout as illustrated inFig. 7 for a 2 × 2 mesh. Again, the results predicted by the

Fig. 7. Cook’s problem: clamped trapezoid under end shear.

Fig. 9. Geometry and material of shear wall.

FE program code written in the present research, agree wellwith the OPT element reported by Felippa [27].

Through these three verification examples, accuracy and con-vergence of the present formulation of OPT element, as wellas the compatibility of the FE code, are demonstrated. In thenext step, the applicability of the developed code is shown byanalyzing shear wall structures.

6. Application of present study element in shear wallstructure

6.1. Cantilever shear wall without opening

The FE program developed in this study is now applied toan analysis of shear wall structures. Fig. 9 shows geometry,

Fig. 8. Results for vertical displacement at C for different subdivision.


loading and material property of a cantilever shear wall. Theshear wall structure was analyzed for different subdivisions, toillustrate the capability of OPT element in analysis of the wallstructures. To show the efficiency and accuracy of the presentelement against conventional FE in the analysis of the shearwall, eight node isoparametric element was used to model theshear wall. The results for horizontal displacement at the topof shear wall, using OPT and eight node element for differentFE discretization, are shown in Fig. 10. This figure shows thatby using finer mesh, the higher lateral deflections of the shearwall occurred in the case of conventional FEs; whereas, usingOPT element, the result was converged to almost similar val-ues, indicating the accuracy and fast rate convergence of OPTelement.

6.2. Shear wall with opening

In order to show the efficiency, suitability, accuracy and su-periority of the OPT element based on the proposed formulationan attempt has been made to analyze shear wall structures with

Fig. 10. Results for horizontal displacement of the top of shear wall.

Fig. 11. Geometry and material of coupled shear wall.

openings. Commercial packages, namely SAP-2000, STAAD-PRO and FE program based on plane stress formulation wereused for comparisons [30]. Fig. 11 shows geometry and materialproperty of an eight story coupled shear wall. The structure wasrepresented by two FE models, namely model a (coarse mesh)and model b (fine mesh).

The lateral displacement of each model at story 2, 4, 6 and8 for all the FE codes has been tabulated in Table 3. The clas-sical eight node plane stress element and standard commercialsoftware [30] reflect a comparable result. The finer the mesh,the analysis converged to greater deflection. However, by appli-cation of the current alternative OPT element, the deflectionsobtained using coarse mesh are very similar to those using finemesh. Moreover, the results indicate that the coarse mesh fromthe OPT element converge to more accurate deflection obtainedfrom the FE by isoparametric eight node analysis using finemesh. Therefore, it could be concluded that the new formula-tion of the OPT element can be considered to be more efficientas it did not require fine mesh in order to get accurate results.

Contour of normal stress distribution, �x calculated bySTAAD-PRO, SAP 2000 and the present study are shown inFig. 12. These plots show that FE code using present OPTelement is capable of predicting almost similar stress distribu-tion in the shear wall and at the connecting beams. Moreover,the stress distribution evaluated by the developed FE codebased on alternative formulation of OPT element with coarseFE mesh, agrees well with the stress distributions given bythe commercial packages where fine mesh was used. Thiscomparison further proves the computational efficiency of theproposed formulation of the OPT element.

7. Conclusion

In this study an alternative formulation of OPT element wasemployed and its computational algorithm has been imple-mented in an FE code. The implemented code was verified,using standard benchmark examples, and was found to be suit-able for further use. The implemented code has been applied


Table 3Comparison of the lateral deflection at different story level

Finite element method Model Lateral displacement at floor level

Floor 2 Floor 4 Floor 6 Floor 8

Eight node isoparametric element Model a 0.56 1.53 2.59 3.62Model b 0.68 1.82 3.02 4.16Difference (%) 21.4 18.9 16.6 14.9

SAP2000 Model a 0.55 1.48 2.54 3.62Model b 0.77 2.06 3.40 4.66Difference (%) 40.0 39.1 33.8 35.5

STAAD-PRO Model a 0.68 1.68 2.78 3.86Model b 0.79 2.08 3.44 4.69Difference (%) 16.1 23.8 23.7 21.5

Present study Model a 0.71 1.91 3.19 4.43Model b 0.74 1.98 3.28 4.51Difference (%) 4.2 3.6 2.82 1.8

Fig. 12. Contour of normal stress for shear wall and connection beam.


to the analysis of shear wall structures with and without open-ings. Based on the results obtained in this study, it could beconcluded that:

(a) The developed code based on the reformulation of the OPTwas found to be reliable as the results obtained from thiswork were found to be similar to those reported by Felippa[26–29].

(b) The OPT based FE code was found to be more efficientand accurate as it displayed greater accuracy for the deflec-tion and stresses of shear wall structures, with and withoutopening, even using coarse mesh in the FE modeling ofthe structures.

Acknowledgments

The authors would like to thank Professor C.A. Felippa ofthe University of Colorado for providing the technical adviceat the initial stage of this research work. The work reportedhere has been supported by University Putra Malaysia. Theirfinancial support is gratefully acknowledged.

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analysis of shear wall structure using optimal membrane triangle element

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