Measurement 50 (2014) 263–269

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Measurement

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An effective quadrilateral membrane finite element based onthe strain approach

0263-2241/$ - see front matter � 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.measurement.2013.12.043

⇑ Corresponding author at: Mechanical Engineering Department, Supe-rior National School of Technology, ENST SNVI National Road N�. 5, ZI,ROUIBA, Algiers, Algeria. Tel./fax: +213 0776 01 18 50.

E-mail addresses: c_rebiai@yahoo.fr (C. Rebiai), belounarl@yahoo.com(L. Belounar).

1 Tel./fax: +213 0666 25 61 45.

Cherif Rebiai b,a,⇑, Lamine Belounar b,1

a Mechanical Engineering Department, Superior National School of Technology, ENST SNVI National Road N�. 5, ZI, ROUIBA, Algiers, Algeriab Numerical Modeling and Instrument in Soil-Structure Interaction Laboratory (MNIISS), University of Biskra, 07000 Biskra, Algeria

a r t i c l e i n f o

Article history:Received 20 July 2013Received in revised form 11 December 2013Accepted 31 December 2013Available online 12 January 2014

Keywords:Strain approachElastoplastic analysisMembrane finite elementDrilling rotation

a b s t r a c t

Based on the strain approach, a new simple and efficient four-node quadrilateral mem-brane finite element with drilling rotation is developed. It can be used for the elastic andelastoplastic analysis. The displacements field of this element is based on the assumedfunctions for the various components of strain which satisfy the compatibility equationand it is developed in some way to improve the element performance in the distorted con-figurations. This finite element has the three degrees of freedom at each of the four nodes(the two translations and the in-plane rotation) and the displacement functions of thedeveloped element satisfy the exact representation of the rigid body modes. Numericalresults show that the proposed strain based element exhibits an excellent behavior forboth regular and distorted mesh over a set of problems in both analyses.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Since the appearance of finite elements method, manyresearchers have adopted the strain based approach forthe development of new finite elements. The advantagesof these elements have been illustrated on several articles[1,2] compared with displacement-based ones. The use ofthis approach was first applied by Ashwell and Sabir, andconcerned only with curved elements [3,4].This approachwas later extended to plane elasticity problems [5,6],to three-dimensional elasticity [7], to cylindrical shells[8–10], and to plate bending [11].

Also recently considerable attention has been given tothe development of simple and efficient rectangular ele-ments having the in-plane rotation as nodal degrees offreedom [12]. The main motivation is the improvementof the accuracy and to provide an ideal membrane element

to form shell element. Indeed in the past several modelssuch as rectangular and triangular plane elasticity ele-ments were developed, among them the elements of Sabir[13] which each of them have three degrees of freedom(DOFs) at each corner node. However these developedstrain based elements with drilling rotation are efficientonly for the regular meshes.

In this context the proposed element in this paper is anew quadrilateral membrane finite element with drillingrotation based on the strain approach named SBQE (StrainBased Quadrilateral Element) able to improve the accuracyand the computation time in the case of regular and dis-torted mesh. Both linear and materially nonlinear analysesare considered. For the purposes of demonstration someselected numerical problems are solved using this devel-oped element.

2. Formulation of the SBQE element

Consider a quadrilateral element SBQE with three de-grees of freedom (Ui, Vi, and in plane rotation hi) at eachof the four nodes which is depicted in Fig. 1.

Fig. 1. SBQE element and coordinate system.

264 C. Rebiai, L. Belounar / Measurement 50 (2014) 263–269

For general plane elasticity problems, the three compo-nents of strain in terms of the displacements are given by

ex ¼@U@x

; ey ¼@V@y

; cxy ¼@U@yþ @V@x

ð1Þ

The components of the strain given in Eq. (1) must sat-isfy an additional equation called the compatibility equa-tion. This equation can be formed by the eliminating U, Vfrom Eq. (1), hence:

@2ex

@y2 þ@2ey

@x2 �@2cxy

@x@y¼ 0 ð2Þ

If these strains given by Eq. (1) are equal to zero, theintegration of these equations allows obtaining the follow-ing expressions:

U ¼ a1 � a3y; V ¼ a2 þ a3x; h ¼ a3 ð3Þ

The terms in Eq. (3) are those representing the rigidbody modes. The present element possesses four nodesand three DOFs (U, V, h) per node. Thus the displacementfield must contain twelve independent constants. Threeof them (a1, a2, a3) are already used for the representationof the rigid body components, thus it remains nine con-stants (a4, a5. . .a12) for expressing the displacement dueto straining of the element. These are apportioned amongthe strains as:

ex ¼ a4 þ a6yþ a7xþ a10y2 þ 2a11xy3

ey ¼ a7 þ a8xþ a9y� a10x2 � 2a11yx3

cxy ¼ 2a5ðyþ 1Þ þ 2a6xþ 2a7xþ 2a8yþ a9yþ 2a12x

ð4Þ

The strains given by Eq. (4) satisfy the compatibilityequation given by Eq. (2). Expressions (4) are equated tothe equations in terms of U, V from Eq. (1) and the resultingequations are integrated, to give

U ¼ a4xþ a5ðyþ y2Þ þ a6xyþ 0:5a7x2 þ 0:5a8y2

þ 0:5a9y2 þ a10xy2 þ a11x2y3

V ¼ a5xþ 0:5a6x2 þ a7ðx2 þ yÞ þ a8xyþ 0:5a9y2 � a10x2y

� a11x3y2 þ a12x2

h ¼ �2a5yþ a7x� a9y� 2a10xy� 3a11x2y2 þ a12x ð5Þ

The final displacement functions are obtained by addingEqs. (3) and (5) to obtain the following:

U ¼ a1 � a3yþ a4xþ a5ðyþ y2Þ þ a6xyþ 0:5a7x2 þ 0:5a8y2

þ 0:5a9y2 þ a10xy2 þ a11x2y3

V ¼ a2 þ a3xþ a5xþ 0:5a6x2 þ a7ðx2 þ yÞ þ a8xyþ 0:5a9y2

� a10x2y� a11x3y2 þ a12x2

h ¼ a3 � 2a5yþ a7x� a9y� 2a10xy� 3a11x2y2 þ a12x ð6Þ

The displacement functions of the developed elementSBQE given by Eq. (6) can be written in matrix form as:

fUg ¼ ½C�fAg ð7Þ

where the U is the nodal displacement vector, A is the con-stant parameters vector {ai} = 1. . .12 and the 12 � 12transformation matrix [C] is given in appendix.

The stiffness matrix [Ke] can be calculated from the wellknown expression:

½Ke� ¼ ½C��TZ Z

½Q �T ½D�½Q �dxdy� �

½C��1 ¼ ½C��T ½K0�½C��1

ð8Þ

The determinant of the Jacobean matrix must also beevaluated because it is used in the transformed integralsas follow:Z Z

dxdy ¼Z þ1

�1

Z þ1

�1det jJjdndg ð9Þ

Thus the matrix [K0] is numerically evaluated, and sincethe matrix [C] of the developed element is not singular, itsinverse can be also numerically evaluated and the elementstiffness matrix [Ke] can be obtained by:

½Ke� ¼ ½C��TZ 1

�1

Z 1

�1½Q �T ½D�½Q �detjJjdndg

� �½C��1 ð10Þ

where the strain matrix [Q] and the elasticity matrix [D]are given in appendix.

3. Linear numerical results from test examples

Before proceeding to the benchmark problems whichare mainly extracted from literature when discussing theelement SBQE with drilling DOFs, a brief notes on the ele-ments to be compared are given:

� Q8: the eight nodes quadrilateral element with sixteendegrees of freedom (DOFs).� Q6: the six node quadrilateral element with twelve

DOFs.� SBRIEIR and SBTIEIR: the four and three node strain

based rectangular and triangular in-plane elementswith in-plane rotation with twelve DOFs [13].� SBT2V: The Improved three node strain based triangular

in-plane element with drilling rotation with nine DOFs[14].� HQ4-9b: Isostatic quadrilateral membrane finite ele-

ment with drilling rotation [15].� P5Sb: Pian’s hybrid element with four nodes [16].� FRQ: Quadrilateral element based on fiber rotation [22].� Quadrilateral element with drilling ITW DOFS [18].� Quadrilateral element with drilling rotation Pimp [19].� Q4: quadrilateral element with four nodes.

C. Rebiai, L. Belounar / Measurement 50 (2014) 263–269 265

3.1. Linear MacNeal beam

To assess the robustness to mesh distortion as well asthe accuracy of the results of the developed element(SBQE), we consider a slender beam of MacNeal [17]. Thegeometrical and materials characteristics of the structureare shown in Fig. 2. The deflection results are listed inTable 1. From the linear deflection results we can see thatthe developed element is insensitive to mesh distortion.For the regular mesh all the results are in good agreementwith the exact solution. Compared with the strain basedelements we can see clearly that the developed elementis more accurate in both cases of loads.

3.2. A simple beam: the higher-order patch test

This problem is shown by Ibrahimobigovic et al. [18]and it is relative to a beam fixed by a minimum numberof constraints. As shown in Fig. 3 the beam is subjectedto a pure bending state. The first load condition is consti-tuted by a unit couple of forces applied at the free end ofthe beam whereas the second load case is still a momentbut it is applied as a concentrated couple at the end. Thegeometrical and mechanical characteristics are as follow:

E ¼ 100; v ¼ 0 P ¼ 1; M ¼ 0:5; L ¼ 10; H ¼ 1; t ¼ 1

Both regular and distorted meshes are considered inthis example. The vertical displacement and the rotationat the point B are computed.

The results presented in Table 2 show the better behav-ior of the developed element and its relative insensitivityto distortion. The results in terms of the drilling rotationsshow a significant improvement with those of [18,19].

3.3. Cantilever beam modeled by five irregular quadrilateralelements

This problem has been treated in Ref. [20] in which thebeam as shown in Fig. 4 is subjected to: (a) a pure bendingunder moment M; (b) linear bending under transverseforce P, and it is modeled by 5 irregular quadrilateral ele-ments. The Young’s modulus E = 1500, Poisson’s ratiov = 0.25. The results of vertical displacement VA at the pointA and the stress rxB at the point B are given in Table 3.These results show that the developed element gives

Fig. 2. MacNeal and Harder patch tests: geom

almost the analytical solution of deflection and stresses,and it is more accurate than the other Elements.

3.4. Plane flexure of cantilever beam

The objective of this problem is to calculate the deflec-tion VA at the free end of a cantilever beam, with uniformcross-section, subjected to uniform vertical load withYoung’s modulus E = 107, Poison’s ratio v = 0.3 as shownin Fig. 5.

This problem has been treated in [21]. Table 4 showsthe results obtained for different meshes for this problem.

The results presented in Table 4 show that the presentelement gives better results than all the other elementsin case of distorted meshes, whereas for the regularmeshes it is similar to the Q8 element but in terms of com-puting times it is more economic.

4. Elastoplastic analysis

In this analysis three different yield criteria are em-ployed. The Tresca and Von Mises laws, which closelyapproximate metal plasticity behavior, are consideredand the Mohr Coulomb criterion, which is applicable toconcrete rocks and soils, is used. This analysis employstwo methods for generating body-loads: visco-plastic(named as initial strain) and initial stress methods to pre-dict the response to loading of an elastic perfectly plasticmaterial. Both methods are given in [12].

4.1. Elastoplastic numerical results

We assessed the performance of the proposed element(SBQE) through two numerical tests and compared the re-sults with those of the Q8 element and to the analyticalsolutions. We selected the test examples so as to evaluateaccuracy and robustness of the developed element in reg-ular and distorted meshes.

4.1.1. Bearing capacity analysis of purely coherent soilThe elastic properties, Young’s modulus, Poisson’s ratio,

the undrained cohesion and the uniform stress as consis-tent with [23], were chosen as E = 105 kN/m2, m = 0.3,Cu = 100 kN/m2, and q = 1 kN/m2 respectively. Fig. 6 showsthe geometrical characteristics and meshing of the flexible

etry, mesh and boundary conditions.

Table 1MacNeal-Harder cantilever beam: Numerical results of deflection for different load cases and mesh geometry.

Element Shear P = 1 Bending M = 10

Regular Parallel Trapezoidal Regular Parallel Trapezoidal

P5Sb 0.1081 0.07848 0.00497 0.268 0.170 0.01404SBTIEIR 0.0050 0.0039 0.000054 0.03186 0.02727 0.00108SBT2V 0.1072 0.0944 0.0900 0.255 0.254 0.257HQ4-9b 0.1072 0.1057 0.1058 0.269 0.266 0.266SBQE 0.1073 0.1057 0.1058 0.268 0.266 0.267Analytical 0.1081 0.270

Fig. 3. Geometry and mesh of a simple beam.

Table 2Vertical displacement and rotation at the point B of the simple beam.

Load case Mesh ITW Ref.[18] Pimp [19] SBQE

Vert. Dis. End rot. Vert. Dis. End rot. Vert. Dis. End rot.

Forces Reg. 1.50 0.60 1.50 0.60 1.49 0.59Forces Dist. 1.14 0.57 1.39 0.54 1.48 0.59Couple Reg. 1.50 0.62 1.51 1.44 1.50 0.60Couple Dist. 1.39 0.49 1.39 1.28 1.49 0.59Analytical 1.50 0.6

Fig. 4. Cantilever beam with five irregular quadrilateral elements.

Table 3The deflection and stresses at point A et B for bending problem of cantileverbeam.

Elements Loads M Loads P

VA rxB VA rxB

Q6 98.4 �2428 100.4 �3354SBRIEIR 98.08 �2850 98.57 �3870Q4 45.7 �1761 50.7 �2448SBQE 99.19 �2970 99.35 �3918Exact 100 �3000 102.6 �4050

266 C. Rebiai, L. Belounar / Measurement 50 (2014) 263–269

strip footing. Bearing failure in this problem occurs when qreaches the Prandtl load [23] given by:

qultime ¼ ð2þ pÞCu ð11Þ

It is usual in problems of this type to make the loadsincrements smaller as the failure load is approached. Atload levels well below failure, convergence should occurin relatively little iteration [23]. The number of iterationsto achieve convergence for each load increment is also ob-tained. The maximum number of iterations that will be al-lowed within any load increment is provided as a data andthis permits the algorithm to stop and no more load incre-ments are applied.

As shown in Fig. 7 the results, found by the elementSBQE have been plotted in the form of a dimensionlessbearing capacity factor q/cu versus centerline displace-ment. These Results shows that SBQE with 12 degrees offreedom and the Q8 element with 16 degrees of freedomhave similar results but the later uses more degrees offreedom.

When the Tresca and the VonMises yield criteria areused the corresponding numerical solutions with both ele-ments are given in Table 5. It can be seen that SBQE is ro-bust and accurate in elasto-plastic analysis.

Fig. 5. Cantilever beam in plane flexure.

Table 4The deflection VA of the beam in plane flexure.

Mesh Elements

FRQ [22] Q8 SBRIEIR SBQERI:2 � 2 EI:3 � 3 AI EI:4 � 4

(a) 2.76a 3.03 2.75 2.92(12)b (16) (12) (12)

(b) 3.44 3.70 3.43 3.64(18) (26) (18) (18)

(c) 3.56 3.84 3.56 3.80(24) (36) (24) (24)

(d) 1.09 0.64 2.84 2.96(e) 1.61 1.76 2.09 3.043

Beams theory VA = 4.03.a VA vertical displacement in A; EI: exact integration; AI: analytical

integration; HP hammer point.b TNDF: Total number of degrees of freedom.

Fig. 6. Geometry and irregular mesh of the flexible strip footing.

0 1 2 3 4 5 6

q/cu

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Dis

plac

emen

t (m

)

SBQE Q8

Prandtl load=5.14

Fig. 7. Bearing stress versus centerline displacement.

Table 5Vertical displacement under the loaded node of coordinate (0,0).

Displacement (m) Displacement (m)Von Mises Tresca

SBQE Q8 SBQE Q8

200 kN �0.006503 �0.006592 �0.006503 �0.06592300 kN �0.01121 �0.01155 �0.01098 �0.01118350 kN �0.01594 �0.01630 �0.01501 �0.01587400 kN �0.02247 �0.02316 �0.02118 �0.02249450 kN �0.03362 �0.03317 �0.03261 �0.03213480 kN �0.04370 �0.04227 �0.04279 �0.04100500 kN �0.05195 �0.05084 �0.05118 �0.04934510 kN �0.05759 �0.05665 �0.05596 �0.05498515 kN �0.06263 �0.06093 �0.06115 �0.05862520 kN �0.07627 �0.07459 �0.06981 �0.06669

C. Rebiai, L. Belounar / Measurement 50 (2014) 263–269 267

4.1.2. Stability of a slope subjected to gravity loadingIn order to check the accuracy of the present element

SBQE, in this example Fig. 8, the geometrical characteris-tics, material properties, criterion and conditions were

chosen as the same of those used in [23]. The factor ofsafety (F) of the slope is to be assessed, and this quantityis defined as the proportion by which tang u (friction an-gle) and Cohesion C must be reduced in order to causefailure.

Fig. 8. Slope subjected to gravity loading (distorted mesh).

1 1.5 2 2.5 3F (Factor of safety)

-14

-12

-10

-8

-6

-4

-2

V maxx 1

0- 5

SBQEQ8

Bishop and Morgenstern 1960

F=2.505

Fig. 9. Maximum of displacement versus Factor of safety.

268 C. Rebiai, L. Belounar / Measurement 50 (2014) 263–269

Results presented in Fig. 9 in terms of the factor ofsafety and the maximum of displacement at convergenceshow that the convergence to the reference solution givenin [24] with the SBQE element is quite rapid and similar tothe Q8 element. We can see also that the SBQE behavesbetter in distortion configurations.

5. Conclusions

This study proposed a new four node membrane quad-rilateral finite element named SBQE based on the strain ap-proach for the elastic and elasto-plastic analysis. Theproblem typically encountered in strain based elementsis to ensure that the element behaves equally well for reg-ular as well as irregular meshes. The present developedelement is found to be numerically insensitive to distortedmeshes. This element is simple and with only twelve de-grees of freedom and contains higher order of polynomialterms. In terms of computing times it can be less expensivethan other displacement elements. It was clearly revealedthat the proposed element could offer a significant effi-ciency and accuracy in both elastic and elastoplasticanalysis.

Appendix A

Components of the matrix [C] of the dimension[12 � 12] for the SBQE are:

½C� ¼1 0 �y x yþy2 xy 0:5x2 0:5y2 0:5y2 xy2 x2y3 00 1 x 0 x x2=2 yþx2 xy 0:5�y2 �x2y �x3y2 x2

0 0 1 0 �2y 0 x 0 �y �2xy �3x2y2 x

264

375

where xi and yi are the coordinates of node i (i = 1, 4), thematrix [C] is given by:

½C� ¼ ½½C1�½C2�½C3�½C4��T

For the case of plane stress problems the elasticity ma-trix [D] is:

½D� ¼ Eð1� v2Þ

1 v 0v 1 00 0 1�v

2

264

375

For the case of plane strain problems the elasticity ma-trix [D] is:

½D� ¼ Eð1þ vÞð1� 2vÞ

ð1� vÞ v 0v ð1� vÞ 0

0 0 ð1�2vÞ2

264

375

The strain matrix is given by:

½Q � ¼0 0 0 1 0 y x 0 0 y2 2xy3 00 0 0 0 0 0 1 x y �x2 �2x3y 00 0 0 0 2ðyþ 1Þ 2x 2x 2y y 0 0 2x

264

375

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