Finite Elements in Analysis and Design 43 (2007) 861–869 www.elsevier.com/locate/finel
Anal An alys ysis is of sh shea earr wa wall ll st stru ruct ctur uree us usin ing g op optim timal al me memb mbra rane ne tr tria iang ngle le el elem emen entt M. Paknahad ∗, J. Noorzaei, M.S. Jaafar, Waleed A. Thanoon Civil Engine Engineering ering Department, Department, Fa Faculty culty of Engine Engineering ering,, Univer University sity Putra Malays Malaysia, ia, 43400 UPM-Se UPM-Serdan rdang, g, Malays Malaysia ia
Received 25 January 2006; received in revised form 9 March 2007; accepted 30 May 2007 Available 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 standard benchmark examples. Numerical studies indicate that the FE idealization, with coarse mesh using this alternative optimal membrane triangle element, produced good results for the analysis of shear wall structures. The results were found to be satisfactory with a wide range of element aspect ratio. ᭧ 2007 Elsevier B.V. All rights reserved. Keywords: Finite elements; High performance element; Drilling degrees of freedom; Shear wall structures with/without opening
(iii) To simpl simplify ify the model modeling ing of conne connection ctionss betwe between en plates, shells and beams.
1. Introducti Introduction on
The idea of including normal-rotation degrees of freedom at corner points of plane-stress finite elements (FEs) (the socalled drilling freedom) is an old one [1–8] [1–8].. Many efforts to develop membrane elements with drilling degrees of freedom were made during the period 1964–1975, which came out with inconclusive results. The classical FE formulation to develop membrane elements with drilling degrees of freedom was unsuccessful successful.. These These unsatisfa unsatisfactory ctory endeav endeavors ors caused caused Irons and Ahmad [9] to view it as futile for any further attempt to develop membrane elements with drilling degrees of freedom. The main motivations behind inclusion of drilling degrees of freedom were: (i) To impro improve ve the eleme element nt performance performance while avoiding avoiding the use of midpoint degrees of freedom. The midpoint nodes have lower valency respect to corner nodes, demand extra effort in FE mesh, and can cause modeling difficulties in nonlinear and dynamics analysis. (ii) To solve the “normal rotation problem” of shells analyzed analyzed with FE programs that carry six degrees of freedom per node. ∗
Corresponding Correspond ing author. author. E-mail address:
[email protected] (M. Paknahad).
0168-874X/$0168-87 4X/$- see front matte matterr doi:10.1016/j.finel.2007.05.010
᭧
2007 Elsevier B.V. All rights reserved.
Numerical techniques based on FEs with drilling degrees of freedom have received attention in recent years. For examples, MacNeal [10–12] reported the processes of formulation, computation and validation of a defect free performance of fournode membrane quadrilateral element, with drilling degrees of freedom based on the Isoparametric principles. Sze et al. [13] proposed proposed a mixed mixed quadrilate quadrilateral ral plane element with drilling drilling degre degrees es of freedo freedom m using using Allman Allman’’s interp interpola olatio tion n scheme scheme,, Hellinger–Reissener functional and assumed stress field. The element stiffness matrix is generated based on numerical integration scheme. The performance of the element has been established by analyzing few standard benchmark examples. Piancastelli [14] introduced a plate-type FE with six degrees of freedom for each node to analyze anisotropic composite materials. materials. A similar similar study on composite composite folded anisotropic anisotropic structures using plate element with drilling degrees of freedom was reported by Lee et al. [15] [15].. Pimpinelli [16] studied a four nodes quadrilateral membrane with drilling degrees of freedom. The formulation was mainly based on minimization of the modified Hu–Washizu 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 weak form, form, the equili equilibri brium um proble problem m associ associate ated d with with the
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boundary values in the presence of independent rotation fields. In the two-dimensional case, these lead to membrane elements with drilling degrees of freedom. Cook [18] developed a 24 degrees of freedom quadrilateral shell element by including the drilling degrees of freedom. The author concluded that numerical results are good but the element is not the best available four-node shell element in all test cases. Chinosi [19] combined a membrane element with plate bending element which lead to a shell elements with six degrees of freedom. Further more, Chinosi and collaborators [20] constructed a new FE with drilling degrees of freedom for linear elasticity problems. They showed that the new element was more efficient compared with the author’s earlier proposed shell element. Zhu et al. [21] discussed the development of a new quadrilateral shell element with drilling degrees of freedom. One point quadrature was used for the analysis of nonlinear geometrical and material problems. Ibrahimbegovic [22–25] presented membrane elements with drilling degrees of freedom based on a variational formulation which employs an independent rotation field. Two new membrane elements, namely MQ2 and MQ3 were developed. These elements exhibited good performance over a set of problems. Furthermore, the author demonstrated the application of these elements for geometrically nonlinear shell theory. Felippa et al. [26] studied the formulation of 3-node, 9-dof membrane elements with the drilling degrees of freedom within the context of parameterized variational principles. The investigation has constructed an element of this type, using the extended free formulation (EFF). They constructed this element within the context of the assumed natural deviatoric strain (ANDES) formulation. The resulting formulation has five free parameters. These parameters are optimized against pure bending by energy balance methods. Furthermore, Felippa [27] compared derivation methods for constructing optimal membrane triangles with corner drilling freedoms. In this report a comprehensive summary of element formulation approaches and the construction of an optimal 3-node triangle (OPT) using the ANDES formulation were presented. Based on the extensive review of the literature on elements with degrees of freedom derived on different principles, it was seen that for two dimensional plane stress problems Felippa and co-workers optimal membrane triangle element based on ANDES formulation, was found to be most efficient and suitable. In the present work, an attempt has been made to further enhance the formulation of Felippa’s work [26–29]. Hence, an alternative formulation of triangle optimal element has been proposed to make the triangle optimal element more attractive as far as its programming effort and computational efficiency are concerned. The objectives of the present work are to: (i) carry out reformulation (or alternate formulation) of OPT element in an alternative form, (ii) implement this reformulation and its computational algorithm 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 formulation (FF) of Bergan and a variant of the assumed natural strain (ANS) method due to Park et al. [1,3]. Extensive formulations of ANS and ANDES were published by Felippa et al. [26–29]. The basic steps of the formulations are summarized in this section. Assuming that the element to be constructed has nodal displacement degrees of freedom collected in vector v , elastic modulus matrix E , and volume V , stiffness matrix is constructed by using the fundamental decomposition of stiffness equations: 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 found by Bergan [6] as part of the FF and is a scaling coefficient ( > 0). The basic stiffness matrix Kb is constructed by the standard procedure (CST element). The main portion of the strains is left to be determined variationally from the constant stress assumptions which are used to develop Kh . 2.1. Element description
The membrane triangle shown in Fig. 1 has straight sides joining 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 = (x2 y3 − x3 y2 ) + (x3 y1 − x1 y3 ) + (x1 y2 − x2 y1 ) = y21 x13 − x21 y13 .
(3)
In addition to the corner nodes 1, 2 and 3, midpoints 4, 5 and 6 shall also be used for derivations, although these nodes do not appear in the final equations of the element. Midpoints 4, 5, 6 are located at the opposite corners 3, 1 and 2, respectively. As shown in Fig. 1, intrinsic coordinate systems are 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
sides [27]. The triangle will be assumed to have constant thickness h and uniform plane stress constitutive properties. The well-known triangle coordinates showed in Fig. 2, are denoted by 1 , 2 and 3 , which satisfy 1 + 2 + 3 = 1. The degrees of freedom are collected in the node displacement vector uR as uR = [ ux1 uy1 1 ux2 uy2 2 ux3 uy3 3 ]T .
(4)
Here uxi and uyi denote the nodal values of the translational displacements along x and y, respectively, and i is the “drilling rotations” about z.
2.4. The higher order stiffness
The ANDES form of higher order stiffness matrix Kh developed in [8], is Kh = CfacT Tu K T u ,
2.2. Natural strains
[] = [ 21 32 13 ]T ,
T e =
1 4A2
⎡ ⎢⎢⎣
[e] = [ exx eyy 2exy ]T ,
Q2 =
2A 3
2A 3
= T e−1 e, (5)
⎡ ⎢⎢ ⎢⎢⎣ ⎡ ⎢⎢ ⎢⎣
2 y23 y13 l21
2 y31 y21 l32
2 y12 y32 l13
2 x23 x13 l21
2 x31 x21 l32
2 x12 x32 l13
2 (y23 x31 + x32 y13 )l21
2 (y31 x12 + x13 y21 )l32
2 (y12 x23 + x21 y32 )l13
Enat = T eT ET e ,
(7)
where Enat is the natural stress–strain matrix defined which is constant over the triangle.
(9)
where K is the 3 × 3 higher order stiffness in terms of the hierarchical rotations and Cfac is a scaling factor. To express K compactly, the following matrices are introduced:
Q1 =
In the derivation of the higher order stiffness by ANDES, natural strains play a key role. Strains along the three side directions were used in [28]. The natural strains are collected in the three vectors. The natural strains are related to Cartesian strains by the “strain gage rosette” transformation:
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Q3 =
2A 3
2.3. The basic stiffness
⎡ ⎢⎢ ⎢⎢ ⎣
1
2
3
2 l21 4
2 l21 5
2 l21 6
2 l32 7
2 l32 8
2 l32 9
2 l13
2 l13
2 l13
9
7
8
2 l21 3 2 l32 6 2 l13
2 l21 1 2 l32 4 2 l13
2 l21 2
⎤ ⎥⎥⎦
,
2 l32 5 2 l13
⎤ ⎥⎥ ⎥⎥⎦ ⎤ ⎥⎥ ⎥⎦
,
,
(6)
5
6
4
2 l21 8
2 l21 9
2 l21 7
2 l32 2
2 l32 3
2 l32 1
2 l13
2 l13
2 l13
⎤ ⎥⎥ ⎥⎥ ⎦
.
(10)
Kb = V −1 LELT ,
Depending on nine free dimensionless parameters; 1 –9 , the scaling by 2A/3 is for convenience in correlating with prior developments. Matrix Qi are evaluated at the midpoints by the following terms:
where V = Ah is the element volume and L is a 3 × 9 matrix that contains a free parameter b [26–29]:
Q4 = 21 (Q1 + Q2 ), Q6 = 12 (Q3 + Q1 ),
An explicit form of the basic stiffness was published by Bergan et al. [6]. It can be expressed as
L = 12 h
⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣
y23
0
0
x23 1 6 b x32 (x31 − x12 )
1 6 b y23 (y13 − y21 ) y31
0 1 6 b y31 (y21 − y32 ) y12
0 1 6 b y12 (y32 − y13 )
0
x13 1 6 b x13 (x12 − x23 )
0
x21 1 6 b x21 (x23 − x31 )
x23 y23 1 3 b (x31 y13 − x12 y21 ) x13 y31 1 3 b (x12 y21 − x23 y32 ) x21 y12 1 3 b (x23 y32 − x31 y13 )
If b = 0, the basic stiffness reduces to the stiffness matrix of the CST element. In this case, the rows and columns associated with the drilling rotations vanish. In the direct fabrication approach, the decomposition is explicitly used to construct the stiffness matrix in two stages, first Kb and then Kh .
⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦
.
Q5 = 12 (Q2 + Q3 ),
(8)
T T K = h(QT 4 Enat Q4 + Q5 Enat Q5 + Q6 Enat Q6 ),
Kh = 43 0 T Tu K T u ,
(11)
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and Eq. (11) can be rewritten as Q∗4 = LL−1 Q4 ,
Q∗5 = LL−1 Q5 ,
Q∗6 = LL−1 Q6
(17)
by substituting the new form of T e and Qi in high order stiffness formulation and expanding it K = h(( LL−1 Q∗4 )T (T e∗ LL)T E(T e∗ LL)( LL−1 Q∗4 ) + ( LL−1 Q∗5 )T (T e∗ LL)T E(T e∗ LL)( LL−1 Q∗5 )
Fig. 2. Natural strains, along side directions.
+ ( LL−1 Q∗6 )T (T e∗ LL)T E(T e∗ LL)( LL−1 Q∗6 )),
K = h((Q∗4T )( LL−T LLT )(T e∗T ET ∗e )( LLLL−1 )(Q∗4 )
Table 1 Dimensionless parameter of OPT element
b
0
3 2
1 2
(18)
1
2
3
4
5
6
7
8
9
1
2
1
0
1
−1
−1
−1
−2
+ (Q∗5T )( LL−T LLT )(T e∗T ET ∗e )( LLLL−1 )(Q∗5 ) + (Q∗6T )( LL−T LLT )(T e∗T ET ∗e )( LLLL−1 )(Q∗6 ))
(19)
from definition of matrix LL it can be written: where 0 is an overall scaling coefficient. So finally KR assumes a template form with 11 dimensionless parameters: b , 0 , 1 , . . . , 9 : KR (b , 0 , 1 , . . . , 9 ) = V −1 LELT + 34 T Tu K T u .
(12)
The free dimensionless parameters are determined from a higher order patch test which tunes up the higher order stiffness of triangular elements. These parameters are collected and tabulated in Table 1 [27]. 3. Alternative formulation of higher order stiffness matrix
In the present study an alternative formulation of the above element is presented, which is more efficient compared to the optimal element constructed by Felippa [27]. In the subsequent discussion these formulations are presented in detail. Let us assume LL matrix as LL =
⎡ ⎣
2 l21
0 2 l32
0 0
0
0 0 2 l13
⎤ ⎦
.
(13)
Then the formulation of Qi and T e (Eqs. (6) and (10)) has been rewritten in the following form: Qi = LL−1 Q∗i ,
T e∗ =
1 4A2
where 2A Q∗1 = 3 Q∗3 =
2A 3
⎡ ⎢⎣
1 4 7 5 8 2
and
∗ = T e∗T ET ∗e , Enat
(14)
y31 y21 x31 x21 y31 x12 +x13 y21
y12 y32 x12 x32 y12 x23 +x21 y32
(15)
T e = T e∗ LL y23 y13 x23 x13 y23 x31 +x32 y13
2 5 8 6 9 3
3 6 9 4 7 1
,
2A Q∗2 = 3
9 3 6
7 1 4
⎤ ⎥⎦
,
8 2 5
∗ ∗ ∗ K = h(Q∗4T Enat Q∗4 + Q∗5T Enat Q∗5 + Q∗6T Enat Q∗6 ).
(20)
Hence the formulation of Kh with new notation is more efficient in view point of computational time and effort compared to that of Felippa [27]. The same formulation is also adapted for calculation of stresses. 4. Computational procedure and development of an FE code
The followings are the major computational steps adopted in implementing the proposed OPT element: Step i: Define geometrical parameter of the OPT element to
calculate 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 employing 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 existing two-dimensional FE analysis program written by Noorzaei et al. [30] has been extensively modified in view of inclusion of triangle optimal element which is based on new formulation presented in this paper [31]. This program is multi-element, multi-degrees of freedom and has dynamically dimensioned features. The program was written in FORTRAN language and works under FORTRAN power station environment. 5. Testing and verification
,
(16)
In order to validate the formulation, computational algorithm and implementation of new formulation of OPT element, three benchmark examples which are available in the literature are considered [19]. Table 2 shows the notations used for previous
M. Paknahad et al. / Finite Elements in Analysis and Design 43 (2007) 861 – 869
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Table 2 Identifier of triangle element instances Name
Description
ALL-3I ALL-3M ALL-LS CST LST-Ret OPT
Allman 88 element integrated by 3-point interior rule Allman 88 element integrated by 3-midpoint rule Allman 88 element, least-square strain fit Constant strain triangle CST-3/6C Retrofitted LST with b = 43 Optimal ANDES template
Fig. 5. Cantilever under end shear.
Fig. 3. Slender cantilever beam under end moment.
Fig. 6. Tip deflection for short cantilever mesh 16 × 4.
Fig. 4. Tip deflection for cantilever beam.
results in literature. In all benchmark examples the used units were consistent. 5.1. Example 1—cantilever beam under end moment
The slender cantilever beam of Fig. 3 is subjected to an end moment M = 100. The exact tip deflection is 100. The geometric data, material properties, boundary conditions, loading and dimension of the beam are also presented in this figure. The beam has been discretized using regular meshes, ranging from 2 × 2 to 32 × 2, with each rectangle mesh unit consisting of four half-thickness overlaid triangles. The element aspect ratios vary from 1:1 to 16:1. Fig. 4 shows computed tip deflections for several element types and five aspect ratios (1, 2, 4, 8, and 16), respectively. It is clear from this plot that the results obtained from this study
are similar to those published by Felippa (using similar OPT element). The figure also indicates that the OPT element is superior 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 been selected as a test problem for plane stress elements by many investigators since it was originally presented in 1966 [29]. The geometrical data, material property, boundary conditions and loading are shown in Fig. 5. The comparison value is the tip deflection c at the center of the end-loaded cross section. An approximate solution derived from two-dimensional elasticity, based on a polynomial Airy’s stress function, gives c = 0.35533. Fig. 6 shows computed deflections for rectangular mesh units with aspect ratios of 1, 2 and 4, respectively. Mesh units consist of four half-thickness overlaid triangles. For reporting purposes, the load was scaled, so that the “theoretical solution” becomes 100. The deflection evaluated from the present investigation is identical as reported by Felippa [26–29].
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5.3. Example 3-cook’s problem
Fig. 8 shows results computed for the plane stress problem defined in Fig. 7. This problem was proposed by Cook [18] as a test case for nonrectangular quadrilateral elements. There is not any known analytical solution, but the OPT results for the 64 × 64 mesh are used for comparison purposes. For triangle tests, quadrilaterals were assembled with two triangles in the shortest-diagonal-cut layout as illustrated in Fig. 7 for a 2 × 2 mesh. Again, the results predicted by the
FE program code written in the present research, agree well with the OPT element reported by Felippa [27]. Through these three verification examples, accuracy and convergence of the present formulation of OPT element, as well as the compatibility of the FE code, are demonstrated. In the next step, the applicability of the developed code is shown by analyzing shear wall structures. 6. Application of present study element in shear wall structure 6.1. Cantilever shear wall without opening
The FE program developed in this study is now applied to an analysis of shear wall structures. Fig. 9 shows geometry,
Fig. 7. Cook’s problem: clamped trapezoid under end shear.
Fig. 8. Results for vertical displacement at C for different subdivision.
Fig. 9. Geometry and material of shear wall.
M. Paknahad et al. / Finite Elements in Analysis and Design 43 (2007) 861 – 869
loading and material property of a cantilever shear wall. The shear wall structure was analyzed for different subdivisions, to illustrate the capability of OPT element in analysis of the wall structures. To show the efficiency and accuracy of the present element against conventional FE in the analysis of the shear wall, eight node isoparametric element was used to model the shear wall. The results for horizontal displacement at the top of shear wall, using OPT and eight node element for different FE discretization, are shown in Fig. 10. This figure shows that by using finer mesh, the higher lateral deflections of the shear wall occurred in the case of conventional FEs; whereas, using OPT element, the result was converged to almost similar values, indicating the accuracy and fast rate convergence of OPT element. 6.2. Shear wall with opening
In order to show the efficiency, suitability, accuracy and superiority of the OPT element based on the proposed formulation an attempt has been made to analyze shear wall structures with
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openings. Commercial packages, namely SAP-2000, STAADPRO and FE program based on plane stress formulation were used for comparisons [30]. Fig. 11 shows geometry and material property of an eight story coupled shear wall. The structure was represented 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 and 8 for all the FE codes has been tabulated in Table 3. The classical eight node plane stress element and standard commercial software [30] reflect a comparable result. The finer the mesh, the analysis converged to greater deflection. However, by application of the current alternative OPT element, the deflections obtained using coarse mesh are very similar to those using fine mesh. Moreover, the results indicate that the coarse mesh from the OPT element converge to more accurate deflection obtained from the FE by isoparametric eight node analysis using fine mesh. Therefore, it could be concluded that the new formulation of the OPT element can be considered to be more efficient as it did not require fine mesh in order to get accurate results. Contour of normal stress distribution, x calculated by STAAD-PRO, SAP 2000 and the present study are shown in Fig. 12. These plots show that FE code using present OPT element is capable of predicting almost similar stress distribution in the shear wall and at the connecting beams. Moreover, the stress distribution evaluated by the developed FE code based on alternative formulation of OPT element with coarse FE mesh, agrees well with the stress distributions given by the commercial packages where fine mesh was used. This comparison further proves the computational efficiency of the proposed formulation of the OPT element. 7. Conclusion
Fig. 10. Results for horizontal displacement of the top of shear wall.
In this study an alternative formulation of OPT element was employed and its computational algorithm has been implemented in an FE code. The implemented code was verified, using standard benchmark examples, and was found to be suitable for further use. The implemented code has been applied
Fig. 11. Geometry and material of coupled shear wall.
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Table 3 Comparison 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 Model b Difference (%)
0.56 0.68 21.4
1.53 1.82 18.9
2.59 3.02 16.6
3.62 4.16 14.9
SAP2000
Model a Model b Difference (%)
0.55 0.77 40.0
1.48 2.06 39.1
2.54 3.40 33.8
3.62 4.66 35.5
STAAD-PRO
Model a Model b Difference (%)
0.68 0.79 16.1
1.68 2.08 23.8
2.78 3.44 23.7
3.86 4.69 21.5
Present study
Model a Model b Difference (%)
0.71 0.74 4.2
1.91 1.98 3.6
3.19 3.28 2.82
4.43 4.51 1.8
Fig. 12. Contour of normal stress for shear wall and connection beam.
M. Paknahad et al. / Finite Elements in Analysis and Design 43 (2007) 861 – 869
to the analysis of shear wall structures with and without openings. Based on the results obtained in this study, it could be concluded that: (a) The developed code based on the reformulation of the OPT was found to be reliable as the results obtained from this work were found to be similar to those reported by Felippa [26–29]. (b) The OPT based FE code was found to be more efficient and accurate as it displayed greater accuracy for the deflection and stresses of shear wall structures, with and without opening, even using coarse mesh in the FE modeling of the structures.
Acknowledgments
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