Analysis of coupled shear walls using high order finite elements

Finite Elements in Analysis and Design 5 (1989) 181-194

181

Elsevier Science Publishers B.V., Amsterdam - - Printed in the Netherlands

ANALYSIS OF C O U P L E D SHEAR WALLS U S I N G H I G H O R D E R FINITE E L E M E N T S Moshe E I S E N B E R G E R * Department of Civil Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

Batia PERRY Department of Civil Engineering, Teehnion -- Israel Institute of Technologr', Technion City 32000, Israel Received June 1988 Revised January 1989

Abstract. In this work finite element solutions for coupled shear walls and other approximate methods are compared. Ten different finite elements, which belong to four different families of elements, are used to solve three examples. For each example several meshes are used for the analysis. It is concluded that the use of high order Lagrange type elements yields the best results, with instant convergence, for the simplest and coarsest mesh possible for the representation of the shear wall geometry and loading.

Introduction In many structures, shear walls are an important component for providing load resistance to horizontal loads due to winds or earthquakes. In many shear walls it is necessary to introduce openings for functional or architectural considerations. The openings weaken the shear wall and complicate the analysis. The most important factor on the deflections of the wall are the lintel beams, which are the beams that connect the two complete walls over the openings. A number of approximate methods have been developed for the determination of the stiffness of coupled shear walls and the internal force distribution within them [5,1,6,9,8]. The finite element method has been used by many researchers [5,3,1,9]. This method enables us to handle variation in wall thickness and material properties, irregular geometry and location of the openings, and more general loading cases. All the researchers to date have used the method with only a few element types. There are several families of finite elements that one can use for plane stress analysis of rectangular shear walls with rectangular openings. These are: the Lagrange family, serendipity elements, serendipity elements with additional internal nodes, and assumed stress field elements. In this work, three examples are solved using the four families of elements, and the results are compared with those from other finite elements and approximate solutions. It is concluded that the use of high order Lagrange type elements yields very good results, using the minimum number of elements that are required to represent the geometry of the coupled shear wall. * On leave from Technion--Israel Institute of Technology, Technion City 32000, Israel. 0168-874X/89/$3.50 © 1989, Elsevier Science Publishers B.V.

M. Eisenberger, B. Perry / Analysis of coupled shear walls

182 Element

types

In this work ten different finite elements that belong to four families were used. The common geometry of coupled shear walls is rectangular openings. This simplifies the formulation of the elements, as implicit integration is possible. Most of the elements that are used are well known and used extensively in commercial finite element programs and by researchers in the field. However, to the best of our knowledge some of the higher order elements are used here for the first time. The elements vary from 8 to 72 degrees-of-freedom (dof).

Lagrange elements The family of Lagrange elements is well known [10]. These are rectangular elements with perimeter and internal nodes, arranged in an equally spaced mesh across the elements. The shape functions for these elements are formulated by multiplication of functions of the x coordinate only by functions of the y coordinate only. The degree of the two polynomials is not necessarily the same, but in this work only functions of the same order in both directions were used. The shape functions are complete polynomials of degree m (as for the unidirectional functions) and incomplete polynomials up to degree 2m. In Fig. 1 four elements are shown: (a) P R 9 - - p l a n e rectangle element with 9 nodes (3 × 3); (b) PR16--plane rectangle element with 16 nodes (4 × 4); (c) P R 2 5 - - p l a n e rectangle element with 25 nodes (5 × 5); and (d) P R 3 6 - - p l a n e rectangle element with 36 nodes (6 × 6). Each node has two dof in the x and y directions; thus the four elements have 18, 32, 50, and 72 dof respectively. The explicit terms for the element stiffness matrices were found using REDUCE--a symbolic algebra computer program [4].

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M. Eisenberger, B. Perry/Analysis of coupledshear walls

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Serendipity elements Serendipity elements are elements with nodes at the corners and along the edges of the element. The shape functions for these elements are at the most complete polynomials up to degree 3, and incomplete polynomials of degree up to m + 1, where m is the number of nodes along an edge. In Fig. 2 four elements of this family are shown: (a) PR4--plane rectangle with 4 nodes (8 dof); (b) PR8--plane rectangle with 8 nodes (16 dof); (c) PR12--plane rectangle with 12 nodes (24 dof); and (d) PR16S-plane rectangle with 16 nodes (32 dof--different from PR16 of the previous section). As for the previous family, explicit terms for the stiffness matrices were found using REDUCE [4]. The shape functions for the less popular elements PR12 and PR16S are given in Appendix 1.

Special elements In this family one element was formulated in order to examine the effect of adding internal nodes to the serendipity elements. In Fig. 3 such an extension of the PR16S element into PR17, with one additional node at the center of the element, is shown. The shape functions for this element are given in Appendix 1.

Assumed stress fieM elements The element that was used in the comparison is the PRH2 element (Fig. 4), which was formulated originally by Pian [7] and reformulated by Cook [2]. Higher order elements of this type were not readily available. The element has four nodes (8 dof) with linearly varying stress

M. Eisenberger, B. Perry / Analysis of coupled shear walls

184

13 14 15 16 1

12 _

I

'8

6

2

3

4

5

1

Fig. 3. Special element.

Fig. 4. Assumed stress field element.

field in both the x and y directions. For the special case of rectangular elements this element is similar to other elements with internal nodes as discussed by Cook [2].

Examples In this section three examples from the literature are solved using the elements that were selected and the results are compared with those from other finite element and approximate methods.

Example 1 This example is taken from the paper by MacLeod [5]. The purpose of this example is to compare the different elements performance for stiffness calculations for various values of the h/L ratios. This is an experimental model of a 7-story coupled shear wall (Fig. 5). The various dimensions are: H = 4.5", B = 3.75", and L = 2.5". The model was loaded at point A z by a horizontal load of 1 lb and the deflection was measured at point A 1. The lintel beams depth was taken such that h/L = 1.05, 0.85, 0.65, 0.45, and 0.25 for five different cases. This example was solved using the ten selected elements using the meshes given in Fig. 6. The number of elements in these meshes is in the range between 35 (mesh a) and 280 (mesh r) modeled by increasing the number of divisions along and through the depth of the lintel beams gradually. The results for the relative error in the horizontal deflection at point A1 with respect to the converged value obtained using mesh c and the highest order element PR36, are given in Fig, 7 AI J

A

-]

lib.

Fq -1 "l

-] _71

Fig. 5. Example 1 : 7 story coupled shear wall [5].

M. Eisenberger, B. Perry / Analysis o/coupled shear walls II11

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for h / L = 0.85. For lower values of this ratio the relative errors were higher. It is evident from the meshes in Fig. 6 and the graphs that meshes with equal numbers of elements yield different results. However, these give an indication of the convergence of the various elements. The three values that were plotted for element PR36 differ only in the fifth significant figure, and represent a practically converged solution for the error calculations. The first five elements ( P R 4 - P R 1 2 ) were used to solve the problem using all the meshes in Fig. 6. The next three (PR16, PR16S, and PR17) were used with 12 meshes (a-l), element PR25 with meshes a - f , and PR36 with meshes a-c. This was due to limitations on the problem size that were imposed by the computer that was used for this study (VAX750).

M. Eisenberger, B. Perry /Analysis of coupled shear walls

186

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~,

,

-2.00 .4.00 -6.00 -8.00 -10.00

"12"0020 1;

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100 110 140! 110 110 2;0 210 210 210 280 # of elements

Fig. 7. Relative error in top deflection for example 1: h / L = 0.85.

From Fig. 7 several observations can be made: (i) Elements PR9 and PR12 are equivalent, and PR16S is very close to them. (ii) Element PR17 yields small improvement compared to PR16S, but it is not as good as PR16. (iii) element PR36 converges instantly. As for the families of elements, the Lagrange family gives the best results. The same qualitative results were obtained for other ratios of h/L. Computation times for these runs were recorded as follows:

Table 1 Example 1: Relative error for top deflection using minimal mesh for different h / L ratios

h/L

PR4

PRH2

PR8

PR9

PR12

PR16S

PR17

PR16

PR25

PR36

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10

74.87 68.95 60.67 52.19 44.86 38.64 33.56 29.45 26.13 23.42 21.20 21.77 17.85 16.58 15.51 14.62 13.86 13.24 12.72 12.29 11.95

49.82 49.61 44.09 38.78 32.09 27.41 23.65 20.65 18.24 16.30 14.70 13.39 12.28 11.35 10.54 9.85 9.24 8.71 8.24 7.81 7.44

49.94 44.40 36.33 28.97 23.60 19.52 16.53 14.30 12.59 11.26 10.19 9.32 8.59 7.98 7.45 6.99 6.60 6.25 5.93 5.65 5.40

49.22 43.37 34.99 27.46 21.85 17.69 14.68 12.47 10.82 9.55 8.56 7.78 7.14 6.61 6.18 5.81 5.49 5.22 4.98 4.77 4.59

5.13 9.41 11.99 12.39 12.81 12.26 11.52 10.74 9.98 9.28 8.63 8.06 7.54 7.07 6.65 6.27 5.92 5.61 5.32 5.06 4.81

4.86 9.01 11.59 12.12 12.51 12.00 11.30 10.55 9.82 9.13 8.51 7.95 7.44 6.98 6.57 6.19 5.85 5.54 5.25 4.99 4.74

3.87 7.93 10.52 11.21 11.39 10.80 10.03 9.21 8.44 7.73 7.11 6.56 6.09 5.67 5.31 4.99 4.70 4.45 4.22 4.02 3.83

2.26 4.34 5.39 5.16 5.33 4.92 4.48 4.06 3.69 3.38 3.11 2.88 2.69 2.53 2.40 2.29 2.20 2.12 2.06 2.00 1.95

1.02 1.81 2.15 2.00 2.06 1.90 1.74 1.59 1.46 1.35 1.25 1.17 1.10 1.04 0.99 0.95 0.91 0.88 0.86 0.84 0.82

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

M. Eisenberger, B. Perry / A nalysis of coupled shear walls

187

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~,////////////, Y~. 20 ft

Fig. 8. Example 2: 6-story coupled shear wall [3,9].

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(I) 444

(J)

(k) 336

264

(i) 210

(h) 156

(g)

(f)

102

(e)

76

64

(d) 38

(c) 36

(b) 32

Fig. 9. Meshes used for Example 2.

(a)

30

188

M. Eisenberger, B. Per~" / A nalysis of coupled shear walls

C P U time for the PR8 element using mesh r (this is the most commonly used element in commercial packages; 1982 equations with band width of 58) was 263 seconds, whereas element PR16 gave better results using mesh a (756 equations with band width of 68) and 134 seconds of CPU time. CPU time for element PR36 using mesh a (1984 equations with band width of 172) was 394 seconds. It is seen that computation time for the higher order elements is roughly the same as for the low order elements with much better accuracy. The relative errors in the top deflection for various ratios of h/L (between 0.1 and 1.1) are given in Table 1 for the solutions using mesh a (the " m i n i m a l " mesh) for all the elements. It is seen that the errors vary as a function of h/L, due to different distribution of the internal forces for the various cases. In general it can be stated that errors of up to 5% are achieved for element PR16, and less than 2% for element PR25. The general conclusion from the results in this example is that it is best to use the PR36 element (the highest order element) with mesh a - - t h e " m i n i m a l " m e s h - - t h a t is a high degree element with the smallest mesh that is required to represent the geometry of the wall. This conclusion will be examined in the following two examples. The authors believe that there is no real justification to move to an higher order element--like P R 4 9 - - a s the results using the PR36 element are excellent.

Example 2 Figure 8 shows a coupled shear wall loaded by six concentrated forces. The numerical data for this problem is: E = 576000 K i p s / f t 2, u -- 0.4, t = 1 ft, and h/L = 0.4. This example was /~----------.-.~

-[__[

I \

[

i-y-=r---

i i ?-

(b) PR9-J

(a) PR4-J

(d) PR25-a

(e) PR36-a

Fig. 10. Stresses for example 2 at 9, 27.5, 37, 47, 57, 67, and 77 ft from the bottom of the wall.

18.176 8.626 82.528

20.560 8.394 83.840

22.116 8.361 83.558

22.473 8.049 80.526

20.229 7.310 73.283

N S M

N S M

N S M

N S M

N S M

2

3

4

5

6

444

20.239 5.682 58.737

22.510 6.503 65.060

22.086 6.803 67.991

20.573 6.878 65.083

20.573 6.878 68.707

12.744 6.003 59.417

PRH2

20.337 5.811 58.253

22.525 6.435 64.371

22.118 6.732 67.275

20.583 6.803 67.954

18.278 6.744 67.335

12.687 5.882 58.157

PR8

444

a N - - a x i a l force; S - - s h e a r force; M - - b e n d i n g moment.

444

12.796 7.409 73.381

N S M

1

Mesh

PR4

Action a

Beam

336

20.333 5.706 57.201

22.518 6.317 63.193

22.110 6.611 66.070

20.573 6.684 66.763

18.250 6.627 66.161

12.723 5.804 57.386

PR9

Table 2 Example 2: Forces and moments at left end of lintel beams for different elements

264

20.354 5.877 58.911

22.545 6.502 65.041

22.137 6.801 67.965

20.603 6.872 68.643

18.282 6.810 67.989

12.624 5.838 57.686

PR12

264

20.304 5.587 56.020

22.507 6.191 61.926

22.108 6.483 64.783

20.573 6.557 65.499

18.240 6.504 64.932

12.742 5.652 55.867

PR16

102

20.393 5.952 59.651

22.545 6.582 65.841

22.137 6.883 68.785

20.593 6.951 69.440

18.293 6.888 68.775

12.631 5.991 59.211

PR16S

102

20.428 5.888 59.003

22.544 6.515 65.165

22.126 6.813 68.089

20.592 6.882 68.747

18.295 6.820 68.097

12.635 5.965 58.961

PR17

76

20.300 5.619 56.316

22.469 6.209 62.108

22.081 6.502 65.695

20.556 6.577 65.695

18.245 6.523 65.129

12.785 5.665 55.999

PR25

64

20.298 5.578 55.910

22.472 6.148 61.504

22.086 6.440 64.356

20.558 6.516 65.083

18.241 6.464 64.531

12.784 5.620 55.547

PR36

O0 '.D

:x

190

M. Eisenberger, B. Perry / Analysis of coupled shear walls" *.

2'

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LS',.

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~ty

L

1

also solved by Saffarini [9] and Girijavallabhan [3]. Twelve finite element meshes, as shown in Fig. 9, were used for the analysis. The number of elements in these meshes is in the range between 30 and 444. The results that were obtained for the deflections are similar to those presented in Example 1. In this example the stresses in the wall and the integrated actions in the lintel beams are compared. Figure 10 shows the stresses that were calculated using several meshes and elements. The stresses were calculated at 17 points across the width of each of the columns and connected by line segments. At boundary lines of elements the values from the two elements were avaraged. It is seen that the stresses that were calculated using mesh a (30 elements) and element PR36 are the same as those that were calculated using finer meshes with lower order finite elements. The results for the forces and moments at the left end of the lintel beams are given in Table 2, where the beams are numbered from top to b o t t o m in the shear wall. This demonstrates again the excellent convergence of the analysis using the PR36 element, for problems of coupled shear walls.

_

(f) 102

I

-..~

__--~

(e) 82

(d) 65

II

(e) 50

(b) 39

Fig. 12. Meshes used for example 3.

(a) 33

3.231 2.063 0.574

7.966 5.185 11.528

S3 M3

N4 S4 M4

N5 S5 M5

108

6.403 3.340 4.754

6.437 3.231 4.574

N3

Mesh

4.376 7.242 6.874

4.292 7.230 6.782

N2 S2 M2

108

7.929 5.193 11.390

3.340 2.097 0.596

0.817 0.687 1.400

0.892 0.736 1.444

N1 S1 M1

PR H 2

PR4

Action

108

7.743 5.073 11.346

3.389 2.145 0.574

6.355 3.389 4.698

4.250 7.052 6.776

0.822 0.691 1.411

PR8

108

7.692 5.049 11.365

3.406 2.163 0.571

6.337 3.406 4.682

5.235 7.001 6.788

0.814 0.687 1.409

PR9

108

7.712 5.071 11.361

3.403 2.151 0.596

6.349 3.403 4.736

4.263 7.027 6.775

0.807 0.684 1.411

PR12

108

7.709 5.071 11.365

3.403 2.152 0.599

6.348 3.403 4.738

4.264 7.025 6.771

0.807 0.684 1.410

PR16S

Example 3: Forces and m o m en ts for different elements compared with the results from [9]

Table 3

PR17

108

7.667 5.050 11.378

3.417 2.170 0.599

6.330 3.417 4.716

4.250 6.986 6.782

0.800 0.680 1.408

PR16

108

7.709 5.071 11.365

3.403 2.152 0.599

6.348 3.403 4.738

4.264 7.025 6.771

1.410

0.807 0.684

82

7.618 5.032 11.397

3.447 2.180 0.581

6.320 3.447 4.736

4.245 6.944 6.801

0.787 0.673 1.405

PR25

50

7.610 5.029 11.403

3.447 2.190 0.585

6.310 3.447 4.710

4.240 6.935 6.875

0.789 0.675 1.406

PR36

8.470 5.060 10.780

3.770 2.390 2.320

6.170 3.770 2.630

4.190 7.700 7.820

0.870 0.780 1.860

Ref. [9]

-n

192

M. Eisenberger, B. Perry. / Analysis of coupled shear walls

Example 3

For the third example a more complex wall was chosen from the work of Saffarini [9]. The wall is square with three irregular openings as shown in Fig. 11. The wall is loaded by four horizontal loads, and all the dimensions are given in the figure. The rest of the data for this problem is: E = 576000 K i p s / f t 2, u = 0.4, t = 1 ft. Six meshes were used for this problem (Fig. 12), as the geometric complexity required a large n u m b e r of elements (33) even for the " m i n i m a l " mesh. In Table 3 the resultant forces and m o m e n t s for the five sections in Fig. 11 are given. The convergence of all the elements is fairly rapid for this example due to the large n u m b e r of elements in the meshes. Several values that are taken from [9] are very far from the converged values in the table, especially the values for M4, N4, $1, MI, and M 2.

Summary and recommendations For all the examples the results from the finite element m e t h o d were more accurate than the results from all other approximate methods. The best solutions were those from elements of the Lagrange family: Similar serendipity elements (PR16 c o m p a r e d with PR16S) are less accurate, even if an internal node is added (PR17). The use of element PR36 is recommended. This element enables one to use the " m i n i m a l " mesh and yet obtain relative errors of less than 1% for all coupled shear walls. This is a very important result: One does not have to make an effort in meshing if he uses the PR36 element for this type of problems. The " m i n i m a l " mesh is unique, and it is easily a u t o m a t e d into the analysis program. As for C P U time it was observed that for the same relative error higher order elements are more efficient. The m e m o r y requirements behave in the same manner. The m e m o r y size for a fixed relative error is smaller for higher order elements. Element PR36 is best for coupled shear wall analysis from all three aspects: Accuracy, c o m p u t a t i o n time, and m e m o r y requirements. Further improvements to its p e r f o r m a n c e could be achieved by condensing the internal nodes in the element, resulting in a PR20 element, with the same accuracy for static problems, and about 60% reduction in c o m p u t a t i o n time. It is also r e c o m m e n d e d that higher order elements be tested for similar problems, such as plate bending and shell analysis.

References [1] L. BRAGADA CRUZ, "Comparative study of shear walls with openings by the finite element method and as a frame structure", in: Proc. Regional Conf. Tall Buildings, Madrid, 1973. [2] R.D. COOK, Concepts and Applications of Finite Element Analysis, Wiley, New York, NY, 2nd edn., 1981. [3] C.V. GIRIJAVALLABHAN,"Analysis of shear walls with openings", J. Struct. Div. 95, pp. 1632-1649, October 1969. [4] A.C. HEARN, REDUCE User's Manual, Rand Publications, vet. 3.1, cp78 (rev. 4/84) edition, 1984. [5] I.A. MACLEOD,"Lateral stiffness of shear walls with openings", in: Proc. Syrup. Tall Buildings, University of Southampton, Pergamon, Oxford, U.K., pp. 223-244, 1966. [6] D. MICHAEL,"The effect of local wall deformations on the elastic interaction of cross walls coupled by beams", in: Proc. Syrup. Tall Buildings, University of Southampton, Pergamon, Oxford, U.K., pp. 253-272, 1966. [7] T.H.H. PIAN,Derivation of element stiffness matrices by assumed stress distributions, AIAA J. 2, pp. 1333-1336, 1964. [8] R. ROSMAN, "Approximate analysis of shear walls subject to lateral loads", J. Am. Concr. Inst. 35, pp. 717-732. 1964. [9] H.S. SAFFARINI, New Approaches in The Structural Analysis of Building Systems. PhD thesis, University of California, Berkley, 1983. [10] O.C. ZIENKXEWICZ,The Finite Element Method, McGraw Hill, New York, NY, 4th edn., 1977.

M. Eisenberger, B. Perry / Analysis of coupled shear walls

193

Appendix 1 The 12 shape functions for element PR12 are given below. They are n u m b e r e d according to the node numbers in Fig. 2.

f, = ~(1 - f)(1 - */)[-10

+ 9(~ :2 + "1/2)]

(1)

f2 = 3~(1 + ~)(1 -- 77)[ -- 10 + 9(~ :2 + */2)]

(2)

f3 = 3~(1 + tj)(1 + */)[--10 + 9(f 2 + */2)]

(3)

f4 = ~(1 -- t~)(1 + */)[--10 + 9(~ 2 + */2)]

(4)

f5 = 9 ( 1 - ~2)(1 - */)(1 - 3tj)

(5)

f6 = 9 ( 1 - ~ 2 ) ( 1 - */)(1 + 3f)

(6)

fv = 9 ( 1 + f ) ( 1 -*/2)(1 - 37/)

(7)

f8 = 9(1 + ~j)(1 - */2)(1 "{'- 3*/)

(8)

./9 = ~2(1 - ~2)(1 + */)(1 + 3~)

(9)

fl0 = 9 ( 1 - ~2)( 1 + */)(1 - 3f)

(10)

f,1 = 9 ( 1 - f)(1 - */2)(1 + 3*/)

(11)

f,2 = 9 ( 1 - f)(1 - */2)(1 - 3*/)

(12)

The 16 shape functions for element PR16S are given below. They are numbered according to the node numbers in Fig. 2. f, = ~ ( 1 - ~)(1 - 7 / ) [ - ~ ( 4 ~

e - 1) - * / ( 4 . / 2 -

1) - 3]

f2 = - 2f( 1 - * / ) ( 1 - ~e)(1 - 2~)

(13)

(14)

f3 =

½(1 - ~ 2 ) ( 1 - 4 t ~ 2 ) ( 1 - */)

(15)

f4 =

~ ( 1 - */)(1 - ~ 2 ) ( 1 + 2 ~ )

(16)

fs = ~ ( 1 + ~)(1 - * / ) [ ~ ( 4 f 2 - 1) - *//(4*/2 - 1) - 3]

(17)

f6

(18)

=

- -

3./(1 + ~)(1 - ,/2)(1 - 2*/)

f7 = ½(1 -*/2)(1 -4./2)(1 + ~ )

(19)

f8 = ~*/(1 + ~)(1

*/2)(1 "4- 2*/)

(20)

f9 = lJ~-(1 + ~)(1 + */) [~(4~ 2 - 1) + */(4*/2 - 1) - 3]

(21)

flo = 2 f ( 1 + */)(1 - ~2)(1 + 2 f )

(22)

½(1 - ~ 2 ) ( 1 -- 4 ~ 2 ) ( 1

(23)

fll =

-

+ */)

f12 = - 2~( 1 + */)(1 - ~2)(1 - 2~)

(24)

f,3 = ~ ( 1 - f ) ( 1

(25)

+ * / ) [ - f ( 4 ~ 2 - 1) +*/(4*/2- 1) - 3]

f14 = 3./(1 - */2)(1 + 2./)(1 - f )

(26)

/~ = ½(1 - n 2 ) ( 1 - 4./2)(1 - ~)

(27)

f , 6 = --

J*/(1 - f)(1 - * / 2 ) ( 1 - 2 n )

(28)

194

M. Eisenberger, B. Perry / Analysis of coupled shear walls

T h e 17 shape functions for element PR17 are given below. T h e y are n u m b e r e d according to the node n u m b e r s in Fig. 3. f, = ~ ( 1 - ~)(1 - 7 ) [ - 4 f ( ~

2 - 1) - 4 7 ( 7 2 -

1) + 3 f 7 ]

(29)

f2 = - ~ ( 1 - 7 ) ( 1 - ~2)(1 - 2~)

(30)

f3 = ½(1 - ~ 2 ) ( _ 7 _

(31) (32) (33) (34) (35) (36) (37) (38) (39) (40)

f4 = 2~( 1 - 7 ) ( 1 f5 = ~ ( 1 + ~)(1

4~2)(1 - 77) ~2)(1 "Jr-2~)

-

-

7)[4~(~

2 -

1)

-

4 7 ( 7 2 - 1) - 3~r/]

f6 = - 27(1 + ~)(1 - 72)(1 - 2 7 ) ]'7 = ½(1 - 7 2 ) ( ~ -

472)(1 + f)

f~ = -~7(1 + ~ ) ( 1 - 7 2 ) ( 1 + 2 7 )

f9 = ~ ( 1 + ~)(1 + 7 ) [ 4 ~ ( ~ 2 -- 1) + 4 7 ( 7 2 - 1) - 3~7] f,o = - ~ ( 1

fu

= ½(1

+ 7 ) ( 1 - ~2)(1 + 2~) -

~2)(

7

-

4~2)(1 + 7 )

f,2 = - 2~( 1 + 7 ) ( 1 - ~2)(1 - 2~) f13 = ~ ( 1 - ~)(1 + 7 ) [ - - 4 ~ ( ~

2 --

£ 4 = -~7(1 - 72)(1 + 2 7 ) ( 1 -

$)

f,5 = ½(1 - 7 2 ) ( - ~ f,6 = - ~7(1 - f ) ( 1

-

f,-/ = ( 1 - - ~ 2 ) ( 1 - - 7 2)

1) + 4 7 ( 7 2 - 1) + 3 f 7 ]

(41) (42)

4 7 2 ) ( 1 - ~)

(43)

27)

(44)

72)(1

-

(45)