The analysis of shear-wall buildings

Build. Sci. Vol. 1. pp. 271-276. Pergamon Press 1966. Printed in Great Britain

The Analysis of Shear-Wall Buildings S. M. A. KAZIMI, B.Sc.(Engng), Ph.D.*

In spite of much interest in shear walls the theoretical side is lagging because of the extreme difficulties in mathematical analysis of the "mixed boundary value problem with multiply-connected regions '. The present paper reviews very briefly existing work done on this subject along with a brief description of the author's own contribution in the form of ' Line Solution ' and ' Grid Analysis' techniques. Specific stress is laid on the use of digital computors for such problems.

A number of assumptions are made, e.g. constant shear stress throughout a panel taking only shear force; and rigid stiffeners, taking only direct force and pin-jointed to each other, etc. The method is tedious and only as accurate as the assumptions but gives a better insight into the behvaiour of shear-walls. In 1956 K. Muto,[3] suggested an empirical formula for the deflection of sheer panels with openings, based on his extensive research work, i.e.

INTRODUCTION RECENTLY there has been a considerable increase in the number of tall buildings, both residential and commercial, because of certain advantages of ' circulation ', ' land utilisation ' and ' planned development' etc., and the present trend appears to be towards tall buildings. Structural frames are quite efficient for vertical loads but their inherent weakness lies in lack of stiffness under lateral loading, caused by wind forces, seismic actions and atomic blast. The advantages of shear walls as primary lateral load resisting elements for blast resistant structures were first recognised by Benjamin and Williams in 1948,[1] and since then considerable work has been done in U.S.A., U.K. and Japan. However, due to complexity of the analysis, the problem is far from completely solved as yet.

Deflection of wall with opening Deflection of wall without opening 1-1"25a/(Area of opening/Gross area of wall) On the experimental side Futami and Fujimoto[4], and Kokinopoulos[5] have done useful work, using the photoelastic method. However, only specific results of limited value were obtained. The second stage of the shear-wall problem is decidedly much more difficult and no attempt has been made so far to provide a mathematically exact or even acceptable solution. The first paper on this topic by Green[6] has given a simple treatment of the load distribution in the bracing walls of multi-storey buildings with openings based on elementary elastic requirements such as stiffness ratios, etc. The paper is very interesting and does not tend to idealize the behaviour of the structure in any set mathematical pattern. Recently there has been a trend of excessive idealization followed by Cardon[7], Shulz[8], Erikson[9], Beck[10] and Rosman[11]. All these papers use the same sort of differential equation to represent the behaviour of complete building with minor changes in finer points of the analysis. The governing equation is in term of N, the shear force in the connecting beams at each storey, and is an ordinary second order linear differential equation with constant coefficients, solved by conventional means. Thus it is seen that the existing methods, are far from the ultimate goal of complete analysis of tall buildings.

P R E V I O U S WORK

Only a few papers are quoted here, out of a mass of technical literature published, to show the modern trends. Broadly speaking, the shear-wall problem could be studied in two stages; i.e. the analysis of isolated, one storey shear-panels and the analysis of complete shear-wall buildings. The first stage of research is comparatively less involved, though mathematically speaking it is a boundary value problem with multiply connected regions and is extremely difficult to solve by exact methods. Experimental methods and approximate theoretical methods have been tried with varying success. Chronologically speaking, Benjamin and Williams first studied the problem experimentally--using fairly large (~}) scale models. Based on such experiments a simple design method was presented in 1958.[1] Later, Otsuki[2] presented his design method in 1950 based on 'shear-flow principle '. * Assistant Professor, Dept. of Civil Engineering, Regional Engineering College, Srinagar (Kashmir). 271

272

S. M . A. K a z i m i

THE SHEAR-WALL P R O B L E M

A. Line solution techniques

As all the forces acting on the shear wall are in its own plane, it may be considered as a two dimensional plane-stress problem in the theory of elasticity. The boundary conditions are generally mixed and it may have ' multiply-connected regions as well if some opening is present in the shear wall. The governing equations in terms of displacements are : CoV~U+,.~.Z+Sx~)~,., = 0 2

8z"

32u

[ }

-&

(1)

]

c0V v + ?--T2 + ?,x@ - c E

]

Where,

Details of the line solution techniques have been presented earlier[12, 131. Briefly speaking, it is a semi-exact mathematical method for the solution of linear partial differential equations with constant coefficients, having two-point boundary conditions. The nth order governing partial differential equation of any two dimensional structure is rendered unidirectional by dividing the structure into a number of strips by a series of lines m in number parallel to. say, the .\ axis, and replacing all the y derivatives with their finite difference equivalents. This gives a set of m ordinary dill ferential equations of nth order which in turn are reduced to a set of mn first order equations by the use of a new set of wlriables as db)~4 d.v - FA,i+ , (k

c° - l + v

.j

~: = 2(1 - v)

. .m-l;

O, 1, =

0,

1.2...n--2)

in matrix form the set may be represented as

and u, v are the displacements in x and y directions in the orthogonal system of co-ordinates; v, pg, and E are the Poisson's ratio, density and modulus of elasticity respectively, V 2 is the Laplacian operator. A simpler formulation based on stresses and displacements which is more useful for mixed boundary conditions, has been recently presented [13] as,

dF dx

-

AF+B

(4)

where A is a square matrix and F and B are column matrices. This first-order matrix differential equation is solved by matrix progression method and standard computer programmes are made available for the computation. B. Grid analysis

,x (?er~r

33'

['cry

~'721

-C.V ~ + v ~('V = + E - -~1,2 = pg

3u

&" 2 ( l + v ) +

(:3'

~ (

o-,. r

.V

?u

&,

7-+v

-,

=

0

~r,. =

0

E

( 1 - v2)

where % and T,:~ are the direct and shear stresses. Exact solutions of these partial differential equations with practical boundary conditions is nearly impossible to achieve. Classical methods like ' conformal mapping techniques ', and ' Galarkin ' o r ' Kantorovich ' methods have been tried without much success. The finite difference and relaxation method have given results for simple cases but are extremely tedious to use for practical problems. Hence a search was made for new methods, preferably based on the application of high speed digital computers. N E W T E C H N I Q U E S FOR ISOLATED SHEAR-WALL PANELS Various mathematical and semi-equivalent methods were tried. As a result of this search, finally two methods were developed for singlestorey shear walls, termed as the line solution and Grid Analysis techniques.

This is an equivalent method[14] for the solution of general plane-stress problems and consists of replacing the actual two-dimensional structure by a hypothetical structure with one-dimensional elements, to form a grid work. Conditions of equilibrium and compatibility are approximately satisfied by specifying desired areas and moments of inertia to the members of this grid. However, this is the only approximation made in this analysis. The resulting grid is exactly solved by the stiffness method (or ttexibility method) and a standard programme for the Ferranti Pegasus computer has been developed. The accuracy of results obtainable by this method is comparable to any of the so-called exact methods a~ shown later. SHEAR-WALL BUILDINGS As discussed earlier, tile second stage of the research programme was concerned with the analysis of multi-storey shear-wall buildings, which is essentially a three-dimensional problem. However, the research was confined to two-dimensional treatment and a method was developed to mathematically reduce the stiffness of floor slabs to an equivalent beam stiffness while treating the resulting structure by grid analysis. A typical shear-wall building of the simplest type was selected for analysis, consisting of a pair of shear walls in each direction connected by floor slabs, Figures 1 and 2 show the general arrangement and equivalent loading on the floor slabs.

The Analysis of Shear-Wall Buildings

273

for these cases have been presented earlier[15] and are: oo

o9

8Pa3~ w -- 7~¢bD/_.~

Z

m = 1,3 fr Y

J

,3 Wl

n = 2,4

~] . nrc[b I q + c • nn/b sin ~ t,}--r/--c) - - q _ c sin -ff t~--r/+ c)J

]!w~z=:=> ]

InZa2\2

. s'nn rcyb

)

(5)

WI

f2

x--

~-| w,

--x for the simply supported case and w, |

tr

pa2[~=e-mny"

We

'(///%6

W = Drc3[ / ~ m3

3"

Front elevation

m = 1,3

Pion

[(1 + mT~y') sinh m~(tl' + c ' ) - mTz(q' + c')

Fig• 1. Skeleton of shear-wall building•

cosh mn(~l' + c')] +

B

+ c ~ e m:~y' ?] -- C.&.-¢/013 m

1,3

[(1 + tony') sinh mn(q' - c ' ) - rnn(q' - c') t

-I'

i

,t, P

'

'

,~

-P'"i

r 5",1

r,,,, ,1

o

B

"Itp

Equivalent Iooding

C Effective wolls

Fig. 2. Loading on the plate.

MATHEMATICAL TREATMENT FOR PLATE STIFFNESSES In this type of building the shear walls are continuous from the foundation to the top floor and withstand all the direct, bending and shear forces, in co-operation with the floor slabs or plates. It is assumed that the mode of displacement of the floor slabs produced by coupled shear walls subjected to lateral loading correspond to that produced by a set of self-equilibrating forces of the form shown in figure 2. At the boundaries the plate is assumed to be simply supported or simply supported on sides AB and CD and free on AD and BC. (Approximated by an infinitely long plate beyond AD and BC.) Mathematical solutions

cosh m n ( n ' - c ' ] t

(6)

for the free edges case; where q, c, a and P are indicated in figure 2, r / ' = q/a etc., D is the flexural rigidity of plate and y is the distance from y . y axis• Standard computer programmes were developed to solve these equations• These plate deflections w were compared with ordinary beam deflections of the same width and equivalent stiffness factors K were obtained for various sizes of floor slabs and shear walls• The results are obtainable in the form of influence diagrams of the type shown in figure 3 for various variables•

GRID ANALYSIS OF COMPLETE BUILDING The equivalent stiffness factor K depends on various factors like the width/length ratio, width of shear wall/length of slab ratio, cross distance of shear walls/length of slabs ratio and Poisson's ratio for the slab material. Influence diagrams have been presented of the type shown in figure 3. Thus for the given dimensions of the slabs and shear wall a suitable value of K is found from the

bOOI-

i 0,751_Fig. 3• Influence of width/length ratio on K.

0.50

Finite plate

0.25

.~--..~--..&,. ~

• ~

,/

Infinite Note"

r

]

2

3 B/L

rotio for slob

]

4

f

.5

274

S. M. A. Kazimi

influence diagrams. Once the equivalent stiffness factor K is known the effective moment of inertia for a slab of width B and thickness t will be

20fl cross was analysed using 162 nodes for precise results. EXPERIMENTAL WORK AND RESULTS

KBt 3

/ --

(7)

12

F'or isolated shear-walls panels with general boundry conditions and multiple-connected regions, the photoelastic method is ideally suited. Hence the results of grid analysis and line solution were verified by photo-elastic models. Figures 5 and 6 show the close agreement between the experimental stresses and those from the two theoretical methods, discussed earlier, for a square shear-wall panel without an opening. This is the simplest case but agreement is equally good for more involved cases as well[I 5]. The photo-elastic method could not be conveniently used for the 8-storey building analysed theoretically. Hence a Perspex model[161 of the building was made at %~(; scale and electrical resistance strain-gauges were fixed at various points for strain measurement. The deflection and strain diagrams for this building are shown in figures 7 and 8. Strain gauges were fixed very near the fixed end where maximum strains occur. The agreement between theoretical and experimental deflections is very good as shown in figure 7. Similarly the strains also (figure 8) are in excellent agreement. Only near the edges of the shear wall the strain gauges could not be fixed and therefore the deep beam effect by theoretical analysis is not quite apparent in the experimental curves.

The grid-analysis techniques discussed earlier may be directly applied to the building now; the floor slabs being replaced by equivalent beams in the grid system as shown in figure 4. ,-[ . . . .

---t

!--k ' I- I I ~

I ' -ir - .T. . . . ~----

'-~-~-~ i ~, !- iTi-+n ---7 !-I fi ....

¢~.

F25 ->'1 -1 .,i . o, .

. . . ' "- . ."~"-I

rT--T L._~_...t ~ i I

I

. .

9I 2

-Sheorwoll$, divided into strips

zz 2~i ~

i-" I, ~ 4

Equivolent gridlines

_j

7 J_ i'°l

3

4

IYL .t 5

6

Fig. 4. Shear-wall building grid.

MK 2 programme for the Ferranti Pegasus computor was modified and improved to consider structures larger than the specified limit of 71 nodes. An experimental building of 80 ft height (8 stories, each 10 ft high) with a floor slab of 30 ft × 40 ft and a pair of l0 ft-wide shear walls at

Line I

I0

0.5

Fig. 5. Shear stress (case 1).

0 '25

0-'5

0-)5

x/l

o

&"

t,

Theoretical stress (line solution) Theoretical stress (stress function) Theoreti¢ol stress (grid onology) Experimental stress

L, e, Fig. 6. Direct stress (case I).

0 25

0-5 x/L

0.75

Fo

1.0

The Analysis of Shear-Wall Buildings

275

Square matrix of co-efficients

CONCLUSION

A

The shear-wall building is evidently the most economical structural form for tall buildings. The research discussed in this paper solves some basic problems in the analysis by presenting new analysis techniques ideally suited for computer application. For simplified shear-wall buildings of the type discussed in this paper, complete stress analysis should not take more than an hour or two on the computer and a few extra hours for preparation of data tapes or cards. The method might be useful for more complicated shear-wall or shear wall-cumframed buildings if a judicious use of simplifying assumptions is made. The accuracy of the methods is demonstrated by the close agreement between experimental and theoretical results.

B

=

p

= Point-load on slab due to forces in shear-wall

D

=

Load vector

Eta/12(1-v2), where t is thickness of plate

~/, c, a and y = Various distances and dimensions shown in figure 2.

g

~___

Equivalent stiffness factor

B

=

Width of plate

I

= Moment of inertia for equivalent beam 0"0465 in

APPENDIX--NOTATION N

= Shear force in the connecting beams

Co

=

=

V

V2

1--v l+v

0.0287 in

2(1-v)

= Poisson's ratio ~2

°I /

02

in 8

-- ~X2d-Sy2

P

= Density of the material

g

= Acceleration due to gravity

E

= Young's Modulus of elasticity

U, l)

-- Displacements in x and y directions

ox

= Direct stress in x direction

%

= Shear stress

F

-- Column matrix of unknown functions

~

4h~

0.01385 in

0.00396in

" +~-*~*

o

Experimental (Jilani) o Theoretical (grid analogy]

I

E

I

I

2

3

I 4

I

~

',

I

5

6

7

8

X 10-2 in

Shearwall-windward side

Fig. 7.

4O0

4O0

300

300

Model deflections for an 8-storey shear-wall building under 8 lb point load.

Shearwall-leaward side

/ Strain gouge location

200

.~

200

:L

I00

I00 'ension

-I00

~

-I00

- 200

"~ o

-zoo

-300

~

-300

/

-400

i~

c

2in

o ~

.s /

Straingauge location

-400 ,2in

-2in

o Experimental (electric resistance strain gauges) ~ Theoretical (gdd analogy)

Fig. 8. B

/

~',

Model strains for 8-storey shear-wall building under 4 lb point load.

ZZ'

276

S. M . A. Kazhni REFERENCES

1. J. R. BENJAMINand H. A. WILLIAMS,The behaviour of one-storey brick shearwalls. Proc. Am. Soc. cir. Engrs. 84, No. ST.4, Paper 1723, 1958. 2. Y. OTSUKI, Stress analysis of walls with rectangular holes. Trans. Inst. Jap. Arch#., No. 41, 72 79. August 1950. 3. K. MuTo, Siesmic analysis of R.C. buildings. Proc. World Conference on Earthquake Engineering, California (1956). 4. H. FUTAMIand M. FUJIMOTO,Experimental study of lateral force distribution in walls by photoelasticity. Trans. Archit. Inst. Japan No. 47, 62-67 (in Japanese), Sept. 1953. 5. E. P. KOKINOPOULOS,Experimental photoelastic determination of stress and deformation condition in earthquake resistant structural elements. Publication of National Technical University Athens, No. 17 (1963). 6. N. B. GREEN,Bracing walls in multi-storey buildings. Proc. Am. Concr. Inst. 49 (3), 233-248 (1952). 7. B. CARDON,Concrete shear-walls combined with rigid frames in multi-storey buildings subjected to lateral loads, Proc. Am, Concr. Inst. 58 (3), (1961). 8. M. SHULZ, Analysis of reinforced concrete walls with openings. Indian Concr. 35 (l 1), (1961). 9. O. ER~CKSSON, Analysis of wind bracing walls in multi-storey housing. Ingenioren (lnternat. Edn.) 5 (4), (1961). 10. H. BECK, Contribution to the analysis of coupled Shearwalls. Proc. Am. Concr. Inst. 59, 1055-1069 (1962). 11. R. ROSMAN, An approximate method of analysis of walls of multi-storey buildings. Cir. Engng. pubL work. Rev. 59 (690), 67 (1964). 12. S. M. A. KAZlMIand S. J. BARI, Further applications of line solution techniques--the plane-stress problem with multiply connected regions. Indian Coner. To be published. 13. S. M. A. KAZIM!and A. COULL, The application of the line solution techniques to the solution of plane-stress problems, lnt. J. Mech. Sci. 6 (5) 391-400 (1964). 14. S. M. A. KAZIMI,Solution of plane-stress problems by grid analysis. Build. Sci. In press. 15. S. M. A. KAZXM!and P. B. MORICE, Co-operation of floor slabs in shear-wall buildings. To be presented at the First International Conference on Shear walls--Southampton (1966). 16. M. M. JILANI, An experimental investigation into the behaviour of shear walls and floor slabs of multi-storey buildings, subjected to lateral loads. M.Sc. thesis, Southampton (1964).

Malgr6 beaucoup d'inter6t pour les Murs Cisaill6s, le c6t6 th6orique est lent ~ cause de l'extr~me difficult6 dans l'analyse math6matique du probl~me des valeurs de limites mixtes avec des r6gions 5. connections multiples. La pr6sente 6tude passe tr6s bri6vement en revue les travaux d6j5. effectu6s sur ce sujet ainsi qu'une br~ve description de la contribution de l'auteur en ce qui concerne les techniques de ' Solution de Ligne ' et 'Analyse de Grille '. Une mention sp6ciale est faite concernant l'emploi des machines b, calculer Dig!tales pour de tels probl~mes. Zufolge ausserordentlich grosser Schwierigkeiten in der mathematischen Analyse des gemischten Randwertproblems mit mehrfach verbundenen Bereichen kann die theoretische Seite mit dem grossen lnteresse an Vertikalschubw~inden nicht Schritt halten. Die vorliegende Abhandlung gibt einen knappen Oberblick iJber bisher geleistete Arbeiten auf diesem Gebiet sowie eine kurze Beschreibung des pers6nlichen Beitrages des Autors in der Form v o n ' L6sung der Linienfiihrung'--und ' Netzanalyse'--methoden. Besondere Bedeutung wird der Verwendung von Digitalrechnern ftir derartige Probleme zugeschrieben.