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Composite action of walls supported on beams
Build. Sci. Vol. 1, pp. 259-270. Pergamon Press 1966. Printed in Great Britain
i
SfB AM UDC 624.07.5
imlI
Composite Action of Walls Supported on Beams A. COULL*
The paper presents an approximate method of analysis of deep walls acting compositely with supporting beams. The stresses in the wall are expressed as power series in the horizontal direction, the coefficients being functions of the height only. After satisfying the equilibrium and boundary conditions for both wall and beam, the remaining coefficients are determined by minimization of the strain energy of the system. The problem reduces to the solution of a fourthorder differential equation subject to given boundary conditions, the influence of the supporting beam appearing only in the latter. The analysis is simplified by dividing the applied load into symmetrical and anti-symmetrical components. Gravitational loads are also included. A number of sample curves are given to indicate the influence on the stresses of the various geometrical and stiffness parameters involved.
INTRODUCTION IN SPITE of their frequent occurrence in civilengineering structures, comparatively little is known about the composite action of walls supported on beams. In the design of framed structures, the stiffening effects of infilling panels have been neglected, and their presence has been treated purely as additional loads on the beams. Although a triangular form is often assumed for the additional loads on beams and linteIs due to the supported panels, it is well-known that this is far from the truth, the ' arching' action in the wall tending to redistribute the loads to the ends of the beams. A similar type of problem is encountered in modern forms of tall-building construction, when shear walls are discontinued at first-floor level to enable a large open space to be available on the ground floor. Wood[l] has reported the results of full-scale tests on the composite action of brick walls supported on reinforced concrete beams, and has suggested an empirical method of moment coefficients for design purposes. Rosenhaupt[2] has used finite-difference techniques to solve the problem; in his work he assumed effectively that no direct forces occur between wall and beam except at the support points, shear forces only being transferred along the interface. The method suffers from the usual disadvantage of a purely numerical technique in that any solution applies only for the given values of wall and beam stiffnesses and dimensions, * Lecturer in Southampton.
A
Civil
Engineering,
University
of
wall density and Poisson's ratio, and applied loading used in the calculations. The present paper presents a simple variational analysis of the problem. The two-dimensional plane stress problem is rendered uni-directional by the assumption that the stresses in the wall may be expressed with sufficient accuracy by a power series in the horizontal co-ordinate, the coefficients of the series being functions of the height only. By choosing the series to satisfy the equilibrium conditions, the vertical edge boundary conditions - - as well as overall statical conditions, the stresses may be expressed in terms of certain unknown functional coefficients. The loads on the beam may be expressed in terms of the stresses at the base of the wall, and the unknown coefficients determined by minimization of the total strain energy of the system. Because of its relevance to the shear wall problem, the form of loading considered is a superposition of a direct load of uniform intensity and a linear distribution having equal and opposite values at the edges of the wall; this corresponds to a combination of an axial and a linear bending stress distribution in the wail. A more elaborate load system, approximating to that in a deep beam, say, could be considered if required. The example given enables load forms which can be described by a cubic power series to be dealt with directly. The selfweight of the wall is incorporated in the analysis. The problem is greatly simplified by splitting any load system into symmetrical and anti-symmetrical components. 259
260
A. Coull
NOTATION The following symbols are used in this paper:--. 0 (u, v) .v, y 1 c t d h
Co-ordinate system Non-dimensional co-ordinates Height of wall Semi-span of wall and beam Thickness of wall Depth of beam Breadth of beam
r
c/I
e p E~, Ey t Exy, G]
d/c Density of wall material Elastic moduli for orthotropic wall material
o-x, %, %y EJ EbA N S M w p Uw, UB F, R
Direct and shear stresses in wall Flexural stiffness of beam Axial stiffness of beam Axial force in beam Shear force in beam Bending moment in beam Direct load intensity on wall Maximum linear load intensity on wall Strain energy of wall and beam Stress functions used in analysis c3t Ebl
k K
ANALYSIS The structure considered is a uniform wall carried on a simply supported beam, and subjected along its upper edge to a vertical load system, as shown in figure 1. The effect of the self-weight of the wall is also considered.
For convenience, a set of non-dimensional coordinates (.v, y) is used, defined with reference to figure 1 as .v = u/l,
~' = . / c
The stresses in the wall obey the usual plane stress equations of equilibrium, which are, in terms of the non-dimensional co-ordinate system, I <2x
c &' + p = 0
1 ~2o~,
1 &,.,.
0' +/
':
t.v
(I)
= 0
where o x and o s, are the direct stresses in the x- and y-directions respectively, %y is the shear stress, and. p is the density of the wall material. It is assumed that the stresses may be expressed as power series in the horizontal co-ordinate y, the coefficients of the series being functions of" the height x only. Hence, by reference to equations (1), statically correct solutions to the equilibrium equations are obtained by the series ~.~
ax =
,
tl
fly =
~rx)'~, i= 0
i
"
2
n i I
• Z'xy = ~, C:y) '~, i=0
t
.
~ Zrv.y~ i
0
" t
(2)
where ' n ' is an integer chosen to give sufficient terms in the series to produce a reasonable approximation to the stress distribution. Considerable simplification is made possible by dividing any load system into symmetrical and antisymmetrical components, with respect to the 0x axis. In the former case, even values only of ' i ' need be used for the direct stresses crx and ay, and odd values o f ' / ' for the shear stresses r,.y, and vice versa for the latter case. In order to allow for possible variations in wall properties in the x- and y-directions, orthotropic stress-strain relations are used. These may be expressed in the form ~r,, = Exe x + Ea.yey
(3)
~. = E_~yex + Eye~, "C,.y = G~/xy
ill
II
w-
_..--:r"U1
P
.£
//////.~
I"
C
~L
c
ld,
"l
a, (=) Fig. 1. Wall supported on beam.
1
I f
~ w
7 q
1
Uw =
0
" / l l l / [ 1
where ex, ey and ?,:,yare the direct and shear strains. The strain energy in the wall then becomes
v- (~)
I
0
( a , a 2 q - A 2 f f ~=- - 2 A
120"xO'y
+Aaz~y)dxdy (4)
where A ~ - EK ,
A2
E<,
Ex -
K"
A12 =
K'
l Aa -G
and
K= In the particular case of an isotropic material, the coefficients reduce to 1 At = A2 = ~ ,
v A12 ='E-w'
Aa -
2(1 + v) Ew
Composite Action of Walls Supported on Beams where E~ and v are Young's modulus and Poisson's ratio, respectively, for the wall material. The loading on the beam is dependent on the magnitudes of the stress resultants N~ (equal to cr~t) and Nxy (z~yt) at the lower edge of the wall. By considering the equilibrium of a small element of o
11"
J N* --
,I-
d$ ~
anti-symmetrical linear distribution with a maximum intensity of ' p ' per unit area (cf. figure 1).
1. Symmetrical loading. Under a symmetrical load system, only even distributions of the direct stresses, and odd distributions of the shear stresses, are possible. If it is assumed that the direct stress a x is represented sufficiently accurately by a parabolic distribution with n = 2 in equations (2), the stresses ay and zxy must be expressed by corresponding fourth- and third-order series respectively. Since the stresses ay and Zxy must vanish along the free edges of the wall at y = ___1, the stress series become
at. l r
'1
~t~-
261
crx ~" crxo-}-crx2Y z
crr ---- (crro +crr2y2)(I _y2) Fig. 2. Force system at wall-beam interface.
zxs = VxylY(l- y2) the beam and wall (figure 2), the conditions of equilibrium at the interface may be shown to be dS dv
Nx
dN dv = N~,
On substituting equations (7) into (1), a set of equilibrium equations is obtained which must be true for all values of the co-ordinate y. On equating corresponding terms, and solving the resulting set of equations, the stresses may finally be expressed in the form
(5)
1
ax = - (w +plx) - ~ F(1 - 3y 2)
dM d d--v = S - NxY -2
1
cry = ~ r Integration of equations (5) yields the bending moment M, shear force S, and axial force N in the beam at any span-wise position. The corresponding strain energy stored in the beam may then be obtained from standard expressions, c
¢
(M2dv
(U2dv
uB= 2 2EJ
+ 32EbA
c
(6)
--c
the strain energy due to transverse shear forces being assumed negligible. If the assumed stress-series (2) are made to obey the equilibrium equations (1) as well as the known boundary conditions at the edges of the wall, a set of relationships between the coefficients of the series is obtained, enabling the stresses to be obtained in terms of chosen functional coefficients. These may be determined by substituting the assumed series into the total strain energy, Uw + Us, the energy integral being minimized by the calculus of variations; the procedure produces a set of linear differential equations in the functional coefficients, together with a complete set of boundary conditions for solution.
(7)
2 dZF(l dx---5
-
y2)2
(8)
1 dF r~y = ~ r-d-xY(1-y2)
where F(x) is an arbitrary functional coefficient, and r = eli. The function F represents a variation from the elementary statical constant-stress solution. The integration constant which arises in the derivation of a~ in equations (8) is eliminated by considering the overall statical requirements for the vertical stresses. On substituting equations (8) into (5), integrating, and putting in the appropriate boundary conditions, the stress resultants in the beam become cat [
,
M = ~---](w +pl) (1 _y2) 1
1
:
S = -ct [(w+pl)y+~F(y-y ct N = -~r
1
a
)]
(9)
dF dx (1-y2)2
Example
where e = d/c.
The particular example is considered of a wall subjected along its upper edge to a symmetrical direct loading of intensity ' w ', together with an
On substituting equations (8) and (9) into (4) and (6), and integrating over the width of wall and span of beam, the total strain energy becomes
A. Coull
262
dF
I
28
u = }tzcf IA,
dx
0
d3F
+A2
x2,
2{~sr2 d2----F[(w+plx)+4F] [
+2A
-
1
dx2
J
(9-~45r 2 idF'i2 dx d x ] ,}
+Aa
fcSt2 [2(w+p/)2 + I EhI 4
+ ]-6-5(w+ pl)
8
] dF~ ( F+ ,,er d.~
63
2. Anti-symmetrical /oading. Under a linear anti-symmetrical loading of intensity py, if the vertical direct stress a~ is represented by an odd series of the third order, the direct and shear stresses, tTy and zxr, must be represented by odd and even series of the fifth and fourth orders, respectively. On satisfying the stress-free edge boundary conditions, the overall statical conditions, and the equilibrium equations (1), the stresses become o-x ---- R y - - 5/3 (p + R)) '3
1 d2R o-y = -]-22 r2 d.v ~ y(I _y2)Z 8
c3t 2 r2 i
~ dx ] j
2835EbA
.,=1 (10)
Minimization of the energy integral (10) by the calculus of variations leads to the following differential equation which must be satisfied by the stress function if the strain energy is to be a minimum d2F
(12)
1 dR %, = ~ r ~-x ( I - 5 y 2 ) ( I - y 2 )
d_F~2t
where the terms in the last set of braces are evaluated at the lower edge, x = 1.
d4F
27
A2r3 ~-fx$-kF = ~- / c ( , r + p / ) - ~ - A,2t,h.
dF 2] +
0
where
R(x)
is the required functional coefficient.
The integration constants which arise in the derivation of equations (12) are evaluated by equating the internal stresses to the applied loads along the top of the wall. No self-weight term is included in equations (12) since the gravitational loading cannot give rise to an anti-symmetrical stress system. The stress resultants in the beam become
62
A~.r4 dx4 + 3(2A~z--As)r2 -dx-2 +-2 A~F = 0
dR
(ll) The minimization technique produces naturally the required boundary conditions, in accordance with the known edge values, as well as the necessary governing equation; in this case, the boundary conditions become dF dx-
0
d2F
1
Atx=0,
F-
At x = I,
A z r Z ~ + ~ k e 2 r-~x+(3Aj2+½ke)F
dF
ct
S = i~[p(1-5y4)-R(I-5)
'2) (1-),2)1
ct dR
N = ~rdxxY(1-y2)2 On following the same procedure as before, and minimizing the total strain energy with respect to R, it is found that the stress function must satisfy the following governing differential equation 4 d4R
d2R
A2r ~x4 +ll(2A~2-A3)rZ ~
495
+-~- AxR
= - ( ? A,2 +2~78ke ) (w+pl)
495
=
dSF
l
--Sf.
dF
A2r 3 ~x 3 +[3(A~z - A3)-½ke]r ~x - k F - 2-~k ( w + p l ) - ? A,zpZr where
(13)
(14) subject to the following boundary conditions: dR
Atx=
0, R = -p,d.x, = 0
C3I
k - Ed
Equation (11) may be solved readily by standard analytical or numerical mathematical techniques. Further simplification is possible if it is assumed that the depth of the beam is small compared to the height of the wall, and that the shear stress zxr may be assumed to vanish at the lower edge. In that case, the boundary conditions reduce to, at x = 1,
A~p
At x = 1,
d2R
A2r2 ~
1
2 dR
+~ ke r-~x +(11A~2 +-}ke)R (55 7 ) = -~-Al,+~ke p
A2r s d3R ~ + [ l l ( A ~ z - A3)_½keJr~x-kR =
7
kp
Composite Action of Walls Supported on Beams In the event of the simpler assumption being made of vanishing shear stress at the lower edge of the wall, the boundary conditions reduce to, at
present theory with the finite-difference solution of Rosenhaupt[2]. The relative properties of wall and beam are defined by the following values of the parameters used in the analysis; r = ¼, e = 0'159, k = 99"3, Eb/Ew = 30, representing a lightweight concrete block wall on a reinforced-concrete beam. Comparisons between distributions of the three stress components trx, a r and t~y, for one-half of the wall are shown in figures 3(a), (b) and (c).
x-----l,
dR -0 dx 3 d3R
A
7 - kR
The complete solution to the problem is obtained by superposition of the symmetrical and antisymmetrical systems. The displacements of the wall and beam may be obtained subsequently by integration of the stress-displacement equations. Since the stress-series used contain terms of the third order in the direct stress ax, an applied loading of this more elaborate form could be considered directly in the analysis presented, and similar equations derived with little extra difficulty.
(ii) Influence of geometrical and stiffness properties on stresses. Calculations were performed for a number of cases to obtain some idea of the influence of the various geometrical and stiffness parameters on the stresses in an isotropic wall. The influence of the span : height ratio r, the wall : beam-depth ratio e, and the relative stiffness of the wall and beam (k) were investigated, particular emphasis being placed on the symmetrical case of a uniform load since this is thought to be the more important case of the two. The influence of the self-weight of the wall has been neglected in order to limit the number of variables involved. In order to provide a concise representation of the results, only the appropriate stress functions and their derivatives have been plotted in figures 4,
Numerical results (i) Compar&on with previous results. Calculations were performed for an isotropic wall subjected to a uniformly-distributed load along the upper edge in order to compare the results of the
0
0
0
LEGEND PRESENT ------
THEORY
ROSENHAUPT'S
THEORY
/# / /
/
/
L
/
/ /
\
/ /
/
/
\
/ 4,'
j,
263
/
\ \
.s
1
4 ~uj
Fig. 3(a),
2
I
0
0
0
264
A. Coull ©
0
0
C ~/.r,..T
,
o'-s
o
0
, I
Fig. 3(b).
A 0"5
Fig. 3(c).
Fig. 3. Comparison between results of present theory and finite-difference solution of Rosenhaupt for (a) vertical stresses, (b) horizontal stresses, (e) shear stresses.
o
,r=-
0
T ID
g
g i
0
2
4
G
0 (a)
Legend e = 1/12 ......
e = 1/9
....
e = 1/6
....
no
shear
at
interface
Fig. 4(a).
2
4
g,-~-
6
©
o
l[o
Composite Action of Walls Supported on Beams
265
o~
Fig. 4(b).
K= 2 0 0 0
_ _ ' ~
I
i
I
I
- - - .~-__.. ~
1
0
-1
Horizontal
0 Stress
- ~ ~ _ . . _ _ 1
Function
2
h2rZd~F
(b) 0
o
\
o~\ b 0
113 b..
K=50
k~,
.K=2000
o O
I1:
Fig. 4(e).
g'l
7:O 'r
Fig. 4. Variations o f stresses with beam depth : span ratio for different wall : beam stiffness ratios. Uniform load on wall.
t.
6
- ---'~'.#/ 0
1 Shear
2 stress
3 function
0
I 1
1/~ r d__¢
(c)
°6|~',f
°t
~:~°
o ,
',
K= 2 0 0 0
~1\" o t ~,,,
~
~o
,,,
~kx, \,,
~
° I/ \ . ~ .\ ' , ,~, " ,
~J
0
2
4 Vertical
6
0 stress
.....
\ " ' ~ . \ ~","' ",, 2
function
Legend r=0.25 r=05 r=l.0
Fig. 5(a).
",,
(a)
4 F, _ /or
(3
2
266
A. Coull
'//tI/t
Fig. 5(b).
Ill
Kso', '~
-2
-1
0
1
-3
-2
-1
0
I
2
3
(b) o
O
k\ ~,
K= 5 0
o
O
O
O r~
z~
LO ~2 O~ :E
Fig. 5(c).
'\, \
32
\')),,,"
O
I
3
0 Shear
Stress
,,';
Fig. 5. Variations o f stresses with beam span : wall-height ratio for different wall : beam-stiffness ratios. Uniform load on wall.
I
4
0
Function
1
2
3
1/3 r d_ff
(c)
©
\
O
~%
.O
~
,
t
On
cr
~t
&
'\',,
0
e=
tO I 0
2
Fig. 6(a). K=50
\N
~,
\
K=2000
%\
i
I
-1
1
0
-1 Stress
© Function
Legend ...... -
-
-- ---
r=l.0 r=05 r = 0.25
(a)
1
%
267
Composite Action o f Walls S u p p o r t e d on B e a m s
ik
0 4-
iI
I I
l
I
i
0
O
'~, 3:
~
I£
2
'\Xi I
o
I
t
1
-2
Horizontal
_ ~ "
~ - -~- . ~~" ~ " ~-J--" -1
Stress
Function
/12
1
o
radaR
(b)
Fig. 6(b).
o
oI
""
' ID
~\
o
I~
N•
•
6
•
/t I
0
0.25 Shear
05
0
Stress Function
o~
0.25 1/ /12 rdR Z'S /
(c)
Fig. 6(c). Fig. 6. Variations of stresses with beam span : wall height ratio for different wall : beam stiffness ratios. Linearlyvarying load on wall.
5 and 6. The corresponding stress components may then be obtained by substitution in equations (8) and (12) for the symmetrical and anti-symmetrical cases respectively. The vertical and shear stresses at the interface between wall and beam are shown in figures 7(a) and (b) respectively, for typical geometrical properties, r = ½ and e = 1/12, for both symmetrical and anti-symmetrical cases. The solutions of equations (11) and (14) were obtained numerically by the use of finite differences. One the governing equations have been set up in
finite difference form as a matrix equation, the influence of different beam properties, which occur in the boundary conditions only, may be readily investigated, since only a few elements in the matrices need be altered. DISCUSSION In this paper, an attempt vide a simple approximate walls supported on beams. used in the assumed series
has been made to promethod of analysis of The number of terms for the stresses in the
268
A. Cou// ~0 i
~9
co r~
E 0 u
~r i
Uniform
loading
~c Q;
Linear applied loading
co co £.
%
i U
>
0
Span
.£ Legend 1
,'i
....
K=50 K= 2 0 0 0
~'=1/2 e = ~12
Fig. 7(a).
Fig. 7(b). Fig. 7. Typical distributions o f vertical and shear loads on beam due to uniform and linearly-varying loads" on top o f the wall.
wall has been restricted to enable all stress components to be expressed in terms of a single stress function, for both the symmetrical and antisymmetrical load cases considered (a uniform load and a linear form of loading along the top of the wall).
The latter form of loading has been included since the problem is encountered in the design of tall buildings, where shear walls are discontinued at first-floor level in order to produce a large uninlerrupted concourse on the ground floor. If the wall is subjected to an axial load and bending moment, which produce stresses of the same forms as the applied loads, at a known distance from the lower edge, the method of analysis may be utilized to investigate the diffusion of stresses in the wall to a supporting beam and columns. The problem finally reduces to the solution of a single fourth-order differential equation (containing even powers only), which may readily be solved analytically or numerically to give the complete stress distribution in the wall and beam. With the simple stress polynomials used, the horizontal and shear stresses have the same form at all levels in the wall, which does not seem likely in practice. The method is put forward as a quick and simple technique for determining approximately the stresses in such structures. The accuracy can always be increased by using more terms in the assumed stressseries, but at the expense of the extra computational difficulty involved in the solution of the resulting set of simultaneous differential equations. For simplicity, the analysis has been restricted to vertical loads only. Horizonlal loads may be considered by incorporating these directly in the assumed series (cf. Ref. 3). By utilizing the conditions of equilibrium at the interface, the loads on the beam may be expressed in terms of the vertical and shear stresses at the lower edge of the wall, and hence in terms of the stress function evaluated at this lower limit. By this means, the stiffness properties of the beam appear only in the boundary conditions for the governing equations, which are obtained on minimization of the total strain energy of the system. Since theJorm of the loading is fixed, the supporting structure may be analysed separately, enabling the method to be extended readily to deal with other support conditions, such as built-in beams, portal frames, etc. The influence of frictional forces at the supports could also be included if required. A comparison has been made between the results of the present theory and those given by Rosenhaupt[2], using a finite-difference method of analysis, which is the only other solution known to the author. Rosenhaupt, in his study of walls subjected to uniform loads along the upper edge, made the assumption that the bending rigidity of the beam is negligible compared to the stiffness of the wall, which effectively restricts normal pressures between the wall and beam to the region immediately over the support. This is obviously a rather extreme assumption, as shown by Wood's tests[l] on brick walls supported on reinforced concrete beams. Because of the different conditions at the lower edge, the stress patterns are quite dissimilar over the lower half of the wall, although reasonable agreement occurs in the upper regions. The present theory indicates that, in the central
Composite Action of Walls Supported on Beams
region, tensile forces exist between the wall and beam [cf. figure 7(a)], under the action of a uniform loading on the upper edge, in the absence of any gravitational effects, or frictional forces at the supports. The method could be used to give a first approximation for a purely numerical solution devised to take account of separation occurring between the wall and beam. For a uniform load applied to the top of the wall, the variation of the three stress components, a~, ay and z~y, with the beam semi-span:wall height ratio, r and with the beam depth : semi-span ratio, e, are shown in figures 4 and 5, respectively, for typical ranges of the parameters concerned (r = ¼, ½, 1, and e = 1/12, 1/9, 1/6). In each case, the variation is shown for two values of the relative stiffness parameter K
of 50 and 2000, representing approximately two limiting cases of a lightweight concrete wall and a reinforced-concrete wall on a reinforced-concrete beam. Although the results have not been shown, calculations were also performed for a value of K of 500; in that case, the results obtained were fairly close to those for a value of K of 2000. For the anti-symmetric linear-load case, the
269
variation of the stress components with the wall height:beam-span ratio only are shown, for the same two values of the relative stiffness K. The simple theory indicates that the magnitudes of the wall stresses are affected much more by the wall height :beam span ratio r and the relative w a l l : b e a m stiffness K, than by the beam depth: wall height ratio e. Comparatively small changes in the stress levels were obtained for the range of values of e investigated. The distribution of stresses a~ and hence the vertical loading on the beam, was found to vary only slightly over the range of parameters considered. The influence of making the assumption that there are no shear stresses at the w a l l : b e a m interface has been investigated, and some results are shown in figure 4. (For simplicity, the corresponding values are not shown in figures 5 and 6, since this would double the number of curves plotted.) The results indicate that the assumption gives reasonably accurate results provided the w a l l : b e a m stiffness ratio K is high (i.e. > 2000). It is of interest to note that, for walls whose height is greater than the span, practically no diffusion of stress occurs in the section above the unit d e p t h : s p a n position. This effect was also noted by Saad and Hendry[4] in connection with the gravitational stresses in deep beams.
REFERENCES
1. R.H. WOOD, Studies in Composite Construction, Part 1. Research Paper No. 13, National Building Studies, H.M.S.O. (1952). 2. S. ROS~NHAUPT,Stresses in Point Supported Composite Walls. Proc. Am. Concr. Inst. 61, 795 (1964). 3. A. COULL, Stress Analysis of Single-storey Shear Walls. Cir. Engng publ. Wks Rev. 60, 1044 (1965). 4. S. SAADand A. W. HENDRY,Gravitational Stresses in Deep Beams. Struct. Engr, 39, 185 (1961).
Ce document prdsente une mdthode approximative d'analyse de murs dpais agissant compositement avec des poutres de support. Les efforts sur l e m u r sont ddcrits comme des sdries de pressions horizontales, les coefficients dtant uniquement en fonction de la hauteur. Apr6s avoir satisfait l'equilibre et les conditions de limite pour le tour ainsi que pour la poutre, les coefficients restants sont ddcrits par la minimisation de l'dnergie de contrainte du syst~me. Le probl6me se rdduit/~ la solution d'une dquation differentielle de quatri~me ordre, soumise ~ des conditions de limite donndes, l'influence de la poutre de support apparaissant uniquement dans cette derni~re. L'analyse est simplifide en divisant le poids appliqud en des composts symmdtriques et antisymmdtriques. Des poids de gravitation sont dgalement inclus. Un certain nombre d'exemples de courbes est donnd afin d'indiquer l'influence sur les contraintes, des divers param6tres gdomdtriques et de raideur impliquds. Es wird eine ann~ihernde Methode fur die Analyse von Tiefw/inden die mit Stiitzbalken zusammenwirken gegeben. Die Beanspruchungen der Wand in der horizontalen Richtung als Kr/iftereihen ausgedruckt, wobei die Koeffizienten ausschliesslich Funktionen der Htihe sind. Nachdem die Gleichgewichts- und Grenzbedingungen fiir Wand und Balken erftillt sind, werden die iibrigen Koeffiizienten durch Minimisation der System-Spannungsenergie bestimmt. Das Problem wird durch die L6sung einer Differentialgleichung vierter Ordnung in Abh~ingigkeit von den Grenzbedingungen gekl/irt; der Einfluss des Stiitzbalkens erscheint bloss in den
270
A. Coull
Grenzbedingungen. Die Analyse wird vereinfacht indem man die angelegte Last in symmetrische und antisymmetrische Komponenten aufteilt. Schwerkraftslasten werden auch berticksichtigt. Eine Reihe von Kurven ist gegeben um den Einfluss der verschiedenen geometrischen und Steifigkeitsparameter auf die Belastung zu zeigen.
i
SfB AM UDC 624.07.5
imlI
Composite Action of Walls Supported on Beams A. COULL*
The paper presents an approximate method of analysis of deep walls acting compositely with supporting beams. The stresses in the wall are expressed as power series in the horizontal direction, the coefficients being functions of the height only. After satisfying the equilibrium and boundary conditions for both wall and beam, the remaining coefficients are determined by minimization of the strain energy of the system. The problem reduces to the solution of a fourthorder differential equation subject to given boundary conditions, the influence of the supporting beam appearing only in the latter. The analysis is simplified by dividing the applied load into symmetrical and anti-symmetrical components. Gravitational loads are also included. A number of sample curves are given to indicate the influence on the stresses of the various geometrical and stiffness parameters involved.
INTRODUCTION IN SPITE of their frequent occurrence in civilengineering structures, comparatively little is known about the composite action of walls supported on beams. In the design of framed structures, the stiffening effects of infilling panels have been neglected, and their presence has been treated purely as additional loads on the beams. Although a triangular form is often assumed for the additional loads on beams and linteIs due to the supported panels, it is well-known that this is far from the truth, the ' arching' action in the wall tending to redistribute the loads to the ends of the beams. A similar type of problem is encountered in modern forms of tall-building construction, when shear walls are discontinued at first-floor level to enable a large open space to be available on the ground floor. Wood[l] has reported the results of full-scale tests on the composite action of brick walls supported on reinforced concrete beams, and has suggested an empirical method of moment coefficients for design purposes. Rosenhaupt[2] has used finite-difference techniques to solve the problem; in his work he assumed effectively that no direct forces occur between wall and beam except at the support points, shear forces only being transferred along the interface. The method suffers from the usual disadvantage of a purely numerical technique in that any solution applies only for the given values of wall and beam stiffnesses and dimensions, * Lecturer in Southampton.
A
Civil
Engineering,
University
of
wall density and Poisson's ratio, and applied loading used in the calculations. The present paper presents a simple variational analysis of the problem. The two-dimensional plane stress problem is rendered uni-directional by the assumption that the stresses in the wall may be expressed with sufficient accuracy by a power series in the horizontal co-ordinate, the coefficients of the series being functions of the height only. By choosing the series to satisfy the equilibrium conditions, the vertical edge boundary conditions - - as well as overall statical conditions, the stresses may be expressed in terms of certain unknown functional coefficients. The loads on the beam may be expressed in terms of the stresses at the base of the wall, and the unknown coefficients determined by minimization of the total strain energy of the system. Because of its relevance to the shear wall problem, the form of loading considered is a superposition of a direct load of uniform intensity and a linear distribution having equal and opposite values at the edges of the wall; this corresponds to a combination of an axial and a linear bending stress distribution in the wail. A more elaborate load system, approximating to that in a deep beam, say, could be considered if required. The example given enables load forms which can be described by a cubic power series to be dealt with directly. The selfweight of the wall is incorporated in the analysis. The problem is greatly simplified by splitting any load system into symmetrical and anti-symmetrical components. 259
260
A. Coull
NOTATION The following symbols are used in this paper:--. 0 (u, v) .v, y 1 c t d h
Co-ordinate system Non-dimensional co-ordinates Height of wall Semi-span of wall and beam Thickness of wall Depth of beam Breadth of beam
r
c/I
e p E~, Ey t Exy, G]
d/c Density of wall material Elastic moduli for orthotropic wall material
o-x, %, %y EJ EbA N S M w p Uw, UB F, R
Direct and shear stresses in wall Flexural stiffness of beam Axial stiffness of beam Axial force in beam Shear force in beam Bending moment in beam Direct load intensity on wall Maximum linear load intensity on wall Strain energy of wall and beam Stress functions used in analysis c3t Ebl
k K
ANALYSIS The structure considered is a uniform wall carried on a simply supported beam, and subjected along its upper edge to a vertical load system, as shown in figure 1. The effect of the self-weight of the wall is also considered.
For convenience, a set of non-dimensional coordinates (.v, y) is used, defined with reference to figure 1 as .v = u/l,
~' = . / c
The stresses in the wall obey the usual plane stress equations of equilibrium, which are, in terms of the non-dimensional co-ordinate system, I <2x
c &' + p = 0
1 ~2o~,
1 &,.,.
0' +/
':
t.v
(I)
= 0
where o x and o s, are the direct stresses in the x- and y-directions respectively, %y is the shear stress, and. p is the density of the wall material. It is assumed that the stresses may be expressed as power series in the horizontal co-ordinate y, the coefficients of the series being functions of" the height x only. Hence, by reference to equations (1), statically correct solutions to the equilibrium equations are obtained by the series ~.~
ax =
,
tl
fly =
~rx)'~, i= 0
i
"
2
n i I
• Z'xy = ~, C:y) '~, i=0
t
.
~ Zrv.y~ i
0
" t
(2)
where ' n ' is an integer chosen to give sufficient terms in the series to produce a reasonable approximation to the stress distribution. Considerable simplification is made possible by dividing any load system into symmetrical and antisymmetrical components, with respect to the 0x axis. In the former case, even values only of ' i ' need be used for the direct stresses crx and ay, and odd values o f ' / ' for the shear stresses r,.y, and vice versa for the latter case. In order to allow for possible variations in wall properties in the x- and y-directions, orthotropic stress-strain relations are used. These may be expressed in the form ~r,, = Exe x + Ea.yey
(3)
~. = E_~yex + Eye~, "C,.y = G~/xy
ill
II
w-
_..--:r"U1
P
.£
//////.~
I"
C
~L
c
ld,
"l
a, (=) Fig. 1. Wall supported on beam.
1
I f
~ w
7 q
1
Uw =
0
" / l l l / [ 1
where ex, ey and ?,:,yare the direct and shear strains. The strain energy in the wall then becomes
v- (~)
I
0
( a , a 2 q - A 2 f f ~=- - 2 A
120"xO'y
+Aaz~y)dxdy (4)
where A ~ - EK ,
A2
E<,
Ex -
K"
A12 =
K'
l Aa -G
and
K= In the particular case of an isotropic material, the coefficients reduce to 1 At = A2 = ~ ,
v A12 ='E-w'
Aa -
2(1 + v) Ew
Composite Action of Walls Supported on Beams where E~ and v are Young's modulus and Poisson's ratio, respectively, for the wall material. The loading on the beam is dependent on the magnitudes of the stress resultants N~ (equal to cr~t) and Nxy (z~yt) at the lower edge of the wall. By considering the equilibrium of a small element of o
11"
J N* --
,I-
d$ ~
anti-symmetrical linear distribution with a maximum intensity of ' p ' per unit area (cf. figure 1).
1. Symmetrical loading. Under a symmetrical load system, only even distributions of the direct stresses, and odd distributions of the shear stresses, are possible. If it is assumed that the direct stress a x is represented sufficiently accurately by a parabolic distribution with n = 2 in equations (2), the stresses ay and zxy must be expressed by corresponding fourth- and third-order series respectively. Since the stresses ay and Zxy must vanish along the free edges of the wall at y = ___1, the stress series become
at. l r
'1
~t~-
261
crx ~" crxo-}-crx2Y z
crr ---- (crro +crr2y2)(I _y2) Fig. 2. Force system at wall-beam interface.
zxs = VxylY(l- y2) the beam and wall (figure 2), the conditions of equilibrium at the interface may be shown to be dS dv
Nx
dN dv = N~,
On substituting equations (7) into (1), a set of equilibrium equations is obtained which must be true for all values of the co-ordinate y. On equating corresponding terms, and solving the resulting set of equations, the stresses may finally be expressed in the form
(5)
1
ax = - (w +plx) - ~ F(1 - 3y 2)
dM d d--v = S - NxY -2
1
cry = ~ r Integration of equations (5) yields the bending moment M, shear force S, and axial force N in the beam at any span-wise position. The corresponding strain energy stored in the beam may then be obtained from standard expressions, c
¢
(M2dv
(U2dv
uB= 2 2EJ
+ 32EbA
c
(6)
--c
the strain energy due to transverse shear forces being assumed negligible. If the assumed stress-series (2) are made to obey the equilibrium equations (1) as well as the known boundary conditions at the edges of the wall, a set of relationships between the coefficients of the series is obtained, enabling the stresses to be obtained in terms of chosen functional coefficients. These may be determined by substituting the assumed series into the total strain energy, Uw + Us, the energy integral being minimized by the calculus of variations; the procedure produces a set of linear differential equations in the functional coefficients, together with a complete set of boundary conditions for solution.
(7)
2 dZF(l dx---5
-
y2)2
(8)
1 dF r~y = ~ r-d-xY(1-y2)
where F(x) is an arbitrary functional coefficient, and r = eli. The function F represents a variation from the elementary statical constant-stress solution. The integration constant which arises in the derivation of a~ in equations (8) is eliminated by considering the overall statical requirements for the vertical stresses. On substituting equations (8) into (5), integrating, and putting in the appropriate boundary conditions, the stress resultants in the beam become cat [
,
M = ~---](w +pl) (1 _y2) 1
1
:
S = -ct [(w+pl)y+~F(y-y ct N = -~r
1
a
)]
(9)
dF dx (1-y2)2
Example
where e = d/c.
The particular example is considered of a wall subjected along its upper edge to a symmetrical direct loading of intensity ' w ', together with an
On substituting equations (8) and (9) into (4) and (6), and integrating over the width of wall and span of beam, the total strain energy becomes
A. Coull
262
dF
I
28
u = }tzcf IA,
dx
0
d3F
+A2
x2,
2{~sr2 d2----F[(w+plx)+4F] [
+2A
-
1
dx2
J
(9-~45r 2 idF'i2 dx d x ] ,}
+Aa
fcSt2 [2(w+p/)2 + I EhI 4
+ ]-6-5(w+ pl)
8
] dF~ ( F+ ,,er d.~
63
2. Anti-symmetrical /oading. Under a linear anti-symmetrical loading of intensity py, if the vertical direct stress a~ is represented by an odd series of the third order, the direct and shear stresses, tTy and zxr, must be represented by odd and even series of the fifth and fourth orders, respectively. On satisfying the stress-free edge boundary conditions, the overall statical conditions, and the equilibrium equations (1), the stresses become o-x ---- R y - - 5/3 (p + R)) '3
1 d2R o-y = -]-22 r2 d.v ~ y(I _y2)Z 8
c3t 2 r2 i
~ dx ] j
2835EbA
.,=1 (10)
Minimization of the energy integral (10) by the calculus of variations leads to the following differential equation which must be satisfied by the stress function if the strain energy is to be a minimum d2F
(12)
1 dR %, = ~ r ~-x ( I - 5 y 2 ) ( I - y 2 )
d_F~2t
where the terms in the last set of braces are evaluated at the lower edge, x = 1.
d4F
27
A2r3 ~-fx$-kF = ~- / c ( , r + p / ) - ~ - A,2t,h.
dF 2] +
0
where
R(x)
is the required functional coefficient.
The integration constants which arise in the derivation of equations (12) are evaluated by equating the internal stresses to the applied loads along the top of the wall. No self-weight term is included in equations (12) since the gravitational loading cannot give rise to an anti-symmetrical stress system. The stress resultants in the beam become
62
A~.r4 dx4 + 3(2A~z--As)r2 -dx-2 +-2 A~F = 0
dR
(ll) The minimization technique produces naturally the required boundary conditions, in accordance with the known edge values, as well as the necessary governing equation; in this case, the boundary conditions become dF dx-
0
d2F
1
Atx=0,
F-
At x = I,
A z r Z ~ + ~ k e 2 r-~x+(3Aj2+½ke)F
dF
ct
S = i~[p(1-5y4)-R(I-5)
'2) (1-),2)1
ct dR
N = ~rdxxY(1-y2)2 On following the same procedure as before, and minimizing the total strain energy with respect to R, it is found that the stress function must satisfy the following governing differential equation 4 d4R
d2R
A2r ~x4 +ll(2A~2-A3)rZ ~
495
+-~- AxR
= - ( ? A,2 +2~78ke ) (w+pl)
495
=
dSF
l
--Sf.
dF
A2r 3 ~x 3 +[3(A~z - A3)-½ke]r ~x - k F - 2-~k ( w + p l ) - ? A,zpZr where
(13)
(14) subject to the following boundary conditions: dR
Atx=
0, R = -p,d.x, = 0
C3I
k - Ed
Equation (11) may be solved readily by standard analytical or numerical mathematical techniques. Further simplification is possible if it is assumed that the depth of the beam is small compared to the height of the wall, and that the shear stress zxr may be assumed to vanish at the lower edge. In that case, the boundary conditions reduce to, at x = 1,
A~p
At x = 1,
d2R
A2r2 ~
1
2 dR
+~ ke r-~x +(11A~2 +-}ke)R (55 7 ) = -~-Al,+~ke p
A2r s d3R ~ + [ l l ( A ~ z - A3)_½keJr~x-kR =
7
kp
Composite Action of Walls Supported on Beams In the event of the simpler assumption being made of vanishing shear stress at the lower edge of the wall, the boundary conditions reduce to, at
present theory with the finite-difference solution of Rosenhaupt[2]. The relative properties of wall and beam are defined by the following values of the parameters used in the analysis; r = ¼, e = 0'159, k = 99"3, Eb/Ew = 30, representing a lightweight concrete block wall on a reinforced-concrete beam. Comparisons between distributions of the three stress components trx, a r and t~y, for one-half of the wall are shown in figures 3(a), (b) and (c).
x-----l,
dR -0 dx 3 d3R
A
7 - kR
The complete solution to the problem is obtained by superposition of the symmetrical and antisymmetrical systems. The displacements of the wall and beam may be obtained subsequently by integration of the stress-displacement equations. Since the stress-series used contain terms of the third order in the direct stress ax, an applied loading of this more elaborate form could be considered directly in the analysis presented, and similar equations derived with little extra difficulty.
(ii) Influence of geometrical and stiffness properties on stresses. Calculations were performed for a number of cases to obtain some idea of the influence of the various geometrical and stiffness parameters on the stresses in an isotropic wall. The influence of the span : height ratio r, the wall : beam-depth ratio e, and the relative stiffness of the wall and beam (k) were investigated, particular emphasis being placed on the symmetrical case of a uniform load since this is thought to be the more important case of the two. The influence of the self-weight of the wall has been neglected in order to limit the number of variables involved. In order to provide a concise representation of the results, only the appropriate stress functions and their derivatives have been plotted in figures 4,
Numerical results (i) Compar&on with previous results. Calculations were performed for an isotropic wall subjected to a uniformly-distributed load along the upper edge in order to compare the results of the
0
0
0
LEGEND PRESENT ------
THEORY
ROSENHAUPT'S
THEORY
/# / /
/
/
L
/
/ /
\
/ /
/
/
\
/ 4,'
j,
263
/
\ \
.s
1
4 ~uj
Fig. 3(a),
2
I
0
0
0
264
A. Coull ©
0
0
C ~/.r,..T
,
o'-s
o
0
, I
Fig. 3(b).
A 0"5
Fig. 3(c).
Fig. 3. Comparison between results of present theory and finite-difference solution of Rosenhaupt for (a) vertical stresses, (b) horizontal stresses, (e) shear stresses.
o
,r=-
0
T ID
g
g i
0
2
4
G
0 (a)
Legend e = 1/12 ......
e = 1/9
....
e = 1/6
....
no
shear
at
interface
Fig. 4(a).
2
4
g,-~-
6
©
o
l[o
Composite Action of Walls Supported on Beams
265
o~
Fig. 4(b).
K= 2 0 0 0
_ _ ' ~
I
i
I
I
- - - .~-__.. ~
1
0
-1
Horizontal
0 Stress
- ~ ~ _ . . _ _ 1
Function
2
h2rZd~F
(b) 0
o
\
o~\ b 0
113 b..
K=50
k~,
.K=2000
o O
I1:
Fig. 4(e).
g'l
7:O 'r
Fig. 4. Variations o f stresses with beam depth : span ratio for different wall : beam stiffness ratios. Uniform load on wall.
t.
6
- ---'~'.#/ 0
1 Shear
2 stress
3 function
0
I 1
1/~ r d__¢
(c)
°6|~',f
°t
~:~°
o ,
',
K= 2 0 0 0
~1\" o t ~,,,
~
~o
,,,
~kx, \,,
~
° I/ \ . ~ .\ ' , ,~, " ,
~J
0
2
4 Vertical
6
0 stress
.....
\ " ' ~ . \ ~","' ",, 2
function
Legend r=0.25 r=05 r=l.0
Fig. 5(a).
",,
(a)
4 F, _ /or
(3
2
266
A. Coull
'//tI/t
Fig. 5(b).
Ill
Kso', '~
-2
-1
0
1
-3
-2
-1
0
I
2
3
(b) o
O
k\ ~,
K= 5 0
o
O
O
O r~
z~
LO ~2 O~ :E
Fig. 5(c).
'\, \
32
\')),,,"
O
I
3
0 Shear
Stress
,,';
Fig. 5. Variations o f stresses with beam span : wall-height ratio for different wall : beam-stiffness ratios. Uniform load on wall.
I
4
0
Function
1
2
3
1/3 r d_ff
(c)
©
\
O
~%
.O
~
,
t
On
cr
~t
&
'\',,
0
e=
tO I 0
2
Fig. 6(a). K=50
\N
~,
\
K=2000
%\
i
I
-1
1
0
-1 Stress
© Function
Legend ...... -
-
-- ---
r=l.0 r=05 r = 0.25
(a)
1
%
267
Composite Action o f Walls S u p p o r t e d on B e a m s
ik
0 4-
iI
I I
l
I
i
0
O
'~, 3:
~
I£
2
'\Xi I
o
I
t
1
-2
Horizontal
_ ~ "
~ - -~- . ~~" ~ " ~-J--" -1
Stress
Function
/12
1
o
radaR
(b)
Fig. 6(b).
o
oI
""
' ID
~\
o
I~
N•
•
6
•
/t I
0
0.25 Shear
05
0
Stress Function
o~
0.25 1/ /12 rdR Z'S /
(c)
Fig. 6(c). Fig. 6. Variations of stresses with beam span : wall height ratio for different wall : beam stiffness ratios. Linearlyvarying load on wall.
5 and 6. The corresponding stress components may then be obtained by substitution in equations (8) and (12) for the symmetrical and anti-symmetrical cases respectively. The vertical and shear stresses at the interface between wall and beam are shown in figures 7(a) and (b) respectively, for typical geometrical properties, r = ½ and e = 1/12, for both symmetrical and anti-symmetrical cases. The solutions of equations (11) and (14) were obtained numerically by the use of finite differences. One the governing equations have been set up in
finite difference form as a matrix equation, the influence of different beam properties, which occur in the boundary conditions only, may be readily investigated, since only a few elements in the matrices need be altered. DISCUSSION In this paper, an attempt vide a simple approximate walls supported on beams. used in the assumed series
has been made to promethod of analysis of The number of terms for the stresses in the
268
A. Cou// ~0 i
~9
co r~
E 0 u
~r i
Uniform
loading
~c Q;
Linear applied loading
co co £.
%
i U
>
0
Span
.£ Legend 1
,'i
....
K=50 K= 2 0 0 0
~'=1/2 e = ~12
Fig. 7(a).
Fig. 7(b). Fig. 7. Typical distributions o f vertical and shear loads on beam due to uniform and linearly-varying loads" on top o f the wall.
wall has been restricted to enable all stress components to be expressed in terms of a single stress function, for both the symmetrical and antisymmetrical load cases considered (a uniform load and a linear form of loading along the top of the wall).
The latter form of loading has been included since the problem is encountered in the design of tall buildings, where shear walls are discontinued at first-floor level in order to produce a large uninlerrupted concourse on the ground floor. If the wall is subjected to an axial load and bending moment, which produce stresses of the same forms as the applied loads, at a known distance from the lower edge, the method of analysis may be utilized to investigate the diffusion of stresses in the wall to a supporting beam and columns. The problem finally reduces to the solution of a single fourth-order differential equation (containing even powers only), which may readily be solved analytically or numerically to give the complete stress distribution in the wall and beam. With the simple stress polynomials used, the horizontal and shear stresses have the same form at all levels in the wall, which does not seem likely in practice. The method is put forward as a quick and simple technique for determining approximately the stresses in such structures. The accuracy can always be increased by using more terms in the assumed stressseries, but at the expense of the extra computational difficulty involved in the solution of the resulting set of simultaneous differential equations. For simplicity, the analysis has been restricted to vertical loads only. Horizonlal loads may be considered by incorporating these directly in the assumed series (cf. Ref. 3). By utilizing the conditions of equilibrium at the interface, the loads on the beam may be expressed in terms of the vertical and shear stresses at the lower edge of the wall, and hence in terms of the stress function evaluated at this lower limit. By this means, the stiffness properties of the beam appear only in the boundary conditions for the governing equations, which are obtained on minimization of the total strain energy of the system. Since theJorm of the loading is fixed, the supporting structure may be analysed separately, enabling the method to be extended readily to deal with other support conditions, such as built-in beams, portal frames, etc. The influence of frictional forces at the supports could also be included if required. A comparison has been made between the results of the present theory and those given by Rosenhaupt[2], using a finite-difference method of analysis, which is the only other solution known to the author. Rosenhaupt, in his study of walls subjected to uniform loads along the upper edge, made the assumption that the bending rigidity of the beam is negligible compared to the stiffness of the wall, which effectively restricts normal pressures between the wall and beam to the region immediately over the support. This is obviously a rather extreme assumption, as shown by Wood's tests[l] on brick walls supported on reinforced concrete beams. Because of the different conditions at the lower edge, the stress patterns are quite dissimilar over the lower half of the wall, although reasonable agreement occurs in the upper regions. The present theory indicates that, in the central
Composite Action of Walls Supported on Beams
region, tensile forces exist between the wall and beam [cf. figure 7(a)], under the action of a uniform loading on the upper edge, in the absence of any gravitational effects, or frictional forces at the supports. The method could be used to give a first approximation for a purely numerical solution devised to take account of separation occurring between the wall and beam. For a uniform load applied to the top of the wall, the variation of the three stress components, a~, ay and z~y, with the beam semi-span:wall height ratio, r and with the beam depth : semi-span ratio, e, are shown in figures 4 and 5, respectively, for typical ranges of the parameters concerned (r = ¼, ½, 1, and e = 1/12, 1/9, 1/6). In each case, the variation is shown for two values of the relative stiffness parameter K
of 50 and 2000, representing approximately two limiting cases of a lightweight concrete wall and a reinforced-concrete wall on a reinforced-concrete beam. Although the results have not been shown, calculations were also performed for a value of K of 500; in that case, the results obtained were fairly close to those for a value of K of 2000. For the anti-symmetric linear-load case, the
269
variation of the stress components with the wall height:beam-span ratio only are shown, for the same two values of the relative stiffness K. The simple theory indicates that the magnitudes of the wall stresses are affected much more by the wall height :beam span ratio r and the relative w a l l : b e a m stiffness K, than by the beam depth: wall height ratio e. Comparatively small changes in the stress levels were obtained for the range of values of e investigated. The distribution of stresses a~ and hence the vertical loading on the beam, was found to vary only slightly over the range of parameters considered. The influence of making the assumption that there are no shear stresses at the w a l l : b e a m interface has been investigated, and some results are shown in figure 4. (For simplicity, the corresponding values are not shown in figures 5 and 6, since this would double the number of curves plotted.) The results indicate that the assumption gives reasonably accurate results provided the w a l l : b e a m stiffness ratio K is high (i.e. > 2000). It is of interest to note that, for walls whose height is greater than the span, practically no diffusion of stress occurs in the section above the unit d e p t h : s p a n position. This effect was also noted by Saad and Hendry[4] in connection with the gravitational stresses in deep beams.
REFERENCES
1. R.H. WOOD, Studies in Composite Construction, Part 1. Research Paper No. 13, National Building Studies, H.M.S.O. (1952). 2. S. ROS~NHAUPT,Stresses in Point Supported Composite Walls. Proc. Am. Concr. Inst. 61, 795 (1964). 3. A. COULL, Stress Analysis of Single-storey Shear Walls. Cir. Engng publ. Wks Rev. 60, 1044 (1965). 4. S. SAADand A. W. HENDRY,Gravitational Stresses in Deep Beams. Struct. Engr, 39, 185 (1961).
Ce document prdsente une mdthode approximative d'analyse de murs dpais agissant compositement avec des poutres de support. Les efforts sur l e m u r sont ddcrits comme des sdries de pressions horizontales, les coefficients dtant uniquement en fonction de la hauteur. Apr6s avoir satisfait l'equilibre et les conditions de limite pour le tour ainsi que pour la poutre, les coefficients restants sont ddcrits par la minimisation de l'dnergie de contrainte du syst~me. Le probl6me se rdduit/~ la solution d'une dquation differentielle de quatri~me ordre, soumise ~ des conditions de limite donndes, l'influence de la poutre de support apparaissant uniquement dans cette derni~re. L'analyse est simplifide en divisant le poids appliqud en des composts symmdtriques et antisymmdtriques. Des poids de gravitation sont dgalement inclus. Un certain nombre d'exemples de courbes est donnd afin d'indiquer l'influence sur les contraintes, des divers param6tres gdomdtriques et de raideur impliquds. Es wird eine ann~ihernde Methode fur die Analyse von Tiefw/inden die mit Stiitzbalken zusammenwirken gegeben. Die Beanspruchungen der Wand in der horizontalen Richtung als Kr/iftereihen ausgedruckt, wobei die Koeffizienten ausschliesslich Funktionen der Htihe sind. Nachdem die Gleichgewichts- und Grenzbedingungen fiir Wand und Balken erftillt sind, werden die iibrigen Koeffiizienten durch Minimisation der System-Spannungsenergie bestimmt. Das Problem wird durch die L6sung einer Differentialgleichung vierter Ordnung in Abh~ingigkeit von den Grenzbedingungen gekl/irt; der Einfluss des Stiitzbalkens erscheint bloss in den
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Grenzbedingungen. Die Analyse wird vereinfacht indem man die angelegte Last in symmetrische und antisymmetrische Komponenten aufteilt. Schwerkraftslasten werden auch berticksichtigt. Eine Reihe von Kurven ist gegeben um den Einfluss der verschiedenen geometrischen und Steifigkeitsparameter auf die Belastung zu zeigen.