NATIONAL BUREAU OF STANDARDS REPORT

4037

SHRINKAGE STRESSES IN MASONRY WALLS

PROGRESS REPORT NO. 2

by

Sven D. Svendsen* D. Watstein and S. Levy

Report to

Office of the Chief of Engineers

Department of the Army

Bureau of Yards and Docks

Department of the Navy

and

Assistant Chief of Staff

Department of the Air Force

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS REPORT

ms PROJECT SIEPOKT

1001-10-l|8l2 April 8, 1955 1+037

SHRINKAGE STRESSES IN MASONRY WALLS

PROGRESS REPORT N0 o 2

by

Sven Do Svendsen, D. Wat stein and S 0 Levy

To

Office of the Chief of Engineers

Department of the Army

Bureau of Yards and Docks

Department of the Navy

and

Assistant Chief of Staff

Department of the Air Force

<nbs>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication,

unless permisslo

2S,D,C. Such |

tally prepared I

Approved for public release by the ln p

8,, « 11 prohibited

Director of the National Institute of standards, Washington

Standards and Technology (NIST) report has been ipecm _ , ~ ,

‘ report for Its own use, on October 9, 201 5.

SHRINKAGE STRESSES IN MASONRY WALLS

PROGRESS REPORT NO, 2

by

Svend D. Svendsen, D, Watstein and S, Levy

Abstract

The current progress report presents the development

of the elastic theory for stresses in a wall which

undergoes shrinkage while its lower edge is restrained

from shrinking by a rigid footing. The solution obtained is represented by an infinite set of equations.

In the numerical solutions presented 9 the finite number

of equations and coefficients selected in each case was

such as to give adequate agreement between the given

boundary conditions and the computed boundary stresses

and displacements.

The numerical solutions were obtained for a wall having

a length of twice its height. Work is in progress for

walls in which the length and height are equal and in

which the length is four times the height. These solutions are based on the assumption that the footing is

sufficiently rigid to prevent all shortening along the

attachment line, A numerical solution is also being obtained for a wall having a length of twice its height,

with non-uniform footing restraint.

The theoretical results are being checked by noting

stresses in plaster and concrete models. The work thus

far has been confined to walls having a length of twice

the height, with two different degrees of restraint provided by footings of different sizes. The agreement between the theoretical and observed values of principal

stress at several selected points was fair. It was noted

that the agreement between the observed and computed

stresses improved for points further removed from the free

edge of the wall.

Further experimental work is being planned with models of

concrete containing internal strain gages to determine

strains in walls undergoing actual shrinkage.
2

1. INTRODUCTION

A newly constructed masonry wall contains as a rule a

considerable amount of moisture * As the wall dries out, it

undergoes shrinkage whose magnitude depends on the type of

construction and the properties of the materials used, A

similar tendency to shorten is caused by thermal contraction

when the temperature on one or both sides of the wall decreases.

In most cases, however, the wall is only a part of a

larger structure and hence is not free to contract. It is

normally restrained by other parts of the construction, for

instance by adjoining walls and floors, and especially by

its own foundation. The foundation, which can be a basement

wall or a footing, is usually more rigid and has less tendency

to contract than the wall itself. The restraint, the magnitude of which primarily depends on the rigidity of the structural parts involved, causes stresses in the wall which

eventually may result in cracks.

The investigation described in this report is the first

part of a study of the distribution and intensity of shrinkage stresses in contracting masonry walls restrained from

shortening at the foundation level. The study falls in two

parts s

(a) An experimental investigation of the stresses

carried out with wall models by the Structural Engineering

Section.

(b) A theoretical analysis of the problem and development of mathematical expressions for the stresses in the

walls. This analysis is being carried out by the Engineering Mechanics Section.

The first steps of the study are described in Progress

Report No. 1, dated June 18, 19

2. EXPERIMENTAL TESTS WITH WALL MODELS

The shortening of walls due to shrinkage and thermal

contraction was assumed to be an entirely elastic deformation in this study. As the models for practical reasons

have to be made considerably smaller than the prototypes

(a scale of about 1/5 is used), it was necessary to employ

materials with a high degree of homogeneity and elasticity

for the models.
3

In order to simplify the tests , they were confined to

those concerned with shrinkage and thermal contraction alone*

No vertical loads which would represent the loads from floors

roof and walls were applied* It was assumed that the effect

of such loads can be estimated and the resultant stresses com

puted through the principle of superposition.

It was further assumed that the foundation is very rigid

in comparison with the wall and that the bond between them is

sufficiently strong to prevent displacement of the wall relative to the foundation.

2*1* Materials and description of models

The first models described in Progress Report No* 1 were

made of a special rubber having a modulus of elasticity of

about 650 psi and a Poisson* s ratio of 0*5« The tests with

these models, however, did not give satisfactory results and

the rubber was accordingly replaced with plaster of Paris and

concrete in all subsequent work*

The plaster of Paris used was a high strength gypsum

known as Hydrocal. The proportions of the mix used in the

models reported herein, were 2*5 parts of plaster to 1 part

of water, by weight, and the compressive strength was about , L|.000 psi. This mix had a modulus of elasticity of 2.15 x 10°

psi and a Poisson's ratio of 0.28, as determined with sonic

resonance tests on 2=- by [}.-in. cylinders* Some preliminary

tests were made with other mixes ranging in proportions from

2.25-3.0 parts of plaster to 1 part of water. The values of

Young's modulus ^determined for these mixes varied from 1*7 x

10° to 2*3 x 10° psi and Poisson's ratio from 0*17 to 0.28.

In order to diminish the expansion of the plaster during the

setting time, some mixes were made with various amounts of a

special retarder added.

Two different mixes of concrete were used for models,

both made with a high early strength portland cement. One

of the mixes was proportioned, by weight, 1?2. 5^2.0 with

maximum size of gravel of 3/8 in .

;

the net water-cement ratio

was 0.58 and the slump in. Determined on three months old

6- by 12-In. cylinders, the compressive strength of this mix

was 8160 psi, the modulus of elasticity 5.82 x 10° psi and

Poisson's ratio 0.19. The corresponding values for the other

mix were? proportions 1 s3«1s 2.6 w/c ratio 0.53* slump

1 l/l| in., compressive strength 5800 psi, modulus of elasticity 5«67 x 10° and Poisson's ratio 0.16. This mix also
kcontained Vinsol Resin in the amount of 0.006 percent by weight

of cement

.

The form and size of the concrete and plaster models are

shown in figure 1. To prevent curvature of the line where the

wall is attached to the foundation strip during the test, the

specimen was made symmetrical with a wall attached to each

side of the foundation strip. The size of the walls was the

same for all models, the length being twice the height, but

the foundation strip was made with two different cross sections,

3 l/2 in. by 4 I/I4

. in. and 6 1/4 in. by 4 1/4 in. The rubber

models had the same form, but they were smaller, with 18 - by

9 “ by 1 -in. walls and a 2 - by 4“ by 22 -in. foundation strip.

2, 2„ Test methods

It can be readily appreciated that a test procedure for

determining stresses in masonry walls caused by actual drying

shrinkage poses a number of experimental difficulties and takes

a long time to carry out. For this reason, the experimental

work in this study was confined to models in which shrinkage

stresses were simulated by a suitable loading procedure. The

analogy between the two testing methods is shown in figure 2 . If the wall is placed on a footing which imposes no restraint

( 2 a), the free shrinkage will cause a contraction which in an

exaggerated manner, is indicated by the dotted lines. A very

rigid foundation, representing 100 percent restraint, will

correspondingly cause the wall to deform in a manner similar

to that in 2b. If the foundation of the wall 2a, which was

permitted to shrink freely, is now replaced with a rigid one,

and this foundation is elongated by applying a tensile load,

the result will be as shown in 2c. For a correctly chosen

load, the longitudinal strains in walls 2b and 2 c will be

equal and the stresses in every point of the two walls will

be the same.

In view of the relatively low extensibility of plaster

and concrete, the models were actually loaded in compression

instead of tension. A high compressive load was applied and

the corresponding strains noted at various points were assumed

to correspond to 11 zero stress’ 1 in the wall. Upon releasing

the load until only a nominal load was imposed on the footing,

the strains were again noted. Under the assumption that all

deformation were within the elastic range, this procedure gave

the same strains and stresses as a tensile load of the same

magnitude as the difference between the initial and final compressive load.
5

The strains and stresses in walls and foundations were

determined with bonded wire strain gages attached to the surface* Both single gages and 3 element rectangular rosettes

were used, the latter giving the magnitude and direction of

the principal stresses* Figure 3 shows one of the plaster

models with some of the gages attached , placed in the testing

machine

.

The rubber models were tested with a tensile load applied to the ends of the foundation* The strains were determined by comparing the dimensions of a rectangular grid on

the strained and unstrained wall* These tests did not give

satisfactory results , partly because the modulus of elasticity of the rubber to a large extent varied with the stress

history of the specimen* Another source of error was the

fact that the grid did not provide markings which were precise enough or well enough defined* The fact that this test

method gave no information about the shearing deformations,

was also a serious disadvantage*

2*3» Wall with 3 l/2- by 4 l/i^-in* foundation (type I)

Several models with a foundation of this size were made,

all of them of plaster of Paris, but only one of the specimens was given a complete test* Figure I), shows the distribution of bonded wire gages on this model, each station representing a gage on both front and back sides* Measurements

were only made on one of the two walls, the several gages on

the opposite wall being used only to check the symmetry of

loading*

Figure 5 gives curves showing the distribution of the

horizontal strain, averaged for both faces of the wall, in

planes 1 to 6 (figure l\.) for a load increment of 19*500 lb*

Curves 1 and 2 show the deformations of the foundation at

its centerline and at the top 3/1+ in* from the face of the

wall. The strains in these two planes agree surprisingly

well but they show that the wall was only partially restrained by the foundation from shrinking, A restraint of

100 percent would have produced uniform strains along the

entire length of the foundation, while in this case the

strains near the center of the wall were only \\$ percent of

the strains at the free ends* The strains given in figure 5

show that the non uniform strain distribution, although quite

marked, persisted only for a short distance from the corner.

The strains 1 in* above the foundation, represented by curve

3 were almost uniform over most of the length. The large
6 , difference between curve 3 and curves 1 and 2 near the ends

of the wall indicates , however^ that the shear stresses in

these areas must be very higho

The magnitude of the strains decreased rapidly with the

height above the foundation, and the variation of the strain

near the ends became less abrupt,, The top of the wall had

no contraction at all and a small expansion was recorded

near the centerline (curve 6),

Horizontal, vertical and principal stresses were computed for points A to G (figure ^), The values given in

table 1 are these stresses divided by E xe m

where E is the

modulus of elasticity for the wall material and e the strain

171 in the foundation at the center of the wall* The values are

in this way made independent of the properties of the material

and (to a certain degree) of the rigidity of the foundation.

All calculations are based on four different tests made over

a period of six months, and each point represents the average

of four different rosettes; a is the angle between the horizontal axis and the principal tensile stress and cx/ the

angle between the axis and the principal shear stress. Tensile stresses were taken as positive, and compressive stresses

as negative.

The magnitude and direction of the principal tensile

and compressive stresses are also shown in figure 6, Even

though there are not enough data for a complete stress picture, the figure gives some interesting indications. The

principal tensile stresses are highest just above the foundation, decreasing rapidly with the height near the edges and

somewhat slower near the center. The stresses are nearly

horizontal at the centerline and approach an upward direction

with the distance from this line. If the stresses become large

enough to cause cracks, they appear first at the bottom of the

wall at right angle to the direction of the principal tensile

stresses,

2»l\.o Wall with 6 1/Ip- hy 4 l/lp-in. foundation (type II)

Attempts were made to increase the restraint in the wall

by increasing the size of the foundation to 6 l/Lp— by I4. I/I4. in.

Several models of this type were cast, both of plaster of Paris

and concrete with the size of the wall being the same as before.

This proved, however, to be a more difficult task than expected,

as most of these models developed cracks before testing or during the loading of the specimen. The cracks occurred most
7 • frequently at the bottom edge of the wall* but they also appeared in the foundation just below the wall*

In the plaster of Paris specimens the cracks were obviously due to the expansion of the material during the setting

period while the strength was still very low* This expansion

increased with the cross section and consequently was much

larger for the foundation than for the wall* This caused a

difference in strain along the boundary * and the wall had a

tendency to crack at the bottom edge where it was sharply restrained from expansion by the mold* Attempts were made to

diminish the expansion by using different amounts of a plaster

retarder in the mix* This did not help* however* as the retarder only postponed the starting time of setting without

decreasing the size of the deformation* Part of the expansion

in plaster of Paris was thermal* and accordingly an attempt

was made to cool the foundation with ice during its initial

setting period* In this way it was possible to prevent cracks

at the bottom edge of the wall* but the cooling seemed to increase the internal stresses and the tendency of cracking inside the foundation.

The cracks in the concrete models were probably due to a combination of thermal expansion during the first day's curing

in the mold and the shrinkage of the specimen after it was

taken out of the fog chamber. The thin wall dried out and

shrank much faster than the foundation* and the shrinkage

stresses in it developed in the same manner as in a restrained

prototype. As shown in figure 6* these stresses may cause

cracks which are almost horizontal at the bottom edge of the

wall below and to the left of point A. In order to decrease

the size of these stresses* models were made of a leaner mix

with lower w/c ratio and given a longer curing time in the fog

chamber* The results were somewhat better in that no visible

cracks were observed* but the strain measurements were quite

erratic and pointed to the existence of cracks in the interior

of the models*

Figure 7 shows the foundation strains for a plaster model

and a concrete model of this type* The irregularities of the

curves* especially the difference in strain between the two

ends of each foundation are probably due to internal cracks.

In figure 8* the degrees of restraint of the two types of

foundations are compared. The strain values are averaged for

the left and right side* and they are given in percent of the
8

strain at the free ends. It will be seen that the restraint

at the centerline of the wall has increased from ]_|_6 percent

for the small foundation to 65 = ?0 percent for the large.

The latter is probably still less than the restraint provided

by the foundation of actual masonry walls.

During the tests , some of the models were inadvertently

loaded to failure , with the rupture occurring in the wall.

The failure was due to cracks following one or both of two

different patterns. One type of cracking developed below

the wall and well within the foundation and was apparently

caused by tensile stresses both in the plane of the wall and

those forming a small angle with that plane. The other type

of crack followed an almost elliptical curve in the wall,

starting from the bottom corners . This type of crack can be

explained from the stress picture shown in figure 6. These

stresses are the result of a tensile load| a compressive

load would change the sign and thus give tensile stresses perTM

pendicular to those caused by a tensile load. A crack due to

compression of the specimen will consequently have a tendency

to follow the direction of the stress arrows in figure 6.

Both of these crack patterns are evident on the nearest

wall of the model shown in figure 9»

2.5. Comparison of computed and observed stresses

Both the theoretical and the experimental part of the

study are far from complete and furnish* at the present time,

no basis for a valuation of results. In table 2, however,

the theoretical and experimental stress values are given for

the three locations A, E, and G, (see figure 6), on a model

with foundation 3 l/2- by l\. l/lp in. Figure 10 shows a comparison of the size and direction of the principal tensile

stress in these points. The theoretical values are computed

for a completely rigid foundation while the restraint during

the experimental tests was only 6 percent. Even so, the

stresses and directions seem to be in good agreement for E and

G. The location of A is very close to both the foundation and

the edge of the wall, and rather large discrepancies were to

be expected for this point.
9o

2 060 Plans for future tests

As it seems to be very difficult to get a greater degree

of restraint by casting the specimens integrally with a sufficiently rigid foundation* other methods will be tried in the

future tests, A possible solution is indicated in figure 11 , The double wall and the foundation strip is cast as a slab

with uniform thickness and with bolt holes along the strip,,

When this slab is cured* the rigidness of the foundation will

be increased by attaching a steel bar to each side of the

strip and bolting slab and bars firmly together,, Special care

has to be taken to prevent any movement between slab and bars

during the test 0 By using this method* it should also be possible to determine stresses due to actual shrinkage , In this case internal

strain gages will be used in the wall and the steel bars have

to be attached while the slab is still saturated with moisture.

The shrinkage stresses can then be determined by readings of

the gages while the slab Is drying. Walls with different ratio

of height to length will be tested.

Several test specimens of the type described above were

cast and tests are in progress.

3 , ELASTIC THEORY FOR STRESS IN WALL

RESTING ON A FOOTING

The construction of the wall is assumed to be such that*

if it were unrestrained* it would shrink uniformly as it dries

out. It is assumed that the footing restrains the shrinkage

of the bottom of the wall. The equations will be derived in

sufficiently general form to apply not only to the case of a

footing sufficiently stiff to prevent all shortening and bending along the attachment line* but also to the case of a footing which is deformed by the shrinkage of the wall.

3,lo Equations for interior wall

The interior of the wall can be considered

tic slab which obeys the customary equations of

We then have the equation*

(See page 25 * dx

eq (26)*

-x ,2 4*A ,

- = dX*cfy*

of ref

6 O

1)

to be an elaselastici ty.

(1)
10

The stress function 4> in equation (1) is a function that,

in a system of rectangular coordinates x and y in the plane of

the wall, gives

crx

- = stress in x - direction

cJ” y

=6 3

(pjdj< 2 = stress in y

- direction

Txy ~ ~ (5

2

<})/dX. dy ~ shear stress

3o2„ Boundary conditions

The boundary conditions at the top of the wall, y

s b,

figure 12, are

foyW -o

( 2 a)

An artificial boundary condition outside the wall along y

» o is included to obtain a convenient separation of even and odd

terms in the final equations

foy )y-o (2b)

Along the sides x « o and x ® a, the boundary conditions are

(bx )/ = o, a ^ *

,) = o ( 3

)

Along the bottom of the wall, y

~ b/2, displacement boundary

conditions apply. For a footing sufficiently stiff to prevent

all shortening and bending, the conditions are

and (V) ^ 6/2 = O

( k )

(5)

where S/E is the jahirt" shortening/ due to shrinkage and E is

Young’s modulus. The vertical displacement is v.
11

3»3 Solution

No single expression satisfying equation (1) and the

boundary conditions , equations (2) and (5)* could be found.

It was decided, therefore, to obtain a solution in the form

of a series of expressions, each of which would identically

satisfy equation (1) and the sum of which could be made to

satisfy the boundary conditions (2) to (5>) to any desired

degree of accuracy. A comparison of the boundary values

computed from the solution with the given values could then

form a ready means of judging the adequacy of the solution.

Numerous expressions can be found to satisfy equation

(1). In order to have a rapidly convergent solution, it is

desirable that the expressions selected separate into groups,

each of which primarily affects the stress along only one

boundary. In each group it is desirable that the first

term give a general picture of the stress, the first two

give a somewhat finer picture, and so on.

The expressions selected for the solution apply every

where in the plane except at x s o, a, 2a, ..<><, and y - o,

b/2, b, ... and can be written as

(6a)

where

,
12

The k, JLf t, p, and s coefficients are determined by the boundary conditions., An alternative form which is useful for determining stresses and displacements in the region o<x<a, b/2<y<b

is

a = y JLt6 km \

£<r>h 2 y S/n mTT*

r l&ma { cosh a mff

A

. & cosA

nr' irb~

j a

M* /, 3... V sr) 9 f

t 3 s.. ^ 2CLCL 2a8d

+ T. f~

s'nA -h^ CosA

^

Q-^) }

~t~ ^ _JT*Q. J £/nh "C ZyL S/n/t ) \ S/n

' a 60*A

1 *

S/n ”2lDL

CL

8b

y- 4ZT-^ — /A y 7Ti - a /_ (a “*) /2y _ A

' /6 ~ Pm

co,*‘ mr£- P (T" )

/ 2cl SenA

' rmrb cost) snrrb.

cos* vshO-j )

co5/> rr^6- SZCL

sen mnjit.. cu

+ rzih. (i-tys-i)**, *PzVt±}] /6ma,

rr>: /} 3... Sin&y 2 mir6__

2o.

Sen

CL

(6b)
13 .

In equation (6a) and (6b) the summations involving the coefficients k

m

and.J?

m

primarily affect the stresses on the top

of the wall, the summations involving the t coefficients pri

marily affect the stresses on the sides of the walls and the

summations involving the coefficients pm and s m primarily

affect the stresses at the bottom of the wallo

The stress in the x direction is obtained from

( 7 )

as

Ox = r /ea* Sin mrrx fst/?A Sn?A zq (/

G> ] cosA s snrrA t> CosA m-rrS

BO. BCL

___ 4-g, cosA*2

^

(/--£*) }

'"V'A cosA 02ZZ& | ACL I

5 ZCA22A

m-t, 3 *.

.

g*

^

\-_siaA^L + 2*. cos*. 6- f)

I S/slAs smrA A $/nA niTTh

££l - ^

eT*b(/- s£-) I mrrb 3, //iA mTfb j BCL >

+ r

7r*-r/a-

/C> A 3 t

+ r 7T4- /6az m-/ 3„..

.

/°r

n sis) -—

gSin .sr>rrg_ cu

nrry

c 03A z /iTTa 2X Q-&)

cz ^osA S/Z&-

<c A

*91 if -cosA

co^A

*%%?(/ - i]

1 cogA cl

ag

u‘0' :t).

CO£A Cl20. rrA.

_ ASo. stnhcLXI

wrrh cosh

^+ y.JL-*m6- o £— /6a3 s'tnA

zCinA,

2.0.

((b"t)^A ^A-&fe~$*nhm

gr

i-(J£ ~k)

. _<:cl

/rtrrb

cosA

sv/rA fj$,. %)+%& co

^

( 8 )

m- >,3. -

.
11;,

The stress in the y direction is similarly

Oy = \ J^rn s/'n snZDL- f- 3tnA /J£P* _ ^y. b/nA lia^

y ^— /6a. 3 c<~

^ cos/ 3 /t/rrA A> c.osA aj jtA

m=/t 2 — Z-CL 2. cl

A y~ stn /r)/rjL.L^AaA IiSSL—^L co^A t^rT

3- (/-rs*’ ) ) /6 a.3 **-

j S///As op rrb b s/aA arrrb I ay-/, 3 ... ,2cl 2d.

/ X- 77*/)ci f a -/’/ry- /

b/I/A / . 2£_ <">/ ~yj

) /— /66> 3 b I cosA

2

cirra. ^ cosA aTCcq

/>-/, 2, 3} ... COSA A/Ca

2b

, A7/a

COSA) 26

///TO.

(/-¥)

cosA nrfcc

2 b

-f- }

Tit & s/n

/6a

2 ^

ni CL At=/f 3. . .

\rr,,.A $g*Q-f*) /4V _/)

1 cosA

3 ///orb v h ' cosA cr>rrb

1 2CL 2CL

2ct ^ o/aA 2a

a?/rb cosA //mb

20.

j- y

77 ^//y 6 g

<,//) ci

• aiax

/6 a3 m=/,3... salA 2 cm/6

2a

fjj l\ s/aA mTrb_ (_£_ _j/ \

\ b 2 ) a. ( 2 b )

( 9)
The shearing stress is obtained from

Tx Y = _

<5/c5 Y

( 10 )

as

TtXY = Y~ ir*/nh- y cos furry. (- cosb^a^ s cosh so. (/~2r) _ s/oA 2a 0 "if)

/6 a 3 rr> \cos,A £ninb_ 6 co^/i m/rb. rrirfb cash mrrb.

2CL 2 CL 2a

rrtffb casb rn rrb. 2d

„ . mKb-/, £¥.)

bci cnsb £,c>- w b / mrrb strib

fMTf y » w7fy~Jt) s. ^ v cosb_d -i- JLy~ sjob^Sd^ill^—

stnb£mr/£> b Slab mrrb.

f dim-/, 2X\

. . y /.^ 2X))

/ > jrfnd_/ rr * a/rr

j

-casbb _j_ 2X cosA £6 (/„*. I— 2b s/f?A.^_6

' /6 6 s b

I cosbs arm a. cosA mra nrfa cosA niro— I r) - /, 2t 3 • • * ' 2b 26 2b >

/. Y-Z± ( s/nA^a* (~t ~f) ^4?*_/) (/-&)

) /6 a

z & ( cosA & mrrb & ' cosA d2JZ£- f m = /t 3 20. 2a I

((%-£) cos/i

^

(§-£W-i-& 7T*/rt6

/6 a 3

m =/, 3 . . .

/vrrx.

cos <2 Stab 3mV

6

2a

~Q— Sir)A

mrrb

mzzb-. (-JL

) ~ Ak~ sinA -d/zA (JL

a ) \ g, 6 / mrrh & K t> 2 /
16

At the lower edge of the wall, it is desirable to know

the vertical displacement v. This was found by substituting

the value of <p from equation (6b) into

V =

(12)

where E is Young's modulus, P is Poisson’s ratio, and the integration function has been taken as zero since it merely represents a rigid body rotation* The resulting expression for

(v) k/2 ab the l°wer e(^ge °T the wall is

(V) y= b/2

/-^i> 7/ 36 sin '22££-{^Qs/i /~J? I £ — /6 0 2 Q- \sinA 2 mrr£> /-/-)) mrrt suoA mm

aa

/6 0‘

/j3..

.

2d 2a.

+4L 77 3a

S66*

n- 2,#.,.

/ COS -^22ZL ./2H6

Sis?A 2> cosA £ /rrra

JjL.

a

. U2L£- /, 2K \ Si/?A 22 ( / a ) 2 0

CO5A

-A-JZA-

' nrra

flTTa _ / , 2.7 \

cosA 26 \ ! a )

cosA s?rrcL 26

+ A- -£L_ ^

st-„ ^

+

/

sn- /» * ' * • 7“ *r*6

/6o z

• <222^- Stsi O- St/)A j22/E£L

2a.

/ / ~7~ \) ~7~ mrrS

s/nA d?rr6-

m -

/, 3.

( 13 )
17

The strain at the lower edge of the wall is found from

the general equation

**~-k ^-^y) - f (£*--

* ff) U4)

Substituting the value (j) from equation ( 6b) into equation

(II4.), the strain (€x )y„ b/2

al°ng lower edge of the wall is.

/2220L

a

4-a

nirrh cosb norrb

~Za3

2Ltna-

/6

# -

.< 3. ,

,

fn

• ,/3*DL 5/nn £»Zb '

. . s

jstnA 2b C / a. cosA nrrcL _ 2b

*/]?& cos/) WO

/?7Ta. cosA c>7Tci_

Zb

+±L3Z1

£: /6 a2 Prn S/s? 6TM32L

CL

(/+?)

cosbs snub

2a.

0- 9) S/siA

rrmb cosA sntrb

2/x

miiL

2a.

1

+

~E

^ -31:, 3a_a Sen S/s? 222ZZ2La

tn= /} 3.

Os)
18

The boundary conditions ((X-) v«.^ - - o, equation (3), A. X— 0^0

(CT ) = <?, equation (2a), and (O' ) = o* equation (2b),

y y- d y= o are automatically satisfied by the expressions used in either

equations (6a) or ( 6b )

„

Substituting the appropriate value

for (T ) in the second of equations (2a), expressing the

xy y=b

resulting equation as a Fourier series involving cosine terms,

and equating the coefficients of the cosine terms to zero

gives a set of equations involving k, A, t, p, and s coefficients. Repeating the above process with the second of equations (2b) and combining the resulting equations with those

obtained earlier gives

0 • s- *>">

n odd

mn />;

(l6)

and,

°= ii*r te ff4 ' COsecA*^gpl Jrr, -h -*§£ cosect^gi-

/

/— cosA /?77r6

za

-j- cosecA 2 Sm 5

I efayLHtf)

ct even

(17)
19

When the boundary condition in the second of equations

(3) is similarly used one obtains from those equations corresponding to even cosine functions , n even,

° = tn-t- A/??=/, 3... ('no -/ a a

( 18) and from those corresponding to odd cosine functions , n odd,

o = tr±aL l-ZA.

/66 s \nrra

sec/i 2 /y7rp- 26

^ 6b \^n+ /- ' /M- /

[r) /Cr> ^ C l) £

/n 4rr. I

07 -/, 3 ...

(19)

When the boundary condition in equation ( Ip ) is applied

and the resulting equation* including the constant term S/E,

is expressed as a Fourier series involving sine terms,

equating the coefficients of the successive sine terms to

zero gives

4S _ j / ,

,

.i) 3/nA fq~~ 4

&

"> 7r /6a 3

] W ' cos.6

z /7)7t6 TMn6 cos>/? /nrrt m 7r 3 $TM

4a.2

.{2±L , , -/) s 7/ £/a/n

(m£/6

/f) z /n z O \

r? a )

(20)
20

When boundary condition equation (5) is used with equation ( 13 ) and the coefficients of the successive sine terms

are equated to zero one gets

mn6 / -) \ L. 2a O-V)

/? 2

/?/7T& rmfh £/r>/) /rjrrt>

&CL SCI

4m*.6 /6a*

f s/o

7/36

/6 a

3

/+)?

sia/i e m?r6

SCI

j- 2d 0-$} /rjJT^ 1 V- 5 f-tj* fa2

cl ~ #//*'§')

7 r»rr6

COTM 2* /^^Z_ na(ma£+/i*&.y

ri = 2, 4-..

It should be noted that each of equations (16) through

(21) represents an infinite set of equations * With a finite

number of coefficients, only a finite number of these equations can be satisfied* In the subsequent numerical solutions, we will include, starting with the lowest, as many of

equations (l6) as we have k coefficients, as many of equations

( 17 ) as coefficients, as many of equations ( 18 ) as even

1 coefficients, as many of equations ( 1 9 ) as odd t coefficients, as many of equations (20) as s coefficients, and as

many of equations (21) as p coefficients. The resulting

solution, of course, will not satisfy the omitted equations

and to that extent will not satisfy the given boundary conditions* By including a sufficient number of coefficients,

however, the difference can be made quite small* A check on

this point can readily be obtained by comparing a plot of

the given boundary conditions with that of the boundary conditions satisfied* Where the difference is negligible from

an engineering point of view, the solution is adequate.

3* Ip, Numerical examples

The computations were done both with conventional handcomputing machines and, where it seemed advisable, with SEAC,

the NBS high speed electronic digital computer. SEAC was

particularly useful in solving the large sets of simultaneous

equations for the values of the coefficients*

Poisson’s ratio has some effect on the answer because of

the displacement boundary conditions at the bottom of the wall.
21

Experience with other related problems indicates that this

effect should be small. In the solution given here the

value l)=0o3 was used.

3*4-. 1* Wall with length twice height

For such a wall, a r b, two solutions were obtained. In the

first of these, 9 coefficients were used k

l* &!> t l

? t29

/, , p-,, s , s , and s^. The corresponding nine equations are

given in table 3 together with the values of the coefficients

resulting from the solution. In the second solution, i|0 coefficients were used. Their values are given in table Lj..

The degree to which these solutions satisfy the given

boundary conditions can be judged from figure 13? for the free

boundaries x- 0, x= a, y=b, and from figure 1

4

- for the restrained boundary, b/2» The agreement appears to be excellent for both solutions with the 4.0-term solution giving some

superiority close to the corner attached to the footing. The

agreement for the IqO-term solution at distances from the corner

exceeding 7 percent of the wall height is within 20 percent of

the shrinkage strain S/E for the strain along the footing,

within 3 percent of the shrinkage shortening Sa/E for the vertical displacement at the footing, and within 13 percent of

the shrinkage stress S along the free walls.

The shearing stress in the wall at the footing for the

14. 0-term solution is shown in figure 15» This stress is zero

at the center, positive on the right half of the wall, and

negative on the left half. In the center half of the wall,

the stress varies nearly linearly. In the end quarters, the

stress increases somewhat more rapidly and then in the last

10 percent of the wall length drops to a low value. The maximum shearing stress is less than 0.9S.

The stresses at three locations near one corner of the

wall are given in table 3 (a) for the 9"term solution and

3 (b) for the l^O-term solution. The maximum stresses for the

two solutions agree within 3 percent for the point at .0312a

and within 1 percent for the points at .14.03a and . 24- 95a. The maximum shear stresses in the two solutions agree within

24- percent for the point at .0312a, within 20 percent for the

point at „l4- 03 a, and within 7 percent for the point at . 24- 95a. The directions of the maximum^ stresses agree within 29° for

the point at .0312a, within 1° for the point at . l4-03a, and

within 5° for the point at . 24- 95a.
22

For all three points of the lj. 0 - terra solution and for the

two points farthest from the corner in the 9 -term solution,

the minimum stress is much smaller than the maximum stress in

absolute value. In no case does the stress exceed the shrinkage stress S in magnitude. The maximum stress is directed

approximately towards the bottom corner of the wall,

3. 4® 2, Wall with length twice height,

non-uniform footing restraint

In the tests it was found that the strain along the footing was larger near the wall ends than it was at the center.

This was attributed to the use in the tests of a finite ratio

of footing cross-sectional area to wall cross-sectional area

rather than the infinite ratio in equation (Iq), The distribution of (£ ) in the test is shown in figure 16, The

y-b/2

numerical solution with 4O coefficients is being repeated with

boundary condition ( Lj. ) replaced by the curve in figure 16, The

computation is about one-half complete,

3 . 4 ® 3 ® Wall with length and height equal

A numerical solution is being obtained for a wall whose

length and width are equal, b- 2 a, The solution makes use

of 60 coefficients. The simultaneous equations for the coefficients have been coded for solution on SEAC, The work in this

case is therefore about one half complete,

3.1|.l4-» Wall with length four times height

The work on this wall has been started and is about 10

percent done.

REFERENCES

1, S, Timoshenko

Theory of Elasticity

McGraw-Hill Book Co,, 1934®
Table 1 . --Principal stresses in points A to G

Plaster Model No. 1 (Type I)

t 3

3 Point 3

! 1 Ox I ! Txy

3

1 Oc

°

1

1 (Tmax

?

? 0

"'/r>//?

t O

; 0,

1

t T max

j b. * 6 m 1 ! * (z n

1

viz * £ rr> \ c. * (z m

t 1

3 A «

t 3

• 445

1

3 .788

3

t

3 . 656

1

3

1

? 51.3

1

3 1.290

t

t

3 - . 066

1

!

3 6.3

3

1

3 . 678

?

t t

3 B 1

» ?

. 91+6

3

1 .054

3

!

3 459

f

1

1

! 22.9

3

3 1.138

3

3 - . 142

!

V

3 -22.1

!

!

3 .641

3

? i

? c

1

? t .878

3

' -.106

i

3

3 .280

3

1

1

5 15.8

i

3 ,978

!

t

3 -.183

s

J

3 -30.2

1

I

3 .581

!

! i

3 D 3

? { .013

j

3 .039

j

1

3 . 06Ll

?

3

)

f 51.0

1

3 .092

t

3

3 -.040

3

J

3 6,0°

1

3

3 . 066

t

r t

• E !

t 3 .308

i

3 „1Ll9

i

1

3 .256

f

3

J 36,4

t

3 .496

t

r

3 - . 042

3

!

3 - 8.6

t

t

3 .269

1

i j

t p t

t i -.012

1

3 -.077

!

f

3 -.005

1

t

1

! 85.1

3

3 -.012

1

1

3 -.077

1

t

3 40.1

1

1

3 .033

1

r t

' G 3

] i .219

!

3 >050

1

i

3 ,100

r

3

!

1 23.4

1

3 ,265

1

t

3 .002

3

1

3 -21.6

t

1

3 .131

1

i

t

i 1

3

!

"f

3

t

t

r

T

't

1

!

'l

r

j

't

T

t

' 3

1

1
Table 2 . --Comparison of computed and experimental values

Plaster Model No. 1 (Type 1)

' Method

Point' of

t t Ox O' Y , Txy

A

E

» determine * ,£"*£/* , CX

5 ~ t U max , \J s?7/n

x e * 6‘ cx,

, T/?7£?X

t J ? ? ! ! J t

t 9 -term » * » t ? ? t 'solution ?s <>256 » » 39 3 ? » Ip 6 ^ ? 49.19 t • 794 ? - . 145 4-19

t ! 1 5 I t

, 40-term ,, ,

0 ,

j solution , t . 042 } o733 ,»160 f 77*56 ?

. » 769

? 006 32.56

6.30

t ft t ? 1 1 t

1 9 - 1 e rm , , , 5 , , t

t solution JJ.269 5.184 t «197 ? 38.96 , . 428 , .025

T

" Yt t s t ”” »*“

t “

- 6.04

40 -term , isolution 55.230 jd39 5.237 39.59 J .426'-. 057 5.41

.470

.381

I ! ! f I ? J C

r Exp e rim. 1

5

.445 '.788 '.656 ' 5l.30 '1.290'-.066

t ? f ? J ? t ! 678

202

.241

1 ? ! { ? J J ?

1 Experim. s

t

. 308 i . 149 '.256 » 36.40 ' .496t-.042 •1 8.60

J ? .269

G

9- term

j j

t !

j t TT

t !

! t t

j solution , 5.183 ,.059 ,.087 , 27.31 5 . 228 { .014 -17.69

I

, 4°-term , , ,

, solution ,j.l94 5.036

? “ " cl —p—

t Experim. t

? .219

t j j

s

t _ I !

, .082 , 23.10 ,

, 229

1

.001 21.90

t ? 050 ' .100 'I 23.40 ' . 265 ' .002 - 21.60

.107

114

131
Table 3. —Wall with length twice height, first solution, 9

coefficients

.

0 s 0.4£8l39k1 y 0o25t 1

y- O 2251+70p 1

0 * 0<,2^k1

-h OJp58l39t 1

y- 0,25px

/- 0.03p

3

0 * 2 <>[]_67l|01p 1

y- lo56886l7 1 y 0.012^66t2 y-1.4l7290s 1

0 = 0.27lpL56p

3 - 0.217495t 2

- 0.288723s

3 o = 0.155848^ -y o.o4t

2

- o.o6o8o7s 1

0 = 0.08^ y 0.200208t2

- 0.08s 1

- 0.035503 s 3

- 0.011891

1.273240Sa2 = - 0.196395^ - 1.727l8lt

1

- 3.745349p x - 7-75l569s 1

0.4244l3Sa2 = 0.503350^ - 1„046393p

3

- 7-75l569s

3

0.254648Sa2 = 0.474500t 1

- 7.75l569s

3

EQUATIONS

SOLUTION

k

x/Sa2 = -0.049880

/-j/Sa2 = -0.071428

t-j/Sa2 * -0.071269

t 2/Sa2 = -0.077826

P x/Sa2 = 0.180376

P

3

/Sa2 3 0.000905

s-j^/Sa2 3 -0.234265

s 3/Sa^ = -0.059486

s^/Sa2 s -0.037214

Remaining coefficients taken as zero.
Table Li.,- -Wall with length twice height , second solution, I4.0 coefficients

.

k

1/Sa2 = “0 059389251 t io/Sa2 = -.011612346

ko/Sa2 = .012303806 t 12/Sa2 = .008939071

k^/Sa2 - .003772472 t^/Sa2 ^ -.005983598

k?/Sa2 = .001304913 t

l 6/Sa2 = .004767099

ip/Sa2 = -.075968510 P1/Sa2 = .188439011

/3/Sa2 = .010713538 P

3

/Sa2 = -.003551489

i^/Sa2 = .005101076 p^/Sa2 = -.163158134

/? /Sa2 = .001912432 P7

/Sa2 = -.354876034

t-j/Sa2 = -.061716726 P 9/Sa2 = ".555787489

t

3/Sa2 = -.000280045 Pn/Sa2 ~ -.759714680

t^/Sa2 = .027934929 Pl 3/Sa2 = -.963912395

t 7 /Sa2 = -.024216724 Pl5/Sa2 = -1.167365113

t

9/Sa2 = .022785816 s 7/Sa2 = -.242707516

t lx/Sa2 = -.018657394 s 3/Sa2 = -.063090690

t 13/Sa2 = .015630606 s^/Sa2 = -.029900727

t 15/Sa2 = -.012683033 s 7

/Sa2 = -.012157563

t 2/Sa2 = -.082028509 s 9/Sa2 = -.001303778

t^/Sa2 = .059449535 s n/Sa2 ~ .005902147

t 6/Sa2 --= -.025981647 s 13/Sa2= .010951224

tg/Sa2 = .019490944 Si5/Sa2 = .014630406

Remaining coefficients taken as zero
MD ! 1—1 o

O' -d cn in -d" o O 1 1—1 o vO o o o o o 0 o S '* 00 in I—1 s 6

-d v£> P P Ph

w on CVJ 1—1 e aj 0 8 1 d d 0 0 p © - °- — — o — “ «- c- <- & CO - — — — O. <» — —1 — X £ P © co m co p i—1

pd

K Pi CO o OJ S \ co pd

p

CO to \ o- o o V- on CO i—i S V- -do oo0 f—1 o ° 0 - - - - - - - - ~ - - - - pd 1—1 o (—i

-d cn o pd pd cn o CO o UD o o o (h o o o e v

C\J o vD

©

ra

d (j o p CVJ 3 Cs 1—1

in vO c Pd in vO & VJ 1 8 ! d © £ II ii 0 03 °H «» — — o- — —

—

O to

p ~ — O.

— c —

———

© c o to d m p CO © p in o TO g in u d S co ; in pd pd © CO O vO in

cap pd OJp

u \ vO sO o ©C b i—i o oo ^ to OO

ino oo p 8 o o 0 TO 00 - - - — c- O' =- ~ <=* — — p 0 a a cO » ~ c- o- c- o~ — — —

—

O

G

p 0 vO I—1 c fd

•H

O

O

Q i—1 O on o P UD o o P P C ^ * o o o O O in in i—i

CO P p s -- cn 00

p P £ . - o o o o d d d x ~d on nj d p 6 V IP cn on O r—H

fd S 0 p ^30

p on CVJ i—1 O p - * — — o. <=- =. C= —< o P-, & U

TO

X

m •rH » —

—

—, — «-

—' —

© co co pd CO aoX

© S t V

Pq o OJ OJ S & cO cr vO o d d K p pd CVJ so g,\ vO OJ CO

<D v/ « o o ©

K P pd CO

p

p1 p

1) VJ o o o P o o c- - «, =_ - —

—

co CO OJ pd \ in P P co on

CO --~- vO cn CC «. S>5 o p co © cO x pd p o oo CO

CO ' V- o o 0 X

p oo o

co©(• - - - - ~ - - - - V“' o o Mp

03© CO o 03 co ra \ on pd o ©

p i to tn> o CO in TO CO on cn in i © L_, d O

on f—1 o TO \ on on on • o o o © ki p i

i o

in©i—i P

CO - <- - - «- - » - - - £ b

co f - ° o o

co CO

m

p \ in vO oo in

CO

X nj CVJ

p CO (— pd i o

Eh

b 0 o o \ pd on cn , X O CVJ i—i ~

C\J

o- <a- =- c- o- a- c c~

b o o o \ CVJ C*~\ in c- c» - Ic- c- C« — C*> c- p 1—1 O cr CVJ

C 8 on -d" pd \ CVJ on in O © o rH oo p i

1 o cn p o o o C 1 on pd pd p t>> O 05 o i

i co

cO - o- C- o- «- OO C- o- ~ p \ o o o o P i>5 o nj cn in cO - o- c- o- — °° — °- o.

—

(P

©

p o cn o \ on -d pd o OO in X o rH 00 (d cO i—1 O cn o o 0 \ on -d" -d

-0-0-0-

— O- Oo e-— cX oo rHo OJo

a- c— ae» c-» «»- oK-Angle

C^/

measured

positively

from

x-axis

counterclockwise,

figure

TYPE i: t

= 3 +

type n: t

= 65"

FIG. I DIMENSIONS AND FORM OF WALL MODELS

1

i

i I

i

i

S 0% RESTRAINT | a. FREE SHRINKAGE

j

1

L

100% RESTRAINT

b. RESTRAINED SHRINKAGE

1 /

/

\

\

\

100% RESTRAINT

C. LOADING TEST

FIG. 2 ANALOGY OF LOADING TEST TO RESTRAINED SHRINKAGE

Fig. 3 - A plaster model with gages in a testing machine

@©

e>

Ll

STRAIN

GAGES

ON

PLASTER

MODEL

NO. I

(TYPE

I)

SIZE

OF

foundation:

<x>

C\J

GO

o

00

00

<x>

o o o o o o o O o o

o O' o o o o o O o

00 <D in ro CJ

1 9

01 X

x 3 ‘Nivais

FIG.

5

HORIZONTAL

STRAINS

IN

PLASTER

MODEL

NO.

I

(TYPE

I)

NOTE!

ENCIRCLED

NUMBERS

REFER

TO

WALL

LEVELS

W

FIG.

4

Vt

FIG. 6 PRINCIPAL STRESSES IN PLASTER MODEL NO. I (TYPE I)

400

to

CvJ

00

00

CVJ

CD

>01 X

x3 ‘nivhis

<£

Li_

HORIZONTAL

STRAIN

IN

FOUNDATION

% ‘NIVU1S

FIG.

8

HORIZONTAL

STRAIN

IN

FOUNDATION

EXPRESSED

AS

PERCENTAGE

OF

THAT

AT

FREE

ENDS.

RESULTS

AVERAGED

WITH

RESPECT

TO

CENTER

LINE.

r Fig. 9 - View of plaster model of type II showing the

crack pattern developed in a compression test.

a. COMPUTED VALUES. 9 -TERM SOLUTION.

b. COMPUTED VALUES. 40-TERM SOLUTION.

C. EXPERIMENTAL VALUES.

C

4

FIG. 10 COMPARISON OF COMPUTED AND EXPERIMENTAL VALUES

OF PRINCIPAL STRESSES AT POINTS A, E, AND G.

FIG. II WALL MODELS WITH STEEL FOUNDATION

y

Fig. I2~ Coordinate sgstem for theoretical

analgsis. Top of wall along y

= b,

bottom along g

= b/2 , one side along

x=0, and the other along x-a.

CJ "Oc

r>o

_o

COco c

CD>o

-Oco

CD

<n

-+-

(0) 4—

c0 4—C

<J~> C c C CD CD a) <d CD o

<J 5 4— 4-

4- 4- 4-

cD 4—

CDo

MCL> 4-

a>

O o o a

a o o o o

a> <T>- - r. b b V-

?< + G

l/)I 

4-0I @I a-I «T®I -0-4-

m 0 ®

©

n 01

iT)

£T

J to

-- - in

4-

Iffl

<a-| a

</}b3 C/>

"6

C

qd>

CTN cncco

cO

*0

Q)Da>

<S) Q>O'

“D

a/

a)

CT' Co

3T\ a

TJ5 o O

-G ^c

4-

O o - cgE

O)

CD

i_

cr CI roalL

"Oco

rr\ u

C5 TJCDO

_D

ini ’

1

Wall

o +

.0 ©

_Q_ „

O O

+

e*E

o

5

L

+

TT~ O.

+

'+ ©

© —o o

Strain

5 a

Vertical displacement

o

G — — — Given boundary

conditions

+ 9 coefficient solution

o 40 coefficient solution

Fig. 14 ~ Agreement of boundary vcdues in solutions

with given boundarg cond it ions along footing.

o 40 -term solution

. 15 - Variation of shear stress in

wall along base of footing.

Wall

Fiq. 16 ~ Strain distribution along footing -

(a) boundary condition for rigid footing,

section 2.4.1; (b) boundary condition for

test footing, section 2.4.2.

THE NATIONAL Bl REAL OE STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of < longress, March

3, 1901, as amended bv Congress in Public Law 619, 1930. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

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