0-169

This is to certify that the

thesis entitled

An Analysis of the Settlement of
Footings on Sand

presented by

Richard Wayne Christensen

has been accepted towards fulfillment
of the requirements for

M. S. degree infliiLEDgineering

W

 

 

 

Major professor

Date May 20, 1.960

 

   
 

  

LIB RA R Y
Michigan State

University

 

AN ANALYSIS OF THE SETTLEMENT OF FOOTINGS ON SAND

BY

RICHARD WAYNE CHRISTENSEN

AN ABSTRACT

Submitted to the College of Engineering of Michigan
State University of Agriculture and Applied
Science in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE
Department of Civil Engineering

1960

Approved W

Richard Wayne Christensen

AN ABSTRACT

In this thesis, an attempt is made to formulate a rational
method of analysis of the settlement of footings on sand.

Present methods of analysis are reviewed.

The problem undertaken is that of determining the stresses
and settlement under a rigid continuous footing on sand. The method
of settlement analysis developed is a numerical procedure based on
the theory of elasticity.

It is assumed that the sand mass is elastic and isotropic
but not homogeneous. A grid system is established for the sand mass
and the displacement equations from theory of elasticity are written
in finite difference form for each of the grid points. On the basis of
triaxial tests, the modulus of elasticity of sand is taken as a function
of the minor principal stress. The minor principal stress is evaluated
and the values of E for each of the grid points are estimated. The
displacement equations are then solved by the digital computer. The
stresses at the grid points are then determined, also by the digital
computer.

It was found that the contact pressure and settlement
obtained in the numerical method agrees very well with the results

of model te sts .

AN ANALYSIS OF THE SETTLEMENT OF FOOTINGS ON SAND

BY

RICHARD WAYNE CHRIST ENS EN

A THESIS

Submitted to the College of Engineering of Michigan
State University of Agriculture and Applied
Science in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE

Department of Civil Engineering

1960

ii

ACKNOWLEDGMENTS

The author wishes to express his most sincere
thanks and appreciation to Dr. T. H. Wu, Department
of Civil Engineering, Michigan State University whose
expert guidance and encouragement helped make this

study pos sible .

iii

TABLE OF CONTENTS

Section Page
ACKNOWLEDGMENTS ii
LIST OF SYMBOLS iv

I. INTRODUCTION 1

II. ANALYTICAL METHODS OF PREDICTING

SETTLEMENTS ' 5

III. DEVELOPMENT OF THE NUMERICAL METHOD 12
IV. SETTLEMENT ANALYSIS 20
V. CONCLUSIONS 24
APPENDIX 42

BIBLIOGRAPHY 44

US

N‘

LIST OF SYMBOLS

compression index

bearing capacity index
deviator stress

void ratio

modulus of elasticity
modulus of elasticity in shear
depth of significant stress
pressure intensity

horizontal displacement
vertical displacement

unit strain

unit weight of soil

settlement

normal stress

major principal stress

minor principal stress
shearing stress

Poisson's ratio

effective angle of internal friction

angle of obliquity

iv

I. INTRODUCTION

Stresses and Displacements

In 1885, the French mathematician Boussinesq (2) applied the
theory of elasticity to the problem of a concentrated load acting on a
semi-infinite solid. He obtained the following solution for the stresses

(Figure 1).

‘B

a——

 

 

///N/////// :-§-1—D3( 2+z2)-5/2
I . z
I “ Z P --1- Z /
l f frail-2N9; -%<rZ-ZZ> 21' 31' z(.2..2,-5 2'}
I \ r r
l Z’ZJIIPJLd-r
|
r 0:\ 3P 2 2 2 -5/2
frz=-Zrz (r +2) [1]

Figure 1: Diagram illustrating
the symbols of the Boussinesq
equations.

These equations are known as Boussinesq's equations. They indicate,

in a general way, the stress distribution that might be expected in a

soil mass when its depth is large compared to the dimensions of the
loaded area. Experimental evidence indicates that this stress distribution

is approximately correct for most soils. [See for instance Jurgenson(5).]

If the theory of elasticity approach is extended to the case of
a circular, uniformly loaded area of radius r, it is found that the

vertical stress beneath the center of the loaded area is given by

3
Z
0244+ 2 23/2 [2]
(r +z)

where p is the intensity of the load.

The settlements produced by the flexible, uniform load on
the surface of a semi-infinite elastic solid have also been computed
by elasticity methods. For a square loaded area, if}: is assumed to

be 0. 5, the settlement is (9)

/0cor = ——B—' 4E b

[3]

,0... = 43192

where f’cor = settlement at the corner of the loaded area

/Ocen = settlement at the center of the loaded area

p intensity of load

b width of loaded area

E: modulus of elasticity of the solid
The general shape of the settlement profile resulting from the appli-
cation of a uniform load is shown in Figure 2. This is the case if the

load is perfectly flexible.

b
p

 

'I /
1111mm

Figure 2: Settlement profile for a flexible, uniform load on a
semi-infinite elastic solid.

If a load is applied to a semi-infinite elastic solid by a per-
fectly rigid footing, the pressure on the base of the footing will not
be uniform. The solution of the problem by the theory of elasticity

results in the pressure distribution shown below.

70

 

 

 

 

/// i 1 I ;

 

 

 

 

 

 

 

 

 

 

 

I
Figure 3: Pressure distribution on a rigid footing on a
semi-infinite elastic solid.

The settlement of the rigid footing is found to be about 7. 3 per cent
smaller than the average settlement of a flexible footing of the same

area, acted on by an equal intensity of load p.

Limitations of Elastic Theory

 

The preceding discussion treats the stresses and displacements
in a semi-infinite elastic solid under the application of various types
of loading. This of course assumes that the material subjected to
stress is elastic, homogeneous, and isotropic. It is well known that
such is not the case with soils, particularly sand. If a sample of
elastic 'material has acting on it the principal stresses ()1, 0—2, and 0'3
and one of the principal stresses is increased, the resulting strain is
independent of the initial principal stresses. However, for sand, the
strain is dependent to a very large extent on the initial principal
stresses. It decreases as the initial principal stresses are increased.
In a mass of sand, the confining pressure increases with depth due to
the weight of the overburden. It may therefore be concluded that a
mass of sand is elastically nonhomogeneous in the vertical direction.

At great depths in a sand mass, where the confining pressure
is large, the variation in the ratio of applied stress to strain is small.
However, near the surface of a sand stratum, the assumption of
proportionality between stress and strain which does not change with
depth is far from correct.

It should also be noted that Poisson's ratio is not constant for
granular soils. It may increase from 0. 2 at low stresses to more
than 0. 5 at high stresses.

In light of these facts, it becomes obvious that the elasticity

solutions which are based on the assumption of isotropy and homo-
geneity of the material are not applicable to the problem of the settle-
ment of footings on sand.

Because of the large number of variables and the mathematical
difficulties involved, the approach to the problem of settlement of
footings on sand in the past has been primarily empirical. This
approach has usually consisted of the accumulation of data relating
the observed settlement of footings to some readily measurable soil
property such as the relative density (or resistance to the penetration
test). (10) The results of these data are presented in the form of
design charts. The charts are very conservative and their use generally
results in a safe design. However, in exceptional cases (such as a
very loose sand) they may give results on the unsafe side. It is there-
fore desirable to develop a more fundamental understanding of the
mechanism of settlement.

II. ANALYTICAL METHODS OF PREDICTING
SETTLEMENT '

Elastic Analysis

 

It is assumed that the stress distribution below a uniformly
loaded, flexible footing is close to that computed from the theory of
elasticity. The elastic constants necessary for settlement calculations

can be determined from the triaxial test.

The vertical unit strain in an elastic material is given by

ACT-1
éyz-E— -%(A0’Z+A03) [4a]

where E = modulus of elasticity and/.1 = Poisson's ratio. In the

triaxial test, (7' = 0’ so
2 3

AG"1 AC)"3
6 = E- ' 2P? W

V

This expression may be written in slightly different form as follows.

A0' A0' .AO~ AU-
6 3—1 ' 2 —‘§' 'l' —-3" " ——2 or
y E f1 E E E
I Z l
: ~— - A -- ACT - AU“ 4
6y (E "El U3+E( 1 3) [C]
Equation [4c] can be expressed as
Ah
= — = C - A ACT - AO" 5
6y h III 03*‘31‘ 1 3) [1
l l 2
where CI — E and CIII — (E - g)

h 2 thickness of layer considered

Ah = change in thickness i. e. settlement of layer considered

The elastic constants Cl'and CIII can be determined by two sets

of triaxial tests. To determine C , the lateral pressure is held constant

(A0}, = O) and the deviator stress (A0"1 - A 03) is increased. Then

éy

C :
-A6
I W. 3)

which is the slope of the deviator stress versus axial

strain curve. To evaluate C the deviator stress is held constant

111’

(AO’l- AO’ = 0). Then G =

1 . .
3 III E , Wthh is the slope of the lateral pressure

(0'3) versus axial strain curve.

As was pointed out earlier, /.1 and E for sand are not constants,
but vary with the confining pressure. In order to account for this fact,
the laboratory specimens should be tested under initial values of 0'1 and
0-3 appropriate for the depth of the points under consideration. The
values of CI and CIII are then representative of that attainable in the
sand at the various depths.

The settlement calculation can be carried out by numerical
summation. The sand mass is divided up into horizontal layers,
throughout which CI and CIII are assumed constant, and the settlement
of each layer is calculated. The total settlement at the centerline

of the footing is then equal to the summation of the settlements of all

the individual layers. That is
= - AG’ C AO’ - A0’ 6
P 2 {C111 3 + 1( 1 3)lh [ ]
where lo = settlement of the footing and h = height of each individual layer.

The changes in the principal stresses AG}, and A01 caused by the

load are determined from Boussinesq's equations.

Plastic Analysis

 

In a thesis presented in 1956, Bond (1) developed a method,
based on plastic theory, of predicting the settlement of circular footings
on sand.

The state of stress on a vertical line beneath the center of a

circular foundation is expressed as

4)
2
or 03"“ 6+3?) [7]

when cpe = partially mobilized angle of internal friction.
The value of (he at any point beneath the center of the footing
is calculated assuming that potential surfaces of failure develop as

shown below.

 

 

Figure 4: Probable shear pattern beneath a rigid, circular footing
on a dry sand. [After Bond (1).]

The value of o is assumed constant along any particular failure

e
surface. The vertical stress along the centerline is calculated by
Boussinesq's equation. Triaxial tests were used to determine the

stress-strain characteristics of a typical sand. The test results

9

were expressed in a plot of (be versus vertical strain for various values
of 01' where (be is determined by equation [7].

Having the values of GI and (be at a particular point, the stress-
strain curves can be used to find the vertical strain 6 at the point.
Integration of the strain along the centerline gives the settlement of
the footing at that point.

Comparison of Bond's theoretical calculation of (be beneath the
center of the footing with experimental results obtained by the
Waterways Experiment Station (13) shows good agreement. However, the
experimental results are not in good agreement with the values of (be
as determined by Boussinesq's equation and equation [7].

Experimental results obtained by Bond show good agreement
between measured vertical stress and the vertical stress as calculated
by Boussinesq's equation. The settlements determined from these
tests agree with theoretical calculations for dense sand but not for
loose sand. It is obvious that the failure pattern assumed in the plastic

analysis does not develop in the case of loose sand. [See Myerhoff (6).]

Consolidation

 

Hough (4) has presented a method, based on volume changes
in one-dimensional compression, for predicting settlements of footings
on any type of soil.

In a one-dimensional consolidation test, a relationship between

the void ratio and the applied pressure is obtained. It is usually plotted

10

on a semi—log graph. (See Figure 5.)

 

 

 

 

 

 

 

 

 

A6
s) M 6" ' A ”9/0
i
W
\
§
AP
fl
pressure , )0 {/09 560/6)
Figure 5: Compression diagram. [After Hough (4).]

The slope of the pressure—void ratio curve is called the compression

index (CC ). The change in void ratio Ae for any increment of pressure

increase Ap is given by the equation

A

Ae = CC log (1 + l) [8]
Pi ‘

initial vertical pressure on the'element. The change in

where Pi -—

thickness Ah of a soil layer with an initial thickness of h is

Ae [9]

l+eO

 

Ah=h

where eO = initial void ratio. Equations [8] and [9] may be combined

to obtain the following expression for the change in thickness of a soil

 

layer
C
C A
Ah=h log(l+-—P) [10]
1+ eO pi

11
For the purpose of simplifying the notation, the following substitutions

are introduced.

l+eO

Let -—C--— = C = bearing capacity index
c

and Ah zlo = settlement
Equation [10] may then be expressed as
res
h
Ap = (10 - 1)pi [11]
By combining the foregoing basic relationships from the consolidation
test, Hough obtained an equation relating the settlement of a footing to
the contact pressure.
£2
_ __Y__ hs 2
pc - 2 hs (10 -1)(hS + 3B) [12]
27B

where hS is defined as the depth of significant stress and is determined

by

Z
2 10B
h5 (h5 + B) = T pC [12a]

and B = width of footing, pC = average contact pressure. These
expressions are based on the assumed pressure distribution shown

in Figure 6.

12

 

 

 

= MAP

 

, ; . /7°"
[70":YfiA

Figure 6: Variation of stress with depth. [After Hough (4).]

Observation of the conditions imposed on the sample in the
consolidation test reveals serious shortcomings in this method of
settlement analysis when applied to sands. In the consolidation test,
the sample is compressed under the condition of zero lateral strain,
while in the case of sand beneath a loaded footing, considerable lateral
strain may develop. Therefore this method completely eliminates
that part of the settlement contributed by lateral displacements.
Settlements calculated by this method are apt to be too low, particularly
in the case of loose sand where the effect of lateral strain is significant.

III. DEVELOPMENT OF THE NUMERICAL
METHOD

Displacement Equations

 

One considers the case of a continuous, rigid footing acting on
a semi-infinite mass of sand. It is assumed for the present that the

sand is elastic, but not homogeneous.

l3

 

Figure 7: - Rigid footing
continuous in the z-

/ Y direction.

Since the footing is continuous in the z-direction the problem can be

 

 

 

 

 

treated as one of plane strain (éz = 0). Then from elastic theory,

Hooke' 8 Law is

 

 

0;: = 7Ie + 2G ex where 7\ = (1+5)?1_2P)
O’yz7le+2G€y [13] e: éx+éy
z"xy : 7:yx : Gny - 2(f:+}1)

In the two-dimensional problem éx = :35 and 6y = 3—3 and ny = ghee—:1,

3U 3U 2V
O'x he+2G 'bx ny yx (3y 2x
_ 9V
(TY - 7le+ZGjé—; [13a]

in which U and V are the displacements in the x and y directions

respectively.

14
The equations of equilibrium for the two-dimensional problem in

elasticity are

 

 

9x + BXY + X = where X = body force in x—direction
[14]

’bcfy zzxy

3y + 2x + Y = 0 Y = body force in y-direction

By taking the partial derivatives of the Hooke's Law equations, the

following expressions are obtained.

 

 

903: ”(M/36) 2 9U
ax ‘ 2x +25§(Gé§)
20’

y_ _3(‘se)+ 2 G2!

—( )
9y 3y 2y 9y

[15]

22’
_1x _ .2. 9U 9V
2x -9x [G(@ +‘9x)]
32’
2U _’&_V
TXY- 731Gb ‘l‘ —)l
y ®y+ 9x

The body forces X and Y are assumed to be zero. Substituting equations

[15] into [14] one obtains the displacement equations for the problem.

.2 29 2X .2. 22 .2 2.11 1"

9x [7)(6x + ’2)!” + 23x(G@x) + 2y[G(—_ 2y +fix —_:)]

.9. 2.9 fl 2. 9U 21" _ 16
r 2x[(7’+2G)2x +7) 2y] + By [CHE—ya” —)] o [ a]

15

.2. LU?! .2 1V _’3 LUZY-
and 2y[7’(9x + 2331+ 2?y(Gay) + 2x[G(2y +9.31" 0
.2 .21 2.11 .2. LU 2.". _
or ay[(7>+zc})2y +22X1+ ax [G(ay + ax)] _ 0 [16b]

Equations [16a] and [16b] may be written in terms of finite differences.
A grid system as shown in Figure 8 is set up for the soil mass beneath

the loaded footing.

 

 

 

 

 

 

 

 

1
fr 57/” //€/7/a/ /m//fl€
\. I 1 '1‘ v]
5' I 4 I 3.1, 2"/ )h /“l
”a
A (L A
5 a 4 z a»; 242 I1] 2
5r} 4 .3 3‘3 2 3 / 3
.4 41’4 3‘4 24 134

 

 

 

 

 

Figure 8: Grid system for the
numerical method.

91
o,
ex
0‘
Lu
0‘
'9.
“I
_ m”

For any point 0 in the grid (Figure 9), the central difference

expressions are as follows.

 

 

 

 

 

 

 

0W n [76
f7
- h
f a W ' 0 6
] Figure 9: Finite difference
notation.
*V
SW 3 56

 

l6

 

 

 

U -U U -U

@ 0U ‘1 ( h 0)‘“e ( oh 6) 1
. W, O 9 O __ __ _ _ _
‘9;[(7)+2G)5‘;] = h — h2[aw., 0(Uw U0) “8 , O(UO Ugll
VS -V VS -V
W nw e ne

2 2V W( 2h )- )4 2h ) 1
flag-V): 2h _fi = 4h2 [hwu/sw-Vnw) _ %e(vse-Vne)]
Similarly,
.2. w -L _ _
2y(Gay _ hZ [G , s(Us U0) - Gn,o(Uo Un):I
’2) ’0V 1
_ — =— v -v - -
was“) h2[GO’ ( w c)) G ,O(vo ve)]

Substitution of the finite difference expressions into equations [16a] and

[16b] gives
1 1 1 l
«2(a ~G)U +—(a +a )U +—(G -G )U +-—(a +a )U +-—(G+G )U
o o o 2 w o w 2 o s s 2 o e e 2 o n n
l 1 l 1
+-(7s+G)V -—(71+G)V +—(2+G)v -—(7, +G)V =0 [17a]
4 w 3 SW 4 e 3 se 4 e n ne 4 w n nw
l l l 1
-2(a.+G)V +—(G+G )V +—(a +a )V +——(G+G)V +—(a +a )V
o o o 2 o w w 2 o s s 2 e o e 2. n o n

1 1 1 l

— + U -— +G U +— +G U -— +G U = 0 17b
+40‘s Gw) sw 4(As e) se 4(hn e) ne 4(hn w) nw [ ]
These two expressions are the displacement equations from theory of elas-

ticity expressed in finite difference form. They apply in general to any point in

the grid.

17

In order to obtain expressions for points on the boundaries of
the grid, it is necessary to consider the boundary conditions imposed
by the problem. First of all, it is assumed that row 5 and column 5
(Figure 8) are far enough from the applied load so that the U and V
displacements at these points are zero.

In row 1, the following situation exists. The footing is allowed
to undergo an arbitrary settlement, say V0. Then V1_1 = VZ-l = V0.
The soil directly beneath the footing is assumed to adhere to the base

of the footing so that no lateral displacement is possible, or

Ul-l = U2_1= 0.

At points 3—1 and 4-1, a; = )e+ zo 6y = 0. So

29+?"

__ 2.1.1.} afl: 0 [16C]
’ax @y

N 9x y

)+2G<—”—‘—’)=7»
av

Or, in finite difference form, at the point 3-1,

U41” 21 V32" 31
71 =
31( 2h ) + a31( h ) 0
If - o 25 th - 3 a d
)1 — . , en 7)— 3, n
- U _ = 17
U41 21 + (“V32 V31) 0 [ C]
Similarly, for point 4-1,
- U 6 - v = 17d
U51 31 + (V42 41) 0 [ 1

18

At the centerline of the footing, U = 0 due to symmetry. It may also
be seen from symmetry that U(right of centerline) = -U(1eft of centerline)
and V(right of centerline) = V(left of centerline).
If equations [17] are applied to all points in the grid, a system of
fifty simultaneous equations in U and V results, twenty-three of which
vanish by virtue of the fact that they are identically equal to zero. The
remaining twenty-seven simultaneous equations are shown in tabular

form in Table 1. These equations can be solved by the digital computer

through routine tape L2.

Determination of Stresses at Grid Points
Once the displacements at the grid points have been determined,
the expressions for Hooke's Law can be used to evaluate the stresses

at the grid points. Equations [13] can be written

 

”E 119U+231+1 E ”U

‘T :[ 5; 0y 11+») '7???

x (19:)(1-2P]

2V

 

 

 

 

 

. “(l-7:2“) 2H 11 ..
or 0; [(l+/u)(1-Z)1)(0x +(1+}1)(1-Z}1)(9y 1E [18a]
03' (Infill-2;» 2x 2y (1+)1) 9y
or =1P““'2“’ 1213+ P (3511: [18b]

0;, (’1 +}1)(1fi-2}1) 2y (1+;il(1-2}1)

19

E _9_U 3V]
xy 2(1+}1) 2y 3x

 

 

or [xy

1 1
[[r—(,+P)1<%§+%x—>[Z(W1<V>]E [18c]

In finite difference form

a; = [k2(Uw- Ue) + k1(VS- Vn)]EO [19a]
= [k2(VS-Vn) + k1(UW- Ue)]Eo [19b]
1;}, = [k1(US- Un) + (vw- vent:O [19c]
where k

 

1 2h(1+}1)(l-2}1)

. P+(1'2P)
Z 2h(1+}1)(1-2p)

In its natural state the sand has vertical and horizontal stresses
0}; and 0" acting on it which must be added to the values obtained

0 YO

from equations [19] to obtain the total horizontal and vertical stress

at a point.

The principal stresses (3'1 and 0}; can then be obtained from the

 

 

formulas
0‘ + o- 0' - 0' 2
crl=—:—:-+/—X—2—Y) +(T )2 [20a]
XY
0'3 = ---—--‘Y--\/(---—-y-)Z + <ny )2 [20b]

A program was prepared to perform the operations indicated in

equations [19] and [20] on the digital computer. A copy of this program

is included in the appendix.

20

IV. SETTLEMENT ANALYSIS

Description of Soil Studied

 

The material used in the laboratory investigations was Ottawa
sand which passed a no. 16 sieve and was retained on a no. 30 sieve.
The sand was tested in a loose and a dense state. To achieve the
loose state, the sand was simply poured into the mold with no compaction.
The dense state was obtained by compacting the material in five layers

as it was placed in the mold.

Elastic Method

 

Triaxial tests. A series of triaxial tests was performed to

 

determine the constants CI and CIII for loose and dense sands. All

specimens were consolidated under a principal stress ratio of 2. 0
(k0 = 0. 5) before application of hydrostatic or deviator stress. A
summary of the tests performed is given in Table 2. The stress-
strain curves obtained from these tests are shown in Figures 11 and
12. From the stress-strain curves, the values of CI and C111 were
determined as described in section II. Curves demonstrating the
relationship between the elastic constants and the minor principal
stress (for compacted sand only) are shown in Figures 13.

Settlement calculations. The settlement analysis was carried

 

out in the following manner. First of all, it was noted that the order

of magnitude of the constant CIII is very small compared to CI.

21

A0" is therefore negligible

The settlement contributed by the term CIII - 3

and was omitted in the calculations.

The actual mechanism of settlement is not perfectly reproduced
in the laboratory tests; that is, in the laborabory tests, the stresses
AC);5 and (A01- A 0;) are applied in two separate stages, whereas in a
loaded soil mass they occur simultaneously. It is therefore necessary

to determine the value of CI that most nearly represents the true

 

condition.
sl
¢e
arc/c 2 —J
are/5 / arc/e 3

 

 

 

L Ad: .J 0'
l

 

 

 

0' “'1 a; 07
0
Figure 10: Mohr's circle representation of stresses in sand.

In Figure 10, circle 1 represents the initial state of stress in

the sand mass. As the footing load is applied, 0'3 and 0‘1 increase by

A0; and A0i so that the final state of stress is represented by circle 3 .

Circle 2 represents the state of stress in a triaxial specimen consolidated

to the final 03 while the principal stress ratio remains unchanged

(0; = O. 501). Then the deviator stress is increased and the Mohr's

circle increases in size until it coincides with circle 3 . The increase

in strain under the increasing deviator stress was used to calculate CI.

22

CI is chosen as that which corresponds to the value of 03 which exists
at the point under consideration in the sand mass after application of
the footing load. Although this choice of CI most nearly represents
the stress changes taking place in the sand mass, it is not strictly
correct. It omits the additional settlement which takes place before
03 builds up to its final value. However, the error due to this
omission is small.

The settlement of a continuous footing 15 feet wide, supporting
a load of 1 kg/cm2 was calculated by the method outlined in section II,
using the results of the triaxial tests. The values of ACT1 and A0}, as
determined by means of Boussinesq's equations are given in Table 3.

As calculated by the elastic method, the settlement is 21. 8 cm at the

centerline of the footing. (See Table 4.)

Consolidation Method

 

A standard consolidation test was performed on a sample of
Ottawa sand in the loose state. The load-settlement curve obtained
is shown in Figure 14.

The settlement under a continuous footing 15 feet wide, supporting
a load of 1 kg/cmz was calculated using equations [12]. The calculated

settlement is 0. 578 cm.

Numerical Method

 

In order to make use of the equations derived in section III,

23
it is necessary to establish the values of E, 7) and a at each of the
grid points. A rather complete set of triaxial test data has been
published by Chen (3) from which one can plot the variation of the
modulus of elasticity in a sample of sand with the minor principal
stress. Figure 17 shows the E versus 0‘3 curves one obtains from
Chen's data.

According to the curves in Figure 15, the relationship between
E and 03 is practically a straight line for low stresses. This can be
expressed as E = k0:-5 where k is some constant. Actually _0'}2 is not

3
exactly constant, but varies somewhat with the deviator stress.
However, the variation is small and may be ignored for simplicity.
The lower portion of the curves was used to determine the value of k
because this portion represents the range of stresses encountered in
the problem.

The value of k was determined from the curves for a medium
sand. The average value of k for these curves is 2500.

With the value of k established, it remains to obtain an estimate
of 0‘3 so that starting values of E, 71 and a can be determined for each
of the grid points. A reasonable estimate of 03 can be obtained by
the use of Boussinesq's equations. 0’3 was evaluated in this manner
for a load of l kg/cm2 acting on the footing. The stresses obtained

and the resultant constants E, 7\ and a for all grid points are given in

Table 3.

24

The values of the constants E, 7) and a from Table 3 were sub-
stituted into the equations given in Table 1. These equations were
solved by the digital computer, giving the values of the displacements
U and V at all points in the grid. Using these values of U and V, the
stresses at the grid points were determined by means of equations
[19] and [20]. This operation was also performed on the digital computer.

After the values of 0:3 have been computed, new values for E
can be obtained from the relationship E = keg and the entire procedure
repeated until the values of 0' converge. In this manner, all values of
stresses and displacements in the grid system can be evaluated for
the given settlement. However, in this investigation it was not possible
to carry out the successive approximation because of the lack of storage
space. Only the first approximation was obtained.

The investigation was made for a rigid, continuous strip footing
15 feet wide, with a given settlement of 1 cm. It was assumed that
)1 = 0.25 and E = 2500 0’3.

The distribution of stresses and displacements obtained from

this investigation is shown in Figures 16 and 17.

V. CONCLUSIONS
The settlements calculated by the different theories are listed
in Table 4. It is obvious that the different methods of settlement

analysis give, very different results.

25

The settlement calculated by the consolidation method is far
too small. Although the sand is very loose the settlement is only
0.578 cm. The answer is unreasonable. As was pointed out, the
conditions of stress and strain in the consolidation test are not con-
sistent with actual conditions. Therefore, the consolidation method
of analysis should not be applied to sands.

The settlement obtained by the elastic analysis is larger than
those commonly observed (10). If one examines the validity of this
method, it is noted that certain inconsistencies exist. Because triaxial
tests, in which 0; equals cg, are used in the calculation of settlement,
the assumption that the footing is circular in shape is inherent in the
development. The principal stresses 0’2 and 0'3 are not equal under a
continuous footing. Rather 0-2 is greater than 0'3, which means that
the application of this method to a continuous footing results in an
overestimation of settlement. This fact was borne out in recent tests
conducted under the direction of Bishop at Imperial College (14).
Bishop has developed an apparatus which is capable of measuring the
stress-strain characteristics of sand under the condition of plane
strain. It has been found that the vertical strain in a sample of sand
is reduced considerably when loaded under plane strain conditions.

Another factor is that the footing is assumed to be perfectly
flexible. For rigid footings, the settlement is much smaller than that

at the center of flexible ones.

26

Field observations of the settlement of circular foundations
resting on loose sand at Drammen, Norway ( 7 ) indicate that the
magnitude of settlement obtained in this investigation is quite reasonable.

It may be concluded therefore, that the elastic analysis is
capable of producing good results provided that the stress-strain
conditions of the tests from which the elastic constants are obtained
are the same as those under the actual footing.

The results obtained by the numerical method developed in this
thesis are quite promising. The contact pressure distribution ob-
tained agrees in general with that observed in model tests. [See for
instance, Bond (1)].

To evaluate the calculated settlement, it is necessary to examine
the value of k adopted in the calculations. The constant k is determined
under the condition that 0’2 equals 0'3. The error involved is the same
as that for the elastic analysis. To be strictly correct, k should be
determined from the stress-strain characteristics of sand under plane
strain conditions. This necessitates the use of a testing apparatus
similar to that developed by Bishop which is, as yet, not in general use.

The reliability of k equal to 2500 can be examined from another
point. Chen's data, from which this k was obtained can be compared with
the elastic constants from the triaxial tests carried out in this investi-
gation. It is possible to compute, from the results of the triaxial tests,

the values of E and}1 of the sand for various 0'3. This was done and

27
the results presented in Figures 18 and 19. If one compares the
values of E obtained from these tests with those obtained by Chen,
one finds a rather large discrepancy. There are two reasons for this.

With Chen's data a k of 2500 was obtained by taking the secant
modulus at 0. 50D. Whereas, in the present series of tests these
constants represent the tangent moduli at stresses considerably above
the principal stress ratio of 2. 0. For a <1) of 300, this principal stress
ratio corresponds to approximately 0. 50D. In order to compare the
moduli at approximately equal shear stresses, k should be taken
from the curve corresponding to 0. 75D in Chen's data. Then one
finds that k = 800 which is in much better agreement with that obtained
in the triaxial tests performed in this investigation. For a k of 800
the settlement is approximately 3 cm for the same contact pressure.
The range in settlement then, is between 1 cm and 3 cm which is in

reasonable agreement with observed results.

TABLE 1

SIMULTANEOUS EQUATIONS IN U AND V

28

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41

APPENDIX

Computer Tape for (TX, 0', z" and 0'

y xy 3
8002840001 0020+
8002840002 OOIF 0011?
1902626000 003F 003F
8002840000 OOZF 002F
L400140001 003F 003F
80028403F6 003F 003F
-5000263L6 001F OOOF
81004263FJ 001F 001F
4000122000 0063F 0063F
7J3LFL4001 L5308F L0306F
40001263L8 L5413F L0401F
L4002263F- 0014F 0015F
423FFL5001 OOOF OOIF
L4000L4000 00200+
263F—L03LL ~5F468L
703LL-53F7 L44L4211L
4000081004 814FL025L
50000L43LL 2210L402F
323FLL43LO L5F661F
423L5L5001 lOlF-JF
L400026000 40FL1F
40002L03L4 401FL52F
423L8233L9 4OFL58L
5000140000 L44L468L
U3F9L43L4 L52F427L
423F9L43L8 L023L32F
423L800027 L524L401F
800080000N 41F814F
40001273L1 L025L402F
465F6604-8 326LL424L
7LLLLLLLL6 4OF814F
223L-00001 501F402F
L025L323L
7525L-5F
401F5025L
75F0039F
L42F2616L
OOOFOOZF
OOFOOSF

000F001 0F

00230+
4OFL527L
-4F4225L
461L402F
92961F366L
L5F366L
92706F226L
92961F5026L
228L461L
7526LL51L
L019L327L
-5F401F
L52F006F
50L3214L
L7F661F
2315L671F
L7F-4F
40F50F
7526L0036F
824F1040F
-410F4OF

L51LL016L
461L009F
3623L92961F
L52FL019L
462F0010F
3216L22F
OOFOOIOF
821013F001F

00260+
401F+5F
428L511F
101F-JF
402F50F
L51F662F
-5FL02F
101F368L
L42F263L
LSZFZZF

0050+

50300F 500L
26200F 242L
50400F 502L
26200F 244L
50490F 504L
26200F 246L
50500F 506L
26200F 248L
50700F, 508L
26200F 24100F
00100+
L5308F L0306F

40398F
7J490F

L5413F

40399F

50398F
40398F
L0401F
50399F

7J491F L4398F
40600F L56L
L425F 406L
L50L L420F
400L L53L
L420}? 403L
L521F LOZOF
4021F 360L
LSOL L422F
4OOL L53L
L422F 403L
L523F 4021F
L524F LOZOF
4024F 360L
L523F 4024F
L5493F 40490F
L5492F 40491F
L528F 400L
L529F 403L
L526}? LOZOF
4026F 360L
L5313F L0301F
40398F 2628L
L5408F L0406F
L4398F 40398F
50398F 7J493F
40632F L531L
L425F 4031L,
L526L L420]?

43

4026L L528L
L4ZOF 4028L
L521F LOZOF
4021F 3626L
L526L L422F
4026L L528L
L422F 4028L
L523F 4021F
L524F LOZOF
40241? 3626L
L56OOF L0616F
L4700F L0716F
40398F 50398F
75398F IOZF
40399}? 41498F
50632F 75632F
L4399F 5050L
26260F 265ZL
40399F L5600F
L4616F L4700F
L4716F 101F
L4399F 403991?
50399F 75500F
40648F 2658L

L557L
40571...
L420F
L549L
4049L
L431F
L553L
4053L
L420F
L554L
4054L

L431F
L530F

L425F
L544L
4044L
L420F
L552L
4052L
L420]?
L545L
4045L
L425F
L556L

4056L
LOZOF

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44

BIBLIOGRAPHY

Bond, Douglas, "The Use of Model Tests for the Prediction of
Settlement under Foundations in Dry Sand. " Ph. D.
thesis, University of London, 1956.

Boussinesq, J. , ”Applications de Potentiels a l'Etude de
I'Equilibre et des mouvement des Solides Elastiques. "
Gauthier-Villars, 1885.

Chen, Liang-Sheng, "An Investigation of Stress-Strain and
Strength Characteristics of Cohesionless Soils by Triaxial
Compression Tests. " Proceedings of the Second Inter-
national Conference on Soil Mechanics and Foundation
Engineering, Vol. 4, 1940.

 

 

 

Hough, B. K. , ”Compressibility as the Basis for Soil Bearing
Value, ” Proceedings of the American SocietLof Civil
Engineers, August, 1959.

 

 

Jurgenson, Leo, "The Application of Theories of Elasticity and
Plasticity to Foundation Problems. " Journal of the Boston
Society of Civil Engineers, July, 1934.

 

 

Myerhoff, G. G., ”The Bearing Capacity of Sand. " Ph. D.
thesis, University of London, 1950.

Simons, N. , "Settlement of Structures on Sand. " Internal Report
No. F. 31, Norges Geotekniske Institute, March, 1956.

Taylor, Donald W. , Fundamentals of Soil Mechanics, John
Wiley 8: Sons, Inc. , 1948.

 

Terzaghi, Karl, Theoretical Soil Mechanics, John Wiley &
Sons, Inc., 1943.

 

Terzaghi, Karl and Peck, Ralph B., Soil Mechanics in Engineering
Practice, John Wiley 8: Sons, Inc. , 1948.

 

Timoshenko, S. and Goodier, J. N., Theory of Elasticity,
McGraw-Hill Book Company, Inc. , 1951.

 

Tschebetarioff, ...Grego.ry P. ,_ Soil Mechanics, Foundations, and
Earth Structures, McGraw-Hill Book Company, Inc. , 1951.

 

 

45

13. Waterways Experiment Station, Investigations of Pressures and
Deflections for Flexible Pavements, Report No. 4,
Homogeneous Sand TesLSection, Technical Memorandum
No. 3-323, December, 1954.

 

 

 

14. Wood, C. C. , ”Shear Strength and Volume Change Characteristics
of Compacted Soil under Conditions of Plane Strain. "
Ph. D. thesis, Imperial College, 1957.

 

 

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