A general solution for active and passive earth pressures
by William Smith Hartsog
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in Civil Engineering
Montana State University
© Copyright by William Smith Hartsog (1968)
Abstract:
The Rankine assumptions were used as a basis to develop equations for active and passive earth
pressures within a slope having an infinite extent. The analysis includes: cohesive and noncohesive
soils; the angle of internal friction of the soil; seepage forces caused by a ground-water table which is
parallel to the ground surface; and the plane on which the stresses are to be found may be at any
inclination.
The equations for active and passive pressures were nondimension alized to reduce the number of
parameters and aid in programming the expressions for solution on an IBM 1620, Model II, digital
computer.
The program systematically varies the parameters in order to develop tables which simplify the
solution for a variety of soil properties and slope geometry. The dimensional equations were also
programmed for solution for a specific set of soil properties and slope geometry. A GENERAL SOLUTION FOR ACTIVE AND
PASSIVE EARTH PRESSURES
by
W I L L I A M SMITH HARTSOG
A thesis submitted to the Graduate Faculty in partial
fulfillment of the requirements for the degree
of
MASTER OF SCIENCE
in
Civil Engineering
Approved:
Head- Major Department
chairman, Examining Committee
MONTANA STATE UNIVERSITY
Bozeman, Montana
August, 1968
ACKNOWLEDGEMENT
The author wishes to express his gratitude to the faculty of the
Civil Engineering and Engineering Mechanics department at Montana State
University for their assistance.
Special appreciation is extended to
Dr. Glen L. Martin for his guidance and assistance in conducting this
study and preparing this thesis.
Appreciation is extended to the staff of the.Intermountain Forest
and Range Experiment Station, especially to Mr. Michael J. Gonsior, for
their assistance and cooperation. .
Appreciation is extended to. M r s . Ruthann Hartsog, the author's
wife, for typing this paper.
.
i
iii
TABLE OF CONTENTS
Chapter
I
II
Page
INTRODUCTION............................... '.................
I
STRESS RELATIONSHIPS ON A PLANE PARALLEL TO THE GROUND
SURFACE WITHIN AN INFINITE SLOPE .......................
2
;
III
IV
V
Infinite S l o p e ..............................................
Elemental Volume of S o i l ................... ■................
Normal and Tangential Stre s s e s ...............................
Neutral S t r e s s .................................... ..........
Effective S t r esses..........................................
Obliquity Angles.............................................
2
IO
10
11
MOHR' S CIRCLE OF STRESSES...... i........................ ..
15
Sign Conventions...................... '......................
Mohr's Coordinates..........................................
Coulomb's E q u a t i o n..... '........ :..........................
Mohr Rupture E n v e l o p e .................... '..................
Active S t r e s s ............ ....................................
Passive S t r e s s ...........................
Origin of P l a n e s ......................................... . • •
15
I5
I5
I5
18
18.
18
ANALYTICAL DEVELOPMENT.................................. ;..
21
2
7
Line Through Point of Known Stress........................
Seepage Force Per Unit A r e a .................... ............
Radius of Stress Circle;....................................
Circle 'of S t r e s s e s ................. ............■...........
Center of Stress Circle........................... '..........
Expressions for Origin of Planesi................... '......
Line Through Origin of Planes .................... '....... ..
Normal Stress on Plane of Investigation............ ......
Tangential Stress on Plane of Investigation......... .....
21
23
24
24
■ 25
26
26
28
■ 29
DISCUSSION...................................................
' 30
Equation Compatability......................................
Critical Depth Definition..................................
Critical Depth in Cohesive Material.......................
Critical Depth in Noncohesive Material.................■• • •
Equations for Critical D e p t h ...............................
iv
30
31
32
32
34
Chapter
VI
Page
NONDIMENSIONAL F O R MULATION....................................
36
Basic Parameters........................................ ■ 36
Nondimensional Equations.....................................
Nondimensional M o h r 1s Di a g r a m ...........................
Critical Depth Nondimensional Equations......................
VII
VIII
36
38
38
R E S U L T S ......................................
Dimensional F o r m .................... ■.......................
Nondimensional F o r m ............................. '........ .
Use of Nondimensional Tables..........................
41
44
CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STU D Y .........
48
A P P ENDIX............................. ......................
49
Appendix A:
44
Dimensional Computer Program....................
LITERATURE 'CONSULTED........................................
v
50
55
LIST OF TABLES
Table
I
II
III
IV
V
- Page
Chronological Summary of the Development of Infinite Slope
Earth Pressures..........................................
2
Active Stresses................................................ . .
4-2
■ Passive S t r esses................................................
43
Nondimensional Active Stresses....................
45
' Nondimensionai Passive Stresses...............................
46
vi
LIST OF FIGURES
Figure
Page
1
Profile of an. Infinite S l o p e ...........................
5
2
An Elemental Volume of Soil, A A 1B B 1, Within an.
Infinite S l o p e ......................................
6
3
Force Components Acting on Plane A - A ' .................
8
4
Stresses Acting on Plane A - A ' ..........................
8
5
Flow. Net Within an Infinite S l o p e .....................
9
6
Stress Relationships for Plane A-A' Above the
Water T a b l e ..................................... ....
12
Stress Relationships for Plane A-A' Below the
^ater Table with the Water Table Coinciding
with the Ground Surface............................
12
Stress Relationships for Plane A-A' Below the
Water Table with the Water Table Below the Ground S u r f a c e ..................... ............... ..
13
Stresses on Plane A-A'
Shown on Mohr's Coordinates...
16
10
Mohr Diagram for Active and Passive Stress Conditions
17
11
Active State of S t r e s s .. . ........................ .
19
12
Passive State of S t r e s s......................... .'.....
19
13
Active and Passive Stress Circles and Their
• Relationship to Mohr's Rupture Envelope...........
22
14
Typical Stress Circle..................................
27
I5
Stress Condition at the Critical Depth When the
Unit Cohesion is Greater. Than Z e r o .... .i
7
8
9
16
'
•33
Stress Condition at the Critical Depth When the
Unit Cohesion Equals Z e r o ....................... .
33
17
Stress Relationships at Critical'Depth...............
35
■18
Nondimensional Mohr D i a g r a m ............................
39
vii
ABSTRACT
The Rankine assumptions were used as a basis to develop equations
for active and passive earth pressures within a slope having an infinite
extent.
The analysis includes:
cohesive and noncohesive soils;
the
angle of internal friction of the soil; seepage forces caused by a ground
water table which is parallel to the ground surface;
and. the plane on
which the stresses are to be found may be at any inclination.•'
The equations for active and passive pressures were nondimensionalized to reduce the number of parameters and aid in programming the
expressions for solution on an IBM 1620, Model II, digital computer.
The program systematically varies the parameters in order to develop
tables which simplify the solution for a variety of soil properties and
slope geometry., The dimensional equations were also programmed for
solution for a specific set of soil properties and slope geometry.
viii
A GENERAL SOLUTION FOR ACTIVE AND
PASSIVE EARTH PRESSURES
CHAPTER I
INTRODUCTION
In i860, Rankine presented the original development for lateral
earth pressures within a slope of infinite extent.
R a nkine1s development
was based on a conjugate stress relationship between vertical stresses and
stresses on a vertical plane and considered a dry cohesionless material
with either a sloping or a horizontal surface.
During the latter part of
the nineteenth century, the Mohr circle of stresses was presented for the
graphic representation of the state of s t ress.
The analytical work of
Rankine was subsequently adapted to a graphical method using the Mohr
circle of stresses.
In 1915? Bell extended Rankine1s work to include cohesive soils.
Bell's graphical procedure made use of Mohr's stress circle, but in an
inconvenient manner.
Martin (1961) derived an analytical expression to
extend R a n kine1s work to include cohesive soils, but the development
includes neither seepage conditions nor a. variable plane of investigation.
The development herein extends Mbrtin1s analytical development to
include stresses on any plane in the slope, and includes seepage stresses
in the analysis.
It is assumed in the following development that seepage
stresses are caused by a water table parallel to the ground surface.
A
chronological summary of the developments based on Rankine's assumptions
is shown in Table I .
The graphical methods presently being used for the determination of
TABLE I
CHRONOLOGICAL SUMMARY OF THE DEVELOPMENT OF INFINITE
SLOPE EARTH PRESSURES
FACTORS CONSIDERED
INVESTIGATOR
angle of
slope
angle of
internal
friction
unit
cohesion of
the soil
APPROACH
angle of the
plane of in­ seepage
vestigation
graphical
vertical
RANKINE
X-
X
BELL
X
X
X
vertical
MARTIN
X
X
X
vertical
HARTSOG
X
X
X-
any
analytical
X
X
X
X
X
-3earth pressures are tedious and time consuming.
The analytical expressions
developed herein would also be time consuming were it not for the advent
of the digital computer.
CHAPTER II
STRESS RELATIONSHIPS ON A PLANE PARALLEL TO THE GROUND
SURFACE WITHIN AN INFINITE SLOPE
The term, infinite sl o p e , is defined as a slope of constant incli­
nation of unlimited e x t e n t .
For practical purposes, a slope that has a
length to depth ratio of twenty or greater has been considered an infinite
slope.
The classical infinite slope analysis requires that soil proper­
ties be constant at any and all vertical sections along the slope.
If
this requirement is met, the slope- can be studied by analyzing the stress
.
conditions within an elemental volume of soil.
Figure I shows the basic assumed geometry of the infinite slope.
In Figure I , Z is the depth to the plane of investigation, Z
. .
is the depth
w
to the-water table, and (Z-Zy ) is the vertical distance from the water
table to the plane of investigation.
The total unit weight of the soil
material ■above the. water table is denoted as
rated unit weight of the soil.
the horizontal is denoted as 0.
Y ,- and” ^
. is the satu- .■
t
sat
The angle the ground surface makes with
It is assumed that the planes representing
the ground surface and the water table are parallel.
The slope profile with an elemental volume of soil (AA'BE') is
shown in Figure 2.
Figure 2, and if AA'
If a unit, dimension is assumed normal to the plane of
is assumed, to be of unit length, the resulting plane
(which shall be referred to as plane A-A') will be of unit area.
The
f o r c e , W, acting on plane A-A', is equal to the weight of the vertical
column of soil above plane A-A'.
■ The equation for the vertical force, W, is found by adding the
GROUND SURFACE
\
GROUNDWATER TABLE
PLANE PARALLEL TO GROUND SURFACE
AT DEPTH OF INVESTIGATION________
S u re
Pr*
of
jAifj
Oj te
s j Or
\
\
\
\
GROUND SURFACE
products of respective unit weights and volumes within the vertical column
of soil:
¥ =
t w
cosO +
V
sat
(Z-Z )cosQ
w
( I )
Equation (I) is valid for the general case; that is,plane A-A'
groundwater t a b l e .
When plane A-A'
is below the
is above the groundwater table, the
quantity (Z-Z ) is zero and Z is replaced with Z .
w
w
The vertical force, ¥,
and tangential
can be, resolved into components normal (N)
(T) to plane A - A ' , as shown in Figure 3.
The expressions-
for the normal force and the tangential force a r e , therefore:
2
% Z cos 0 +
t w
N =
T =
■
p
Y
• (Z-Z ) cos 0
sat
w
Z sinOcosO +
t w
^
sat
.
.
( 2 )■ .
(Z-Z )sinOcosQ .
w
. -
( 3 )
The normal stress, (T , shown in Figure 4? is equal in magnitude to
the force N, because plane A-A'is of unit a r e a : ■
It follows,
from Equation;.
(2), that:
'
(T = X Z cos^O + X
a
t w •
sat
(Z-Z )cos^Q
w
Similarly, the tangential stress,
tC , is equal in magnitude to the force T.
a
F r o m Equation (3), the equation for
lP =
a
X Z sinOcosO +
t w
-Because the analysis
(4 )
-
%
tC can be written as:
a
,(Z-Z )sinOcosQ
sat w
( 5 )
will be on the basis of effective stresses,
it is necessary to determine the neutral stress, u.
found b y use of a flow net, as shown in Figure $.
The neutral stress is
The pressure head along
-3-
B'
Figure 3.
Force Components acting on plane A-A'.
Figure 4.
Stresses acting on plane A - A ' .
GROUND
SURFACE
\ \ \ \ \ T \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \
\ \
EQlU IFO TE m iA L LIUES
GROUNDWATER IABEE
FLOW LIUES
A'
PLANE FARALLEL TO GROUND SURFACE
AT DEPTH OF INVESTIGATION
N0 1e :
6 5.
eat j
Wc
sW g
Oet
Ojte
W o 'Pe.
-10an equipotential line is equal to (Z-Z )cos^O.
plane A-A'
The neutral stress on
is the pressure head multiplied b y the unit weight of water
u =
(6 )
y (Z-Z )cos^Q
W
W
The effective stress, (T , is found b y subtracting the neutral
stress from the total stress:
+ %Lat(Z-Zw)c°s^Q - ^(Z^Zjcos^G
Because the buoyant unit weight, Y ,,
D
( 7 )
is equal to ( Y- Y ) , Equation (?)
SSlj w
can be written as:
a
(8 )
= YZ cos^O + Y (Z-Z jcos^Q
t w
b
w
Because water cannot resist shear stress at negligible velocities, the
shear s t r e s s , 1C , is not affected b y the neutral stress: ' Equations. (.5.) -■
a
.
and (S) are' the basic relationships for the effective'stresses at a given
depth and on a given plane of investigation.
With this information, the
stress condition (Q^, lCq ) may be adapted to a Mohr diagram.
Stresses cannot be added vectorially because they are not vector
quantities, but because they are' acting on the unit area of plane A - A ' ,
t h e 'normal and tangential stresses can be used interchangeably with normal
and tangential forces.
The subsequent development makes use of this
resolving of stresses in order to define obliquity angles
(the angle
between the resultant and the normal effective stress).
There are three different conditions for which the obliquity angle
-11can be defined:
Case I.
Plane A - A 1 above the water ta b l e .
When plane A-A'
is
above the water table the angle of obliquity is equal to the slope angle,
0, as shown in Figure 6.
Case II.__Plane A - A 1 below the water table with the water table
coinciding, with the ground surf ace.
The obliquity angle, (3, is defined in
terms of effective stresses as shown in Figure 7.
The tangent of the
obliquity angle is found b y dividing the tangential stress by the effective
normal stress:
tan(3
=
% s a t (Z-Zw)SinQcosQ
--------------5-----—
cos^Q
( 9 )
Solving Equation (9) for the angle of obliquity, (3, yields:
( 10 )
(3 = tan 1
Case III.
Plane A - A 1 below the water table with the water table-
below the ground surface.
For this general case, shown'in Figure 8, the
tangent of the obliquity angle, Y ,
is the quotient of the tangential stress
and the effective normal stress:
Z sinQcosO +
tan1
/'
v VJ
a^cos^O
/
,(Z-Z )sinOcosO
S Elu
W
4- a % ( Z - Z j cos^G
( 11 )
-12-
B'
Figure 6.
Stress relationships for plane A - A 1
above the water table.
with the water table coinciding with the ground s urface,
-13-
STRESSES CAUSED BY
SOIL ABOVE THE WATER
TABLE
RESULTANT
STRESSES CAUSED BI
SOIL BELOW THE WATER
TABLE
Figure 8.
Stress relationships for plane A-A' below the water table
with the water table below the ground surface.
-1 4-
Rewriting Equation (11) and solving for the obliquity angle yields
tan
^ a t (Z-Zw>
-I
■tf.z
t W
+
yb<z-v
tanO
( 12
CHAPTER III
M O H R ’S CIRCLE OF STRESSES
The Mohr circle of stresses Is used to find the state of stress on
any plane perpendicular to the plane of elemental volume of soil, A A 1B B 1,
shown in Figure 2.
The sign convention for stresses
is that commonly'
used in soil m e c hanics, normal stresses in compression are positive and
counterclockwise shear stresses are positive.
When the effective stresses,
(Tq and
are plotted on Mohr's coordi
nates, t h e .obliquities of the various components are as shown in Figure 9.
Note that the stress condition on plane A-A'
state of stress
diagram,
The
( ( T 3 tC ) will lie on line DE if plane A-A' is above the
a '
a
water table, or on line EF if plane A-A'
state of stress
is a function of depth.
is below the water table.
The
( CT , tC ) shown in Figure 9 can be adapted to a Mohr
a
a
as shown in Figure 10.
The strength characteristics of a soil material may be defined by
Coulomb's e q u a t i o n : ■
s = c + (Ttan0
where:
(13)
s = the shear strength of the soil material;
• c = the unit cohesion;
CT = the normal effective stress;
and
0 = the angle of internal friction.
Equation (13) is shown graphically in Figure 10, and this graphic illus­
tration of Equation (13) is k n o w n 'as the Mohr rupture envelope.
The
leaves of the Mohr rupture envelope, MN and M' N' , shown in Figure 10,
(+)
STRESSES CAUSED BY SOIL ABOVE THE
WAT E R TABLE
Figure 9.
Stresses on plane A-A'
STRESSES CAUSED BY
SOIL BELOW THE
WATER TABLE
shown on Mohr's coordinates
S=C'
■ACTIVE STRESS CIRCLE
-
17
-
PASSIVE STRESS CIRCLE
Figure 10.
Mohr diagram for active and passive stress conditions.
-18re present the strength of the soil based on Coulomb's equation.
A state
of stress above MN or below M'N 1 will cause failure of the slope, whereas
a state of stress that lies within the envelope may exist in a stable slope.
The leaves of the envelope define impending failure conditions.
The main objective of this development is to find equations for two
limiting values of stresses,
of.stress.
commonly called the active and passive states
When the soil is on the verge of failure it will be in a state
of plastic- equilibrium.
■In the active case the lateral pressure is relieved and elongation
occurs parallel to the direction of pressure release, as shown in Figure
11.
The. lateral pressure, P , shown in Figure 11, is the minimum which
will maintain stability.
This state of stress is shown by the active
stress circle in Figure 10.
In the passive case the lateral pressure is increased and compres­
sion occurs parallel to the direction of pressure increase, as shown in
Figure 12.
The lateral pressure, P , shown in Figure 12, is the maximum
which will maintain stability.
This state of stress is shown by the pas­
sive stress circle in Figure 10.
Figure 10 shows that three conditions must be met for a soil to be
in an active or passive state of stress.
These conditions a r e :
stress circle must be tangent to the Mohr, failure envelope,- 2)
of the stress circle must lie on the T a x i s , and 3)
•must pass through the point of known stress
I ) . The
the center
the stress circle
((T , ¥ ! ) •
a a
A modification of the Mohr's circle of stresses will be used herein.
This modification, the origin of planes, allows a better visualization of
-19-
Figurs 11.
Active state of stress
Figure 12.
Passive state .of stress
-20the stresses and the orientation of planes on which the stresses act than
the conventional Mohr's circle method.
The origin of planes, OP, is shown
in Figure 10 and is obtained as follows:
1)
Through the point of known stress condition, draw a line
oriented identically to the plane on which the known stress a c t s .
2)
The intersection of this line and the circle of stresses
locates the origin of planes
3)
•
The stress condition on any plane can be found by constructing
a line oriented identically to the plane on which the stresses are to be
f ound.
4)
The point where this line again intersects the circle of
stresses yields the desired stress condition
The stresses for the active case
^
•
((T , Tl )ac^. and the passive case
((T ,1C )
are shown for planes oriented 45° clockwise from the horizonc ’ c pass
^
tal.
CHAPTER IV
ANALYTICAL DEVELOPMENT
The problem is to develop expressions for the normal and tangen­
tial stresses for the active and passive states of stress, with the angle
of the i n v e s t i g a t i o n , b e i n g one of the variables.
these expressions are:
inclination, 0;
Other variables in
the known stress condition ((^,Va );
the slope
and the strength characteristics of the soil,
c and 0.
There will be no need to develop separate equations for
(<f
C
,1C
)
C p 3.S S
a c t, and-
because the development results in a quadratic equation that
yields solutions for both the active and passive states of stress.
The equation for a line parallel to the ground surface and passing
through point (Of ,1^), as shown in Figure 13, is:
tC
=
0" tan9 + b
( 14
where b is the intercept.
Substituting the known stress condition ((Tq ,Yq) , given b y Equations
(5)
and (8), into Equation (l4) yields:
y.Z sinQcosQ +
t w
^ (Z-Z )sinOc'osO
sat
w
P
= Y z cos^QtanO +
t w
.
.
2 -
(15
X (Z-Z ) cos OtanO + b
b'
w
.Solving for b from Equation (15):
b =
(JTsat- ^ j) ^(Z-Zw ) sinQcosoj
Recalling that Jfy =
(^gat""X)^ ’ scIuatiori (1&) may be written as:
b = X sin0(Z-Z )cos0
W
( 16
W
'
(17
D ’ b pass
Figure 13.
Active and passive stress circles and their
relationship to Mohr's rupture envelope.
-23.Equation (17) is recognized as the equation-for the seepage fohce per
unit a r e a , S: ■
,
S =
.■
'
= % sinG(Z-Z )cosG
where:
'
- ( 18 )
i = sin9 (the hydraulic gradient as. shown in FigureL5);
V = '(Z -Z ) cos'O (the volume of the column 'of',soil between
W
-' •
•*,
the water" table and plane A - A ’ shown in Figure 2).•
'The intercept,, b, is therefore equal to.the seepage force per unit area, S.
Substituting the' expression.-' for b, from Equation (17) , into Equation (14) , .
yields the equation for a line through point (fa j.^a )? w i t h :a slope'/equal'
to tanO:
1C
'. '
'
•.
- TtanO +. Xw (Z-Zw ) sinOcosO = Tta n O + S
"'
,
■
( 19 )
From Equation (13) and Figure 1-3, t h e .equation for a leaf of the
Mohr rupture envelope can .be written as-:
( 20 )
Wj - Ttan^ ~ c ~ 0
The normal form.of Equation (20) is the expression for all lines .perpendic­
ular t o ■the -leaf of the failure envelope:'
,
■
.
Ttan^.
= -0
/l+tan^. vn+tan^^
( 21 J
i/l+tarfV
Because the stress circle is tangent.to the failure envelope,., the radius ■
of the stress circle, r, is perpendicular to the failure envelope at the
point of tangency.
Noting that V
equals zero, the length of the radius is.
-24found by substituting the coordinates for the center of the circle of
stresses
((T03V0) into Equation (21):
r =
(LtanjZf
c
- ---■■■
—
—
y.1+tan^0 y/l+tan^0
(
The general equation for the circle of stresses passing through point
((Ta JtCa ) is, therefore:
(rotan0+c)^
------ L = o
( 23 )
1+tan20
The origin of planes,
(^-,/Iq3) , is defined by the intersection of the
circle of stresses and the line parallel to the ground surface that passes
through the point (^a ^ a) •
The general equation for this intersection is
found by substituting the expression for tC from Equation (19) into Equation
(23).:
._
9
„
(<rotan0+c)2
(<T-rnr + (CtanGH-S)Z -------- = O
l+tan^#
( 24 )
Equation (24) can be rewritten to yield a quadratic in (T:
^ 2 ( U t a n 2O) +
f (-2fo+2tan9S)
-2 ' 2
(Gotan0+c)^
. -Hf + S2 ----------- = 0
0
( 25 )
I+tan20
Equation (25) will lead to two solutions for (T:
one solution ((T ) will
-25correspond to the point of known stress
( ( ^ , T );
(l
Tb ) will correspond to the origin of planes
and the other solution
(l
Tb
•
Equation (25) can
be solved b y means of a "sum of roots" solution (Rosenback, et al, 1958 )with the following result:
2(T0 -2tan9S
<T„ + (Tv
( 26 )
I+tan^Q
B y substituting 0“
for (T, Equation (24) can' be rewritten as a
quadratic in T , in which T
O
o
is the only unknown:
+ Tp(-2fa-2Tatan^0-2tan0c)
"2
2
2
2
2
+ T" (I+tan 0+tan 9+tan 0tan 9)
a
( 27 )
+ 2ftan9S(l+tan^0) + S^fl+tan^) a
= 0
-
Solving Equation (27) b y means of the "quadratic equation" and r e ­
arranging, yields:
2
(I+tan 0) + tan0c T
a
i/
p
'
] - f (I+tan 0 ) -tan0cV
Lk a
J
( 28 ).
- ( f (I+tan^0+tan^9+tan^0tan^9)
,i
+ 2(Tatan9S(l+tan^0) + S' (I+tan^0)
- c^
The positive root defines the center of the passive stress circle and
the negative root defines the center of the active stress circle.
Solving for (T
Equation (26), yields:■
-262 2tan9S
__o__________
- r
(l+tan^Q)
where:
( 29 )
a
(Tq is defined b y Equation (28) ;
S is defined b y Equation (l8 );
and
is defined b y Equation (8 ).
Because (f has tyo r o o t s , it follows that (f will also have two roots, one
o
b
for the active origin of planes and the other for, the passive origin of
planes, as shown in Figure 13.
An equation for H can be developed, in
b
terms of the defined quantity, (f , by substituting (f into Equation (19):
b
b
T, = (TtanO + ^ (Z-Z ) sin9cos9 = (f tanO + S
D D
W W
*j^,.
( 30 )
With the expressions, Equations(29) and (30), defining the origin of planes,
the state of stress,
be obtained.
((T^j tC ) , on any plane of investigation, oJ, can now
The conventions and notation for the typical stress circle
are shown in Figure I A.
/
■
The general equation for the slope of the line, GE, that passes
through point (G^,!^), as shown in Figure 14, is:
tanoC = ----^
Therefore,
( 31 )
b
the equation of the line that passes through point (if ,Y ),
b b
and has a slope of <%, i s :
lC = tC + ((T-IT Hanot
b
b
( 32 )
H
REFERENCE
HORIZONTAL
Figure 14.
Typical stress circle.
-28the state .of stress on a plane that is oriented at an angle c*£ can be
obtained b y developing an equation for the intersection of the stress
circle and line GE.
The equation for the intersection is found by substi­
tuting the expression for
T
from Equation (32) into Equation (23):
((Totan 0 +c)
+
( 33 )
1.+tan^0
Equation (33) can be rewritten to yield a quadratic in (f:
(T2O h ta n 2Oc) +
(r( - 2 f + 2 'r btanc=c-2fb ta n 2-c)
+ (<T + 1C -21? (T, tanoc+T ta n oc)
o
b
b b
b
( 34 )
2
(crota n ^+ c)
■I+tan^0
The two roots of Equation (34) are determined b y -the "sum of roots"
technique.
One root is the normal stress at the origin of planes, (T^,
while the other root is the normal stress on the plane of investigation, (T
c
2<T -ZiC tan^+2(T tan2=<
(T
= — 2___ ^______ _ b _______ ff=-
C
.I+tan^c
( 35 )
.b
Because all terms on the right-hand side of Equation (35) have' been
previously defined, the only u n known, (T , is expressed in terms of defined
c
para m e t e r s .
An alternate form of Equation (35) can be written by .substi­
tuting the expression for-
from Equation (30) into' Equation (35) :
-29-
2
—
_
2<t + (T (-I +tan-2tan°£.tan9)-2tan«;S
r = - 2 ----- ^ ------ 5--------------------c
I+tanid
( 36 )
An equation for 1C can be developed by substituting (f , into Equation .('32)
c
c
T
c
= %
b
+ (f-C)tand.
Equations
c
b
( 37 )
(36) and (37) are general equations for the normal and
tangential stresses on a plane at any inclination, when the soil within
an infinite slope is at the active or passive state of stress.
CHAPTER V
DISCUSSION
EQUATION COMPATABILITY
As a means of verifying this analytical development, the equations
presented will be compared to those developed b y other investigators.
Martin (1961)■'has shown that his solution for active and passive pressures
on a vertical plane agrees with developments b y various other investigators
Because this paper is an extension of Martin's work, demonstrating that
the equations presented herein agree with Martin's equations would also
imply agreement with developments by other investigators.
The proof of compatibility is accomplished b y showing that Equation
(35), for the normal stress (T , reduces to Martin's equation for normal
stress;
therefore the restrictions
vertical plane)
(no seepage forces and stresses on a
on Martin's development must be applied to Equation (35).
Recalling the trigonometric identity,
(35)
(l+tan^) = (—
, Equation
can be rewritten:
(f = 2(T^cos^L - 2 f ^sih^cosoi + 20^3in^ -
( 38 )
The normal stress on a vertical plane is found by setting cl equal to 90°
in Equation (38 ):
Therefore the normal stress on a vertical plane is equal to the normal
stress at the origin of planes, (Pb in Equation (29).
CTb can be adapted
to the no seepage force condition by setting S equal to zero in Equation
(29):
-31% (Tp
( 40 )
l+tan^O
Substituting the expression for
and recalling that (Tq equals
_
from Equation (28) into Equation (40),
for this special case:
20^ + 2fl^tan20 + 2ctan0
I+tan^O
(-2ffa -25L
a tan20-2ctan0)2
(l+tan20)2 '
4 |^(r^(l+ tan20+tan20+tan 0tan2O) - c^j
(I+tan2 O)2
-^a
Equation (41) is exactly; the same
(except for notation) as Martin's equation
for the normal stress on a vertical plane. ■ Therefore the general equationsgiven in this analytical development agree with equations, given by other
investigators for their more restrictive conditions.
CRITICAL DEPTH
The soil properties, slope of ground surface, depth of the water
table, and depth to the plane of investigation will all affect the stress
condition (Oua ,1Cq ) .
point
Active and-passive stress circles can be drawn through
(^a J1Ca ) if, and only if, this stress condition remains within the
Mohr envelope.
The Mohr diagram, with the stress conditions
('Ta J1C l) super­
imposed, is used to s h o w .conditions where failure is impending.
The crit­
ical depth can be defined as the depth at which the stress condition on
-32plane A-A' is equal to the strength of the soil.
tion,
drawn.
When the stress condi­
lies on the Mohr envelope, only one stress circle may be
For this condition, the active and passive stress circles coin­
cide, and failure is impending.
The stress on plane A-A',
(lTa >Yg ) , at the critical depth, in a
cohesive material, is shown in Figure 15.
I)
no critical depth exists;
2)
a critical depth exists;
3)
(3ZQ>0, a critical depth exists.
If:
and
.
.
.
The stress on plane A-A' , (^a J1Cg ) , at the critical depth, in a
cohesionless material; is shown in Figure 16.
If:
1)
0»(3>O, no critical depth exists;
2)
0=(3>Q, no critical depth exists;
3)
0=(3=O, a.failure condition exists at all depths;
4)
(3>0 ?O, a critical depth exists;'
5)
(3?O?0, a failure condition exists at all d e p t h s .
and
The equations for critical depth are found b y equating stress
conditions to strength conditions.
Equations
Equating the right-hand sides of
(19) and (13), yields:
(TtanO + S = CTtanjZf + c
Substituting the expressions for (f and S from Equations
a
( 42 )
(8) and (18)
into Equation (42) and solving for the critical depth, Z c r , yields:
-33(+)
C
___L
Figure 15.
Stress condition at the critical depth when
the unit cohesion is greater than zero.
Figure 16.
Stress condition at the critical depth
when the unit cohesion equals zero.
—34-—
c Z
cr
= Z +
w
Y.Z cos
(tanO-tan0)
----- §----- --------------------- 4
X cos O(tanO-tan0) + V sinOcosQ
D
W
For the special case of no ground water (Z-Zw ) is zero.
tions
(8) and (l8) for Z
Z
( 43 )
Solving Equa­
:
c
= — --- -------------cr
Xj.cos^9 (tanO-tan0)
( 44 )
which is the critical depth for the special case of no ground water.
Note that in all the equations presented, a slope will be stable as ■
long as the depth of the soil material is less than the critical depth.
Figure 17 shows the stress condition at the critical depth for a
typical slope.
This figure is also helpful for visualizing the expres­
sions presented in previous chapters for the stresses acting on plane
A-A' .
I,
r
y+zWcos^o
U
Figure 17.
Stress relationships at critical depth.
CHAPTER VI
NONDIMENSIONAL FORMULATION
In order to reduce the number .of variables,
the equations presented
in the analytical development were nondimensionalized.
Equation (36) for
(Tc was expanded and divided b y c, the unit cohesion, to determine the basic
dimensionless parameters:
1)
Y =
2)
Y1 =
3)
X =
L tz-zO
= Y' —
y.
b
The development o'f the nondimensional expressions parallels the ana­
lytical development.
It was necessary to define additional parameters for
the equations which lead to the final nondimensional expressions for the
normal and tangential stresses.
Because each nondimensionalized equation
has an analog in the analytical development, the equations are presented
with few details.
The nondimensional f o r m of the effective normal stress on plane A-A'
(T, is obtained from Equation (8):
N" = —
a
= Ycos2Q + Y 1COs2Q
( 45 )
c
The nondimensional form of the seepage force per unit area, S, is obtained
-37f r o m Equation (18)
v = - =
c
XsinOcogO =
("0
sinOcosO
( 46 )
The nondimensional form of the tangential stress on plane A - A 1 ,
tC , is
obtained from Equation (5) :
T" = — = N"tanO + XsinOcosO
a
.c
a
( 47 )
The- nondimensional form of the center of the stress circle,
is obtained
from Equation (28):
N" = — = N M (l+tan^0) + tanjZl
°
c
a
+ i ^ n 2 ( i + tan^0)^ + 2N" (1 +tan^0) tan0 + t a n ^ j
-N" ^ (I+tan^jZf+tan^O+tan^Zftan^O)
a
- 2N"vtanO (I+tan^0)
a
( 48
-v (I+tan 0) + I;
The nondimensional form of the normal stress at the origin of planes, (T ,
b
is obtained from Equation (29) :
(Th
2
= — =. 2cos 0 (r-vtan-O)
- M”
( 49 )
-38The nondimensional f o r m of the tangential stress at the origin of planes',
1C ,
is obtained from. Equation (30):
b
V
= N^tanO + v
( 50 )
The nondimensional form of the normal stress on a plane at the depth of
investigation oriented at angle
(T, is obtained f r o m Equation (36):
2
2
N"
c
COS <k. [2-N"+N" C-I +tan =c ■-2tanoCtanO)
( 51 )
The nondimensional form of t h e .tangential stress on a plane at the depth
of investigation oriented at angle °C, lC j, is obtained from Equation. (37) :
V
T" = —
C
g
= T" + (N"-N")taneC
b
( 52 )
c b
Figure 18 shows the notation on a nondimensional Mohr's diagram,
which is analogous to Figure 13.
\
The equations for the nondimensional analogy of the critical depth
were developed in the same manner as Equations
(42) through (44)•
The
critical depth in terms of the nondimensional parameters is:.
I - XsinQcosO
= --------------
(I + Y')
cr
where:
,
.
( 53 )
cos^Q(tanQ-tanjZi)
(Y + Y')
is the nondimensional form of the critical depth
cr
b act
Figure 18.
Nondimensional Mohr diagram.
-40For the special case of no groundwater table, the nondimensional critical
depth is:
I
Y
cr
( 54 )
co s ^ S (tan9-tan0)
CHAPTER VII
RESULTS
DIMENSIONAL F O R M
The equations presented in the analytical development were pro­
grammed for solution on an IBM 1620, Model II, digital computer.
The
slope inclination 0, the depth to the water table Zw , the depth of inves­
tigation Z, and the soil properties 0, c,
and YL were assigned appro­
priate values and the active and passive stresses acting on a plane at
inclination <*. were determined.
The angle <*. was incremented from O0 to 360°
in order to solve for the stress on selected planes.
A listing of this
computer program is shown.in Appendix A and typical results are shown in
Tables II and III.
—4-2—
TABLE LI
ACTIVE STRESSES
unit cohesion,
c = 1000.O p s f
depth to water table, Z
angle of internal
f r i c t i o n , .0 = 30°
ground surface
inclination, 0 = 20°
soil unit weight above the
water table,
100.0 pcf
bouyant unit
weight, ^ = 65.0 pcf
Z
DEPTH TO PLANE' OF
INVESTIGATION
10
..
OL
ANGLE OF PLANE
OF INVESTIGA­
TION*
1 5 .0
"
_£=
NORMAL EFFECTIVE
STRESS ON PLANE
OF INVESTIGATION
-
3 0 .0
4 5 .0
6 0 .0
7 5 .0
9 0 .0
3 0 0 .0
1 7 2 .0
5 8 6 .4
8 4 3 .7
8 7 4 .9
6 7 1 .6
2 8 8 .4
- 1 7 2 .0
- 5 8 6 .4
- 8 4 3 .7
- 8 7 4 .9
- 6 7 1 .6
- 2 8 8 .4
- 7 9 2 .4
= 8 2 3 .6
- 6 2 0 .3
- 237 .1
2 2 3 .3
6 3 7 .7
8 9 5 .0
1 5 .0
3 0 .0
4 5 .0
6 0 .0
7 5 .0
9 0 .0
2 8 5 .0
3 1 5 .0
3 3 0 .0
3 4 5 .0
3 6 0 .0
9 2 6 .1
7 2 2 .9
3 3 9 .7
- 535.1
3 0 0 .0
3 1 5 .0
3 3 0 .0
3 4 5 .0
3 6 0 .0
20
V 5
TANGENTIAL EFFECTIVE
STRESS-ON PLANE OF
INVESTIGATION
-120.7
105.0
1 8 6 1 .7
1483.1
9 0 0 .4
2 6 9 .6
- 2 4 0 .0
- 4 9 2 .2
- 4 1 9 .2
-
-
= 2 0 . 0 ft.
'
4 0 .6
542.1
1 1 7 2 .8
1 6 8 2 .6
1 9 3 4 .7
is measured counterclockwise from the horizontal.
4 5 1 .6
9 6 1 .3
1 2 1 3 .5
1 1 4 0 .5
7 6 1 .9
179.1
- 4 5 1 .5
- 9 6 1 .3
- 1 2 1 3 .5
- 1 1 4 0 .5
- 7 6 1 .9
- 179.1
TABLE III
/
PASSIVE STRESSES
unit cohesion,
c = 1000.0 psf
depth to water tab l e , Zw = 20.0 ft.
angle of internal,
friction, 0 = 30°
ground surface
inclination, 0 = 20°
soil unit weight above the
water t a b l e , ^ = 100.0
bouyant unit
weight,
= 65.0 pcf
OL
Z
DEPTH TO PLANE OF
INVESTIGATION
ANGLE OF PLANE
OF INVESTIGA­
TION*
NORMAL EFFECTIVE
STRESS ON PLANE
OF INVESTIGATION
240.0
-
20
2 5 5 .0
2 7 0 .0
2 8 5 .0
3 0 0 .0
3 1 5 .0
150.0
1 6 5 .0
180.0
1 9 5 .0
2 1 0 .0
' 2 2 5 .0
2 4 0 .0
2 5 5 .0
2 7 0 .0
2 8 5 .0
3 0 0 .6
3 1 5 .0
TANGENTIAL EFFECTIVE
STRESS ON PLANE OF
INVESTIGATION
4 2 2 1 .8
1 5 0 .0
1 6 5 .0
1 8 0 .0
1 9 5 .0
2 1 0 .0
2 2 5 .0
10
1C =
eL
-
2 8 7 9 .4
1 6 9 2 .0
9 7 7 .9
9 2 8 .3
1 5 5 6 .6
2 6 9 4 .4
4 0 3 6 .8
5 2 2 4 .2
5 9 3 8 .3
5 9 8 7 .8
5 3 5 9 .6
. ,
.
6 3 6 2 .2
4 5 9 1 .9
2 9 6 9 .2
1 9 2 8 .9
1 7 4 9 .8
2 4 7 9 .8
3 9 2 3 .4
5 6 9 3 .7
7 3 1 6 .4
8 3 5 6 .7
8 5 3 5 .8
7 8 0 5 .8
* oc ig measured counterclockwise from the horizontal.
.
2480.1
2 5 2 9 .7
1 9 0 1 .4
7 6 3 .6
- 5 7 8 .7 .
- 1 7 6 6 .0 .
- 2480.1
- 2 5 2 9 ,7 ..
- 1 9 0 1 .4
- 7 6 3 .6
5 7 8 .7 .
1 7 6 6 .0
3 2 1 3 .8
3 3 9 3 .0
2 6 6 2 .9
1 2 1 9 .4
- 5 5 0 .9
- 2 1 7 3 .6
- 3 2 1 3 .8
- 3 3 9 3 .0
- 2 6 6 2 .9
- 1 2 1 9 .4
5 5 0 .8
2 1 7 3 .5
-44NONDIMENSIONAL F O R M
The equations presented in the nondimensional formulation were
programmed for an IBM 1620, Model II, digital computer. ' The parameters
0, 0, Y, and Y 1, were assigned various levels, and the nondimensional
values representing the active and passive stresses, acting on a plane
at inclination o^, were determined.
The angle «.was incremented from
0° to 360° in order to solve for the stress on selected planes.
The
parameters 0, 9, Y, and Y ' , were then systematically varied to provide
a comprehensive tabulation of results.
Typical results are presented
in Tables IV and V.
As an e x a m p l e ,'the resulting nondimensional tables' may.be used as
f ollqws:
Let:
0=0°
-
G = 20°
■ Z' = 4.0 ft.
■
W
Z = 6.3 ft. . '
'
'
c = 320 psf
= 41.60 p cf
Kt = 80 p cf
OL= 75 °
It is desired that the active stress for the' above conditions be deter­
mined.
The nondimensional parameters Y and Y' are calculated as follows
SLz
Y = — ^
c
(80)(4.0)
= ---------
320
= I .0
-45TABLE IV .
NONDIMENSIONAL ACTIVE STRESSES
angle of internal
friction, 0 = 0 °
,
0
)
ground surface
inclination, 0 = 20°
y , (2 )
T" (5)
N" (4 )
otO)
C
C
I .00
15.0
0 .3 0
3 0 .0
.
4 5 .0
6 0 .0
7 5 .Q
9 0 .0
2 8 5 .0
.
3 0 0 .0
0 .6 0
1 5 .0
.7313
1 .1 3 2 3
1 .3 1 6 0
'
3 0 .0
4 5 .0
6 0 .0
7 5 .0
2 7 0 .0
2 8 5 .0
3 0 0 .0
3 1 5 .0
3 3 0 .0
3 4 5 .0
3 6 0 .0
^ Y is a nondimensional parameter
^Y'
.4103
.8113
.9949
- .9119 '
.5846 "
.1006
-<4103 .
'- . 8 1 1 3
- .9 9 4 9
- . 9119 . '
.
.2204
3 1 5 .0
3 3 0 .0
3 4 5 .0
3 6 0 .0
I .00
1 .2 3 3 0
.9056
.4217
- .0 8 9 6
- .4 9 0 2
- .6 7 3 8
- .5 9 0 8
- .2 6 3 5
(Y = .
-
-.5 8 4 6
-.1006
1 .5 4 3 3
1.1021
. 5846
.1296
-.-1411
r .1550
.0916
L5328 .'
1 .0 5 0 3
1 .5 0 5 4
I .7761
1 .7 9 0 0
'
I
.6878
.9586
.9725
.7258
.2846
- .2 3 2 8
- .6 8 7 8
-.9 5 8 6
-.9 7 2 5
- .7 2 5 8
-.2 8 4 6
.2328
/c).
is a nondimensional parameter (Y' = ^ ( Z - Z w )/ c ) .
3
^ is the angle of the plane of investigation as measured counter­
clockwise from the horizontal.
is the nondimensional form of the effective stress on the plane
of investigation.
is the nondimensional form of the tangential effective stress on
the plane of investigation.
-46TABLE V
NOWDIMENSIONAL PASSIVE STRESSES
angle of internal
friction, 0 = 0 °
ground surface
inclination, 0 = 20°
y , (2 )
Y (1 )
I .00
-
0 .3 0
\
..
.«<
NM(4 )
3)
C
1 6 5 .0
1 8 0 .0
1 9 5 .0
2 1 0 .0
2 2 5 .0
2 .2 2 0 4
1 .7 0 2 9
1.2581
. 1 .0 0 5 4
' 1 .0 1 2 4
1 .2 7 7 2
1 .7 2 9 0
2 .2 4 6 6
2 .6 9 1 3
2.9441
2.9371
2 .6 7 2 2
2 40 .0
2 5 5 .0
2 7 0 .0
2 8 5 .0
3 0 0 .0
3 1 5 .0
3 3 0 .0
I '.00
0 .6 0
1 9 5 .0
2 1 0 .0
2 2 5 .0
240:0
2 5 5 .0
2 7 0 .0
2 8 5 .0
3 0 0 .0
3 1 5 .0
3 3 0 .0
3 4 5 .0
3 6 0 .0
is a _nondimensional parameter ( Y =
^ Y 1 is a nondimens ional parameter
Tm( 5 )
.
1 .5 6 1 4
1 .1 7 3 9
1 .0 0 9 9
1 .1 1 3 5
1 .4 5 6 7
1 .9 4 7 7
2 .4 5 4 9
2 .8 4 2 3
3 .0 0 6 3
2 .9 0 2 8
2 .5 5 9 5
2 .0 6 8 5
-
.9693
.9623
.6 9 7 4
.2457
- .2 7 1 8
- .7 1 6 6
-.9 6 9 3
-.9 6 2 3
- .6 9 7 4
- .2 4 5 7
.2718
.7166
.8946
.5 5 1 4
.0 6 0 4
-.4 4 6 7
- .8 3 4 2
-.9 9 8 1
- .8 9 4 6
- .5 5 1 4
- .0 6 0 4
.4467
.8342
.9981
^ Z w/ c ) ;
(Y' = ^-J3(Z-Zw)Zc)..
is the angle of the plane of investigation as measured counter­
clockwise from the h o r izontal.
4
N" is the nondimensional form of the effective stress on the plane
o£ investigation.
is the nondimensional form of the tangential effective stress on
the plane of investigation.
-478. ( Z-Z
I
' =
(4 1 .6 0 ) ( 6 .3 - 4 .0 )
,)
— ----- —
=
c
---------------------
»
0 .3
320
The upper portion of Table IV is applicable for the' example values of
0,
6, Y, and Y' ; ■ therefore, values are obtained for parameters N" and
T" from the line corresponding to the d value of 75°:
c
.
jp = -0.4902
T"
=
c
0 . 5,846
''
Since
ffc
= — , then (JT- = cN^ =' (320) (-0.4902) — -I 57.0: pounds, per square.■
c
■
■
foot, which is the active effective normal stress on a plane inclined at
vC
an angle od of 75°.
Since T^ = — , then
c
1C1
c
= cT" = (320) (0.5846].= 186.8 '
. . . .
pounds per square foot, which is the tangential stress on a plane inclined
at an angle
pc of 75°.
'
CHAPTER VIII
CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER S T U D Y .
CONCLUSIONS
When the Rankine assumptions are to be used in the determination
o f earth pressures, the equations presented herein can be applied to yield
the active and passive earth pressures, on a plane at any inclination,
within an infinite slope having a water table which is parallel to the
.
ground s u r face.
The solutions presented herein do not include pressures due- to
freezing or swelling of soil, hydrostatic water conditions, et cetera.
The solutions are documented for use when the Rankine assumptions•are,
valid, however, the author does not advocate the use of R ankine1s theory
in those cases where conjugate, stresses do not exist.
RECOMMENDATIONS FOR FURTHER STUDY •
A useful extension of this study would- be to develop nomographic ■■
solutions for the equations developed herein.
Another useful extension would be to integrate the expressions
presented herein for earth pressures in order to solve for earth thrusts.
Other possibilities are:
1)
Letting the Mohr envelope be of cur­
vilinear f o r m in order to simulate nonlinear soil strength with respect to
depth;
and 2)
defining the shape of the complete failure" surface.
APPENDIX
APPENDIX A
DIMENSIONAL COMPUTER PROGRAM
51
**********0 IMENS IONa L COMRUTER PROGRAM***********
C
STRESS ON ANY PLANE -INFINITE SLOPE THEORY
W. HARTSOG
c * * * * * * -X--X--X--X--X--X-jX-X-X-X-XrX-X-X-X-X X-X X-X X-X-X-X X X X X--X-X-X X X-X-X X-X-X X-X X X-X X X X X X X X-X-X-X-X X-X X-X X X X-X X- X-
READ I , COHES * PHID, GAMMA, THETAD
C OH ES = C OH ES ION P.S.F.
PH ID= FR IC T ION ANGLE IN DEGREES
GAMMA=UNIT WEIGHT OF SOIL ABOVE W.T. I N P . C . F .
THETAD=INCLINATI ON OF PLANE IN DEGREES
- I FORMAT (AFlOel)
PRINT 2, COHES, PHID, GAMMA, THETAD
20FORMAT (I O X ,IOHCOHESI ON =,F6.1,4H P S F »7X ,6HPHID =,F6.1»5H DEG. ,
I 7 X ,7HGAMMA =,F6.1,4H PCF,7X,8HTHETAD = , F 6 . I,BH D E G . )
READ 3 ,G A M M A B , ZW
C GAMMAB=BUOYANT UNIT WEIGHT,
ZW=VERTICAL DEPTH TO W.T.
3 FORMAT (2 F IO . I )
READ 33 ,N
C N IS INCREMENT OF ALPHA IN DEGREES
33 FORMAT ( H O )
READ 34, L,M
C L IS INCREMENT OF DEPTH
M IS MAXIMUM
DEPTH
34 .FORMAT (2110)
PRINT 4, N ,G A M M A B ,ZW
■
.
4 FORMAT (39 X ,3FIN = , I3 ,I 5'X ,8 HGAMMAB =,F6.1,4H PCF , I IX »4HZW .= ,F6 . I ,
I4H FT.)
PRINT 9
9 FORMAT (lH0'19X»46H**********************************************)
P 1=3.14159
PH I= (PH ID * P I)/180.
THETA= THETAD*PT/180.
GAMMAW=62.4
A = C O S F (T H E T A )
A2=A*A
B=SINF(THETA)
C
C
C
Ta NT=BZA
TANT2 = T ANT x TANT
. .H = SINFtPHI )/C O S F (P H I )
D= I .+H*H
GA2=GAMMA*A2
. GA2D=GA2*D
COHESH=COH ES*H
7-2 DO 51 IZ = L, M, L'
' Z= IZ
C Z=TOTAL DEPTH TO PLANE UNDER STUDY
IF(Z-ZW)20, 20, 30
. 20 Zl=Z
Z2=0.
'
'
52
GO TO 5
. 30 Zl=ZW
C
Zl=DEPTH OF SOILS ABOVE W 6T e
ZZ=Z-ZW
C ■Z2=DEPTH OF SOILS b e l o w W 0T e
5 s IGl=(Z 1*GAMMA+Z2*GAMMAB)*A2
C S i g i =NORMAL INTERGRANULAR STRESS o n PLANE INCLINED AT T h e T a
S=B*GAMMAW*Z2*A
'■
■
....
T I= S I G l * IA N T + S
c TI=S h e a r i n g s t r e s s o n p l a n e i n c l i n e d a t t h e t a
G O = S IG1*D+C0HESH
BOO=
( (-S IG1 *D -C 0HE SH )* * 2 )-( {SI G1 * S I G 1 )*(D + TA N T 2 + H * H * T A N T 2 )
1+2.*SIG1*TANT*S*D+S*S*D-COHES*COHES)
IF (BOO) 1001,1002,1002
1001 PRINT 1003,Z
1003 FORMAT (I H l ,3H. Z = ,E S . I ,68H**# ■FAILURE CONDITIONS EXIST AT THIS DEP
ITH WITH ABOVE PROPERTIES ***)
1004 GO TO 1005
1002 B O= SQ RT F (BOO)
SIGOA=GO-BO
C SIGOA=CENTER OF ACTIVE STRESS CIRCLE
S IGOP = GO + BO
C SIGOP = CENTER OF PASSIVE STRESS. CIRCLE
5IG2A=((2.*STG0A-2.*TANT*5)/(1.+TANT2))-SIG1
C S IG2A = NORMAL INTERGRANULAR STRESS AT ORIGIN OF PLANES FOR ACTIVE CASE
S IG 2 P =( (2•* S IGO P — 2•* TA N T * S )/(Ic + T AN T 2) J-SIGl
C S IG 2 P = N O R M A L ■INTERGRANULAR STRESS AT ORIGIN OF PLANES FOR PASSIVE CASE
T2 A = S IG2 A* T ANT + S
C T2A=SHEAR AT'O.P. (ACTIVE CASE)
T2P=SIG2P*TANT+S
C T 2 P=SHEAR AT O.P. (PASSIVE CASE)
C -.FOR ABOVE EQUATIONS REFER TO NOTES
DELA = ATANFI (-S IG 2 A + S IG O A )/THA )
IF(DELA) 50,59,60
59 IF (T 2 A ) 50,60,60
" '
50. DDELA = I80. + ( (I 8 0 ./P I )*DELA)
C DDELA=ANGLE TANGENT MAKES WITH HORIZONTAL (ACTIVE) -NOTES 13-17
GO TO 201
60 DD EL A=( I 80•/P I)*DELA
201 D E LP =A T AN FI (- S IG2P+5IG0P)/T2P)
.IF(DELP) 100,209,200
209 IF (T 2 P ) 500,400,400
200 IF (T 2P ) 300,400,400
4.00 DDELP=I 180./PI)*DELP
C DDELP = ANGLE TANGENT MAKES-WITH HORIZONTAL (PASSIVE) -NOTES 13-17
GO TO 80
300 DDELP=180,+ ( (I 8 0 oZ P I )* D E L P )
GO TO 80
100 if (T 2 P ) 500,600,600
53
'500 DD EL P=I 8 0 c+ ( (I 80»/P I )*DEL P )
GO TO 80"
600 DDELP = 360»+( (I 80 «/ P I )*DELP)
80 PRINT 6 » Z,DDELA
6 FO R M A T {IH l »3H Z = »F5. I »11 X »I3HDELTA ACTIVE=,F 6 .I I
PRINT 7
7 F O R M A T (I H O / / / ?I9X,A H S I G l ,13X,2H T I ,I 3X ,5 H S IG O A »11X ,5HS IG 2 A »I 2 X ,
I3 H T 2 A )
PRINT 21, SIG1»T1»SIG O A ,S IG2 A ,T 2A
21 FORMAT (9 X ,5 F I 6 oI )
PRINT 23
23 FORMAT (I HO ,2 5X-,5‘HALPHA ,2 I X ,5HS IG3 A ,12 X ,3HT3 A ,I 3X ,3HP3'A )
DO 24 IALPHA= N ,3 6 0 ,N
AL PH A D = !ALPHA
C ALPHAD=ANGLE OF PLANE FOR WHICH STRESS CONDITIONS ARE TO BE FOUND
IF (ALPHAD-DDELA) 15,15,16
16 ALPHAD=ALPHAD+loO.
IF(ALPHAD-360.) 15,15,92
92 GO TO 1O 5
15 ALPHA =( ALP HAD^PI)/180.
TANA = SINFf AL PHA )/COSFfALPHA)
‘~
"
T a NA2=T a NA*T a NA
IF (AL P H A D - 9 0 .) 44,40,44
44 IF (ALPHAD-270.) 48,40,48
48 5 I G 3 A = ( (2.*SIG0A-2.*T2A*TANA + 2.*5I G2 A* TA NA 2)/(I »+ T AN A 2) J-SIG2A
C S IG3A = NORMAL STRESS ON PLANE AT ANGLE ALPHA -(ACTIVE)
T3A=T2A+(S IG3A-SIG2A)*TANA
C T3 A= SHEAR STRESS ON PLANE AT ANGLE ALPHA - ( A C T I V E ) ■
GO TO 41
'
.
r
4 0 S IG3A=SJ G2 A
.
T3 A = - T2 A
41 P3 A = S Q R T F (S IG 3 A * S IG3A+ T3A*T3 A ) 'C
P3A.= RESULTANT STRESS ON PLANE AT ANGLE ALPHA' -(ACTIVE)"
24 PRINT 25 ,A L P H A D ,S IG 3 A ,T 3 A ,P 3 A '
25 FORMAT (IHO ,2 4 X ,F 6 . I ,9 X ,3 F I6 .1 ) '
•
105 PRINT 6 6 ,Z,DDELP
' 66 FORMAT (1H1,3H Z = ,F 5 . I ,11 X ,14 HD ELTA PASS IV E = ,F 6 . I )
PRINT 70
70' FORMAT (I HO// / ,I 9X ,4HS IGl , 13X,2HT1 ,1 3 X ,5HS IGO P ,11 X ,5HS IG2 P ,I 2 X ,
13HT2P)
PRINT 21, S I G l ,Tl,SIG0P,SIG2P,T2P
PRINT: 230
2 30 F O R M A T (I H O ,25 X ,5HALPHA ,2 I X ,5 H 5 IG 3 P ,I 2X ,3HT3P ,I 3X ,3 H P 3 P )
IF (S IGO P-S IG2 P ) -81,82,82
82 DO 78 IALP HA =N ,360,N
■ AL P H A D = !ALPHA
IF (ALPHAD-DDELP) 115,115,116
116 AL P h a d = AL P H A D + I 80.
54
IF(ALPHAD-360.) 115,115,71
71 GO TO 51
115 ALPHA ={ ALPHAD-x-p I )/180.
- TANA = SINFf ALPHA)/COSF{ALPHA)
TANA2=TANA*TANA
C
IF (ALPHAD-90.)76,74,76
76 IF (AL P HA D- 2 70 .) 73,74,73
73 S I G S P = I (2.*SIGOP-2.*T2P*TANA+2.*SIG2P*TANA2)/ (1.+TANA2))-SIG2P
S IG3 P = NORMAL STRESS ON PLANE AT ANGLE ALPHA - ( PASSIVE)
C
T3P=T2P+(5IG3P-SIG2P)*TANA
TSP=SHEAR STRESS ON PLANE AT ANGLE ALPHA -(PASSIVE)
GO. TO 75
74 SIGSP=SIGZR
TSP=-TZP
75 PSP=SQRTFfSIG3P*SIG3P+T3P*T3P)'
C
P3P = RESULTANT STRESS ON PLANE AT ,ANGLE ALPHA -(PASSIVE)
78 PRINT 25 ,AL P H A D ,S IG 3 P ,T3 P ,P3P
GO TO 51
81 DO 84 IALPHA =N,360,N
87 AL-PHAD= IALPHA
IF (ALPHAD+180*-DDELP) 84,85,85
85 IF (ALPHAD-DDELP) 86,51,51
86 A L PHA=(ALPHAD*PI)/180.
TANA=SINFf ALP HA)/COSF (ALPHA)
- •
TANAZ=TANA-X-TANA
IF (A L P H AD - 90 .)91,99,91
91 IF (ALPHAD-270.) 93,99,93
93 SIGSP=I (Z 0jHSIGOP-Z o*12 P--T AN A +2..*S IGZ P+T ANAZ )/ (.1.-+ TANAZ ))-S IGZ-P
T s P = T Z P + (S IG3P-SIGZP)*TANA
GO TO 94
'
99 S IG 3 P=S IGZP
_
'
. T 3.P = -T 2 P
94 Ps P = SQRT F (S IG S P * S IGSP+ TsP* Ts P )
95 PRINT Z5,ALPHAD,SIG3P,T3P,P3P
84 CONTINUE
51 CONTINUE
1005 CALL EXIT
END
LITERATURE CONSULTED
Andersen, P., Substructure Analysis and D e s i g n , The Ronald Press Com­
pany, New York, 1956.
Capper, P. L. and Cassie, W. F., The Mechanics of Engineering So i l s ,
E. and F. N. S p o n , London, England, i960.
M a r t i n , G . L ., A General Solution for Active and Passive Pressures
on a Vertical Plane in a Sloping Earth Mass of Infinite Length,
Unpublished Master's Thesis, Oregon State College, June 1961.
M a r t i n , G. L.,
Thoughts on Rankine Earth Pressure," Proceedings of
the Fourth Annual Engineering Geology and Soils Engineering
.S y m posium, Moscow, Idaho, April 1966.
Rider, P. R.', Analytic G e o m e t r y , Macmillan Company, New York,. 1947.
Rosenbach, J. B., Whitman,. E. A., Meserve, B . E., and Whitman, P. M.,
Essentials of College A l g e b r a , Ginn and Company, Boston, 1958.
Taylor, D. W., Fundamentals of Soil Mechanics, John Wiley and Sons.,
New York, 1965.
Terza g h i , K., Theoretical Soil Mechanics, John Wiley and Sons-, New
- Ybrk, 1959. '
MONTANA
O I/b2 10014206 4
9
-
H378
H258
cop.2
*
Hartsog, W.S.
A general solution
for active and pas­
sive earth pressures
NAME
?.? /-'
7f
'x -
. :
AND
ADDRESS
.'• r-- ,T-T^-' „
I!
A- 17-1/
A
-
'UJiJfW
N 3 T %