Stress Paths | Soil Mechanics - Civil Engineering (CE) PDF Download

 Page 1


  STRESS ANALYSIS AND STRESS PATHS 
 
 
 THE MOHR CIRCLE 
 
 The discussions in Chapters 2 and 5 were largely concerned with vertical stresses.  A 
more detailed examination of soil behaviour requires a knowledge of stresses in other directions 
and two or three dimensional analyses become necessary.  For the graphical representation of the 
state of stress on a soil element a very convenient and widely used method is by means of the 
Mohr circle.  In the treatment that follows the stresses in two dimensions only will be considered. 
 
 Fig. 7.1(a) shows the normal stresses s
y
 and s
x
 and shear stresses t
xy
 acting on an 
element of soil.  The normal stress s and shear stress t acting on any plane inclined at ? to the 
plane on which s
y
 acts are shown in Fig. 7.1(b).  The stresses s and t may be expressed in terms 
of the angle ? and the other stresses indicated in Fig. 7.1(b).  If a, b and c represent the sides of the 
triangle then, for force equilibrium in the direction of s: 
 
  s a = s
x
 b sin ? +t
xy
 b cos ? + s
y
 c cos ? + t
xy
 c sin ? 
 
  s = s
x
 sin
2
 ? + t
xy
 sin ? cos ? + s
y
 cos
2
 ? + t
xy
 cos ? sin ? 
 
  s = s
x
 sin
2
 ? + s
y
 cos
2
 ? + t
xy
 sin 2 ? (7.1) 
 
Similarly if forces are resolved in the direction of t 
 
  t a = s
y
 c sin ? - t
xy
 c cos ? - s
x
 b cos ? + t
xy
 b sin ? 
 
  t = s
y
 sin ? cos ? - t
xy
 cos
2
 ? - s
x
 sin ? cos ? + t
xy
 sin
2
 ? 
 
   = 
s
y
 - s
x
2
  sin 2 ? - t
xy
 cos 2 ? (7.2) 
 
Equation (7.1) can be further expressed as follows 
 
  s  -  
(s
x
 + s
y
)
2
  =  
(s
y
 - s
x
)
2
 cos 2 ?  +  t
xy
 sin 2 ? 
 
This equation can be combined with equation (7.2) to give 
Page 2


  STRESS ANALYSIS AND STRESS PATHS 
 
 
 THE MOHR CIRCLE 
 
 The discussions in Chapters 2 and 5 were largely concerned with vertical stresses.  A 
more detailed examination of soil behaviour requires a knowledge of stresses in other directions 
and two or three dimensional analyses become necessary.  For the graphical representation of the 
state of stress on a soil element a very convenient and widely used method is by means of the 
Mohr circle.  In the treatment that follows the stresses in two dimensions only will be considered. 
 
 Fig. 7.1(a) shows the normal stresses s
y
 and s
x
 and shear stresses t
xy
 acting on an 
element of soil.  The normal stress s and shear stress t acting on any plane inclined at ? to the 
plane on which s
y
 acts are shown in Fig. 7.1(b).  The stresses s and t may be expressed in terms 
of the angle ? and the other stresses indicated in Fig. 7.1(b).  If a, b and c represent the sides of the 
triangle then, for force equilibrium in the direction of s: 
 
  s a = s
x
 b sin ? +t
xy
 b cos ? + s
y
 c cos ? + t
xy
 c sin ? 
 
  s = s
x
 sin
2
 ? + t
xy
 sin ? cos ? + s
y
 cos
2
 ? + t
xy
 cos ? sin ? 
 
  s = s
x
 sin
2
 ? + s
y
 cos
2
 ? + t
xy
 sin 2 ? (7.1) 
 
Similarly if forces are resolved in the direction of t 
 
  t a = s
y
 c sin ? - t
xy
 c cos ? - s
x
 b cos ? + t
xy
 b sin ? 
 
  t = s
y
 sin ? cos ? - t
xy
 cos
2
 ? - s
x
 sin ? cos ? + t
xy
 sin
2
 ? 
 
   = 
s
y
 - s
x
2
  sin 2 ? - t
xy
 cos 2 ? (7.2) 
 
Equation (7.1) can be further expressed as follows 
 
  s  -  
(s
x
 + s
y
)
2
  =  
(s
y
 - s
x
)
2
 cos 2 ?  +  t
xy
 sin 2 ? 
 
This equation can be combined with equation (7.2) to give 
7-2 
 
 
 
 
Fig. 7.1  Stress at a Point 
 
 
 
 
 
 
 
Fig. 7.2  The Mohr Circle 
 
Page 3


  STRESS ANALYSIS AND STRESS PATHS 
 
 
 THE MOHR CIRCLE 
 
 The discussions in Chapters 2 and 5 were largely concerned with vertical stresses.  A 
more detailed examination of soil behaviour requires a knowledge of stresses in other directions 
and two or three dimensional analyses become necessary.  For the graphical representation of the 
state of stress on a soil element a very convenient and widely used method is by means of the 
Mohr circle.  In the treatment that follows the stresses in two dimensions only will be considered. 
 
 Fig. 7.1(a) shows the normal stresses s
y
 and s
x
 and shear stresses t
xy
 acting on an 
element of soil.  The normal stress s and shear stress t acting on any plane inclined at ? to the 
plane on which s
y
 acts are shown in Fig. 7.1(b).  The stresses s and t may be expressed in terms 
of the angle ? and the other stresses indicated in Fig. 7.1(b).  If a, b and c represent the sides of the 
triangle then, for force equilibrium in the direction of s: 
 
  s a = s
x
 b sin ? +t
xy
 b cos ? + s
y
 c cos ? + t
xy
 c sin ? 
 
  s = s
x
 sin
2
 ? + t
xy
 sin ? cos ? + s
y
 cos
2
 ? + t
xy
 cos ? sin ? 
 
  s = s
x
 sin
2
 ? + s
y
 cos
2
 ? + t
xy
 sin 2 ? (7.1) 
 
Similarly if forces are resolved in the direction of t 
 
  t a = s
y
 c sin ? - t
xy
 c cos ? - s
x
 b cos ? + t
xy
 b sin ? 
 
  t = s
y
 sin ? cos ? - t
xy
 cos
2
 ? - s
x
 sin ? cos ? + t
xy
 sin
2
 ? 
 
   = 
s
y
 - s
x
2
  sin 2 ? - t
xy
 cos 2 ? (7.2) 
 
Equation (7.1) can be further expressed as follows 
 
  s  -  
(s
x
 + s
y
)
2
  =  
(s
y
 - s
x
)
2
 cos 2 ?  +  t
xy
 sin 2 ? 
 
This equation can be combined with equation (7.2) to give 
7-2 
 
 
 
 
Fig. 7.1  Stress at a Point 
 
 
 
 
 
 
 
Fig. 7.2  The Mohr Circle 
 
7-3 
                              
2
2
2
2
2 2
xy
y x y x
t
s s
t
s s
s +
?
?
?
?
?
?
?
?
-
= +
?
?
?
?
?
?
?
?
+
-                                  (7.3) 
 
 This is the equation of a circle with a centre at 
 
  s = 
s
x
 + s
y
2
 ,           t = 0 
 
and a radius of 
 
  
?
?
?
?
?
? (s
y
 - s
x
)
2
2
  + t
xy
 
2
 
1/2
 
 
 This circle known as the Mohr circle is represented in Fig. 7.2.  In this diagram point A 
represents the stresses on the s
y
 plane and point B represents the stresses on the s
x
 plane.  The 
shear stresses are considered as negative if they give a couple in the clockwise direction.  In 
geomechanics usage the normal stresses are positive when compressive. 
 
 Point C represents the stresses t and s on the ? plane.  The location of point C may be 
found by rotating a radius by an angle equal to 2? in an anticlockwise direction from the radius 
through point A. 
 
 Alternatively point C may be found by means of the point O
P
 known as the “origin of 
planes” or the “pole”.  This point is defined as follows:  if any line O
P
 X is drawn through the 
origin of planes and intersects the other side of the Mohr circle at point X then point X represents 
the stresses on the plane parallel to O
P
 X.  In other words line O
P
 A in Fig. 7.2 is parallel to the 
plane on which the stress s
y
 acts and line O
P
 B is parallel to the plane on which the stress s
x
 acts.  
To find the point on the circle representing the stresses on the ? plane, line O
P
 C is drawn parallel 
to that plane to yield point C.  Both of the constructions just described for the location of point C 
may be verified by means of equations (7.1) and (7.2). 
 
 From Fig. 7.2 the major and minor principal  stresses s
1
 and s
3
 and the inclinations of 
the planes on which they act may also be determined.  A more detailed treatment of the Mohr 
circle may be found in most books on the mechanics of solids. 
 
EXAMPLE  
 
 Major and minor principal stresses of 45kN/m
2
 and 15kN/m
2
 respectively act on an 
element of soil where the principal planes are inclined as illustrated in Fig. 7.3(a). 
Page 4


  STRESS ANALYSIS AND STRESS PATHS 
 
 
 THE MOHR CIRCLE 
 
 The discussions in Chapters 2 and 5 were largely concerned with vertical stresses.  A 
more detailed examination of soil behaviour requires a knowledge of stresses in other directions 
and two or three dimensional analyses become necessary.  For the graphical representation of the 
state of stress on a soil element a very convenient and widely used method is by means of the 
Mohr circle.  In the treatment that follows the stresses in two dimensions only will be considered. 
 
 Fig. 7.1(a) shows the normal stresses s
y
 and s
x
 and shear stresses t
xy
 acting on an 
element of soil.  The normal stress s and shear stress t acting on any plane inclined at ? to the 
plane on which s
y
 acts are shown in Fig. 7.1(b).  The stresses s and t may be expressed in terms 
of the angle ? and the other stresses indicated in Fig. 7.1(b).  If a, b and c represent the sides of the 
triangle then, for force equilibrium in the direction of s: 
 
  s a = s
x
 b sin ? +t
xy
 b cos ? + s
y
 c cos ? + t
xy
 c sin ? 
 
  s = s
x
 sin
2
 ? + t
xy
 sin ? cos ? + s
y
 cos
2
 ? + t
xy
 cos ? sin ? 
 
  s = s
x
 sin
2
 ? + s
y
 cos
2
 ? + t
xy
 sin 2 ? (7.1) 
 
Similarly if forces are resolved in the direction of t 
 
  t a = s
y
 c sin ? - t
xy
 c cos ? - s
x
 b cos ? + t
xy
 b sin ? 
 
  t = s
y
 sin ? cos ? - t
xy
 cos
2
 ? - s
x
 sin ? cos ? + t
xy
 sin
2
 ? 
 
   = 
s
y
 - s
x
2
  sin 2 ? - t
xy
 cos 2 ? (7.2) 
 
Equation (7.1) can be further expressed as follows 
 
  s  -  
(s
x
 + s
y
)
2
  =  
(s
y
 - s
x
)
2
 cos 2 ?  +  t
xy
 sin 2 ? 
 
This equation can be combined with equation (7.2) to give 
7-2 
 
 
 
 
Fig. 7.1  Stress at a Point 
 
 
 
 
 
 
 
Fig. 7.2  The Mohr Circle 
 
7-3 
                              
2
2
2
2
2 2
xy
y x y x
t
s s
t
s s
s +
?
?
?
?
?
?
?
?
-
= +
?
?
?
?
?
?
?
?
+
-                                  (7.3) 
 
 This is the equation of a circle with a centre at 
 
  s = 
s
x
 + s
y
2
 ,           t = 0 
 
and a radius of 
 
  
?
?
?
?
?
? (s
y
 - s
x
)
2
2
  + t
xy
 
2
 
1/2
 
 
 This circle known as the Mohr circle is represented in Fig. 7.2.  In this diagram point A 
represents the stresses on the s
y
 plane and point B represents the stresses on the s
x
 plane.  The 
shear stresses are considered as negative if they give a couple in the clockwise direction.  In 
geomechanics usage the normal stresses are positive when compressive. 
 
 Point C represents the stresses t and s on the ? plane.  The location of point C may be 
found by rotating a radius by an angle equal to 2? in an anticlockwise direction from the radius 
through point A. 
 
 Alternatively point C may be found by means of the point O
P
 known as the “origin of 
planes” or the “pole”.  This point is defined as follows:  if any line O
P
 X is drawn through the 
origin of planes and intersects the other side of the Mohr circle at point X then point X represents 
the stresses on the plane parallel to O
P
 X.  In other words line O
P
 A in Fig. 7.2 is parallel to the 
plane on which the stress s
y
 acts and line O
P
 B is parallel to the plane on which the stress s
x
 acts.  
To find the point on the circle representing the stresses on the ? plane, line O
P
 C is drawn parallel 
to that plane to yield point C.  Both of the constructions just described for the location of point C 
may be verified by means of equations (7.1) and (7.2). 
 
 From Fig. 7.2 the major and minor principal  stresses s
1
 and s
3
 and the inclinations of 
the planes on which they act may also be determined.  A more detailed treatment of the Mohr 
circle may be found in most books on the mechanics of solids. 
 
EXAMPLE  
 
 Major and minor principal stresses of 45kN/m
2
 and 15kN/m
2
 respectively act on an 
element of soil where the principal planes are inclined as illustrated in Fig. 7.3(a). 
7-4 
 
 
 
 
 
 
 
 
 
Fig. 7.3 
 
 
 
(a) Determine the inclination of the planes on which the maximum shear stresses 
act. 
Page 5


  STRESS ANALYSIS AND STRESS PATHS 
 
 
 THE MOHR CIRCLE 
 
 The discussions in Chapters 2 and 5 were largely concerned with vertical stresses.  A 
more detailed examination of soil behaviour requires a knowledge of stresses in other directions 
and two or three dimensional analyses become necessary.  For the graphical representation of the 
state of stress on a soil element a very convenient and widely used method is by means of the 
Mohr circle.  In the treatment that follows the stresses in two dimensions only will be considered. 
 
 Fig. 7.1(a) shows the normal stresses s
y
 and s
x
 and shear stresses t
xy
 acting on an 
element of soil.  The normal stress s and shear stress t acting on any plane inclined at ? to the 
plane on which s
y
 acts are shown in Fig. 7.1(b).  The stresses s and t may be expressed in terms 
of the angle ? and the other stresses indicated in Fig. 7.1(b).  If a, b and c represent the sides of the 
triangle then, for force equilibrium in the direction of s: 
 
  s a = s
x
 b sin ? +t
xy
 b cos ? + s
y
 c cos ? + t
xy
 c sin ? 
 
  s = s
x
 sin
2
 ? + t
xy
 sin ? cos ? + s
y
 cos
2
 ? + t
xy
 cos ? sin ? 
 
  s = s
x
 sin
2
 ? + s
y
 cos
2
 ? + t
xy
 sin 2 ? (7.1) 
 
Similarly if forces are resolved in the direction of t 
 
  t a = s
y
 c sin ? - t
xy
 c cos ? - s
x
 b cos ? + t
xy
 b sin ? 
 
  t = s
y
 sin ? cos ? - t
xy
 cos
2
 ? - s
x
 sin ? cos ? + t
xy
 sin
2
 ? 
 
   = 
s
y
 - s
x
2
  sin 2 ? - t
xy
 cos 2 ? (7.2) 
 
Equation (7.1) can be further expressed as follows 
 
  s  -  
(s
x
 + s
y
)
2
  =  
(s
y
 - s
x
)
2
 cos 2 ?  +  t
xy
 sin 2 ? 
 
This equation can be combined with equation (7.2) to give 
7-2 
 
 
 
 
Fig. 7.1  Stress at a Point 
 
 
 
 
 
 
 
Fig. 7.2  The Mohr Circle 
 
7-3 
                              
2
2
2
2
2 2
xy
y x y x
t
s s
t
s s
s +
?
?
?
?
?
?
?
?
-
= +
?
?
?
?
?
?
?
?
+
-                                  (7.3) 
 
 This is the equation of a circle with a centre at 
 
  s = 
s
x
 + s
y
2
 ,           t = 0 
 
and a radius of 
 
  
?
?
?
?
?
? (s
y
 - s
x
)
2
2
  + t
xy
 
2
 
1/2
 
 
 This circle known as the Mohr circle is represented in Fig. 7.2.  In this diagram point A 
represents the stresses on the s
y
 plane and point B represents the stresses on the s
x
 plane.  The 
shear stresses are considered as negative if they give a couple in the clockwise direction.  In 
geomechanics usage the normal stresses are positive when compressive. 
 
 Point C represents the stresses t and s on the ? plane.  The location of point C may be 
found by rotating a radius by an angle equal to 2? in an anticlockwise direction from the radius 
through point A. 
 
 Alternatively point C may be found by means of the point O
P
 known as the “origin of 
planes” or the “pole”.  This point is defined as follows:  if any line O
P
 X is drawn through the 
origin of planes and intersects the other side of the Mohr circle at point X then point X represents 
the stresses on the plane parallel to O
P
 X.  In other words line O
P
 A in Fig. 7.2 is parallel to the 
plane on which the stress s
y
 acts and line O
P
 B is parallel to the plane on which the stress s
x
 acts.  
To find the point on the circle representing the stresses on the ? plane, line O
P
 C is drawn parallel 
to that plane to yield point C.  Both of the constructions just described for the location of point C 
may be verified by means of equations (7.1) and (7.2). 
 
 From Fig. 7.2 the major and minor principal  stresses s
1
 and s
3
 and the inclinations of 
the planes on which they act may also be determined.  A more detailed treatment of the Mohr 
circle may be found in most books on the mechanics of solids. 
 
EXAMPLE  
 
 Major and minor principal stresses of 45kN/m
2
 and 15kN/m
2
 respectively act on an 
element of soil where the principal planes are inclined as illustrated in Fig. 7.3(a). 
7-4 
 
 
 
 
 
 
 
 
 
Fig. 7.3 
 
 
 
(a) Determine the inclination of the planes on which the maximum shear stresses 
act. 
 
 (b) Determine the inclination of the planes on which the following condition is 
satisfied 
  t = ± s tan 45° 
 
 (c) On how many planes are shear stresses having a magnitude of 5kN/m
2
 acting? 
 
 In Fig. 7.3(b) the Mohr circle has been drawn, A and B representing the major and minor 
principal stresses respectively.  By drawing line A O
P
 parallel to the major principal plane the 
origin of planes O
P
 may be located. 
 
(a) The maximum shear stress may be calculated from equation (7.3) or it may simply be 
read from the Mohr circle. 
 
 Clearly t max = 
s
1
 - s
3
2
 
 
   = 15 kN/m
2
 
 
 The points of maximum shear stress are represented by C and D.  Therefore the planes 
on which these stresses act are parallel to lines O
P
 C and O
P
 D respectively.  As shown 
on the figure these planes are inclined at 45_ to the principal planes.  This will always be 
the case regardless of the inclination of the principal planes. 
 
(b) The lines representing the relationship 
 
  t = ± s tan 45° 
 
 have been drawn in Fig. 7.3(b).  Since the circle touches neither of these lines there are 
no planes on which the relationship holds. 
 
(c) The points on the circle representing a shear stress of 5kN/m
2
 are E,F, G and H so there 
are four planes on which this shear stress acts.  These planes are parallel to the lines O
P
 
E, O
P
 F, O
P
 G and O
P
 H respectively. 
 
 
STRESS PATHS 
 
 When the stresses acting at a point undergo changes, these changes may be conveniently 
represented on a plot of shear stress against normal stress.  Such a situation is illustrated in Fig.