Lecture 19 Stresses (1)
Force
Force is a kind of mechanical action between different
objects,it tends to change the shape,volume or
movement state of the object with a force upon it.
Force = mass × acceleration (kg m s-2) [Newton][N]
Force is a vector quantity,and thus possesses both magnitude
and direction; it can be represented by an arrow whose length
specifies the magnitude and whose orientation specifies the
orientation of the force.
F
Unit,Newton
1 Newton = 1 kilogram meter per second squared
vector scalar (only magnitude)
Resolution and resultant of forces
A Force F resolved into two components F1 and F2,
B Two forces F1 and F2 represented by resultant F
F1
F2
F F1
F2
F
A B
Surface Forces and Body Forces
Surface forces,the forces acting on the contact
surface between adjacent parts of rock system,
between adjacent blocks or adjacent lithosphere
plates,The contact surface may be or may be not
a visible material boundary,It can be a
imaginary surface inside the object considered,
Body forces,the forces can work at a distance and
depend on the amount of material affected,so,we can
call body forces distant forces,Gravitational force is
an example of body forces,The gravitational force on
a rock body of mass m is
F = mg
where g is the acceleration of gravity,g varies with
depth in the earth and with position on the earth’s
surface,but for the purpose of structural geology,it is
a constant 9.8m/sec2.
Body forces
Uniform forces
Nonuniform forces
External forces
Imaginary plane
Uniform
Internal
Forces
a b
a
F F
F
?
N=F
?=N/A=F/A
Internal forces and stresses
Stress on a plane,internal forces acting on unit area of the
given plane within the considering body.
x
p
dF
dp
F
P
F
??
?
?
??
l i m
0
m
I n t e r n a l f o r c e a r e a
F
P
F —P —
External
forces
Internal
forces area stress
Stress acting at a point m on a plane n is a vector,it can be
resolved into two components ? and ?,? is normal to the
plane,called normal stress,? is tangential to the plane,called
shear stress.
Normal stress and
shear stress
Magnitude of stress
Stress = Force / Area,limit Area approaching zero
Units of stress
[ Newton / m2 or a ‘Pascal’],or simply say ‘Pa’
That is 1 pascal = 1 newton per square meter.
1 newton = 1 kilogram meter per second squared (1 kg m s-2)
A more commonly used unit is the bar or the kilobar,
Where:
1 bar = 105 pascals = 0.1 MPa
Magnitude and Units of Stress
Normal and shear stresses at a fault plane (A) and
a bedding plane during flexural slip folding (B)
p p p
x
y
z ?x
?y
Resolution of stress in two dimensions (A)
and in tree dimensions (B)
Stress components in three dimensions
Infinitesimal cube
?xy=?yx
?yz=?zy
?zx=?xz
Nine components of stress at a point in matrix form
Since ?xy=?yx ?yz=?zy ?zx=?xz,only six stress
components left,three normal stresses ?x,?y,?z
and three shear stresses ?xy,?y z,?zx
For an arbitrarily chosen set of orthogonal axes x,y,z,six
independent quantities are necessary to specify completely
the state of stress at a point.
(5-2)
Principal stresses,principal planes,principal
axes and principal directions
The components vary as the orientation of the orthogonal
coordinate axes change,It can be proved that at any point
we can find an orientation for the cube such that the shear
stresses on the surfaces of the cube are all zero,That is:
?xy =?y z = ?zx = 0
The only three normal stresses left are called principal
stresses,The three mutually perpendicular planes on
which the shear stress is zero are called principal stress
planes,The axes are called principal stress axes,The
directions along the principal stress axes are called
principal stress directions.
The conventional notation for principal stresses are ?1,
?2,?3,where ?1 ≥?2 ≥ ?3,or called greatest,
intermediate and least principal stresses,respectively,
If the three principal stresses at a point are determined,
the state of stress at the point is completely specified.
Greatest,intermediate and
least principal stresses
中面（m i d d l e p l a n e )
Thin plate
Middle plane
Stress ellipse defined by tails of stress vectors acting
across various planes at point p,The two stress vectors
?ABand ?BA associated with a particular plane p are
identified,(after W.D,Means,1976)
Stress ellipse defined by heads of stress vectors across
various planes at a point,In this case the normal
component of each stress is tensile (not compressive),
(after W.D,Means,1976)
(A) General triaxial stress ellipsoid in perspective
view.(B)Views normal to each of the principal planes
of the ellipsoid,(After W.D,Means,1976)
? 1 ?
1
? 1
? 2 ?
2
? 3
? 3
? 3
? 2
( A)
( B )
?1> ?2 > ?3
What are body forces? What are surfaces forces?
What are stresses?
What is the condition for the resolution of stress?
When three stresses components can determine the state of
stress at a point?
Why six components of stress can specify the state of stress
at a point?
What are principal stresses? Principal direction? Principal
Planes?
Is stress at a point a vector? Is stress on a plane a vector?
Problems
Plans
? We are going to do some field works in the Saturday
morning,Please bring the compass,hammer,lens,
field notebook,geological bag,pencil and meet at
the north gate of the university campus,Write a field
report.
? Select one topic,read some references,write a
review and make a presentation.
Force
Force is a kind of mechanical action between different
objects,it tends to change the shape,volume or
movement state of the object with a force upon it.
Force = mass × acceleration (kg m s-2) [Newton][N]
Force is a vector quantity,and thus possesses both magnitude
and direction; it can be represented by an arrow whose length
specifies the magnitude and whose orientation specifies the
orientation of the force.
F
Unit,Newton
1 Newton = 1 kilogram meter per second squared
vector scalar (only magnitude)
Resolution and resultant of forces
A Force F resolved into two components F1 and F2,
B Two forces F1 and F2 represented by resultant F
F1
F2
F F1
F2
F
A B
Surface Forces and Body Forces
Surface forces,the forces acting on the contact
surface between adjacent parts of rock system,
between adjacent blocks or adjacent lithosphere
plates,The contact surface may be or may be not
a visible material boundary,It can be a
imaginary surface inside the object considered,
Body forces,the forces can work at a distance and
depend on the amount of material affected,so,we can
call body forces distant forces,Gravitational force is
an example of body forces,The gravitational force on
a rock body of mass m is
F = mg
where g is the acceleration of gravity,g varies with
depth in the earth and with position on the earth’s
surface,but for the purpose of structural geology,it is
a constant 9.8m/sec2.
Body forces
Uniform forces
Nonuniform forces
External forces
Imaginary plane
Uniform
Internal
Forces
a b
a
F F
F
?
N=F
?=N/A=F/A
Internal forces and stresses
Stress on a plane,internal forces acting on unit area of the
given plane within the considering body.
x
p
dF
dp
F
P
F
??
?
?
??
l i m
0
m
I n t e r n a l f o r c e a r e a
F
P
F —P —
External
forces
Internal
forces area stress
Stress acting at a point m on a plane n is a vector,it can be
resolved into two components ? and ?,? is normal to the
plane,called normal stress,? is tangential to the plane,called
shear stress.
Normal stress and
shear stress
Magnitude of stress
Stress = Force / Area,limit Area approaching zero
Units of stress
[ Newton / m2 or a ‘Pascal’],or simply say ‘Pa’
That is 1 pascal = 1 newton per square meter.
1 newton = 1 kilogram meter per second squared (1 kg m s-2)
A more commonly used unit is the bar or the kilobar,
Where:
1 bar = 105 pascals = 0.1 MPa
Magnitude and Units of Stress
Normal and shear stresses at a fault plane (A) and
a bedding plane during flexural slip folding (B)
p p p
x
y
z ?x
?y
Resolution of stress in two dimensions (A)
and in tree dimensions (B)
Stress components in three dimensions
Infinitesimal cube
?xy=?yx
?yz=?zy
?zx=?xz
Nine components of stress at a point in matrix form
Since ?xy=?yx ?yz=?zy ?zx=?xz,only six stress
components left,three normal stresses ?x,?y,?z
and three shear stresses ?xy,?y z,?zx
For an arbitrarily chosen set of orthogonal axes x,y,z,six
independent quantities are necessary to specify completely
the state of stress at a point.
(5-2)
Principal stresses,principal planes,principal
axes and principal directions
The components vary as the orientation of the orthogonal
coordinate axes change,It can be proved that at any point
we can find an orientation for the cube such that the shear
stresses on the surfaces of the cube are all zero,That is:
?xy =?y z = ?zx = 0
The only three normal stresses left are called principal
stresses,The three mutually perpendicular planes on
which the shear stress is zero are called principal stress
planes,The axes are called principal stress axes,The
directions along the principal stress axes are called
principal stress directions.
The conventional notation for principal stresses are ?1,
?2,?3,where ?1 ≥?2 ≥ ?3,or called greatest,
intermediate and least principal stresses,respectively,
If the three principal stresses at a point are determined,
the state of stress at the point is completely specified.
Greatest,intermediate and
least principal stresses
中面（m i d d l e p l a n e )
Thin plate
Middle plane
Stress ellipse defined by tails of stress vectors acting
across various planes at point p,The two stress vectors
?ABand ?BA associated with a particular plane p are
identified,(after W.D,Means,1976)
Stress ellipse defined by heads of stress vectors across
various planes at a point,In this case the normal
component of each stress is tensile (not compressive),
(after W.D,Means,1976)
(A) General triaxial stress ellipsoid in perspective
view.(B)Views normal to each of the principal planes
of the ellipsoid,(After W.D,Means,1976)
? 1 ?
1
? 1
? 2 ?
2
? 3
? 3
? 3
? 2
( A)
( B )
?1> ?2 > ?3
What are body forces? What are surfaces forces?
What are stresses?
What is the condition for the resolution of stress?
When three stresses components can determine the state of
stress at a point?
Why six components of stress can specify the state of stress
at a point?
What are principal stresses? Principal direction? Principal
Planes?
Is stress at a point a vector? Is stress on a plane a vector?
Problems
Plans
? We are going to do some field works in the Saturday
morning,Please bring the compass,hammer,lens,
field notebook,geological bag,pencil and meet at
the north gate of the university campus,Write a field
report.
? Select one topic,read some references,write a
review and make a presentation.
