REVIEW OF STRESS
Stress components in three dimensions
Infinitesimal cube
?xy=?yx
?yz=?zy
?zx=?xz
(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
STRESSES (2)
Classes of Stress States
1
2
0
321
???
3
1
2
0
321
???Triaxial compressive stresses
Biaxial compressive stresses
1
??
1
??
2
0
3
??
2
3
1
?
1
??
2
0
3
??
1
Hydrostatic stresses
Uniaxial stresses
3
1 321
0 ???
3
1 ? 3 ? 21
Plane stresses
Pure shear stresses
Stresses acting on a given plane
1
2
?
t
?
??
1
?2
A?
A? cos? A
? sin?
?
?
?
?
A triangle element
Equilibrium equations:
?F? = 0 ?Ft = 0
-?A? + ?1 A? cos? cos?+ ?2 A?*
*sin?sin?=0
?A? -?1 A? cos? sin?+ ?2A? *
*sin?cos?=0
? = ?1 cos 2(?) + ?2 sin 2(?) (1)
? = ?1 cos?sin?－ ?2 sin?cos? (2)
Mohr’s Circle Stress Equations
Using the triangle formulae:
cos2 ? ? (1/2)(1+cos2 ?) sin 2? ? (1/2)(1-cos2 ?)
We can rewrite equation (1) as follows:
? = 1/2 (?1 + ?2 ) + 1/2 (?1 - ?2 ) cos2? (3)
Introduce the triangle formula
sin2? =2cos?sin?
into equation (2),we can obtain:
? = 1/2 (?1 -?2 ) sin2? (4)
? reach a maximum value when ? is 45 degrees and that is
?max= 1/2 (?1 -?2 ) (5)
??
??
2?
?1?3
+
-
Left-lateral
Right-lateral
?? = [?1 - ?3] sin(2?)
2
?? = [?1 + ?3] + [?1 - ?3] cos(2?)
2 2
Normal
Stress Axis
MOHR STRESS CIRCLE OR GRAPH
??
??
Mohr’s circles in three dimensions
A.D1represents the plane on which the maximum
principal stress acts；
B,The relationship between plane D1 and the
principal directions
Mohr’s circles for some three dimensional
stress states
Uniaxial stresses Hydrostatic stresses
Plane stresses Pure shear stresses
Triaxial compressive
stresses
Biaxial compressive
stresses
A B C
D
F
E
IMPORTANT TERMS FOR STRESS ANALYSIS
? Mean Stress,?m= 1/2[?1 + ?3]
? Differential Stress,[?1 - ?3]
Deviatoric stresses,?1 -?m =1/2 [?1 - ?3]
?3 -?m =-1/2 [?1 - ?3]
?Compressive Strength
?
?Tensile Strength
? Effective Normal Stress,[?n – Pp ]
Effect of hydrostatic and deviatoric stresses (shown in two
dimensions)
(A) volume change (B ) shape change
Stress field
To every point in any rock body there corresponds
a state of stress,and this spatial array of states of
stress at an instant is called a stress field.
Stress field in uppermost cubic kilometer of the
earth（ after Means,1976) 1bar=105Pa
trajectories
trajectories
Surface of earth
Axial diameter scale of stress ellipse
Stress trajectories
Distinguish between plane stresses and pure shear stresses
What does hydrostatic stresses mean?
Can you express different stress states by diagrams?
How to derive the equations for normal and shear stresses on a
given plane?
Where is the maximum shear stress?
What is hydrostatic stress?
What is deviatoric stress?
What is differential stress?
What is stress field?
What are stress trajectories?
Problems