Engineering Rock Mechanics

Stress analysis 401

Remember the notation:

normal stress, 7 x,---- acting on a plane normal to the x-axis

shear stresses, acting on a plane normal to the x-axis

0

-Tqc

cacting in the y-direction

and also, remember the sign convention:

for normal stresses,

for shear stresses,

compression is positive;

positive stresses act in

positive directions on

negative faces.

The shear stress convention is difficult to remember. It is made easier by
realizing there are always two pluses and a minus:

on a + face a + stress acts in a - direction

on a + face a - stress acts in a + direction

on a - face a + stress acts in a + direction.

If we consider moment equilibrium of the cube around the three axes,
we find that

• 
  
  
  
  • rxy - zyx ryz - rzy zzx = rxz
  
  
  
  

and so the state of stress at a point (as 6 is reduced to zero) is defined by

six independent qualities

01, oy, a,, qy, zyz, zzx.

These stresses are usually written in matrix form:

1: 2 stress tensor.

Note that because of the complementary shear stresses (Le. zxy = zyx, etc.)
the stress tensor is symmetric about the leading diagonal.

Generul stress field in three dimensions

It often happens that each component of the stress tensor varies in

magnitude from point to point within a body-they are functions of x, y

and z. If this is the case, then an element will be in equilibrium if