Page 56 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

The co-ordinate transformation is defined by the equations.

2 1/2
2
r = (x + y )
y

= arctan
x
and
x = r cos
y = r sin

If R,
, Z are the polar components of body force, the differential equations of equi-
librium, obtained by considering the condition for static equilibrium of the element
shown in Figure 2.11, are

∂ rr 1 ∂ r
∂ rz rr −
+ + + + R = 0
∂r r ∂
∂z r
∂ r
1 ∂

∂
z 2 r
+ + + +
= 0
∂r r ∂
∂z r
∂ rz 1 ∂
z ∂ zz zz
+ + + + Z = 0
Figure 2.11 Cylindrical polar coor- ∂r r ∂
∂z r
dinate axes, and associated free-body
diagram. For axisymmetric problems, the tangential shear stress components, r
and
z ,
and the tangential component of body force,
, vanish. The equilibrium equations
reduce to

∂ rr ∂ rz rr −
+ + + R = 0
∂r ∂z r
∂ rz ∂ zz rz
+ + + Z = 0
∂r ∂z r
For the particular case where r,
,z are principal stress directions, i.e. the shear stress
component rz vanishes, the equations become

∂ rr rr −
+ + R = 0
∂r r
∂ zz
+ Z = 0
∂z
Displacement components in the polar system are described by u r , u
, u z . The
elements of the strain matrix are defined by

∂u r
ε rr =
∂r
1 ∂u
u r
ε

= +
r ∂
r
∂u z
ε zz =
∂z
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