Page 186 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

It is instructive to follow the procedure developed by Airy (1862) and described by
Timoshenko and Goodier (1970), in establishing a particular form of the field equation
for isotropic elasticity and plane strain. The differential equations of equilibrium in
two dimensions for zero body forces are

∂ xx ∂ xy
+ = 0 (6.1)
∂x ∂y
∂ xy ∂ yy
+ = 0
∂x ∂y
or
2 2 2
∂ xy ∂ xx ∂ yy
=− =− (6.2)
∂x∂y ∂x 2 ∂y 2
For plane strain conditions and isotropic elasticity, strains are defined by
1

ε xx = ( xx − yy )
E
1

ε yy = ( yy − xx ) (6.3)
E
1
xy = xy
G

2(1 + )
= xy
E
where
E

E =
1 − 2

=
1 −
The strain compatibility equation in two dimensions is given by
2 2 2
∂ ε yy ∂ ε xx ∂ xy
+ = (6.4)
∂x 2 ∂y 2 ∂x∂y
Substituting the expressions for the strain components, (equations 6.3) in equation
6.4, and then equations 6.2 in the resultant expression yields
2
2
2
2
2
1 ∂ yy ∂ xx
1 ∂ xx ∂ yy
(1 + ) ∂ xy

− + − = 2
E ∂x 2 ∂x 2 E ∂y 2 ∂y 2 E ∂x ∂y
2
2
(1 + ) ∂ xx ∂ yy

=− +
E ∂x 2 ∂y 2
which becomes, on simplification,
2 2 2 2
∂ xx ∂ xx ∂ yy ∂ yy
+ + + = 0
∂x 2 ∂y 2 ∂x 2 ∂y 2
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