Engineering Rock Mechanics

41 6 Appendix A: Stress and strain analysis

Collecting these expressions together:

Ex = -au/ax, E, = -av/ay

r

These are the equations of strain in terms of displacement.

Example. Say that

1

u = -ql+ 2xy2 + 3x2y

1000

and

x +y.

'=-( 1 2)

1000

What are the strains and rigid body rotation at (1, l)? Taking the necessary

derivatives and substituting x = 1 and y = 1:

--- au- (2yz+6xy)=-^8

ax 1000 1000

7

4xy+3x2 =-

-=-( au 1 )

ay 1000 1000

av 1 2

ax 1000 1000

• = - kX) = -

av - I 2

ay 1000 (24 = G'

Substituting into the analytical formulae:

= -0.002

av 2

ay 1000

& =--=--

Y xy =--+- [:; 3 =-- [ +4)=--=-0.009 9

1000 1000 1000 -

Q=- 'La' "1 =- '[ )= -- =-0.0025rad.

2 ax ay 2 1000 1000 1000