Page 204 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

Solution for 4 , 5 , 6 yields an interpolation function for u y identical to equation
6.38, with u yi replacing u xi , etc. The variation of displacements throughout an element
is therefore described by

u x e e
[u] = = [N][u ] = [N i I, N j I, N k I][u ] (6.39)
u y
where N i = (a i + b i x + c i y)/2 , with similar expressions N j , N k and I isa2 × 2
identity matrix.
By defining the displacement field in an element in terms of the nodal dis-
placements, the interpolation procedure ensures continuity of the displacements
both across an element interface with an adjacent element, and within the element
itself.
Once the displacement field in an element is defined, the state of strain can be
established from the strain–displacement relations. For plane strain problems, a strain
vector may be defined by

⎡ ⎤ ⎡ ⎤
∂u x ∂
0
⎡ ⎤ ⎢ ∂x ⎥ ⎢ ∂x ⎥
ε xx
⎢ ⎥ ⎢
⎥ "
#
⎥ u x
⎢ ⎥ ⎢
⎢ ⎥ ⎢ ∂u y ∂ ⎥
⎢
⎥
[ε] = ⎢ ε yy ⎥ = ⎢ ⎥ = ⎢ 0 ⎥
⎢ ∂y
⎣ ⎦
⎢
⎥
∂y ⎥
u y
⎢ ⎥ ⎢ ⎥
xy ⎣ ∂u x ∂u y ⎦ ⎣ ∂ ∂ ⎦
+
∂y ∂x ∂y ∂x
or
[ε] = [L][u] (6.40)
Since displacements are specified by equation 6.39, equation 6.40 becomes
e
e
[ε] = [L][N][u ] = [B][u ]
where
⎡ ⎤
∂N i ∂N j ∂N k
0 0 0
∂x ∂x ∂x
⎢ ⎥
⎢ ⎥
⎢ ⎥
∂N i ∂N j
[B] = ⎢ 0 0 0 ∂N k ⎥
⎢
∂y ∂y ⎥
⎢ ∂y ⎥
⎢ ⎥
⎣ ∂N i ∂N i ∂N j ∂N j ∂N k ∂N k ⎦
∂y ∂x ∂y ∂x ∂y ∂x
For the case of linear displacement variation, the terms ∂N i /∂ x , etc., of the B matrix
are constant, and thus the strain components are invariant over the element.
6.6.2 Stresses within an element
The state of total stress within an element is the sum of the induced stresses and the
initial stresses. Ignoring any thermal strains, total stresses, for conditions of plane
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