Page 205 - Rock Mechanics For Underground Mining

THE FINITE ELEMENT METHOD

strain, are given by

⎡ 0 ⎤

⎡ ⎤
1 /(1 − ) 0
⎡ ⎤ ⎡ ⎤ xx
xx ε xx ⎢ ⎥
E(1 − ) ⎢ ⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥
⎣ yy ⎦= ⎢ /(1 − ) 1 0 ⎥ ⎣ ε yy ⎦+ ⎢ yy ⎥
(1 + )(1 − ) ⎣ ⎦ ⎢ 0 ⎥
⎣ ⎦
0 0 (1 − 2 )/2(1 − )
xy xy xy
or
0
[ ] = [D][ ] + [ ]
e
0
= [D][B][u ] + [ ] (6.41)
0
where [ ] is the vector of total stresses, [D] is the elasticity matrix, and [ ]isthe
vector of initial stresses.
6.6.3 Equivalent nodal forces
The objective in the finite element method is to establish nodal forces q xi , q yi etc.,
equivalent to the internal forces operating between the edges of elements, and the
body force

b x
[b] =
b y
operating per unit volume of the element. The internal nodal forces are determined
e
by imposing a set of virtual displacements [ u ] at the nodes, and equating the in-
ternal and external work done by the various forces in the displacement field. For
e
imposed nodal displacements [ u ], displacements and strains within an element
are
e
e
[ u] = [N][ u ], [ ] = [B][ u ]
e
The external work done by the nodal forces [q ] acting through the virtual displace-
ments is
e T
e
e
W = [ u ] [q ]
and the internal work per unit volume, by virtue of the virtual work theorem for a
continuum (Charlton, 1959) is given by

T
T
i
W = [ ] [ ] − [ u] [b]
T
e
e
T
= ([B][ u ]) [ ] − ([N][ u ]) [b]
T
T
e T
= [ u ] ([B] [ ] − [N] [b])
Integrating the internal work over the volume V e of the element, and equating it with
the external work, gives
% %
e T T
[q ] = [B] [ ]dV − [N] [b]dV
V e V e
187