Page 207 - Rock Mechanics For Underground Mining

THE DISTINCT ELEMENT METHOD

Equilibrium at any node requires that the applied external force at the node be in bal-
ance with the resultant internal equivalent nodal force. Suppose the internal equivalent
nodal force vector for elements a and b is given by

a
a
a T
a
a
a

[q ] = q q q q q q a
x1 y1 x2 y2 x3 y3
b
b
b
b
b
b T

[q ] = q q q q q q b
x2 y2 x4 y4 x3 y3
The nodal equilibrium condition requires
a
for node 1: r x1 = q , r y1 = q a
x1 y1
b
for node 2: r x2 = q a + q , r y2 = q a + q b
x2 x2 y2 y2
with similar conditions for the other nodes. The external force–nodal displacement
equation for the assembly then becomes
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎡ ⎤
r x1 a u x1 f x1
K 00
r y1 ⎢ ⎥ u y1 f y1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 00⎥
r x2 u x2 f x2
⎢ ⎥ ---------- ⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ------------ ⎥
r y2 a b u y2 f y2
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ = ⎢ K + K ⎥ ⎢ ⎥ + ⎢ ⎥
r x3 ⎢ ⎥ u x3 f x3
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ------------ ⎥ ⎢ ⎥ ⎢ ⎥
r y3 ⎢ b ⎥ u y3 f y3
⎢ ⎥ --------- ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎣ 00 K ⎢ ⎥ ⎢ ⎥
⎣ r x4 ⎦ ⎦ ⎣ u x4 ⎦ ⎣ f x4 ⎦
00
r y4 u y4 f y4
a
b
where appropriate elements of the stiffness matrices [K ] and [K ] are added at the
common nodes. Thus assembly of the global stiffness matrix [K] proceeds simply by
taking account of the connectivity of the various elements, to yield the global equation
for the assembly
g
g
g
[K][u ] = [r ] − [f ] (6.44)
g
Solution of the global equation 6.44 returns the vector [u ] of nodal displacements.
The state of stress in each element can then be calculated directly from the appropriate
nodal displacements, using equation 6.41.
In practice, special attention is required to render [K] non-singular, and account
must be taken of any applied tractions on the edges of elements. Also, most finite ele-
ment codes used in design practice are based on curvilinear quadrilateral elements and
higher-order displacement variation with respect to the element intrinsic co-ordinates.
For example, a quadratic isoparametric formulation imposes quadratic variation of
displacements and quadratic description of element shape. Apart from some added
complexity in the evaluation of the element stiffness matrix and the initial load vector,
the solution procedure is essentially identical to that described here.
6.7 The distinct element method
Both the boundary element method and the finite element method are used exten-
sively for analysis of underground excavation design problems. Both methods can
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