Neither the classical mechanics of continua nor the mechanics of discontinua can be adapted to some situations encountered in rock mechanics. In par- ticular, problems where the half-space is not isotropic, or with complex boundary conditions, require a more sophisticated mathematical approach, and a method which is easily programmed for computer analysis. The recently developed 'finite element' method of numerical stress analysis can be extended to deal with particular forms of anisotropy, mainly in rock mechanics (Zienkiewicz and co-workers, from 1965 to 1968). For example, problems can be solved concerning orthotropic rock tunnels, where the modulus of elasticity E1 in the direction x differs from the modulus E2 in a direction at right angles to x, or tunnels in curved non-homogeneous strata (fig. 7.28).
Dam foundations on nonhomogeneous rock masses, cut by faults or weak
The finite element method 187
( [J • J
Fig. 7.28 Configuration and loading problems analysed with the finite element method: (a) isotropic: Ex = E2, vx = v2. (b) orthotropic: Ex = 1, E2 = E3 =£ Ex. (c) curved, non-homogeneous strata (after Zienkiewicz, Mayer & Cheung, 1966).
strata can be investigated; stress calculations where rock tensile strength is nil can be compared to that when it is not nil.
A case of anisotropy, generally referred to as transverse isotropy, is where the material is isotropic in the yz plane but non-isotropic with respect to directions normal to this plane (fig. 7.29). In a completely general three- dimensional, anisotropic, elastic situation the six stress components and the
Fig. 7.29 A stratified (transversely isotropic) material.
six strain components can be related by a six by six matrix of coefficients.
Using cartesian co-ordinates:
f2
7xy
7l/3
> =
011 012
021 022
013 014
023
0 1 5 016 "
<
ay
rxy yz
(1)
This matrix is symmetrical and therefore 21 elastic constants are sufficient to explain the behaviour of any material. In an isotropic material, the number of constants is reduced to two: E and v.
Referring to fig. 7.29 which shows the orientation of axes assumed in a stratified material, it can be shown that the independent constants remaining in the strain-stress relationship are (Zienkiewicz, 1965).
012 012 0 0 0 f Gx
022 023 0 0 0 a22 0 0 0
yXy 044 0 0
- 023) o 044 J
Alternatively, using more conventional definitions of elastic constants, by analogy with the isotropic case, we can write
1 x
=:'FGx~'W Gy~~"W
(2)
Yxy
yyz
v± v2 1
~~~FGx~~~FGy + ~F(
EJX EJ2 £J2
1
(1 + v2)
7zx = -pr
(3)
When considering only plane strain problems (see Zienkiewicz, Cheung &
Stagg, 1966):
= yyz = yX2 = 0 and - ? = n, (4) the a2 becomes
Gx + V2Oy,
The finite element method 189 resulting in the following relationships in the plane xy:
Vxv = 7T
(5)
It is more convenient to express the stresses in terms of strains, therefore:
- v 2 - E2
{(1 - v2)ey}
v2)(l - v 2 - v2)ex
(6)
Equation (6) can be conveniently written in matrix form
(7)
(In the case of linear visco-elasticity the matrix (D) is one of differential or integral operators.)
To analyse the stress in a two-dimensional body it is first sub-divided into small triangular (or rectangular) elements (fig. 7.30) which are assumed to
Fig. 7.30 Subdivision of a two-dimensional body into finite elements.
be connected to each other only at nodal points corresponding in this instance with the apices of the triangles. If at each node, such as /, the dis- placements in directions of the x and y co-ordinates are listed as
(8)
and if the displacements of an element /,/, m, are defined as
{<5e} = (9)
then it is possible to associate theoretical displacement forces which act at the nodes as
{Re} = [Ke]{de}. (10)
It is possible to set up a series of simultaneous equations at each node to ensure equilibrium. Then, for example, at a node i we have
(11) in which {Ret} is the internal force contributed by an element, {Ff} is the external force contributed by an element and summation concerns all the leements meeting at a particular node. The external load {Ff} may be due either to concentrated loads acting at such a point or may be caused by distributed load acting through the element. As all the internal forces are dependent on the nodal displacements the equilibirum conditions result in a system of simultaneous equations:
\ • /
• = i (12)
from which the displacements and hence the stresses (equation (7)) can be determined.
Displacements are given by equations (8) and (9). The displacements within the element are assumed to vary linearly as
"1 x y 0 0 0"
0 0 0 1 x v
(13)
The finite element method 191 The six constants a can be determined by equating the six nodal displace- ment components with x and y co-ordinates taking on appropriate values as
{<5e} <
ô i
v,
By equation (13):
1 0 1 0 1 .0
Xi
0
*y
0 xm
0 0 Jy 0 ym
0 0 l 0 1 0 1
0
0 xt
0 xm
0 yt
0 Jy 0 ym.
(14)
p i ; 0 0 If]
~ L0 0 0 1 x y\
The strains are defined as
\7xy) du
Jy
dx dv
Ty
+
dv dx
0 1 0 0 0 0' 0 0 0 0 0 1 0 0 1 0 1 0
The stresses are given by
{a} = [DM. (7)
The finite element method is being extensively used for the analysis of strain-stress patterns around large tunnels and underground excavations.
Residual rock stresses, the fc-value, rock jointing, rock faults and progressive fissuring of the rock mass can be introduced in the programs. Similarly the stability of rock slopes and many other problems of rock mechanics can be solved with the finite element method.
In sections 10.2, on special problems for mining and tunnel engineers, 10.5.3, on Fenner's equation, 10.10, on the estimate of the required rock support, 10.11 on underground hydro-electric power stations and 16.3 on Waldeck II, the use of the finite element method for solving special problems will be amply discussed.
Cundall (1971; 1974) has developed a computer program that can model behaviour of assemblages of rock blocks and visually display this behaviour on the screen of a cathode-ray oscilloscope. There is no restriction in block
shapes and no limits to the magnitude of displacements and rotations that are allowed. The rock geometry is specified by the user, who draws lines on the screen of the cathode-ray oscilloscope which is connected to a mini- computer. The programme allows the blocks to move relative to one another, under the action of gravity and forces specified by the user, as a function of time. Joint properties may be specified.
8 Interstitial water in rock material and rock masses