This paper presents the results of the studies on the strength of rock masses with discontinuities from the viewpoint of complexity.
Various discontinuities in rock masses are mostly distributed self-similarly and they can be considered as fractal.
Assuming that rock masses which have morphological self-similar discontinuities, are also mechanical self-similar, a numerical formula is derived as follows.
San = (1-K
n) S
jn+K
nS
in (1) where, San = apparent strength of rock masses which have discontinuities of n-orderSjn = strength of discontinuity of n-orderSin = strength of intact rock of n-orderKn = coefficient of discontinuity of n-order (area ratio of discontinuities to the rock masses) Similarly, the following formula is derived for the apparent strength of rock masses which have discontinuities ofhigher order ((n+1) -order). S
an+1 = (1-K
n+1) S
jn+1+K
n+1S
in+1 (2)
On the assumption described above, the apparent strength of n-rder (S
an) can be replaced with the apparent strength of (n+1) -order (S
an+1) by the nesting effect.
The strength of discontinuity can be considered almost similar in any order, that is S
an=S
in+1, S
jn≅S
jn+1, formula (3) is derived from formulas (1) and (2).
S
an+1 = (1-K
nK
n+1) S
jn+K
nK
n+1S
in (3)
Formula (3) is generalized to formula (4)
S
an+m= (1-K
n…K
n+m) S
jn+K
n…K
n+mS
in (4) Formula (4) shows that the apparent coefficient of discontinuity of the rock masses which have discontinuities of higher order, Ka = KnKn+m, therefore, if the Ladanyi's theory applied, the apparent strength of rock masses which have discontinuities of higher order is obtained from formula (5). S
an+m=σ (1-K
n…K
n+m (V+tanφ
b) + K
n…K
n+mτ
r/1- (1-K
n…K
n+m) Vtanφ
b (5)
In the later half of this paper,
1) the coefficients of discontinuities are obtained from the results of block shear tests in situ
2) the strength of rock masses which was calculated using the coefficients of discontinuities as above and ones by the Ladanyi's theory are compared, and
3) the relation of the hierarchy of discontinuities and the strength of rock masses is studied.
View full abstract