Stress in a rock mass near an unsupported working with random irregularities of form
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IN A ROCK
WORKING V. T . G. T .
WITH
MASS
RANDOM
NEAR AN UNSUPPORTED IRREGULARITIES
OF F O R M UDC 622.268 (477.60)
Glushko, A . G. P r a v d a , R u b e t s , a n d N. T . B o b r o
Predictions of the strength of umupported mine workings require determinations of the stress in the surrounding rock. The stress depends both on natural geological and m a n - m a d e factors [I]. To formulate a mathematical model approximating the actual state, the factors forming the stressed state of the rock must be taken fuUy into account and described. In particular, in investigating the stressed state of the rock surrounding an unsupported working, one must take account of such factors as random deviations of the working outline from the planned outline, the nonunfformity of the mechanical properties of the rock mass, etc. In the given case the use of applied methods of theory of random functions [2, 3] is an important theoretical basis for the solution of such problems. Let us examine a horizontal unsupported working with circular planned cross sectiorL Deviation of the actual crom-sectionai outline from the zero circle of radius R o m a y be represented as a s t e a d y - s t a t e random function, the spectral expansion o f which has the form (0) = ~ (uz cos kO + vz sin kS), g=0
(1)
where uk and v k are random values with zero values and pairwise equal dispersions, determined by statistical proceasing of the results of field measurements of the working outline [3]. Let us soDe the problem of determining the stress concentration for an infinite weightless e l a s t i c p l a n e with an outline p (o) = Ro + ~ (o).
(2)
The p(O) boundary is free from stress i.e., the normal N and t a n g e n t i a l T components of the load at the outline are equal to zero. In contrast with [3], the stressed state at infinity is introduced as follows:
(3) which enables us to obtain a solution simultaneously for the whole range of i n i t i a l stresses in the rock mass : from hydrostatic (p = cO to plane stress (p = 0 or q = 0). The minus sign denotes compressive stresses. The solution is found in a polar system of coordinates. Using the procedure adopted for such cases [4], we write the conditiom for a zero circumference (7 = Rd: I d~ (0) _ O;
7 Or.to
d8
(4)
1 d~ (0) = 0.
7
dO
Institute of Geotechnical Mechanics, A c a d e m y of Sciences o f the Ukraine, Dnepropetrovsk. Translated from Hziko--Telchnicheskle Problemy Razrabotki Poleznykh Iskopaemykh, No. 2, pp. 119-122, March-ApriL 1972. Original a r t i c l e submitted July 5, 1971. 9 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission of the publisher. A copy o[ this article is available from the publisher for $I5.00.
222
The boundary conditions at infinity 7 "* ~ohave the form: %
=
p..arq -p--q T-
T
--
--
cos
20;
P+q +-~cos20;
[ I
(8)
I
P -- q "LY0 ~
Z
SIFI
I
I
2O.
Following this procedure, the
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