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An elastic treatment of ground movement due to mining—Corrigendum
AN
ELASTIC
TREATMENT OF GROUND LINING-CORRIGENDUM
Department
~OVE~~~
By IL S. BERRY of Mining Engineering, University
DUE
TO
of Nottingham
(Received 28th March, 1963) IT HAS come to light that the subsidence curve used as an example in the second paper of this series (BERRY and SALES 1961) was not taken from actual measurements, but was a forecast hased on initial movements and past experience. Subsequently the true subsidence curve (Fig. 1) has been examined by the author and new estimates of the elastic parameters kI and ks have been made. The values (kl = 14, fez = 0) based on the wrong curve had already come under suspicion for two reasons, First, the values led to d~p~po~ionately large horiaontal components of the surface displacement. Secondly, it had not been found possible to obtain these values of 4 and L from the relations
using physically plausible values for the moduli and Poisson’s ratios (El, E2, M, v~, YJ. In fact, it may be shown that a consequence of the positive-definiteness of the strain-energy function and the positiveness of the maduli and Poisson’s ratios is the inequality E2fEl > $klz : from physical considerations it does not seem likely that E2fE1 will be much larger than unity, and the very high value necessary when kx = 14 is hardly credible. Initial estimates of 4 = l~2 = 6 were made from the true practical curve by means of the formulae derived from the approximate expression for surface subsidence, - us (a) (BERRY and However, poor correspondence with the practical curve was obtained when these SALES 1901). values were substituted in the exact expression for subsidence due to complete closure (represented in the theoretical model by a uniform discontinuity, t, in vertical displacement). The approximation was eventually abandoned and the curve was fitted by trying various values of k, and ks in the exact expression. It was found that a considerable range of pairs of values gave almost equally good correspondence between theoretical and practical subsidence curves, but preference was given to the pair (kl = 2, kz = 10) in which kl was smallest, in order to keep Ez/El small (Fig. 1). Furthermore, when the horizontal strain at the surface (3uO/3;c) was examined, this pair of constants allowed a good fit of the theoretical curve to the measured values {Fig. 2), taking into account the erratic nature of the latter, while the pairs with larger kl made the strain maxima and minima too large in magnitude. In order to make the theoretical subsidence at the central point agree with the measured value of 17 cm, the uniform displacement discontinuity (t) at the excavation was given the value 44 cm, about f of the excavated thickness, and the same value was used in the strain computations. The new estimates were used to calculate the surface movement, assuming the type of nonuni$orm discontinuity at the excavation which would occur if there were no contact of roof and floor (non-closure) and the free surface were at an infinite distance. The theoretical expression has as a factor the product of the primitive stress, p, and an elastic constant, $2 22 -
1 E2 -
v22 _ (1 E, - m 373
V,“)
’
(2)
FIG.
1. l
-)
compared with theoretical cnrve for kg = 2, ka = 10 (------+
FIG. 2. Horizontal strain at surface : practical points and theoretical curve (h = 2, ks = 10).
Measured subsidence (-- . -
Corrigendum
375
and agreement with the practical subsidence curve at in = 0 was obtained by putting ps;; = O,~Q. The fit of both subsidence and strain curves to the field data was almost as good as that of the curves based on the complete closure model. The similarity allowa us to assume that the intermediate state of partial closure would give at least as good agreement. If there were no closure, the adoption of the likely value for p would lead to the approximate result 822 22 = 3.5 x IO-’ (lb/in*)-l, but as there is actually some closure (in the sense of appreciable t~nsmi~ion of stress between roof and floor) sgg must be larger. In either ease a maximum for El may be found with the help of equation (2) and is given by KC1< 7.14 x lo6 lb/in*, corresponding to y1 = 0. Further computations based on the new estimates will be found elsewhere (BERRY lQf38).
ACKNOWLEDGMENT
The author wishes to thank the National Coal Board for a grant in aid of this Iesearch but the views expressed are those of the author 8nd not necessarily those of the Roard.
REFERENCES &EU~Y, D. S.
BEBRY,D. S. and SALES,T. W.
1963 1061
Min. Engr. (to be published). 3. Mech. Phgs. s’olids 9, 52.
ELASTIC
TREATMENT OF GROUND LINING-CORRIGENDUM
Department
~OVE~~~
By IL S. BERRY of Mining Engineering, University
DUE
TO
of Nottingham
(Received 28th March, 1963) IT HAS come to light that the subsidence curve used as an example in the second paper of this series (BERRY and SALES 1961) was not taken from actual measurements, but was a forecast hased on initial movements and past experience. Subsequently the true subsidence curve (Fig. 1) has been examined by the author and new estimates of the elastic parameters kI and ks have been made. The values (kl = 14, fez = 0) based on the wrong curve had already come under suspicion for two reasons, First, the values led to d~p~po~ionately large horiaontal components of the surface displacement. Secondly, it had not been found possible to obtain these values of 4 and L from the relations
using physically plausible values for the moduli and Poisson’s ratios (El, E2, M, v~, YJ. In fact, it may be shown that a consequence of the positive-definiteness of the strain-energy function and the positiveness of the maduli and Poisson’s ratios is the inequality E2fEl > $klz : from physical considerations it does not seem likely that E2fE1 will be much larger than unity, and the very high value necessary when kx = 14 is hardly credible. Initial estimates of 4 = l~2 = 6 were made from the true practical curve by means of the formulae derived from the approximate expression for surface subsidence, - us (a) (BERRY and However, poor correspondence with the practical curve was obtained when these SALES 1901). values were substituted in the exact expression for subsidence due to complete closure (represented in the theoretical model by a uniform discontinuity, t, in vertical displacement). The approximation was eventually abandoned and the curve was fitted by trying various values of k, and ks in the exact expression. It was found that a considerable range of pairs of values gave almost equally good correspondence between theoretical and practical subsidence curves, but preference was given to the pair (kl = 2, kz = 10) in which kl was smallest, in order to keep Ez/El small (Fig. 1). Furthermore, when the horizontal strain at the surface (3uO/3;c) was examined, this pair of constants allowed a good fit of the theoretical curve to the measured values {Fig. 2), taking into account the erratic nature of the latter, while the pairs with larger kl made the strain maxima and minima too large in magnitude. In order to make the theoretical subsidence at the central point agree with the measured value of 17 cm, the uniform displacement discontinuity (t) at the excavation was given the value 44 cm, about f of the excavated thickness, and the same value was used in the strain computations. The new estimates were used to calculate the surface movement, assuming the type of nonuni$orm discontinuity at the excavation which would occur if there were no contact of roof and floor (non-closure) and the free surface were at an infinite distance. The theoretical expression has as a factor the product of the primitive stress, p, and an elastic constant, $2 22 -
1 E2 -
v22 _ (1 E, - m 373
V,“)
’
(2)
FIG.
1. l
-)
compared with theoretical cnrve for kg = 2, ka = 10 (------+
FIG. 2. Horizontal strain at surface : practical points and theoretical curve (h = 2, ks = 10).
Measured subsidence (-- . -
Corrigendum
375
and agreement with the practical subsidence curve at in = 0 was obtained by putting ps;; = O,~Q. The fit of both subsidence and strain curves to the field data was almost as good as that of the curves based on the complete closure model. The similarity allowa us to assume that the intermediate state of partial closure would give at least as good agreement. If there were no closure, the adoption of the likely value for p would lead to the approximate result 822 22 = 3.5 x IO-’ (lb/in*)-l, but as there is actually some closure (in the sense of appreciable t~nsmi~ion of stress between roof and floor) sgg must be larger. In either ease a maximum for El may be found with the help of equation (2) and is given by KC1< 7.14 x lo6 lb/in*, corresponding to y1 = 0. Further computations based on the new estimates will be found elsewhere (BERRY lQf38).
ACKNOWLEDGMENT
The author wishes to thank the National Coal Board for a grant in aid of this Iesearch but the views expressed are those of the author 8nd not necessarily those of the Roard.
REFERENCES &EU~Y, D. S.
BEBRY,D. S. and SALES,T. W.
1963 1061
Min. Engr. (to be published). 3. Mech. Phgs. s’olids 9, 52.