THE
LIMITING Yu.
A.
SPAN
OF A N U N S U P P O R T E D
Modestov
and
Yu.
V.
Shuvalov
FISSURED
ROOF
UDC 622.831.24
The country rocks of most c o a l deposits are split by fissures of various origimo The density and size of the fissures and their cohesion depend on the particular geological conditions. C a l c u l a t i o n of the l i m i t i n g permissible exposure of such roof rock is of importance for mining practice. Borisov and Kuznetsov [1, 2] have discussed the fundamental laws of the mechanics of fissured rocks and the nature of their interaction with the supports. These authors compare the action of the individual roof beds with that of a t h r e e - j o i n t e d arch. This theory is valid and is confirmed by numerous laboratory and field investigations. The condition of equilibrium of a t h r e e - j o i n t e d arch with no supports is governed by the e q u a l i t y of the thrust in the middle joint, due to the action of the weight of the rock within the given bed, to the reaction of the rock under compression and crushing. The horizontal thrust is determined by the usual formulas of the strength of m a terials, and depends on the span of the working, the density of the rock, the thickness of the bed, the d i s p l a c e m e n t of the roof, and the nature of the applied load. The area of the middle joint, on which depends the horizontal thrust, is d e t e r m i n e d e m p i r i c a l l y [2], and Borisov [1] finds the height of the region of interaction of the blocks, which corresponds to the area of the joint per unit length of the working, starting from their geometry. The authors assumed the height of the region of contact of the blocks at the abutments of the arch to be twice as great as the height of the region of contact of the blocks at the joint. The expressions derived require some adjustment, because they are based on the assumption that the blocks are tightly pressed against one another, and that as the middle joint sinks the region of the contact decreases from half the stratum thickness to zero. In this case the roof has m a x i m u m supporting c a p a c i t y with m i n i m u m d e f o r m a tion. However, field and laboratory observations show that the supporting c a p a c i t i e s of fissured roofs, when they are deformed within certain limits, begin by increasing. An illustration is afforded by the deposits of the north east, where the fissure gape of the c o a l seams and rock strata is very large and reaches 0.5-1.5 m [ 3 ] . In m a n y cases the fissures are filled with ice, c l a y , etc. We s o m e times encounter open fissures, which are due to degradation of frozen ground at a given depth or to its thawing by the a c c e s s of warm air through the working. In the C h u l ' m a n pit ( C h u l ' m a n coalfield) a c o a l seam about 3.0 m t h i c k lies almost horiL .;K z o n t a l l y (0-4 deg). The roof rocks are aleurolite and sandstone. The thickness of individual beds of the i m m e d i a t e roof varies from 0.2 to 1.0-1.5 m. There is often a false roof of aleurolite, 0.2 m thick. The country rocks are split up by fissures perpendicular to the stratification. The density of fissures is from 0.2 to 3.0 per meter. The gape of the cracks is 0.2-0.5 cm.
F
A
;t
~
_
_.
_
_I~
Fig. 1. Scheme for calculating area of contact of middle joint of fissured roof.
Mining operations are carried out in the zone of the n e a r - z e r o temperatures, and therefore the fissures are partly f i l l e d with loose ice and partly open. Despite the high strength of the i m m e d i a t e - r o o f rock (the compressive strength of aleurolite exceeds 400 kgf/cm2and that of sandstone exceeds 700 k g f / c m 2 ) , t h e l i m i t i n g spans of the unsupported
Leningrad Mining Institute. Translated from F i z i k o - T e k h n i c h e s k i e Problemy Razrabotki Poleznykh Iskopaemykh, No. 5, pp. 8-13, S e p t e m b e r - O c t o b e r , 1969. Original article submitted December 29, 1967.
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 for any purpose whatsoever without permission of the publisher. ,4 cop)" of this article is available from the publisher for $15.00.
494
TABLE 1. Comparison of Model and Theoretical Data ~:
Size of crack, K m m
Index 0.5 Maximum
2,0
4,o
size of region
of contact of blocks, C j, m m calculated . . . . . . . . measured in model Breaking load, kg:
6,6 5--7
calculated . . . . . . . . measured in model
3,5 2--5
,2
'_86
12 I2--14
0,5 1,0
roof are short. A block of aleurolite caves in working slightly more than one meter wide, while for sandstone the stable span is 5-6 m. The presence of free fissures of certain dimensions in the roofs indicates that the schemes suggested in [1, 2] are only particular cases of a more general scheme of the action of a fissured roof. Figure 1 shows the scheme of action of a three-jointed arch made of two separated blocks, split up by fissures of width K. The action of the blocks is the same in the left-hand and right-hand parts of the arch if they are of the same size, and therefore, we shall consider only one (the left-hand) side of the arch. The width L/2 of the block is practically equal to the half-span of the working; h, the height of the block, corresponds to the thickness of the rock bed; and y is the sag of the middle joint or the deformation of the roof at the center of the working. When the blocks rotate about axes at the sides of the working, the m a t e r i a l at the joint (crown) and abutments of the arch becomes crumpled. We can determine the size of the zone of crumpling by starting with the geometrical construction shown in Fig. 1 (line ENF). ENF = EN + NF;
(1)
E N = (7' N tg .~ N C' E;
(2)
C' N
NF I-N C'E
=
.
tg ~ N F C ' "
= L~ N F C "
:
~ DA
(3)
(4)
D " = a,,,
whence
EN :
(5)
C'Ntg%;
N F =
c" N ;
(6)
tg a n ENF
C'N:
-
C' N t g % +
MNC'--MN=
MNC"
= AC'
tg
sin .~ M A C "
=
C'N ---- 2 - - ;
+
:
(7) :
sin 2 a n
a n
MNC'--
ACt :: AC--
MNC'
C'N
A C " sin (% +
l/ AB~+BC 2 :
L+K.
=MNC'--
hs +
. V -n 2 + -L4- ~ , sin ( % + % ) ;
L2
--
4
..... 2
%);
.
'
'
(8)
(9)
(10)
(11)
495
C'N
--
h'+T
2 E NF
=
[r
L2 ,
h2+
L+K
Bin ( % + % ) - - - - 2
--
sin(~o+~n)
4
;
(12)
--
(13)
sin 2 a n
L % = a r c t g 2t~ ;
(14)
a n is the angle of rotation of the block an :: arcsin
2y
L
(15)
The angle of rotation of the block, which determines the period of action of the t h r e e - j o i n t e d arch, m a y vary from 0 deg to some value at which the projection of the diagonal AC' of the block on the horizontal axis MNC' will be equal to (L + K / 2 : (16)
MNC" = MN;
--
L~
ks+-~-
L +K
sin(%+%)
=
(17)
2
whence L+K
% = arcsln
_
2
l/
- - %.
(18)
h~ + --~-
-
When the angle of rotation increases further, there is no cohesion between the two blocks forming the arch, and the latter cannot form. Since the blocks crumple not only at the crown but also at the abutments of the arch (see Fig. 1, point A), the region of contact of the blocks (cj = ENF) falls to half at the same angle of rotation of the block. Allowing for crumpling of the blocks at the abutments of the arch, the size of the region of contact is
C3 ---~
C
LZ -~-+h~
sin(ao + a n ) - -
L
K
2
2
(19)
sin 2 an
Here we must take account of the deformation of the blocks themselves under compression, which in fact has the same effect as a crack b e t w e e n blocks of equal width. The m a x i m u m deformation of the blocks under c o m pression can be roughly e s t i m a t e d as k c o m p - - - - - L%'s L~
(20)
J
where Oc. s is the compressive strength of the rock in k g f / c m ~ a n d E is the modulus of e l a s t i c i t y in k g f / c m B. The expression for the size of the region of contact of the blocks at the crown, allowing for their d e f o r m a tion under compression, is Cj =
/
L-" ~ - + h~ sin (ao + an) sin 2 a n
496
L
K
2
2
L %.s .
2 E h
(21)
The v a l i d i t y of this formula for the size of the region of contact of the blocks at the crown of the arch was tested by comparing the t h e o r e t i c a l data with e x p e r i m e n t a l results obtained by flexing bars of a sand-paraffin mixture. We used a lever press to test the bars. This apparatus enabled us to test bars with cracks of various widths, which was achieved by vary! ing'the distance between the ends of the bar and the walls of the c a s sette in which it was placed. We tested specimens 12 cm long, 3 c m wide, and 1.8 c m high. A load was applied to the center of the bar Fig. 2. Scheme for calculating rise and smoothly increased until a crack appeared. The force required of arch. to cause a crack was recorded, and then the load was again smoothly increased from zero to a value at which the bar parted e n t i r e l y . We simultaneously observed the sag of the middle joint and the size of the region of contact of the blocks at the crown. 84
1
Table 1 lists the e x p e r i m e n t a l results on the breaking load and the region of contact of the blocks at the crown for given crack size. It also gives the values of these variables predicted a n a l y t i c a l l y . The size of the region of contact of the blocks was determined from (21), and the breaking load a p p l i e d to the center was c a l c u l a t e d from a formula given in [1]: p _ 2 %.s --
q (n - - cj + Y) L ~
(22)
where Oc. s is the compressive strength of the m a t e r i a l from which the b e a m is m a d e . T h e c a l c u l a t e d data on the size of the region of contact of the blocks and the breaking loads agree with those ,,measured on the models, showing that the proposed scheme of c a l c u l a t i o n is correct and was correctly applied. Knowing the area of contact of the blocks cj, we can find the rise of the arch f t which is equal to the distance between the resultants of the horizontal thrusts at the crown and abutments of the block [1, 2]. It depends on the sag of the crown y, the size of the region of contact of the blocks cj, and the nature of the stress distribution in the hinges. Taking the distribution of the stresses in the joints to be nearly triangular for brittle rocks (Fig. 2) and fz to be nearly rectangular for plastic rocks,
f, = h
2 c3 j
Y;
(23)
f 2 "~- h - - cj - - y,
(24)
where y is the sag of the middle joint or the d i s p l a c e m e n t of the roof, in meters.
TABLE 2. Size of Region of Contact of Blocks in Crown for False Roof of Seam D Span, m
T o t a l exposure of fissures, m Compression of block, m Size of region of contact for roof displacements (in meters) of: V~-0,05 y=0,1 y=0,2
1,o
0,02 0,004
0.028 0.046 0,036
t
2,0
3.0
0,04 0,008
0,06 0,012
m
49"/
TABLE 3. Size of Region of Contact of Blocks in Crown for I m m e d i a t e Roof of Seam D
Spans m
5,0
7,0
I0,0
0,06 0,034
0,08 0,048
0,12 0,068
0,231 0,25
0,048 0,152
3,0
T o t a l size of fissures,
I
]
m . . . . . . . . . . . . . .block, m I 0,020 0,04 Compressionof Size of region of contact for roof displacements (~n~,leters) of:
0,246 0,356 0,345
y=0,3 g=0,5
The horizontal thrust is
T~ -~-
q L~
q L2
T2 == 8 ( h - c j--y)
tons;
tons,
(25)
(26)
where q = 7h is the load on unit length of the span in tons per meter, and the condition of stability of the arch is
~C,S
~c.s
_--
q L2
k g f / c m z,
. . . . . 2_q_L_~_ ~.... k g f / c m z. - - 8 cj (h - - cj - - y)
(2"/)
(28)
Equations (27) and (28) give the m a x i m u m span at which the system can be in equilibrium i . e . , the l i m i t i n g span of an unsupported roof. We have to c a l c u l a t e the number of values of cj for various spans, and, substituting them into (27) or (28), we can verify which of them satisfy the given condition of equilibrium. By the above method we c a l c u l a t e d the l i m i t i n g spans of unsupported roofs for several models d e v e l o p e d in the Rock Pressure Laboratory of the Leningrad Mining Institute. The c a l c u l a t e d results agree w e l l with the e x p e r i m e n t a l data. Thus, for a m o d e l of sand-paraffin mixture (Oc.s = 1 2 k g f / c m 2 , h = 1.0 cm), the l i m i t i n g span is 37 cm for a sag of 1.5-2.0 m m of the middle joint (the width of the crack corresponded to compression of the block). From this m o d e l it was found that the limiting unsupported span was 40-42 cm. From formula (21) we c a l c u l a t e d the size of the region of contact of the blocks for the false and i m m e d i a t e roofs of seam D of the C h u l ' m a n pit (Tables 2 and 3). The l i m i t i n g spans c a l c u l a t e d from (27) with the data in Tables 2 and 3 lie within the ranges 1-2 m for the false roof and 7-10 m for the i m m e d i a t e roof; this agrees adequately with the actual data. Thus a comparison of the c a l c u l a t e d values with the results of field observations and laboratory investigations leads us to consider that the suggested method of c a l c u l a t i n g the l i m i t i n g spans of unsupported roofs is suitable for the solution of p r a c t i c a l problems. LITERATURE CITED I.
2. 3.
498
A. A. Borisov, Calculations of Rock Pressure in the Faces of F l a t - L y i n g Seams [in Russian], Nedra, Moscow (1964). G. N. Kuznetsov, "Determination of the t o t a l supporting c a p a c i t y of the roofs of underground workings," in: VNIMI Symposium, No. XXII, Ugletekhizdat, Moscow (1950). A. F. Zil'berbord, "Influence of natural factors on the conditLons of underground m i n e r a l mine workings in the permafrost region," in: T h e r m a l : a n d M e c h a n i c a l Processes in Mining [in Russian], Nedra, Moscow (1965).
