CC BY
36
y^K 622
Wang Weiming
POSSIBILITY CALCULATION MODEL OF THE BOLT-LENGTH IN FRACTURED
ROCK ROADWAY
1. Introduction
The roof of roadway surrounding rock becoming unstable and falling of ground is a kind of frequent dangerous accident. So fixing the bolt-length has important engineering significance by studying the stability of roadway surrounding rock roof. We all know that the essential and abundant condition of falling ground in fractured rock roof in roadway is
s • sign(r • hi )tjj > 0 (1)
where, s is displacement direction of dangerous rock; r is external forces on dangerous rock; ni is outer unit normal of .ith group structural plane.
Roadway surrounding rocks in fractured rocks can be considered as a blocks set cut by many different attitudes’ structural planes. The block is called dangerous rock when meeting movement condition. The displacement directions of blocks and the dominant external force directions are identical when fall of ground. If the dominant external force is its weight then the block displacement direction is vertical. So on the roadway roof in vertical direction and no geometry restricting convex blocks can be seen as dangerous rocks. The support bolt-length is the first max falling height. Dangerous rocks existence has close relation with structural plane attitude. Structural plane attitude and distribution in rocks are random. The length of bolt used to consolidate the surrounding rocks is also random variable. So the theory of possibility can be used to set up the conjectural model.
2. Set up possibility space
Q is been constituted by inter-set of structural plane traces. Event field. T is been seen as a closed range surrounded by excavation plane and structural plane trace possibility. A set function P in this event area can be defined as the possibility space (Q ,T ,
P).
Giving, on the roadway transverse section two groups of traces are distributed and traces in the same structural plane are parallel. The space between traces obey minus exponent distribution, two groups parallel traces form quadrilateral set. The roadway sections are variable randomly to the net (Fig. 1). Supposing in the first group there are K1 traces on specimen plane. In the second group there are K2 traces. So there are (K1-1)x (K2-1) closing subsets are formed called ^. Roadway transverse sections dropping into every subset has the same possibility. So
Q e [£i
/i = (kj -1) x (k2 -1)]
(2)
As for the falling event the 9 conditions at the position which roadway section drops into the traces nets in fig.1 may occur, only ©N ©n ©n ©n ® may lead to falling.0 represents impossible event. So the falling event subset has the range from 0 to some maximum (called £max). And the event filed:
r e [D ^Si < Smax/i = (k1 - j) x (k2 -1)]
(3)
Fig. 1. The relativity position of roadway cross section on the net of structural plane trace
Now we define the falling possibility as:
(4)
i=1
Thus, falling event has uniform distribution in specimen space. Its possibility value equalizes to the value of possibility, which the max height point is located on the max area made up by structural plane traces and excavation plane traces.
The event filed is the cutting set of two kind traces (such as fig.2).
3. The analysis of event field
Imagine there are two intersect traces and their length are l1 and l2 The max falling height is hmax. Given Z is random falling height. Zmax is the max falling height; x is projection height of trace li , y is projection height of trace l2. B is the trace length of excavation plane. We can find:
Fig.2. The distribution of falling
X = \l^sina1,Y = \l21 sin a2, hmax = B/ r (5) where
r = \ctga\ + |ctga21;
a1, a2 are respectively the intersecting angles between two structural plane trances and excavating plane trance
[1 , X , Y] [^Max/hMax , X/hMax , Y/hMax]
(6)
Analyzing the factor of event field we find that 3 cases may occur:
(1) X>1,Y>1 then
ZMax=min[1,X,Y]=1 , Ze[0,1]
(7)
(2)X<YandX<1 , Y<1 then
ZMax=min[1,X,Y]=X ,Ze [0,X]
(8)
(3) y<x,and x<1,y<1 then
ZMax=Y, Ze [0,Y] (9)
4. Condition possibility
(1) if x>1 y>1,0 < z < 1 then the possibility of falling unit height h<Z is (fig. 3)
1 2
-r(1 - Z)2 2
p(h < z/x,y) = 1 -2-----------------= 2Z -Z2(10)
r / 2
(2) if x<y,x<1 ,y<1, 0 < z < x then the
possibility of falling unit height h<Z is (fig.4)
Fig.3. Event field and specimen space offirst case
Fig.4. Event field and specimen space of the second case
Fig.5. Event field and specimen space of the third case
p(h < z / x, y ) =
1 2 -(r - z r) = 2z - z2
1(2rr - x2r) 2x - x2
(11)
(3) if y<x , x<1, y<1, 0 < z < x then the possibility of falling unit height h<Z is (fig.5)
2z - z2
p(h < z/x,y) =-------------— (12)
2y - y2
5. Bolt length possibility
Supposing the distributions of x and y are independent respectively. So the falling height possibility may be described:
p(h <z) = j jp(h/x,y)f1(x)f2(y)dxdy (13) xy
where f1(x) and f2(x) are the distribution function of x and y respectively. Relational study indicate that structural plane trace obey minus exponent distribution. So, x and y are also obey minus exponent distribution respectively. The possibility needed bolt-length L<Z can be deduced from those condition. We obtain with the figure integration
3
p(L < z) = (2z - z2 )e-W1 + 2wiz^1 x
3(x + y) [f1(z) + f2(z) + f3(z), 0 < z < 1;
p(L<0)=0, P(L<1)=1, (14)
where
Structural plane attitude and traces average
The number of planes Dip angle Dip direction angle Trace length average ( m )
P1 67 90 1.523
P2 57 80 1.381
P3 65 99 2.355
1
, 1 1+z
1 —~ z ----~
f , ) * yr e x , 8e 2x
f1(z) =—e s[-----------------+-------------
6 z(2 - z) (1 + z)(3 - zy
- WLd+z)
, e-w1z 16e 2
f2(z) =--------------^t +
f3(z) = -
6(2z - z2)2 3(1 + z)2(3 - z)2 (15)
z 1+z
-(x+—) x 2y
3(2z - z2)(1 + z)(3 - z)
-[2-+W2(1+z)] 1024e 2y
+
2
3(1 + z)(9 - z* )(2 - z) w1 and w2 are distributing synthesize parameters.
6. Engineering example
A roadway, the dip angle is a =87.7° dip direction angle is /3 =277° . The height of transverse section of roadway is 4m and the span is 5m. According to geology explore and the result analyzed the structural planes can be divided into 3 groups in roadway surrounding rocks. Each group has a dominant direction. Every dominant directions,attitude and the length average value of structural plane trace in specimen windows are shown in Table .
Divided into three groups and calculation the result shows in fig.6.The possibility of needed bolt length <=1.775m is 0.95.The possibility of bolt length<=0.385 is 0.382 designed bolt length is 2m.
7. Conclusion
(1) The bolt length deduced from the model belongs to upper value.
(2) The bolt length deduced from this model is more convenient than the model of continuous medium mechanics model.
(3) As long as the basic parameter of structural plane and roadway geometry size are given the bolt length can be deduced primarily. With engineering advancing, the measuring statistic model and parameters will make perfect continuously and the result will trend to more and more precise.
MAIN REFERENCES
1. Wang Weiming. Dangerous rocks forecasting and application in surrounding rocks. Coal industry publishing house.
2. Priest S.D. ,Hudson J.A. Estimation of Discontinuity spacing and trace length using scanline surveys // Inter. J. Rock Mech. Min. Sci. and Geomech.Abstr.,1981, P.183-197.
3. Kulatilake P H. S. W., Wu T. H. The density of discontinuity trace in sampling windows // Int. J. Rock Mech.Nin.Sci.$Geomech.Abstr.,1984,Vol.21,No.6, P.345-374.
4. Shaoxiang H.,Weiming W,Xianwei Li. The statistics of the discontinuities in rock mass in coal mines and their application, Proc.,99 Inter. Symp. on Min. Sci. and Tech.,China. A. A. Balkema,Printed in the Nehterlands,1999, P.227-230.
□ Автор статьи:
Ван Вэй-мин
- доктор, профессор Шаньдунского научно-технического университета
КНР
