Arch. Min. Sci., Vol. 54 (2009), No 1, p. 135–143

Arch. Min. Sci., Vol. 54 (2009), No 1, p. 135–143

Arch. Min. Sci., Vol. 54 (2009), No 1, p. 135–143
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Electronic version (in color) of this article is available: http://mining.archives.pl
ANDRZEJ KWINTA*
TRANSITIVITY POSTULATE EFFECT ON FUNCTION OF INFLUENCES RANGE RADIUS
IN KNOTHE THEORY
WPŁYW POSTULATU TRANZYTYWNOŚCI NA FUNKCJĘ PROMIENIA
ZASIĘGU WPŁYWÓW W TEORII KNOTHEGO
This paper presents the analysis of the transitivity postulate effect on the function of variability form
in the radius of main influences range in Knothe theory. As a result of performed considerations one should
find that the prerequisite is that the influence range radius function in rock mass for Knothe theory should
be monotonically increasing.
Keywords: prediction of deformations, function of influences
Teoria Knothego służąca do prognozowania deformacji wywołanych podziemną eksploatacją górniczą
pierwotnie została opracowana dla powierzchni terenu, a dopiero później została rozwinięta na przestrzeń
górotworu. Teoria prognozowania powinna spełniać postulaty matematycznego modelu ośrodka. Pomimo,
że rzeczywisty górotwór odbiega swoją budową od wyidealizowanego ośrodka teoretycznego, to stosowana
teoria prognozowania deformacji powinna spełniać postulaty matematycznego modelu ośrodka. Jednym
z postulatów jest postulat tranzytywności (1), który determinuje własności funkcji zmienności promienia zasięgu wpływów głównych w górotworze r(z). Postulat ten można scharakteryzować następująco:
wynik obliczeń przemieszczeń na zadanym poziomie nie zależy od tego czy obliczenia te wykonuje się
bezpośrednio dla tego poziomu, czy też w trakcie obliczeń wyznacza się przemieszczenia na pośrednich
poziomach obliczeniowych. W tabeli 1 przedstawiono zbiór wybranych poglądów na postać funkcji
zasięgu wpływów głównych w górotworze, proponowane przez różnych autorów. W literaturze pojawiają się „dziwne” modele bądź też modyfikacje teorii Knothego, co do których można mieć poważne
zastrzeżenia formalne. Ponieważ ciągle trwają prace nad uszczegółowieniem opisu przejścia deformacji
przez górotwór, należy określić dopuszczalne formy modyfikacji teorii, które byłyby zgodne z postulatami
matematycznymi ośrodka. W niniejszej pracy przeprowadzone rozważania teoretyczne oparto na „elementarnej” eksploatacji. Schemat rozmieszczenia eksploatacji i horyzontów obliczeniowych przedstawiono
na rysunku 2. W wyniku przeprowadzonych obliczeń uzyskano zależność (20), która wskazuje, że tylko
funkcja niemalejąca daje zadość postulatowi tranzytywności. Fakt ten w istotny sposób zmniejsza ilość
rozwiązań możliwych do zaakceptowania od strony formalnej.
Słowa kluczowe: prognozowanie deformacji, funkcja wpływu
*
UNIVERSITY OF AGRICULTURE, DEPARTMENT OF GEODESY, BALICKA 253A, 30-198 KRAKÓW, POLAND
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1. Introduction
The mining results in displacements and deformations both on the surface of and
inside the medium. The universally used theory of forecasting of deformations was
originally developed for the area surface and only then it was expanded upon the rock
mass space. One of the basic elements of the Knothe theory (1950) is the main influences range parameter, which results from the model parameterisation. To make the
theory of forecasting applicable it should satisfy postulates of the mathematical model
of medium. This model was given by Litwiniszyn (1962) and one of the postulates is
the transitivity postulate that determines the properties of the function of variability in
radius of the range of main influences in rock mass r(z) (Jędrzejec, 1986).
New attempts to make the description of transition of the deformation process
through rock mass (Janusz, 2003; Paleczek, 2007; Prusek & Jędrzejec, 2008) more detailed are being made constantly; therefore it is necessary to define the possible forms
of the function of variability in radius of the range of main influences inside rock mass
in Knothe theory.
2. Views on variability in radius of main influences
range in rock mass
Due to lack of the appropriate set of geodesic measurements results inside rock
mass, the works on construction of the r(z) function are mainly based on theoretical
considerations (Budryk, 1953) and model tests (Drzęźla, 1979; Kołodziejczyk et al.,
2000; Krzysztoń, 1962). Some attempts to verify the obtained results were being made
on the random observation material (Kowalski 1984). Table 1 presents the selected forms
of the function of the range of main influences r(z) given by different authors.
From the analysis of considerations made by different authors one may draw the
following conclusions:
• Lack of the appropriate measuring material in rock mass for the need of full
verification of the existing solutions,
• There is an ambiguity in description of the course of variability in radius of the
main influences range in rock mass,
• Based on the volume conservation law we obtain the n parameter values that are
much higher than unity (Budryk),
• As a result of the model tests the n parameter values lower than unity were
obtained (Krzysztoń, Drzęźla), which was confirmed by the analysis of the real
limited measuring material (Kowalski),
• The value of radius of the influences range on the level of the bed being exploited
results from having adopted the specific form of r(z) function and not from theoretical considerations.
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TABLE 1
Views on the form of function of the range of main influences in rock mass
Author
Budryk, 1953
Knothe, 1984
Gromysz, 1977
Function
æzæ
r (z) = r (H ) ç ç
èH è
Parameter
n = 2p tg b » 5
n
n=1
n = 0.61
Drzęźla, 1979
æ z + z0 æ
çç
r (z) = r (H ) çç
è H + z0 è
Kowalski, 1984
æzæ
tgb (z, H ) = cH m ç ç
èH è
Kot 1981
æzæ
r (z) = k ç ç
èH è
n
n Î 0.405 , 0.735
n = 0.665
Hm
z0 =
m -1
1- n
n
tgβH = cHμ , 1 – n = 0.34; 0.52; 0.45
ìk = 548
í
îk = 407
n = 0.405
n = 0.344
The attempts to change the form of the function of radius of the range of influence
in rock mass, which would take the geomechanical properties of rocks into consideration, are being made. In his work, Paleczek (2005) introduces the relationship between
the radius of the range of influences in rock mass and the rock firmness coefficient, but,
unfortunately, the author has made some formal mistakes. He assumed that the transitivity
postulate must be satisfied for the function of radius of the influences range, while such
a formal requirement refers to the form of the influences function. Such an incorrect
assumption negates almost all of the solutions in table 1. In the function of radius of the
range in rock mass proposed by him, the author introduces the exploitation coefficient,
which, after all, is an independent parameter of the prediction theory, and therefore the
two theory parameters become dependent on each other.
The issue related to the form of the function of propagation of influences within rock
mass was dealt with by Janusz in his publication (2003). As a result of the considerations performed he suggests that variability in convexity of the function of curve of the
influences range in rock mass should be introduced. The author proposes the introduction of two zones in rock mass where the deformation process would run differently.
According to this author, it is possible that radius of the range in rock mass decreases
with the growing distance from the exploitation area, which seems to be rather strange
idea that has no confirmation in the reality.
In his considerations, Jędrzejec (1986) found the independence between the transitivity of depressions in Knothe-Budryk theory and the form of the function of radius
of the range of influences. The investigations of transitivity of the theory were carried
out by Bydłosz (1997), however he made a disastrous logical error in his considerations
that disavowed these considerations. Therefore, the author of this paper thinks that it is
necessary to check the permissible form of the function of variability in radius of the
138
main influences range inside rock mass. Based on the transitivity postulate, the detailed
analysis of this issue with reference to the vertical displacements of Knothe theory will
be carried out below.
3. Transitivity postulate
The transitivity postulate is one of the postulates of the mathematical model of
medium. This model was given by Litwiniszyn (1962) who determined what conditions
the prediction theory of the mining influences should comply with.
By limiting to the vertical displacements, the transitivity postulate in the operational
notation can be presented as follows:
w (III) = F[w (I) ; I , III] = F{F[w (I) ; I , II]; II , III}
(1)
where:
I, II, III — designations of levels in rock mass,
w(i) — vertical displacement on the i th calculation level,
F[.] — operator (the transition function),
The transitivity postulate can be characterised as follows:
The result of calculations of transitions at the preset level does not depend on
whether these calculations are carried out directly for this level or displacements on the
intermediate calculation levels are determined during the calculations.
For schematic ideas of the transitivity postulate see figure 1.
Level III
W(III)
Level II
W(II)
Level I
W(I)
Fig. 1. Diagram of execution of transitivity postulate
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4. Form of function r(z) for knothe theory
Let us carry out the analysis concerning the form of the function of the range of
main influences provided that the transitivity postulate is satisfied. One can put a questions whether the function of influences based on the Gaussian curve always satisfies
the transitivity postulate or not. To simplify, the analysis will be carried out for a flat
state of displacements with the assumption of having exploited the elementary volume
out. Let us consider the situation as in figure 2.
z=H
x
z = zII
rIII
rII
P
WP
x
WII
z = zI
rI
x
WI
z=0 x=0
x
Fig. 2. Diagram of exploitation distribution and calculation levels
The elementary mine was assumed in the origin of coordinates. On the specified
levels in rock mass I and II the subsidence occur. Let us choose any point P on the second
calculation level, which is within the range of influences of elementary mine.
Let us make the following assumptions and designations:
– vertical displacements on level I and II, respectively: wI, wII
– vertical displacements of the calculation point P(xp, zII):
• at transition from level z = 0 to level z = zII / wp2 /
• at transition from level z = zI to level z = zII / wp3 /
– radius of the main influences range: rI = r(z = zI – 0), rII = r(z = zII – 0),
rIII = r(z = zII – zI)
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– exploitation coefficient – a
– exploitation thickness – g
In accordance with the Knothe theory, for elementary mine in a flat state of deformation we can note:
w(x) =
æ
a×g
x2 æ
exp ç- p 2 ç dx
ç
r
r çè
è
(2)
if we assume that a . g . dx = 1, than for level I we have:
wI (x) =
æ
1
x2 æ
exp çç- p 2 çç
rI
rI è
è
(3)
wII (x) =
æ
1
x2 æ
exp çç- p 2 çç
rII
rII è
è
(4)
the same for level II
vertical displacement of point P at the transition from level z = 0 to level z = zII will be
described as follows:
æ
x 2p æ
1
ç
wp2 =
exp - p 2 ç
ç
rII
rII çè
è
(5)
The calculation of vertical displacement of point P at transition of calculations
through level I causes that the subsidence occurred on this level is assumed as the reason for the occurrence of vertical displacements on level II. Therefore, to calculate the
displacements one should use the integral formula with the assumption that individual
depressions are thicknesses of individual elements
æ
(x p - x)2 æç
wI (x)
ç
wp3 =
exp - p
dx
ç
r
rIII2 çè
- ¥ III
è
+¥
(6)
In order to calculate the integral in the above formula we will use the following
mean-value theorem for definite Riemann integral (Leja 1971):
b
b
f (x) × h (x) dx = f (x ) h (x) dx
a
a
(7)
141
This theorem is true when the function f (x) is continuous and the function h(x) has
got constant sign within the interval <a,b>.
Using the above-mentioned theorem, we obtain:
æ
(x p - x )2 æç +¥
1
ç
wp3 =
exp - p
wI (x)dx
ç
ç
rIII
rIII2
è
è -¥
(8)
At the same time we know that
+¥
æ
1
x2 æ
exp çç- p 2 çç dx = 1
r
rI è
è
-¥ I
(9)
æ
(x p - x )2 æç
1
ç
wp3 =
exp - p
ç
ç
rIII
rIII2
è
è
(10)
that is
in order for the transitivity postulate to be satisfied the following equality must be
true:
wp 2 = wp 3
(11)
Using the formulas (5) and (10) with reference to (11) we obtain:
æ
æ
x 2p æ
(x p - x )2 æç
1
1
ç
ç
ç
exp - p 2 =
exp - p
ç
ç
ç
rII
rII çè rIII
rIII2
è
è
è
(12)
let us arrange the equation (12)
æ x 2p
(x p - x )2 æç
rIII
= exp çp 2 - p
ç rII
ç
rII
rIII2
è
è
(13)
upon taking the logarithm of and converting equation (13) we obtain:
x 2 - 2 xp x +
rIII2
r
r2 - r2
ln III - xp2 III 2 II = 0
rII
p
rII
(14)
Equation (14) is the quadratic equation relative to variable ξ . In order for the equation to have at least one solution its discriminate should be higher than or equal to 0,
that is:
142
ér 2 æ r æ
r2 - r2 ù
D = 4 xp2 - 4 ê III ln çç III çç - xp2 III 2 II ú ³ 0
rII úû
êë p
è rII è
(15)
let us arrange the inequity (15):
xp2
rII2
-
1 æ rIII æ
ln ç ç ³ 0
p çè rII çè
(16)
Inequity (16) is satisfied for each xp when:
ær æ
ln çç III çç £ 0
è rII è
(17)
and thus
ær
ln çç III
è rII
æ
çç £ ln 1
è
(18)
and then
r (z II ) ³ r (z II - z I )
(19)
z II ³ z II - z I
(20)
we also know that
The following conclusion results from inequities (19) and (20):
The transitivity postulate in Knothe theory is satisfied when the function describing
the change of the main influences range radius in rock mass is monotonically increasing.
4. Summary
In spite of the fact that structure of the real rock mass differs from the idealised
theoretical medium, the applied theory of forecasting of deformations should satisfy
postulates of the mathematical model of medium. In the literature, there occur “strange”
models or modifications of Knothe theory about which one can have serious formal
reservations.
This paper presents the considerations concerning the form of the variability function
of main influences radius range in rock mass for Knothe theory. Only the non-decreasing
143
function satisfies the transitivity postulate. This fact to an essential extent reduces the
number of formally acceptable solutions.
At present, the mining carried out in the protecting pillars of the shafts being liquidated allows the verification of the form of the variability function of main influences
radius range in rock mass in situ based on the geodesic observations. The results of
measurements taken comprehensively in mines in which mining is being carried out
would allow many interesting observations of the course of the deformation process to
be obtained.
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Received: 28 May 2008