Archives of Mining Sciences 50, Issue 4 (2005) 401–415

Archives of Mining Sciences 50, Issue 4 (2005) 401–415

Archives of Mining Sciences 50, Issue 4 (2005) 401–415
401
STANISŁAW KNOTHE*
ASYMMETRIC FUNCTION OF DISTRIBUTION OF MINING EXPLOATATION INFLUENCES
IN THE MEDIUM WITH CHANGING PROPERTIES
ASYMETRYCZNA FUNKCJA ROZKŁADU WPŁYWÓW EKSPLOATACJI GÓRNICZEJ
W OŚRODKU ZMIENIAJĄCYM SWOJE WŁASNOŚCI
The model elaborated in the fifties (Knothe, 1951, 1953, 1957) for prediction of rock mass (surface)
displacements, based on the normal distribution of mining exploitation influences [1], allows to achieve
the theoretical profile of a subsidence trough with a satisfactory accuracy for practice [2]. This profile as
well as the distributions of displacements and deformations of a terrain, determined at the assumption (4),
reveal the symmetry in relation to the origin of coordinate system or to the ordinate axis (Fig. 1 and 2a).
The real profile of the trough is usually asymmetric (Fig. 2b). The approach of the predicted (theoretical)
profile of the trough to the real one can be obtained by introduction of the asymmetric curve of mining
exploitation influences to the mathematical model (Fig. 3) taking into consideration the changes of rock
mass properties caused by mining. For construction of the asymmetric curve two functions of normal
distribution f1(x) and f2(x) were assumed. These functions have different parameters of the influences
dispersion r1 and r2 in outer and inner part of the subsidence trough (Fig. 5). The function (13) describing
asymmetric profile of subsidence trough gives the better compatibility of the theoretical subsidence trough
profile (Fig. 6) with the real one and allows for prediction of more accurate values of displacements and
deformations.
Keywords: mining damages, displacements and deformations of the surface
Opracowany przed ponad pięćdziesięciu laty model (Knothe, 1951, 1953c, 1953d, 1984) prognozowania przemieszczeń górotworu (powierzchni) oparty na normalnym rozkładzie wpływów eksploatacji
górniczej (1) pozwala na ogół z wystarczającą dla praktyki dokładnością uzyskiwać teoretyczny profil
niecki osiadania (2). Profil ten, jak i rozkłady przemieszczeń i odkształceń terenu wyznaczone przy
uwzględnieniu założenia (4), wykazują symetrię względem początku układu współrzędnych lub względem
osi rzędnych (rys. 1 i 2a). Rzeczywisty profil niecki zwykle jest asymetryczny (rys. 2b). Zbliżenie prognozowanego (teoretycznego) profilu niecki do profilu rzeczywistego może być uzyskane przez wprowadzenie
do matematycznego modelu prognozy wpływów w miejsce symetrycznej krzywej wpływów, opisywanej
*
INSTYTUT MECHANIKI GÓROTWORU PAN, UL. REYMONTA 27, 30-059 KRAKÓW, POLAND
402
krzywą Gaussa, asymetrycznej krzywej rozkładu wpływów eksploatacji (rys. 3), uwzględniającej zmiany
właściwości górotworu pod jej wpływem. Za punkt wyjścia dla konstrukcji tej krzywej przyjęto dwie
funkcje rozkładu normalnego f1(x) i f2(x) o różnych parametrach rozproszenia wpływów r1 i r2 w części
zewnętrznej i w części wewnętrznej niecki (rys. 5). Po pomnożeniu wartości rzędnych tych funkcji przez
–
–
odpowiednie wartości współczynników a1 i a2 (11) otrzymuje się funkcję f1(x) i f2(x) (12). Otrzymana
krzywa wpływów f(x), złożona z dwóch gałęzi opisanych równaniami (12) jest funkcją asymetryczną, ciągłą
i różniczkowalną klasy II, opisującą losowy rozkład wpływów eksploatacji, pozwalającą uzyskać profil
niecki obniżeniowej w jej części zewnętrznej w1(x) oraz w części wewnętrznej w2(x). Po uwzględnieniu
przesunięcia punktu przegięcia profilu niecki w kierunku pola eksploatacji o wielkość p asymetryczny
profil niecki obniżeniowej opisują równania (13). Równania te pozwalają na uzyskanie lepszej zgodności teoretycznego profilu niecki osiadania z rzeczywistym (rys. 6) oraz dokładniejszych spodziewanych
wartości przemieszczeń i odkształceń terenu.
W szczególnym przypadku, gdy wartości parametrów rozproszenia wpływów są sobie równe
(r1 = r2) wartości współczynników a1 i a2 są sobie równe (a1 = a2 = 1) i asymetryczna krzywa wpływów
przechodzi w symetryczną krzywą rozkładu Gaussa.
Słowa kluczowe: szkody górnicze, przemieszczenia i deformacje powierzchni
Introduction
Geometric-integral models of the strata’s displacements caussed by underground
mining operations base on the so called influences functions (curves) describing their
distribution in the rocks medium induced by extraction of an elementary volume of the
deposit. In 1951 S.Knothe in his doctor’s thesis, devoted to the influence of underground
exploitation upon the surface accepted, on the basis of the field observations, that his
distribution is a normal one(Gauss). Considering vertical displacements of the strata in
a vertical plane perpendicular to a considerably long, straight line exploitation front he
assumed that a Gauss function of normal distribution of influences at the horizon of the
seam being exploited corresponds to each point of the strata with a co-ordinate of x = s
(origin of co-ordinate system x = 0 at the edge of exploitation) located at the height
z = H above the seam.
f (x, s) =
é- p (x - s)2 ù
1
exp ê
ú
r
r2
êë
úû
(1)
where:
r=
H
— parameter of influences dispersion, so called radius of main influences,
tg b
β — angle of main influences range,
H — depth of extraction.
With this assumption, identifying co-ordinate s with co-ordinate x and introducing
changeable of integration λ, the profile of the border part of subsidence trough is described by equation:
403
w(x) = wmax
-x
é - pl 2 ù
1
exp ê 2 ú dl
r
ë r û
-¥
ò
(2)
where:
wmax — is absolute value of maximum possible lowering in the bottom in a determined complete subsidence trough.
Inclinations T(x) and curvatures K(x) of the trough’s profile are determined by derivadw(x)
d 2 w(x)
tives of the equation (2)
and
.
dx
dx 2
In the already mentioned doctor’s thesis besides the assumption (1) the assumption
dw(t)
to the difference between the
of proportionality of velocity of point subsidence
dt
value of a final subsidence wk(t), corresponding to the exploitation situation at the t
moment, and the value of the point subsidence w(t) to which the point was subjected
at the t moment
dw(t)
= c[wk (t) - w(t)]
dt
(3)
The method of description and prediction of exploitation influence elaborated on the
basis of assumptions (1) and (3) allows for calculations of final vertical displacements
of the strata after termination of extraction as well as transient displacements during the
course of exploitation. The method was published and presented at Congresses in the
fifties (Knothe, 1953a, 1953b, 1953c, 1953d, 1957, 1959) and developed in later years
(Knothe, 1984).
The method of prediction of exploitation influences was supplemented by W. Budryk
(1953a, 1953b) suggesting determination of horizontal displacements u, on the basis of
Awierszyn’s assumption, their proportionality to T inclinations
u (x) = - B
dw(x)
dx
(4)
and determining the value B for the conditions of the Upper Silesian Coal Basin
B=
r
@ 0.4r
2p
(5)
du ( x)
.
dx
The profile of a boarder part of a determined subsidence trough obtained with the
assumptions of (1) and (4) is symmetrical in relation to the inflexion point located above
which also eanbles to determine the horizontal displacements e (x) =
404
the edge of exploitation. Symmetries respective to it indicate the curves of distributions
for inclinations T(x), curvatures K(x), horizontal displacements u(x) and horizontal deformations ε(x), as shown in fig.1.
e
0.4 r
wmax
e ma x = 0.6 r
0.4 r
wmax
e max = – 0.6 r
x
u
umax = – 0.4 wmax
x
K
Kmax = – 1.52
Kmax = 1.52
wmax
r2
wmax
x
r2
0.4 r
0.4 r
T
wmax
Tmax = r
x
z
r
wmax
r
x)
w(
H
b
b
x
Fig.1. Theoretical distribution of vertical displacements w(x) and horizontal ones u(x),
inclinations T(x), curvatures K(x) and horizontal deformations ε(x) in a subsidence trough
in case of a symmetric curve of normal distribution of exploitation influences
Rys.1. Teoretyczne rozkłady przemieszczeń pionowych w(x) i poziomych u(x), nachyleń T(x),
krzywizn K(x) oraz odkształceń poziomych ε(x) w niecce obniżeniowej w przypadku
symetrycznej krzywej rozkładu normalnego wpływów eksploatacji
405
The profiles of troughs determined by field measurements (fig. 2b) and recorresponding displacements and deformations distributions are to a lesser or greater extent different from a theoretical profile of a trough created in such a way (fig. 2a).
The attention to those differences has been paid since long time, among others by
A. Jung (1960) in his lecture in 1959 during the XI Mining–Metallurgy Day in Freiberg,
in which, on the basis of field measurements done in Saar Basin was stated, an asymmetry
of a subsidence trough is determined (differences in the influence range r1 towards rockmass and r2 towards exploitation area), displacement of inflexion point of the trough’s
profile towards gobs, value of lowering in the inflexion point of the trough greater than
half of maximum lowering (0.5wmax). Besides the said observations as a rule greater
absolute values of curvatures and horizontal strains have been stated in the internal part
of the trough (above exploitation) than in the external part (above the rockmass).
a)
r
r
0
s
x
0'
wmax
w (x)
H
b
b)
b
r2
0
x
r1
s
x
0'
wmax
H
p
b2
b1
Fig. 2. Theoretical (2a) and real (2b) profile of a subsidence trough
Rys. 2. Teoretyczny (2a) i rzeczywisty (2b) profil niecki obniżeniowej
x
406
The occurrence of the mentioned differences is obvious. The properties of the strata
are to a lesser or greater extent different from the medium properties accepted in the
mathematical model, in which a loose medium, horizontally homogenous, not changing
its properties under the influence of exploitation was assumed and in which the distribution of influences is normal. Sand or the strata made of loose, not large rock elements
can, for instance, be such a mdium.
A. Jung saw the possibility of improvement in accuracy of a theoretical forecast of
exploitation influence in the application of two different curves of normal distribution of
influence for external and internal part of a subsidence trough with parameters of r1 and r2.
In case of a normal distribution that led to discontinuity of derivatives of the trough’s
profile and thus to various values of inclination, curvatures, horizontal displacements
and horizontal deformations in the point of its profile’s inflexion. He also took the
account of a possibility of introduction of additional functions correcting the displacements distribution in the profile of the trough. The proposals were not, anyhow, within
the framework of consequently built up mathematical model of displacements based
on properties of the medium undergoing random processes. They were just directed
towards selection of functions (two or more) which were to describe most accurately
the measurements results.
Irrespective of a number of comments – regarding some differences between the
real and expected influences of exploitation determined by a forecast method, assuming
normal distribution of influences – it found, both in Poland and abroad, a common application because of its merits. The method is consequently built up, is transparent, easy
to be applied and enables to forecast exploitation influences with a satisfactory accuracy.
The method allows, what is also not without a meaning, for an accurate description of
displacements at least in the case of one real medium (sand).
Asymmetry of the curve of influences
Differences occurring between forecasted profiles of a subsidence trough obtained
with the assumption of normal distribution of influences and real profiles determined
by levelling surveys can be removed or reduced considerably at least by acceptance of
the asymmetric curves of influences. The curve, taking the account of advantages as
provided by now by the assumptions of normal distribution of exploitation influences,
should refer to curves of normal distribution.
Subsidence w(s) of any point A(s) with co-ordinate s, to which asymmetric curve of
influences f(x – s) is appropriated, can be determined with the use of influences function
f(x) corresponding to the point A(s = 0) located beyond the edge of exploitation (fig. 3).
w(s) = wmax
s
ò
-¥
f (x - s) dx = wmax
-s
ò f (x)dx
-¥
(6)
407
-s
s
A (s)
A (s= 0)
0
w(s)
ó
= ô f (x – s)dx
õ
-¥
-s
ó
= ôf (x)dx
õ
-¥
f (x – s)
f (x)
x
Fig. 3. Determination of subsidence of any point A(s) with the use of curve of influence
corresponding to point A(s = 0)
Rys. 3. Wyznaczanie obniżenia dowolnego punktu A(s) przy pomocy krzywej wpływu
odpowiadającej punktowi A(s = 0)
The equality of values of both determined integrals is visible in the figure 3. After
identification of coordinate s with coordinate x and introduction of integration changeable λ one obtains a profile of a subsidence trough.
w(x) = wmax
-x
ò f (l)dl
(7)
-¥
Dependence between function w(x) describing a profile of a subsidence trough and
dw(x)
the function of influence f(x), which is symmetric to function
in relation to the
dx
axis of ordinates (fig. 4)
f (x) =
dw(- x)
dx
(8)
enables to determine a profile of a subsidence trough with the knowledge of the function
of influence or determine functions of influences on the basis of respective documentation of trough originated in the terrain.
Asymmetric curve of influences f(x), enabling to bring closer a theoretically determined profile of a complete subsidence trough to a real profile with maintenance of
advantages as provided by now by assumption of normal distribution of exploitation
influences should meet the following conditions:
– should be a curve constructed with the greatest possible utilization of normal
distribution,
– the range of exploitation influence towards the rockmass should be, taking the
account of geological conditions, greater towards the goaf and thus the parameters
408
r2
r1
0
s
(x)
x)
w(
wmax
H
f (x) = T(–x)
dw (x )
T(x) = dx
x
Fig. 4. Profile of the trough w(x) and influences function f (x) = T(–x) corresponding
to the point located above the edge line of exploitation
Rys. 4. Profil niecki w(x) i odpowiadająca punktowi powierzchni położonemu
ponad granicą eksploatacji funkcja wpływów f (x) = T(–x)
of dispersion of influences above the rockmass r1 and the gobs r2 should meet the
condition r1 > r2;
– lowering in the trough’s profile inflexion point should be greater than half of the
maximum subsidence (0.5wmax);
– function of influence should be continuous and differentiable, and its integral in
the range of –∞, +∞ equals 1;
– point of the trough’s profile inflexion at which maximum inclination occurs should
be displaced from edge line of exploitation towards gobs.
Asymmetrisation of the curve of normal distribution of exploitation influences is possible provided the changes of properties of the medium during extraction are taken into
account. The course of these changes depends on a number of factors and most of all on
location in reletion to the advancing exploitation front and the method of exploitation. It
is possible to consider in a theoretical model of the strata’s displacements with far going
simplifications. The assumption that a change of the medium’s properties takes place
above the exploitation front can be such an initial simplification. With this assumption
the medium is divided, by a vertical plane perpendicular to axis x and running across
the edge of exploitation, into two areas above the rockmass and above the exploitation
area. The values of dispersion parameters of exploitation influences in these areas are
different. Usually the range of main influences r1 above the rockmass is greater than the
range of r2 above the exploitation area.
The above mentioned assumptions are the first steps for accomplishment of asymmetric function of normal distribution. Functions of normal distribution (fig. 5) are the
initials of this structure:
409
s
x
f (x)
_
f1 (x)
P2
_
P2 = a2 P2
_
f2 (x)
P1
_
P1 = a1 P1
f1 (x)
r1
f2 (x)
x
r2
Fig. 5. Diagram of a structure of asymmetric curve of influences of normal distribution f (x)
Rys. 5. Schemat konstrukcji asymetrycznej krzywej wpływów rozkładu normalnego f (x)
æ p x2 æ
1
exp çç- 2 çç
r1
è r1 è
for
æ p x2 æ
1
f 2 (x) = exp çç- 2 çç
r2
è r2 è
for
f1 (x) =
x£0
(9)
x³0
where the dispersion parameters of influence r1 and r2 correspond to the primary properties of the strata above the rockmass and its altered properties above the exploitation
areas under the influence of exploitation. The surface areas of fields P1 and P2 limited
by these functions and by the x-axis are equal
0
P1 = P2 =
-¥
f1(x) dx =
¥
f2(x) dx = 0.5
(10)
0
and their sum is equal to 1
P1 + P2 = 1
The application of discontinuous curve of influence the branches of which are described by functions f1(x) and f2 (x), for the analysis of strata displacements is certainly
unacceptable at least from such a reason that strata properties change suddenly (jump
410
like) above the edge of exploitation. The above disadvantage can be removed provided
–
–
the functions f1(x) and f2(x) are replaced by the functions f 1 (x) and f 2 (x) obtained by
their multiplication by respective coefficients a1 and a2. The values of those coefficients
should be such so that:
–
–
– maximum values of ordinates (at point x = 0) of functions f 1 (x) and f 2 (x) are
equal,
and
—
—
– the sum of areas of fields P1 and P2 limited by these functions is equal to the sum
of areas of fields P1 and P2 limited by the functions f1(x) and f2(x).
Thus the conditions must be met:
a1
1
1
= a2
r1
r2
and
0.5a1 + 0.5a2 = 1
The conditions are met by the values of coefficients:
a1 =
2
r
1+ 2
r1
a2 =
and
2
1+
(11)
r1
r2
The curve of influences f (x) obtained this way comprised of two branches described
by equations
æ -p x 2 æ
1
f1 (x) = a1 f 1 (x) = a 1 exp çç 2 çç
r1
è r1 è
for
x£0
and
(12)
f 2 (x) = a 2 f 2 (x) = a2
æ -p x
1
exp çç 2
r2
è r2
2
æ
ç
ç
è
for
x³0
is an asymmetric, continuous and differentiable function (class II) describing a random
distribution of exploitation influences in the medium the properties of which are changing under influence of mining excavation.
The equations (12) conformant with the formula (7) enable to obtain a profile of
a subsidence trough w(x) in its external part w1(x) for x ≥ 0 and its internal part w2(x)
for x ≤ 0 with the assumption that the point of the profile inflection is located above
the edge line of exploitation in the origin of coordinates system. Taking the account of
displacement of the profile inflection point, and thus the whole subsidence trough by
411
the value of p towards exploitation area one receives the profile of a subsidence trough
w(x + p) described by equations:
-x
é- pl2 ù
1
exp ê 2 ú dl
r
ë r1 û
-¥ 1
w1 (x + p) = wmax a1
for
x ³ –p
and
(13)
w2 (x + p ) = 0.5wmax a1 + wmax a 2
é - pl2 ù
1
exp ê 2 ú d l
r
ë r2 û
-p 2
x
for
x £ –p
The equations (13) enable to determine:
– inclinations distribution
æ -p x 2 æ
dw( x + p )
1
= wmax a1 exp çç 2 çç
dx
r1
è r1 è
for
æ -p x 2 æ
dw( x + p )
1
T2 (x + p) =
= wmax a 2 exp çç 2 çç
dx
r2
è r2 è
for
T1 (x + p) =
x ³ –p
(14)
x £ –p
– curvature distribution
æ -p x 2 æ
2p
d 2 w( x + p )
ç 2 ç
w
a
x
exp
=
max 1 3
ç r ç
dx 2
r1
è 1 è
for
æ -p x 2 æ
2p
d w( x + p )
ç 2 ç
w
a
x
exp
=
K2 (x + p ) =
max 2 3
ç r ç
dx 2
r2
è 2 è
for
K1 (x + p ) =
x ³ –p
(15)
2
x £ –p
– horizontal displacements distribution
u1 ( x + p ) = - BT1 ( x + p )
for
x ³ –p
u 2 ( x + p ) = - BT2 ( x + p )
for
x £ –p
e 1 ( x + p ) = - BK1 ( x + p)
for
x ³ –p
e 2 ( x + p ) = - BK 2 ( x + p)
for
x £ –p
(16)
– and horizontal deformations
(17)
412
Maximum values of those magnitudes and their location in relation to the edge line
of exploitation are illustrated in the form of a set in table 1 and shown in figure 6.
e
e2max = – B1.52 a2
0.4r+p
wmax
r22
e1max = B 1.52 a1
0.4r-p
wmax
r12
x
p
u
1
umax = - B a1 r wmax
1
x
p
K
p
K2max = 1.52 a2
wmax
r22
0.4r+p
K1max = - 1.52 a1
wmax
r12
0.4r-p
T
T max = a1
wmax
wmax
r1
x
p
z
r2
x
r1
)
w (x
H
b2
b1
x
Fig. 6. Theoretical distribution of vertical displacements w(x) and horizontal ones u(x), inclinations
of terrain T(x), curvatures K(x) and horizontal deformations ε (x) in a subsidence trough in case
of an asymmetric curve of normal distribution of exploitation influences
Rys. 6. Teoretyczne rozkłady przemieszczeń pionowych w(x) i poziomych u(x), nachyleń terenu T(x),
krzywizn K(x) oraz odkształceń poziomych ε (x) w niecce obniżeniowej w przypadku asymetrycznej
krzywej rozkładu normalnego wpływów eksploatacji
413
The suggested asymmetric function of the function of exploitation influences distribution composed of the two functions (12) of different parameters of influences dispersion
allows to obtain an asymmetric profile of a subsidence trough (13). The profile meets
all the five above stated conditions and reflects, to a much greater extent, the reality
than the profile determined on the basis of a symmetric function of normal distribution
of influences.
Introduction of asymmetric curve of influences, the asymmetry of which is dependent
on changes of the medium under the influence of exploitation, is a further, consequent
step in adjustment of the mathematical model of forecasting the exploitation influences
as comperd to reality.
Final comments
The utilization of asymmetric curve of influence in the mathematical model of
forecasting the exploitation influences is justified, most of all in the cases, where the
problem can be limited to a two-dimensional state, in which the exploitation influences
are analysed in vertical cross-sections, perpendicular to straight-line exploitation border
of respectively long sections or to advancing exploitation front of a greater length. The
model can also be used, with some limitations for forecasting or describing of influences
of exploitation areas with the outline close in shape to a rectangle. In case of exploitation
carried out in irregular, not large, and distributed workings it is more advisable to use
the area of influences of normal distribution with symmetric cross-sections.
The application of asymmetric curve of influences allows for approximation of the
results of a theoretical forecast to reality. The properties of the medium accepted in the
mathematical model and properties of the real medium (the strata) will, anyhow, stay
different. Thus the results of forecasts must still be different than the reality but to a
lesser extent than in the cases of application of a symmetric curve of normal distribution, what is the case at present.
In equations regarding maximum values of horizontal displacements and horizontal
deformations, as placed in table 1, the value of the B coefficient, determined by equation (5) was not introduced. This value raises the greatest number of reservations, show
considerable variations and is, as a rule, smaller than stated and different in various
coalfields.
In cases where the values of the main influences range r and angles of the range of
these influences β are determined in relation to the edge of exploitation one should take
the account of the value of displacement of inflexion point of the trough’s profile p.
414
TABLE 1
Values and location of maximum magnitudes of displacements
and deformations of the terrain
TABLICA 1
Wartości i położenie maksymalnych wielkości przemieszczeń i deformacji terenu
Value
(indicator)
Value and location in relation to theedge of exploitation
In external part of the trough
a
a1
wmax = 2 wmax
r2
r1
T(x)max
inclination
K(x)max
curvature
(x = – p)
a1wmax
( x1 =
2p
e
r1
2p
u(x)max
horizontal
displacement
ε(x)max
horizontal
deformation
In internal part of the trough
w
1
= -1.52 a1 max
r12
r12
a2 wmax
- p = 0.4 r1 - p )
( x2 = -
- Ba 1
2p
e
w
1
= 1.52a2 max
r22
r22
r2
- p = -0.4 r2 - p)
2p
1
1
w = - Ba2 wmax
r1 max
r2
( x = - p)
Ba1wmax
( x1 =
2p
e
r1
2p
w
1
= 1.52 Ba1 max
2
r1
r12
- p = 0.4 r1 - p )
Ba 2 wmax
( x2 = -
2p
e
w
1
= - 1.52 Ba2 max
r22
r22
r2
- p = - 0.4 r2 - p )
2p
Remark: value of B co-efficient according to W. Budryk (1953) does not exceed
B = r » 0.4r
2p
REFERENCES
B u d r y k , W., 1953a. The determination of values of horizontal deformations of the ground surface. Bull. Acad. Pol.
Sc. Cl. IV, vol. 1, is. 1-2, p. 50-51.
B u d r y k , W., 1953b. Wyznaczanie wielkości poziomych odkształceń terenu (The determination of values of horizontal
deformations of the ground surface). Archiwum Górnictwa i Hutnictwa, vol. 1, is. 1 p. 63-74.
J u n g , A., 1960. Möglichkeiten der praktischen Anwendung der Vorausberechnungverfahren nach Knothe und Akimov.
Freiberger Forschungshefte A. 146, 195-206.
K n o t h e , S., 1951. Wpływ podziemnej eksploatacji na powierzchnię z punktu widzenia zabezpieczenia położonych
na niej obiektów (Influence of underground mining on surface with regard to safety). Ph.D. thesis, unpublished.
Library of AGH.
K n o t h e , S., 1953a. Abbaugeschwindigkeit und Bodendeformationen. Bergakademie 5, 12, 513-518.
415
K n o t h e , S., 1953b. The curvature of the profile of the subsidence basin. Bull. Acad. Pol. Sci., Cl. IV, vol. 1, is. 1-2,
p. 41-45.
K n o t h e , S., 1953c. Równanie profilu ostatecznie wykształconej niecki osiadania (A profile equation for a definitely
shaped subsidence through). Archiwum Górnictwa i Hutnictwa, vol. 1, is. 1, p. 22-38.
K n o t h e , S., 1953d. Wpływ czasu na kształtowanie się niecki osiadania (Influence of time on formation of subsidence
through). Archiwum Górnictwa i Hutnictwa, vol. 1, is. 1, p. 51-62.
K n o t h e , S., 1957. Observations of surface movements under influence of mining and their theoretical interpretation. Proocedings of the European Congress on Group Movement held at the University of Leads, April 1957, p. 210-218.
K n o t h e , S.,1959. Observations et interpretation theorique des mouvements de surface. Rev. de l’Ind. Minerale, vol. 41,
is. 1, p. 964-974.
K n o t h e , S., 1984. Prognozowanie wpływów eksploatacji górniczej (Forecasting of the influence of underground
mining.) Katowice, Śląsk, p. 160.
REVIEW BY: PROF. DR HAB. INŻ. ANDRZEJ Z. SMOLARSKI, KRAKÓW
Received: 08 October 2005