Archives of Mining Sciences 51, Issue 3 (2006) 453–470

Archives of Mining Sciences 51, Issue 3 (2006) 453–470
453
ANDRZEJ NIEROBISZ*
THE MODEL OF DYNAMIC LOADING OF ROCKBOLTS
MODEL OBCIĄŻENIA UDAROWEGO KOTWI
The article uses the results of tests carried out in the framework of the Targeted Project of the State
Committee for Scientific Research (KBN) entitled “Rockbolting systems for roadway workings threatened
with tremors” (registr. No 6T122003C/06233)for verification of a rockbolt dynamic loading model. This
model assumes that a tremor or rockburst can be considered as a hit of conventional weight on a rockbolt
bar, which on one side is ended with a washer and nut, and on the other side is fixed at the hole bottom
(Fig. 1). Omitting the longitudinal vibrations of the rockbolt bar accompanying the hit and its mass we
can assume that the rockbolt bar supports a conventional rock block with mass m1, which is hit by mass
m2 > m1 with kinetic energy Ek. Assuming a plastic model of collision of masses m1 and m2, the kinetic and
potential energy, which the system of masses m1 and m2 possesses after plastic collision, was calculated.
The total impact energy Eu defined by the relation (5) was compared with the rockbolt work Wk, which
quantitatively is equal to the figure field under the curve determined by the function F(p), showing the
displacement of the rockbolt (relation 6) (Fig. 2). For the verification of theoretical assumptions of the
model, a physical model with mass m1 was created, consisting of mass m1 that was loading a rockbolt
which was grouted into a test cylinder (simulating the rock mass). On the mass m1 one has dropped from
a suitable height the impact mass m2, observing the rockbolt behaviour and measuring by means of apparatus the rockbolt loading and its displacement (Fig. 6, 7). The obtained results were collected in Table 1
and their statistical analysis was carried out according to the following procedures:
1. The absolute difference between the impact energy Eu and rockbolt work Wk for each test was
calculated. When the difference exceeded 5%, the result was rejected.
2. It has been checked if the number of results determined when using the method mentioned above
is sufficient.
3. With the aid of a test of difference rank signs the hypothesis was checked assuming that the
theoretically calculated impact energy Eu is not essentially different from the measured rockbolt
work Wk.
As a result of the analysis the random sample of results which met the assumed significance criteria
was selected. Afterwards by help of the test of difference rank signs it has been indicated that there is no
reason to reject the hypothesis that the total impact energy Eu calculated by means of theoretical relations
is not essentially different from the rockbolt work Wk calculated from measurement results.
Keywords: safety, mining, rockbursts, roofbolting
*
GŁÓWNY INSTYTUT GÓRNICTWA, PL. GWARKÓW 1, 40-166 KATOWICE, POLAND
454
W artykule wykorzystano wyniki badań stanowiskowych przeprowadzonych w ramach Projektu
Celowego KBN pt. ”Systemy kotwienia dla wyrobisk korytarzowych zagrożonych wstrząsami” (nr rejestr. 6T122003C/06233) do zweryfikowania modelu obciążenia udarowego kotwi. Model ten zakłada,
że wstrząs albo tąpnięcie można traktować jako uderzenie umownego ciężaru na żerdź kotwi z jednej
strony zakończonej podkładką i nakrętką a z drugiej zamocowanej na dnie otworu (rys. 1). Pomijając
towarzyszące uderzeniu drgania podłużne żerdzi oraz jej masę można założyć, że żerdź kotwi podtrzymuje
pewien umowny blok skalny o masie m1 w który uderza masa m2 > m1 o energii kinetycznej Ek. Przyjmując model plastyczny zderzenia mas m1 i m2 obliczono energię kinetyczną i potencjalną jaką posiada
po zderzeniu plastycznym układ mas m1 i m2. Sumaryczną energię udaru Eu określoną zależnością (5),
porównano z pracą kotwi Wk, która ilościowo równa jest polu figury pod krzywą określoną funkcją F(p)
obrazującą przemieszczenie kotwi (zależność (6), rys. 2).
p2
Wk =
F( p)dp
p1
Dla sprawdzenia założeń teoretycznych modelu zbudowano model fizyczny składający się z masy
m1 obciążającej kotew wklejoną do walca badawczego (symulującego górotwór). Na masę m1 z odpowiedniej wysokości opuszczano masę udarową m2 obserwując zachowanie się kotwi oraz mierząc za
pomocą aparatury obciążenie kotwi i jej przemieszczenie (rys. 6, 7). Uzyskane wyniki zebrano w tabl. 1
i przeprowadzono ich analizę statystyczną według następujących procedur:
1. Obliczono różnicę bezwzględną energii udaru Eu i pracy kotwi Wk dla każdej próby. Jeżeli ta
różnica była większa niż 5% to takie wyniki odrzucano.
2. Sprawdzono czy liczba wyników określona powyższym sposobem jest wystarczająca.
3. Za pomocą testu rangowych znaków różnic sprawdzono hipotezę zakładającą, ze obliczona
teoretycznie energia udaru Eu nie różni się istotnie od zmierzonej pracy kotwi Wk.
W wyniku przeprowadzonej analizy została wyselekcjonowana próba losowa uzyskanych wyników
spełniająca założone kryteria istotności. Następnie za pomocą testu rangowych znaków różnic wykazano,
że nie ma podstaw do odrzucenia hipotezy, że sumaryczna energia udaru Eu obliczana za pomocą zależności
teoretycznych nie różni się istotnie od pracy kotwi Wk obliczanej z wyników pomiarów.
Przeprowadzone rozważania pozwoliły na przedstawienie następujących wniosków:
1. Aby kotew nie uległa zniszczeniu w trakcie udarowego jej obciążenia, całkowita energia udaru
Eu (zależność (5)), nie może być większa od pracy wykonanej przez kotew Wk (zależność (7)).
Powinna być spełniony warunek:
Eu ≤ Wk
2.
3.
W warunkach rzeczywistych (pod ziemią) zależności powyższe powinny się sprawdzać pod
warunkiem, że na skutek udaru nie nastąpi zniszczenie kotwi (zerwanie żerdzi, nakrętki, przeciągnięcie podkładki) lub jej wypchnięcie do wyrobiska
Na podstawie przeprowadzonych analiz wykazano, że nie ma podstaw do odrzucenia hipotezy
o równości energii udaru Eu i pracy kotwi obliczanej i mierzonej według zależności (5) i (7).
Mogą one być pomocne przy projektowaniu wzmocnienia wyrobisk zagrożonych tąpaniami za
pomocą kotwi, ponieważ znając z badań stanowiskowych charakterystykę dynamiczną pracy
kotwi można je zabudować w wyrobisku dobierając ich zagęszczenie w zależności od spodziewanej energii udaru.
Słowa kluczowe: bezpieczeństwo, górnictwo, tąpania, obudowa kotwowa
455
1. Introduction
Roofbolting is conventionally applied in Polish copper ore mines. Expanding anchors, resin-grouted rockbolts and rope-bolts are used. When using annually more than
1.2 million pieces of bolts, 71% are expanding anchors and 29% resin-grouted bolts. In
practice all types of bolts work in conditions of dynamic loading, which are caused by
rock mass tremors with seismic energy up to 109 J (Butra et al., 2005). In most cases
after tremors no noticeable changes in roof and support were observed. However, there
occur cases when as a result of a tremor follows destruction of roofbolts (Report, 2003).
These problems took up among others: Siewierski (1975), Piechota (1998), Kidybiński
(1999), Nierobisz (2005).
In Polish hard coal mines the rockburst hazard occurs in 60% of mines. Long-standing
work caused that the number of rockbursts decreased (Konopko, 2005). These results
were obtained due to the development of assessment methods of sources and state of
hazard as well as application of active methods of rockburst prevention. Significant is
also the reduction of coal production. In rockburst prevention in many aspects were
engaged among others: H. Filcek, Z. Kłeczek, A. Zorychta (1984), A. Goszcz (1999),
J. Dubiński, W. Konopko (2000), S. Szweda (2001), J. Drzewiecki (2001).
Considering the cases of rockbursts that occurred within the last ten years we can
observe a constant tendency of increasing their effects in the form of support damage
in roadway workings, where the fundamental method to protect these workings against
the effects of rockbursts is the use of more stronger sections of frames, their concentration and reinforcement by means of crown runners supported by Valent or SV props.
Therefore in the framework of the Targeted Project of the State Committee for Scientific
Research (KBN) entitled “Rockbolting systems for roadway workings threatened with
tremors” (registr. No 6T122003C/06233) work was undertaken aiming at the application
of appropriate rockbolts, which in connection with standing support would constitute
better protection of roadway workings against the effects of rockbursts.
Basing on results of tests carried out using testing rigs realised in the framework
of the project mentioned above, an attempt was made to develop a model of dynamic
loading of rockbolts, which was verified using statistical inference methods. The results
of this work were presented below.
2. Analysis of dynamic resistance of rockbolts
A tremor or rockburst can be treated as a hit of a conventional weight on a rockbolt
bar, which on one side is ended with a washer and nut, and on the other side is fixed at
the hole bottom. Omitting the longitudinal vibrations of the rockbolt bar accompanying
the hit and its mass we can assume that the rockbolt bar supports a conventional rock
block with mass m1, which is hit by mass m2 > m1 with kinetic energy Ek (Fig. 1).
456
seismic tremor
m2
gap
m1
Ek
Ek
Ep
Fig. 1. The model of dynamic loading of rockbolts
Rys. 1. Model obciążenia udarowego kotwi
A plastic model of collision of masses m1 and m2 was adopted. The kinetic energy
which the system of masses m1 and m2 possesses after the plastic collision we can determine from the formula:
Ek =
(m1 + m 2) × Vp2
2
(1)
where:
Vp — common velocity of masses m1 and m2 after collision determined from the
relation:
m2
(2)
Vp =
× V1
m1 + m 2
V1 — mass m2 velocity which can be expressed by help of the relation:
V1 =
2×g × h
(3)
g — acceleration of gravity; m/s2,
h — height of mass m2 fall; m.
Substituting the formulae (2) and (3) for the relationship (1) we have obtained:
Ek =
m 22 × g × h
m1 + m 2
(4)
457
The total impact energy acting on the rockbolt is the sum of the kinetic energy Ek
and potential energy Ep:
E u = E k + Ep =
m 22 × g × h
+ (m1 + m 2 ) × g × p ; kJ
m1 + m 2
(5)
where:
p — rockbolt displacement (movement from the hole, bar elongation); m.
The result of acting of dynamic loads modelled by the impact of falling mass can be
the following effects:
• elastic deformation of rockbolt bar and return to the initial state,
• bar rupture, rupture of rockbolt connection with the rock mass, destruction of
washer, nut,
• plastic deformation of rockbolt.
In order to avoid the destructure of the rockbolt in consequence of a hit, the impact
energy Eu should be balanced with the rockbolt work Wk, which can be determined from
the relation:
p2
Wk =
(6)
F( p)dp
p1
where:
F(p) — function describing the rockbolt loading according to its displacement.
In terms of quantity the rockbolt work is equal to the figure field under the curve
determined by the function F(p) in the p1 = 0 to p2 = 440 mm bracket (Fig. 2). Because
the determination of the equation illustrating the displacement course of the rockbolt
in the function of its loading can be labour-intensive, we propose to divide the displacement course into elementary intervals and to calculate the rockbolt work from the
relation (Fig. 3):
p1
Wk =
p2
F1 (p) dp +
0
p3
F2 (p)dp +
p1
F3 ( p)dp + × × × +
p2
pn
Fn (p)dp for n = 1, 2, 3, ..., n
(7)
p n -1
The relations mentioned above should prove correct under the stipulation that in
consequence of impact the destruction of rockbolts will not follow (bar rupture, washer
protraction) or its pushing into the working.
458
160
140
Load F, kN
120
100
80
60
440
40
Rockbolt work Wk =
20
0
F( p)dp
0
0
50
100
150
200
250
300
350
400
450
500
Displacement of rockbolt bar p, mm
Fig. 2. Performance characteristic of DAP rockbolts during static load capacity tests
Rys. 2. Ilustracja pracy kotwi typu DAP w trakcie statycznych badań nośności
160
F1
140
Load F, kN
120
100
F2
F3
F4
F5
80
F6
60
40
20
0
0
50
p1
100
p2
150
p3
200
250
300
350
Displacement of rockbolt bar p, mm
400
p4
450
500
p5 p6
Fig. 3. Method of rockbolt work calculation
Rys. 3.Sposób obliczania pracy kotwi
3. Rockbolt work performance characteristic
As regards the notion of rockbolt performance characteristic, we understand graphic presentation of displacement of the rockbolt (rockbolt movement from the hole,
bar elongation) mounted in the rockmass (or its model) according to the exerted load.
459
Considering the method of rockbolt loading we can speak about a static or dynamic
rockbolt performance characteristic.
It should be mentioned that the form of the performance characteristic of rockbolts
mounted in the rockmass by help of resin-grouted cartridges depends on a number of
factors, such as:
1. Rockbolt construction:
a. material, of which the rockbolt bar is carried out,
b. method of rockbolt bar ribbing,
c. form of rockbolt end,
d. construction of washer and nut,
e. construction of bar thread,
2. parameters of binder that binds the rockbolt with the rock:
a. compressive strength, tensile strength,
b. binder rigidity,
c. binder viscosity,
d. gelation time,
3. technology of rockbolt hole performing:
a. diameter of rockbolt hole and its size in relation to bar diameter,
b. method of hole drilling,
c. cleanness of hole walls,
d. hole rectilinearity,
e. shape of hole walls,
f. hole length,
4. technology of rockbolt mounting:
a. time of resin-grouted cartridges mixing,
b. distribution accuracy of cartridge on the assumed hole length,
c. appropriate destruction of resin-grouted cartridges foils in the mixing phase,
d. thickness of binder layer between the bar and rockbolt hole wall,
e. length of rockbolt bar resin-grouting,
5. rock mass properties:
a. compressive strength,
b. lithological structure of rock mass,
c. rock mass cracks,
d. rock mass soaking ability.
The mentioned above list consisting of 24 parameters indicates that the rockbolt
work is extremely difficult for testing. Therefore for each type of rockbolt the performance characteristic will be different, dependent on factors mentioned above. By way of
example in order to illustrate the mentioned above problems were presented two static
characteristics of DAP rockbolts
(Fig. 4, 5), which were mounted in coal and sandstone (Nierobisz et al., 2005).
460
180
160
rupture of nut
140
rupture of connection grout – rock mass
120
Load, kN
rupture of connection grout – rock mass
100
80
60
40
20
0
0
20
40
60
80
100
120
140
160
180
Displacement, mm
Fig. 4. Performance characteristic of DAP bolts mounted in coal
Rys. 4. Charakterystyka pracy kotwi zabudowanych w węglu
200
measurement interrupted
180
160
Load, kN
140
120
measurement interrupted
100
measurement interrupted
80
60
40
20
0
0
20
40
60
80
100
120
140
Displacement, mm
Fig. 5. Performance characteristic of DAP rockbolts mounted in sandstone
Rys. 5. Charakterystyka pracy kotwi typu DAP zabudowanych w piaskowcu
160
461
4. Balance of impact energy and rockbolt work
The truthfulness of relations (5) and (7) was proved, creating a physical model consisting of mass m1 loading the rockbolt resin-grouted into the testing cylinder (rock mass
simulation). On mass m1 from an appropriate height the impact mass m2 was dropped;
at the same time the rockbolt behaviour was observed and its loading and displacement
measured (Fig. 6).
Impact mass m 2
Mass m 1 supported
by rockbolt
Test cylinder
Force sensor
Rockbolt bar
Rockbolt washer
Fig. 6. Scheme of testing rig for dynamic tests fixed in test cylinder
Rys. 6. Schemat stanowiska do badań dynamicznych kotwi zamocowanych w walcu badawczym
The rockbolt grouted into the testing cylinder carried out in conformity with the
standard (Polish Standard, 1999) was based on a force sensor and upper plate of the rig.
On the bottom part of the rockbolt, ended with a washer, was based a moveable bottom
plate with four rods topped with an upper plate, on which the mass m1 was based. Each
rockbolt was once loaded through a free fall of impact mass m2 on mass m1. The load
to the rockbolt was transmitted through rods leading to the moveable bottom plate, and
next to the rockbolt washer and nut (Fig. 7).
462
Impact mass m 2
Mass m 1 supported by rockbolt
Moveable upper plate
Leading rods
Test cylinder with resin-grouted rockbolt
Force sensor
Stable plate of testing rig
Rockbolt bar
Moveable bottom plate
Rockbolt washer with nut
Displacement sensor
Fig. 7. View of testing rig for dynamic tests prepared to carry out impact test of IR-4W rockbolts
Rys. 7. Widok stanowiska do badań dynamicznych kotwi przygotowanego
do wykonania próby udarowej kotwi typu IR-4W
The fundamental element of the measuring system used in these tests was a fourchannel measuring amplifier of DMCplus type of the HMB firm, connected with a tensometric force sensor, used to measure the effect of mass impact on the tested rockbolt,
and with a resisting displacement sensor, applied to measure the bockbolt movement
from the testing cylinder (Fig. 7).
The obtained testing results were presented in Table 1. The dynamic performance
characteristics of selected rockbolt types were presented in Figures 8÷10 (rockbolt movement from the testing cylinder with the minus sign was assumed).
463
TABLE 1
List of results of rockbolt testing carried out by help of falling mass impact
TABLICA 1
No.
Rock-bolt type
Impact mass
m2, kg
Height of mass
fall m2, h, m
Kinetic load
energy EK, kJ
Rockbolt
displacement
p, m
Potential
energy Ep, kJ
Impact energy
Eu, kJ
Max. dynamic
force Fd, kN
Rockbolt work
Wk , kJ
Zestawienie wyników badań kotwi przeprowadzonych za pomocą udaru spadającej masy
1
2
3
4
5
6
7
8
9
10
1
2
RS
5000
0.30 10.511 0.119
5000
0.20
4000
0.70 18.312 0.146
7.007
0.082
Remarks
Visual inspection of rockbolts
after testing
11
Rockbolt was not ruptured
8.172 18.683 175.1 17.085 – bar elongation by 119 mm
took place m1 = 2000 kg
5.631 12.638 163.4 11.505
Rockbolt was not ruptured
– bar elongation by 82 mm
took place m1 = 2000 kg
Rockbolt was not ruptured
3
8.594 26.909 207.2 25.432 – bar elongation by 146 mm
took place m1 = 2000 kg
4
4000
0.50 12.658 0.070
Rockbolt movement from
the test cylinder by 70 mm.
Rupture of rockbolt rope bar.
m1 = 2200 kg
4.258 16.916 255.6 11.430
Rockbolt performace
characteristic presented
in Fig. 8
4000
0.30
2.919 10.514 275.0 10.600
Rockbolt movement from
the test cylinder by 48 mm.
Rockbolt was not ruptured
m1 = 2200 kg
2.007 8.336 254.9 9.688
Rockbolt movement from
test cylinder by 33 mm.
Rockbolt was not ruptured
m1 = 2200 kg
AGl
5
6
4000
7
0.25
7.595
6.329
0.048
0.033
4000
0.25
6.329
0.190 11.556 17.885 137.8 18.770
8
4000
0.25
6.329
0.058
3.528 9.857 165.1 7.945
9
4000
0.30
7.595
0.088
5.352 12.947 174.8 13.350
AGp
Rockbolt movement from
test cylinder by 190 mm.
Rockbolt was not ruptured
m1 = 2200 kg.
Rockbolt performance
characteristic presented
in Fig. 9
Rockbolt movement from
test cylinder by 58 mm.
Rockbolt was not ruptured.
m1 = 2200 kg
Rockbolt movement from
test cylinder by 88 mm.
Rockbolt was not ruptured.
m1 = 2200 kg
464
1
2
10
3
4000
11 IR-4W 4000
12
4000
4
5
6
7
8
9
10
11
0.40 10.126 0.300 18.247 28.373 223.4 13.638
Total rockbolt movement
from test cylinder.
Cutting off connection
between rockbolt hole
wall and binder.
Rockbolt was not ruptured.
m1 = 2200 kg
0.30
0.500 30.411 38.006 49.8 11.450
Total rockbolt movement
from test cylinder.
Cutting off connection
between rockbolt hole
wall and binder.
Rockbolt was not ruptured.
m1 = 2200 kg.
Rockbolt performance
characteristic presented
in Fig. 10
0.280 17.030 22.093 106.5 5.870
Total rockbolt movement
from test cylinder.
Cutting off connection
between rockbolt hole
wall and binder.
Rockbolt was not ruptured.
m2 = 2200 kg
0.20
7.595
5.063
300
250
Load, kN
200
150
100
50
0
-450
-50
-400
-350
-300
-250
-200
-150
-100
Displacement, mm
Fig. 8. Dynamic performance characteristic of AGl rockbolts
Rys. 8. Charakterystyka dynamiczna kotwi typu AGl
-50
0
465
200
180
160
Load, kN
140
120
100
80
60
40
20
0
-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
Displacement, mm
Fig. 9. Dynamic performance characteristic of AGp rockbolts
Rys. 9. Charakterystyka dynamiczna pracy kotwi typu AGp
250
Load, kN
200
150
100
50
0
-350
-300
-250
-200
-150
-100
-50
0
Displacement, mm
Fig. 10. Dynamic performance characteristic of IR-4W rockbolts
Rys. 10.Charakterystyka dynamiczna pracy kotwi typu IR-4W
5. Analysis of results
The results presented in Table 1, obtained during trials of rockbolt loading with
impact mass constituted the subject of an analysis, aimed at comparing of theoretical
calculations of impact energy Eu (relation (5), column 8 in Table 1) with the obtained
rockbolt work results Wk (relation (7), column 10 in Table 1). In the calculations it was
assumed that the rig frame has the features of a rigid construction. For the realisation of
the task mentioned above the statistical analysis according to the following procedures
was used (Daniek, 1997; Bobrowski, 1986):
466
1. The absolute difference between the impact energy Eu and rockbolt work Wk for
each test was calculated. When the difference exceeded 5%, the result was rejected.
2. It has been checked if the number of results determined when using the method
mentioned above is sufficient.
3. With the aid of a test of rank difference signs the hypothesis was checked assuming
that the theoretically calculated impact energy Eu is not essentially different form
the measured rockbolt work Wk.
5.1. Calculation of absolute differences
The absolute difference of measuring results is the absolute value of difference between the values of two variables. On the basis of the definition mentioned above for
the needs of the present analysis was assumed:
R E = Eu - Wk ×
1
100
(8)
where:
RE — absolute difference between impact energy and rockbolt work, %,
Eu — total impact energy of falling mass acting on the rockbolt, kJ,
Wk — rockbolt work, kJ.
It has been assumed that the absolute difference of impact energy Eu calculated by
help of the theoretical relation and the measured rockbolt work Wk will not exceed 5%.
Otherwise the test was rejected as doubtful.
On the basis of data included in Table 1 the calculations of RE were carried out. The
obtained results were drawn up below:
Test No
RE, %
1
1.6
2
1.1
3
1.5
4
5.5
5
0.1
6
1.4
7
0.9
8
1.9
9
0.4
10
14.7
11
26.6
12
16.2
From the above Table it is visible that the tests 4, 10, 11, 12 do not meet the assumed criterion RE ≤ 5%. Therefore they should be rejected as doubtful. The remaining 8
tests meet the assumed criterion. Moreover, it can be noticed that in test 4 followed
the rupture of the rope bar, and in the tests No 10, 11, 12 the rockbolts moved totally
from the testing cylinder (Table 1). The conclusion is that in cases of destruction of the
rockbolt or its grout connection with the surroundings the rockbolt work Wk decreases
impetuously – the given relations do not prove true.
467
5.2. Checking of test size
It was assumed that the set of results of absolute differences RE of impact energy and
rockbolt work constitutes a random test of some general population. As general population was understood the entirety of results possible to obtain in the same conditions.
In order to prove the reliability of results obtained in such a way, the necessary number
of measurements for a test was determined from the relation:
nk =
ta2 × S 2
d2
(9)
where:
nk — necessary number of measurements for the test
tα2 — critical values of Student’s distribution for the assumed significance degree
α and number of independent variables r,
S2 — test variance,
d — accuracy of test measurements.
After assuming α = 0.05, r = 8 from the Student’s distribution table tα2 = 5.59 was
read off. Under the assumed measurement accuracy of absolute differences d = 0.5%
and calculated test variance S2 = 0.38, the necessary measurement number of absolute
differences from the relation (9) was calculated. The obtained number of necessary
measurements nk = 5 is lower than the number of measurements performed n = 8, thus
the test size is sufficient to draw conclusions for the general population.
5 . 3 . Te s t o f d i f f e r e n c e r a n k s i g n s
Let x1, x2, , xi, xn and y1, y2, ,yi, yn to be random realisation sequences of random
variables X and Y; at the same time realisations with the same index create pairs (x1,y1),
(x2,y2), (xi, yi), (xn, yn) connected through testing conditions. A random variable x is
understood as a set of results x1, x2, xi, xn of the calculated total impact energy Eu, and
the random variable Y is understood as a set of results y1, y2, yi, yn of measured rockbolt
work Wk.
The null hypothesis was assumed
H0: F(x) = G(y)
(10)
H1: F(x) ≠ G(y)
(11)
towards the alternative hypothesis
where F(x) means the distribution function of continuous random variable X, whereas
G(y) is the distribution function of continuous random variable Y. Speaking otherwise
468
the hypothesis (10) assumes that the results obtained by means of theoretical relations
do not differ essentially from results obtained from measurements.
In order to verify the hypothesis (10), the differences of results di = xi – yi (i = 1, 2,
…, n) of total impact energy Eu and rockbolt work Wk were calculated. The obtained
difference sequence di was transformed into a non-decreasing sequence:
|d1| ≤ |d2| ≤ ≤ |di| ≤ |dn|
(12)
Next to each difference di was assigned the rank Ti– = (i) when di < 0 or rank Ti+ = (i)
when di > 0. The hypothesis (10) is checked using the following test characteristic:
Tn = min{T –, T +}
(13)
where:
T–=
åT ,T = åT
–
+
i
i
+
i
i
The obtained from the formula (13) value Tn is compared with the critical value Tn,α,
such one that if true is the null hypothesis (10) then P(Tn ≤ Tn,α) = α. The values Tn,α can
be read off from tables (Bobrowski, 1986).
If the inequality
Tn ≤ Tn,α
(14)
is fulfilled, then the null hypothesis is rejected in favour of the alternative hypothesis,
and the probability that the decision is false does not exceed the assumed significance
level α.
However, if
Tn ≥ Tn,α
(15)
then there are no grounds to reject the null hypothesis.
For data from Table 1 calculations were carried out according to procedures presented
above (Table 2). The following test characteristic was obtained:
T 8 = min {T – = 11, T + = 25} = 11
At the assumed significance level α = 0.05 from tables were read off T8;0.05 = 4.
Because T8 = 11 > T8;0.05 = 4, so there are no grounds to reject the hypothesis H0: F(x)
= G(y). Speaking otherwise, there are no grounds to reject the hypothesis that the total
impact energy Eu calculated by help of the relation (5) does not differ essentially from
the rockbolt work Wk calculated from measurement results from relation (7).
469
TABLE 2
Auxiliary calculations to check the null hypothesis of difference rank sign tests
TABLICA 2
Obliczenia pomocnicze dla sprawdzenia hipotezy zerowej testu rangowych znaków różnic
i
xi = Eu
yi = Wk
di = xi – yi
|di|
(i)
1
2
3
4
5
6
7
8
18.683
12.638
26.909
10.514
8.336
17.885
9.857
12.947
17.085
11.505
25.432
10.600
9.688
18.770
7.945
13.350
1.598
1.133
1.477
–0.086
–1.352
–0.885
1.912
–0.403
1.598
1.133
1.477
0.086
1.352
0.885
1.912
0.403
7
4
6
1
5
3
8
2
–
–
–
–
–
Ti–
Ti+
7
4
6
1
5
3
8
2
i= 8
å (i)
11
25
i =1
6. Conclusions
The carried out considerations allow to present the following conclusions:
1. In order that the rockbolt would not be destructed during its impact loading, the
total impact energy Eu (relation 5) cannot be higher than the work carried out by
the rockbolt Wk (relation (7)). The following condition should be fulfilled:
Eu ≤ Wk
2. In real conditions (underground) the above relations should come true under the
stipulation that in consequence of the impact no destruction of the rockbolt will
occur (rupture of bar, nut protraction of washer) or its pushing into the working.
In order to obtain such a certainty, underground tests should be carried out (e.g.
simulating impact by means of explosive detonation).
3. On the basis of carried out analyses was pointed out that there are no grounds to
reject the hypothesis about the equality of impact energy Eu and rockbolt work
calculated and measured according to the relations (5) and (7). They can be helpful
when designing the reinforcements of workings threatened with rockbursts by help
of rockbolts, because knowing from rig tests the dynamic performance characteristic of rockbolts they can be mounted in the working, selecting their concentration
according to the expected impact energy.
470
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REVIEW BY: PROF. DR HAB. INŻ. ANDRZEJ WICHUR, KRAKÓW
Received: 03 February 2006