Impact testing for rockbolt design in rockburst conditions

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 31, No. 6, pp. 671-685, 1994 Copyright © 1994 Elsevier Science Lid 0148-9062(94)E0004-L Printed in Great Britain. All rights reserved 0148-9062/94 $7.00 + 0.00

Pergamon

Impact Testing for Rockbolt Design in Rockburst Conditions X. YIt P. K. KAISERt Weight-drop tests on steel rods simulating the impact loading of mechanical bolts by rock blocks ejected during rockbursts were performed in the laboratory to study the dynamic response of these bolts. The steel rods were loaded to deform either elastically or plastically. Cushions of rubber and wood and a slip mechanism were investigated as a means to reduce the elastic stress or plastic strain in steel rods. It is found that the elastic stress or the plastic strain can be determined using the energy balance approach. While the stress can be used as the controlling parameter for rockbolt design within the elastic range, the plastic strain parameter must be employed to exploit the energy absorbing potential of the bolt if plastic deformation of the bolt is acceptable. Rockbolt design examples are given to illustrate the concepts of elastic and plastic designs. It is found that soft cushions placed between mesh and head plate can be employed to absorb wave energy and protect weak connections in a rockbolt-mesh support system, effective slip mechanisms must be designed for fully grouted bolts, and frictional bolts are ideal for excavation walls. Finally, it is pointed out that if the accumulation of plastic strains is anticipated due to repeated seismic loadings, additional support elements must be installed with time.

1. N O M E N C L A T U R E

a = am= 6y = T = Uc = v~= V =

_+ = A = Ac = Am= B to H = c = d = dc =

indicates the s t a n d a r d d e v i a t i o n cross-sectional area o f a steel rod surface area o f a c u s h i o n a m p l i t u d e of a strain wave 2-8 layers o f r u b b e r wave velocity in a steel rod plastic d e f o r m a t i o n per d r o p for steel rods u l t i m a t e yielding d i s p l a c e m e n t o f a yielding cushion or slip m e c h a n i s m E = statistical m e a n E = kinetic energy of ejected rock block, E = M v ~ / 2 Ee, %, ~ = elastic, plastic and total strains = strain rate Ed = d y n a m i c elastic m o d u l u s o f steel E E E = energy e q u i v a l e n t elastic m o d u l u s o f r u b b e r F c y = d y n a m i c yield load o f a yielding cushion or slip mechanism F m = m a x i m u m load F s = slip load Fy = d y n a m i c yield load for steel rods F'y = d y n a m i c yield load for steel rods in s t r a i n - h a r d e n i n g region g = g r a v i t a t i o n a l acceleration k, k ' = stiffness o f steel rod in elastic a n d in s t r a i n - h a r d e n i n g regions k c, K, K ' = stiffness o f cushion, K = k k ~ t a n d K ' = k 'k ¢- 1 L = free length o f steel rod M = m a s s o f d r o p - w e i g h t or ejected rock block m = modification factor for r u b b e r p = density o f steel

stress in r u b b e r c u s h i o n a m p l i t u d e o f stress wave d y n a m i c yield p o i n t for steel rods h a l f period o f wave frictional energy d i s s i p a t e d d u r i n g slip i m p a c t velocity statistical variance

2. I N T R O D U C T I O N

As mining is carried out at deeper levels in hard rocks, rockbursts are frequently encountered. One of the most severe rock mass failure modes during rockburst is the ejection or expulsion of rocks into underground openings [1,2, 3]. Rockbolts used for support are subject to impact loading by the ejected rock blocks. Hedley and Whitton [4] investigated failures of mechanical bolts due to rockbursts at Quirk Mine, Elliot Lake, Canada. Bolts designed for rockburst conditions must hold the rock blocks, and a design model called the rock ejection model can be used. This model generally consists of four components: an anchor, a steel rod, a bearing plate and an ejected rock block. The concept of rock block ejection was employed by Wanger [5] to assess support requirements for rockbursts in South African gold mines. The same approach was adopted by Hedley [6] to develop a conceptual chart for support selection based on source magnitude and distance from the source. The objective

t G e o m e c h a n i c s Research Center, L a u r e n t i a n University, Sudbury, O n t a r i o , C a n a d a P3E 2C6. 671

672

YI and KAISER:

I M P A C T T E S T I N G R O C K B O L T DESIGN

of this paper is (i) to evaluate the effectiveness of elastic and yielding cushions for the protection of steel rods and (ii) to investigate steel failure phenomenon due to impact-loading. The results may be used to improve existing rockbolt support systems and to develop design methods for rockburst conditions. This paper is composed of two parts, including both laboratory tests and data analyses using the energy balance approach. Laboratory weight-drop tests were employed to simulate impact loading of mechanical bolts by ejected rock blocks in underground mines with rockburst hazards. In Part I, the stresses in the steel rods were kept below the elastic limit, and rubber layers from an old conveyer belt (nylon reinforced), soft wood blocks and an artificial wooden plate made of wood wool were tested as cushions. The soft wood blocks and the artificial wooden plates are currently being used in some Canadian mines. A small steel rod of 6.35 mm dia and a large steel rod of 15.88 mm dia were tested. With the large steel rod, soft wood cushions yielded before the steel did. The results obtained in this part are applicable to rockbolts for permanent underground openings where plastic or permanent rockbolt deformations are not allowed in the design. In Part II, small steel rods with rubber cushions were loaded into the plastic deformation range. In addition, a slip mechanism was employed for the bottom bearing nut to study its.effect on reducing the plastic deformation of the steel rods. The results of this part are applicable to rockbolts for mining drifts and slopes where substantial plastic deformations are acceptable as long as the bolts do not reach the ultimate strain limit within the design life.

3. STATIC TESTS OF CUSHIONS AND STEEL RODS

Static tests were performed to determine the mechanical properties of the rubber and wood cushions. Rubber cushions with 1, 2, 4 and 8 layers were tested. Each rubber layer had dimensions of 100 x 100 x 13.5 mm (thickness) and a mass of 141 g (approx. 1044 k g m -3 density). These rubber elements had been used in the dynamic impact tests before they were tested statically. An Instron materials testing machine was employed to perform the uniaxial compressive tests with a loading rate, in terms of strain, of around 2 x 10-3 s-~. Two series of tests with different areas of loading on the top and bottom faces of a cushion, reflecting a slightly different dynamic test procedure, were carried out. In series No. 1, a rectangular steel plate which has dimensions of 80 x 37 x 6 mm (thickness) and was used as a bearing plate in the dynamic tests, was placed underneath and a steel cylinder of 76.2 mm dia was placed on top of a cushion. In series No. 2, the rectangular steel plate was replaced with a steel clamp which has a circular top surface of 76.2 mm dia and was used to simulate a yielding plate and nut in the dynamic tests. In series No. 2, a soft wood block of 90 x 90 mm in surface area and 39mm in thickness (185 g mass) was also tested. This wood block was of the same material and size as the soft

25 t i 4 I

20 Serie

10

/

/

Series No. 1

5 t

t

0

I

I

I

I

2

3

Displacement (ram) Fig. 1. Load~lisplacement plot for one layer of rubber from series No. I and No. 2 static tests.

wood cushions used in the dynamic tests and was loaded normal to the grains. The load-displacement plot for one layer of rubber in series No. I tests is shown in Fig. 1. The curve becomes clearly non-linear (stiffer) above a load of 15 kN. The stiffness values of rubber cushions with various layers and at different load levels were measured from similar load-displacement plots for different layers of rubber. The load-displacement plot for one layer of rubber in series No. 2 tests is also shown in Fig. 1. It is relatively linear due to the larger loading surface area on the cushion. The stiffness values as well as the maximum loads and displacements during tests were obtained from similar plots for various layers of rubber. The elastic modulus calculated from the stiffness remains essentially constant for various cushion thicknesses below a load level of about 20 kN. In order to determine the stress versus strain relations at higher stresses, a two-layer rubber cushion was loaded to a much higher load level and a wood cushion to a large displacement (Fig. 2). The rubber and wood cushions demonstrate quite different mechanical behavior. The rubber has initially a low elastic modulus, but stiffens at higher stresses. Conversely, the wood has a 2O

5

0

10

20

30

40

50

Strain (%) Fig. 2, Stress-strain plot from series No. 2 static tests of two layers of rubber and a soft wood block.

YI and KAISER: IMPACT TESTING ROCKBOLT DESIGN

high initial elastic modulus at low load levels, but eventually yields. It therefore has an essentially elastoplastic behavior. The stress-strain curve for the rubber becomes non-linear if the stress level is higher than 4 MPa. For the analytical studies, it was desirable to linearize the portion of the curve between zero and the maximum stress involved. A line was drawn such that the underlying area was equal to the area under the original curve. The slope of this line is called the "energy equivalent elastic modulus" (EEE) of the cushion. A linear regression analysis was performed for the measured EEE data for different stress levels and the following relation was obtained (with an R-squared correlation coefficient of 0.98) [7]: EEE = 23.1 _+ 2.1a

673

I Topclamp Straingauge

Concrete beam

i

Steelrodinsidea protectiontube

Drop weight

(1)

where EEE and the stress in cushion tr are measured in MPa. The stress-strain relation for the soft wood block can be approximated by an elasto-plastic model with an initial elastic modulus of 200 MPa and a yield stress of 7 MPa. The steel rods were made of Grade 1045 steel, with supplier specified mechanical properties of 530 MPa for the yield point, 630 MPa for the ultimate strength and 12% for the ultimate strain (measured on 50 mm length). Static tensile tests of steel rod specimens were performed. The middle portion of a steel specimen was 50 mm long, and machined down from a diameter of 6.35 to 4 mm. A specimen was made with dimensions conforming to the ASTM standard and five specimens were tested. The loading rate was around 2 x 10 -4 S-1. From the tests, the means and standard deviations for the ultimate strength, the strain at ultimate strength and the ultimate strain were 719 + 14 (MPa), 6.0 +_ 0.4 (%) and 8.2 _+ 0.9 (%) respectively (_-_+indicates the standard deviation). This ultimate strength is about 100MPa higher than the supplier's specification whereas the ultimate strain is about 4% lower. The fact that the two ends of a specimen were not allowed to rotate in the static tests and nearly all the specimens failed at the small to large diameter conjunction may explain why the obtained ultimate strain is lower than the supplier specified value. For rockbolt applications, an ejected rock block is normally restrained by surrounding rocks, and therefore, the ends of a rockbolt are not free to rotate during dynamic loading. 4. PART I--TESTS IN THE ELASTIC RANGE

4.1. Dynamic Tests on a Small Steel Rod with Cushions

The experimental set-up is shown in Fig. 3 with a rubber or soft wood cushion inserted on top of the bearing plate as a cushion. A small steel rod of 6.35 mm diameter and 2.44 m length (with a 2.28 m free length) was employed. It was protected by a concentric steel tube from bending caused by a dropping weight. A test was denoted by three digits and a letter, e.g. "11B-3". The first digit gives the weight number (No. 1 and 2 for 18.36 and 48.494 kg, respectively), the second the height number (No. 1 and 2 for 0.1 and 0.3 m, respectively), and

Bearingplatei andnuts

Cushion

Fig. 3. Schematic of weight-drop test set-up.

the third after the hyphen the repeated drop number (3 repeated drops were performed for each combination of weight and height). The letter indicates the number of rubber layers with "B, D and H " for two, four and eight layers, respectively. The letter " W " was used to denote a soft wood cushion. Weight No. 1 has a conical base with a circular bottom of 76.2 mm dia. This matches the steel clamp used to simulate a yielding plate and nut. It was lifted using a pulley assembly during the weight drop tests. Weight No. 2 was built up on No. 1 with two additional steel plates secured to the top. A strain gauge was installed immediately under the top clamp to record the strain wave in the steel rod. This strain gauge measured 1.57 mm in the direction of strain and 3.05 mm in the perpendicular direction. One strain gauge was sufficient to accurately measure the tensile strain because (i) only the first half period of a strain wave was of interest (see Fig. 4) and (ii) concentricity of loading was ensured by the protection tube (see Fig. 3). Two recorded strain waves from test Nos 21 (without cushion) and 21D (with 4-layer or 54 mm thick rubber cushion) are presented in Fig. 4. Without any cushion, a "rider wave" rides on a lower frequency sinusoidal "carrier wave". This phenomenon was studied in details by Yi and Kaiser [8] and it was found the rider wave was due to wave reflections up and down the steel rod and the carrier wave due to rod elongation. With rubber cushions, strain waves were smoothed out, the wave amplitude A mdecreased and the half period T increased. The strain wave with a soft wood block (not shown) is similar to the strain wave without any cushion. The soft wood block has essentially no stress reduction effect for the small steel rod, since the stiffness of the former is too large compared to that of the latter. Figure 5(a) and (b) shows plots of statistical values ( m e a n + s t a n d a r d deviation) for two characteristic parameters, i.e. the

674

YI and K A I S E R :

IMPACT TESTING ROCKBOLT DESIGN

2.5 ~,[

/

2

"~

the cross-sectional area A of the steel rod. It follows from the above equation that with decreasing cushion stiffness kc, the stress amplitude decreases while the half period of wave increases.

NO cushion shion

1.5

4.2.2. Comparison of experimental and theoretical results .['or rubber cushions

0.5

I*' 0

'*t

" T

5

10

15

20

25

Time (milli-second) Fig. 4. Typical e x p e r i m e n t a l strain waves from test No. 21 w i t h o u t cushion and No. 21D with four layers of r u b b e r cushion.

strain amplitude A m and the half period T for three combinations of the drop weight and height. The wave propagation velocity for the steel rod equals the distance between two points divided by the wave propagation time, which was determined to be c = 5.16 + 0.05 km s -~ [7]. The dynamic elastic modulus of the steel rod is therefore: E d = c2p = 206 _+ 4 GPa (with a density p = 7750 kg m-3 for the steel rod). The stress equals the strain multiplied by Eo and the stiffness of the small steel rod was k = EoAL-' = 2.86 MN m - ' , where the standard deviation of Ed is ignored since the corresponding coefficient of variation is only 2%. A and L are the cross-sectional area and the free length of the small steel rod, respectively, with A = 31.65 mm 2 and L = 2.28 m.

4.2. Analysis of Dynamic Test Results 4.2.1. Theoreticalformulation In an elastic dynamic test, the sum of masses of the steel rod, bottom plate and cushions was much less than the mass of the steel weight M, and the gravitational force Mg (g being the gravitational acceleration) was much less than the maximum dynamic force Fm in the steel rod. Therefore, the maximum force (or force amplitude) and the half period T of a wave can be obtained from energy balance analysis as follows [7]:

The static stiffness values k¢ of both rubber and wood cushions were determined previously and are assumed to be equal to the dynamic stiffness values, since the dynamic values could not be determined with the available test equipment. Equation (2) can be employed to predict the maximum stress G, and the half period T for the dynamic tests described in the previous section. The cushion stiffness values k~ at different load levels and for different rubber layers were determined from series No. 1 static tests, and the standard deviation of the measured k~ was estimated to be 0.1 MN m J. The standard deviation of the drop-heights for both height No. i and 2 are assumed to be 3 mm. For a non-linear function of multiple independent variables, F = f ( x ~ , x2 . . . . . x,), the following equation may be used for evaluating the mean or expectation E(F) and the variance V(F) of the above function [10]: E(F) = f ( E ( x t ) , E(x2) . . . . . E(x,)) and V(F)=Y~(ff/6xi)2V(xi) (i = 1 to n), where the partial derivatives 6f/&x; are evaluated at E(x;). 3

Amplitude A m

2.5 \

2

\

\1.

1.5 <

1

~

0.5

k

o

\

I

I l t t l

I I I I

Test No. 25

(b) Half period T /1' /

rm = ~ kMv~ I+K

~- 20 / /

,,f 15

and T = nk/(1 +K)M k

(a)

/ / b /I ¢

~d (2)

where K = kk ~ ~, k and k¢ are the stiffness values of steel rod and cushion, respectively, and M and v~ are the mass and velocity (immediately before impact) of the drop weight. From the above equation, (FmTr-~)T = Mvl, where the term in parentheses is the average load of a sine wave over time T. This is a result of the law of momentum conservation. The maximum stress in the steel rod rrm is equal to the maximum load Fm divided by

.r-

10

/ /"

/

/

.¢

.v

¢

5

o

i

l

t

l

l

l

l

t

l

l

l

Test No. Fig. 5. Plot of m e a n s and s t a n d a r d d e v i a t i o n s from d y n a m i c tests on a small steel rod for (a) a m p l i t u d e A m and (b) half period T of wave.

YI and KAISER:

IMPACT TESTING ROCKBOLT DESIGN

675

Table 1. Theoretical parameters for dynamic tests on small steel rod with cushions

Test No.

21 21B 21D 21H 21W 11 liB llD IlH 12 12B 12H

M a x i m u m load from tests, F m (kN)

15.0+ 1.2 13.2+0.4 12.1 +0.1 10.6 + 0.2 15.5+0.4 9.7 + 0.2 7.9+0.1 7.2+0.0 6.3 +0.1 17.3 +0.2 14.6+0.4 11.1 +0.3

Cushion stiffness, k c (MN m - i )i"

3.8(1 +0.03) 2.2(1 +0.05) 1.2 (1 + 0.08) 23.4 3.5(1 +0.03) 2.0(1 +0.05) 1.1 (1 +0.09) 3.9(1 +0.03) 1.2(1 +0.08)

Rod to cushion stiffness ratio, K:~

Theoretical m a x i m u m stress, a m (MPa)

Theoretical half period, T (msec)

0.001 0.753(I+0.03) 1.300(1 +0.05) 2.383 (I + 0.03) 0.001 0.001 0.817 (1 +0.03) 1.430(1 +0.05) 2.600(!+0.09) 0.001 0.733(1+0.03) 2.383(1+0.08)

520.67 (! +0.02) 393.49 (i +0.02) 343.49(! +0.02) 283.21 (I + 0.03) 520.67(1 +0.02) 320.37 (1 + 0.02) 237.78(I +0.02) 205.62 (I +0.02) 168.94(1 +0.04) 554.93(1 + 0.01) 421.75(1 + 0.01) 301.86(1 +0.03)

12.9(1 + 0.00) 17.1 (1 + 0.01) 19.6 (1 + 0.01) 23.8 (1 + 0.03) 12.9(1 + 0.00) 8.0 (1 + 0.00) 10.7(1 + 0.01) 12.4 (1 +0.02) 15.1 (1 +0.03) 8.0(1 + 0.00) 10.5(1 +0.01) 14.6(1 +0.03)

t C u s h o n stiffness at different loads are determined from series No. 1 static tests. Data are presented in the form of mean (I + coefficient of variation). ~ K = k k ~-~, with k = 2.86 M N m -l for the steel rod. For tests without cushions, K is set to 0.001.

Theoretical predictions of stress amplitude trm and half period T are presented in Table 1. Comparisons between the measured values and the theoretical predictions can be made in terms of a relative difference parameter. The difference between two statistical values is determined at the 95% confidence level using the following formula for three data points: (meant - mean2) + 1.603(s~ + s~)°5, where meant and mean2, and st and s2 are means and standard deviations of experimental and theoretical parameters, respectively [11]. The relative difference equals this difference divided by the theoretical mean value mean2. The relative differences for rubber cushions in the m a x i m u m stress o"m vary from 9 + 15% (test No. 21) to 19 + 9% (test No. 21H) with an average of 7% for the means, and the relative differences in the half period T vary from - 13 +__5% (test No. 12H) to 6 ___6% (test No. 21) with an average of - 2 % for the means [7]. These relative differences are considered to be small. However, it was found that the relative differences in trm (or T ) for rubber cushions increase (or decrease) somewhat with increasing cushion thickness for each of the three different combinations of weights and heights. This indicates that the experimental maximum stress tends to be higher than the theoretically predicted value for thicker rubber cushions, i.e. equation (2) slightly overestimates the effects of thicker cushions. This will be discussed later. A wood cushion has a stiffness of (23.4 M N m - ' ) and a yield load of 31.9 kN. This stiffness value is much larger than that of the steel rod ( 2 . 8 6 M N m - t ) or K = 0.12. According to equation (2), this soft wood block had little effect on the reduction of elastic stress.

were: (i) the top end of the large steel rod was screwed into the top steel clamp rather than clamped, and the bottom plate was allowed to move upward on the steel rod; and (ii) it was not possible to use a steel tube to protect the large steel rod, instead, two strain gauges were mounted on diametrically opposed sides to record bending effects. Weight No. 2 was dropped from height Nos 2 (0.3 m) and 3 (1.2 m). Rubber cushions of one (A), (a)

No. wSt

2.5

~" =

2 1.5

g •N

1 0.5

-0.5

I 4

I 8

12

Time (milli-second)

(b) From test No. L23W

2.5

4.3. Dynamic Tests on a Large Steel Rod with Cushions 4.3.1. Experimental technique In dynamic tests presented in this section, a large steel rod 15.88mm in diameter and 2.24m in length was employed. The purpose was (i) to find out whether the test results are different for a larger steel rod having the same geometric dimension as a mechanical rockbolt and (ii) to study how wood cushions yield under dynamic loading. The experimental set-up was similar to the one for the small steel rod (Fig. 3). The main differences

~

0.5 0 -0.5 0

I 4

I 8

12

Time (milli-second) Fig. 6. Strain waves from two strain gauges on diametrically opposite sides of a large steel rod with cushions of (a) an artificial wood plate of wood wool and (b) a soft wood block for the same combination of drop weight (48.5 kg) and height (1.2 m).

676

YI and KAISER:

I M P A C T T E S T I N G R O C K B O L T DESIGN

two (B), four (D), six (F) and eight (H) layers as well as soft wood blocks (W) identical to those described previously were used. Test No. L23F, for example, denotes the test on the large steel rod with weight No. 2 and height No. 3 and with a rubber cushion of 6 layers. In addition, an artificial wood plate made of wood wool (denoted as AW) was tested as an elastic cushion. It had a geometry of 125 x 125mm (about 12mm thickness) and was designed for use in a rockbolt-mesh support system in a Canadian mine. The bottom steel plate measured 100 x 100 x 12.7 mm (thickness) and weighed 1.012 kg.

theoretical calculations, and the dynamic elastic moduli of thicker rubber cushions must have been under-estimated. Since the dynamic elastic moduli of rubber cushions could not be obtained with available testing facilities, equation (2) may be modified empirically. According to the above discussion, only terms involving K should be modified, and the following modified expressions are introduced:

4.3.2. Comparison of experimental and theoretical results Rubber cushions. The test results are similar to those

T = n(l + K ) " ~ f - ~

for the small steel rod [7]. For example, with one layer of rubber (13.5mm thick) the wave amplitude A m decreased by about 23% whereas the half period T increased by about 18%, and with two layers of rubber the corresponding percentages are 36 and 29%, respectively. Theoretical parameters are evaluated with the energy equivalent elastic modulus EEE calculated from equation (1) for the rubber cushions. Again, the relative differences in Am increase whereas those in T decrease with increasing cushion thickness, i.e. equation (2) overestimates the effect of thicker cushions. This issue will be discussed in the next sub-section. Wood cushions. Figure 6(a) and (b) show the strain waves for tests with the artificial wood plate and with a soft wood block, respectively. The tests were conducted with the same combination of drop weight (48.5 kg) and height (1.2 m). No damage of the artificial wood was visible, but the soft wood block was severely damaged. The sinusoidal wave shape for the artificial wood [Fig. 6(a)] indicates an essentially elastic response, but the strain waves for the soft wood block [Fig. 6(b)] with relatively flat portions near the peaks indicate cushion yielding. For the artificial wood plate, the strain waves were smoothed out and the amplitudes reduced slightly (by about 16%). The dynamic yield point of the soft wood block is about 15 MPa at a strain rate of about 1 s-~. This strain rate is approximately three orders of magnitude higher than for static testing, and the dynamic yield point is approximately twice the static yield point (Fig. 2). With the soft wood blocks (39 mm thick), the stress amplitudes in the steel rod were decreased by about 30%.

where m is the modification factor and in equation (2), the equivalent value for m is 0.5. For a reduced value of m = 0.366, the relative differences between experimental and theoretical parameters for rubber cushions are plotted in Fig. 7(a) and (b) for tests on the small steel rod. The relative differences are now reduced and they do not demonstrate a consistent trend. In other words, the effect of cushion thickness is essentially absent. The same conclusion can be reached for tests on the large steel rod [7].

1 Fro- (1 + K ) ~ ~

and (3)

(a) 30 Stress Amplitude ~r,, 20

8

10

e-

.~ -10 -20 I

-30 t'q

I eq

I ~

I

I

I

I

I

I

I

e'q

Test No. 20

(b) Half

Period

T

10 5

4.3.3. Modification of equation (2) Equation (2) over-estimated slightly the effects of thicker rubber cushions for tests on both small and large steel rods. For the same steel rod, the steel rod to cushion stiffness ratio K increases with decreasing cushion stiffness or increasing cushion thickness. The fact that equation (2) over-estimated the effects of thicker rubber cushions indicates that the over-estimation occurred when the stiffness ratio K was increased. This overestimation is probably due to the fact that the static elastic moduli of rubber cushions were used in the

~-10 -15 -20

I l t l l l l l l l

Test No. Fig. 7. Relative differences between experimental and theoretical parameters in (a) stress amplitude a m and (b) half period T of wave for dynamic tests on the small steel rod employing equation (3).

YI and KAISER: 5. P A R T I I - - T E S T S

IMPACT TESTING ROCKBOLT DESIGN

677

300

IN THE PLASTIC RANGE

201 mm

5.1. Tests with Cushions The experimental set-up is shown in Fig. 3. A weight of 48.494kg was dropped from a height of 1.4m (5.24m s -] impact velocity) for all tests. Tests with cushions and with a slip mechanism were performed. During a test with a cushion, the steel rod underwent an incremental plastic deformation upon each impact. The weight was repeatedly dropped until the steel rod eventually broke. Eight, four and two layers of rubbers, cut from an old conveyor belt, denoted by "H, D and B" and having thicknesses of 108, 54 and 27 mm, respectively, and a 39 mm thick soft wood cushion were used as cushions. Tests without cushions are denoted by "N". Test No. H 1-1, for example, denotes the first drop (last digit after the hyphen) on No. 1 steel rod (first digit) with rubber cushion H. The bottom bearing plate was replaced with a steel clamp called the "bottom clamp" with a surface diameter of 76.2 mm and a mass of 0.857 kg. For each type or thickness of cushion, three steel rods were tested in the same manner in order to obtain representative data. For repeated drops, the weight was dropped from the same altitude, initially 1.4 m above the top of the cushion, despite the fact that the actual height of drop increased slightly with incremental plastic deformation of a steel rod. In other words, the kinetic energy of the weight, immediately before its impact on the cushion (or directly on the bottom clamp), became slightly greater with each subsequent drop. 5. I. 1. Plastic elongation from direct measurements In order to obtain a crude measure of the effectiveness of different cushions on the protection of steel rods, the total plastic deformation and the number of repeated drops were recorded for each rod after it broke. The locations of breakage on the 13 steel rods are plotted in Fig. 8, which shows that the positions of breakage were quite evenly distributed over the length of steel rods. The total plastic deformations of the individual steel rods varied considerably from 145 to 253 mm (Fig. 9). Yet, the statistics for groups of tests with different cushions were not significantly different from one another, varying from 196.3 _+ 34.5 mm (3 data) with a 54 mm thick rubber cushion, to 205.5 _+ 26.5 mm (4 data) with a 108 mm thick rubber cushion. The overall average and standard deviation of the total plastic deformations for the 15 data was 201_+29mm. Therefore, the total amount of plastic deformation at the time of breakage

.~o 250

15o @ [...

100

zzzzz

=~

~=~

~ =

~

Steel Rod No. Fig. 9. Total plastic deformations from direct elongation measurements after breakage (means and standard deviations for each cushion are shown).

seems to be a material property independent of the cushion effects. The average ultimate strain of the steel rods tested was 8.8 + 1.3%. Since this ultimate strain was measured on a 2280 mm length, the localized strain due to steel necking is essentially not included and this ultimate strain may be considered to be the strain at ultimate strength. The supplier specified that static ultimate strain is about 12% which is measured on a 50 mm length and therefore includes the necking strain. The corresponding strain at ultimate strength may be estimated to be 9-10% (see end of Section 3). Hence, the dynamic strain value is not very different from the static value. The average plastic deformation per drop was essentially the same for the tests with the same cushion, but the statistics for different cushions were distinctively different from each other. In other words, the parameter of the average plastic deformation per drop shown in Fig. 10 is a good indicator of the effectiveness of a cushion. It decreased from 16.1 + 0.6 mm per drop for tests without any cushion, to 10.6 + 0.3 mm per drop for tests with a 108 mm thick rubber cushion. The total 20

e~

16

12 I

w=l~*

a-

2440 mm long steel rods Top end

Bottom end

IIIII

................

J l l l l l l l l l J l l J l l

z z z z z =~=~ ~88= ~ I

0

I

I

500

1000

I

1500

I

2000

I

2500

Positions of Breakage (ram) Fig. 8. Locations o f breakages on steel rods in the plastic tests with cushions.

Steel Rod No. Fig. 10. Average plastic deformation per drop from direct elongation measurements after breakage (statistics for each cushion are also shown) with "N, B, D and H" representing no cushion, 27, 54 and 108 mm thick rubber cushions and " W " a 39 mm thick soft wood cushion.

Y I and K A I S E R :

678

14

(a)

6i

m

4

E

Ep £e

2

5

10

20

15

25

30

35

Time (milli-second) 14

(b) Test No. H3-1

12 t-

~,

10

8

r= 6

4

I

I

I

I

I

5

10

15

20

25

30

Time (milli-second) 14

(e) Test No.

12 10

~,

8

~

6 4

0

I

I

I

I

I

5

10

15

20

25

I)ESIGN

looking upward. The total rotation after a rod broke was about 100-200 depending on the total plastic deformation which the rod sustained. Detailed measurements were performed for tests with the two-layer (27 mm thick) rubber cushion. Steel rod Nos B I to 3 sustained plastic rotations of 209, 174 and 155 at the 14th, 12th and l lth drops, respectively. The average plastic rotation per drop for these three rods was 14.5 4- 0.4 . It seemed that this plastic rotation of a steel rod was directly associated with its plastic elongation, but the mechanism of rotation is not fully understood.

'l'est No. N3-I

12 i

g

IMPACT TESTING ROCKBOLT

30

Time (milli-second) Fig. 11. Strain waves from first d r o p s in plastic tests for test Nos (a) N3-1 w i t h o u t cushion, (b) H3-1 with a 108 m m thick thick r u b b e r cushion and (c) WI-1 with a 39 m m thick soft w o o d cushion.

number of drops needed to break a steel rod increased from 12.8 + 2.6 to 19.5 4-2.9 reflecting an increase of approx. 52%. Since the soft wood block could not yield before the steel rod did, the plastic deformation is approximately equal to that for a test without any cushion. During the tests, it was observed that the bottom clamp or the bottom end of the steel rod sustained permanent rotations with respect to the top clamp (or the top end of the steel rod) after each impact. The direction of the rotations was consistently clockwise if

5.1.2. Elastic and plastic strains .[rom strain gauge measuremen ts

The strain wave in a steel rod was monitored using a strain gauge located about 70 mm below the top clamp (Fig. 3). Figure 1 l(a) shows the strain wave from the first drop for steel rod No. N3 without any cushion. The strain wave is not completely elastic and it does not return to zero at the end of the test, indicating a plastic or irreversible deformation of the steel rod. The maxim u m elastic or recoverable strain ~e is equal to the maximum total strain ~ minus the plastic strain c0, that is, ~e = e - %. The rider wave (Section 4.1) is present only on the elastic portion of the strain wave. The strain waves from subsequent drops for the same steel rod had similar shapes. Figure l l(b) presents the strain wave from the first drop for rod No. H3 with a 108 mm thick rubber cushion. As compared to Fig. l l(a), the rider wave is essentially absent. Strain waves from subsequent drops for the same steel rod and cushion had similar shapes. Furthermore, strain waves from tests with all rubber cushions having thicknesses from 27 (D) to 108 mm (H) had similar shapes. Figure 1 l(c) shows the strain wave from the first drop for rod No. WI with a soft wood cushion. Since this soft wood cushion was relatively stiff and did not yield completely, a rider wave can still be seen on the elastic portion of the strain wave but is less pronounced than for tests without any cushion. Strain waves such as those in Fig. 1 l(a)-(c) could have been obtained for each drop until the breakage of a steel rod, if the strain gauge could withstand all the incremental plastic strains. Unfortunately, strain gauges became debonded after a plastic strain of about 3%. Statistics of measured parameters including the plastic strain ~p and the maximum total strain E for strain waves from the first three drops were obtained. The elastic strains Ee were calculated from the plastic and the total strains as explained previously and are presented in Fig. 12. The elastic strain for the 3rd drop without any cushion is not shown and could not be obtained since the limit of the strain measurement was exceeded. The following two observations can be made from the plot: (i) for the same drop number (e.g. 1st drop) the elastic strains for different steel rods (with different cushions) are not significantly different from each other and (it) for the same steel rod, the elastic strain increases with each subsequent drop. The first observation indicates that the elastic strain parameter reflects the elastic material prop-

YI and KAISER: IMPACT TESTING ROCKBOLT DESIGN 6

679

24 £e

19 5

/~

!

t/t

41 3

I I I I I I I [ I -e~ -~,,~ - e q ~

!

t)

o e.,,.q

14

9

I

I I I I I , I -e~,~ "7~,~,

Drop No. Fig. 12. Statistics of elastic strains for the first three drops from strain gauge measurements with "N, B, D and H" representing no cushion, 27, 54 and 10g mm thick rubber cushions and "W" a 39 mm thick soft wood cushion.

erty of the steel rods. The second observation implies either a strain hardening behavior or a decreasing unloading-reloading modulus with each subsequent drop. The yield point may be estimated by multiplying the maximum elastic strain Ee for the first drop by the initial dynamic elastic modulus for steel: Ed = 206 (GPa) (Part I). The mean and standard deviation for the first drops on all steel rods is calculated to be: tTy= 881 _+ 53 (MPa). This yield point is 66% higher than the supplierspecified static yield point of 530 MPa and even higher than the static ultimate strength of 719 MPa by about 20%. The average strain rate in an impact test can be estimated to be 1 s-' [peak strain divided by rise time, Fig. 1l(a)].

5.1.3. Comparison between experimental and analytical results Analytical results. In this sub-section, the average plastic strain per drop will be roughly estimated by assuming that the mechanical behavior o f the steel rod is linearly elastic and perfectly plastic (Fig. 13). This assumption o f perfect plasticity results in an underestimation o f the energy consumed in the strain-hardening process. On the other hand, the higher imparted energy o f the drop-weight in each subsequent drop, due to incremental plastic elongations o f the steel rod, is ignored in the analysis. Considering the energy balance between the consumed and imparted energies, the ignored a m o u n t o f the imparted energy tends to compensate for the neglected strain-hardening.

N

I

I

B

I

D

H

W

Cushion Type Fig. 14. Theoretical values of the average plastic deformation per drop for different cushions with "N, B, D and H" representing no cushion, 27, 54 and 108 mm thick rubber cushions and "W" a 39 mm thick soft wood cushion.

Since the gravitational stress in a steel rod due to a dropped weight was negligible compared to the yield stress, the plastic deformation d of a steel rod upon a single impact can be obtained from energy balance analysis as follows [7]:

d=MV~ 2Fy

Fy (K + I)

(4)

2k

where Fy is the yield load o f the steel rod with Fy = ayA. F o r the tests with cushions, M = 4 8 . 4 9 4 k g , v~ = 5 . 2 4 m s - ' , Fy = 881(1 _+ 0.06) M P a x 31.65 m m 2 = 27.88(1 _+ 0.06) k N [data presented in the form o f mean (1 + coefficient o f variation)] and k = 2.86 M N m - ' . For cushions, the dynamic elastic moduli could not be obtained with available facilities and the static elastic moduli o f the rubber and w o o d are used instead in the calculation o f cushion stiffness values. W h e n the yield load o f the steel rod Fy was reached in a plastic test, the stress in the cushion was FyA ~' = 6(1 _+ 0.06) M P a (Ac being the surface area o f the b o t t o m steel clamp, Ac = 4558 mm2). The elastic modulus o f rubber is approx. 35 M P a at this stress level (Section 3). The fact that the stress level o f 6 M P a in a cushion was below the 6 4

Using Equation(4)

Usingmodified Equation(5)

e-

2

-2 tYy= (881 + 53) MPa

e~

-4 ¢-

-6 =(206 + 4) GPa

-8

I

I

I

I

I

I

I

N B D H W N B D H W Cushion Type Fig. 13. Assumed linearly elastic and perfectly plastic material property for steel rods.

Fig. 15. Differences in the average plastic deformation per drop parameter between directly measured and theoretical values.

680

YI and KAISER: IMPACT TESTING ROCKBOI_T DESIGN

Test No.t SI- I S1-2 SI-3 SI-4 S1-5

Table ~. ~ Measured parameters from slip tests Slip distance Plasticstrain Slip load and (mm) (milli-strain) distancelevels+ + Remarks 12 7.95 high. small 0 9.32 0 9.32 35 3.29 high, medium clamp loosened before drop 19 8.23 high, small

$2-2

77

2.03

high. large

clamp moved up after first drop

$3-1 35 5.76 low, medium $3-2 34 4.39 high, medium clamp moved up after first drop $3-3 38 1.10 low, large clamp moved up after second drop t"s" indicates slip tests, the first digit is the steel rod number and the second is the drop number; :p'high" or "low" indicate that the frictional resistance of the bottom clamp is higher or lower than the elastic limit of steel rods; "small", "medium" or "large" are rough indicators of slip distances. static yield stress of 7 MPa for wood explains why the wood cushion did not yield and behaved elastically with an elastic modulus of about 200MPa. Employing equation (4), the plastic deformations per drop for steel rods with different cushions are calculated and plotted in Fig. 14. The mean and variance of d are again calculated using the Taylor Series Approximation Method (Section 4.2). Comparison of theoretical and measured values. Comparing analytical results in Fig. 14 to the directly measured results in Fig. 10, a good agreement exists in the trend that the plastic deformation per drop decreases with increasing cushion thickness for tests with rubber cushions and that the soft wood block does not have any effect. Quantitative comparisons are presented below. The differences in the average plastic deformation per drop between experimentally measured and theoretical results are calculated and plotted on the left-hand side of Fig. 15. The difference between two independent statistical data, a _ _ A a and b_+Ab, is ( a - b ) _ + (Aa 2 + Ab 2)o5. The directly measured values are slightly larger than the theoretical values. The differences vary from - 2.9 4- 1.8 (without any cushion) to 1.0 __ 3.7 m m (with a 108 m m thick rubber cushion). Considering the uncertainty of the dynamic behavior for the steel rods and the cushions, this agreement between predictions and measurements is considered to be good. However, there exists a clear trend that the differences between theoretical and experimental values increase with increasing cushion thickness, i.e. the effects of thicker rubber cushions are over-estimated by equation (4). Similar to equation (3), this problem can be solved by employing the following modified equation:

d=MV~ 2Fy

Fr (g + l) 2m

(5)

2k

where m = 0.366. The differences between the experimental and theoretical values using this modified equation are plotted on the right-hand side of Fig. 15. The differences between experimental and predicted values are now independent of cushion thickness. In conclusion, the energy balance approach is applicable to the tests with cushions for predicting plastic deformations of steel rods. Using the modified equation, equation (5), the agreement between experimental and

theoretical values is improved. Since the ultimate plastic deformation of a steel rod at breakage under impact loading is a material property (Fig. 9), the plastic deformation parameter may be used as the controlling parameter for evaluating failures of steel rods under impact loading.

5.2. Slip Tests The experimental set-up is the same as for tests with cushions except that the lower end of the protection tube was cut by about 90 mm (Fig. 3). During a slip test, the bottom clamp was located at about 90 m m above the bottom nut at the bottom end of the steel rod, and the bottom clamp was only loosely tightened so that it could slip once the frictional resistance of its interface with the steel rod was overcome. The nut was provided to stop the sliding bottom clamp. The frictional resistance on the interface of the steel clamp with the steel rod could only be controlled qualitatively, since the clamping force could not be measured. Three steel rods, namely rod Nos S1, $2 and $3 ("S" represents slip tests) were tested. For subsequent drops with the same steel rod, the bottom steel clamp was often loosened and moved upwards to its initial position if a large amount of slip occurred in the previous drop (note: the clamping force may have changed). For each weight drop, the slip distance of the bottom clamp was directly measured and the strain wave in the steel rod recorded using a strain gauge installed immediately under the top steel clamp.

5.2.1. Experimental results Table 2 presents the measured slip distances of the bottom steel clamp and the plastic strains in the steel rods from all slip tests. Figure 16(a)-(d) presents typical strain waves from slip tests. The elastic strain limit of a steel rod is about 5 milli-strains (Fig. 12). In Table 2, the frictional resistances of the bottom clamp are classified as "high" or "low" depending on whether they are higher or lower than the elastic limit of the steel rod. The smallest plastic strain in the steel rod was achieved with test No. $3-3 (the digit after the hyphen indicates the 3rd drop with steel rod No. $3), where the frictional resistance of the bottom clamp was lower than the elastic limit

YI and K A I S E R :

IMPACT TESTING ROCKBOLT

(a)

DESIGN

681

(b)

16

Strain wave for

Strain wave for drop No. SI-4 12 10

g

8

.9 5 4

5

10

15

20

25

30

35

L 5

0

Time (milli-second)

I 10

I 15

I 20

I 25

30

Time (milli-second)

(c)

6

(d)

Strain wave for drop No. $2-2

Strain w.ave _for

~ ' ~

6

4

2

I 5

1 10

I 15

I 20

I 25

I 30

-2 35

Time (milli-seeond)

0

[ 5

I 10

I 15

I 20

t 25

I 30

35

Time (milli-second)

Fig. 16. Typical strain waves from slip tests (drop N o s SI-4, SI-5, S2-2 and S3-3).

of the steel rod and the slip distance was 83 mm (large). Only in this test, the bottom clamp reached the bottom nut and was stopped [Fig. 16(d)]. In all other tests, the bottom clamp stopped by itself before reaching the nut. The strain waves demonstrate that a dynamic slip process can generally be described as a stick-slip phenomenon, but the slip process for each drop was different. A slip process may consist of one large slip or numerous small sudden slips which appear to be semicontinuous (test No. $3-3). A general quantitative model for the strain-time relationships could not be established. The strain waves show that one part of the dynamic energy is consumed during slip (in the forms of surface friction and steel deformation) and the rest in the additional stretching of the steel rod once the bottom clamp either stops by itself or is stopped by the bottom nut. The following general observations can be made: (i) the plastic deformation of a steel rod is less if slip occurs (Table 2), (ii) energy is consumed in both forms of friction and plastic deformation during slip if the strain amplitude is slightly higher than the elastic limit for the steel rod (test No. S2-2) and (iii) energy is consumed only in the form of friction and the slip mechanism is most effective if the strain amplitude during slip is much less than the elastic limit (test No. $3-3).

5.2.2. Analysis of test No. $3-3 with low slip load and large slip distance For test No. $3-3, the peaks of the strain waves in the stick-slip process are more or less constant and are much lower than the elastic limit of about five milli-strains for the steel rod. An equivalent constant strain can be obtained by dividing the area under the strain wave (168.4 milli-strain msec) by the time period of the slip process (15 msec). It has a value of 0.79(1 _ 0.06) millistrains (a standard deviation of 0.05 milli-strain is assumed for its determination), about 16% of the elastic limit. This strain can be converted to an equivalent constant load which can then be multiplied by the slip distance to obtain the frictional energy dissipated during the slip process. This slip mechanism is analogous to a cushion which yields, and the following equation can be derived from energy balance analysis for determining the plastic deformation d which the steel rod sustains after the slip is arrested [7]:

d = MV~ 2Fy

ry 2k

Vc Fy

(6)

where Uc is the frictional energy dissipated during slip. Since a steel rod has an elastic modulus of about

692

YI and KAISER:

IMPACT TESTING ROCKBOLT DESIGN

206 GPa and a cross-sectional area of 31.65 mm 2, the equivalent constant frictional force for test No. $3-3 is 5151(1+0.06) N and the dissipated energy U~= 427.4(1 + 0.06)J. From equation (6), the plastic deformation d = 3.7 _+ 1.2 (mm) (the standard deviation of d is determined using the Taylor Series Approximation Method mentioned previously) and the corresponding plastic strain is 1.6 _+ 0.5 milli-strains. This agrees very well with the experimental plastic strain of 1.10 millistrains (Table 2). Therefore, equation (6) can be applied to situations where the frictional force during slip is below the elastic limit for the steel rod. For test No. $2-2 (second drop with steel rod No. $2), the peaks of the strain waves in the stick-slip process are more or less constant but they are slightly higher than the elastic limit of about 5 milli-strains for the steel rod. Strictly speaking equation (6) is not applicable because the slip process occurred while the steel rod underwentsome plastic deformation. In order to verify this, the same type of calculation as presented above was performed. The equivalent constant strain during slip is about 65% of the elastic limit, and the plastic deformation is found to be - 4 6 . 7 _+ 2.3 (ram) with the negative value indicating that there should have been no plastic deformation in the steel rod for the drop. On the contrary, a plastic strain of 2.03 milli-strains was measured for the drop. This plastic strain did not occur after the slip was arrested, but it occurred during the slip process. 5.2.3. Comparison with tests with cushions

No slip occurred for test Nos S1-2 and S1-3, and the plastic strain was 9.32 milli-strains per drop in both tests (Table 2). This magnitude of plastic strain is comparable to those obtained in the tests without cushions [Fig. l l(a)]. The least plastic strain of 5 6 milli-strains was achieved in the tests with a 108 mm thick rubber cushion. But, the plastic strains for slip test Nos S1-4, $2-2, $3-2 and $3-3 were all less. Therefore, if the slip distance is from medium to large, the slip mechanism is more effective than a 108 mm thick rubber cushion in reducing the steel rod plastic deformation. The least plastic deformations were obtained with the large slip distances and with either low or high frictional resistances. Therefore, a large slip distance must be ensured in order for a slip mechanism to be very effective.

are the only two parameters considered lbr a rockbolt, the strain hardening portion of the stress-strain curve of the rockbolt is essentially ignored, i.e. the failure criterion is based on stress and ultimate strength. It is implied that if the ultimate strength is exceeded, a rockbolt undergoes plastic deformation and then breaks. In rockburst conditions, however, a rockbolt can be subject to both static and dynamic loads. In the case of dynamic loading, the proximity to failure can only be determined if the entire load~teformation behavior is known and the failure criterion must be based on the accumulated plastic strain in the bolt and the ultimate strain limit of steel (Part II). A rockbolt may undergo certain plastic deformation without breaking if the imparted energy is not sufficient to cause complete rupture. In certain applications such as permanent underground installations, a rockbolt may not be allowed to sustain any plastic deformation, and must be designed to function in the elastic range. The design method is similar to the conventional static design in that the yield criterion can be based on elastic stress and yield point (Part I). In other applications such as underground mining, rockbolts may be permitted to undergo substantial plastic deformation without rupture during a prescribed access time period. In other words, rockbolts may be designed to function in the plastic range so as to utilize the energy absorbing capability of the steel. In this type of bolt design, the plastic strain parameter must be employed as the controlling parameter in the failure criterion (Part II). The maximum stress in an elastic design and the plastic strain in a plastic design can be determined employing the energy balance approach in which the sum of imparted energies of an ejected rock block equals the sum of consumed energies. The imparted energies include the kinetic and the gravitational potential energies of the rock block. The consumed energies are the energies dissipated in the friction between the surfaces of the ejected rock block and the surrounding rockmass, absorbed by deformations (elastic and plastic) of the rockbolt and its cushion, and dissipated in the friction at the interface of rockbolt and other materials (borehole wall, grout, yielding nut, etc.). Ignoring the friction between the ejected rock block and the surrounding rock for simplicity and conservative design, the following expressions for the maximum load Fm (elastic design) were derived for elastic and yielding cushions (or slip mechanisms) respectively [7]:

6. DESIGN EXAMPLES

Fm= M g +

6.1. Design Methodology

Both empirical and analytical methods have been used for the design of rockbolts under static loading conditions. As described by Hock and Brown [12], empirical methods include the simple rockbolt design method by Lang T. A. and the more sophisticated support design method developed at the Norwegian Geotechnical Institute, and analytical methods include wedge analysis and rock-support interaction analysis. In these analytical methods, the Young's modulus and the ultimate strength

?

,

Mg)2 + 2 k E (K + l )

and F m = M g + x / ( M g ) 2 + 2 k E - 2k (F~.y- Mg)dc

(7)

where E is the kinetic energy of the ejected rock block, and FCy and de are dynamic yield load and ultimate yielding displacement of a yielding cushion or slip mechanism. The first expression is similar to equations (2) except that the gravitational effect is added. Similarly, the following equations for determining the plastic

YI and KAISER: IMPACT TESTING ROCKBOLT DESIGN deformation of steel, d, (plastic design) were derived for elastic and yielding cushions (or slip mechanisms) respectively:

=

E E

Fy(K + 1) , 2k (Fy - 2Mg)

1

d 2 + (Fy - Mg)d f

t

where Fy is the steel yield load during strain-hardening, whose value depends on the plastic strain history. If the rockbolt is in an elastic state before dynamic loading is applied, F~ = Fy. The above equations are similar to equations (4) and (6), respectively, except that (i) the gravitational effect is added and (ii) the loaddeformation relation for steel is assumed to be bi-linear with slopes of k and k '. As seen from equation (8), for a given rock block size and velocity, support elements with either a small stiffness (k), a soft elastic cushion (low k~) or an energy dissipating yielding cushion or slip mechanism (indicated by Fcydc) sustain lower elastic stresses and plastic deformations. It must be born in mind that equations (7) and (8) are applicable only if an effective anchor (point, bond, or friction) and an effective face plate and nut exist, i.e. there is an arresting mechanism. Otherwise, the rock block would fly off without causing much elastic stress or any plastic strain to the bolt, i.e. the mode of failure would not be the rupture of the bolt itself but the detachment of the ejected rock block with (in the case of anchor failure) or without (in the case of face plate and nut failure) the bolt. Thus, for frictional bolts, such as Split Sets and Swellex, the maximum load in the bolt is simply the frictional resistance. In order to apply the second formulae of these equations, cushion yielding or slip must occur while the steel is within the elastic limit. Further steel elongation (elastic and plastic) may then occur after the yielding or slip is arrested. A yielding mechanism with this characteristic was shown to be the most effective (Section 5.2). The first formula of equation (7) must always be used first to check if cushion yielding or slip occurs. If it does, the second formulae of equations (7) and (8) are then used. In elastic design, however, if the calculated F m (second equation) is less than the yield load of the cushion, the latter should be taken as the maximum load in the rockbolt. Negative values of d imply that the bolt has not sustained any plastic deformation. If equations (7) and (8) are applied to rockbolts in excavation walls, g must be set to zero. If pre-tensioning is applied to bolts or static loads exist before dynamic loading, the static elastic energies stored in the bolts (0.5F2/k, with F~ being the pre-tension load) or cushions (0.5F~/k¢) must be added to the kinetic energy E in the above equations [7].

683

6.2. Material Properties

Rockbolts produced in Canada are made of three types of steels, namely the regular strength (grade 30), the high strength (grade 55) and the extra high strength (grade 75) steels [13]. The densities of these three types of steel are essentially constant at about 7800 kgm -3, and the Young's moduli vary little between 190 and 210 GPa [14, 15]. For a grade 55 high strength steel bar, the static yield point and the ultimate strength are 380 and 585MPa, respectively, and the ultimate strain (tested on 200 mm long specimens) is 12% [13]. Generally, the yield point and the ultimate strength of steel increase with increasing strain rate, whereas the ultimate strain and the slope of the strain-hardening curve decreases with increasing strain rate [16]. The impact tests (Section 5.1) indicated that the dynamic yield point for the low-carbon steel rods at a strain rate of 1 s- ~was about 65% higher than the static yield point and about 20% higher than the static ultimate strength (obtained at a strain rate of about 2 x 10 -3 s-1). For a rockbolt elastically loaded by an ejected rock block, the average strain rate may be expressed as [7]: = 2v~(rrL)- ~. For example, with vi = 1 ms- l, the strain rate is 0.3 s ~ for an effective free length of 2 m. It is anticipated that the dynamic yield point is about 60% higher than the static yield point. Thus, it is estimated that a rockbolt has a dynamic yield point of 627 MPa, an ultimate strength of 780 MPa, an ultimate strain of 8%, a Young's modulus of 200 GPa and a density of 7800 kg m -3. The static stress-strain relations for the rubber and the soft wood cushions were described in Section 3. For the rubber cushion used in the test program, elastic modulus can be calculated from equation (1) and the terms of (K + 1) and (K' + 1) in equation (7) must be replaced with (K + 1)z~ and (K' + 1) 2m (m = 0.366), respectively [equations (3) and (5)]. We assume that the wood cushion has a Young's modulus of 200 MPa (the same as the static value, Section 3), a dynamic yield point of 15MPa (twice the static value, Section 4.3) and an ultimate strain of 20%. Typical rock density is 2600 kg m -3. 6.3. Examples Case No. 1: elastic design for mechanical bolts For a typical rockbolt design in Canadian mines, a mechanical bolt has a diameter of 16 mm and a length of 2 m. The metal bearing plate on the rock surface has an area of 100 x 100mm, which equals a potential cushion surface area. The yield and ultimate loads of the bolt shank is calculated to be 126 and 157 kN, and the yield load of the soft wood is 150 kN. The soft wood cannot yield before the steel does. Let us suppose that the bolt anchor is the weakest link in a mechanical bolt and slip occurs at 50% of the shank yield load or at 63 kN, the bolt is pre-tensioned to a load of 5 kN, and the bolt spacing is 1 × 1 m which is taken as the surface area of an ejected rock block. For a 0.2 m thick rock block, the dynamic factor of safety (dynamic

684

YI and KAISER: 2

IMPACT TESTING ROCKBOLT DESIGN

(a) l Rockbolt shank v~ ..................

1.5 f

i Ejected rock block ~

1

•

=

~///J//JJ'/J././/'./~ •"ll" Head plate Head nut Fig. 18. Illustration of mesh installed on a mechanical rockbolt with cushions.

0.5

t

I

I

50

100

150

200

more than 50 mm thickness, anchor slip can be avoided and the mechanical bolt would function below the elastic limit.

Cushion Thickness (mm)

(b) 25 20

r~

E .=

15 10

r~

.~

5

0

50

100

150

200

Slip Distance (mm) 120

(c)

100 m

80

E

N 6o '~ 4o ~

Mesh wires

C u shion "

0

,-

I

2nd cushion (optional)-~---___~'~'~'~'x~,

rL

Case No. 2: plastic design .]'or grout anchored bolts with a slip nut Rockbolts with a slip mechanism may be designed, based on the slip tests (Section 5.2), such that the equivalent constant frictional resistance or slip load equals about 50% of the bolt yield point or Fs = 0.5Fy = 63 kN. Other parameters being the same as for case No. 1, the thickness of the ejected rock block is increased from 0.2 to 1.0m and the bolt spacing is increased to 1.3 m in order to make the bolt deform plastically. The gravitational acceleration g is set to zero for excavation walls. Figure 17(b) demonstrates that, compared to the roof, the slip mechanism is much more effective for rockbolts in the walls. This implies that slip-mechanism based rockbolts or frictional bolts, such as Split Sets, are ideal for supporting excavation walls, but they may not be used alone for roof support. This conclusion agrees well with practical experience with frictional wall support in rockburst-prone ground. As a matter of fact, it can be seen from the second formulae of equations (7) and (8) that a slip mechanism is effective only if the slip load is higher than the gravitational load of the ejected rock block or E~ > Mg.

2o

-', " 0

50

=

~

1oo

-

= I:

150

=

200

Slip Distance (mm) Fig. 17. Rockbolt design examples for dynamic loading. (a) Shows the dynamic factor of safety versus cushion thickness plots in an elastic design of mechanical bolts for both cushions of a soft wood and a conveyor-belt rubber, (b) plots the plastic strain normalized to the ultmate strain of steel vs slip distance relations for the roof and walls of an excavation in the plastic design o f grout anchored bolts with a yielding nut and (c) is a plot of the plastic strain normalized to the ultimate strain of steel versus slip distance relation for a fully grouted bolt.

yield load divided by maximum dynamic load) for bolt anchor failure has been computed for both cushions of soft wood and conveyor-belt rubber at different thicknesses [Fig. 17(a)]. Without any cushion or with a soft wood cushion of up to 200 mm thickness, anchor slip would occur. However, with a rubber cushion of

Case No. 3: plastic design for fully grouted bolts The effective free length of a fully grouted bolt, such as grouted rebar or cable, can be very small. Supposing that a grouted rebar has an effective free length of 0.2 m with all other parameters the same as in case No. 2, Fig. 17(c) shows that if bond slip on the interface of steel and grout is not allowed to occur, the bolt would have ruptured. With only 20 mm of slip, the rupture can be avoided. From these three case studies, the following observations can be made: (i) rubber cushions may be used in an elastic design for reducing bolt stress; (ii) the energy absorbing capability of the rockbolt itself should be exploited, e.g. by employing increased free length or using better energy absorbing material, in a plastic design; (iii) in situations where complete opening collapse must be avoided but partial closure is acceptable, slip potentials for fully grouted bolts and cables must be

YI and KAISER: IMPACT TESTING ROCKBOLT DESIGN exploited, e.g. by employing smooth bars or utilizing debonding techniques; and (iv) frictional bolts are ideal for excavation walls and should be used together with other support elements with greater energy absorbing capabilities. 7. DISCUSSION Reduction o f mesh damage by use o f cushions and slip mechanisms

In mining applications, meshes are installed on the heads of rockbolts and are often in direct contact with the metal plates. After a rockburst, it is frequently observed that the mesh is torn off at the metal plate, the head plate is pushed off the rockbolt or premature bolt anchor failure occurs before the rockbolt itself fails. Figure 18 shows schematically the loading of a mechanical rockbolt in a bolt-mesh support system. Since the mesh wires, the head plate, the head nut and the rockbolt are mechanically connected in series, they bear equal loads. Because elastic stresses or plastic strains in rockbolts can be reduced by use of cushions and slip mechanisms, damage of the components in the support system can also be alleviated. Furthermore, cushions on the rock-mesh contact eliminate stress concentrations and improve load transfer. Ideally a second cushion should be placed before the mesh is installed as shown in the figure. However, this may not be economically warranted and is recommended only if the specific failure mode has been identified. Effects o f repeated seismic loading

I f a bolt is plastically strained during multiple impact loading, plastic strains accumulate and rupture of the bolt will eventually occur. Rockbolts, which experience static elastic or plastic deformations before impact loadings due to adjacent excavations and time-dependent rockmass behavior, are particularly susceptible to rupture in rockburst conditions. 8. CONCLUSIONS The following conclusions can be drawn from the experiments and analyses: (I) The elastic stress or plastic strain in a simulated rockbolt subject to impact loading can be determined employing the energy balance approach. While the stress can be used as the controlling parameter f~': rockbolt design in the elastic range, e.g. for permanent underground openings, a plastic strain criterion must be used if a plastic deformation is acceptable or desirable during the design life, e.g. for temporary mining excavations. By the plastic design method, the energy absorbing capability of a rockbolt itself can be exploited. (2) Mesh tearing-off and premature bolt anchor failure of mechanical bolts may be reduced by use of soft cushions placed between the mesh RMMS 3116~H

685

and the metal plate. Effective slip mechanisms must be designed for fully grouted bolts. Frictional bolts are ideal for excavation walls and should be used together with other support elements with greater energy absorbing capabilities for excavation roof. (3) If plastic strains accumulate as a result of repeated seismic loadings or of time-dependent static loading, additional support elements must be installed with time to ensure safety during the design life of a support system. Acknowledgements--Laboratory assistance by S. Maloney and G.

McDowell and financial support provided by the Natural Sciencesand Engineering Research Council (NSERC) of Canada are gratefully acknowledged. Accepted for publication I1 April 1994.

REFERENCES

I. Ortlepp W. D. Invited lecture: The design of support for the containment of rockburst damage in tunnels--an engineering approach. In Proceedings of the International Symposium on Rock Support (Edited by Kaiser P. K. and McCreath D. R.), pp. 593-609, 16-19 June, Sudbury, Canada. Balkema, Rotterdam (1992). 2. Ontario Ministry of Labor Interpretation 34a of Regulation 694. Occupational Health and Safety Act and Regulationsfor Mines and Mining Plants. Occupational Health and Safety Division, Toronto,

Canada (1990). 3. Yi X. Mechanisms of rockmass failure and prevention strategies in rockburst conditions. In Proceedings of the 3rd International Symposium on Rockbursts and Seismicity in Mines (Edited by Young P.), pp. 141-145, 16-18 August, Kingston, Canada. Balkema, Rotterdam (1993). 4. Hedley D. F. G. and Whitton N. Performance of bolting systems subject to rockbursts. In Symposium on Underground Support Systems (Edited by Udd J.), pp. 73-78, 19-21 September. Canadian Institute of Mining and Metallurgy (1983). 5. Wagner H. Support requirements for rockburst conditions. In Proceedings of the 1st International Symposium on Rockbursts and Seismicity in Mines (Edited by Gay N. C. and Wainwright E. H.),

pp. 209-218, Johannesburg, 1982. SAIMM (1984). 6. HedleyD. G. F. Rockburst Handbook for Ontario Hardrock Mines. CANMET Special Report SP92-1E, Canada Communication Group-Publishing, Ottawa, Canada (1992). 7. Yi X. Dynamic response and design of support elements in rockburst conditions. Ph.D. thesis, Department of Mining, Queen's University, Kingston, Ontario, Canada (1993). 8. Yi X. and Kaiser P. K, Elastic stress waves in rockbolts subject to impact loading. Int. J. Numer. Analyt. Meth. Geomech. 18, 121-133. (1994). 9. Brady B. G. H. and Brown E. T. Rock Mechanicsfor Underground Mining. Allen & Unwin, London (1985). 10. Hart M. E. Reliability-based Design in Civil Engineering. McGraw-Hill, New York (1987). 1I. Box G. E. P., Hunter W. G. and Hunter J. S. Statistics for Experiments. Wiley, New York (1978). 12. Hoek E. and Brown E. T. Underground Excavations in Rock. The Institution of Mining and Metallurgy. Austin, London (1982). 13. CSA (Canadian Standards Association). Roof and rock bolts, and accessories. CAN/CSA-M430-90, Rexdale, Toronto, Ontario, Canada (1990). 14. ASM (American Society for Metals). Metals Handbook, 9th Edn, Vol. 1. Metals Park, OH (1978). 15. GereJ. M. and Timoshenko S. P. Mechanics of Materials, 3rd Edn. PWS-KENT, Boston, MA (1990). 16. ASM (American Society for Metals). Metals Handbook, 9th Edn, Vol. 8. Metals Park, OH (1985).