- Email: [email protected]
Influence of geometry and material properties on the axial vibration of a rock bolt
ARTICLE IN PRESS
International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951 www.elsevier.com/locate/ijrmms
Influence of geometry and material properties on the axial vibration of a rock bolt Ana Ivanovic´, Richard D. Neilson School of Engineering, College of Physical Sciences, King’s College, Fraser Noble Building, University of Aberdeen, AB24 3UE, UK Received 17 July 2007; received in revised form 18 October 2007; accepted 20 October 2007 Available online 4 December 2007
Abstract A continuous dynamic model for the axial vibration of a rock bolt system is presented. The model comprises three sections: the fixed length, bonded into the rock, the free length, which is not coupled to the rock, and the protruding length, which extends beyond the rock. The head assembly is modelled as a discrete mass and a spring, and a further discrete mass is included, representing a testing device that can be attached to the protruding end. Each section is modelled as a continuous elastic rod governed by the wave equation, with suitable compatibility conditions applied between the sections and boundary conditions, which also account for the effect of the discrete components, applied at the ends. Solutions in non-dimensional form are substituted into the boundary conditions to allow the natural frequencies to be calculated, and it is shown that two possible solutions for the mode shapes can be used for the fixed length—an exponential solution or the classical sinusoidal solution—depending on the stiffness of the grout relative to that of the bar. The conditions for which the two solutions are valid are developed, and changes in the frequency ratio with changes in length ratio, and the stiffness ratios of the grout and the anchor head relative to the stiffness of the fixed length of the anchorage are examined. Generally, the state of a bolt after installation is unknown and this does not provide proper assurance of the safety of the structure for which the bolts are used. The model provides a viable tool for helping to assess the condition of the bolt by using the natural frequencies associated with areas of the bolt of particular interest, e.g. the free length. The results show how the changes in the stiffness and/or length ratios affect the dynamics associated with fixed length of the bolt and the quality of the bonding installation. A case study is presented showing how the model can be used effectively to interpret real data. r 2007 Elsevier Ltd. All rights reserved. Keywords: Rock bolts; Vibration; Numerical modelling and analysis
1. Introduction Millions of rock bolts are installed worldwide as the main support for structures such as tunnels and mines. The main features of a rock bolt system, as shown in Fig. 1, are the protruding, free and fixed anchor lengths, denoted as subsystems I, III and IV, respectively, the anchor head assembly, denoted as subsystem II, and the interfaces between the tendon and grout and grout and surrounding rock mass, denoted as subsystems V and VI, respectively. The installation of most rock bolts is usually achieved by first drilling a borehole. A cable or bar is then bonded Corresponding author. Tel.: +44 1224 273 265; fax: +44 1224 272 497.
E-mail address: [email protected] (A. Ivanovic´). 1365-1609/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2007.10.003
along part of its length into the borehole using either resin or cement grout. Thereafter, the installation procedure for both cases continues by placing a bearing plate and nut, in the case of a rock bolt, or barrel and wedges, in the case of a cable bolt, at the top of the borehole, and finishes by loading the tendon up to the required level by either tightening the nut or using a stressing jack. Research undertaken by Benmokrane et al. [1] suggests that the level of load in rock bolts or ground anchorages is very likely to change during their lifetime from that initially applied. Furthermore, changes in the free length are shown to be very common at early stages of loading due to cracking of the grout at the top end of the fixed length, i.e. at the boundary between fixed and free length [2–4]. This results in changes in the free fixed length ratio, which may
ARTICLE IN PRESS 942
A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
VI III
V
II
IV
Fig. 1. Rock bolt system.
influence greatly the static and dynamic response of the bolt. In addition, in mining applications lateral ground movement can sever an anchorage and therefore reduce its fixed length and load capacity and therefore its effectiveness as a support. Since many rock bolts are installed in safety critical applications, a method of assessing all of these aspects is needed if long term monitoring of such bolts is to be achieved. To this end a non-destructive testing technique, GRound ANchorage Integrity Testing (GRANIT) has been developed, which comprises an impact device, attached to the free end of the rock bolt by which an axial impulse of a low magnitude is applied. The resulting axial vibration is captured by an accelerometer attached to the impact device and is then further processed using an artificial neural network, the details of which can be found in [5]. Although lateral or bending vibrations could also be examined, the use of near full encapsulation in mines and secondary grouting for corrosion protection in tunnels limits the freedom of the bolt to bend whereas axial vibration is possible. Alongside the development of the non-destructive testing system, a lumped parameter dynamic model simulating the axial vibration of a rock bolt system was developed by one of the authors [6], in order to better understand what influences the static and dynamic response of bolts and also to validate the non-destructive technique. Validation of the lumped parameter model was achieved by comparing the results obtained from the model with those obtained from experiments undertaken on a laboratory rock bolt anchorage to an accuracy of 3% [7]. Investigation of the distribution of the load along the fixed anchor length was also undertaken with the model and the results obtained, which showed an exponential distribution, were in agreement with previous theoretical [8–11] and laboratory studies [12–15]. Unlike previous models [16] this lumped parameter model is able to replicate the effect of prestress load in the rock bolt when the anchor head is modelled and incorporated in
the model [17]. The model improved the performance of the GRANIT system and is currently used successfully as part of it. However, in order to undertake a parametric study, the lumped parameter model would need to be run for a large number of times for different geometries with each parameter changed separately. In addition, the number of modes, which can be identified, is determined by the number of masses included in the model and the accuracy of representation of the higher modes and their associated natural frequencies is limited. If the system is transformed into a non-dimensional form then the number of parameters used in the analysis is reduced. This can be achieved more conveniently, with a continuous system than with the lumped parameter system. In addition, the accuracy of representation of the higher modes and frequencies is then not limited. The system can be presented in a generic form using ratios of parameters and then solved. These nondimensional ratios provide scaling laws [18,19]. As a consequence, rock bolts, which have the same parameter ratios, will exhibit the same dynamic characteristics even if the actual values of the parameters are substantially different (e.g. bolts with the same free/fixed length ratio will respond in a similar way). This allows the general response of the rock bolt system to be explored conveniently. In this paper such a continuous dynamic model of a rock bolt system for axial vibration is presented and appropriate solutions for the natural frequencies and mode shapes developed. The aim is to identify modes that provide information on the free length and load and whether any modes can be excited, which provide information about the fixed length, with a view to ascertaining either the bond quality or the length of the bonded section. This information can then be used in interpreting the vibration data gathered from rockbolts in the field to provide insights into their condition. A case study in which the model is applied to data from a mine is included at the end of this paper.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
2. Continuous model
u(x,t) ¼ U(x)T(t), are given by
A continuous model was developed in which the bar is represented as an elastic continuum, described by the wave equation, with additional discrete elements used to represent the anchor head and the GRANIT impact device as shown in Fig. 2. M1 and M2 in Fig. 2 represent the mass of the impact device and mass of the head assembly, respectively, K represents the linearised head stiffness and k, the stiffness per unit length of the grout which was obtained using thick plate theory; the details of this can be found in [6]. The findings from a parametric study undertaken previously by the authors indicate that with an increase of the Young’s modulus of the grout or a decrease of the borehole the shear stiffness of the grout annulus is greater [6]. Although the model described in this paper contains both continuous elastic and discrete elements, it will be referred to as the continuous model in this paper to differentiate it from the previous lumped parameter model.
x¼0:
2.1. Boundary conditions for the model The continuous model was developed with boundary and compatibility conditions, which allow for the dynamic effects of the mass of the testing device, the mass and stiffness of the anchor head and the grout stiffness. The rock stiffness is assumed to be infinite and the steel bar is assumed to be made of a homogeneous, isotropic and linear elastic material. The assumption of infinite rock stiffness is made as a first approximation based on the fact that the rock stiffness is high relative to the grout stiffness. The grout stiffness therefore controls the dynamic response. Although damping is a major concern in some similar dynamic systems, e.g. the driving of piles [20], it will be shown in this study that for many cases of interest related to the detection of load, most of the vibrational energy is concentrated in the free length and so damping has been neglected in this paper. The effect of this will be the subject of future investigations. For the continuous model shown in Fig. 2, the boundary and compatibility conditions, after separation of variable
AE
dU 1 ðxÞ d2 TðtÞ TðtÞ ¼ M 1 U 1, dx dt2
943
(1)
dU 1 ðxÞ dU 2 ðxÞ TðtÞ KU 1 ðxÞTðtÞ þ AE TðtÞ dx dx d2 TðtÞ ¼ M2 U 1, ð2Þ dt2
x ¼ l 1 : AE
x ¼ l 1 : U 1 ðxÞ ¼ U 2 ðxÞ,
(3)
x ¼ l1 þ l2 :
dU 2 ðxÞ dU 3 ðxÞ ¼ , dx dx
(4)
x ¼ l1 þ l2 :
U 2 ðxÞ ¼ U 3 ðxÞ,
(5)
dU 3 ðxÞ ¼ 0, (6) dx where x is the position along the bolt, measured from the protruding length, A is the cross-sectional area of the bar, E is Young’s modulus of the steel bar, U(x) is the mode shape of the length and T(t) is the principal coordinate. The subscripts denote the section of the bolt being modelled with subscripts 1, 2 and 3 denoting elements relating to the protruding, free and fixed lengths, respectively. x ¼ l1 þ l2 þ l3 :
2.2. Wave equation for prismatic bars Separate solutions for the axial vibration of the system can be written for each section of the bolt. Although the wave equation can be shown to apply for each section, there are two different conditions along the bar that need to be considered. The first case is where the bar is unrestrained, the condition which applies along the protruding and free lengths. The second case is where the bar is elastically restrained by the grout which is the case along the fixed length. 2.2.1. Unrestrained bar For the free and protruding lengths, the classical wave equation q2 u 1 q2 u ¼ qx2 c2 qt2
Fig. 2. Continuous model.
(7)
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
944
holds, where u is the longitudinal displacement of the element at axial position x. The solution uðx; tÞ is then given as UðxÞTðtÞ where T(t) is the solution of the free vibration problem given as TðtÞ ¼ A cos ot þ B sin ot and the mode shapes U(x) are defined as UðxÞ ¼ C cosðox=cÞ þ D sinðox=cÞ. The constants A* and B* depend on the initial conditions, C and D depend on the boundary conditions the ends of the bar, o is an angular pat ffiffiffiffiffiffiffiffiffi frequency and c ¼ E=r is the wave speed, where r is the density of the steel bar. 2.2.2. Restrained bar When an elastically restrained bar is considered, the system is described by the following wave equation, in which k is the stiffness per unit length of the grout: 2
2
q u 1q u k u. ¼ 2 2þ 2 qx c qt EA
(8)
After separation of variables, this produces 2 q2 U o k þ U ¼ 0. qx2 c2 EA
(9)
For ðo=cÞ2 4k=EA, the roots of the frequency equation are a complex conjugate pair, and the solution is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 k o2 k x þ D sin x, (10) UðxÞ ¼ C cos 2 2 EA EA c c whereas for ðo=cÞ2 ok=EA, real roots are obtained and the solution becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 (11) UðxÞ ¼ C 3 e k=EAðo =c Þx þ D3 e k=EAðo =c Þx . These wave models and solutions were used in the derivation of the continuous system. The solutions U1, U2 and U3, with the corresponding constants C1, D1, C2, D2, C3 and D3, respectively, are taken in the form shown in Eqs. (12), (13), (14) or (15). The solution for T(t), with the corresponding constants A* and B*, is given in Eq. (16). U 1 ðxÞ ¼ C 1 cos U 2 ðxÞ ¼ C 2 cos
o o x þ D1 sin x, c c o o ðx l 1 Þ þ D2 sin ðx l 1 Þ, c c
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 k ðx l 1 l 2 Þ U 3 ðxÞ ¼ C 3 cos 2 EA c rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 k ðx l 1 l 2 Þ þ D3 sin 2 EA c
(12)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
k=EAðo =c Þðxl 1 l 2 Þ
TðtÞ ¼ A cos ot þ B sin ot.
(16)
2.3. Non-dimensional form of the continuous model In order to obtain the continuous model in non^ dimensional form, new time and length scales, t and x, are defined: t t ¼ on t ) t ¼ , (17) on x^ ¼
x ^ 3. ) x ¼ xl l3
(18)
The solutions are then written in the form ^ 3. ^ 3 ¼ TUl u ¼ ul
(19)
The length by which the other lengths are scaled is chosen to be the fixed length l3, since the influence of the stiffness of the grout compared to the stiffness of the whole system is of interest in this paper particularly for ascertaining if the quality of the bond is satisfactory. Similarly, the first axial natural frequency on of an elastically unrestrained bar of length l3, with free–free boundary conditions, as defined in Eq. (20), was used as the reference: pc on ¼ . (20) l3 This choice allows the examination of the effect of grout stiffness, and also produces convenient expressions. A frequency ratio, o r¼ , (21) on can then be defined, and the solutions written in the form ^ ¼ C 1 cos rpx^ þ D1 sin rpx, ^ U 1 ðxÞ (22) ^ ¼ C 2 cos rpðx^ l^1 Þ þ D2 sin rpðx^ l^1 Þ U 2 ðxÞ and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 p2 k^ðx^ l^1 l^2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ D3 sin r2 p2 k^ðx^ l^1 l^2 Þ
(23)
^ ¼ C 3 cos U 3 ðxÞ (13) or ^ ¼ C3e U 3 ðxÞ ð14Þ
or U 3 ðxÞ ¼ C 3 e
and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þ D3 e k=EAðo =c Þðxl 1 l 2 Þ (15)
pffiffiffiffiffiffiffiffiffiffiffi
^ 2 p2 ðx ^ l^1 l^2 Þ kr
and ^ ¼ A cos rt þ B sin rt, TðtÞ
pffiffiffiffiffiffiffiffiffiffiffi ^ 2 2 ^ l^1 l^2 Þ þ D3 e kr p ðx
ð24Þ
(25)
(26)
where l^1 ¼ l 1 =l 3 and l^2 ¼ l 2 =l 3 . After substituting Eqs. (17)–(26) into non-dimensional forms of the boundary conditions Eqs. (1)–(6), the following
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
945
set of equations is obtained: 2
^ 1 r2 p2 m
6 6 rp sin rpl^1 K^ cos rpl^1 6 6 6 ^ 2 r2 p2 cos rpl^1 þm 6 6 6 cos rpl^1 6 6 6 6 0 6 6 6 0 4 2
C1
3
0
rp
0
0
0
0
rp
0
sin rpl^1
1
0
0
0
rp sin rpl^2
rp cos rpl^2
0
0
cos rpl^2
sin rpl^2
0
0
0
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 p2 k^ sin r2 p2 k^
rp cos rpl^1 K^ sin rpl^1 ^ 2 r2 p2 sin rpl^1 þm
0
7 7 7 7 7 7 7 7 0 7 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 r2 p2 k^ 7 7 7 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 r2 p2 k^ cos r2 p2 k^ 0
7 6 6 D1 7 7 6 7 6 6 C2 7 7 6 6 7 ¼ 0. 6 D2 7 7 6 7 6 6 C3 7 5 4 D3
ð27Þ
The matrix for the whole system, when the exponential solution is used, is 2 ^ 1 r2 p2 m rp 0 0 6 6 rp sin rp l 1 K^ cos rpl^1 rp cos rpl^1 K^ sin rpl^1 l3 6 6 0 rp 6 2 ^ 2 r2 p2 sin rpl^1 þm ^ 2 r p2 cos rpl^1 þm 6 6 6 6 cos rpl^1 sin rpl^1 1 0 6 6 6 0 0 rp sin rpl^2 rp cos rpl^2 6 6 6 0 0 cos rpl^2 sin rpl^2 6 4 0 0 0 0 3 2 C1 7 6 6 D1 7 7 6 7 6 6 C2 7 7 6 6 7 ¼ 0, 6 D2 7 7 6 7 6 6 C3 7 5 4 D3
^ K, ^ m ^ 1 and m ^ 2 are non-dimensional parameters, where k, defining the ratio of the stiffness of the grout to the axial stiffness of the bar, the stiffness of the head to the axial stiffness of the bar, the mass of the impact device to the mass of the bar and the mass of the head to the mass of the bar, respectively, and are given by k^ ¼ kl 23 =AE, ^ 1 ¼ M 1 =Al 3 r and m ^ 2 ¼ M 2 =Al 3 r. K^ ¼ Kl 3 =AE, m 2.4. Discussion As shown in Eqs. (24) and (25), there are two solutions for U3, which depend on whether the expression under the square root in Eq. (25) is positive or negative. The choice of
3
0 0 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k^ r2 p2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ^ 2 2 k^ r2 p2 e kr p
0
3
7 7 7 7 0 7 7 7 7 7 0 7 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 2 2 ^ kr p 7 7 7 1 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 5 2 2 ^ k^ r2 p2 e kr p
ð28Þ
the appropriate solution depends on the relative size of the ^ If the stiffness ratio k^ is large relative to terms r2p2 and k. r2p2, then the exponential solution is valid. If k^ is small then the sinusoidal solution is valid, a result underpinned by the case k^ ! 0, which is effectively an unrestrained bar. The value of the stiffness ratio k^ is governed by the stiffness of the fixed length of the bar and the grout. However, it would be beneficial to know which solution is more appropriate for a particular geometry and material properties of the grout before trying to find the natural frequency of the system and therefore any conclusion regarding the grout condition. This will be addressed later in the paper.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
946
3. Validation of the continuous model 3.1. Exponential solution In order to validate the model with the exponential solution, the results obtained from the continuous system are compared with the results obtained from the lumped parameter model, two simple analytical models and laboratory experiments. The properties of the laboratory rock bolt rig are described in Tables 1 and 2 and were used in the models. The rock bolt used in the experiments was built initially for a feasibility study of the GRANIT system in the late 1990s. From Table 1 it can be seen that the protruding length is not typical of mining or tunnelling practice and the bore hole of this bolt of a somewhat larger diameter than usually found. However, the experiments allow for the general concepts of the model to be validated. The two simple analytical models are based on a bar with one end fixed and the other free, and a bar with one end fixed and a mass attached at the other end. In the following discussion, these analytical models are denoted as case a for the plain bar with the total length comprising the
protruding and free lengths and case b for the plain bar with a mass attached at its free end and with its total length comprising the protruding and free lengths. Case b models the case when the impact device of the GRANIT system is attached to a rock bolt. The results are presented in terms of both the frequency ratio and actual frequency. From Table 3 it can be seen that all the frequency ratios, r are in very good agreement. The frequency ratio obtained from the simple analytical model for case a is 0.184 while the value of r obtained from the lumped parameter model is 0.180. The frequency ratio obtained from the continuous system is similar to the frequency ratio obtained from the lumped parameter model and the experiments. The lower frequency ratios obtained from the numerical model and the continuous model compared to the simple analytical model are due to the fact that, in both cases, the fixed length has some flexibility, i.e. they are not completely constrained. Similar results are found for case b where the mass of the impact device is attached at the free end of the bar. The results show excellent agreement between the values obtained from the previously validated lumped parameter and the continuous model. 3.2. Sinusoidal solution
Table 1 Geometry of the experimental rock bolt rig
Validation of the sinusoidal solution was undertaken by comparing the continuous model with the lumped
Rock bolt Bolt diameter (mm) Bore hole diameter (mm) Protruding length (mm) Free length (mm) Fixed length (mm)
25 45 1140 2780 1440
Table 4 First and second frequency ratio obtained using exponential and sinusoidal solutions, respectively
Table 2 Properties obtained from the laboratory experiments
Solution used
Lumped parameter model [Hz]
Continuous model [Hz]
exp
0.121 [218.0] 0.343 [615.9]
0.123 [220.5] 0.347 [622.46]
sin 1825 12.95 0.383 0.04
Steel Density (kg/m3) Young’s modulus (GPa) Poisson’s ratio Damping ratio
7895 210 0.3 0.03
1 first mode (l.p.m) vs exp solution (c.m)
0.5
Table 3 Frequency ratios of the first natural frequency [first natural frequencies] of the rock bolt using different models Case Lumped parameter model [Hz]
Simple analytical Continuous model [Hz] system [Hz]
Experimental data [Hz]
a
0.184 [329.9] 0.113 [202.2]
0.185 [332] 0.109 [195.7]
b
0.180 [322.5] 0.110 [197.7]
Amplitude
Resin Density (kg/m3) Young’s modulus (GPa) Poisson’s ratio Damping ratio
0
-0.5
second mode (l.p.m.) vs sin solution (c.m.)
l.p.m. - first mode c.m. - exp. solution l.p.m. - second mode c.m. - sin. solution
-1
-1.5 0.182 [326.1] 0.112 [200.1]
0
0.5
1
1.5
2
2.5
3
3.5
4
Length ratio Fig. 3. First and second mode shapes obtained from the lumped parameter model (lpm) and continuous model (cm).
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
parameter model with lower values for the stiffness of the grout. The geometry used is the same as before but the stiffness of the grout was reduced from the original value of 12.9 GPa to 12 MPa to represent a weakly bonded bolt with a poor quality of grout. As shown in Table 4 and Fig. 3, the first and second frequencies and the corresponding mode shapes from the continuous model, respectively, are in good agreement with the lumped parameter model. The first mode was obtained using the exponential solution while the second mode was obtained when the sinusoidal solution was used. The data used to validate the model are only one particular case with a particular free/fixed length ratio and grout stiffness. The length ratio l^2 , however, may be substantially smaller than examined so far, for example in rock bolts used in the mining industry [21,22] or the stiffness of the grout may be lower than that used in the validation. The following section examines the conditions under which the sinusoidal (Eq. (24)) and exponential (Eq. (25)) solutions, for the fixed anchor length, hold. 4. Analysis and discussion The frequency ratios, r, were found by expanding the determinants of the matrices in Eqs. (27) and (28) and then finding the roots. The corresponding mode shapes were then calculated and examined. As an initial analysis, the frequency ratios for a range of different grout stiffness ratios k^ were calculated in order to establish the point where transition between the exponential and sinusoidal frequencies occurs. In order to observe the influence of changes between the free and fixed lengths only, all the elements, apart from length ratios l^1 and l^2 and the stiffness ^1 ¼ m ^ 2 ¼ K^ ¼ 0 in the ratio k^ were set to zero, i.e. m matrices presented in Eqs. (27) and (28). Finally, the influence of the anchor head stiffness was examined
9
4.1. Effect of grout stiffness ratio k^ The solutions obtained for l^ ¼ l^1 þ l^2 ¼ 0:5 and a range ^ of grout stiffness ratio 0oko1000 are shown in Fig. 4. ^ This spans a range of k which includes typical resin bonded mine rock bolts down to very weak bonds. The value k^ 875 corresponds to perfect bonding using a grout with high Young’s modulus of 12.9 GPa. The figure shows that as the grout stiffness decreases, the number of modes determined with the exponential solution in the fixed length also decreases. For lower stiffness ratios more sinusoidal modes are present and these have smaller frequency spacings between them than the exponential modes. For higher stiffness ratios, the stiffness has generally little effect on the frequency ratios of the exponential modes. However, very close to the boundary, reduction in the stiffness ratio does result in a slight drop in frequency ratio for the exponential modes. As discussed before, the boundary between the appropriate solutions is given by k^ r2 p2 ¼ 0.
(29) ^ This boundary line relating k and r is plotted in Fig. 4. If the boundary for a rock bolt with a particular length is required, a bar of length corresponding to the protruding and free length part of the whole system can be considered, as shown in Fig. 5. The natural frequencies of the system represented in the figure are ð2i 1Þpc ; i ¼ 1; 2; . . . ; 1. (30) 2l The frequency ratios, with respect to the frequency of the fixed length, l3, of this system can then be obtained oi ¼
pcð2i 1Þl 3 ð2i 1Þ 1 . ¼ 2 pc2l l^
7
2 2
6 5 4
Exponential
3
ð2i 1Þ p k^i ¼ . (32) 4l^ This relationship defines the critical value of k^i which occurs at the boundary between the exponential and sinusoidal solutions for any mode i and length ratio l^
2 1 0 0
(31)
When the frequency ratio, developed in Eq. (31), is introduced into Eq. (29), the stiffness ratio as a function of l^ can be obtained as follows:
Sinusoidal
8 Frequency Ratio r
by varying the head stiffness ratio k^ while keeping the length ratios l^1 and l^2 constant.
ri ¼
10
947
100 200 300 400 500 600 700 800 900 1000 Grout Stiffness Ratio
Fig. 4. Frequency ratio r vs. grout stiffness ratio k^ for all possible solutions of the continuous system for l^ ¼ 0:5 (dotted line: exponential solution, solid line: sinusoidal solution, * approximate boundary).
Fig. 5. A bar representing free+protruding length.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
examined using Fig. 4. The ‘*’ in Fig. 4 denote points calculated using this relationship for l^ ¼ 0:5. As can be seen the points clearly define the transition curve. This means that if only the free length is known, the relationship may be used to establish which solution is appropriate for a ^ then the solution is particular case. If k^i 4ð2i 1Þ2 p2 =4l, below the transition line in Fig. 4, and the exponential solution holds. Although in field testing it is not possible to determine whether a mode is sinusoidal or exponential per se, if the sinusoidal modes can be excited and detected, then there should be a sudden change in the spacing between consecutive modes at the transition between the exponential and sinusoidal modes. The higher is the frequency ratio, at which this transition occurs, the better is the bonding of the fixed length to rock. If the sinusoidal modes cannot be excited, then instead of a change in frequency spacing between the consecutive modes, the frequency ratio at which no further modes can be detected will determine the transition point. Again, the higher this point, the better the bonding.
10 9
Sinusoidal
8 Frequency Ratio r
948
7 6 5 4 3
Exponential
2 1 0 0
0.2
0.4
0.8
0.6
1
1.2
1.4
1.6
1.8
2
l1 + l2 Length Ratio Fig. 7. Frequency ratio r vs. length ratio l^ for k^ ¼ 500 (dashed line: exponential solution, solid line: sinusoidal solution).
2.5
4.2. Effect of length ratio
2 free length fixed length
1.5 1 Amplitude
In order to examine the influence of the length ratios l^1 and l^2 on the frequency ratios, all the parameters in the ^ were set to zero. Changes in system, other than l^1 , l^2 and k, the l^ ¼ l^1 þ l^2 length ratio were made and all frequency ratios up to r ¼ 10 were calculated using both solutions. The length ratios were varied in the range 0–2, and the corresponding values of r are presented in Figs. 6 and 7 for values of k^ ¼ 50 and k^ ¼ 500, respectively. Figs. 6 and 7 confirm that the higher the stiffness of the grout along the fixed length, the more exponential solutions are obtained for a given length ratio. The higher length ratios result in more exponential modes, the
0.5 0 -0.5 1st mode (exp) 2nd mode (exp) 3rd mode (sin) 4th mode (sin)
-1 -1.5 -2 0
0.5
1
1.5
Position along the bar 10
Fig. 8. Mode shapes for free length ratio l^ of 0.5 and k^ ¼ 50 using both exponential and sinusoidal solutions.
9
Frequency Ratio r
8 7 Sinusoidal 6 5 4 3 2 Exponential
1 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
l1+l2 Length Ratio Fig. 6. Frequency ratio r vs. length ratio l^ for k^ ¼ 50 (dashed line: exponential solution, solid line: sinusoidal solution).
frequency spacing between which is smaller. For both the sinusoidal and exponential solutions there is a trend of increasing frequency ratio as the length ratio decreases although for the exponential solution this is more apparent than for the sinusoidal solution. Once the frequency ratios were calculated as described previously, the constants C1, D1, C2, D2, C3, D3 were found from the equations in matrix form allowing the mode shapes of the system to be generated. Figs. 8 and 9 show the mode shapes obtained for the first frequencies for l^ ¼ 0:5 and k^ ¼ 50 and the first eight frequencies for l^ ¼ 0:5 and k^ ¼ 500, respectively. The number of modes detected for each solution is the same as can be identified from Figs. 6 and 7 for the length ratio of 0.5.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951 15 solid line - exp dotted line - sin
12.5
Amplitude
10
free length
first sinusoidal mode
fixed length
7.5 5 2.5 0 -2.5
949
The findings from the analyses suggest that a device able to excite the higher frequencies would provide information on the state of the installed bolts and their fixed length. The current impact device used in the GRANIT system however has been tuned to provide information about the lower modes and therefore does not allow these higher frequencies to be excited or identified. This study therefore gives an encouragement for further development of the device so that the behaviour of rock bolts used in practice is better understood. 4.3. Effect of anchor head stiffness ratio
-5 1
1.5
Position along the bar
Fig. 9. Mode shape for a length ratio l^ of 0.5 and k^ ¼ 500 using both exponential and sinusoidal solution.
Examination of the first sinusoidal mode, for the higher stiffness ratio of k^ ¼ 500, reveals a higher amplitude in the fixed length compared to the free length. The amplitudes for higher modes then become more uniform along the anchorage. The trend of larger amplitudes in the fixed end for the sinusoidal modes is also apparent for the lower stiffness in Fig. 8, although this is less pronounced. The results confirm that there are two groups of frequency ratios—the lower ones defined by the exponential solution and dominated by motion in the free length and those defined by the sinusoidal solution and dominated by motion in the fixed length. The highest frequency exponential solution has a shape within the fixed length of the anchorage, which is close in form to that of the first sinusoidal solution. A number of practical outcomes can be inferred from the results. In the dynamic testing of an anchorage, it should be possible to estimate a number of parameters from the spacing between consecutive natural frequencies. Firstly, the modes below the transition are exponential modes and the spacing between two consecutive frequencies allows the free length to be assessed by following equation: l¼
pc oi oi1
or
l^ ¼
p . ri ri1
(33)
Secondly, the sinusoidal modes appear to be related to the total length of the anchorage with the spacing between the higher frequencies tending to that present between the frequencies of a free–free bar of total length l^ þ l^3 . This can be seen in Fig. 6, for k^ ¼ 50, where for the higher sinusoidal frequency ratios, at smaller length ratios, the spacing tends to a value of frequency ratio of 1, which is the increment between frequencies for the axial vibration of a free–free bar of length l3 ¼ 1. At l^ ¼ 0:5 the total length is 1.5 and the frequency ratio is about 0.7, which corresponds closely to the frequency ratio for a free–free bar of this length. For the stiffness ratio of k^ ¼ 500, this trend would be true for frequency ratios higher than those presented in Fig. 7.
The effect of load in a rock bolt has been shown to generally result in an increase in the stiffness of the anchor head assembly. This in turn results in an increase in natural frequency with increasing load [17]. To investigate what effect this would have on the different solutions, the model was run for a range of head stiffness ratios K^ with different ^ values of length ratios l^1 and l^2 , and stiffness ratio k. Fig. 10 shows an example of the rise in first natural frequency (an exponential mode) associated with the increase of head stiffness, which generally occurs for increasing load in the rock bolt. The figure also shows that the higher exponential modes are less affected by the increase in head stiffness ratio and in the case of the two highest frequency modes for this case (k^ ¼ 500) there is no appreciable change in frequency ratio while none of the sinusoidal modes are modified. These results provide a basis for determining the load in a rock bolt from the first natural frequency measured by dynamic testing. The fact that the higher frequencies do not change with variations in the head stiffness resulting from changes in load may provide a means of measuring the free (unbonded) length independently from the test data. This is important, as the first natural frequency is affected by both 10 9
Sinusoidal
8 Frequency Ratio r
0.5
0
7 6 Exponential
5 4 3 2 1 0 0
2
4
6
8
10
12
14
16
18
20
Head Stiffness Ratio
^ ratio for k^ ¼ 500, Fig. 10. Frequency ratio r vs. head stiffness ratio K, l^1 ¼ 0:1 and l^2 ¼ 0:4, dashed line: exponential solution, dotted line: sinusoidal solution
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
the free length and the load. If the higher exponential frequencies can be measured to determine the free length then this can be used along with the first natural frequency to estimate the actual load. This again suggests that a device capable of exciting higher frequencies would give more information on the condition of the rock bolt. 5. Case study With the exploration of various theoretical scenarios using the model completed the interpretation of data from both laboratory tests and field tests has been made viable. In 2001, dynamic test data were taken from a number of rockbolts in a UK coal mine using the GRANIT system. The data showed that most bolts, deemed as working effectively, had natural frequencies around 850–900 Hz. However, a number of bolts were identified with frequencies of 600 Hz or lower. HSE documentation for mining support systems recommends fully bonded bolts [21]. However in practice this is seldom achievable and some unbonded length occurs behind the head assembly. The practical design values for the geometry of the bolts tested were therefore l 1 ¼ 0:1 m, l 2 o0:3 m, l 3 42:0 m with a total length of 2.4 m, and a bolt diameter of 22 mm. This geometry provides a long fixed length to strengthen the strata and a short free length to ensure that the surface strata are well supported; the latter minimises the likelihood of shallow roof falls. Using the values noted above l^1 ¼ 0:05 was obtained and the continuous model was therefore run over the range 0:05ol^2 o0:95, i.e. 0,1 mol2o1.0 m. The mass of the testing device used, M1 was 8.5 kg which, using a density of 7850 kg/ ^ 1 ¼ 1:42. Values of m3 for the bolt, gives a mass ratio of m K^ ¼ 1 and k^ ¼ 500 were calculated as being representative of a ‘‘good’’ bolt, i.e. a bolt for which the grouting is strong and which has some load at the bolt head (typically about ^ assuming 80 kN). The theoretical maximum value of k, perfect bonding is 875 as noted in Section 4.1 and so 500 was taken as an acceptable achievable bond within a working mine environment. K^ ¼ 5 and k^ ¼ 500 were then used as being representative of a well bonded bolt with higher load at the upper end of the acceptable range. The other permutations shown in Fig. 11 were chosen as cases with either poor bond quality ðk^ ¼ 50Þ or low load ðK^ ¼ 0Þ. In Fig. 11, a maximum acceptable length ratio of l^2 o0:15 corresponding to an unbonded or free length of 0.3 m was set and the intersections with a line projected from the horizontal axis and the frequency curves identified. The projections to the frequency ratio axis give an ‘acceptable’ region highlighted by the shaded rectangle shown in the figure. For the case of a ‘‘good’’ bolt (l^2 ¼ 0:1, K^ ¼ 1 and k^ ¼ 500), the same procedure gives rE0.63. If this value is mapped onto 900 Hz, then 600 Hz corresponds to rE0.42. Projecting the latter value onto the frequency curves (marked with dots) and then projecting these intersections onto the horizontal axis shows that all possible scenarios have free length ratios in excess of
0.8 Khat=1 , khat=500 Khat=0 , khat=500 Khat=1 , khat=50 Khat=0 , khat=50 Khat=5 , khat=500
0.7 Good bolt
Frequency Ratio r
950
0.6
Acceptable Range
0.5 0.4 0.3 0.2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I2 Length Ratio
Fig. 11. Frequency ratio r vs. length ratio l^2 for different scenarios of a rockbolt, with the shaded rectangle denoting bolts with acceptable characteristics.
l^2 ¼ 0:15. This shows that no matter how these bolts with rE0.42 are modelled, with low load, or poor bond quality, they are not within specification. This was confirmed when these bolts were exhumed by UK coal mine practitioners. A cross over of the two curves (K^ ¼ 0 and k^ ¼ 500) and ^ (K ¼ 1 and k^ ¼ 50) can be noticed in the figure. Despite the fact that the K^ ¼ 0 and k^ ¼ 500 curve, defining a non prestressed well bonded bolt, gives a lower frequency range for higher length ratios than the K^ ¼ 1 and k^ ¼ 50 curve, defining a prestressed, poorly bonded bolt, the former was used to define the acceptable region. The reason for this is that the mining industry is more inclined to accept well bonded bolts with no prestress than prestressed poorly bonded bolts since the former may act as dowels, taking the load caused by the horizontal movements which are commonly found in mines. Projecting from the K^ ¼ 5 and k^ ¼ 500 curve onto the r axis for l^2 ¼ 0:15 gives an upper bound on the ‘acceptable’ frequency range. Projecting from the K^ ¼ 0 and k^ ¼ 500 case onto the r axis for l2 ¼ 0.15 gives a lower bound on the ‘acceptable’ frequency range. It can be seen that the bolts at r ¼ 0.42 fall well outside this range, again identifying them as unacceptable. Although the graph has been plotted here in nondimensional form this could be redrawn for any geometry of rockbolt with actual lengths or stiffnesses on the axes, providing mining or tunnelling engineers with a quick look up table given a first quick check of the ‘‘quality’’ of a bolt from the axial vibration. The use of the findings with regard to the spacing between the consecutive frequency ratios, allowing the total and free length detection was not possible in this study since the current method used for the assessment of rock bolts has been tuned to obtain the first natural frequency only. This, however, gives an encouragement for further development of the system where total length may be estimated by looking at the higher frequencies.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
6. Conclusions This paper presents a new continuous model for the axial vibration of a rock bolt system and highlights the importance of assuming appropriate solutions for the fixed length of the system in order to obtain a valid representation of the dynamics of the rock bolt system. The nondimensional formulation of the model allows the effects of changes in the ratios between different components of the system, e.g., lengths, stiffness and mass characteristics, to be examined conveniently. From the model developed it has been shown that there are two possible solutions that can be used to represent the fixed length of the bolt, an exponential solution or a sinusoidal solution, the validity of which depends primarily on the geometry of the rock bolt and the stiffness characteristics of the fixed length. The transition point, in non-dimensional form, pffiffiffi between solutions is governed by the relationship r ¼ k^=p, where values of the frequency ratio r lower than the critical value require use of the exponential solution. The number of modes with an exponential expression for the fixed length and the spacing of these modes are determined by the stiffness of the grout and the free length of the bolt respectively. The spacing between the exponential solutions has the potential to provide an estimate of free length as the higher modes are relatively unaffected by stiffness of the anchor head. The spacing of the sinusoidal solutions is related primarily to the total length of the rock bolt. The model allows a map of curves to be produced which can be used to show different bolt conditions. An ‘acceptable’ region has been identified for a set of mining bolts tested in practice and shows that the frequencies found during testing correspond well to the unbonded length of the bolt and the load. The main practical outcome of this work is the need to develop a device capable of exciting the higher frequencies shown to be significant for the estimation of the total length and free length independently from the load. References [1] Benmokrane B, Chekired M, Xu H. Monitoring behaviour of grouted anchors using vibrating-wire gauges. J Geotech Eng 1995;121(6):466–75. [2] Yap LP, Rodger AA. A study of the behaviour of vertical rock anchors using the finite element method. Int J Rock Mech Min Sci 1984;21(2):47–61.
951
[3] Dunham RK. Anchorage tests on strain gauged resin bonded bolts. Tunnels Tunnelling 1976:73–6. [4] Weerasinghe RB, Littlejohn GS. Load transfer and failure of anchorages in weak mudstone. In: Proceedings of the international conference on ground anchorage and anchored structures. London: 1997. p. 34–44. [5] Starkey A, Ivanovic´ A, Neilson RD, Rodger AA. Use of neural networks in the condition monitoring of ground anchorages. Adv Eng Software 2003;34:753–61. [6] Ivanovic´ A, Neilson RD, Rodger AA. Lumped parameter modelling of single tendon ground anchorage systems. Geotech Eng ICE 2001;149(2):103–13. [7] Ivanovic´ A. The dynamic response of ground anchorage systems. PhD thesis, University at Aberdeen, Aberdeen, UK, 2001. [8] Coats DF, Yu YS. Three dimensional stress distributions around a cylindrical hole and anchor vol. 2. In: Proceedings of the second international conference on Rock Mechanics Belgrade; 1970. p. 175–82. [9] Nitzsche RN, Hass CJ. Installation induced stresses for grouted roof bolts Int J Rock Mech Min Sci 1976;13:17–24. [10] Aydan O¨, IchikawaY, Kawamoto T. Load bearing capacity and stress distributions in/along rockbolts with inelastic behaviour of interfaces. In: Proceedings of the fifth international conference on numerical methods in geomechanics. vol 2. 1985. p. 1281–92. [11] Farmer IW. Stress distribution along a resin grouted rock anchor. Int J Rock Mech Min Sci Geomech Abstr 1975;12:347–51. [12] Benmokrane B, Chennouf A, Mitri HS. Laboratory evaluation of cement based and grouted rock anchors. Int J Rock Mech Min Sci 1995;32(7):633–42. [13] Dunham RK. Field testing of resin anchored rock bolts Colliery Guardian 1974;May:146–51. [14] Mothersille DKV. The influence of close proximity blasting on the performance of resin bonded bolts. PhD thesis, University of Bradford, Bradford, UK, 1989. [15] Xu H. The dynamic and static behaviour of resin bonded rock bolts in tunnelling, PhD thesis, University of Bradford, Bradford, UK, 1993. [16] Tannant DD, Brummer RK, Yi X. Rockbolt behaviour under dynamic loading: field tests and modelling. Int J Rock Mech Min Sci 1995;32(6):537–50. [17] Ivanovic A, Neilson RD, Rodger AA. Influence of prestress on the dynamic response of ground anchorages. J Geotech Geoenviron Eng 2002;128(3):237–49. [18] Nayfeh AH. Perturbation techniques. New York: Wiley; 1981. [19] Taylor RN, editor. Geotechnical centrifuge technology. Glasgow: Blackie; 1995. [20] Smith EAL. Pile driving analysis by the wave equation. J Soil Mech Found Eng Div ASCE 1960;86(4):35–61. [21] Deep Mines Industry Advisory Committee. Guidance on the use of rockbolts to support roadways in coal mines. Sudbury, UK: HSE Books; 1996. [22] British Standards Institute. BS7861-1 Strata reinforcement support system components used in coal mines. Specification for rockbolting. BSI London, 1996.
International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951 www.elsevier.com/locate/ijrmms
Influence of geometry and material properties on the axial vibration of a rock bolt Ana Ivanovic´, Richard D. Neilson School of Engineering, College of Physical Sciences, King’s College, Fraser Noble Building, University of Aberdeen, AB24 3UE, UK Received 17 July 2007; received in revised form 18 October 2007; accepted 20 October 2007 Available online 4 December 2007
Abstract A continuous dynamic model for the axial vibration of a rock bolt system is presented. The model comprises three sections: the fixed length, bonded into the rock, the free length, which is not coupled to the rock, and the protruding length, which extends beyond the rock. The head assembly is modelled as a discrete mass and a spring, and a further discrete mass is included, representing a testing device that can be attached to the protruding end. Each section is modelled as a continuous elastic rod governed by the wave equation, with suitable compatibility conditions applied between the sections and boundary conditions, which also account for the effect of the discrete components, applied at the ends. Solutions in non-dimensional form are substituted into the boundary conditions to allow the natural frequencies to be calculated, and it is shown that two possible solutions for the mode shapes can be used for the fixed length—an exponential solution or the classical sinusoidal solution—depending on the stiffness of the grout relative to that of the bar. The conditions for which the two solutions are valid are developed, and changes in the frequency ratio with changes in length ratio, and the stiffness ratios of the grout and the anchor head relative to the stiffness of the fixed length of the anchorage are examined. Generally, the state of a bolt after installation is unknown and this does not provide proper assurance of the safety of the structure for which the bolts are used. The model provides a viable tool for helping to assess the condition of the bolt by using the natural frequencies associated with areas of the bolt of particular interest, e.g. the free length. The results show how the changes in the stiffness and/or length ratios affect the dynamics associated with fixed length of the bolt and the quality of the bonding installation. A case study is presented showing how the model can be used effectively to interpret real data. r 2007 Elsevier Ltd. All rights reserved. Keywords: Rock bolts; Vibration; Numerical modelling and analysis
1. Introduction Millions of rock bolts are installed worldwide as the main support for structures such as tunnels and mines. The main features of a rock bolt system, as shown in Fig. 1, are the protruding, free and fixed anchor lengths, denoted as subsystems I, III and IV, respectively, the anchor head assembly, denoted as subsystem II, and the interfaces between the tendon and grout and grout and surrounding rock mass, denoted as subsystems V and VI, respectively. The installation of most rock bolts is usually achieved by first drilling a borehole. A cable or bar is then bonded Corresponding author. Tel.: +44 1224 273 265; fax: +44 1224 272 497.
E-mail address: [email protected] (A. Ivanovic´). 1365-1609/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2007.10.003
along part of its length into the borehole using either resin or cement grout. Thereafter, the installation procedure for both cases continues by placing a bearing plate and nut, in the case of a rock bolt, or barrel and wedges, in the case of a cable bolt, at the top of the borehole, and finishes by loading the tendon up to the required level by either tightening the nut or using a stressing jack. Research undertaken by Benmokrane et al. [1] suggests that the level of load in rock bolts or ground anchorages is very likely to change during their lifetime from that initially applied. Furthermore, changes in the free length are shown to be very common at early stages of loading due to cracking of the grout at the top end of the fixed length, i.e. at the boundary between fixed and free length [2–4]. This results in changes in the free fixed length ratio, which may
ARTICLE IN PRESS 942
A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
VI III
V
II
IV
Fig. 1. Rock bolt system.
influence greatly the static and dynamic response of the bolt. In addition, in mining applications lateral ground movement can sever an anchorage and therefore reduce its fixed length and load capacity and therefore its effectiveness as a support. Since many rock bolts are installed in safety critical applications, a method of assessing all of these aspects is needed if long term monitoring of such bolts is to be achieved. To this end a non-destructive testing technique, GRound ANchorage Integrity Testing (GRANIT) has been developed, which comprises an impact device, attached to the free end of the rock bolt by which an axial impulse of a low magnitude is applied. The resulting axial vibration is captured by an accelerometer attached to the impact device and is then further processed using an artificial neural network, the details of which can be found in [5]. Although lateral or bending vibrations could also be examined, the use of near full encapsulation in mines and secondary grouting for corrosion protection in tunnels limits the freedom of the bolt to bend whereas axial vibration is possible. Alongside the development of the non-destructive testing system, a lumped parameter dynamic model simulating the axial vibration of a rock bolt system was developed by one of the authors [6], in order to better understand what influences the static and dynamic response of bolts and also to validate the non-destructive technique. Validation of the lumped parameter model was achieved by comparing the results obtained from the model with those obtained from experiments undertaken on a laboratory rock bolt anchorage to an accuracy of 3% [7]. Investigation of the distribution of the load along the fixed anchor length was also undertaken with the model and the results obtained, which showed an exponential distribution, were in agreement with previous theoretical [8–11] and laboratory studies [12–15]. Unlike previous models [16] this lumped parameter model is able to replicate the effect of prestress load in the rock bolt when the anchor head is modelled and incorporated in
the model [17]. The model improved the performance of the GRANIT system and is currently used successfully as part of it. However, in order to undertake a parametric study, the lumped parameter model would need to be run for a large number of times for different geometries with each parameter changed separately. In addition, the number of modes, which can be identified, is determined by the number of masses included in the model and the accuracy of representation of the higher modes and their associated natural frequencies is limited. If the system is transformed into a non-dimensional form then the number of parameters used in the analysis is reduced. This can be achieved more conveniently, with a continuous system than with the lumped parameter system. In addition, the accuracy of representation of the higher modes and frequencies is then not limited. The system can be presented in a generic form using ratios of parameters and then solved. These nondimensional ratios provide scaling laws [18,19]. As a consequence, rock bolts, which have the same parameter ratios, will exhibit the same dynamic characteristics even if the actual values of the parameters are substantially different (e.g. bolts with the same free/fixed length ratio will respond in a similar way). This allows the general response of the rock bolt system to be explored conveniently. In this paper such a continuous dynamic model of a rock bolt system for axial vibration is presented and appropriate solutions for the natural frequencies and mode shapes developed. The aim is to identify modes that provide information on the free length and load and whether any modes can be excited, which provide information about the fixed length, with a view to ascertaining either the bond quality or the length of the bonded section. This information can then be used in interpreting the vibration data gathered from rockbolts in the field to provide insights into their condition. A case study in which the model is applied to data from a mine is included at the end of this paper.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
2. Continuous model
u(x,t) ¼ U(x)T(t), are given by
A continuous model was developed in which the bar is represented as an elastic continuum, described by the wave equation, with additional discrete elements used to represent the anchor head and the GRANIT impact device as shown in Fig. 2. M1 and M2 in Fig. 2 represent the mass of the impact device and mass of the head assembly, respectively, K represents the linearised head stiffness and k, the stiffness per unit length of the grout which was obtained using thick plate theory; the details of this can be found in [6]. The findings from a parametric study undertaken previously by the authors indicate that with an increase of the Young’s modulus of the grout or a decrease of the borehole the shear stiffness of the grout annulus is greater [6]. Although the model described in this paper contains both continuous elastic and discrete elements, it will be referred to as the continuous model in this paper to differentiate it from the previous lumped parameter model.
x¼0:
2.1. Boundary conditions for the model The continuous model was developed with boundary and compatibility conditions, which allow for the dynamic effects of the mass of the testing device, the mass and stiffness of the anchor head and the grout stiffness. The rock stiffness is assumed to be infinite and the steel bar is assumed to be made of a homogeneous, isotropic and linear elastic material. The assumption of infinite rock stiffness is made as a first approximation based on the fact that the rock stiffness is high relative to the grout stiffness. The grout stiffness therefore controls the dynamic response. Although damping is a major concern in some similar dynamic systems, e.g. the driving of piles [20], it will be shown in this study that for many cases of interest related to the detection of load, most of the vibrational energy is concentrated in the free length and so damping has been neglected in this paper. The effect of this will be the subject of future investigations. For the continuous model shown in Fig. 2, the boundary and compatibility conditions, after separation of variable
AE
dU 1 ðxÞ d2 TðtÞ TðtÞ ¼ M 1 U 1, dx dt2
943
(1)
dU 1 ðxÞ dU 2 ðxÞ TðtÞ KU 1 ðxÞTðtÞ þ AE TðtÞ dx dx d2 TðtÞ ¼ M2 U 1, ð2Þ dt2
x ¼ l 1 : AE
x ¼ l 1 : U 1 ðxÞ ¼ U 2 ðxÞ,
(3)
x ¼ l1 þ l2 :
dU 2 ðxÞ dU 3 ðxÞ ¼ , dx dx
(4)
x ¼ l1 þ l2 :
U 2 ðxÞ ¼ U 3 ðxÞ,
(5)
dU 3 ðxÞ ¼ 0, (6) dx where x is the position along the bolt, measured from the protruding length, A is the cross-sectional area of the bar, E is Young’s modulus of the steel bar, U(x) is the mode shape of the length and T(t) is the principal coordinate. The subscripts denote the section of the bolt being modelled with subscripts 1, 2 and 3 denoting elements relating to the protruding, free and fixed lengths, respectively. x ¼ l1 þ l2 þ l3 :
2.2. Wave equation for prismatic bars Separate solutions for the axial vibration of the system can be written for each section of the bolt. Although the wave equation can be shown to apply for each section, there are two different conditions along the bar that need to be considered. The first case is where the bar is unrestrained, the condition which applies along the protruding and free lengths. The second case is where the bar is elastically restrained by the grout which is the case along the fixed length. 2.2.1. Unrestrained bar For the free and protruding lengths, the classical wave equation q2 u 1 q2 u ¼ qx2 c2 qt2
Fig. 2. Continuous model.
(7)
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
944
holds, where u is the longitudinal displacement of the element at axial position x. The solution uðx; tÞ is then given as UðxÞTðtÞ where T(t) is the solution of the free vibration problem given as TðtÞ ¼ A cos ot þ B sin ot and the mode shapes U(x) are defined as UðxÞ ¼ C cosðox=cÞ þ D sinðox=cÞ. The constants A* and B* depend on the initial conditions, C and D depend on the boundary conditions the ends of the bar, o is an angular pat ffiffiffiffiffiffiffiffiffi frequency and c ¼ E=r is the wave speed, where r is the density of the steel bar. 2.2.2. Restrained bar When an elastically restrained bar is considered, the system is described by the following wave equation, in which k is the stiffness per unit length of the grout: 2
2
q u 1q u k u. ¼ 2 2þ 2 qx c qt EA
(8)
After separation of variables, this produces 2 q2 U o k þ U ¼ 0. qx2 c2 EA
(9)
For ðo=cÞ2 4k=EA, the roots of the frequency equation are a complex conjugate pair, and the solution is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 k o2 k x þ D sin x, (10) UðxÞ ¼ C cos 2 2 EA EA c c whereas for ðo=cÞ2 ok=EA, real roots are obtained and the solution becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 (11) UðxÞ ¼ C 3 e k=EAðo =c Þx þ D3 e k=EAðo =c Þx . These wave models and solutions were used in the derivation of the continuous system. The solutions U1, U2 and U3, with the corresponding constants C1, D1, C2, D2, C3 and D3, respectively, are taken in the form shown in Eqs. (12), (13), (14) or (15). The solution for T(t), with the corresponding constants A* and B*, is given in Eq. (16). U 1 ðxÞ ¼ C 1 cos U 2 ðxÞ ¼ C 2 cos
o o x þ D1 sin x, c c o o ðx l 1 Þ þ D2 sin ðx l 1 Þ, c c
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 k ðx l 1 l 2 Þ U 3 ðxÞ ¼ C 3 cos 2 EA c rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 k ðx l 1 l 2 Þ þ D3 sin 2 EA c
(12)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
k=EAðo =c Þðxl 1 l 2 Þ
TðtÞ ¼ A cos ot þ B sin ot.
(16)
2.3. Non-dimensional form of the continuous model In order to obtain the continuous model in non^ dimensional form, new time and length scales, t and x, are defined: t t ¼ on t ) t ¼ , (17) on x^ ¼
x ^ 3. ) x ¼ xl l3
(18)
The solutions are then written in the form ^ 3. ^ 3 ¼ TUl u ¼ ul
(19)
The length by which the other lengths are scaled is chosen to be the fixed length l3, since the influence of the stiffness of the grout compared to the stiffness of the whole system is of interest in this paper particularly for ascertaining if the quality of the bond is satisfactory. Similarly, the first axial natural frequency on of an elastically unrestrained bar of length l3, with free–free boundary conditions, as defined in Eq. (20), was used as the reference: pc on ¼ . (20) l3 This choice allows the examination of the effect of grout stiffness, and also produces convenient expressions. A frequency ratio, o r¼ , (21) on can then be defined, and the solutions written in the form ^ ¼ C 1 cos rpx^ þ D1 sin rpx, ^ U 1 ðxÞ (22) ^ ¼ C 2 cos rpðx^ l^1 Þ þ D2 sin rpðx^ l^1 Þ U 2 ðxÞ and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 p2 k^ðx^ l^1 l^2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ D3 sin r2 p2 k^ðx^ l^1 l^2 Þ
(23)
^ ¼ C 3 cos U 3 ðxÞ (13) or ^ ¼ C3e U 3 ðxÞ ð14Þ
or U 3 ðxÞ ¼ C 3 e
and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þ D3 e k=EAðo =c Þðxl 1 l 2 Þ (15)
pffiffiffiffiffiffiffiffiffiffiffi
^ 2 p2 ðx ^ l^1 l^2 Þ kr
and ^ ¼ A cos rt þ B sin rt, TðtÞ
pffiffiffiffiffiffiffiffiffiffiffi ^ 2 2 ^ l^1 l^2 Þ þ D3 e kr p ðx
ð24Þ
(25)
(26)
where l^1 ¼ l 1 =l 3 and l^2 ¼ l 2 =l 3 . After substituting Eqs. (17)–(26) into non-dimensional forms of the boundary conditions Eqs. (1)–(6), the following
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
945
set of equations is obtained: 2
^ 1 r2 p2 m
6 6 rp sin rpl^1 K^ cos rpl^1 6 6 6 ^ 2 r2 p2 cos rpl^1 þm 6 6 6 cos rpl^1 6 6 6 6 0 6 6 6 0 4 2
C1
3
0
rp
0
0
0
0
rp
0
sin rpl^1
1
0
0
0
rp sin rpl^2
rp cos rpl^2
0
0
cos rpl^2
sin rpl^2
0
0
0
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 p2 k^ sin r2 p2 k^
rp cos rpl^1 K^ sin rpl^1 ^ 2 r2 p2 sin rpl^1 þm
0
7 7 7 7 7 7 7 7 0 7 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 r2 p2 k^ 7 7 7 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 r2 p2 k^ cos r2 p2 k^ 0
7 6 6 D1 7 7 6 7 6 6 C2 7 7 6 6 7 ¼ 0. 6 D2 7 7 6 7 6 6 C3 7 5 4 D3
ð27Þ
The matrix for the whole system, when the exponential solution is used, is 2 ^ 1 r2 p2 m rp 0 0 6 6 rp sin rp l 1 K^ cos rpl^1 rp cos rpl^1 K^ sin rpl^1 l3 6 6 0 rp 6 2 ^ 2 r2 p2 sin rpl^1 þm ^ 2 r p2 cos rpl^1 þm 6 6 6 6 cos rpl^1 sin rpl^1 1 0 6 6 6 0 0 rp sin rpl^2 rp cos rpl^2 6 6 6 0 0 cos rpl^2 sin rpl^2 6 4 0 0 0 0 3 2 C1 7 6 6 D1 7 7 6 7 6 6 C2 7 7 6 6 7 ¼ 0, 6 D2 7 7 6 7 6 6 C3 7 5 4 D3
^ K, ^ m ^ 1 and m ^ 2 are non-dimensional parameters, where k, defining the ratio of the stiffness of the grout to the axial stiffness of the bar, the stiffness of the head to the axial stiffness of the bar, the mass of the impact device to the mass of the bar and the mass of the head to the mass of the bar, respectively, and are given by k^ ¼ kl 23 =AE, ^ 1 ¼ M 1 =Al 3 r and m ^ 2 ¼ M 2 =Al 3 r. K^ ¼ Kl 3 =AE, m 2.4. Discussion As shown in Eqs. (24) and (25), there are two solutions for U3, which depend on whether the expression under the square root in Eq. (25) is positive or negative. The choice of
3
0 0 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k^ r2 p2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ^ 2 2 k^ r2 p2 e kr p
0
3
7 7 7 7 0 7 7 7 7 7 0 7 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 2 2 ^ kr p 7 7 7 1 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 5 2 2 ^ k^ r2 p2 e kr p
ð28Þ
the appropriate solution depends on the relative size of the ^ If the stiffness ratio k^ is large relative to terms r2p2 and k. r2p2, then the exponential solution is valid. If k^ is small then the sinusoidal solution is valid, a result underpinned by the case k^ ! 0, which is effectively an unrestrained bar. The value of the stiffness ratio k^ is governed by the stiffness of the fixed length of the bar and the grout. However, it would be beneficial to know which solution is more appropriate for a particular geometry and material properties of the grout before trying to find the natural frequency of the system and therefore any conclusion regarding the grout condition. This will be addressed later in the paper.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
946
3. Validation of the continuous model 3.1. Exponential solution In order to validate the model with the exponential solution, the results obtained from the continuous system are compared with the results obtained from the lumped parameter model, two simple analytical models and laboratory experiments. The properties of the laboratory rock bolt rig are described in Tables 1 and 2 and were used in the models. The rock bolt used in the experiments was built initially for a feasibility study of the GRANIT system in the late 1990s. From Table 1 it can be seen that the protruding length is not typical of mining or tunnelling practice and the bore hole of this bolt of a somewhat larger diameter than usually found. However, the experiments allow for the general concepts of the model to be validated. The two simple analytical models are based on a bar with one end fixed and the other free, and a bar with one end fixed and a mass attached at the other end. In the following discussion, these analytical models are denoted as case a for the plain bar with the total length comprising the
protruding and free lengths and case b for the plain bar with a mass attached at its free end and with its total length comprising the protruding and free lengths. Case b models the case when the impact device of the GRANIT system is attached to a rock bolt. The results are presented in terms of both the frequency ratio and actual frequency. From Table 3 it can be seen that all the frequency ratios, r are in very good agreement. The frequency ratio obtained from the simple analytical model for case a is 0.184 while the value of r obtained from the lumped parameter model is 0.180. The frequency ratio obtained from the continuous system is similar to the frequency ratio obtained from the lumped parameter model and the experiments. The lower frequency ratios obtained from the numerical model and the continuous model compared to the simple analytical model are due to the fact that, in both cases, the fixed length has some flexibility, i.e. they are not completely constrained. Similar results are found for case b where the mass of the impact device is attached at the free end of the bar. The results show excellent agreement between the values obtained from the previously validated lumped parameter and the continuous model. 3.2. Sinusoidal solution
Table 1 Geometry of the experimental rock bolt rig
Validation of the sinusoidal solution was undertaken by comparing the continuous model with the lumped
Rock bolt Bolt diameter (mm) Bore hole diameter (mm) Protruding length (mm) Free length (mm) Fixed length (mm)
25 45 1140 2780 1440
Table 4 First and second frequency ratio obtained using exponential and sinusoidal solutions, respectively
Table 2 Properties obtained from the laboratory experiments
Solution used
Lumped parameter model [Hz]
Continuous model [Hz]
exp
0.121 [218.0] 0.343 [615.9]
0.123 [220.5] 0.347 [622.46]
sin 1825 12.95 0.383 0.04
Steel Density (kg/m3) Young’s modulus (GPa) Poisson’s ratio Damping ratio
7895 210 0.3 0.03
1 first mode (l.p.m) vs exp solution (c.m)
0.5
Table 3 Frequency ratios of the first natural frequency [first natural frequencies] of the rock bolt using different models Case Lumped parameter model [Hz]
Simple analytical Continuous model [Hz] system [Hz]
Experimental data [Hz]
a
0.184 [329.9] 0.113 [202.2]
0.185 [332] 0.109 [195.7]
b
0.180 [322.5] 0.110 [197.7]
Amplitude
Resin Density (kg/m3) Young’s modulus (GPa) Poisson’s ratio Damping ratio
0
-0.5
second mode (l.p.m.) vs sin solution (c.m.)
l.p.m. - first mode c.m. - exp. solution l.p.m. - second mode c.m. - sin. solution
-1
-1.5 0.182 [326.1] 0.112 [200.1]
0
0.5
1
1.5
2
2.5
3
3.5
4
Length ratio Fig. 3. First and second mode shapes obtained from the lumped parameter model (lpm) and continuous model (cm).
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
parameter model with lower values for the stiffness of the grout. The geometry used is the same as before but the stiffness of the grout was reduced from the original value of 12.9 GPa to 12 MPa to represent a weakly bonded bolt with a poor quality of grout. As shown in Table 4 and Fig. 3, the first and second frequencies and the corresponding mode shapes from the continuous model, respectively, are in good agreement with the lumped parameter model. The first mode was obtained using the exponential solution while the second mode was obtained when the sinusoidal solution was used. The data used to validate the model are only one particular case with a particular free/fixed length ratio and grout stiffness. The length ratio l^2 , however, may be substantially smaller than examined so far, for example in rock bolts used in the mining industry [21,22] or the stiffness of the grout may be lower than that used in the validation. The following section examines the conditions under which the sinusoidal (Eq. (24)) and exponential (Eq. (25)) solutions, for the fixed anchor length, hold. 4. Analysis and discussion The frequency ratios, r, were found by expanding the determinants of the matrices in Eqs. (27) and (28) and then finding the roots. The corresponding mode shapes were then calculated and examined. As an initial analysis, the frequency ratios for a range of different grout stiffness ratios k^ were calculated in order to establish the point where transition between the exponential and sinusoidal frequencies occurs. In order to observe the influence of changes between the free and fixed lengths only, all the elements, apart from length ratios l^1 and l^2 and the stiffness ^1 ¼ m ^ 2 ¼ K^ ¼ 0 in the ratio k^ were set to zero, i.e. m matrices presented in Eqs. (27) and (28). Finally, the influence of the anchor head stiffness was examined
9
4.1. Effect of grout stiffness ratio k^ The solutions obtained for l^ ¼ l^1 þ l^2 ¼ 0:5 and a range ^ of grout stiffness ratio 0oko1000 are shown in Fig. 4. ^ This spans a range of k which includes typical resin bonded mine rock bolts down to very weak bonds. The value k^ 875 corresponds to perfect bonding using a grout with high Young’s modulus of 12.9 GPa. The figure shows that as the grout stiffness decreases, the number of modes determined with the exponential solution in the fixed length also decreases. For lower stiffness ratios more sinusoidal modes are present and these have smaller frequency spacings between them than the exponential modes. For higher stiffness ratios, the stiffness has generally little effect on the frequency ratios of the exponential modes. However, very close to the boundary, reduction in the stiffness ratio does result in a slight drop in frequency ratio for the exponential modes. As discussed before, the boundary between the appropriate solutions is given by k^ r2 p2 ¼ 0.
(29) ^ This boundary line relating k and r is plotted in Fig. 4. If the boundary for a rock bolt with a particular length is required, a bar of length corresponding to the protruding and free length part of the whole system can be considered, as shown in Fig. 5. The natural frequencies of the system represented in the figure are ð2i 1Þpc ; i ¼ 1; 2; . . . ; 1. (30) 2l The frequency ratios, with respect to the frequency of the fixed length, l3, of this system can then be obtained oi ¼
pcð2i 1Þl 3 ð2i 1Þ 1 . ¼ 2 pc2l l^
7
2 2
6 5 4
Exponential
3
ð2i 1Þ p k^i ¼ . (32) 4l^ This relationship defines the critical value of k^i which occurs at the boundary between the exponential and sinusoidal solutions for any mode i and length ratio l^
2 1 0 0
(31)
When the frequency ratio, developed in Eq. (31), is introduced into Eq. (29), the stiffness ratio as a function of l^ can be obtained as follows:
Sinusoidal
8 Frequency Ratio r
by varying the head stiffness ratio k^ while keeping the length ratios l^1 and l^2 constant.
ri ¼
10
947
100 200 300 400 500 600 700 800 900 1000 Grout Stiffness Ratio
Fig. 4. Frequency ratio r vs. grout stiffness ratio k^ for all possible solutions of the continuous system for l^ ¼ 0:5 (dotted line: exponential solution, solid line: sinusoidal solution, * approximate boundary).
Fig. 5. A bar representing free+protruding length.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
examined using Fig. 4. The ‘*’ in Fig. 4 denote points calculated using this relationship for l^ ¼ 0:5. As can be seen the points clearly define the transition curve. This means that if only the free length is known, the relationship may be used to establish which solution is appropriate for a ^ then the solution is particular case. If k^i 4ð2i 1Þ2 p2 =4l, below the transition line in Fig. 4, and the exponential solution holds. Although in field testing it is not possible to determine whether a mode is sinusoidal or exponential per se, if the sinusoidal modes can be excited and detected, then there should be a sudden change in the spacing between consecutive modes at the transition between the exponential and sinusoidal modes. The higher is the frequency ratio, at which this transition occurs, the better is the bonding of the fixed length to rock. If the sinusoidal modes cannot be excited, then instead of a change in frequency spacing between the consecutive modes, the frequency ratio at which no further modes can be detected will determine the transition point. Again, the higher this point, the better the bonding.
10 9
Sinusoidal
8 Frequency Ratio r
948
7 6 5 4 3
Exponential
2 1 0 0
0.2
0.4
0.8
0.6
1
1.2
1.4
1.6
1.8
2
l1 + l2 Length Ratio Fig. 7. Frequency ratio r vs. length ratio l^ for k^ ¼ 500 (dashed line: exponential solution, solid line: sinusoidal solution).
2.5
4.2. Effect of length ratio
2 free length fixed length
1.5 1 Amplitude
In order to examine the influence of the length ratios l^1 and l^2 on the frequency ratios, all the parameters in the ^ were set to zero. Changes in system, other than l^1 , l^2 and k, the l^ ¼ l^1 þ l^2 length ratio were made and all frequency ratios up to r ¼ 10 were calculated using both solutions. The length ratios were varied in the range 0–2, and the corresponding values of r are presented in Figs. 6 and 7 for values of k^ ¼ 50 and k^ ¼ 500, respectively. Figs. 6 and 7 confirm that the higher the stiffness of the grout along the fixed length, the more exponential solutions are obtained for a given length ratio. The higher length ratios result in more exponential modes, the
0.5 0 -0.5 1st mode (exp) 2nd mode (exp) 3rd mode (sin) 4th mode (sin)
-1 -1.5 -2 0
0.5
1
1.5
Position along the bar 10
Fig. 8. Mode shapes for free length ratio l^ of 0.5 and k^ ¼ 50 using both exponential and sinusoidal solutions.
9
Frequency Ratio r
8 7 Sinusoidal 6 5 4 3 2 Exponential
1 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
l1+l2 Length Ratio Fig. 6. Frequency ratio r vs. length ratio l^ for k^ ¼ 50 (dashed line: exponential solution, solid line: sinusoidal solution).
frequency spacing between which is smaller. For both the sinusoidal and exponential solutions there is a trend of increasing frequency ratio as the length ratio decreases although for the exponential solution this is more apparent than for the sinusoidal solution. Once the frequency ratios were calculated as described previously, the constants C1, D1, C2, D2, C3, D3 were found from the equations in matrix form allowing the mode shapes of the system to be generated. Figs. 8 and 9 show the mode shapes obtained for the first frequencies for l^ ¼ 0:5 and k^ ¼ 50 and the first eight frequencies for l^ ¼ 0:5 and k^ ¼ 500, respectively. The number of modes detected for each solution is the same as can be identified from Figs. 6 and 7 for the length ratio of 0.5.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951 15 solid line - exp dotted line - sin
12.5
Amplitude
10
free length
first sinusoidal mode
fixed length
7.5 5 2.5 0 -2.5
949
The findings from the analyses suggest that a device able to excite the higher frequencies would provide information on the state of the installed bolts and their fixed length. The current impact device used in the GRANIT system however has been tuned to provide information about the lower modes and therefore does not allow these higher frequencies to be excited or identified. This study therefore gives an encouragement for further development of the device so that the behaviour of rock bolts used in practice is better understood. 4.3. Effect of anchor head stiffness ratio
-5 1
1.5
Position along the bar
Fig. 9. Mode shape for a length ratio l^ of 0.5 and k^ ¼ 500 using both exponential and sinusoidal solution.
Examination of the first sinusoidal mode, for the higher stiffness ratio of k^ ¼ 500, reveals a higher amplitude in the fixed length compared to the free length. The amplitudes for higher modes then become more uniform along the anchorage. The trend of larger amplitudes in the fixed end for the sinusoidal modes is also apparent for the lower stiffness in Fig. 8, although this is less pronounced. The results confirm that there are two groups of frequency ratios—the lower ones defined by the exponential solution and dominated by motion in the free length and those defined by the sinusoidal solution and dominated by motion in the fixed length. The highest frequency exponential solution has a shape within the fixed length of the anchorage, which is close in form to that of the first sinusoidal solution. A number of practical outcomes can be inferred from the results. In the dynamic testing of an anchorage, it should be possible to estimate a number of parameters from the spacing between consecutive natural frequencies. Firstly, the modes below the transition are exponential modes and the spacing between two consecutive frequencies allows the free length to be assessed by following equation: l¼
pc oi oi1
or
l^ ¼
p . ri ri1
(33)
Secondly, the sinusoidal modes appear to be related to the total length of the anchorage with the spacing between the higher frequencies tending to that present between the frequencies of a free–free bar of total length l^ þ l^3 . This can be seen in Fig. 6, for k^ ¼ 50, where for the higher sinusoidal frequency ratios, at smaller length ratios, the spacing tends to a value of frequency ratio of 1, which is the increment between frequencies for the axial vibration of a free–free bar of length l3 ¼ 1. At l^ ¼ 0:5 the total length is 1.5 and the frequency ratio is about 0.7, which corresponds closely to the frequency ratio for a free–free bar of this length. For the stiffness ratio of k^ ¼ 500, this trend would be true for frequency ratios higher than those presented in Fig. 7.
The effect of load in a rock bolt has been shown to generally result in an increase in the stiffness of the anchor head assembly. This in turn results in an increase in natural frequency with increasing load [17]. To investigate what effect this would have on the different solutions, the model was run for a range of head stiffness ratios K^ with different ^ values of length ratios l^1 and l^2 , and stiffness ratio k. Fig. 10 shows an example of the rise in first natural frequency (an exponential mode) associated with the increase of head stiffness, which generally occurs for increasing load in the rock bolt. The figure also shows that the higher exponential modes are less affected by the increase in head stiffness ratio and in the case of the two highest frequency modes for this case (k^ ¼ 500) there is no appreciable change in frequency ratio while none of the sinusoidal modes are modified. These results provide a basis for determining the load in a rock bolt from the first natural frequency measured by dynamic testing. The fact that the higher frequencies do not change with variations in the head stiffness resulting from changes in load may provide a means of measuring the free (unbonded) length independently from the test data. This is important, as the first natural frequency is affected by both 10 9
Sinusoidal
8 Frequency Ratio r
0.5
0
7 6 Exponential
5 4 3 2 1 0 0
2
4
6
8
10
12
14
16
18
20
Head Stiffness Ratio
^ ratio for k^ ¼ 500, Fig. 10. Frequency ratio r vs. head stiffness ratio K, l^1 ¼ 0:1 and l^2 ¼ 0:4, dashed line: exponential solution, dotted line: sinusoidal solution
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
the free length and the load. If the higher exponential frequencies can be measured to determine the free length then this can be used along with the first natural frequency to estimate the actual load. This again suggests that a device capable of exciting higher frequencies would give more information on the condition of the rock bolt. 5. Case study With the exploration of various theoretical scenarios using the model completed the interpretation of data from both laboratory tests and field tests has been made viable. In 2001, dynamic test data were taken from a number of rockbolts in a UK coal mine using the GRANIT system. The data showed that most bolts, deemed as working effectively, had natural frequencies around 850–900 Hz. However, a number of bolts were identified with frequencies of 600 Hz or lower. HSE documentation for mining support systems recommends fully bonded bolts [21]. However in practice this is seldom achievable and some unbonded length occurs behind the head assembly. The practical design values for the geometry of the bolts tested were therefore l 1 ¼ 0:1 m, l 2 o0:3 m, l 3 42:0 m with a total length of 2.4 m, and a bolt diameter of 22 mm. This geometry provides a long fixed length to strengthen the strata and a short free length to ensure that the surface strata are well supported; the latter minimises the likelihood of shallow roof falls. Using the values noted above l^1 ¼ 0:05 was obtained and the continuous model was therefore run over the range 0:05ol^2 o0:95, i.e. 0,1 mol2o1.0 m. The mass of the testing device used, M1 was 8.5 kg which, using a density of 7850 kg/ ^ 1 ¼ 1:42. Values of m3 for the bolt, gives a mass ratio of m K^ ¼ 1 and k^ ¼ 500 were calculated as being representative of a ‘‘good’’ bolt, i.e. a bolt for which the grouting is strong and which has some load at the bolt head (typically about ^ assuming 80 kN). The theoretical maximum value of k, perfect bonding is 875 as noted in Section 4.1 and so 500 was taken as an acceptable achievable bond within a working mine environment. K^ ¼ 5 and k^ ¼ 500 were then used as being representative of a well bonded bolt with higher load at the upper end of the acceptable range. The other permutations shown in Fig. 11 were chosen as cases with either poor bond quality ðk^ ¼ 50Þ or low load ðK^ ¼ 0Þ. In Fig. 11, a maximum acceptable length ratio of l^2 o0:15 corresponding to an unbonded or free length of 0.3 m was set and the intersections with a line projected from the horizontal axis and the frequency curves identified. The projections to the frequency ratio axis give an ‘acceptable’ region highlighted by the shaded rectangle shown in the figure. For the case of a ‘‘good’’ bolt (l^2 ¼ 0:1, K^ ¼ 1 and k^ ¼ 500), the same procedure gives rE0.63. If this value is mapped onto 900 Hz, then 600 Hz corresponds to rE0.42. Projecting the latter value onto the frequency curves (marked with dots) and then projecting these intersections onto the horizontal axis shows that all possible scenarios have free length ratios in excess of
0.8 Khat=1 , khat=500 Khat=0 , khat=500 Khat=1 , khat=50 Khat=0 , khat=50 Khat=5 , khat=500
0.7 Good bolt
Frequency Ratio r
950
0.6
Acceptable Range
0.5 0.4 0.3 0.2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I2 Length Ratio
Fig. 11. Frequency ratio r vs. length ratio l^2 for different scenarios of a rockbolt, with the shaded rectangle denoting bolts with acceptable characteristics.
l^2 ¼ 0:15. This shows that no matter how these bolts with rE0.42 are modelled, with low load, or poor bond quality, they are not within specification. This was confirmed when these bolts were exhumed by UK coal mine practitioners. A cross over of the two curves (K^ ¼ 0 and k^ ¼ 500) and ^ (K ¼ 1 and k^ ¼ 50) can be noticed in the figure. Despite the fact that the K^ ¼ 0 and k^ ¼ 500 curve, defining a non prestressed well bonded bolt, gives a lower frequency range for higher length ratios than the K^ ¼ 1 and k^ ¼ 50 curve, defining a prestressed, poorly bonded bolt, the former was used to define the acceptable region. The reason for this is that the mining industry is more inclined to accept well bonded bolts with no prestress than prestressed poorly bonded bolts since the former may act as dowels, taking the load caused by the horizontal movements which are commonly found in mines. Projecting from the K^ ¼ 5 and k^ ¼ 500 curve onto the r axis for l^2 ¼ 0:15 gives an upper bound on the ‘acceptable’ frequency range. Projecting from the K^ ¼ 0 and k^ ¼ 500 case onto the r axis for l2 ¼ 0.15 gives a lower bound on the ‘acceptable’ frequency range. It can be seen that the bolts at r ¼ 0.42 fall well outside this range, again identifying them as unacceptable. Although the graph has been plotted here in nondimensional form this could be redrawn for any geometry of rockbolt with actual lengths or stiffnesses on the axes, providing mining or tunnelling engineers with a quick look up table given a first quick check of the ‘‘quality’’ of a bolt from the axial vibration. The use of the findings with regard to the spacing between the consecutive frequency ratios, allowing the total and free length detection was not possible in this study since the current method used for the assessment of rock bolts has been tuned to obtain the first natural frequency only. This, however, gives an encouragement for further development of the system where total length may be estimated by looking at the higher frequencies.
ARTICLE IN PRESS A. Ivanovic´, R.D. Neilson / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 941–951
6. Conclusions This paper presents a new continuous model for the axial vibration of a rock bolt system and highlights the importance of assuming appropriate solutions for the fixed length of the system in order to obtain a valid representation of the dynamics of the rock bolt system. The nondimensional formulation of the model allows the effects of changes in the ratios between different components of the system, e.g., lengths, stiffness and mass characteristics, to be examined conveniently. From the model developed it has been shown that there are two possible solutions that can be used to represent the fixed length of the bolt, an exponential solution or a sinusoidal solution, the validity of which depends primarily on the geometry of the rock bolt and the stiffness characteristics of the fixed length. The transition point, in non-dimensional form, pffiffiffi between solutions is governed by the relationship r ¼ k^=p, where values of the frequency ratio r lower than the critical value require use of the exponential solution. The number of modes with an exponential expression for the fixed length and the spacing of these modes are determined by the stiffness of the grout and the free length of the bolt respectively. The spacing between the exponential solutions has the potential to provide an estimate of free length as the higher modes are relatively unaffected by stiffness of the anchor head. The spacing of the sinusoidal solutions is related primarily to the total length of the rock bolt. The model allows a map of curves to be produced which can be used to show different bolt conditions. An ‘acceptable’ region has been identified for a set of mining bolts tested in practice and shows that the frequencies found during testing correspond well to the unbonded length of the bolt and the load. The main practical outcome of this work is the need to develop a device capable of exciting the higher frequencies shown to be significant for the estimation of the total length and free length independently from the load. References [1] Benmokrane B, Chekired M, Xu H. Monitoring behaviour of grouted anchors using vibrating-wire gauges. J Geotech Eng 1995;121(6):466–75. [2] Yap LP, Rodger AA. A study of the behaviour of vertical rock anchors using the finite element method. Int J Rock Mech Min Sci 1984;21(2):47–61.
951
[3] Dunham RK. Anchorage tests on strain gauged resin bonded bolts. Tunnels Tunnelling 1976:73–6. [4] Weerasinghe RB, Littlejohn GS. Load transfer and failure of anchorages in weak mudstone. In: Proceedings of the international conference on ground anchorage and anchored structures. London: 1997. p. 34–44. [5] Starkey A, Ivanovic´ A, Neilson RD, Rodger AA. Use of neural networks in the condition monitoring of ground anchorages. Adv Eng Software 2003;34:753–61. [6] Ivanovic´ A, Neilson RD, Rodger AA. Lumped parameter modelling of single tendon ground anchorage systems. Geotech Eng ICE 2001;149(2):103–13. [7] Ivanovic´ A. The dynamic response of ground anchorage systems. PhD thesis, University at Aberdeen, Aberdeen, UK, 2001. [8] Coats DF, Yu YS. Three dimensional stress distributions around a cylindrical hole and anchor vol. 2. In: Proceedings of the second international conference on Rock Mechanics Belgrade; 1970. p. 175–82. [9] Nitzsche RN, Hass CJ. Installation induced stresses for grouted roof bolts Int J Rock Mech Min Sci 1976;13:17–24. [10] Aydan O¨, IchikawaY, Kawamoto T. Load bearing capacity and stress distributions in/along rockbolts with inelastic behaviour of interfaces. In: Proceedings of the fifth international conference on numerical methods in geomechanics. vol 2. 1985. p. 1281–92. [11] Farmer IW. Stress distribution along a resin grouted rock anchor. Int J Rock Mech Min Sci Geomech Abstr 1975;12:347–51. [12] Benmokrane B, Chennouf A, Mitri HS. Laboratory evaluation of cement based and grouted rock anchors. Int J Rock Mech Min Sci 1995;32(7):633–42. [13] Dunham RK. Field testing of resin anchored rock bolts Colliery Guardian 1974;May:146–51. [14] Mothersille DKV. The influence of close proximity blasting on the performance of resin bonded bolts. PhD thesis, University of Bradford, Bradford, UK, 1989. [15] Xu H. The dynamic and static behaviour of resin bonded rock bolts in tunnelling, PhD thesis, University of Bradford, Bradford, UK, 1993. [16] Tannant DD, Brummer RK, Yi X. Rockbolt behaviour under dynamic loading: field tests and modelling. Int J Rock Mech Min Sci 1995;32(6):537–50. [17] Ivanovic A, Neilson RD, Rodger AA. Influence of prestress on the dynamic response of ground anchorages. J Geotech Geoenviron Eng 2002;128(3):237–49. [18] Nayfeh AH. Perturbation techniques. New York: Wiley; 1981. [19] Taylor RN, editor. Geotechnical centrifuge technology. Glasgow: Blackie; 1995. [20] Smith EAL. Pile driving analysis by the wave equation. J Soil Mech Found Eng Div ASCE 1960;86(4):35–61. [21] Deep Mines Industry Advisory Committee. Guidance on the use of rockbolts to support roadways in coal mines. Sudbury, UK: HSE Books; 1996. [22] British Standards Institute. BS7861-1 Strata reinforcement support system components used in coal mines. Specification for rockbolting. BSI London, 1996.