determination of equivalent mohr-coulomb criterion from
TRANSCRIPT
Determination of Equivalent Mohr-CoulombCriterion from Generalised Hoek-Brown CriterionQingqing Yang
Southwest Jiaotong UniversityFei Cai ( [email protected] )
Gunma University: Gunma Daigaku
Research Article
Keywords: equivalent Mohr-Coulomb parameter, Hoek-Brown criterion, least squares method, tensilestrength
Posted Date: November 15th, 2021
DOI: https://doi.org/10.21203/rs.3.rs-1073021/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License
1
Determination of Equivalent Mohr-Coulomb Criterion from 1
Generalised Hoek-Brown Criterion 2
3
Qingqing Yang1,2, Fei Cai3,* 4
5
1 Faculty of Geosciences and Environmental Engineering, Southwest 6
Jiaotong University, Chengdu, 610031, China 7
2 MOE Key Laboratory of High-speed Railway Engineering, Southwest 8
Jiaotong University, Chengdu, 610031, China 9
3 Department of Environmental Engineering Science, Gunma University, 10
Kiryu, 376-8515, Japan 11
12
*Corresponding Author: Fei Cai, PhD 13
Gunma University, Japan 14
Tel & Fax: +81(277) 30 1621 15
Email: [email protected] 16
ORCID number: 0000-0002-8810-2567 17
18
First Author: Qingqing Yang, PhD 19
Southwest Jiaotong University, China 20
Email: [email protected] 21
ORCID number: 0000-0001-7775-0844 22
2
Abstract 23
A new analytical solution is presented for determining equivalent Mohr-24
Coulomb (MC) shear strength parameters over an arbitrary interval of 25
minor principal stress σ3 from the generalised Hoek-Brown (HB) criterion 26
using least squares method. Comparison with several published examples 27
demonstrates that the proposed solution had a capacity to accurately 28
determine equivalent MC parameters over a given interval of σ3, as well as 29
instantaneous MC parameters by using a very small interval of σ3. EMC 30
parameters depended heavily on the interval of σ3, which highlighted the 31
importance of intervals of σ3. A calculation case shows that the equivalent 32
internal friction angle and cohesion over the interval of σ3 from tension cut-33
off σcut-off to maximum minor principal stress σ3max were approximately 12% 34
smaller and 10.3% larger than those over an interval from tensile strength 35
to σ3max, respectively. The proposed solution offers great flexibility for the 36
application of the HB criterion with existing methods based on the MC 37
criterion for rock engineering practice. 38
39
Keywords: equivalent Mohr-Coulomb parameter; Hoek-Brown criterion; 40
least squares method; tensile strength 41
42
3
1 Introduction 43
The linear Mohr-Coulomb (MC) and nonlinear Hoek-Brown (HB) 44
criteria are the most widely used failure criteria in slope stability analysis, 45
and most of the currently used geotechnical codes are based on the 46
application of the MC parameters. Consequently, it is necessary to convert 47
the HB criterion to the MC criterion without significantly affecting the 48
results of analysis. However, the HB criterion is expressed major and minor 49
principal stresses σ1 and σ3, and its conversion to the relationship between 50
shear strength τf and normal stress σn is not straightforward. 51
Generally, the HB failure envelope is replaced by a tangent line and 52
instantaneous MC (IMC) parameters are determined by locating the 53
tangent of the HB envelope at a specified σn. The nonlinearity of the HB 54
criterion causes IMC parameters to vary with σ3, consequently, the method 55
theoretically yields the most accurate MC parameters but only for a specific 56
stress state. An exact solution for intact rock was proposed by Hoek (1983), 57
and IMC parameters can be obtained numerically for rock masses (Kumar 58
1998; Carranza-Torres 2004; Priest 2005; Yang and Yin 2006; Shen et al. 59
2012a; Lee and Pietruszczak 2017). Alternatively, another method provides 60
equivalent MC (EMC) parameters by best fitting an averaged line 61
equivalent to the HB criterion over an interval of σ3. EMC parameters are 62
more appropriate to evaluate the overall strength of rock mass, especially 63
for preliminary design purposes. EMC parameters were determined by 64
4
linear regression analysis for intact rock (Hoek and Brown, 1997) and rock 65
masses (Hoek et al., 2002). Note that the solution proposed by Hoek et al. 66
(2002) is valid only for the artificial interval from biaxial tensile strength 67
σt2 to maximum minor principal stress σ3max. No accurate analytical 68
solution for determining EMC parameters over an arbitrary interval of σ3 69
is available yet. 70
A new analytical solution was proposed to determine EMC parameters 71
over an arbitrary interval of σ3 using least squares method. The reliability 72
and performance of the proposed method were verified based on several 73
published examples. The importance of the intervals of σ3 was highlighted 74
by comparing EMC parameters over different intervals of σ3. 75
76
2 Overview of the Equivalent Hoek-Brown and Mohr-Coulomb 77
criteria 78
The HB failure criterion for intact and fractured rock masses was first 79
proposed in 1980 to describe a nonlinear empirical relationship between σ1 80
and σ3. Since then, it has been widely accepted by the international rock 81
mechanics, and updated several times in response to experience gained 82
with its applications (Hoek and Brown 2019). The HB criterion in the latest 83
version, often referred to as the generalised Hoek-Brown criterion, is 84
expressed as (Hoek and Brown 1997): 85
5
31 3
a
bci
ci
ms
(1) 86
where, σ1 and σ3 are the major and minor principal stresses, and σci is the 87
uniaxial compressive strength of the intact rock mass. mb, s, and a are the 88
constants of the HB criterion. Hoek et al. (2002) proposed the following 89
revised empirical expressions for mb, s, and a. 90
100exp
28 14b i
GSIm m
D
(2) 91
100exp
9 3
GSIs
D
(3) 92
1 1 20exp exp
2 6 15 3
GSIa
(4) 93
where GSI, mi, and D is the Geological Strength Index, rock type-94
dependent parameter, and the disturbance factor, respectively. 95
The equations provided by Hoek et al. (2002) for determining EMC 96
parameters over an artificial interval [σt2, σ3max] are: 97
1
3max
1
3max
6
arcsin
2 2 1 6
a
bb
ci
a
bb
ci
mam s
ma a am s
(5) 98
1
3max 3max
1
3max
(1 ) 1 2
6
2 1 12 1
a
b bci
ci ci
a
bb
ci
m ma a s s
c
mam s
a aa a
(6) 99
Note that the value of σ3max has to be determined for each case. Hoek et 100
al. (2002) suggested empirical formulae to determine the value of σ3max for 101
6
deep tunnels and slopes, respectively. 102
103
3 Proposed solution to determine EMC parameters 104
A new analytical solution was formulated by least squares 105
approximation to determine EMC parameters over an arbitrary interval of 106
σ3, of which the lower limit is not necessary to be σt2. That is, for an 107
arbitrary interval of σ3 defined by [σ3a, σ3b], we search for a best linear 108
approximation expressed as the MC criterion of the generalised HB 109
criterion in the sense of least squares, as illustrated in Fig. 1. 110
The generalised HB is expressed as 111
31HB 3
a
bci
ci
ms
(7) 112
where σ1HB is the major principal stress of the HB criterion. 113
The equivalent MC criterion can be expressed using σ1 and σ3 114
1MC 3p q (8) 115
where σ1MC is the major principle stress of the equivalent MC criterion, and 116
2 cos
1 sin
cp
(9) 117
1 sin
1 sinq
(10) 118
where φ is internal friction angle, and c is cohesion. 119
In the sense of least squares, choose p and q to minimize the integral 120
3
3
2
1MC 1HB 3
b
a
Q d
(11) 121
A minimum of Q gives 122
7
2 2
3 3 3 3 1 3 3 2
3
3 3
4 6b b a a b a
b a
S Sp
(12) 123
3 3 1 2
3
3 3
6 12b a
b a
S Sq
(13) 124
where, 125
1 12
2 2 3 31 3 3
1
2 1
a a
ci b b b ab a
b ci ci
m mS s s
a m
(14) 126
1
3 3
33 3
2 3 3 2 1
3 3
11
3 2 11
a
b b b b
ci cicib a a
bb a b a
ci ci
m ms a s
Sa a m m m
s a s
(15) 127
Using Eqs. (12) and (13), φ and c can be expressed as 128
1arcsin
1
q
q
(16) 129
2
pc
q (17) 130
Noted that, substituting the lower and upper limits σ3a = -sσci/mb and σ3b 131
= σ3max into Eqs. (12) and (13), p and q can be obtained. After p and q 132
obtained, using Eqs. (16) and (17), the expressions of φ and c exactly 133
identical to Eqs. (5) and (6), can be obtained. 134
Although Hoek et al. (2002) declared that the fitting process involves 135
balancing the areas above and below the MC envelope, the expressions of 136
φ and c can also be obtained using least squares approximation. It should 137
be pointed out that -sσci/mb is biaxial tensile strength σt2 because 138
substituting σ3 = -sσci/mb into Eq. (1) gives σ1 = -sσci/mb. 139
140
8
4 Validation of the proposed solution 141
4.1 Example 1 142
The first example is adopted from Hoek et al. (2002). This is a typical 143
example in which σci = 50000 kPa, GSI = 45, and mi = 10. For an 144
undisturbed in-situ rock mass (D = 0) surrounding a tunnel at a depth of 145
100 m, σ3max was calculated by the formula suggested by Hoek et al. (2002) 146
for tunnels, assuming γ = 27 kN/m3, while the lower limit of σ3 was 147
assumed to be σt2 = -sσci/mb. For the rock mass with the same parameters 148
but in highly disturbed slope (D = 1) with 100 m height, σ3max was 149
calculated by the formula suggested by Hoek et al. (2002) for slopes, while 150
the lower limit of σ3 was also assumed equal to σt2. Table 1, an extract from 151
a basic Microsoft Excel spreadsheet, illustrates that the calculated EMC 152
parameters using the present solution agreed very well with those reported 153
by Hoek et al. (2002). This verified the reliability and performance of the 154
proposed solution. 155
156
4.2 Example 2 157
The proposed least squares solution can also be used to approximately 158
calculate IMC parameters using a small interval of σ3 centered at the 159
specified σ30. If σn was specified, a process of Kumar et al. (2020) should 160
be used to calculate σ3 corresponding to a given σn. 161
The second example is chosen from the works of Hoek et al. (2002). The 162
9
associated parameters are as follows: σci = 30000 kPa, σn = 800 kPa, GSI = 163
15, D = 0.7, and mi = 16. At a given σn of 800 kPa, the specified σ30 of 164
480.5339 kPa was obtained using Newton-Raphson method. Thus, the 165
interval of σ3 was assigned to [0.999σ30, 1.001σ30]. Table 2 shows the IMC 166
parameters at the specified σn of 800 kPa, and the results obtained from this 167
study had a good consistence with those reported by others. 168
More examples published in the literature (e.g. Hoek and Brown 1997; 169
Priest 2005; Shen et al. 2012b) were also employed for comparison, which 170
is not presented here for space limit. The comparison demonstrates that 171
there was a very close agreement in the values of MC parameters with the 172
published results. This implies that the proposed solution had the capacity 173
to accurately determine EMC parameters for different rock types over a 174
wide range of stress levels, as well as IMC parameters at specified stress 175
level using a small interval of σ3. 176
177
5. Discussions 178
5.1 EMC parameters over the interval from uniaxial tensile strength 179
to maximum minor principal stress 180
As mentioned in Section 3, the method proposed by Hoek et al. (2002) 181
is valid only for the interval [σt2, σ3max]. For practical slope stability 182
problems, uniaxial tensile strength σt1 or tension cut-off σcut-off is a more 183
realistic lower limit of σ3 than σt2, although Hoek (1983) declared that σt2 184
10
is equal to σt1 for brittle materials. Actually, the two tensile strength is not 185
necessarily equal to each other (Michalowski and Park 2020); this is 186
dependent on the curvature of initial segment of HB envelope. 187
Unfortunately, σt1 cannot be represented by a convenient close-form 188
expression in Eq. (1); it can be obtained by solving the HB criterion under 189
the condition σ1 = 0 using the Newton-Raphson method. The calculation 190
shows that a good initial guess for σt1/σci is 0.999σt2/σci, and the Newton-191
Raphson method can converge in 4-5 iterations with correct digits larger 192
than 8. 193
Example 1 was used to investigate the difference in the values between 194
σt2 and σt1 respected to HB constants. Fig. 2 illustrates that the difference 195
in estimated values between σt1 and σt2 was remarkable at a small mi. For 196
extremely small values of mi, σt2 was 10-20% larger than σt1. However, the 197
difference between the two became insignificantly at a large mi. This 198
coincided with the conclusion drawn by Sari (2010). The EMC parameters 199
over the intervals [σt2, σ3max] and [σt1, σ3max] were also determined for 200
Example 1. Calculations demonstrate that the discrepancy in c and φ over 201
the intervals [σt2, σ3max] and [σt1, σ3max] was less than 2%, even the 202
maximum difference in values of σt1 and σt2 was up to 16.18% when GSI = 203
10, mi = 3, and D = 1. The slight difference in EMC parameters over the 204
two intervals was attributed to the small values of σt1 and σt2 when GSI and 205
mi was small; with the increasing GSI and mi, the values of σt1 and σt2 206
11
increased but the difference in the two values reduced and was negligible 207
in a practical sense. 208
209
5.2 EMC parameters over the interval from tension cut-off to 210
maximum minor principal stress 211
Triaxial extension test results of Ramsey and Chester (2004) were 212
reproduced in Fig. 3, combined with the HB envelope. The HB envelope 213
projects back in tensile regime to give an intercept of σ3 = σt1 = -17.2 MPa 214
for σ1 = 0; the calculated σt2 = -17.5 MPa; this does not correspond to the 215
tensile failure data which gives an average tensile strength σ3 = -7.75 MPa. 216
In other words, the HB criterion has no provision for predicting the tensile 217
strength which was highlighted by the ‘tension cut-off’ (broken line in Fig. 218
3) (Hoek and Martin 2014). Hoek and Brown (2019) recommended that 219
the MC parameters, derived from the HB criterion, should not be used 220
without a tension cut-off. For practical slope stability problems, σt1 or σcut-221
off is a more realistic lower limit of σ3 than σt2. For tunnel problems, zero 222
should be a better estimate for the lower limit of σ3. 223
The EMC parameters over different intervals of σ3 were determined for 224
the rock mass tested by Ramsey and Chester (2004), for which the 225
associated parameters by back analysis are as follows: σci = 129300 kPa, 226
mb = 7.37, s = 1, and a =0.5, respectively. Table 3 lists the EMC parameters 227
over different intervals represented with tensile strength and slope height, 228
12
to which the corresponding value of σ3max was calculated by using the 229
empirical formula given by Hoek et al. (2002). It is evident that the EMC 230
parameters for the interval of σ3 from σcut-off to σ3max, as well as those over 231
the interval [0, σ3max], were markedly different from those over the interval 232
[σt2, σ3max] and [σt1, σ3max]. In this case, φ over the intervals [σcut-off, σ3max] 233
and [0, σ3max] was approximately 12% and 16.5% smaller than that over the 234
interval [σt2, σ3max], respectively; c over the intervals [σcut-off, σ3max] and [0, 235
σ3max] was approximately 10.3% and 15.7% larger than that over the 236
interval [σt2, σ3max], respectively. Thus, the practical interval of σ3 should be 237
carefully determined when calculating EMC parameters. 238
239
6 Conclusions 240
The study provided a new analytical formulation of equivalent MC shear 241
strength parameters over an arbitrary interval of σ3 for the HB criterion 242
based on least squares method. Comparison with published results 243
demonstrates that the proposed solution had the capacity to accurately 244
determine the EMC parameters over a broad range of values of GSI, D, mi, 245
and σci. Some conclusions can be drawn as follows: 246
(1) The proposed solution was valid to determine EMC parameters over 247
an arbitrary interval of σ3. The solution for the EMC parameters provided 248
by Hoek et al. (2002) is a special case of the proposed solution in case that 249
the lower limit of σ3 equals to the biaxial tensile strength σt2. 250
13
(2) The present solution can also be used to approximately calculate IMC 251
parameters at a specified σn using a very small interval of σ3. The 252
comparison demonstrates that there was a close agreement in the values of 253
IMC parameters published in the literature. 254
(3) The calculated EMC parameters heavily depended on the specified 255
interval of σ3. The discrepancy in φ and c over the intervals [σt2, σ3max] and 256
[σt1, σ3max] was negligible in a practical sense. A calculation case 257
demonstrates that the equivalent φ over the intervals [σcut-off, σ3max] and [0, 258
σ3max] was approximately 12% and 16.5% smaller than that over the 259
interval [σt2, σ3max], respectively; the equivalent c over the intervals [σcut-off, 260
σ3max] and [0, σ3max] was approximately 10.3% and 15.7% larger than that 261
over the interval [σt2, σ3max], respectively. 262
263
Acknowledgements 264
This research was supported by the Second Tibetan Plateau Scientific 265
Expedition and Research (STEP) program (Grant No. 2019QZKK0905), 266
the National Natural Science Foundation of China (41402244, 41877226, 267
41877237). 268
269
270
271
272
14
Competing interests 273
The authors declare that they have no known competing financial 274
interests or personal relationships that could have appeared to influence the 275
work reported in this note. 276
277
15
List of Notations 278
a Hoek-Brown material constant 279
c Cohesion of equivalent Mohr-Coulomb criterion (kPa) 280
D Disturbance factor 281
GSI Geological Strength Index of rock mass 282
Hs Height of slope (m) 283
Ht Depth of tunnel below surface (m) 284
mb Hoek-Brown material constant of rock mass 285
mi Hoek-Brown material constant of intact rock 286
p Auxiliary variable 287
q Auxiliary variable 288
Q Integral of squared residuals over an interval of minor principal stress 289
s Hoek-Brown material constant 290
S1 Auxiliary variable 291
S2 Auxiliary variable 292
φ Internal friction angle of equivalent Mohr-Coulomb criterion (°) 293
σ1 Major principal stress (kPa) 294
σ1HB Major principal stress of Hoek-Brow criterion (kPa) 295
σ1MC Major principal stress of equivalent Mohr-Coulomb criterion (kPa) 296
σ3 Minor principal stress (kPa) 297
σ30 Specified minor principal stress (kPa) 298
σ3a Lower limit of minor principal stress (kPa) 299
16
σ3b Upper limit of minor principal stress (kPa) 300
σ3max Maximum minor principal stress (kPa) 301
σci Uniaxial compressive strength of intact rock (kPa) 302
σcm Rock mass strength (kPa) 303
σcut-off Tension cut-off (kPa) 304
σn Normal stress (kPa) 305
σt Tensile strength (kPa) 306
σt1 Uniaxial tensile strength (kPa) 307
σt2 Biaxial tensile strength (kPa) 308
τ Shear strength (kPa) 309
γ Unit weight of rock mass (kN/m3) 310
311
17
References: 312
Carranza-Torres C (2004) Elasto-plastic solution of tunnel problems using 313
the generalised form of the Hoek-Brown failure criterion. International 314
Journal of Rock Mechanics and Mining Science 41: 629-639 315
Hoek E (1983) Strength of jointed rock masses. Géotechnique 33(3): 187-316
223 317
Hoek E, Brown ET (1997) Practical estimates of rock mass strength. 318
International Journal of Rock Mechanics and Mining Science 34(8): 319
1165-1186 320
Hoek E, Brown ET (2019) The Hoek-Brown failure criterion and GSI – 321
2018 edition. Journal of Rock Mechanics and Geotechnical Engineering 322
11: 445-463 323
Hoek E, Carranza-Torres C, Corkum B (2002) Hoek-Brown failure 324
criterion -2002 Edition. In Proceedings of 5th North American Rock 325
Mechanics Symposium and 17th Tunnelling Association of Canada 326
Conference (Hammah R,; Bawden W,; Curran J,; Telesnicki M. (eds)). 327
Toronto: pp. 267-273 328
Hoek E, Martin CD (2014) Fracture initiation and propagation in intact 329
rock – A review. Journal of Rock Mechanics and Geotechnical 330
Engineering 6: 287-300 331
Kumar P (1998) Shear failure envelope of Hoek-Brown criterion for 332
rockmass. Tunnelling and Underground Space Technology 13(4): 453-333
18
458 334
Kumar V, Burman A, Himanshu N, Barman B (2020) Solution of minor 335
principal stress for generalised Hoek-Brown rock material in two 336
dimension using Newton-Raphson method. Geotechnical and 337
Geological Engineering 38: 1817-1837 338
Lee YK, Pietruszczak S (2017) Analytical representation of Mohr failure 339
envelope approximating the generalized Hoek-Brown failure criterion. 340
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99 342
Michalowski RL, Park D (2020) Stability assessment of slopes in rock 343
governed by the Hoek-Brown strength criterion. International Journal 344
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Priest (2005) Determination of shear strength and three-dimensional yield 346
strength for the Hoek-Brown criterion. Rock Mechanics and Rock 347
Engineering 38(4): 299-327 348
Ramsey J, Chester F (2004) Hybrid fracture and the transition from 349
extension fracture to shear fracture. Nature 428: 63-66 350
Sari M (2010) A simple approximation to estimate the Hoek-Brown 351
parameter ‘mi’ for intact rocks. In ISRM International Symposium - 352
EUROCK 2010 (Zhao J.; Labiouse V.; Dudt JP.; Mathier JF. (eds)). 353
London: Taylor & Francis Group, pp. 169-172 354
Shen JY, Priest SD, Karakus M (2012a) Determination of Mohr-Coulomb 355
19
shear strength parameters from Generalized Hoek-Brown criterion for 356
slope stability analysis. Rock Mechanics and Rock Engineering 45: 123-357
129 358
Shen JY, Karakus M, Xu CS (2012b) Direct expression for linearization of 359
shear strength envelops given by the Generalized Hoek-Brown criterion 360
using genetic programming. Computers and Geotechnics 44: 139-146 361
Yang XL, Li L, Yin JH (2004) Seismic and static stability analysis for rock 362
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367
20
List of Figures 368
Fig. 1 Equivalent MC criterion for HB criterion 369
(Single column, Color online only) 370
Fig. 2 Difference in the values between the biaxial and uniaxial tensile 371
strength 372
(Single column, Color online only) 373
Fig. 3 Combined plot of HB criterion with tension cut-off 374
(Single column, Color online only) 375
The figures in this work were created by MSOffice. 376
377
List of Tables 378
Table 1 Equivalent MC parameters over the interval [σt2, σ3max] for 379
Example 1 380
Table 2 Instantaneous MC parameters related to a specified normal stress 381
for Example 2 382
Table 3 Equivalent MC parameters over different intervals of σ3 383
represented with slope height and tensile strength for the rock mass tested 384
by Ramsey and Chester (2004) 385
The tables in this work were created by MSOffice. 386
Figures
Figure 1
Equivalent MC criterion for HB criterion (Single column, Color online only)
Figure 2
Difference in the values between the biaxial and uniaxial tensile strength (Single column, Color onlineonly)
Figure 3
Combined plot of HB criterion with tension cut-off (Single column, Color online only)