Parameter determination for nonlinear stress criteria using a simple

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Parameter determination for nonlinear stress criteria using a simple regression tool Li Li, Michel Gamache, and Michel Aubertin

Abstract: In geotechnique, laboratory test results are often employed to estimate the parameter values required for the application of criteria commonly used to define specific stress states such as yielding, failure, and residual strength. Many of these criteria rely on a nonlinear formulation relating the different stress tensor components (or the corresponding invariants). There are many regression techniques that can be selected to obtain corresponding parameters based on experimental results. Some of these can be fairly complex but, considering the intrinsic variability of the available data on geomaterials, relatively simple methods are usually deemed sufficient. In that regard, one should favor methods directly available from common software packages. In this paper, the authors present a simple analysis procedure based on Microsoft Excel® which can be applied to different two- and three-dimensional stress criteria. As an example, the technique is applied to the failure strength of rock samples using two nonlinear criteria. Key words: rock mechanics, failure criterion, curve fitting, regression analyses, parameter determination. Résumé : En géotechnique, on utilise fréquemment des données d’essais du laboratoire afin d’estimer la valeur des paramètres requis pour l’application de critères utilisés afin de définir certains états spécifiques, tel que la limite élastique, la rupture, et la résistance résiduelle. Plusieurs critères sont basés sur des formulations non linéaires qui relient les contraintes principales (ou leurs invariants). Il existe de nombreuses techniques de régression permettant d’obtenir les paramètres de tels critères à partir des résultats d’essais. Certaines de ces techniques sont assez complexes mais compte tenu de la variabilité intrinsèque des données disponibles sur les géomatériaux, on peut en général utiliser des méthodes relativement simples. Certaines sont d’ailleurs aisément accessibles à partir de codes commerciaux. Dans cet article, les auteurs présentent une procédure d’analyse basée sur le chiffrier Microsoft Excel®, qui peut être utilisée avec divers critères bi- et tri-dimensionnels. À titre d’exemple, la technique est appliquée à deux critères non linéaires décrivant la rupture des roches. Mots clés : mécanique des roches, critère de rupture, régression, détermination de paramètres.

Introduction In soil and rock mechanics, it is customary to express specific material states in stress space, using mathematical expressions to define curves or surfaces known as criteria. These are used in constitutive models and structural analysis to define particular conditions such as yielding associated with the onset of inelastic behavior, failure corresponding to peak stress on the stress–strain curves, and residual strength reached at relatively large strain beyond the peak. In the case of linear expressions, such as the classic Coulomb criterion expressed in the Mohr (shear stress τ – normal stress σ) plane, the corresponding parameters (i.e., for Coulomb, c the cohesion and φ the friction angle) are usually fairly easy to obtain from common linear regression analyses. For the different nonlinear expressions that have been developed for geomaterials over the years (e.g., Desai and Siriwardane 1984; Chen and Baladi 1985; Lade 1993; Sheorey 1997), a linear regression technique may not be apReceived February 8, 2000. Accepted May 12, 2000. Published on the NRC Research Press website on December 14, 2000. L. Li, M. Gamache, and M. Aubertin. Department of Civil, Geological and Mining Engineering, École Polytechnique de Montréal, C.P. 6079, Succursale Centre-ville, Montréal, QC H3C 3A7, Canada. Can. Geotech. J. 37: 1332–1347 (2000)

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propriate, however. In such instances, one can use a trial and error procedure (e.g., Carter et al. 1991), but more systematic methods have been proposed. For instance, Shah and Hoek (1992) have developed a simplex reflection method for such nonlinear criteria. There are, in fact, many optimization techniques available to obtain material parameters by minimizing the error summation between experimental and computed data. In recent years, some very sophisticated schemes have been developed and used, particularly in the case of complex nonlinear equations of plasticity and viscoplasticity (e.g., Tarantola 1987; Cailletaud and Pilvin 1994; Bruhns and Anding 1999). Nevertheless, in the case of nonlinear stress criteria for geomaterials, the limited number of data and natural dispersion do not, in most cases, warrant the use of very elaborate methods. Instead, the authors consider that fairly simple procedures, readily available in common commercial codes, can be well suited to solve the problem at hand. In the following, the authors show how to use such a simple technique. The proposed procedure is presented and applied to the failure of laboratory rock samples, using the recently developed Mises-Schleicher and Drucker-Pager unified (MSDPu) criterion. After recalling the main equations used to define the limit (failure) surface in stress space, the regression procedure is introduced and applied to test results taken from the literature. A brief comparison is then made with the results of Shah and Hoek (1992) obtained © 2000 NRC Canada

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with the simplex reflection technique applied to the Hoek and Brown (1980) criterion. A short discussion follows.

Fig. 1. Representation of the MSDPu criterion (a) in the J21/2–I1 space; and (b) in the π plane. CTC, conventional triaxial compression; RTE, reduced triaxial extention.

MSDPu failure criterion for rocks The procedure described below can be applied to a variety of constitutive equations, including most failure criteria used for geomaterials. It was initially developed for a threedimensional (3D) stress criterion proposed for rocks. Since the pioneering work of Griffith (1924), many nonlinear criteria have been presented for rocks. Most of these criteria were initially formulated using the same assumption as that of Griffith, hence neglecting the influence of the intermediate principal stress. However, it has been shown that such two-dimensional (2D) expressions may have serious limitations and drawbacks. Accordingly, various 3D failure criteria have been developed over the years (e.g., Pan and Hudson 1988; Lade 1993; Wang and Kemeny 1995). Aubertin and Simon (1998) have proposed the MisesSchleicher and Drucker-Prager (MSDP) multiaxial failure criterion. This relatively simple criterion provides good results when used to describe the failure strength of rocks and rock-like materials submitted to compressive and (or) tensile loads. A more general (unified) mathematical version of this criterion has been elaborated by Aubertin et al. (1999a, 1999b). For low-porosity rocks, the ensuing multiaxial criterion (MSDP unified or MSDPu) can be expressed as [1a]

J21/ 2 − F0 Fπ = 0

in a general form, or [1b]

J2 1 / 2 =

[α 2 (I 12 − 2a1I1) + a 22 ]1/ 2

b [b2 + (1 − b2 ) sin2 (45° −1.5θ)]1/ 2

in an explicit manner, where F0 gives the shape of the failure surface in the J21/ 2 – I1 plane, and Fπ is the controlling function in the octahedral (π) plane perpendicular to the hydrostatic axis. The material parameters are defined as follows: [2]

α=

3

1/ 2

2 sin φ (3 − sin φ) 2

 σt  −  σ −σt  b a1 = c − 2 2 6α (σ c + σ t ) σ 2c

[3]

[4]

   a2 =   

  σt  σc + 2  2 b − α  σcσt   3 (σ c + σ t )   

1/ 2

     

where σc (= C0) and σt (= |T0|) are the uniaxial strength in compression and tension, respectively; the value of φ in eq. [2] is equal to φb; b is used to define the criterion shape in the π plane (usually b ≅ 0.75); I1 is the first invariant of the Cauchy stress tensor σij; J2 is the second invariant of the

deviatoric stress tensor Sij; and θ is the Lode angle defined in the octahedral (π) plane. Figure 1 shows the criterion in the J21/ 2 – I1 space and the Fπ in the π space. This criterion has also been extended to the long-term strength of rocks and failure of rock masses (e.g., Aubertin et al. 1999b), and it could equally be used for soils and concrete; these aspects will not be introduced here. The four parameters of MSDPu (i.e., a1, a2, α (or σc, σt, and φb), and b) can be obtained from simple and independent laboratory tests, including uniaxial compression tests, uniaxial (direct or indirect) tensile tests, and tilt tests on plane surfaces. Triaxial compression and extension tests can also be used to obtain or confirm parameter values. © 2000 NRC Canada

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In many practical situations, one or more of these parameters has not been directly measured and must be inferred from indirect results. This would be the case, for instance, if φb had not been evaluated (by tilt tests on plane surfaces); the values of φb could then be deduced from the slope α (see Fig. 1) determined from triaxial compression test results. If sufficient experimental data are available (see Discussion), an optimization curve-fitting process can then be applied. In the next section, a simple procedure to optimize the parameter values, based on a nonlinear programming technique combined with a preprocessing of the input data, is presented and applied to MSDPu.

as mentioned earlier. The function to be optimized then becomes ERR(A, B) = Σ{Y[Xi(xi, yi), A, B] – Yi(xi, yi)}2. Theoretically, the optimal values of parameters A and B could be obtained by using the same procedure as that in the linear case. In practice, however, this technique does not work well, in part because Y and X are two interdependent variables. It should also be mentioned that some nonlinear relations cannot be transformed into a linear relation. Further details on the disadvantages of this approach are given by Shah and Hoek (1992). In the more general case, the nonlinear relation can be written

Regression procedure

[6]

The method of least squares, which minimizes the sum of the squares of the deviations, is well known and often used for fitting a set of k paired data (xi, yi, i = 1 to k) to the common linear equation y = a′x + b′ (where a′and b′ are the linear slope and intercept, respectively). If a linear relation exists between y (dependent variable) and x (independent variable), good results can be obtained using the method of least squares. When y does not vary linearly with x, one can modify the method to obtain a pseudolinear equation Y(x, y) = AX + B (where A and B are the linear slope and intercept, respectively, in the X–Y system), but this approach is not always satisfactory (for reasons explained later). Alternately, one can use other, more elaborate methods. For instance, Shah and Hoek (1992) have used a simplex reflection technique to estimate the Hoek and Brown (1980, 1988) criterion parameters. This technique is based on heuristics and somewhat intuitive concepts of nonlinear minimization for a multivariable function; it can give excellent results, but it may also imply some nontrivial programming for the user. Instead, a nonlinear programming approach can be used. This approach might be simpler to understand and to apply, as many commercial software programs offer such optimization tools. Before entering into the nonlinear fitting scheme, it is useful to recall the linear regression procedure (without too much mathematics). Suppose x and y are related by the linear equation y = a′x + b′. The determination of parameters a′ and b′ involves the limit values of err(a′, b′) = Σ [ y(xi, a′, b′) – yi]2, where err represents the error of regression for a given pair of parameters a′ and b′. The optimal value of parameters a′ and b′ can then be obtained by resolving the following two equations:

where y is the dependent variable; x is the independent variable; and α1, α2, …, αn are n parameters. The function to be optimized then becomes ERR(α1, α2, …, αn) = Σ[ y(xi, α1, α2, …, αn) – yi]2. If there is no constraint on any parameters, then the optimal solutions can be deduced by solving the following equation system:

[5a]

∂ err (a ′ , b′ ) = 0 ∂a ′

[5b]

∂ err (a ′ , b′ ) = 0 ∂b′

Equations [5a] and [5b] constitute a linear equation system. If the solution exists for a given series of data, a unique solution can be obtained directly, since the surface err(a′, b′) in the error (err), a′, b′ space is a convex function. In this case, the limit solution generally corresponds to the appropriate solution. In the case of a nonlinear relation, the latter can be changed into the pseudolinear form Y(x, y) = AX(x, y) + B,

[7]

y = f (x, α1, α2, …, αn)

∂ ERR(α 1, α 2 , ...., α n) = 0 ∂α j

j = 1 to n

Contrary to eq. [5], however, these n equations are nonlinear. Because of that, more than one solution can exist. Of course, the optimal solution (α1*, α2*, …, αn*) should still be the one corresponding to the minimum error (or deviation) Min[ERR(α1, α2, …, αn)]. If eq. [7] can be solved analytically, there would be no problem with the regression procedure. In reality, this is rarely the case and an iterative method must be applied to solve the equation system. Since there may exist a series of local solutions corresponding to local minima, the obtained results could well depend upon the given initial values. Figure 2 illustrates this phenomenon with a one-parameter (αj) function y = f(x, αj); here, one can distinguish the local and global optimization situations (see Luenberger 1984 for more details). This inconvenience in the procedure can appear with any nonlinear regression analyses (e.g., Bruhns and Anding 1999) and may produce some confusing results to the uninformed or unsuspicious user. To overcome this difficulty, the authors propose a preoptimization scheme, named PREO hereafter. The PREO scheme consists of finding a set of parameter values that are close to the global optimization point. To do so, a plausible range of values must be chosen for each parameter. The problem then becomes constrained and nonlinear. Let αj be one of the parameters, where j = 1, 2, …, n. For this parameter, a constraint condition is introduced: Lj ≤ αj ≤ Uj, where Lj and Uj represent the lower and the upper bounds, respectively, of the value of the jth parameter αj. A heuristic approach based on an iterative procedure can be used where each parameter is increased from αj = Lj to αj = Uj by an increment ∆α j (see Fig. 2). Then, this heuristic n

approach will evaluate ∏ [(Uj − L j )/ ∆α j + 1] possible soluj =1

tions (where ∏ is the multiplication operator). At each step, the solution is recorded. At the end, the best solution is used as a starting or initial value for the nonlinear programming © 2000 NRC Canada

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Fig. 2. Characteristic of a nonlinear error function.

procedure. The optimal result (α 1*, α 2*, …, α n*) can then be deduced by a nonlinear programming technique. In principle, the pre-optimization scheme PREO can be combined with any constrained nonlinear programming procedure to find the global optimal solution. Here, the authors combine the pre-optimization scheme with the nonlinear program SOLVER included in the very commonly used commercial software Microsoft Excel®. In this commercial code, SOLVER uses a code of nonlinear optimization identified as Generalized Reduced Gradient (GRG2), produced by Leon Lasdon at the University of Texas (Austin) and Allan Waren at Cleveland Sate University (Lasdon and Waren 1982). The proposed implanted scheme is then called PREO-SOLVER. The package also contains a linear programming algorithm named Simplex; however, the Simplex method can only be used to solve linear problems and is completely different from the simplex reflection algorithm proposed by Shah and Hoek (1992).

Application of PREO-SOLVER to MSDPu The proposed procedure can be applied to the determination of material parameters for any nonlinear criteria. Here, it is used for the MSDPu criterion using laboratory test results on intact rock samples. The goal is to find the values of parameters σc, σt, φ, and b which correspond to the minimum value of ERR(σc, σt, φ, b) = Σ[J 12/ 2 (I1i, σc, σt, φ, b) – J 12/i2 ]2, where I1i and J2i are the first stress invariant and second deviatoric stress invariant, respectively, corresponding to the ith pair of experimental results taken from the set. The preoptimization scheme and the recall of SOLVER are preprogrammed using Microsoft Visual Basic language, and incorporated into Microsoft Excel® using the “macro” command program (see Appendix 2). As shown in this macro program, a plausible range of parameter values is first assumed. In the case of MSDPu, the parameters all have a physical sense, so this range can be easily established from values published on relevant materials (e.g., for rocks see Barton and Choubey 1977; Goodman 1980; and Charmichael 1982). Once the applied pre-optimization scheme calculations are completed,

the nonlinear programming procedure SOLVER can be executed with appropriately selected constraints; in this case, the following have been used σc ≥ 0, σt ≥ 0, 0 ≤ φ ≤ 60°, 0.75 ≤ b ≤ 1, and 5 ≤ σc /σt ≤ 50. Figure 3 shows an example of the procedure in the form of a Microsoft Excel® output sheet, assuming that only the value of b was known beforehand (b = 0.75, when required). For a given set of experimental results obtained under conventional triaxial compression (CTC: σ1 > σ2 = σ3, where σ1, σ2, and σ3 are the major, intermediate, and minor principal stresses, respectively), the pre-optimization procedure (PREO) gave σc = 160 MPa, σt = 20 MPa, φb = 25°, and ERR = ∑( J21/2cal. – J21/2exp.)2 = 21.1628 (see Fig. 3a). The results are further refined by using SOLVER (see Fig. 3b), which gives σc = 156.6 MPa, σt = 3.1 MPa, φb = 23°, and ERR = ∑( J21/2cal. – J21/2exp.)2 = 10.5355 (see Fig. 3c). One can recall that under CTC conditions (θ = 30°), the value of b does not play a role in defining the MSDPu failure curve (i.e., Fπ = 1 in eq. [1]). Other typical results are presented to further illustrate the procedure and the good correlation between MSDPu and failure strength of rocks. Figures 4–8 show comparisons between experimental data and the MSDPu criterion with the calculated parameters for different rock types, from sandstone to granite, subjected to CTC loading conditions. These figures show that the MSDPu criterion can provide a good description of experimental data for different rocks, with parameters providing fairly small error values. The procedure can successfully be applied when very few data are available (Fig. 6), or when the data bank is relatively large (Fig. 5). Thus, the procedure is very flexible when it comes to the number of tests required for its application. Figures 9–11 present the MSDPu criterion drawn from the optimal parameters obtained by the PREO-SOLVER procedure for rock samples subjected to loading conditions with various Lode angles in the π plane. Again, the relative error remains reasonable. Figure 12 shows the MSDPu criterion applied to rock-like material subjected to different loading conditions with various Lode angles. One can see that the MSDPu criterion adequately represents these experimental © 2000 NRC Canada

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Fig. 3. Example of a PREO-SOLVER calculation sheet: (a) execution of the PREO program; and (b) Microsoft Excel® sheet for SOLVER.

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Fig. 3. (concluded). (c) Optimal results from PREO-SOLVER.

Fig. 4. Estimation of MSDPu parameters for Pennant sandstone submitted to CTC (θ = 30°) loading conditions (experimental data from Franklin and Hoek 1970). ε, the difference between calculated J21/2cal. and measured J21/2exp.

results when parameters are obtained from the above procedure. Note that in determining the MSDPu parameters for curves shown in Figs. 9–12, b was fixed at 0.75 because it is usually the value observed for low-porosity rocks. This reduces the number of iterations with the PREO scheme.

In the above calculations with the MSDPu criterion, it was considered that σc, σt, and φb had not been measured directly, so they had to be determined from the procedure. Of course, knowing one or more of these values beforehand reduces the number of parameters to be back-calculated. © 2000 NRC Canada

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Fig. 5. Estimation of MSDPu parameters for Portland limestone submitted to CTC (θ = 30°) loading conditions (experimental data from Franklin and Hoek 1970).

Fig. 6. Estimation of MSDPu parameters for Webatuck dolomite submitted to CTC (θ = 30°) loading conditions (experimental data from Brace 1964).

Application to the Hoek-Brown 2D criterion As shown in the previous section, the PREO-SOLVER procedure can be used to determine the parameters for the MSDPu nonlinear criterion. The proposed procedure can also be used for any other nonlinear criteria. To illustrate such capabilities, the procedure is applied to the well-known HoekBrown 2D criterion, which can be written as follows: [8]

(σ 1 – σ 3) – (mσ 3 σ c + sσ c2)1/2 = 0

where m and s are material parameters (s = 1 for intact rocks), and σc (= C0) is the uniaxial compressive strength of intact rock (Hoek and Brown 1980). For intact rocks, the analysis consists of finding the values of parameters σc and m. For rock masses, σc must be known beforehand and parameters m and s must be estimated. The object functions to be minimized are ERR(σc, m) = Σ[σ 1(σ3i, σc, m) – σ1i]2 for intact rocks, and ERR(m, s) = Σ[σ1(σ3i, m, s) – σ1i]2 for rock masses. Here, σ1i and σ3i are the principal stresses corresponding to the ith pair of experimental results. © 2000 NRC Canada

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Fig. 7. Estimation of MSDPu parameters for Carrara marble submitted to CTC (θ = 30°) loading conditions (experimental data from Franklin and Hoek 1970).

Fig. 8. Estimation of MSDPu parameters for Blackingstone Quarry granite submitted to CTC (θ = 30°) loading conditions (experimental data from Franklin and Hoek 1970).

Table 1 shows the results obtained for data supplied by Shah and Hoek (1992). The PREO-SOLVER values and corresponding errors are also compared with those obtained with the simplex reflection method on Tennessee marble. In the PREO-SOLVER scheme, the range of σc was first assumed to vary from σc min = 0 to σc max = 200, and m was varied from mmin = 0 to mmax = 100; these are extreme values selected to illustrate that the imposed constraints do not have to be very restrictive. The initial stepping value for both parameters was deliberately large (a value of 10) in PREO. The PREO program gave σc = 110 and m = 10 with a sum of error squares ERR = Σ(σ1 cal. – σ1 exp.)2 = 2180. Using these

as starting values, with the constraints σc ≥ 0 and m ≥ 0, SOLVER gave σc = 135.05 and m = 5.481, a solution almost identical to that obtained by Shah and Hoek (1992) with the simplex reflection procedure (see Table 1). Table 2 shows a comparison between the results from the simplex reflection method provided by Shah and Hoek (1992) and those from the PREO-SOLVER regression procedure for a basalt rock mass. In PREO-SOLVER, the uniaxial compression strength of the intact rock, σc = 305 MPa (a value supplied by Shah and Hoek 1992, see their Table 7), was used as the initial value of SOLVER. The range of parameter m was from 0 to 50, with steps of 10, and parameter © 2000 NRC Canada

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Fig. 9. Estimation of MSDPu parameters for dolomite submitted to different loading conditions with various Lode angles θ (experimental data from Mogi 1971): (a) in the J21/2–I1 plane; and (b) in the π plane.

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Fig. 10. Estimation of MSDPu parameters for trachyte subjected to different loading conditions with various Lode angles θ (experimental data from Hoskins 1969): (a) in the J21/2–I1 plane; and (b) in the π plane.

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Fig. 11. Estimation of MSDPu parameters for granite submitted to different loading conditions with various Lode angles θ (experimental data from Mazanti and Sowers 1965): (a) in the J21/2–I1 plane; and (b) in the π plane.

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Fig. 12. Estimation of MSDPu parameters for concrete submitted to different loading conditions with various Lode angles θ (experimental data from Mills and Zimmerman 1970): (a) in the J21/2–I1 plane; and (b) in the π plane.

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Can. Geotech. J. Vol. 37, 2000 Table 1. Comparison between simplex reflection method and the proposed procedure for determination of the Hoek-Brown parameters for intact Tennessee marble. Experimental results

Simplex reflection method (Shah and Hoek 1992) (σc = 135.03, m = 5.482)

PREO-SOLVER (σc = 135.05, m = 5.481)

σ3 exp.

σ1 exp.

σ1 cal.

(σ1 cal. – σ1 exp.)2

σ1 cal.

(σ1 cal. – σ1 exp.)2

0.00 3.52 7.03 14.06 21.09 28.12 35.16 49.22

137.10 147.65 158.20 189.84 200.38 217.96 246.09 288.27

135.03 147.88 160.12 183.30 205.06 225.73 245.54 283.03

4.28 0.05 3.69 42.82 21.89 60.33 0.30 27.45 160.82

135.05 147.89 160.14 183.31 205.07 225.74 245.55 283.04

4.21 0.06 3.76 42.61 22.04 60.55 0.29 27.31 160.83

Total

Table 2. Comparison between simplex reflection method and the proposed procedure for determination of the Hoek-Brown parameters for a basalt marble rock mass.

Experimental results

Simplex reflection method (Shah and Hoek 1992) (σc mass = 54.81, m = 21.77, s = 0.03227)

PREO-SOLVER (σc = 305.6, m = 21.74, s = 0.03215)

σ3 exp.

σ1 exp.

σ1 cal.

(σ1 cal. – σ1 exp.)2

σ1 cal.

(σ1 cal. – σ1 exp.)2

0.00 3.05 6.10 9.15 12.20 15.26 18.31 21.36 24.41 27.46 30.51

61.00 146.10 200.90 257.20 304.80 335.60 371.50 408.20 434.10 460.00 485.50

54.81 155.57 214.72 261.69 302.09 338.31 371.32 401.99 430.78 458.03 484.00

38.32 89.68 190.88 20.20 7.34 7.35 0.03 38.58 11.02 3.86 2.25 409.51

54.80 155.58 214.73 261.71 302.11 338.34 371.35 402.02 430.81 458.07 484.04

38.39 89.85 191.28 20.37 7.22 7.48 0.02 38.21 10.81 3.73 2.14 409.50

Total

s varied from 0 to 1 with steps of 0.001. The PREO program gave m = 20 and s = 0.055 and ERR = Σ(σ1 cal. – σ1 exp.)2 = 1899.91. Using these parameters as initial values, with the constraints m ≥ 0 and 1 ≥ s ≥ 0, Microsoft Excel® SOLVER again produced the same results as the simplex reflection method. Hence, the procedure proposed here, although somewhat simpler and less intuitive, can provide results with the same level of accuracy as those of the simplex reflection method.

Discussion The PREO-SOLVER procedure can be used with a small number of test results to estimate the value of material parameters included in the selected nonlinear criterion (i.e., σc, σt, φb, and b for MSDPu). It is well known, however, that a minimum number of data is required to properly evaluate the behavior of geomaterials. In that regard, some organizations have, for instance, suggested using at least 10 valid results to determine the value of a given parameter (like σc). It can be shown, using standard statistical tools (e.g., Obert and Duvall 1967; Lee et al. 1983; Jury 1986), that the minimum number of tests required can be related explicitly to the ob-

served mean and standard deviation values. The range obtained for the “true” property value, defined as a function of the observed values, provides the margin of error. The user then has to decide whether this margin is acceptable or not, and the number of data (or test results) required is established. This technique is commonly used in statistical analyses of small sample sizes. The type of tests to be performed may vary with the criterion expression. For instance, with a 2D formulation there is generally no need to do both conventional triaxial compression (CTC) and reduced triaxial extension (RTE) tests, as these two loading conditions cannot be properly distinguished with a criterion that uses only the extreme values (σ1 and σ3) of the principal stress tensor (unless two sets of parameters are used for these two different loading conditions). With 3D criteria, however, the possibility of explicitly defining the failure surface in the J21/2–I1 plane for various Lode angles (in the π plane) makes it relevant to perform both types of tests (i.e., CTC and RTE). As mentioned earlier, because the geometric function in the π plane (Fπ with MSDPu) goes from 1 at θ = 30° to b at θ = –30°, assuming a value of b a priori (e.g., b ≅ 0.75) reduces the requirements to have both types of results. © 2000 NRC Canada

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This shows that the formulation of the criterion itself gives an indication of the types of tests that are best suited to deduce material parameters. In the case of MSDPu, the explicit and independent use of σc, σt, and φb strongly supports testing under uniaxial compression, uniaxial tension, and shear on plane surfaces; nevertheless, one can also obtain equally valuable information by using diametral compression tests instead of uniaxial tension tests (the two are not exactly equivalent to MSDPu), and CTC tests at relatively high confining pressure instead of shear tests (although tilt tests are easier to perform). The number of independent parameters influences the type of tests to be performed. For instance, with the HoekBrown criterion (2D and 3D versions) for intact rocks, only two parameters are required: σc and m. Once σc is known, the value of m can be obtained from tensile test results (i.e., σt value) or from CTC test results; however, the value of m may differ significantly depending on the tests chosen for its determination. This is why the authors believe that it is preferable not to link directly the results obtained under CTC and tensile loading and have introduced σt as a distinct material parameter in the criterion. This aspect was discussed recently by Aubertin et al. (1999a). As for the specific results shown above, it is interesting to observe that when the back-calculated value of σc model is compared with the average measured value σc exp. on a given rock type, the former is usually larger. This may be a consequence of the influence of damage created by coring and sample preparation which can be reduced when a confining pressure is applied. Hence, if triaxial test results (with radial confined stress σrad ≥ 0) are used to back-calculate σc using MSDPu, the value of σc can be overestimated when compared with the average measured values. An interesting question then arises, but which is somewhat beyond the scope of this paper: what is the real in situ uniaxial strength when only natural flaws exist in the rock? A partial answer was provided by Eberhardt (1998), who showed that the strength of rock samples tested in the laboratory tends to increase when the core depth decreases or when the in situ stress (causing the damage during coring) decreases. This may well indicate that the in situ strength of rock, at the sample scale, might sometimes be higher than expected from laboratory measurements under uniaxial compression. A physically based criterion such as MSDPu, combined with a systematic parameter determination procedure such as that proposed here, can greatly help in finding answers to such questions. It also highlights the necessity of doing enough tests to properly characterize (from a statistical point of view) rock properties. It should be recalled that proper application of any type of limiting criterion should not be based solely on mean values, but should also take into account the variability of material properties.

Conclusions In this paper, the authors have presented a regression procedure that is a combination of an iterative pre-optimization scheme and a nonlinear programming method available in Microsoft Excel® and thus readily accessible. The procedure can be applied to estimate the parameters of any nonlinear

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criteria. As a first illustration, the proposed procedure was applied to evaluate MSDPu parameters for different types of rocks submitted to various loading conditions. The procedure was also applied to the 2D Hoek-Brown criterion. The results show that the simple procedure can provide excellent results, which are comparable to those obtained from a somewhat more complex and more intuitive method.

Acknowledgements Part of this work was financed by grants from the Natural Sciences and Engineering Research Council of Canada (OGP 0089749 and RGPIN 205014-98) and Institut de Recherche en Santé et en Sécurité du Travail du Québec, whose contributions are gratefully acknowledged.

References Aubertin, M., and Simon, R. 1998. Un critère de rupture multiaxial pour matériaux fragiles. Canadian Journal of Civil Engineering, 25(2): 277–290. Aubertin, M., Li, L., Simon, R., and Khalfi, S. 1999a. Formulation and application of a short-term strength criterion for isotropic rocks. Canadian Geotechnical Journal, 36: 947–960. Aubertin, M., Li, L., Simon, R., and Khalfi, S. 1999b. A unified representation of the damage and failure criteria for rocks and rock masses. In Proceedings of the 9th International Congress on Rock Mechanics, Paris, 25–28 August 1999. Edited by G. Vouille and P. Berest. A.A. Balkema, Rotterdam, The Netherlands, Vol. 2, pp. 843–848. Barton, N., and Choubey, V. 1977. The shear strength of rock joints in theory and practice. Rock Mechanics, 10: 1–54. Brace, W.F. 1964. Brittle failure of rocks. In Proceedings of the International Conference on State of Stress in Earth’s Crust, Santa Monica, 13–14 June 1963, pp. 111–178. Bruhns, O.T., and Anding, D.K. 1999. On the simultaneous estimation of model parameters used in constitutive laws for inelastic material behaviour. International Journal of Plasticity, 15: 1311– 1340. Cailletaud, G., and Pilvin, P. 1994. Identification and inverse problems related to material behaviour. In Inverse Problems in Engineering Mechanics, Proceedings of the 2nd International Symposium on Inverse Problems, ISIP’94, Paris, 2–4 November 1994. Edited by H.D. Bui, M. Tanaka, M. Bonnet, H. Maigre, E. Luzzato, and M. Reynier. A.A. Balkema, Rotterdam, The Netherlands, pp. 79–86. Carter, B.J., Scott Duncan, E.J., and Lajtai, E.Z. 1991. Fitting strength criteria to intact rock. Geotechnical and Geological Engineering, 9: 73–81. Charmichael, R.S. 1982. Handbook of physical properties of rocks. Vol. II. CRC Press, Boca Raton, Fla. Chen, W.F., and Baladi, G.Y. 1985. Soil plasticity — theory and implementation. Elsevier, New York. Desai, C.S., and Siriwardane, H.J. 1984. Constitutive laws for engineering materials with emphasis on geological materials. Prentice-Hall, Inc., Englewood Cliffs, N.J. Eberhardt, E. 1998. Brittle rock fracture and progressive damage in uniaxial compression. Ph.D. thesis, University of Saskatchewan, Saskatoon. Franklin, J.A., and Hoek, E. 1970. Developments in triaxial testing technique. Rock Mechanics, 2: 223–228. Goodman, R. 1980. Introduction to rock mechanics. John Wiley & Sons, New York. © 2000 NRC Canada

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Appendix 1: List of symbols a′, b′, A, B a1, a2, b c C0 F0

constants of a function material parameters used in the MSDPu criterion cohesion uniaxial compressive strength of rocks function defining the shape of the MSDPu surface in the J21/2 –I1 plane

Fπ function controlling the shape of the criterion in the π (octahedral) plane I1 first invariant of stress tensor σij (I1 = tr(σij)) J2 second invariant of deviatoric stress tensor Sij (J2 = 1/2Sij Sij) Lj lower bound on the value of jth parameter α j m, s Hoek-Brown criterion parameters Sij deviatoric stress tensor T0 uniaxial tensile strength (negative value) of rocks Uj upper bound on the value of jth parameter α j x, y, X, Y variables of a function xi, yi (i = 1, 2,…, k) set of k pairs of experimental data α slope of the linear portion of the MSDPu criterion in the J21/2–I1 plane α 1, α 2, n parameters included in the nonlinear function y = f(x, …, α n α 1, α 2, …, α n) α 1*, α 2*, …, α n* optimal solution for fitting a set of experimental data ∆α j increment of parameter α j ε J21/2cal. – J21/2exp. φ friction angle φ b basic friction angle σ normal stress σ1, σ2, σ3 major, intermediate, and minor principal stresses, respectively σc uniaxial compressive strength parameter used in MSDPu (σc = C0) σij Cauchy stress tensor σ rad radial confined stress in conventional triaxial compressive tests σ t uniaxial tensile strength parameter used in MSDPu (σt = |T0|) τ shear stress 3( 3)1 / 2 J 3 1 θ Lode angle, where θ = sin −1 3 2( J 23 )1 / 2 = tan −1

σ1 + σ 3 − 2 σ 2 , − 30 ≤ θ ≤ 30 ° 31 / 2 ( σ1 − σ 3 ) n

∏ multiplication operator, where

∏ (U j ) = U1 U2 ... Un j =1

Appendix 2: Macro program for estimating material parameters of the MSDPu criterion Sub OptimMSDPu() ‘ ‘ OptimMSDPu Macro ‘ Short key: Ctrl+Maj+A ‘ ‘Definition of variables: Dim C0 As Variant Dim C0R As Variant Dim T0 As Variant Dim T0R As Variant Dim phi As Variant Dim phiR As Variant Dim MinErr As Variant

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Dim Err As Variant MinErr = 999999999999.9 ‘ = = = = = = = = = = = = = Beginning of PRE-OPTIMIZATION procedure = = = = = = = = = = = = = = = = = For C0 = 0 To 200 Step 10 For T0 = 0 To 50 Step 10 For phi = 0 To 60 Step 5 Range(“B5”).Select ActiveCell.FormulaR1C1 = C0 Range(“B6”).Select ActiveCell.FormulaR1C1 = T0 Range(“B7”).Select ActiveCell.FormulaR1C1 = phi Err = ActiveSheet.Cells(11, 8).Value If Err < MinErr Then C0R = C0 T0R = T0 phiR = phi MinErr = Err ‘Record of the best values: Range(“H5”).Select ActiveCell.FormulaR1C1 = C0R Range(“H6”).Select ActiveCell.FormulaR1C1 = T0R Range(“H7”).Select ActiveCell.FormulaR1C1 = phiR Range(“H8”).Select ActiveCell.FormulaR1C1 = MinErr End If Next phi Next T0 Next C0 ‘ = = = = = = = = = = = = = End of PRE-OPTIMIZATION procedure = = = = = = = = = = = = = = = = = = = = = ‘To attribute the best (recorded) values as SOLVER’s initial values for the final optimization: Range(“B5”).Select ActiveCell.FormulaR1C1 = C0R Range(“B6”).Select ActiveCell.FormulaR1C1 = T0R Range(“B7”).Select ActiveCell.FormulaR1C1 = phiR ‘ = = = = = = = = = = = = = Beginning of SOLVER nonlinear regression procedure = = = = = = = = = = = = = = = = ‘To execute the nonlinear optimization program SOLVER: SolverOk SetCell: = “$H$11,” MaxMinVal: = 2, ValueOf: = “0,” ByChange: = “$B$5,$B$6,$B$7” SolverAdd CellRef: = “$B$5,” Relation: = 3, FormulaText: = “0” SolverAdd CellRef: = “$B$6,” Relation: = 3, FormulaText: = “$B$5/50” SolverAdd CellRef: = “$B$6,” Relation: = 1, FormulaText: = “$B$5/5” SolverAdd CellRef: = “$B$7,” Relation: = 3, FormulaText: = “0” SolverOk SetCell: = “$H$11,” MaxMinVal: = 2, ValueOf: = “0,” ByChange: = “$B$5,$B$6,$B$7” SolverSolve Solverdelete CellRef: = “$B$5,” Relation: = 3, FormulaText: = “0” Solverdelete CellRef: = “$B$6,” Relation: = 3, FormulaText: = “$B$5/50” Solverdelete CellRef: = “$B$6,” Relation: = 1, FormulaText: = “$B$5/5” Solverdelete CellRef: = “$B$7,” Relation: = 3, FormulaText: = “0” ‘ = = = = = = = = = = = = = End of SOLVER nonlinear regression procedure = = = = = = = = = = = = = = = = = = = = = End Sub

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