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It is now recognized that many geomaterials have nonlinear failure envelopes. This non-linearity is most marked at lower stress levels, the failure envelope being of quasi-parabolic shape. It is not easy to calibrate these nonlinear failure envelopes from triaxial test data. Currently only the power-type failure envelope has been studied with an established formal procedure for its determination from triaxial test data. In this paper, a simplified procedure is evolved for the development of four different types of nonlinear envelopes. These are of invaluable assistance in the evaluation of true factors of safety in problems of slope stability and correct computation of lateral earth pressure and bearing capacity. The use of the Mohr-Coulomb failure envelopes leads to an overestimation of the factors of safety and other geotechnical quantities.

Comprehensive literature reviews on nonlinear envelopes can be found in [

envelope is the power-type envelope τ = τ o ( 1 + σ ′ σ ′ t ) 1 / m or τ = ( a + b σ ′ ) n where

τ = shear stress on failure arc; σ ′ = effective normal stress on failure arc = σ − u; σ = total normal stress; u = pore water pressure; σ_{t} = tensile strength of geomaterial; τ_{o} = cohesion of geomaterial; m is a constant. In the alternative representation a, b, and n are constants with the case of a = 0 being common and denotes a purely granular geomaterial. This had been used in several studies on slope stability [

a > [ b 2 n ( 1 − 2 n ) ] 1 2 ( 1 − n ) (1)

prior to this discovery the power-type equation was affirmed to be a valid envelope for n > 0.5 only [

Additionally, it was discovered that the quadratic equation

τ = ( a + b σ ′ ) 0.5 (2)

can only be a legitimate failure envelope if

a ≥ b 2 4 (3)

a and b are parameters in the equation of quadratic failure envelope.

The aims of this manuscript are to

1) Calibrate the modified Maksimovic nonlinear failure envelope from triaxial test data;

2) Develop a simplified procedure for calibrating the polynomial type failure envelope from triaxial test data;

3) Develop a simple methodology for calibrating the power-type failure envelope from triaxial test data;

4) Produce a simpler procedure for calibrating the power-type failure envelope equivalent to the Hoek-Brown failure envelope using triaxial test data;

5) Compute the lateral earth pressure for a material with modified Maksimovic failure criterion;

6) Determine the factor of safety of a slope made of material with modified Maksimovic criterion;

7) Determine the factor of safety of a slope made of material with Hoek-Brown criterion.

This work has never been presented in the literature before. As had been mentioned real soil envelopes are nonlinear and the calibration of these nonlinear envelopes will help in determining the true response of soils. The accurate calibration of the Maksimovic failure envelope from triaxial test data has never been attempted before. The influences of nonlinearity on lateral earth pressure and factor of safety of slope with material made of modified Maksimovic law have never been attempted before and are quite difficult to implement.

The calibration of a nonlinear failure envelope from triaxial test data is a problem of considerable difficulty [

They are:

σ ′ = σ ′ 3 + σ ′ 1 − σ ′ 3 1 + ∂ σ ′ 1 ∂ σ ′ 3 (4)

τ = σ ′ 1 − σ ′ 3 1 + ∂ σ ′ 1 ∂ σ ′ 3 ∂ σ ′ 1 ∂ σ ′ 3 (5)

where σ' and τ have their previous meanings, σ ′ 1 = effective major principal stress at failure, σ ′ 3 = effective minor principal stress at failure.

Given a set of experimental determined ( σ ′ 3 , σ ′ 1 ) values then the normal stress and shear stress on the failure plane can be calculated from Equations (4) and (5) respectively.

It could be shown that

∂ σ ′ 1 ∂ σ ′ 3 = N φ = 1 + sin φ ′ 1 − sin φ ′ (6)

where φ ′ = the effective instantaneous friction angle.

These authors deduced that the effective instantaneous cohesion c' is given by

c ′ = 1 2 [ σ ′ 1 ∂ σ ′ 3 ∂ σ ′ 1 − σ ′ 3 ∂ σ ′ 1 ∂ σ ′ 3 ] (7)

Equation (7) does not seem to exist in the literature. Fu and Liao (2010) seems to have derived the effective instantaneous cohesion for Hoek-Brown criterion that requires iteration to obtain.

Maksimovic’s [

According to Srbulov [

Maksimovic [

τ = σ ′ tan ( a 1 + 1 a 2 + a 3 σ ′ ) (8)

which has three parameters that needs to be determined. a_{1}, a_{2} and a_{3} are related to the Maksimovic’s parameters by φ_{B} = a_{1}, Δφ = 1/a_{2}, P_{N} = a_{2}/a_{3}. The determination of a_{1}, a_{2} and a_{3} is not straight forward at all. The parameters were determined for the experimental data presented in test No. O of Holtz and Gibbs [

Therefore a modified Makumovic hyperbolic law, that provided an improved match, was proposed namely

τ = σ ′ tan ( a 1 + σ ′ a 2 + a 3 σ ′ ) (9)

For a drained test on a granular material σ ′ = σ . By using Equations (4) and (5) values of σ and τ are computed in

By substituting for the largest values of σ and τ in

a 1 + σ m a 2 + a 3 σ m = arctan ( τ m σ m ) (10)

with σ_{m} = 1105.69 and τ_{m} = 814.85.

σ_{3} KNm^{−2} | σ_{1 } KNm^{−2} | ∂σ_{1}/∂σ_{3} | σ KNm^{−2} | τ KNm^{−2} |
---|---|---|---|---|

24.2 | 114.8 | 4.37 | 41.02 | 35.16 |

44.9 | 204.90 | 5.14 | 70.94 | 59.07 |

89.0 | 447.8 | 3.76 | 164.35 | 146.14 |

174.6 | 692.8 | 3.63 | 286.49 | 213.21 |

345.7 | 1380 | 3.90 | 556.77 | 416.84 |

690.0 | 2703 | 3.84 | 1105.69 | 814.85 |

To obtain the best values ofa_{2} anda_{3} the method of least squares is used. To implement the least squares method Equation (9) should be expressed as

σ arctan ( τ σ ) − a 1 = y = a 2 + a 3 σ (11)

Applying the method of least squares it is obtained that

a 2 N + a 3 ∑ σ i = ∑ y i (12a)

a 2 ∑ σ i + a 3 ∑ σ i 2 = ∑ σ i y i (12b)

where subscripti will run from 1 to N = number of data points which is 6 in this case. Equation (12) can be solved to obtain

a 3 = N ∑ σ y − ∑ σ ∑ y N ∑ σ 2 − ( ∑ σ ) 2 , (13a)

a 2 = ∑ y − a 3 ∑ σ N (13b)

The best fit values of a_{2} and a_{3} should be obtained as follows: A certain value of a_{1} near one is chosen and then values of a_{2} and a_{3} are calculated as shown above. Equation (10) is used to calculate another value of a_{1}. if the chosen and the calculated a_{1} are equal then the correct solutions have been obtained. If they are different, then, another iteration should be done.

When the above routine is implemented it is obtained that the parameters for the modified Maksimovic failure law for the triaxial data given in

a 1 = 1.0 , a 2 = − 64.35 , a 3 = − 2.6837 .

The Mohr-Coulomb approximation to

τ = 11.42 + 0.7269 σ (14a)

with a SEE of 28.73. SEE = the standard error of estimate calculated from

SEE = 1 N ∑ ( σ 1 p r e d − σ 1 ) 2

The modified Maksumovic approximation to

τ = σ tan ( 1.0 − σ 64.35 + 2.6837 σ ) (14b)

with a slightly lower SEE of 28.04 and the envelope passing through τ = 0 as required. The modified Maksimovic failure envelope would give a better factor of safety when the stability of geomaterial of shallow depth is considered. In

τ = 31.95 + 0.7542 σ (15a)

with a SEE of 31.82.

The modified Maksimovic approximation to

σ_{3} KNm^{−2} | σ_{1 } KNm^{−2} |
---|---|

22.8 | 192.5 |

45.5 | 280.1 |

88.3 | 523.7 |

173.9 | 825.9 |

347.0 | 1572 |

691.0 | 2886 |

τ = σ tan ( 1.0 − σ 254.55 + 2.7241 σ ) (15b)

with a lower SEE of 23.78. This time around the difference is clear. The two envelopes are compared in

Yuanming et al. [_{3} − σ_{1} triaxial test data shown in

Yuanming et al. [_{1} − σ_{3} relation (Equation (16)) that enables an analytical determination of ∂σ_{1}/∂σ_{3} but these authors used a finite difference approximation for ∂σ_{1}/∂σ_{3}.

σ 1 = ( K o ) σ 3 / P a t m σ c ( 1 + σ 3 σ T ) b o (16)

σ_{3} MNm^{−2} | σ_{1 } MNm^{−2} | σ_{3} MNm^{−2} | σ_{1} MNm^{−2} |
---|---|---|---|

0.0 | 2.285 | 5.0 | 11.877 |

0.3 | 3.289 | 6.0 | 12.924 |

0.6 | 4.155 | 8.0 | 14.924 |

0.8 | 4.726 | 10.0 | 17.161 |

1.0 | 5.308 | 12.0 | 19.381 |

2.0 | 7.392 | 14.0 | 20.795 |

3.0 | 9.541 | 16.0 | 22.571 |

4.0 | 11.140 | 18.0 | 24.953 |

where σ_{c}and σ_{T} are the uni-axial compressive and tensile strengths of the specimen respectively.

K_{o}and b_{o} are experimental parameters. Equation (16) is not needed in this study.

Using Equations (4) and (5) on

Let the cubic polynomial failure envelope be

τ = b 1 + b 2 σ + b 3 σ 2 + b 4 σ 3 (17)

Applying the method of least squares regression directly to this the following 4 equations are obtained

∑ τ = N b 1 + b 2 ∑ σ + b 3 ∑ σ 2 + b 4 ∑ σ 3 (18a)

∑ σ τ = b 1 ∑ σ + b 2 ∑ σ 2 + b 3 ∑ σ 3 + b 4 ∑ σ 4 (18b)

∑ σ 2 τ = b 1 ∑ σ 2 + b 2 ∑ σ 3 + b 3 ∑ σ 4 + b 4 ∑ σ 5 (18c)

∑ σ 3 τ = b 1 ∑ σ 3 + b 2 ∑ σ 4 + b 3 ∑ σ 5 + b 4 ∑ σ 6 (18d)

where N is number of data points and the subscripts have been deleted without loss of meaning.

When Equations (18a) to (18d) are solved for the constants b_{1} to b_{4} it is obtained that

b 1 = 0.6549 , b 2 = 0.6690 , b 3 = − 0.04815 , b 4 = 1.0773 × 10 − 3

Yuanming et al. [

The two sets of constants can be seen to be approximately the same.

According to this paper the polynomial envelope is τ = 0.6549 + 0.6690 σ − 0.04815 σ 2 + 1.0773 × 10 − 3 σ 3

The standard error of estimate was determined to be SEE = 0.3727. The Mohr-Coulomb envelope is determined to be τ = 2.036 + 0.0969 σ with an SEE = 1.1870. Thus, the Polynomial envelope provides a better fit to the experimental data. It is believed that the procedure used to do the calibration here is much more simpler and readily understood than the method of Yuanming et al.’s. A Q basic computer program [

The power-type failure envelope takes the form τ = ( a + b σ ) n . A somehow complicated procedure for calibrating the power-type failure envelope has been given in Baker [

Equations (4) and (5) are used to obtain the corresponding σ and τ values at failure. The power-type equation is then expressed as

a + b σ = τ 1 / n (19)

By assuming several values of n Equation (19) is subjected to least squares regression for each value of n. The value of n that yields the minimum SEE is chosen as the correct failure envelope. This has been programmed in the afore mentioned QBASIC 4.5 program for the automatic determination of n, a and b. For the data of

Hoek and Brown [

σ_{3} KNm^{−2} | σ_{1 } KNm^{−2} |
---|---|

69.0 | 225.8 |

138 | 383.7 |

276 | 664.4 |

strength of rock mass. The 2002 version of this equation [

σ 1 = σ 3 + σ c i ( m b σ 3 σ c i + s b ) η (20)

where m b = m i exp ( GSI − 100 28 − 14 D ) , s b = exp ( GSI − 100 9 − 3 D )

η = 1 2 + 1 6 [ exp ( − GSI / 15 ) − exp ( − 20 / 3 ) ]

where GSI is the Geological Strength Index, D is the disturbance factor, σ_{ci} = the compressive strength of the intact rock, m_{i} is a material constant of the intact rock.

The Hoek-Brown criterion can be expressed simply as

σ 1 = σ 3 + ( κ σ 3 + β ) η (21)

where κ = m b σ c i 1 η − 1 and β = s b σ c i 1 η

Letting F = κ σ 3 + β and d e n = κ η + 2 F 1 − η (22a, b)

then it can be shown that

σ = σ 3 + F d e n and τ = F 1 + η 2 d e n d e n − F 1 − η (23a, b)

Equations (23a, b) are the exact parametric (implicit) equations of the Mohr envelope of the H-B criterion. These equations with σ_{3} as the paramaeter are not very much useful in practice because the value of τ are not easily calculated from that of σ, vice versa. Hence, it is necessary to determine a single equation connecting τ and σ.

The H-B criterion utilized herein have the following constants s_{ci}= 40 MN/m^{2}, m_{i} = 10, GSI = 45, D = 0.9 with the rest calculated to be m_{b} = 0.281, η = 0.508, s_{b} = 1.616 × 10^{−4}.

The Hoek-Brown equation cannot be directly used in slope stability analysis because it is defined in terms of principal stresses. Considerable difficulty is encountered when Hoek-Brown equation is used directly in strength-reduction finite element type of slope stability analysis [

σ_{3} KNm^{−2} | σ_{1 } KNm^{−2} |
---|---|

0.0 | 474.2 |

20.0 | 671.6 |

40.0 | 831.1 |

80.0 | 1095.5 |

160.0 | 1519.8 |

300.0 | 2114.8 |

440.0 | 2619.1 |

614.0 | 3176.5 |

a SEE = 123.94. The envelopes derived in Anyaegbunam [^{2} and φ = 38.7˚ with a SEE = 131.77 can be seen to be in excellent agreement. The method used in this paper can be seen to be much simpler and readily understood than Anyaegbunam [

This shall be illustrated for a 2.0 m smooth high wall that is a portion of a wall embedded in homogeneous soil of unit weight γ = 18 KN/m^{3}. The vertical stress, which is the minimum principal stress, is given by σ_{3} = γz(i) where z(i) = vertical distance from the ground surface. Assume that the wall height is divided into a number of divisions such that σ_{3} can be denoted by σ_{3}(i) and that the number of divisions is num. On the basis of a Mohr-Coulomb law the passive pressure at a given point is given by σ 1 M C ( i ) = σ 3 ( i ) N φ + 2 C N φ . where C and φ are the

M-C shear strength parameters and N φ = tan 2 ( π 4 + φ 2 ) . C = 31.95 KN/m^{2}, tanφ = 0.7542, a_{1} = 1.0, a_{2} = −254.55, a_{3} = −2.7241.

The pseudo code for evaluating the passive pressure according to the modified Maksimovic law will be as follows:

Dimension the variables

set i = 1: eps = 0.001

450 compute σ_{3}(i): set σ_{3} = σ_{3}(i)’ 450 is line numbering in the code

compute σ_{1MC}(i): set σ 1 M C = σ 1 M C ( i ) (24)

Let σ_{n} = normal stress on the failure arc. Estimate σ_{n} as σ_{n}_{1}.

σ n 1 = 0.45 [ ( σ 1 M C + σ 3 ) − ( σ 1 M C − σ 3 ) sin φ M C ] _{ }

where φ M C = M-C friction angle φ

500 τ = σ n 1 tan ( a 1 + σ n 1 a 2 + a 3 σ n 1 )

Let d e r = d τ d σ

d e r = τ σ n 1 + a 2 σ n 1 ( a 2 + a 3 σ n 1 ) 2 [ 1 + τ 2 σ n 1 2 ]

t = τ 1 + d e r 2

From

σ n 2 = σ n = σ 3 + t − t 2 − τ 2

If i = num + 1 then END

If | σ n 2 − σ n 1 | > e p s then σ n 1 = σ n 2 : goto 500

If | σ n 2 − σ n 1 | ≤ e p s then σ 1 ( i ) = σ 3 + 2 t : i = i + 1 : goto 450

In

The M-C soil has a passive force and moment of 401.4 KN/m and 353.1 KNm/m at the base of the wall. The M.M. soil has a passive force and moment of 284.3 KN and 203.5 KNm/m at the base of the wall which are 29.2% and 42.4% less than the M-C values respectively. This is usually the case for short walls that are less than 6.0 m in height.

It is proposed to determine the factor of safety of a 45˚ homogeneous slope of

height 10 m via a Maksimovic material model and M-C material model using the Bishop simplified technique [

The material models were those derived from the triaxial data of ^{2}, ϕ = 37.02˚ and for the M. M. model: a_{1} = 1.0, a_{2} = −254.55, a_{3} = −2.7241.

After the soil mass is divided into vertical slices the Bishop method for M-C material is given by

F s = 1 ∑ γ h i sin α i ∑ c ′ i + ( γ h i − u i ) tan φ ′ cos α i ( 1 + tan α i tan φ ′ F s ) (25)

where F_{s} = factor of safety, h_{i} = midheight of slicei, α_{i} = inclination of base of slice i, u = pore pressure on base of slicei.

The Bishop method for modified Maksimovic (M.M.) material is given by

F s = 1 ∑ γ h i sin α i ∑ σ ′ i cos α i tan [ a 1 + σ ′ i a 2 + a 3 σ ′ i ] (26)

where σ ′ i is obtained for each slice from the equation

σ ′ i + tan α i σ ′ i F s tan [ a 1 + σ ′ i a 2 + a 3 σ ′ i ] − ( γ h i − u i ) = 0 (27)

The factor of safetyF_{s} is determined by assuming an initial value of F_{s} and calculating σ ′ i from Equation (27). Thereafter σ ′ i are substituted into Equation (26) to re-calculate F_{s}. The calculation of F_{s} using the Bishop method on a M.M. material model does not exist in the literature.

Using the M-C material model and 10 slices, the following results are obtained: F_{s} = 2.40 with the critical toe circle of radius 14.75 m centered at (−0.64, 14.74) with a central angle of 68.8˚. Using the M.M. material model the F_{s}= 1.64

H | β | Deng’s method | This paper’s method |
---|---|---|---|

15 | 30˚^{ } | 2.570 | 2.586 |

15 | 45˚ | 1.905 | 1.903 |

15 | 60˚ | 1.482 | 1.453 |

with the critical toe circle of radius 16.87 m centered at (−4.46, 16.27) with a central angle of 52.8˚. The use of the M.M. material model is seen to have a strong influence on the calculated factor of safety which is seen to be smaller.

Unfortunately the M.M. law has not been used for stability calculations by other authors and comparison of the results herein cannot be compared with previous results.

Deng et al. [

The H-B parameters of the rock slope are D = 0, GSI = 100, m_{i} = 10, σ_{ci} = 140 KNm^{−2}, α = η = 0.5. m_{b} and s_{b}are calculated from Equation (20). The rock unit weight is γ = 23 KN/m^{3}.

Also, σ c m = σ c i [ m b + 4 s b − α ( m b − 8 s b ) ] [ m b 4 + s b ] α − 1 2 ( 1 + α ) ( 2 + α ) (28)

σ 3 max = 0.72 σ c m ( σ c m γ H ) − 0.91 (29)

σ 3 values are listed from values of 0 to σ 3 max and the H-B relation is used to calculate corresponding values of σ 1 . From these the power-type equivalent to this Hoek-Brown law is deduced by section (3.4) to be τ = ( 134.53 + 14.431 σ ) 0.643 . The derived SEE = 1.26. In

Four Mohr failure envelopes have been presented and the methods of for deriving them from experimental triaxial test data have been explained. The method for computing the passive resistance for granular soil with M.M. (modified Maksimovic) envelope is presented. In addition, the influence of a M.M. envelope on the factor of safety of slope is presented and it is shown that the use of a M.M. material model results in a significant reduction in the calculated factor of safety. Also, it is shown that the factors of safety of rock slope obtained by this paper’s power-type approximation and LEM are close to Deng et al.’s factors of safety using H-B approximation and LEM. A computer program written in QBASIC version 4.5 for doing all the calibrations has been developed by the authors and is available on request from the corresponding author. This program is a useful contribution to geotechnical engineering practice.

The authors wish to thank the University of Nigeria for the payment of the 2019 earned allowances to its staff that helped in funding this research.

The authors declare no conflicts of interest regarding the publication of this paper.

Anyaegbunam, A.J. and Okafor, F.O. (2021) Calibration of Four Nonlinear Failure Envelopes from Triaxial Test Data and Influence of Nonlinearity on Geotechnical Computations. Geomaterials, 11, 42-57. https://doi.org/10.4236/gm.2021.112003