FLAC3D Theory and Background • Factor of Safety

Examples • Verification Problems

Simple Slope in Hoek-Brown Material

Problem Statement

Note

The project file for this example is available to be viewed/run in FLAC3D.[1] The project’s main data files are shown at the end of this example.

Verification exercises are performed to validate the factor of safety calculation using Hoek-Brown material in FLAC3D. The exercises test the strength reduction calculation based upon shear strength, \(\tau\).

The factor of safety with respect to Hoek-Brown shear strength is calculated for a simple slope geometry and compared to results based upon other methods to calculate a safety factor for Hoek-Brown material (generalized Hoek-Brown, equivalent Mohr-Coulomb, and Bishop and Spencer limit equilibrium methods) reported by Hammah et al. (2005). The rock slope for this comparison calculation has an inclination of 45° and a height of 10 m. The rock is represented as a Hoek-Brown material with the following properties:

\(E\)	=	5000 MPa
\(\nu\)	=	0.3
\(\rho\)	=	2500 kg/m3
\(m_b\)	=	0.067
\(s\)	=	0.000025
\(a\)	=	0.619
\(\sigma_{ci}\)	=	30 MPa

The FLAC3D model mesh used for this test is shown in Figure 1. By default, when model factor-of-safety is executed for a FLAC3D model with zone cmodel assign hoek-brown, the factor of safety calculation is performed for Hoek-Brown material with respect to shear strength. The calculated factor of safety for this test is 1.16. The failure surface is shown by the shear strain contour plot in Figure 2. The result compares well with the results reported by Hammah et al. (2005). Table 2 summarizes the safety factors reported for this test.

Figure 1: Slope model mesh.

Figure 2: Factor of safety and failure surface calculated for simple slope in Hoek-Brown material.

Table 2: Factor of Safety Results for Hoek-Brown Slope

Method	Factor of Safety
FLAC3D using Hoek-Brown	1.16
Generalized Hoek-Brown strength reduction*	1.15
Equivalent Mohr-Coulomb strength reduction*	1.15
Bishop’s simplified limit equilibrium*	1.153
Spencer’s limit equilibrium*	1.152

* from Hammah et al. (2005)

Hammah et al. (2005) also report the results for the case in which a horizontal layer of Mohr-Coulomb material is located at the toe of the slope. The layer is 1 m thick and has zero cohesion and 25° friction. The slope with the Mohr-Coulomb layer is shown in Figure 3.

When model factor-of-safety is issued, the strength reduction method is performed concurrently for Hoek-Brown material, as described in Hoek-Brown Material, and for Mohr-Coulomb material, as described in Mohr-Coulomb Material. The factor of safety calculated for this model is 0.96. The results are shown in Figure 4.

Table 3 compares the FLAC3D result with results from other methods reported by Hammah et al. (2005).

Figure 3: Simple slope in Hoek-Brown material with a Mohr-Coulomb layer.

Figure 4: Factor of safety and failure surface calculated for simple slope in Hoek-Brown material with Mohr-Coulomb layer.

Table 3: Hoek-Brown Slope with Mohr-Coulomb Layer

Method	Factor of Safety
FLAC3D using Hoek-Brown and Mohr-Coulomb	0.96
Generalized Hoek-Brown strength reduction*	0.95
Bishop’s simplified limit equilibrium*	0.934
Spencer’s limit equilibrium*	0.963

* from Hammah et al. (2005)

Reference

Hammah, R. E., et al. “The shear strength reduction method for the generalized Hoek-Brown criterion,” ARMA/USRMS 05-810, 2005.

Data Files

HoekBrownSlope.dat

;************************************************************************
;                  FoS wrt Shear Strength for HB Material
;*************************************************************************
model new
model large-strain off
; --- geometry ---
zone create brick point 0 ( 0,0,0) point 1 (40,0, 0) ...
                  point 2 ( 0,1, 0) point 3 ( 0,0, 8) size 80 1 16
zone create brick point 0 (15,0,8) point 1 (40,0, 8) ...
                  point 2 (15,1, 8) point 3 (25,0,18) ...
                  point 4 (40,1,8) point 5 (25,1,18) ...
                  point 6 (40,0,18) point 7 (40,1,18) size  50 1 20
zone face skin ; Label model boundaries
; --- Assign model and properties
zone cmodel assign hoek-brown
zone property density 2.5e-3 young 5000 poisson 0.3 tension 1e10
zone property constant-mb 0.067 constant-s 2.5e-5 ...
              constant-a 0.619 constant-sci 30
zone property flag-evolution 1 flag-fos 0
; --- boundary conditions ---
zone face apply velocity-normal 0 range position-x 0
zone face apply velocity-normal 0 range position-x 40
zone face apply velocity-normal 0 range position-y 0
zone face apply velocity-normal 0 range position-y 1
zone face apply velocity-normal 0 range position-z 0
; --- settings ---
model gravity 10
model save 'initial'
; --- solution ---
model factor-of-safety bracket 1.1 1.2 convergence 1 filename 'HBSlope'
model save 'initial'

HoekBrownSlope2.dat

;************************************************************************
;                  FoS wrt Shear Strength for HB Material
;                      with a Mohr-Coulomb Layer
;*************************************************************************
model new
model large-strain off
; --- geometry ---
zone create brick point 0 ( 0,0,0) point 1 (40,0, 0) ...
                  point 2 ( 0,1, 0) point 3 ( 0,0, 8) size 80 1 16
zone create brick point 0 (15,0,8) point 1 (40,0, 8) ...
                  point 2 (15,1, 8) point 3 (25,0,18) ...
                  point 4 (40,1,8) point 5 (25,1,18) ...
                  point 6 (40,0,18) point 7 (40,1,18) size  50 1 20
zone face skin ; Label model boundaries
zone group 'layer' range position-z 8 9
; --- Assign model and properties
zone cmodel assign hoek-brown
zone property density 2.5e-3 young 5000 poisson 0.3 tension 1e10
zone property constant-mb 0.067 constant-s 2.5e-5 ...
              constant-a 0.619 constant-sci 30
zone property flag-evolution 1 flag-fos 0
; --- mohr-coulomb layer ---
zone cmodel assign mohr-coulomb                range group 'layer'
zone property young 5000. poisson 0.3          range group 'layer'
zone property cohesion 0 friction 25 tension 0 range group 'layer'
; --- boundary conditions ---
zone face apply velocity-normal 0 range position-x 0
zone face apply velocity-normal 0 range position-x 40
zone face apply velocity-normal 0 range position-y 0
zone face apply velocity-normal 0 range position-y 1
zone face apply velocity-normal 0 range position-z 0
; --- settings ---
model gravity 10
model save 'initial2'
; --- solution ---
model factor-of-safety bracket 0.9 1.0 convergence 1 filename 'HBSlope2'
model save 'initial2'

Endnote

[1]	To view this project in FLAC3D, use the program menu. Help ▼ Examples…   ⮡   FLAC     ⮡   TheoryAndBackground       ⮡   FactorOfSafety         ⮡   HoekBrownSlope           ⮡   HoekBrownSlope.prj