FLAC3D Theory and Background • Factor of Safety

Strength Reduction Procedure in FLAC3D and 3DEC

The strength reduction method can be applied to essentially any material failure model to evaluate a factor of safety based upon the reduction of a specified strength property or property group. The method has been used extensively in the context of Mohr-Coulomb material and, principally, the simultaneous reduction of cohesion and frictional strength. From FLAC3D Version 5.0, in addition to Mohr-Coulomb strength properties (assigned with zone cmodel assign mohr-coulomb), the method is also applied automatically to ubiquitous-joint strength properties (assigned with zone cmodel assign ubiquitous-joint), and to Hoek-Brown strength properties (assigned with zone cmodel assign hoek-brown) when the model factor-of-safety command is given.

The strength reduction method can also be applied when model factor-of-safety is executed for interface cohesion and friction in FLAC3D (assigned via zone interface node property), or joint cohesion and friction in 3DEC (assigned via block contact property).

The procedure for implementing the strength reduction technique via the model factor-of-safety command is as follows.

First, the code finds a “characteristic response time,” which is a representative number of steps (denoted by \(N_r\)) that characterizes the response time of the system. \(N_r\) is found by setting the material strength (for Mohr-Coulomb material, the cohesion and tensile strength) to a large value, making a large change to the internal stresses (by default, a perturbation factor of 2 is applied to the stress state), and finding how many steps are necessary for the system to return to equilibrium.

A maximum limit of 50,000 is set for \(N_r\) by default. If the model does not reach equilibrium within 50,000 steps, the run will stop and the factor-of-safety solution cannot be completed. If this happens, the user should review the parameters selected for the model. For example, if the user has selected structural support with a high value for Young’s modulus, this may affect the solution convergence time.

It is also possible to set the value for \(N_r\) manually by using the characteristic-steps keyword to specify a value for \(N_r\). Alternatively, the initial perturbation to the internal stresses can be changed by specifying a different perturbation factor using the perturbation keyword. Note that these manual controls should be used with caution.

After \(N_r\) is determined, for a given strength reduction factor, \(F\), \(N_r\) steps are executed. If the unbalanced force ratio [1] is less than 10-5 after \(N_r\) steps, then the system is in equilibrium. If the unbalanced force ratio is greater than 10-5, then another \(N_r\) steps are executed, exiting the loop if the force ratio is less than 10-5. The value of force ratio, at the end of the current span of \(N_r\) steps, is compared with the value of force ratio at the end of the previous \(N_r\) steps. If the difference is less than 10%, the system is deemed to be in nonequilibrium, and the loop is exited with the new nonequilibrium, \(F\). If the above-mentioned difference is greater than 10%, blocks of \(N_r\) steps are continued until: (1) the difference is less than 10%; or (2) 6 such blocks have been executed; or (3) the force ratio is less than 10-5. The justification for case (1) is that the mean force ratio is converging to a steady value that is greater than that corresponding to equilibrium; the system must therefore be in continuous motion.

The following information is displayed during the solution process:

  1. the operation currently being performed;
  2. the number of calculation steps completed to determine a given value of \(F\);
  3. twice the characteristic response time, \(N_r\) (note that the change in unbalanced force ratio is first checked at two times \(N_r\));
  4. unbalanced force ratio;
  5. the number of completed solution stages (i.e., tests for equilibrium or nonequilibrium); and
  6. current bracketing values of \(F\).

The factor of safety solution stops when the difference between the upper and lower bracket values becomes smaller than 0.005 times the mean value. (This resolution limit can be changed with resolution, an optional keyword to the model factor-of-safety command.)

The bracketing solution approach invoked with the model factor-of-safety command may perform a large number of (stable and unstable) solutions before determining a factor of safety. If an approximate range for the factor is known, then the number of solutions (and total solution time) can be reduced by specifying the starting bracket values. This can be done with bracket v1 v2, an optional phrase to the model factor-of-safety command. If the calculated factor falls outside the specified brackets, a warning message will be issued. It is also possible to test whether a specified factor is above or below the actual factor, by setting bracket v1 v2.

The following conditions should be noted when using model factor-of-safety.

  1. The model state must be saved before a model factor-of-safety calculation is performed.
  2. The initial stress state can either be at a zero stress state or at stress equilibrium for the model factor-of-safety calculation. If the model is at a zero stress state, only gravity loading is applied to determine \(N_r\).
  3. The factor of safety calculation is performed in small-strain calculation mode when model factor-of-safety is issued.
  4. If a factor of safety calculation is performed for a coupled fluid flow-mechanical model (with model configure fluid specified), the fluid flow calculation will be turned off and fluid bulk modulus will be set to zero when model factor-of-safety is issued.
  5. The factor of safety calculation assumes nonassociated plastic flow with model factor-of-safety. The keyword associated can be added for an associated plastic flow calculation.

Three files are saved during the model factor-of-safety calculation: the initial-condition state, the latest stable factor of safety calculation, and the latest unstable factor of safety calculation. By default, the initial state is saved to a file named “FOS-Init.sav”, the stable state is saved to a file named “FOS-Stable.sav”, and the unstable state to a file named “FOS-Unstable.sav”. Each factor of safety calculation stage starts from the “FOS-Init.sav” state.

A different file name can be specified for each of these three files (see the model factor-of-safety filename command).

Strength Reduction Properties

The strength properties that can be reduced when using model factor-of-safety are described in the following sections.

Mohr-Coulomb Material

If the Mohr-Coulomb model failure criterion is prescribed, cohesion, \(c\), and friction angle, \(\phi\), are selected by default to be included in the safety factor calculation when executing model factor-of-safety. The reduction equations for these properties are

(1)\[c^{\rm trial}\ =\ {1 \over {F^{\rm trial}}}\ c\]
(2)\[\phi^{\rm trial}\ =\ \arctan\biggl({1 \over {F^{\rm trial}}}\tan\phi\biggr)\]

with the strength reduction procedure. These strengths can optionally be excluded from the model factor-of-safety calculation with the keyword phrase exclude cohesion or exclude friction.

Tensile strength, \(\sigma^t\), can also be included with the optional phrase include tension. The trial properties for tensile strength are calculated in a manner similar to that used for material friction and cohesion. The reduction equation for the tensile strength is

(3)\[\sigma^{t (trial)}\ =\ {1 \over {F^{trial}}}\ \sigma^t\]

Ubiquitous-Joint Material

If the ubiquitous-joint model is used, strength values for the intact material, \(c\) and \(\phi\), and strength values for the ubiquitous joints, \(c_j\) and \(\phi_j\), are included by default in the model factor-of-safety calculation. Tensile strengths, \(\sigma^t\) and \(\sigma^t_j\), can also be selected for reduction by adding the keyword phrases include tension and include joint-tension, respectively. The reduction equations for the intact material are the same as Equations (1) through (3), and for the ubiquitous joints are

(4)\[c_j^{trial}\ =\ {1 \over {F^{trial}}}\ c_j\]
(5)\[\phi_j^{trial}\ =\ \arctan\biggl({1 \over {F^{trial}}}\tan\phi_j\biggr)\]
(6)\[\sigma_j^{t (trial)}\ =\ {1 \over {F^{trial}}}\ \sigma_j^t\]

Ubiquitous-joint cohesion and friction can be excluded from the safety factor calculation with include joint-cohesion and exclude joint-friction, respectively.

Hoek-Brown Material

The Hoek-Brown model supports factor of safety calculations with model factor-of-safety. Strength reduction is performed with respect to shear strength flag-fos = 0).

Note that, although the softening/hardening capabilities of the Hoek-Brown model can be activated before the factor of safety calculation is performed, they should be disabled (by removing the table property assignment) during the strength reduction procedure because the value of the evolution parameter is then ill-defined.

Factor of Safety with respect to Shear Strength, \(\tau\)

The Hoek-Brown criterion can be approximated locally by a Mohr-Coulomb criterion:

(7)\[\tau = \sigma^{\prime} \tan{\phi_c} + c_c\]

where apparent cohesion and friction are given in terms of the local value of \(\sigma_3\) by

(8)\[\phi_c\ =\ 2 \tan^{-1} \sqrt{N_{\phi_c}}\ -\ 90^\circ\]
(9)\[c_c\ =\ {\sigma^{ucs}_c \over {2\sqrt{N_{\phi_c}}}}\]

where (for compressive stresses positive) if \(\sigma_3 \ge 0\):

(10)\[N_{\phi_c}\ =\ 1 + am_b {\Bigl(m_b {\sigma_3 \over \sigma_{ci}} + s \Bigr)}^{a-1}\]
(11)\[\sigma^{ucs}_c =\ \sigma_3 (1 - N_{\phi_c}) + \sigma_{ci} {\Bigl(m_b {\sigma_3 \over \sigma_{ci}} + s \Bigr)}^a\]

and, if \(\sigma_3 < 0\):

(12)\[N_{\phi_c}\ =\ 1 + am_b (s)^{a-1}\]
(13)\[\sigma^{ucs}_c =\ \sigma_{ci} (s)^a\]

A pragmatic approach to evaluate a factor of safety for slopes (based on the strength reduction technique) is used, whereby local cohesion, \(c_c\), and friction coefficient, \(\tan{\phi}_c\), are divided by a factor until active slope failure is detected. The factor directly applies to the maximum allowable value of shear stress \(\tau_{max}\) (see Equation (7)). The reduction factor at the verge of slope collapse is defined as the FOS based on the proposed (local strength reduction) technique.

Although, in theory, it is possible to find a best fit to match the reduced envelope with a Hoek-Brown type equation (see, e.g., Hammah et al. 2005), this step is not required with this particular model implementation because the logic relies on the direct use of envelope tangent (there is no need to define a curve and then the tangent when the tangent is available in the first place—see above). Also, the proposed local strength reduction technique provides a means by which to quantify the shear stress allowance to collapse, as one would expect. In this case, the reduction factor does not apply directly on model parameters (there is no absolute reason why it should).

Factor of Safety with Respect to Unconfined Compression Strength, \(\sigma_{ci}\)

The Hoek-Brown model supports an alternative factor of safety calculation with model factor-of-safety; strength reduction is performed with respect to unconfined compression strength by setting flag-fos = 1. The intact unconfined compressive strength is reduced by a reduction factor until active failure is detected to allow comparison with stability charts for simple slopes obtained by Li et al. (2008), using limit analysis.

Interfaces (FLAC3D)

Interface strengths can be included in the safety factor calculation by adding interface include propname to model factor-of-safety. For the interface strength values \(c_i\) and \(\phi_i\), the equations are

(14)\[c_i^{trial}\ =\ {1 \over {F^{trial}}}\ c_i\]
(15)\[\phi_i^{trial}\ =\ \arctan\biggl({1 \over {F^{trial}}}\tan\phi_i\biggr)\]

Coulomb Joints (3DEC)

Joint strengths can be included in the safety-factor calculation by assigning Coulomb joint material with block contact jmodel assign mohr. By default, joint cohesion and friction angle are included when model factor-of-safety is issued. The strength reduction equations for these properties are

\[c_i^{trial}\ =\ {1 \over {F^{trial}}}\ c_i\]
\[\phi_i^{trial}\ =\ \arctan\biggl({1 \over {F^{trial}}}\tan\phi_i\biggr)\]

Joint strength properties can be included or excluded with the keywords include/exclude cohesion, friction.

Endnotes

[1]The unbalanced force is the net force acting on a gridpoint. The ratio of this force to the mean absolute value of force exerted by each surrounding zone is the unbalanced force ratio. The limiting value for the unbalanced force ratio can be changed with the optional keyword ratio to the model factor-of-safety command.