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Tensile behaviour in the Hoek-Brown model

Application	PLAXIS 2D PLAXIS 3D
Version	PLAXIS 2D CONNECT Edition V22.02 PLAXIS 3D CONNECT Edition V22.02
Date created	October 2022
Date modified	October 2022
Original author	Micha van der Sloot - Bentley User Success Group

The Hoek-Brown model is a well-known elastoplastic constitutive rock model that can simulate the behaviour of homogeneous and isotropic rock ranging from intact to heavily jointed rock mass. The Hoek-Brown criterion provides a non-linear approximation of the rock strength using a continuous formulation of the shear strength and the tensile strength.

The tensile limit of the Hoek-Brown criterion σ_t for the rock mass is defined as:

Hoek Brown equaiton

Hoek-Brown failure criterion in σ'1- σ'3 plane with indication of σt

Figure 1. Hoek-Brown failure criterion in σ'₁- σ'₃ plane with indication of σ_t

Tension limitation

There could be situations in which you would like to limit the tensile capacity of the rock mass, for instance when a rock cut is created by blasting, and one cannot, or should not, rely on the tensile capacity due to generated (micro)fractures.

This can directly be controlled in PLAXIS by activating the tension cut-off option and defining an upper limit for the tensile stresses: the tensile strength. In the case of not allowing any tension in the rock mass for this material, the tensile strength can be set to a value of 0 kN/m².

Hoek-Brown material definition (left) and failure criterion in in σ'1- σ'3 plane (right)

Figure 2. Hoek-Brown material definition (left) and failure criterion in σ'₁- σ'₃ plane (right)

Note: this tension cut-off option is introduced in PLAXIS Version 22.02

Tensile behaviour in Safety analysis

In a Safety analysis, the PLAXIS calculation process will reduce the shear strength with increasing ΣM_sf. For the Hoek-Brown model, it means that the "slope" of the Hoek-Brown criterion in p'-q plane will be reduced with a fixed apex point [Benz, et al. (2007)]. This implies that during this strength reduction method, the tensile limit remains at σ_t if no additional limitation for the tensile strength is specified.

Strength reduction in the Hoek-Brown model: original definition (left) and reduced strength line (right)

Figure 3. Strength reduction in the Hoek-Brown model: original definition (left) and reduced strength line (right)

With the introduction of the tension cut-off option in PLAXIS V22.02, it is now also possible to reduce the tensile strength during a Safety analysis. Users can specify a value for the tensile strength equal to σ_t or any other user-defined value by explicitly activating the tension cut-off option. During the Safety analysis, the current tensile strength is then also updated as a function of the strength reduction factor as:

current tensile strength = user-defined tensile strength / ΣM_sf

Hoek-Brown model with tensile strength defined (left) and ΣMsf-factorized values during a Safety analysis (right)

Figure 4. Hoek-Brown model with tensile strength defined (left) and ΣM_sf-factorized values during a Safety analysis (right)

Example strength reduction

A simplified case is demonstrated here: a stressless block with a Hoek-Brown material, see geometry in Figure 5 and Hoek-Brown material properties in Table 1, is loaded in extension.

Hoek-Brown unit cell loaded in extension

Figure 5. Hoek-Brown unit cell loaded in extension

Parameter	Symbol	Value	Unit
Model		Hoek-Brown
Weight	γ_unsat=γ_sat	0	kN/m³
Stiffness	E_rm	2.4 x 10⁶	kN/m²
Poisson's ratio	ν	0.2
Uni-axial compressive strength intact rock	\|σ_ci\|	3380	kN/m²
Intact rock parameter	m_i	7
Geological Strength Index	GSI	65	0
Disturbance	D	0
Rock mass: strength in compression	σ_c	-479.9	kN/m²
Rock mass: strength in tension	σ_t	34.50	kN/m²

Table 1. Hoek-Brown parameters for unit cell extension test

The calculation procedure for this plane strain model is as follows:

Initial phase is a weightless K0 procedure. This gives a model where all stresses are equal to zero.
In the next plastic phase, the block is loaded in extension with a load of 30 kN/m/m in the upward direction.
A subsequent phase is set to a Safety analysis.
Rerun the phases with an activated tension cut-off and the tensile strength set to 34.50 kN/m².

Without the tension cut-off value, the calculation process is not able to reduce the tensile strength, and the ΣM_sf value increases to large values without leading to an upper limit. When activating the tensile strength equal to σ_t, the strength reduction factor ΣMsf should reach:

tensile strength / active load = 34.5 kN/m² / 30 kN/m² = 1.15

This is also the value we see in the calculation results when activating the tensile strength, see Figure 6 below.

Evolution of ΣMsf vs. displacements without tension cut-off (blue) and with tension cut-off (red)

Figure 6. Evolution of ΣM_sf vs. displacements without tension cut-off (blue) and with tension cut-off (red)

Conclusion

PLAXIS allows more control of the tensile behaviour with the activation of the tension cut-off option in the Hoek-Brown model. Also, when reducing the shear strength in a PLAXIS Safety analysis, we can now also reduce this tensile strength with the same factor, providing more realistic safety factors when developing failure mechanisms are related to these Hoek-Brown materials.

Note: The tensile behaviour in the Hoek-Brown with Softening model (UDSM) is defined in a different manner. Please see the documentation for more details.

Reference

Benz, T., Schwab, R., Vermeer, P.A., Kauther, R.A. (2007). A Hoek-Brown criterion with intrinsic material strength factorization. Int.J. of Rock Mechanics and Mining Sci., 45(2), 210–222.
doi: https://doi.org/10.1016/j.ijrmms.2007.05.003

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