Rock Mechanics and Rock Engineering

, Volume 51, Issue 11, pp 3537–3561 | Cite as

Development of New Three-Dimensional Rock Mass Strength Criteria

  • Mohammad Hadi Mehranpour
  • Pinnaduwa H. S. W. KulatilakeEmail author
  • Ma Xingen
  • Manchao He
Original Paper


Two new three-dimensional rock mass strength criteria are developed in this paper by extending an existing rock mass strength criterion. These criteria incorporate the effects of the intermediate principal stress, minimum principal stress and the anisotropy resulting from these stresses acting on the fracture system. In addition, these criteria have the capability of capturing the anisotropic and scale dependent behavior of the jointed rock mass strength by incorporating the effect of fracture geometry through the fracture tensor components. The new criteria are proposed after analyzing 284 numerical modeling results of the polyaxial, triaxial and biaxial compression tests conducted on the jointed rock blocks having one or two joint sets by the PFC3D software. Some of these simulation results were compared with experimental results to validate the developed PFC3D model that was used for numerical modeling of jointed blocks. In this research to have several samples with the same properties a synthetic rock material that is made out of a mixture of gypsum, sand and water was used. Altogether, 12 joint systems were chosen; some of them had one joint set and the rest had two joint sets. Joint sets have different dip angles varying from 15° to 45° at an interval of 15° with dip directions of 30° and 75° for the two joint sets. Each joint set also has three persistent joints with the joint spacing of 42 mm in a cubic sample of size 160 mm. The minimum and intermediate principal stress combination values were chosen based on the uniaxial compressive strength (UCS) value of the modeled intact synthetic rock. The minimum principal stress values were chosen as 0, 0.2, 0.4 and 0.6 of the UCS. For each minimum principal stress value, the intermediate principal stress value varies starting at the minimum principal stress value and increasing at an interval of 0.2 of the UCS until it is slightly lower than the strength of the sample under the biaxial loading condition with the same minimum principal stress value. To express the new rock mass strength criteria, it was also necessary to determine the intact rock strengths under the same confining stress combinations mentioned earlier. Therefore, the intact rock was also modeled for all three compression tests and the intact rock strengths were found for 33 different minimum and intermediate principal stress combinations.


Discrete element method (DEM) Particle flow code (PFC) Rock mass strength Polyaxial compression test Intermediate principal stress Fracture tensor 

List of Symbols


Disk area

\(a\), \({a_2}\), \({a_3}\), \(b\), \({b_2}\), \({b_3}\)

Empirical coefficients


An empirical constant


Constant coefficient of the modified smooth-joint contact model


Joint cohesion

Dmin, Dmax

Minimum and maximum particle diameters

\({E_{\text{c}}}\), \({\bar {E}_{\text{c}}}\)

Contact and bond Young’s modulus of the linear parallel bond model, respectively

\({F_{ij}}\), \(F_{{ij}}^{r}\), \(F_{{ij}}^{k}\)

Fracture tensor, fracture tensor of the rock mass and fracture tensor of the kth joint set, respectively

\({F_{11}}\), \({F_{22}}\), \({F_{33}}\)

Fracture tensor components in the maximum, intermediate and minimum principal stress directions, respectively

\(f\), \({f_2}\), \({f_3}\)

Monotonically decreasing functions

\(K_{n}^{J}\), \(K_{s}^{J}\)

Joint normal and shear stiffnesses, respectively

\(k_{n}^{J}\), \(k_{s}^{J}\)

Joint normal and shear stiffnesses of the modified smooth-joint contact model, respectively


Minimum joint normal stiffness of the modified smooth-joint contact model

kr,\({\bar {k}_r}\)

Ratio of the normal to shear stiffnesses of the contact and bond for the linear parallel bond model, respectively


Number of parameters to be estimated

\({m^{\left( V \right)}}\)

Number of fracture centers


Total number of joint sets


Total number of data sets


Normal vector

\({n_i}\), \({n_j}\)

Projection of the normal vector in the directions of i and j, respectively

p, \(~{p_2}\), \({p_3}\), q, \({q_2}\), \({q_3}\)

Empirical coefficients


Coefficient of determination


Equivalent radius


Strength ratio between the jointed rock mass and intact rock strengths


Assumed volume

X, Y, Z

Cartesian coordinates

\(\lambda\), \({\lambda _0}\), \({\lambda _2}\), \({\lambda _3}\)

Empirical coefficients

\(\bar {\lambda }\)

Bond radius fraction of the Linear Parallel Bond Model


Friction coefficient of the Linear Parallel Bond Model


Joint friction coefficient of the Modified Smooth-Joint Contact model

\({\sigma _1}\), \(~{\sigma _2}\), \(~{\sigma _3}\)

Maximum, Intermediate and Minimum principal stresses, respectively

\({\sigma _c}\)

Uniaxial compressive strength

\({\bar {\sigma }_c}\), \({\bar {\tau }_s}\)

Bond tensile and shear strengths of the Linear Parallel Bond Model, respectively

\({\sigma _J}\), \({\sigma _I}\)

Jointed rock mass and intact rock strengths, respectively

\(\sigma _{{J,i}}^{P}\)

Predicted jointed rock block strength from the new rock mass strength criterion for data set i

\(\sigma _{{J,i}}^{{{\text{PFC}}}}\)

Strength of the jointed rock block from the PFC3D modeling for data set i

\(\overline{\sigma } _{{J,i}}^{{{\text{PFC}}}}\)

Average strength value of all the PFC3D data

\({\sigma _n}\)

Normal stress

\(\sigma _{n}^{J}\)

Normal stress on the Modified Smooth-Joint Contact

\({\varphi _J}\)

Joint friction angle



China University of Mining and Technology, Beijing


Discrete element method


Joint sides checking approach


Linear parallel bond model


Linear variable differential transformer


Modified smooth-joint contact model


National Institute for Occupational Safety and Health


Particle flow code


Three-dimensional particle flow code


Smooth-joint contact model


Uniaxial compressive strength


Three-dimensional distinct element code



The research was funded by the National Institute for Occupational Safety, and Health (NIOSH) of the Centers for Disease Control, and Prevention (Contract No. 200-2011-39886).


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • Mohammad Hadi Mehranpour
    • 1
    • 2
  • Pinnaduwa H. S. W. Kulatilake
    • 1
    Email author
  • Ma Xingen
    • 3
  • Manchao He
    • 3
  1. 1.Rock Mass Modeling and Computational Rock Mechanics LaboratoriesUniversity of ArizonaTucsonUSA
  2. 2.HPT-Laboratory, Faculty of GeosciencesUtrecht UniversityUtrechtThe Netherlands
  3. 3.State Key Laboratory for Geomechanics and Deep Underground EngineeringChina University of Mining and TechnologyBeijingChina

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