HOEK-BROWN MATERIAL MODEL

HOEK-BROWN MATERIAL MODEL

This document verifies that the Hoek-Brown model is correctly implemented in PLAXIS.For that purpose the elasto-plastic behaviour of rock mass is studied via a circularopening subjected to in situ stress in generalized Hoek-Brown material (α ≥ 0.5).PLAXIS results are verified against the closed form solution presented byCarranza-Torres (2004).

Used version:

• PLAXIS 2D - Version 2015.02

Geometry: In PLAXIS 2D a plane strain model is used with 15-noded elements. Takingadvantage of the model's symmetry, only one-quarter of the geometry is modeled, i.e. 25m × 25 m. The radius R of the opening equals 2 m. Far-field stress σ0 equal to 15000kPa is applied to the top and right boundaries of the model with Line loads. A meshrefinement circular zone with radius equal to 5 m is defined around the opening. Figure 1illustrates the model geometry.

Two separate cases are studied with respect to the support pressure inside the openingpi :

Study case 1: pi = 0 kPa

Study case 2: pi = 2500 kPa

In study case 1 the opening is unsupported. To simulate study case 2, a Line loadperpendicular to the circular surface is used with magnitude equal to 2500 kN/m/m,pointing outwards.

Figure 1 Model geometry and generated mesh (PLAXIS 2D)

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Material: The Hoek-Brown material model is used with unit weight γ equal to zero. Theadopted material parameters are:

Soil: E ' = 5.7 × 106 kN/m2 ν ' = 0.3 σci = 30 × 103 kN/m2 mi = 10

GSI = 50 D = 0 ψmax = 0◦

Meshing: The Fine option is selected for the Element distribution. A Coarseness factorequal to 0.1 is used inside the opening and at the mesh refinement zone in the vicinity.The generated mesh is illustrated in Figure 1.

Calculations: In the Initial phase zero initial stresses are generated by using the K0procedure (γ = 0). To generate far-field stress conditions, the line loads acting on the topand right model boundaries are activated in a separate phase (Phase 1) and thecalculation type is set to Plastic analysis. Both top and right boundaries are set to beFree, while both bottom and left boundaries are set to Normally fixed.

Two separate phases are introduced starting from Phase 1 and the calculation type is setto Plastic analysis as well. Phase 2 simulates the first study case (pi = 0 kPa). Phase 3simulates the second study case (pi = 2500 kPa) and the line load acting on the openingis activated. In both phases, the quarter circle area representing the opening isdeactivated. The Reset displacements to zero option is activated in Phases 2 and 3. Toobtain more accurate results the Max load fraction per step is set equal to 0.1.

Output: Figures 2 and 3 illustrate the plastic points in study cases 1 and 2. The radiusof the plastic zone Rpl equals 3.80 m and 2.58 m respectively.

Figure 2 Plastic points in study case 1 (PLAXIS 2D)

Verification: The Hoek-Brown failure criterion (Hoek & Brown, 1980) is used todetermine the shear strength of intact rock. Its generalized form is used for rock masses:

σ1 = σ3 + σci

(mb

σ3

σci+ s)α (1)

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Figure 3 Plastic points in study case 2 (PLAXIS 2D)

in which σ1 and σ3 are the major and minor principal stresses, and σci is the unconfinedcompressive strength of the intact rock material.

The coefficients mb, s and α are defined as (Hoek, Carranza-Torres & Corkum, 2002):

mb = mi exp(

GSI − 10028− 14D

)(2)

s = exp(

GSI − 1009− 3D

)(3)

α =12

+16

(e−GSI/15 − e−20/3

)(4)

in which GSI is the Geological Strength Index (Marinos & Hoek, 2000), taking values from0 to 100, and D is a factor which expresses the degree of disturbance of the rock mass,varying between 0 and 1.

Carranza-Torres (2004) presents a closed form solution for the problem of excavating acircular tunnel in an elasto-plastic material subjected to uniform loading. The solutionsatisfies the generalized Hoek-Brown criterion (Eq. (1)). Considering elasticperfectly-plastic behaviour, the transformed far-field S0 and internal Pi stresses areobtained as:

S0 =σ0

m(1−α)/αb σci

+s

m1/αb

(5)

Pi =pi

m(1−α)/αb σci

+s

m1/αb

(6)

The critical internal stress pcri is the internal pressure below which plastic zone with

radius Rpl is developed. The transformed critical internal pressure Pcri is calculated as:

µPcrα

i + 2Pcri − 2S0 = 0 (7)

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in which the parameter µ is defined as:

µ = m(2α−1)/αb (8)

The exact solution of the Eq. (7) can be approximated using the Newton-Raphson orother similar numerical methods. The critical internal stress pcr

i is then calculated as:

pcri =

[Pcr

i − s

m1/αb

]m(1−α)/α

b σci (9)

The radius of the plastic zone Rpl is obtained as:

Rpl = R exp

[Pcr1−α

i − P1−αi

(1− α)µ

](10)

The radial σplr and hoop σpl

θ stresses in the plastic region (R ≤ r < Rpl ) are obtained as:

σplr =

[Spl

r −s

m1/αb

]m(1−α)/α

b σci (11)

σplθ =

[Splθ −

s

m1/αb

]m(1−α)/α

b σci (12)

in which the transformed radial Splr and hoop Spl

θ stress are found as:

Splr =

[Pcr1−α

i +(1− α)µ ln

(r

Rpl

)] 11− α (13)

Splθ = Spl

r + µSplαr (14)

For the elastic region (r ≥ Rpl ), based on Lamé's solution, the transformed radial Selr and

hoop Selθ take the following form:

Selr = S0 −

(S0 − Pcr

i)(Rpl

r

)2 (15)

Selθ = S0 +

(S0 − Pcr

i)(Rpl

r

)2 (16)

The actual values of the radial σelr and hoop σel

θ stresses in the elastic region areobtained from Eqs. (11) and (12) by substituting Spl

r and Splθ with Sel

r and Selθ .

Based on Eq. (10) the radius of the plastic zone is 3.794 m and 2.582 m for the studycases 1 and 2 respectively. The divergence between analytical and PLAXIS 2D results is0.2% and 0.1%.

The radial σr and hoop σθ stresses are calculated based on Eqs. (11) and (12) for boththe plastic and the elastic region. Figures 4 and 5 present the comparison betweenPLAXIS 2D and analytical results for the study cases 1 and 2 correspondingly. In studycase 1, radial stress equals 0 kPa at distance 2 m to the tunnel axis, while in study case 2equals 2500 kPa (supporting stress). Both radial and hoop stresses converge to the

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value of the far-filed stress (15000 kPa) as the distance to the opening axis increases. Itis concluded that the results are in good agreement and the Hoek-Brown model iscorrectly implemented in PLAXIS.

Figure 4 Radial and hoop stress in study case 1 (PLAXIS 2D)

Figure 5 Radial and hoop stress in study case 2 (PLAXIS 2D)

REFERENCES

[1] Carranza-Torres, C. (2004). Elasto-plastic solution of tunnel problems using thegeneralized form of the hoek-brown failure criterion. International Journal of RockMechanics and Mining Sciences, 41, 629–639.

[2] Hoek, E., Brown, E. (1980). Empirical strength criterion for rock masses. Journal ofGeotechnical and Geoenvironmental Engineering, 106, 1013–35.

[3] Hoek, E., Carranza-Torres, C., Corkum, B. (2002). Hoek-brown failure criterion -2002 edition. Proceedings of NARMS-TAC, 1, 267–273.

[4] Marinos, P., Hoek, E. (2000). Gsi: a geologically friendly tool for rock mass strengthestimation. In ISRM International Symposium. International Society for RockMechanics.

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