DEFORMABILITY MODULUS OF JOINTED ROCKS,

LIMITATION OF EMPIRICAL METHODS, AND

INTRODUCING A NEW ANALYTICAL APPROACH

Mahdi Zoorabadi

Introduction

Rock Mass

http://www.leeds.ac.uk/StochasticRockFractures/

http://www.ukgeohazards.info/pages/eng_geol/landslide_geohazard/eng_geol_landslides_rockslide_index.htm

Introduction

The commission of Terminology, symbols and graphic representation of the International Society for Rock

Mechanics ISRM ) ISRM, 1975 )

Modulus of elasticity or Young’s modulus (E) : The ratio of stress to corresponding strain below the proportionality limit

of a material.

Modulus of deformation of a rock mass (Em) : The ratio of stress (p) to corresponding strain during loading of a rock

mass, including elastic and inelastic behavior

Modulus of elasticity of a rock mass (Eem) : The ratio of stress (p) to corresponding strain during loading of a rock mass,

including only the elastic behavior

Introduction

Deformability

Modulus

Direct Methods (In situ Tests)

Indirect Methods

Empirical Equations (Based on rock mass classification systems)

Back Analysis

Direct Methods: Borehole Expansion Tests

Interfels DilatometerCambridge institute

Dilatometer

Goodman Jack

(http://www.slopeindicator.com/pdf/goodman%20jack%20datasheet.pdf)

𝐸 = 𝐶.∆𝜎

∆𝑑

𝐶 = 1 + 𝜗 . 𝐷0

Direct Methods: Plate Loading Tests

𝐸 = 𝐶.∆𝜎

∆𝑑

𝐶 =𝑎

2[2 1 − 𝜗2 𝑐𝑜𝑡−1 𝑧 + 1 + 𝜗

𝑧

1 + 𝑧]

Direct Methods: Extra Large Flat Jack

𝐸 = 𝐶.∆𝜎

∆𝑑

𝐶 = (1 − 𝜗2)𝑘𝑖

Rock Mass Classification

Rock Mass Classification

(Serafim and Pereira, 1983)

(Hoek and Diederichs,2006)

Rock Mass Classification

Parameter study on Hoek and Brown (1997) and Hoek and Diederichs (2006) equations (Zoorabadi 2010)

Stress dependency of deformability modulus which was not considered in

empirical equation

An applied normal stress on a rock fracture causes the fracture to close

and decreases the aperture.

Deformability of rock mass containing

discontinuities would have different values

at different depth or stress fields

Stress Dependency of Deformability Modulus

Stress Dependency of Deformability Modulus

New Procedure

(Li, 2001)

(Ebadi et al., 2011)

𝑘𝑛 = 𝑘𝑛𝑖 1 −𝜎𝑛

𝑉𝑚𝑘𝑛𝑖 + 𝜎𝑛

−2

(Bandis et al.,1983)

Stress Dependency of Deformability Modulus

New Procedure

𝑘𝑛𝑖 = −7.15 + 1.75𝐽𝑅𝐶 + 0.02(𝐽𝐶𝑆

𝑎𝑗

𝑎𝑗 =𝐽𝑅𝐶

5(0.2

𝜎𝑐𝐽𝐶𝑆

− 0.1

𝑉𝑚 = 𝐴 + 𝐵 𝐽𝑅𝐶 + 𝐶𝐽𝐶𝑆

𝑎𝑗

𝐷

(Bandis et al.,1983)

(Barton and Choubey 1977)

(Milne et al. 1991)

(Barton 1982)

(Barton and De Quadros 1997)

Case Study

Joint set Dip Dip/Dir Spacing [m]

JRC JCS

A 85 113 2.03 13 30

B 64 41 1.77 13 30

C 80 331 3.83 13 30

Bedding plane

24 156 4 10 30

Elastic modulus of 16 GPa

Case Study

From Measurement:

• Maximum stress orientation: NW

• Ratio between maximum horizontal stress and minimum horizontal stress is 𝜎𝐻𝜎ℎ = 1.5

(Nemcik et al. 2005)

(Zoorabadi et al. 2015)

Case Study - Results

• Deformability modulus at the ground surface (zero acting normal stress was

assumed) was calculated to be 7.2 Gpa (around 0.45% of elastic modulus of intact

rock) .

• Deformability modulus increases significantly with depth increase: 0.78% of the

elastic modulus of intact rock at depth of 50 m.

• For depths deeper that 200 m, deformability modulus of a this rock mass would be more that 90% of the elastic modulus of intact rock.

Case Study - Results

(Snomez and Ulusay 1999).

𝐽𝑣 =

𝑗=1

𝑛1

𝑆𝑗

Joint surface condition of Fair/Good

GSI value for this case would be between

60-70 with average of 65.

Deformability modulus of rock mass would be

around 10 GPa using Hoek and Diederichs

(2006) and 15 GPa by Hoek and Brown (1997).

Conclusions

• Deformability modulus is a stress dependent parameter and increases as applied stress increases.

• All well-known empirical formulations do not consider this property of deformability modulus.

• A new procedure is proposed to quantify the stress dependency of deformability modulus.

• For this case study it was found that for depths higher than 200 m it approaches to the elastic modulus of intact rock.