Deformability modulus of jointed rocks is a key parameter for stability analysis of underground structures by numerical modelling techniques. Intact rock strength, rock mass blockiness (shape and size of rock blocks), surface condition of discontinuities (shear strength of discontinuities) and confining stress level are the key parameters controlling deformability of jointed rocks. Considering cost and limitation of field measurements to determine deformability modulus, empirical equations which were mostly developed based on rock mass classifications are too common in practice. All well-known empirical formulations dismissed the impact of stress on deformability modulus. Therefore, these equations result in the same value for a rock at different stress fields. This paper discusses this issue in more detail and highlights shortcomings of existing formulations. Finally it presents an extension to analytical techniques to determine the deformability modulus of jointed rocks by a combination of the geometrical properties of discontinuities and elastic modulus of intact rock. In this extension, the effect of confining stress was incorporated in the formulation to improve its reliability

1. DEFORMABILITY MODULUS OF JOINTED ROCKS,
LIMITATION OF EMPIRICAL METHODS, AND
INTRODUCING A NEW ANALYTICAL APPROACH
Mahdi Zoorabadi

2. Introduction
Rock Mass
http://www.leeds.ac.uk/StochasticRockFractures/
http://www.ukgeohazards.info/pages/eng_geol/landslide_geohazard/eng_geol_landslides_rockslide_index.htm

3. 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

4. Introduction
Deformability
Modulus
Direct Methods (In situ Tests)
Indirect Methods
Empirical Equations (Based on
rock mass classification
systems)
Back Analysis

5. Direct Methods: Borehole Expansion Tests
Interfels Dilatometer
Cambridge institute
Dilatometer
Goodman Jack
(http://www.slopeindicator.com/pdf/goodman%20jack%20datasheet.pdf)
𝐸 = 𝐶.
∆𝜎
∆𝑑
𝐶 = 1 + 𝜗 . 𝐷0

6. Direct Methods: Plate Loading Tests
𝐸 = 𝐶.
∆𝜎
∆𝑑
𝐶 =
𝑎
2
[2 1 − 𝜗2
𝑐𝑜𝑡−1
𝑧 + 1 + 𝜗
𝑧
1 + 𝑧
]

7. Direct Methods: Extra Large Flat Jack
𝐸 = 𝐶.
∆𝜎
∆𝑑
𝐶 = (1 − 𝜗2)𝑘𝑖

8. Rock Mass Classification

9. Rock Mass Classification
(Serafim and Pereira, 1983)
(Hoek and Diederichs,2006)

10. Rock Mass Classification
Parameter study on Hoek and Brown (1997) and Hoek and Diederichs (2006) equations (Zoorabadi 2010)

11. 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

12. Stress Dependency of Deformability Modulus
New Procedure
(Li, 2001)
(Ebadi et al., 2011)
𝑘 𝑛 = 𝑘 𝑛𝑖 1 −
𝜎 𝑛
𝑉𝑚 𝑘 𝑛𝑖 + 𝜎 𝑛
−2
(Bandis et al.,1983)

13. Stress Dependency of Deformability Modulus
New Procedure
𝑘 𝑛𝑖 = −7.15 + 1.75𝐽𝑅𝐶 + 0.02(
𝐽𝐶𝑆
𝑎𝑗
𝑎𝑗 =
𝐽𝑅𝐶
5
(0.2
𝜎𝑐
𝐽𝐶𝑆
− 0.1
𝑉𝑚 = 𝐴 + 𝐵 𝐽𝑅𝐶 + 𝐶
𝐽𝐶𝑆
𝑎𝑗
𝐷
(Bandis et al.,1983)
(Barton and Choubey 1977)
(Milne et al. 1991)
(Barton 1982)
(Barton and De Quadros 1997)

14. 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

15. Case Study
From Measurement:
• Maximum stress orientation: NW
• Ratio between maximum horizontal stress and minimum horizontal stress is 𝜎 𝐻
𝜎ℎ = 1.5
(Nemcik et al. 2005)
(Zoorabadi et al. 2015)

16. 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.

17. Case Study - Results
(Snomez and Ulusay 1999).
𝐽𝑣 =
𝑗=1
𝑛
1
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).

18. 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.