Deformability modulus of jointed rocks, limitation of empirical methods, and introducing a new...
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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
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.