Chapter 6
STRENGTH AND DEFORMATION
Intact rock Index properties
-
longitudinal velocity and degree of fissuring -Uni-axial compression
-
Point load strength -Shore hardness -Schmidt hardness -Brazilian test
-
4-Point beam test
- Uni-axial tension test
-Rock quality designation
Strength and deformation
1-Uniaxial compression
Uni-axial compressive strength is the ultimate stress a cylindrical rock specimen under axial
load. It is the most important mechanical properties of rock material, used in design analysis
and modeling. Along with measurements of load, axial and lateral deformations of the
specimen are also measured.
o
Stage I-the rock is initially stressed, in addition to deformation, existing micro cracks is
closing causing an initial non- linearity of the curve.
o
Stage II – The rock basically has a linearly elastic behavior with linear stress – strain
curves , both axially and laterally .
o
Stage III – The rock behaves near – linear elastic. The axial stress – strain curve is
near – linear and is nearly recoverable.
o
Stage IV – the rock is undergone a rapid acceleration of micro cracking events and
volume increase.
o
Stage V - The rock has passed peak Stress, but is still intact, even though the internal
Structure is highly disrupt. The specimen is undergone Strain softening (failure)deformation .
o
Stage Vl - The rock has essentially parted to form a Series of blocks rather than an
intact structure.
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-Uni-axial stress-strain at and after peak
Rocks generally fail at a small strain, typically around 0.2 to 0.4%. Brittle rocks, typically
crystalline rocks, have low strain at failure, while soft rock, such as shale and mudstone, tend
to have relatively high strain at failure.
Most rocks, including all crystalline igneous, metamorphic and sedimentary rocks, behave
brittle under uniaxial compression. A few soft rocks, mainly of sedimentary origin, behave
ductile
-Point load index:
Granite
5 - 15
Point load test is a simple index test for
Gabbros
6 - 15
rock material.
Andesite
10 – 15
It gives the standard point load index,
Basalt
9 – 15
is (50).
Sandstone
1–8
Mudstone
0.1 – 6
Limestone
3–7
Gneiss
5 – 15
schist
5 – 10
Slate
1–9
Marble
4 – 12
Quartzite
5 - 15
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-Measurement of the point load strength Is and of the indirect Co
A rock core is loaded diametrically between the tips of tow hardened steel cones, causing
failure through the development of tensile cracks parallel to the loading direction.
The load at failure P peak is recorded and the point load strength is calculated from:Is=ppeak/D2
Where D is the distance between the tow cone tips.The uniaxial compressive strength Co is
then indirectly obtained by using the empirical relationship: C0 = 24 * ls( 50 )
Where Is(50) is the point load strength of 50 mm (2 in.) diameter cores.
2-Triaxial compression
At depth, rock is subjected to axial and lateral stresses (triaxial), and compressive strength is
higher in triaxial condition.
True triaxial compression means 3 different principal stresses. It is often simplified by
making 2 lateral stresses equal to minor principal stress (axisymmetric triaxial test).
The behavior of rock in triaxial compression changes with increasing confining pressure:
a) peak strength increases;
b) Post peak behavior from brittle gradually changes to ductile.
Stress-strain behavior at
elastic region appears
the same as uniaxial
compression.
Axial strain, Ea (%)
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-Effect of intermediate principal stress
Axi symmetric triaxial test gives strength without considering the effect of intermediate
principal stress ( σ2 ),and generally under-estimate the strength.
Rock triaxial compressive strength generally increases with σ2 with a fixed σ3 .
When σ2 is excessively greater than σ3 , the strength may start to decrease.
3-Tensile strength
Rock material generally has a low tensile strength, due to the pre-existing micro cracks in the
rock material. The existence of micro cracks may also be the cause of rock failing suddenly in
tension with a small strain.
Rock material type's tensile strength can be obtained from several types of tests. The most
common tensile test is the Brazilian tests.
(a) Brazilian (indirect) test
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This is an indirect measurement of the tensile strength of rock.
A rock disk of uniform thickness is cut from a rock core, and is loaded diametrically between upper
and lower flat (or rounded) platens in a compression testing machine. Thus, a compressive line-load
is applied to the disk .When the peak load P peak is reached the disk will typically split along the
loaded diameter.
Theoretical stresses in Rock during Brazilian test
Stress parallel to load is compressive at the edges and
tensile in the middle.
Stress perpendicular to the load is greater but rocks are
stronger in compression than tension so the rock splits in
tension.
Theoretical analysis shows that uniform tensile stress
develops this diameter. The tensile strength TB can be
obtained from the elastic solution:
𝟐𝐏𝐩𝐞𝐚𝐤
𝛑 ×𝐃 ×𝐭
Where P peak is the load at failure, t and D are the thickness and diameter of the disk
𝐓𝐁 =
respectively the ratio of t/D is 1/4 to 1/2.
-Four-point-beam modulus of rupture(tensile strength) TMR :
In this test a long rock core is flexured to failure. There are four contacting points dividing
the core in to three sections of equal length. The middle section is under pure bending. The
maximum tensile stress occurs in the bottom layer of the middle section of the core, and this is
where failure typically occurs as P reaches P peak. This test determines the modulus of
rupture (TMR ) (beam flexural tensile strength). From the theory of beams∶
𝐓𝐌𝐑 =
𝟏𝟔 ×𝐏𝐩𝐞𝐚𝐤 ×𝐋
𝟑 ××𝛑 ×𝐃𝟑
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-Direct Uniaxial Tension test Setup:
Difficult to attach sample ends and not bend the sample
4- shear strength:
Rock resists shear stress by tow internal mechanisms, cohesion and internal friction. Cohesion
is a measure of internal bonding of the rock material. Internal friction is caused by contact
between particles, and is defined by the internal friction angle.
Shear strength of rock material can be determined by direct shear test and by tri-axial
compression tests.
Shear strength from triaxial tests
From a series of triaxial tests, peak
stresse(σ1)Are obtained at various lateral
stresses (σ3) . By Plotting mohr circles, the shear
envelope is defined and gives the cohesion and
internal friction angle.
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Test
σ3(MPa)
σ1(MPa)
1
0
41.2
2
1
52.2
3
3
74.1
4
5
90.3
5
10
122
6
15
151
7
20
172
ROCK
UC Strength (MPa)
Tensile Strength (MPa)
Granite
100 – 300
7 - 25
Dolerite
100 – 350
7 – 30
Gabbro
150 – 250
7 – 30
Basalt
100 – 350
10 – 30
Sandstone
20 – 170
4 – 25
Shale
5 – 100
2 – 10
Dolomite
20 – 120
6 – 15
Limestone
30 – 250
6 – 25
Gneiss
100 – 250
7 – 20
Slate
50 – 180
7 – 20
Marble
50 – 200
7 – 20
Quartzite
150 – 300
5 – 20
Compressive, Tensile and shear strengths:
Tensile and shear strengths are important as rock fails mostly in tensile and in shear, even the
loading may appears to be compressive strength so failure in pure compression is not
common.
Theoretically, the three strength are related. This will be discussed in strengths criteria.
5- young's modulus and Poisson's Ratio
can be experimentally determined from the stress-strain curve. They seem to be unaffected by
change of confining pressure.
High strength rock also tend to have high young's modulus, depending on rock type and other
factors.For most rocks, the poisson's ratio is between 0.15 and 0.4.
Uniaxil compressive strength
- 66 -
,
Testing mechanical properties Uniaxial compression tests
Co = P_peak/A
E= σ1 /
Co : Uniaxial compressive strength , E : Modulus of Elasticity , V : poisson's ratio
6- Measuring longitudinal and shear velocities (Vl;Vs) in the lab:
-Testing setup:
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-Compressive and shear waves
 Compressive waves
Particle motion is parallel to the wave
direction

shear waves
particle motion is perpendicular to the
wave direction
-Testing physical properties :
Longitudinal velocity V1 and the Degree of Fissuring Df
The specific longitudinal Velocity V*l of rocks with zero Pores/fissures is given by:
𝟏
𝐂𝐢
∗ =∑ ∗
𝐕𝐥
𝐕𝐋,𝐥
𝐢
(𝑽∗𝒍 , 𝒊
is Longitudinal velocity of The i-th mineral constituent, Whose volume proportion is Ci)
-Different ways to average velocities Of constituent minerals
𝟏
-Reuss :
𝐕𝐥∗
𝐂
= ∑𝐢 ∗𝐢
𝐕𝐋,𝐥
-Hill :
∗
𝐕𝐥,𝐯𝐨=
∑𝐥 𝐂𝐥 𝐕𝐋,𝐥
-Voight :
𝐕𝐥∗,𝐇𝐢𝐥𝐥 =
𝐕𝐫𝐞𝐮𝐬𝐬+ 𝐕𝐯𝐨𝐢𝐠𝐡𝐭
𝟐
The index of continuity( IC ) of Rock is defined as: IC ( % ) =
𝐕𝐥
𝐕𝐥∗
× 𝟏𝟎𝟎
( whereV1 is the actual Measured velocity of rock) :
Spherical porosity np yields the empirical relationship (based on experiments):
ICp ( % ) = 100 - 1.6 np %
The Degree of Fissuring Df is
Obtained from a plot of
IC(%) versus n. If the
ICB of sample is given
by A in Fig ,then D f
Is found from the
Following relationship,
𝐃𝐟 ( % ) =
𝐈𝐂𝐩 − 𝐈𝐂𝐁
𝐈𝐂𝐩
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Where ICp (= (C in plot)
Ic for spherical pores
Only, and ICB (=A)is for both pores and fissures.
Classification
Degree of fissuring (%)
Non-fissured
0
Slightly fissured
0 – 10
Moderately fissured
10 – 25
Strongly fissured
25 – 50
Very strongly fissured
50 – 75
Example problem
-n=20%
-Mineralogy – quartz Vl.c. compression=6060 m/s
-Bedding perpendicular sample:
Vl compression measured = 2400m/s.
-Bedding parallel sample:
Vl compression measured = 3000 m/s .
-Calculate the index of continuity
-Degree of fissuring
-Explain the differences
7- Hardness
Hardness is the characteristic of a solid material to resist permanent deformation. Rock
material hardness depends on several factors, including
Mineral composition and density. A typical measure is the Schmidt rebound hardness
number. Hardness also used as indirect Uniaxial Compressive Strength test .Graduated
window showing the rebound of pellet or spring
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-Measurement of the shore hardness :
Shore hardness (Hshore ) is measured as the extent of rebound of a steel bullet dropped from a
specific height onto the surface of a rock specimen. The harder the rock, the higher the
bounce.
An empirical correlation between rock hardness and its uniaxial compressive strength has
been obtained based on a large number of tests on different rocks, and is given by:
Log Co ( psi ) = 0.000066 × γdry ( 1b/ft 3) ×Hshore +3.62
-Measurement of the Schmidt hardness :
The Schmidt Hammer is a portable tool, similar in principle to the shore scleroscope . It is
used exclusively for rock and rock-like materials and is easy of use in the field. It measures
the rebound off the surface of rock of a spring-driven steel pellet.
An empirical correlation between the Schmidt rock hardness and uniaxial compressive
strength has been obtained based on a large number of tests on different rocks:
Log Co ( psi ) = 0.000014 × γdry ( 1b/ft 3) ×Hsmdt +3.16
8-8---
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8-Abrasivity:
Abrasivity measures the abrasiveness of a rock materials against other materials, e.g., steel.
Abrasivity is highly influenced by the amount of quartz mineral in the rock material. The
higher quartz content gives higher abrasivity.
Abrasivity are measured by tests, e.g., cerchar
test
gives cerchar Abrasivity index (CAI).
Cerchar Abrasivitiy index (CAI)
Granite
4.5-5.3
Diorite
4.2-5.0
Andesite
2.7-3.8
Basalt
2.0-3.5
Sandstone
1.5-3.5, 2.8-4.2
Shale
0.6-1.8
Limestone
1.0-2.5
Gneiss
3.5-5.3
Siate
2.3-4.2
Quartzite
4.3-5.9
9- Fracture Toughness:
Fracture toughness of rock materials the effectiveness of rock fracturing. It is typically
measured by a toughness test. There are three fracture mode: (mode I), (mode II) (mode III).
Correspondingly, there are three fracture Toughness, K I , KII , and KIII .
In rock mechanics, mode I is associated with the crack initiation and propagation in rock
material
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-Rock Mass Classifications
 Rock Mass Rating (See tabals ( 6-6 )
-
Strength of the rock
-
Drill core quality ( RQD )
-
Groundwater conditions
-
Joint characteristics
-
Add values from the 5 tabales to get total score
Rock Strength
Tabla 6.6 rock mass rating Increments
For compressive strength of the rock
Unconfined
Point lode
Index (MPa)
compressive
OR
Strength (MPa)
rating
>10
>250
15
4-10
100-250
12
2-4
50-100
7
1-2
25-50
4
Don't use
10-25
2
Don't use
3-10
1
Don't use
<3
0
-ROCK Quality Designation
Applies to NX core (2.125" diameter) Hi RQD – few fractures in core LOW RQD – many
fractures in core
- 72 -
Joint spacing
 Can use drill core, if available
 Should consider joint set most likely Influence work
Table 6.7
Table 6.8 Rock
Increments of Rock mass Rating for spacing of
Mass Rating
Joints of most Influential set
Increments for
Drill core
Joint
Quality
spacing
(m)
rating
>2.0
20
0.6-2.0
15
0.2-.6
10
0.06-0.2
8
<0.06
5
RQD (%)
Rating
Joint condition -:
Table 6.9
Rock mass rating
Increments for Joint condition
Very rough surfaces of limited
Extent; hard wall rock
30
Slightly rough surfaces; aperture
Less than 1 mm; hard wall rock
25
Slightly rough surfaces; aperture
Less than 1 mm; hard wall rock
20
Smooth surfaces, OR gouge filling
1-5 mm thick, OR aperture of
1-5 mm; joints extend more than
Several meters
10
Open joints filled with more than
5 mm of gouge, OR open more
Than 5 mm; joints extend more
Than several meters
0
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>90
20
75-90
17
50-75
13
25-50
8
<25
3
Ground water condition:
Table 6.10 Increments of Rock mass rating
Due to Groundwater condition
Joint water
Inflow per 10 m
Tunnel Length
pressure Divided
OR
by major principal
(L/mm)
OR
General
stress
condition
0
completely dry
Rating
None
15
<10
0.0-0.1
Damp
0.1-0.2
Wet
0.2-0.5
Dripping
>0.5
Flowing
10
10-25
7
25-125
4
>125
0
Table 6.11 increments of Rock masses rating
Class
Description of
RmR sum of
Rock mass
Rating from5
Categories
I
Very Good
81-100
Rock
II
Good Rock
61-80
III
Fair Rook
41-60
IV
Poor Rock
21-40
V
Very Poor Rock
0-20
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