FAILURE CRITERIA OF ROCKS

1
FAILURE CRITERIA OF ROCKS
(1) Coulomb-Navier Criterion
Navier ???????????????????????????????? Coulomb
(Max. shear stress principle) Normal stress
?????????????????? Failure plane ???? ??????????
Shear resistance ????????????????????????????????
? ????????????????????????? shear plane
???????????
2
Two Dimensional Case . Normal Stress
?? ( ? 1 ? 3)/2 (? 1 - ? 3)/2 . cos 2 ?
. Shear Stress
?? (?1 - ? 3)/2 . sin 2?
??? ?? S0 - ? ? ?
???????? ???? S0 ?????? ??? ?? ?????? ??
?????????? ?????
S0 ? ?? ?? ?( ? 1 ?3)/2 ( ? 1-
?3)/2 . (sin 2 ? ? cos 2?) ---- (1)
3
??? S? Max. shear strength d
S0/ d? 0 ,(take diff equation (1)) ???
tan 2? 1/ ? . (2)
??? (1) and (2) ???????? (2) ?? (1) ?????????
So ?1 /2 ? ( u2 ?1)1/2 ?3 /2
?-(?2 ?1)1/2 (3)
??? ? 0 ????? (1) ????? ? 45o ?????
So 1/2( ?1 - ?3) ??? ? gt 0 , ? lt 45 o
?????? ? ? ????? ? 0 ?
????????????????? shear strength
????????????????????????? Tensile , To , ???
Compression , Co , ??? ?????? ?1 To ???
?3 0 (zero) ???????? (3) ????????? -
4
Failure in tension
So To / 2 ? (?21)1/2
(A) ????? (A) ?????????? shear strength
????????? tensile strength ?????? ? 3 -
C o ??? ? 1 0 ????????? - So C o / 2
( ?2 1)1/2 - ? .. (B)
Failure in compression

• (B) ?????????
• (Co / To) ? (?2 1)1/2/(?2 1)1/2 - ?

5
???? Coulomb Criterion
Failure envelop
For 2-D case ? n 1/2 (? 1 ? 3)
1/2 (? 1 - ?3) . cos 2? Friction (?) ? /
? n Therefore , ? w sin ? / w cos? tan
? ? 1/2 (? 1- ? 3) .sin 2? 1/2 (?
1 - ?3) sin 2? ? o ?1/2 (? 1 ?3)1/2 (?
1- ? 3).Cos 2? ???? ? 1 2 ? o
?3sin 2? ?(1-cos 2?) ?1 2 ? o
? 3(1?2)1/2?/(1?2)1/2-? 2 ? o cos ?
?3(1sin ?) / (1-sin ?)
6
tan ? (1sin ?)/(1-sin ?) ?c 2?o
cos?/(1-sin ?) Ka (1-sin?)/(1sin?) (Active
earth pressure) Ko (1-sin?) (Earth pressure
at rest) Kp (1 sin ?)/(1-sin ?) (Passive
earth pressure) (See Soil Mechanics and
Foundations)
7

• MOHRS THEORY OF FAILURE(1900)
• ????????????????????????????????? , ? ,
  ?????????????????????????????????
  ?????????????????????????????? ????????????
  Normal Stress ????????????????? ???????
• ? f (?n)
• ??????????????????? tensile principal stress
  ????????????????? ???????
• ?1 T0
• ?1 tensile principal stress
• To tensile strength

8
A? t ?o B envelope of the Mohrs circle for
? 3 ? 1 at failure. ????????? Mohrs
envelope ??????????? ????? ?????????????????
Mohr ????????????????? Coulomb ???????
9
? ?( ?0 ?n tan ?)
Cohesion
??? tan ? cot 2? ? ????????? ?o
? ( ? o ? ? ?) ????????????? Coulombs Law
10
3. GRIFFITHS THEORY (1921)
Griffiths criterion of failure
????????????????????? ?????? microscopic cracks
??????????????? ??????????????????? stress
concentration ????????????? ???????????????? ????
????????????????????? ?????????????????????
??????????????????????????????
?max 2 ?0 (C/R)1/2
?max 2 ?0 (C/R)1/2
?max 2 ?0 (C/R)1/2
11
s1
Micro-cracks ??????????????????? fracture ?????
tensile stress ?????????????????
micro-cracks???????????? tensile strength ????????
s3
s3
s1
4 st s1 (1 tan2 f )1/2 - tan f - s3
(1 tan2 f )1/2 tan f
st Tensile strength
12
see details in Handbook on Mechanical
Properties of Rocks. Vol. I , V.S. Vutukuri ,
R.D. Lama, S.S. Salaja.
13
Microcracks development when the rock specimen is
subjected to stress.
Steps in a real density of microcracks
Increasing stress
14
Extension fracture (1 bar)
Brittle shear fracture (300 bar)
Plastic failure (1 kbar)
Types of failure observed in experimentally
deformed limestone at varying confining pressure
15
End of Lecture 5
664332 Engineering Geology I L. Archwichai ,
GTE, KKU.