12 - Mohr Circles and Failure Envelopes for Students/12 - Mohr Circles and... · Title: Microsoft...
Transcript of 12 - Mohr Circles and Failure Envelopes for Students/12 - Mohr Circles and... · Title: Microsoft...
Mohr Circles
F1
F2
F3
F4
F5
H
V
N
T
H
V
N
T
Area = sin
Area = cos
H
V
F H T cos Nsin 0
F V T sin Ncos 0
y cos
1
1
Area = sin
Area = cos
x
y
sin cos sin 0
cos sin cos 0
x sin
x y x y
x y
cos22 2
sin22
Double Angle Formulas
x y
2
x y
2
2
,
Mohr Circle Examples
(32,0) (52,0)(12,0) 35
(39,18.6)
-20
(52,0)
(39,18.6)
+15
(32,20)
(6,2)
(-4,-2)
123
Mohr-Coulomb
1
3
½13) ½13)
(ffff)
Pole f f
1
3
(ffff)
Pole f
(ff–ff)
–f
+f
–f
(ffff)
1
3
Pole f
f
(ffff)
1
3
Pole
ff = cc
Obliquity Relationships
D
R
cotc
3 f 1 fff
c
Cohesive Soil
1 3 1 3
1 31 3
12sin 1 2 cotcot
2
f f f f
f ff f
RD cc
Cohesive Soil
D
R
3 f 1 fff
Cohesionless Soil
1 3 1 3
1 31 3
12sin 12
f f f f
f ff f
RD
Cohesionless Soil
1 2
3
1 sintan 45
1 sin 2
f
f
3 2
1
1 sintan 45
1 sin 2
f
f
Obliquity Relationships
R
Angle of Obliquity
Angle of Obliquity
Example• A conventional triaxial compression test is conducted on
a medium-dense sandy silt with = 32.• If the cell pressure (3) is 40 kPa, what is the major
principal stress at failure?• What is the normal stress on the failure plane at failure?
2 21 3 tan 45 40 tan 61 130 kPa
2
f f
85
45
ff
Cohesionless Soil
85 45sin 32 61 kPa ff
40 13032°