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°