Mohr Circles F 3 F 2 T N V F 4 F 1 H F 5 N Area = sin H T - PDF document

Mohr Circles

F 3 F 2  T N V F 4 F 1 H F 5

N Area = sin  H T  Area = cos         F H T cos Nsin 0 H        F V T sin Ncos 0 V V

   1 Area = sin   x  sin     1  Area = cos           sin cos sin 0   x          cos sin cos 0   y  y  cos 

Double Angle Formulas           x y x y cos2  2 2       x y sin2  2

    x y 2     ,    2     x y 2

Mohr Circle Examples

(52,0) (39,18.6) (32,0) 35  (12,0)

(52,0) (39,18.6) +15  -20  (32,20)

(6,2) (-4,-2)

   3  2  1

Mohr-Coulomb

 

 (  ff  ff )  3   f   f   1 Pole ½  1  3 ) ½  1  3 )

 (  ff  ff )  3   f  –  f  1 Pole (  ff  –  ff )

+  f –  f

  ff  f  3   f   1  f Pole

  ff  f  3   f   1  f Pole

  ff  f  3   f   1  f Pole

 (  ff  ff )  ff = c c  3    1 Pole

Obliquity Relationships

Cohesive Soil  R c      3 f ff 1 f  c cot D

Cohesive Soil   1         R 1 f 3 f 2     1 f 3 f sin          1 D 2 cot c      c cot 1 f 3 f 1 f 3 f 2

 Cohesionless Soil 1 f  R ff  3 f  D  

Cohesionless Soil   1         R 1 f 3 f 2     1 f 3 f sin        1 D    1 f 3 f 1 f 3 f 2

Obliquity Relationships         1 sin     1 f 2   tan 45        1 sin 2 3 f         1 sin     3 f 2   tan 45        1 sin 2 1 f

Angle of Obliquity  R  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 2   3 tan 45 40tan 61 130 kPa 1 f f   2

 130 Cohesionless Soil 45 61kPa ff     45sin 32 40  85  85 32°   ff 