The Delft Sand, Clay & Rock Cutting Model, 2019a

Basic Soil Mechanics.
Squaring equations (2-54) and (2-58) gives:
2 2
v h
v h
2
cos 2
2
2
(2-59)
And:
2
2 v
h
2
sin 2
2
(2-60)
Adding up equations (2-59) and (2-60) gives:
2 2
v h
2 v h
2 2
sin 2 cos 2
2
2
(2-61)
This can be simplified to the following circle equation:
2 2
v h
2 v h
2
2
(2-62)
If equation (2-62) is compared with the general circle equation from mathematics, equation (2-63):
2 2 2
C
C
x x y y R
(2-63)
The following is found:
x
v
h
x C
2
y (2-64)
yC
0
v
h
R
2
Figure 2-46 shows the resulting Mohr circle with the Mohr-Coulomb failure criterion:
c tan
(2-65)
The variable c is the cohesion or internal shear strength of the soil. In Figure 2-46 it is assumed that the cohesion
c=0, which describes the behavior of a cohesion less soil, sand. Further it is assumed that the vertical stress σv
(based on the weight of the soil above the point considered) is bigger than the horizontal stress σh. So in this case
the horizontal stress at failure follows the vertical stress. The angle α of the plane considered, appears as an angle
of 2·α in the Mohr circle. Figure 2-47: shows how the internal friction angle can be determined from a number of
tri-axial tests for a cohesion less soil (sand). The 3 circles in this figure will normally not have the failure line as a
tangent exactly, but one circle will be a bit too big and another a bit too small. The failure line found will be a best
fit. Figure 2-48 and Figure 2-49 show the Mohr circles for a soil with an internal friction angle and cohesion. In
such a soil, the intersection point of the failure line with the vertical axis is considered to be the cohesion.
Copyright © Dr.ir. S.A. Miedema TOC Page 57 of 454