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

cos sin cos sin
1
1
sin cos sin cos
Basic Soil Mechanics.
cos sin sin cos
1
sin cos sin cos
cos sin sin cos
1
sin cos sin cos
(2-104)
sin
1 sin cos
Substituting this result in the equation for the force on the retaining wall gives:
sin
1
2
F s
gh 1 w
2
sin cos
(2-105)
When the denominator in the term between brackets has a maximum, also the whole equation has a minimum. So
we have to find the maximum of this denominator.
f sin cos
F minimum if f maximum
(2-106)
The first derivative of this denominator with respect to the shear angle is:
df
cos2
d
(2-107)
The second derivative of this denominator with respect to the shear angle is:
2
df
d
2
2sin2
(2-108)
The first derivative is zero when the shear angle equals 45 degrees minus half the internal friction angle:
df 1
0 =
d
4 2
(2-109)
Substituting this solution in the equation for the second derivative gives a negative second derivative which shows
that a maximum has been found.
2
df 1
2 for =
2
d
4 2
(2-110)
Substituting this solution for the shear plane angle in the equation for the force on the retaining wall gives:
1 1 sin 1
F gh w g h w
K
2 1 sin
2
2 2
s
s p
(2-111)
The factor Kp is often referred to as the coefficient of passive failure, which is larger than 1. In the case of a 30
degrees internal friction angle, the value is 3.
Copyright © Dr.ir. S.A. Miedema TOC Page 67 of 454