Civil Engineering Reference
In-Depth Information
s
2
s
1
Space
diagonal
F
mc
(6.12)
s
1
s
3
s
2
s
3
Π
plane
Figure 6.6
Mohr-Coulomb failure criterion
of principal stresses from the geometry of Mohr's circle, thus (assuming compression-
negative):
F
mc
=
σ
1
+
σ
3
2
φ
−
σ
1
−
σ
3
2
sin
−
c
cos
φ
(6.11)
Substituting for
σ
1
and
σ
3
from (6.5) gives the function:
cos
θ
√
3
−
sin
θ
sin
φ
F
mc
=
σ
m
sin
φ
+
σ
−
c
cos
φ
(6.12)
3
which shows that the Mohr-Coulomb criterion depends on all three invariants (
σ
m
, σ, θ
).
φ
u
=
The Tresca criterion is obtained from (6.12) by putting
0togive:
F
t
=
σ
cos
θ
√
3
−
c
u
(6.13)
This criterion is preferred to von Mises for applications involving undrained clays because
(6.10) is always satisfied at failure, regardless of the value of
σ
2
. In principal stress space
the Tresca criterion is a regular hexagonal cylinder tangential to the von Mises cylinder
defined by (6.7) and circumscribed by the one defined in (6.9) as shown in Figure 6.5.
6.5 Generation of body loads
Constant stiffness methods of the type described in this chapter use repeated elastic solutions
to achieve convergence by iteratively varying the loads on the system. Within each load
increment, the system of equations
}
i
=
{
}
i
{
[
K
m
]
U
F
(6.14)
