Civil Engineering Reference
In-Depth Information
coordinates that result held in the one-dimensional array
gc(1)
and
gc(2)
respectively. Only the
with the
x
and
y
y
coordinate is required in this case, and the vertical stress
σ
y
is obtained after multiplication by the unit weight held in
gamma
. The horizontal
effective stresses
σ
x
and
σ
z
are obtained by multiplying
σ
y
by the “at rest” earth pressure
K
o
held in
k0
.
Data relating to the geometry and boundary conditions follow a familiar course. After
the stiffness matrix formulation, the nodes and senses of the freedoms, which are to receive
prescribed displacements are read, followed by the plastic convergence tolerance
tol
,the
iteration ceiling
limit
, the number of constant displacement increments that are to be
applied
incs
and the magnitude of the displacement increment held in
presc
.Itmay
be noted that the iteration ceiling does not need to be as high as when using load control.
Convergence is quicker when using displacement control, especially as failure conditions
are approached, since unconfined flow cannot occur. The “penalty” technique is used to
implement the prescribed displacements, as described in Section 3.6.
The program follows familiar lines until the calculation of the failure function. Initially,
the failure function
fnew
is obtained after adding the full elastic stress increment to
those stresses existing previously. If
fnew
is positive, indicating a yielding Gauss point,
then the failure function
f
is obtained using just those stresses existing previously. The
scaling parameter
fac
is then calculated as described in equation (6.35). The plastic stress-
strain matrix [
D
p
] for a Mohr-Coulomb material is formed by the subroutine
mcdpl
(if
implementing the von Mises criterion, the subroutine
vmdpl
should be substituted) using
stresses that have been factored to ensure they lie on the failure surface. The resulting matrix
pl
is multiplied by the scaling parameter
fac
andthenbythetotalstrainincrementarray
eps
to yield the “plastic” stress increment array
elso
. This is simple “forward Euler”
integration of the “rate” equations. “Implicit” versions are described in the next sections.
Integrals of the type described by equation (6.34) then follow and the array
bdylds
is
accumulated from each element. It may be noted that in the algorithm presented here,
the body loads vector is completely reformed at each iteration. This is in contrast to the
viscoplasticity algorithm presented in Programs 6.1, 6.2, and 6.3, in which the body loads
vector was accumulated at each iteration.
At convergence, the stresses must be updated ready for the next displacement (load)
increment. This involves adding, to the stresses remaining from the previous increment, the
one-dimensional array of total stress increments
sigma
minus the one-dimensional array
of corrective “plastic” stresses
elso
.
The example problem shown in Figure 6.19 represents a “sand” with strength param-
eters
coefficient
30
◦
, subjected to prescribed displacements
along the left face. The displacement increments are applied to the
30
◦
,
φ
=
c
=
0, and dilation angle
ψ
=
-components of dis-
placement at the nine nodes adjacent to the hypothetical smooth, rigid wall shown hatched.
The initial stresses in the ground are calculated assuming the unit weight
x
20 kN/m
3
γ
=
1.
Following each displacement increment, and after numerical convergence, the resultant
force on the wall is calculated in two ways. Firstly, the force on the wall is computed
by averaging the
and “at rest” earth pressure coefficient
K
o
=
stresses at the eight Gauss points closest to the wall, and this result
is held in
pav
. Secondly, the nodal reactions are back-figured from the converged stress
σ
x
