Civil Engineering Reference
In-Depth Information
back substitution performed at each iteration for each new right-hand side (using subroutine
spabac
) become clear.
Material properties in Program 4.5 must now include both elastic properties and plastic
moment values for all members. For 1D beams or 2D frames, only one plastic moment
(
) is read, thus
nprops
is 2 in 1D and 3 in 2D. For 3D space frames, however, three
plastic moments, (
M
p
M
py
,
M
pz
,and
M
px
in that order) are read, thus
nprops
is 7, where
y
-and
M
py
and
M
pz
represent respectively, the limiting bending moment about the local
z
-axes of the member, and
M
px
represents the limiting torsional moment about the long
axis of the member.
Loads are applied in
incs
increments to the nodes and the magnitude of each incre-
ment is read into the vector
dload
. The loading remains proportional as is customary in
plastic hinge analysis, so the relative magnitudes of the nodal loads are read by
node
and
val
,where
node
holds the node numbers and
val
holds the load weightings on each
freedom.
Following assembly of the global stiffness matrix and factorisation by subroutine
sparin
, the program enters the load increment loop. For each iteration counted by
iters
,
the external load increments are added to the redistributive loads vector
bdylds
. The equi-
librium equations are solved using subroutine
spabac
and the resulting nodal displacement
increments compared with their values at the previous iteration using subroutine
checon
.
This subroutine observes the relative change in displacement increments from one iteration
to the next. If the change is less than
tol
then the logical variable
converged
is set to
.TRUE
and convergence has occurred. Alternatively,
converged
is set to
.FALSE
and
another iteration is performed.
At each iteration, each element is inspected and its
action
vector computed from
the product of its nodal displacements and the element stiffness matrix. The subroutine
hinge
adds the
action
vector to the values already existing from the previous load
step (held in
holdr
) and checks both nodes to see if the plastic moment value has been
exceeded. If the plastic moment value has been exceeded, the self-equilibrating vector
react
is formed. In Figure 4.30(a), a typical 2D element is shown in which a particular
load increment has pushed the moment value at both nodes over their plastic limit. The
correction vector applies a moment to each node equal to the amount of overshoot of the
plastic moment values, however to preserve equilibrium, a couple is required as shown
in the local coordinate system in Figure 4.30(b). Finally as shown in Figure 4.30(c), the
couple is transformed into global coordinate directions before being assembled into the
bdylds
vector. Only those elements that have moments in excess of the plastic limits will
contribute any loading to
bdylds
.
If, at any load step, the algorithm fails to converge within the prescribed iteration ceiling
limit
then “collapse” of the structure is indicated, because the algorithm has been unable
to satisfy equilibrium without violating the plastic moment values.
The first example shown in Figure 4.31 is a two-bay portal frame subjected to propor-
tional loading. After each increment, the output shown in Figure 4.32 gives the loading
factor
(equal to the accumulated values of
dload
) together with the loaded nodal dis-
placements and the iteration count to achieve convergence. To reduce the volume of output,
only the displacements of loaded nodes (2, 3, and 6) are given. In Figure 4.33, the horizon-
tal movement of point A is plotted against
λ
λ
, indicating close agreement with the theoretical
value of
λ
f
=
1
.
375 given by Horne (1971) for this problem.
