Civil Engineering Reference
In-Depth Information
Equation (6.4), (6.6) or (6.7). The
rst and the second nodes of members and
the positive sign convention for member-end forces are de
fi
fi
ned in Fig. 6.3.
6.7 Multi-stage loading
The problem stated in Section 6.3 can be solved when the analysis for the
time-dependent changes between time
t
j
and time
t
k
are required for the e
ff
ect
of events 1 to
j
, with the last event
j
occurring at
t
j
, with events 1 to (
j
1)
occurring at earlier instants,
t
1
,
t
2
, . . . ,
t
j
−1
. We recall, the term 'event' refers
to the application of forces, the introduction of prestressing, the casting a new
member or the removal or the introduction of a support. The two computer
runs as discussed in Section 6.5 are to be applied, di
−
ff
ering only in the calcula-
tion of the
xed-end forces {
A
r
(
t
k
,
t
j
)} to be included in the input of Com-
puter run 2. These forces are to be determined by a summation to replace
Equation (6.6). The summation is to superimpose the e
fi
ect of creep due to
the forces introduced at
t
1
,
t
2
, . . . ,
t
j
, as well as due to the gradual changes in
internal forces in the intervals (
t
2
ff
t
j
−1
). As example,
Equation (6.11) gives contribution to {
A
r
(
t
k
,
t
j
)}
creep
of the loads introduced at
time
t
i
; where
t
i
<
t
j
<
t
k
:
−
t
1
), (
t
3
−
t
2
), . . . , (
t
j
−
{
A
r
(
t
k
,
t
j
)}
creep, load introduced at ti
E
c
(
t
k
,
t
j
)
E
c
(
t
i
)
=
−
[
φ
(
t
k
,
t
i
)
−
φ
(
t
j
,
t
i
)]{
A
D
(
t
i
)}
(6.11)
The vector {
A
D
(
t
i
)} is to be determined by Equation (6.3) using the results
of a computer run having an input that includes the modulus of elasticity
E
c
(
t
i
) and the loading introduced at
t
i
.
When the structure is subjected to more than one or two events, several
computer runs are required. In this case it is more practical to apply the step-
by-step procedure discussed in Section 5.8, employing a specialized computer
program (see e.g. note 3, page 175).
6.8 Examples
The following are analysis examples of structures subjected to a single or two
events and it is required to determine the change(s) in displacements and/or
internal forces or stresses between time
t
j
and a later time
t
k
.
Example 6.1: Propped cantilever
The cantilever
AB
in Fig. 6.7(a) is subjected at time
t
0
to a uniform load
q
. At time
t
1
a simple support is introduced at
B
, thus preventing the
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