Civil Engineering Reference
In-Depth Information
The approach to an ideal state is achieved when the variables in Equation 11.2
are as close to the desired values as possible. The minimum square error
method is one of the most effective ways to obtain the optimal
Z
, in which
the resulting cable stresses
S
can be written as
s
m
T
=
{
}
S
s
s
(11.3)
1
2
where
m
is total number of cables to be tuned.
By analyzing the response of a unit stress applied at each pair of tuning
cables, the influence values of all targets can be obtained, and the influence
matrix
A
can be written as
a
a
a
11
12
1
m
a
a
a
21
22
2
m
A
=
(11.4)
a
a
a
n
1
n
2
nm
where
a
ij
is the response at target
i
due to a unit stress at cable
j
. Thus, their
relationship can be written as
A S Z
× =
(11.5)
If the number of tuning cables is the same as the number of targets, cable
stresses can be obtained by solving the linear equation (11.5). In this case,
engineering experience is required in selecting cables and the targets. A bad
or contradictory tuning of cables and targets may cause matrix
A
not to
be a diagonal dominant matrix or Equation 11.5 in ill condition. If, as in
most cases,
m
is less than
n
, cable stresses can be optimized by minimizing
the error between the desired state and the state that can be reached.
D
,
which has the same form as
Z
, is the desired target value. The error
E
can
be written as
(11.6)
E D Z
=
−
The optimization goal is to minimize Ω
,
which is the square of
E
, and can
be written as
D Z
2
(11.7)
Ω =
(
−
)
From the variation principle, it is known that the condition to have Ω mini-
mized is
∂
∂
Ω
S
=
0
,
i
=
1 2 3
,
,
,
m
(11.8)
i
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