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n
rule in differentiation, the sensitivity of lateral
load distribution becomes as follows:
the value of
∂
i
can be computed in an ac-
cumulative process. It is now sufficient to substi-
tute for
∂
F d
i
and
∂
P d
i
in Eq.(30) to obtain
the sensitivity of displacement vector to change
in a design variable.
F
∂
d
∂
∂
C
d
=
∂
∂
C
k
⋅
∂
∂
k
T
⋅
∂
∂
T
d
vx
vx
(31)
i
i
A Sensitivity Analysis procedure for
nonlinear reinforced Concrete frame
If Eq.(24) is used for finding the sensitivity
of natural period of the structure, and the change
of the mass of the structure due to change in the
design variables is ignored, one can easily find:
In this section the sensitivity analysis proposed by
Habibi and Moharrami (2010), will be described
to show a different sensitivity analysis procedure
and emphasize on the fact that formulation of
any design sensitivity analysis has to be derived
based on assumptions made on the corresponding
nonlinear analysis.
As per any nonlinear analysis of structures that
requires the stiffness-displacement relation for as-
sembling Eq.(14), and requires force-displacement
relation for solution of nonlinear Eq.(14) in the
form of Eq.(27), it is necessary to define a non-
linear moment-curvature and a stiffness-curvature
relation for concrete structural elements. Habibi
adopted a tri-linear moment-curvature relation,
proposed by Park and Ang (1985) as shown in
Figure 5. This assumption helps to achieve the
amount of curvature for a given moment and
amount of the moment for a given curvature.
The first line represents the without-crack
situation. The second line stands for post-crack
to yielding state and the third line corresponds to
yield to the ultimate state. With the help of these
definitions, the distribution of curvature of a beam
under applied loads can be obtained and plotted
as per Figure 6a. The distribution of curvature
makes it possible to create a nonlinear model at
the element level. Park et al. (1987) proposed a
nonlinear model for R/C members (used in IDARC
software) in which the flexural stiffness in any
section is related to its curvature. Since both
flexural deformation and flexibility have recipro-
cal relation with stiffness of element, there will
be an analogy between flexibility of the section
∂
∂
T
d
T
3
∂
∂
K
d
= −
ϕ
T
ϕ
(32)
8
π
2
i
i
The second term in the parenthesis of Eq.(30),
however, is the variation of internal forces due to
a change in design variable when there is no
change in external loads. This is in fact, the main
challenge in nonlinear sensitivity analysis. To
calculate this quantity, Gong pointed out that
internal forces in members are the accumulation
of their incremental forces,
δF
, induced in mem-
bers during incremental analysis procedure i.e.:
n
n
n
∂
∂
K
d
l
∂
∂
F
d
∂
∂
δ
F
d
∑
∑
=
=
T
δ
∆
(33)
l
l
=
1
l
=
1
i
i
i
Therefore, the sensitivity of internal force can
be accumulatively obtained at every load step as
follows:
n
n
−
1
+
∂
∂
K
d
n
∂
∂
F
d
=
∂
∂
F
d
T
δ
∆
(34)
n
i
i
i
Noting that
K
n
is the tangential stiffness ma-
trix at iteration n and is an explicit function of
plasticity index that in turn is a function of inter-
nal forces in members, one can easily calculate
the value of
∂
n
i
. Having
δ
∆
n
calculated
from analysis at the iteration n, the second term
in the right hand side of Eq.(34) is known, and
K
∂
d
T
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