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[
]{ }
=
{ }
K d
( ,
∆ ∆
P
(14)
the frequencies and modal shapes is the follow-
ing equation:
That is the stiffness matrix itself becomes a
function of displacement. Provided that the non-
linear relation of the stiffness matrix with the
displacement vector is explicitly known, theo-
retically saying, the equation of equilibrium can
be solved by mathematics and the specific dis-
placement vector can be obtained in terms of
design variables. However, if the relation is not
known explicitly, the calculation of displacement
vector Δ and its derivative
∂ ∆
d
i
has to be fol-
lowed in a different manner. Differentiating the
above equation with respect to any design variable
d
i
, results in the following complicated equation:
K M
−
ω ϕ
2
=
0
(21)
Differentiating the above equation with respect
to the design variable,
d
i
, and rearranging the terms
results in the following equation.
∂
ϕ
∂
∂
ω
∂
∂
M
d
−
∂
∂
K
d
2
2
K M
−
ω
=
2
M
ω
+
ω
ϕ
∂
d
d
i
i
i
i
(22)
If the above equation is pre-multiplied by
ϕ
T
and noted that
∂
K d
d
( ,
∆
)
ndf
∂
K d
( ,
∆
)
×
∂
∂
∆
∂
∂
∆
∂
∂
P
d
{ }
+
[
=
∑
]
T
+
r
∆
K d
( ,
∆
)
=
ϕ
K M
−
ω
2
K M
−
ω ϕ
2
=
0
,
∂
∂
∆
d
d
i
r
=
1
r
i
i
i
(25)
then the following equation is obtained:
Obviously the above equation cannot be eas-
ily solved for
∂ ∆
d
i
; therefore, a different
strategy has to be followed for the sensitivity of
structural nonlinear behavior. There are a number
of papers that have paid attention to this problem.
For the sake of illustrations, the sensitivity cal-
culations in moment resisting steel frames by
Gong (2003) is explained and compared to those
of reinforced concrete frames by Habibi and
Moharrami (2010).
∂
∂
ω
ϕ
∂
∂
M
d
∂
∂
K
d
ω
T
M
ϕ ω ϕ
+
2
T
ϕ ϕ
−
T
ϕ
=
2
0
d
i
i
i
(23)
If the vector
f
is normalized in such a way that
M
=1
, then the sensitivity of frequency can
be obtained easily from the following expression:
ϕ
T
ϕ
∂
∂
ω
∂
∂
K
d
∂
∂
M
d
1
2
T
=
ω
ϕ
−
ω
2
ϕ
(24)
d
i
i
i
A Sensitivity Analysis Procedure
for Nonlinear Steel Frame
Nonlinear Sensitivity Analysis
In his research for performance-based design,
Gong used a special nonlinear static (pushover)
analysis scheme proposed by Hasan et al.(2002).
In this nonlinear analysis, the progressive degrada-
tion of stiffness of a frame structure is referenced
to its plastification. As the bending moment, M,
goes beyond the yielding moment M
y
, a partial
plasticity (Plastification) develops in the section.
As the plastification increases, the rigidity of the
In the sensitivity calculations in Eqs.(18-20), since
the structure is in its linear behaviour state, the
stiffnesses of structural members do not depend
on the deformation, Δ, and therefore,
∂
K d
i
and consequently
∂ ∆
d
i
are easily calculated.
However, if the structure is in nonlinear behavior
state, the equation of equilibrium becomes as in
Eq.(14) that is repeated here.
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