Geology Reference
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section decreases. Hasan et al. simulated the partial
plasticity in a section with a semi-rigid connection
in which the rigidity of connection is progres-
sively decreased as the moment in the section is
increased. In particular, a potential plastic-hinge
section was simulated with a semi-rigid connec-
tion whose stiffness variation is measured by a
plasticity-factor
p
that ranges from unity, for ideal
elastic, to zero, for fully plastic. If this degrada-
tion is monitored, its influence on the nonlinear
behavior of the member and the overall behavior
of a structure under increasing lateral load can be
traced. Hasan et al. assumed an elliptic moment-
curvature relation for post elastic behavior of a
section. Then he replaced the plasticity factor
with rigidity factor and performed the nonlinear
analysis in an incremental scheme. As the lateral
load is gradually increased in infinitesimal steps,
the stiffness matrix of the structure is deteriorated.
This deterioration is considered by modifying the
C
s
and C
g
coefficient matrices in the following
formula based on plasticity factors.
This is to say that sum of internal forces at a
node is equal to external loads. Differentiating
Eq.(27) with respect to any design variable
d
i
results in the following relation.
∂
∂
F
d
+
∂
∂
F
⋅
∂
∂
∆
=
∂
∂
P
d
(28)
∆
d
i
i
i
Noting that
∂
F
∆
is actually the global
tangential stiffness matrix, Eq.(28) can be written
as:
⋅
∂
∂
∆
=
∂
∂
P
d
−
∂
∂
F
d
K
(29)
T
d
i
i
i
As per above explanations, the tangential stiff-
ness matrix,
K
T
, for a given displacement vector
is easily calculated. Therefore, the sensitivity of
displacement can be obtained from the following
equation that is similar to Eq.(20) except that the
calculation of
∂
P d
i
and
∂
F d
i
requires
further consideration.
K S C
= ⋅
+ ⋅
G C
(26)
S
g
P
d
F
d
∂
∂
∆
d
∂
∂
−
∂
∂
K
=
⋅
(30)
−
1
T
where, S is the standard elastic stiffness matrix;
C
S
is a correlation matrix expressed in terms of
plasticity factors; G is the standard geometric
stiffness matrix and
C
g
is corresponding correction
matrix, formulated in terms of plasticity factors.
In this way, they could obtain the magnitude of
overall deformation of the whole-structure as well
as all members.
The sensitivity analysis that Gong proposed
for his problem is defined in this framework of
analysis. Since the sensitivity of displacement is
the basis of evaluation of sensitivities of other
structural behaviours, it is usually found first. To
obtain sensitivity of displacement vector, Eq. (14)
was written in the following form.
i
i
i
In the above equation, the first term in pa-
renthesis is indeed the change in the external
earthquake forces. Recalling Eq.(1), this may be
interpreted in two ways. First, the change in the
base shear V, that may occur due to change in the
stiffness of the structure, as a result of change in
design variables. The second, the change in the
vertical distribution factor C
vx
. However, since
in the nonlinear pushover analysis, the load level
is invariant of design variables, the sensitivity
of lateral load depends on sensitivity of vertical
distribution of lateral loads that is a function of
parameter
k
in Eq.(2) and
k
is given in terms of
natural period of the structure. See FEMA356 or
ASCE41 in this regard. Therefore, utilizing chain
F d
( ,
∆ =
)
P d
( )
(27)
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