Geology Reference
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Habibi and Moharrami (2010) adopted the
Park's nonlinear model and modified it to take
the axial force into account and formulated the
nonlinear stiffness matrix of the nonlinear concrete
elements based on these assumptions. For a rein-
forced concrete frame under gravity and lateral
loads, it was assumed that prior to execution of
lateral loads, the structure behaves linearly. The
nonlinear behavior starts when the lateral loads
are applied and increased to a certain level.
Similar to nonlinear analysis of steel frames, it is
assumed that in the reinforced concrete frame,
the unloading does not happen. Habibi (2008)
used the Modified Newton Raphson method for
nonlinear analysis. In this form of nonlinear
analysis, the internal forces in members are ob-
tained using tangential stiffness matrix. At the
beginning of every load step, as shown in Figure
3f the tangential stiffness matrix,
K
l
, is updated,
and the displacement vector is obtained assuming
an elastic behavior for the structure. With the
displacement found, the internal forces in all
members are obtained from the following equation
and the equilibrium is checked in all nodes.
negligible. The following equations are used
to find the displacement vector in the nonlinear
Modified Newton Raphson method.
1
1
)
(37)
1
K
l
δ
∆
j
=
P
l
−
F
j
−
⇒
δ
∆
j
=
K
l
−
(
P
l
−
F
j
−
T
T
∑
∑
l
j
l
∆
=
δ
∆
;
D
=
∆
(38)
j
l
where,
K
l
is the tangential stiffness matrix at the
beginning of load step l ;
δ
∆
j
is the incremental
displacement due to the unbalanced force
(
l
−
−
1
at the jth iteration of the lth load step;
∆
l
is the subtotal displacement for the load step
l and
D
P
F
)
=
∑
∆
is the total displacement for all
load steps, up to load level l.
The sensitivity analysis proposed by Habibi
and Moharrami (2010) is based on the foregoing
nonlinear analysis. Assuming linear behavior
during every unbalanced force analysis, the
Eq.(37) can be utilized to produce the sensitivity
of incremental displacement
δ
∆
j
with respect to
any design variable
d
i
as follows:
l
l
l j
,
=
l
∆
1
j
δF
K
(35)
e
e
e
j
l
j
−
1
l
T
∂
δ
∆
∂
P
∂
F
∂
K
−
1
l
T
j
=
K
−
−
δ
∆
(39)
∂
d
∂
d
∂
d
∂
d
i
i
i
i
where,
∆
j
−1
is the displacement sub-vector for
the element at (j-1)
th
iteration of load step
l
. Real-
izing that the internal forces in members at any
analysis stage is the sum of increments, the fol-
lowing equation is used.
Noting that during any load segment, the prop-
erties of tangential stiffness matrix and the load
level presumably do not change, their derivatives
remain unchanged within any load step. Therefore,
Eq.(30) can be used during repeated analysis in
any particular load segment, and the sensitivity
of displacement vector at any level of loading can
be obtained as follows:
l j
,
l
−
1
l j
,
F
=
F
+
δ
F
(36)
e
e
e
If equilibrium is not satisfied in any node, the
unbalanced force in the node is calculated and the
structure is analyzed for unbalanced forces in all
nodes. This gives the increment in the displace-
ment vector. This process is repeated several times
until the unbalance forces in all nodes become
l
∂
∂
∆
j
∂
∂
δ
∆
j
×
∂
∂
P
d
l
∂
∂
F
j
−
1
−
∂
∂
K
d
(40)
∑
−
1
∑
=
=
K
l
n
l
−
T
∆
j
T
d
d
d
i
l
i
i
l
i
i
where,
n
l
is the number of iterations of N-R pro-
cess at the relative load step
l
. The sensitivity
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