Civil Engineering Reference
In-Depth Information
Ysi
ε
s
Yi
Y_si
H
Y_i
ε
c
t
ε
s
wi
FIGURE 7.11
Using finite-layer approach in analysis.
7.3.2.4 Numerical Procedure
The column cross section is divided into a finite number of thin layers, as seen in
Figure 7.11. The force and moment of each layer is calculated and stored. The bars
are treated as discrete objects in their actual locations. The advantage of that is to
precisely calculate the internal forces induced by steel bars and concrete layers in the
column cross section. The cross section analyzed is loaded incrementally by main-
taining a certain eccentricity between the axial force
P
and the resultant moment
M
R
.
Since increasing the load and resultant moment causes the neutral axis and centroid
to vary nonlinearly, the generalized moment-of-area theorem is devised. The method
is developed using an incremental iterative analysis algorithm, a secant stiffness
approach, and proportional or radial loading. It is explained in the following steps:
1. Calculate the initial section properties.
Elastic axial rigidity,
EA:
∑∑
EA
=
Ewt
+
(
E
−
EA
)
(7.61)
c
ii
s
c
si
i
i
where
E
c
= initial modulus of elasticity of the concrete and
E
s
= initial
modulus of elasticity of the steel bar.
The depth of the elastic centroid position from the bottom fiber of the section
Y
c
:
∑
∑
EwtH Y
(
−+ −
)
(
E
EAHY
)
(
−
)
ci i
i
s
c
si
si
(7.62)
Y
=
i
i
c
EA
Elastic flexural rigidity about the elastic centroid,
EI
:
∑
∑
2
2
EI
=
EwtH YY
(
− −+ −
)
(
EEAH YY
)
(
−
−
)
(7.63)
c
ii
i
c
s
c
si
si
c
i
i
H
Typically, ==
YY
c
G
2
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