Civil Engineering Reference
In-Depth Information
ed
sections on a member, Equations (13.28) and (13.29) can be used to deter-
mine the de
When the curvatures
ψ
have been determined at a number of speci
fi
fl
ection
D
C
; where
D
C
is the transverse distance between the chord
and the de
ected member at any section
C
(Fig. 13.5(c) ). The chord is the
straight line joining the nodes O
1
and O
2
in their displaced position. Again,
the contribution of the curvature over a typical spacing,
s
can be calculated
separately and the contributions of all spacings can be summed up to
give
D
C
.
Contribution of spacing
AB
to
D
C
(Fig. 13.5(c) ):
fl
l
−
x
C
s
ψ x dx
=
l
−
x
C
s
6
(2
ψ
A
x
A
+
ψ
A
x
B
+
ψ
B
x
A
+
2
ψ
B
x
B
)
l
l
when
x
B
x
C
(13.28)
x
C
l
s
ψ
(
l
−
x
)
dx
=
x
l
ls
(
ψ
A
+
ψ
B
)
2
s
−
6
(2
ψ
A
x
A
+
ψ
A
x
B
+
ψ
B
x
A
+
2
ψ
B
x
B
)
when
x
A
x
C
(13.29)
where
x
C
is the distance between node O
1
and the point considered.
13.9 Iterative analysis
The analysis described below applies the displacement method in iterative
cycles. Each cycle starts with known values of the parameters
σ
O
,
γ
,
c
and
ζ
at
each section of individual members; where
σ
O
is the stress at reference point
O;
is the slope of the stress diagram (Fig. 13.3(b) );
c
here means depth
of the part of the section in which the concrete is not ignored; thus, for a
cracked section,
c
is the depth of the compression zone, but for an uncracked
section,
c
γ
=
h
, with
h
being the full height of the section;
ζ
is the interpolation
coe
cient. In each iteration, these values are updated. For the analysis of a
non-prestressed reinforced concrete frame, the initial stresses are assumed
null and the sections are assumed uncracked; thus at all sections,
σ
O
=
0;
γ
=
0;
c
=
h
;
ζ
=
0. For a prestressed frame, the initial stresses are de
fi
ned by the given
parameters
σ
Oin
and
γ
in
and again the sections are assumed uncracked; thus
c
0. The cycles of analysis are repeated until the residual vector
{
F
}
residual
becomes approximately equal to {0}; generation of the vector
{
F
}
residual
is explained below. The analysis cycle is completed in three steps:
=
h
and
ζ
=
Step 1
Determine by conventional linear analysis the nodal displacements
and the member end forces. This involves: generation of sti
ness matrices,
[
S
*] of individual members, transformation of [
S
*] from local member direc-
tions to global directions, assemblage of the transformed matrices ([
T
]
T
[
S
*]
[
T
]) to obtain the sti
ff
ff
ness matrix, [
S
], of the structure, adjustment of the
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