Civil Engineering Reference
In-Depth Information
7.4.1 Remarks on determination of neutral axis position
Equations (7.9) or (7.10) can be used to determine the position of the neutral
axis, and thus the depth
c
of compression zone, for any section having a
vertical axis of symmetry. Equation (7.9) applies when the section is subjected
to a moment,
M
without a normal force. Equation (7.10) applies when
M
is
combined with a normal force, N.
For a section of arbitrary shape, a trial value of the coordinate
y
n
of the
neutral axis is assumed, the integral in Equation (7.9) or the two integrals in
Equation (7.10) are evaluated, ignoring concrete in tension. By iteration a
value
y
n
, between
y
t
and
y
b
, is determined to satisfy one or the other of the
two equations; where
y
t
and
y
b
are the
y
coordinates of the top and bottom
fi
bres, respectively.
Both Equations (7.9) and (7.10) are based on the assumption that the
extreme top and bottom
bres are in compression and in tension, respectively.
Thus the equations apply when:
fi
σ
t1
0
while
σ
b1
0
(7.11)
where
σ
is stress at concrete
fi
bre; the subscripts t and b refer to top and
bottom
bres and the subscript 1 refers to state 1 in which cracking is
ignored. When the extreme top and bottom
fi
bres are in tension and com-
pression, respectively, Equation (7.9) or (7.10) applies when the direction
of the y-axis is reversed to point upwards and the symbol
y
t
in the equa-
tions is treated as coordinate of bottom
fi
fi
bre. It is here assumed that at
least one of
σ
t1
and
σ
b1
exceeds the tensile strength of concrete, causing
cracking.
When a section is subjected to a moment, without a normal force, solution
of Equation (7.9) gives the position of the neutral axis at the centroid of the
transformed section, with concrete in tension ignored. In this case, the equa-
tion has a solution
y
n
between
y
t
and the
y
-coordinate of the extreme tension
reinforcement. However, when a section is subjected to a normal force, N
combined with a moment, M, the neutral axis can be not within the height of
the section; in which case Equation (7.10) has no solution for
y
n
that is
between
y
t
and
y
b
. The following are limitations on the use of Equation
(7.10), depending upon the values of
M
and
N
. It is here assumed that the
compression zone is at top
fi
bre:
(1)
When
N
is compressive, both
σ
t1
and
σ
b1
are compressive when:
I
1
−
y
t
B
1
M
I
1
−
y
b
B
1
N
(7.12)
B
1
−
y
t
A
1
B
1
−
y
b
A
1
where
A
1
,
B
1
and
I
1
are area of transformed uncracked section (state 1)
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