Civil Engineering Reference
In-Depth Information
M
E
c
I
1
ψ
1
=
(8.23)
M
E
c
I
2
ψ
2
=
(8.24)
where
I
1
and
I
2
are the moments of inertia of a transformed uncracked and
fully cracked section about an axis through their respective centroids.
E
c
E
ref
is the modulus of elasticity of concrete, the value used as a reference elasticity
modulus in the calculation of
I
1
and
I
2
. The use of Equation (8.21) is demon-
strated in Example 8.2.
=
8.4.1 Provisions of codes
The interpolation between states 1 and 2 to calculate the mean curvature as
done in Equation (8.21) is adopted in MC-90 and EC2-91.
4
The EC2-91
allows use of the same coe
to calculate, by the same equation, mean
values for deformation parameters such as curvature, strain, rotation or
de
cient
ζ
fl
ection.
The MC-90 considers that the
M
-
relation shown by the lines ABCD in
Fig. 8.5 is most representative of actual practice with the exception of the
part EBC. This part is replaced by the dashed line, which is an extension of
the curve CD (Equation (8.21) ) until it intersects AB at point E. Thus, for
practical app
l
ication, the
M
ψ
−
ψ
relation follows the straight line AE when 0
M
(
M
r
√
β
); where (
M
r
√
β
) represents a reduced value of the cracking
moment;
β
=
β
1
β
2
.
relation is the non-linear part ED,
following hyperbolic Equation (8.21); where
M
y
is the moment which pro-
duces yielding of the reinforcement. If the concrete is in a virgin state and the
loading is of short-term character, the
M
When (
M
r
√
β
)
M
M
y
, the
M
−
ψ
relation is more closely pre-
sented by the lines ABCD. Replacement of the part EBC by EC takes into
consideration the behaviour of a member which has been cracked due to
loads, shrinkage and temperature variations during construction.
The MC-90 also di
−
ψ
ff
ers in the value of the coe
cient
β
2
, which is
considered equal to 0.8 (instead of 1.0) for
fi
rst loading.
ection of members can be calculated most accurately by numerical
integration of the curvatures at various sections (see Appendix C). The
EC2-91 allows, for simplicity, to calculate the de
The de
fl
ection twice, assuming the
whole member to be in uncracked and fully cracked condition in turn (states
1 and 2), and then to employ Equation (8.21), substituting the de
fl
fl
ection
values for the curvatures.
ACI318-01
5
also allows a similar interpolation between the moment of
inertia of a gross concrete section neglecting the reinforcement and the
moment of inertia of a transformed fully cracked section to calculate an
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