Page 132 - Riem-Vol6_nº3

477 IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 3
R. N. F. do Carmo | J. Valença | D. Dias-da-Costa
(1/r)II - curvature at a fully cracked section
z - distribution coefficient which take into account the tension stiff-
ening effect
In a reinforced concrete member, the tensile reinforcement strain
is variable along the beam axis. Consequently, the plastic rotation
has also a discontinuous variation, depending essentially on the
curvature of the cracked sections and, in less extent, on the cur-
vature of the sections between cracks. In Figure 2, the diagonally
dashed area corresponds to the integral of the plastic curvature
along the plastic hinge length, i.e., to the plastic rotation.
In Figure 2 it is also observed that the plastic curvature tends
to localize in cracked sections. Based on this, Bachmann, 1967
[10] proposed a straightforward method for computing the rota-
tion in certain regions of beams due to cracks (rotation between
the two opposite surfaces of the crack). The main characteristic
of this method is the fact of not obtaining the rotation from the
curvature. To apply this model it is necessary to know the num-
ber of cracks in the region of interest and the width and depth of
the neutral axis at each crack [11-12]. In this case, photogram-
metry and image processing are excellent techniques to obtain-
ing all required data.
The model is based on a discrete analysis of the reinforced con-
crete member, being the tangent to the deformed beam discontinu-
ous at each crack. Figure 3 exemplifies this procedure for a region
with negative moments, where the total rotation is equal to the sum
of the rotations in the ‘n’ existing cracks.
(3)
(4)
(1)
Ɵpl - plastic rotation capacity
lpl - length of the plastic hinge
1/r - total curvature
1/ry - yielding curvature
es - total strain of steel reinforcement
esy - yielding strain of steel reinforcement
d - effective depth of a cross-section
x - neutral axis depth
Computing the rotation in critical regions can be a difficult task
since the curvature has a discontinuous development along the
beam axis due to bending stiffness difference between cracked
sections and uncracked sections. On the other hand, in the plastic
hinge region and near failure, the assumption of plane sections is
not valid. This makes difficult determining the rotation by integrat-
ing the curvature along the beam axis.
The contribution of the concrete between cracks on the tensile
strength originates a significant variation of the bending stiffness,
known as tension stiffening effect. Disregarding this effect may
lead to unrealistic predictions, i.e., if only the curvature in a fully
cracked section is considered, a value greater than the actual rota-
tion would be obtained (Figure 1).
According to EC2 [9] and others codes, the mean curvature must
be computed by considering both uncracked and entirely cracked
states, i.e., by applying Equation 2.
(2) (1/r)m = .(1/r)II + (1 -  ).(1/r)I
(1/r)m - mean curvature
(1/r)I - curvature at an uncracked section
Figure 3 – Bachmann's method for computing the rotation (example of a region over a support) [11-12]