Page 139 - Riem-Vol6_nº3

484 IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 3
Plastic rotation and tension stiffening effect analysis in beams using photogrammetry
cessing, since all required data can be easily measured after the
test, thus not requiring to stop the test to perform readings (Tables
2 and 3). However, this method has a drawback: in stages before
steel yielding, when cracks are relatively small, is not possible to
achieve the required accuracy to measure the cracks width.
Table 4 summarizes the plastic rotations in pure bending region
obtained by the three methods mentioned. It is found that the val-
ues obtained using methods 1 and 2 are very similar, with differ-
ences lower than 5%, and with the values smaller than those ob-
tained using method 3, particularly at step 3, where the differences
can reach 40%.
5.4 Tension stiffening effect evaluation
The beam flexural stiffness (EI) decreases as the applied load in-
creases, initially due to concrete cracking and, at a later stage, due
to the reinforcement yielding. Theoretically, flexural stiffness in a
specific section can be determined in state I and II: in state I the
whole cross-section of concrete and steel is considered, whereas
in state II only the concrete under compression and, obviously, the
steel area are considered. The flexural stiffness experimentally
measured should be between these two limits due to the tension
stiffening effect. Figure 14 presents the evolution of flexural stiff-
ness (EI) with the applied load, P. The latter parameter is written
using the dimensionless ratio P/Py (where P y is the load required to
the reinforcement yielding). The stiffness was measured using the
relation between bending moment applied and mean curvature, in
this case measured using the horizontal LVDTs. Initially, stiffness
assumes high values because the curvature is very low and there-
fore very sensitive to any reading variation from the LVDTs.
The tension stiffening effect is particularly important in the analysis
of deformations in concrete structures under serviceability condi-
tions, as recommended in design codes for concrete structures.
As mentioned in Section 2, the distribution coefficient z considers
the contribution of the tension stiffening effect. An analysis of this
effect is provided below to stage 1, where the applied load is ap-
Table 2 – Rotation computed in each
identified crack
Crack x (mm)i
ΣƟ (×10-3 rad) =
i 63.2
w (mm)i
-3 Ɵ (×10 rad)
i
Stage 3
1
2
3
4
5
6
7
8
86
92
100
87
99
91
109
120
1.60
1.20
1.00
1.00
1.20
0.80
0.80
1.00
1.60
1.20
1.00
1.00
1.20
0.80
0.80
1.00
Table 3 – Total rotation obtained
by Bachmann method
Stage -3 Ɵ (×10 rad)
i
-
19.5
63.2
123.5
1
2
3
4
-3 Table 4 – Plastic rotation (x 10 rad)
Stage Method 1 Method 2 Method 3
3
4
31.7
90.1
30.8
95
43.7
104
Figure 14 – Flexural stiffness vs. applied load
Figure 15 – Average curvature in the monitored
region (stage 1)