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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
Reliability of buildings in service limit state for maximum horizontal displacements
Tables 6 and 7 also show sensitivity coefficients of the problems
random variables. These coefficients show the relative importance
of each random variable towards failure. As expected, the horizon-
tal wind action has the largest contribution towards this displace-
ment failure. Uncertainty in concrete strength, which by way of Eq.
(7) affects the Young´s modulus, has minimal contribution. The
model error random variables have a significant contribution (from
21 to 34%) in the evaluated failure probabilities.
It is significant to note that reliability indexes obtained using the
simplified stiffness-reducing models 70/70 and 80/40 are similar.
This result is, in part, consequence of incorporating model error
random variables in the analysis. In the next section, it is verified
whether these reliability indexes agree with results of a rigorous
physical and geometrical non-linear analysis.
4.4 Reliability analysis using rigorous physical
and geometrical non-linear analysis: results
Tables 8 and 9 show results of the rigorous reliability analysis, us-
ing the 50-year extreme live load (Table 8) and the 50-year ex-
treme wind load (Table 9). Results in both tables correspond to
reliability analysis performed using rigorous physical and geometri-
cal non-linear analysis.
As in the simplified analysis, sensitivity coefficients show the same
behavior, with the wind load being the most important variable for
this horizontal displacement failure mode.
It is observed that reliability indexes obtained with the rigorous
analysis are significantly larger than those using the simplified
stiffness-reducing models. For the combination involving 50-year
extreme live loads (less relevant), reliability indexes obtained in
the rigorous analysis were larger than for the simplified analysis.
For the combination involving 50-year extreme wind loads, differ-
ent results were obtained for the three frames studied. For the four
and twelve floor buildings, larger reliability indexes were obtained.
For the eight floor building, a smaller reliability index was obtained
in the rigorous analysis. This result may be a particularity of the
frames studied. However, since the rigorous physical analysis is
more precise, one can conclude that the stiffness-reducing simpli-
fied models can be used for design, but cannot be used for reliabil-
ity analysis (even if model errors are considered).
Since reliability indexes found in the rigorous analysis are all
larger than β=1.5, one concludes that the design criteria of ABNT
NBR6118:2003 [1] with respect to the service limit state for hori-
zontal displacements (masonry damage) are conservative.
5. Concluding remarks
This article presented a study of model errors for the simplified
stiffness-reducing models proposed in ABNT NBR6118:2003 [1] for
evaluation of horizontal displacements of reinforced concrete plane
frames. A limited analysis composed of 42 plane frames of four,
eight and twelve floors has shown that the 70/70 model is more pre-
cise than the 80/40 (column/beam stiffness reduction) model.
Reliability analyses for service limit state of horizontal displace-
ments (masonry damage) were made using the simplified stiffness-
reducing models and using rigorous physical non-linear analysis.
It was observed that the simplified models are appropriate for a
verification of the structural design, but are not suitable for reliabil-
ity analyses (even if model errors are considered).
It was found that the load combination involving the 50-year ex-
treme live load is not relevant for the limit state of horizontal dis-
placements, even when geometrical non-linear effects are consid-
ered. The combination involving the 50-year extreme wind and the
arbitrary-point-in-time live load is more critical and leads to smaller
reliability indexes. Since these reliability indexes are larger than
EUROCODE-recommended values, it is concluded that the design
and verification criteria of Brazilian code ABNT NBR6118:2003 [1]
for horizontal displacements (Eq. 13 and the maximum allowed
Table 8 – Results for 50-year extreme live load combined with annual maximum wind load
N. Floors
b
rigorous
P
f
Sensitivity coefficients
f
c
D
L
W
4
4.957
3.58E-07
0.079
0.000
0.000
-0.921
8
5.016
2.64E-07
0.267
-0.004
-0.002
-0.727
12
5.129
1.46E-07
0.050
0.000
0.000
-0.950
Table 9 – Results for 50-year extreme wind load combined with arbitrary-point-in-time life load
N. Floors
b
rigorous
P
f
Sensitivity coefficients
f
c
D
L
W
4
2.747
3.00E-03
0.088
0.000
0.000
-0.912
8
2.293
1.09E-02
0.000
0.000
0.000
-1.000
12
2.955
1.56E-03
0.057
0.000
0.000
-0.943