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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
A. G. B. CORELHANO | M. R. S. CORRÊA | A. T. BECK
equation (Eq. 13), Φ( ) is the cumulative standard Gaussian distri-
bution function and β is the reliability index. In this article, equation
(14) is solved by the FORM method (MELCHERS, [13]), using the
StRAnD software (BECK [4]). In the FORM method, the original
problem is transformed to the standard normal space, and solved
as a restricted optimization problem: the reliability index is the
smallest distance between the limit state equation and the origin
of the standard normal space. The reliability index is related to the
failure probability by means of:
(15)
1
( )
f
P
b
-
= - F
4.3 Reliability analyses using simplified stiffness
reducing models: results
Tables 6 and 7 show results of the reliability analysis, for the
50-year extreme live load combined with annual maximum wind
load (Table 6) and using the 50-year extreme wind load com-
bined with the arbitrary-point-in-time live load (Table 7). Results
refer to reliability analysis using the simplified models with stiff-
ness reduction and non-linear geometrical analysis. It can be ob-
served that the load combination involving the 50-year extreme
wind load (Table 7) leads to larger “failure” probabilities than the
combination involving the 50-year extreme live load. This is to
be expected, as the wind load acts directly in the direction of the
calculated displacements.
The term “failure”, in this context, is used between quotes, as it
represents failure in respecting the constraint of maximum dis-
placement (H/500) which, in theory, corresponds to a state of
masonry damage. This damage is an irreversible limit state. As
a reference, annex C of the EUROCODE [2] suggests a target
reliability index of β
target
=1.5 for irreversible limit states and for
50 years reference. The reliability indexes found in this article,
which correspond to very flexible structures designed following
ABNT NBR6118:2003 [1], are slightly larger that this target value.
Hence, failure probabilities can be considered acceptable. These
results show that the horizontal displacement verifications of
Brazilian code ABNT NBR6118:2003 [1] (Eq. 13), together with
the maximum allowed displacement of H/1700 (for frequent load
combinations) are conservative.
Table 6 – Results for 50-year extreme live load combined with annual maximum wind load
N. Floors
Model
b
aprox
P
f
Sensitivity coefficients
E
M
f
c
D
L
W
4
70/70
4.019
2.92 E-5
0.305
0.025
0.0
0.0
-0.669
80/40
4.265
9.97 E-6
0.301
0.027
0.0
0.0
-0.672
8
70/70
4.292
8.84 E-6
0.344
0.026
0.0
0.0
-0.630
80/40
4.331
7.43 E-6
0.308
0.030
0.0
0.0
-0.662
12
70/70
4.116
1.92 E-5
0.302
0.023
0.0
0.0
-0.675
80/40
4.159
1.60 E-5
0.292
0.020 0.0 0.0 -0.688
Table 7 – Results for 50-year extreme wind load combined with arbitrary-point-in-time life load
N. Floors
Model
b
aprox
P
f
Sensitivity coefficients
E
M
f
c
D
L
W
4
70/70
2.127
1.60 E-2
0.226
0.029
0.0
0.0
-0.745
80/40
2.369
8.90 E-3
0.216
0.030
0.0
0.0
-0.754
8
70/70
2.441
7.33 E-3
0.238
0.030
0.0
0.0
-0.732
80/40
2.453
7.08 E-3
0.218
0.032
0.0
0.0
-0.750
12
70/70
2.235
1.27 E-2
0.232
0.022
0.0
0.0
-0.746
80/40
2.253
1.21 E-2
0.214
0.023
0.0
0.0
-0.763