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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
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0.3 0.3
ser
n
n
n
F D W L
= + +
where:
ser
F
: combined action value for service limit states;
n
D
: nominal value of dead load;
n
W
: nominal value of wind action;
n
L
: nominal value of life action.
Table 3 summarizes the characteristic values (
f
ck
) and nominal
load values (
D
n
, L
n
, W
n
) used to verify the frames in the service limit
state. Table 4 shows the results obtained, in terms of the horizontal
displacements at the top of the studied buildings. It can be ob-
served in this table that the representative frames were designed
for maximum flexibility.
4.2 Data for reliability analysis
For the service limit state related to horizontal displacements, the
“failure” condition is given by a displacement at the top of the build-
ing larger than H/500. This displacement is related to damage of
the masonry. This limit, indicated by ACI 435.3R-68(1984) [10], is
virtually equivalent to the H/1700 limit considered in the Brazilian
code [1], when the load combination factor for wind load is 0.3 and
the structural response is linear. Hence, the (service) limit state
equation for horizontal displacements is:
(13)
(
)
(
)
evaluated
g , , , ,
.
, , ,
H / 500
M c
M
c
E f D L W E u
f D L W
=
-
where
E
M
,
f
c
,
D
,
L
e
W
are the random variables of the problem,
described in Table 5. The parameters and probability distribution
functions of actions (
D
,
L
e
W
) are evaluated as indicated in Table
5, using the nominal values shown in Table 3.
The reduced loads for service verification (Eq. 12) correspond to
frequent load combinations, to which the structure will be exposed
during its design life. In principle, the reliability analysis for service
limit state could be performed for frequent loads, by combining the
arbitrary-point-in-time life load with the annual maximum of the
wind load. This analysis would result in an annual failure prob-
ability, which would have to be compared with the annual target for
irreversible service limit state (β
target
=2.9 following the EUROCODE
[2]). Alternatively, the 50-year maximum of these actions can be
considered, in order to evaluate reliability for the same period (de-
sign life of the structure). In this case, the target reliability follow-
ing the EUROCODE [2] is β
target
=1.5 (for irreversible service limit
states). In the first case, the probability being evaluated is the prob-
ability that the limit state will occur any year during the structure´s
life. In the second situation, one calculates the probability of the
limit state occurring at least once during the building›s design life.
In this article, the second situation is adopted, as it is considered
to be more representative of the desired situation for a building (no
damage to masonry during the structures lifetime)
Reliability analyses, considering extreme actions, are made for two
load combinations: the first considers the 50-year extreme of the
life load, combined with the annual maximum of the wind load;
the second considers the 50-year extreme wind combined with
the arbitrary-point-in-time of the life load (the value at any point in
time). These combinations are usual, when time-dependent reli-
ability problems are converted in time-independent problems (EL-
LINGWOOD et al.[11], BECK & SOUZA JR, [12]). The parameters
and probability distributions of: the 50 year extreme live and wind
loads, the annual maximum wind loads and the arbitrary-point-in-
time life load are presented in Table 5.
Following the First-Order Reliability Method (FORM), failure prob-
abilities are evaluated by:
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( ) 0
( )d ( )
f
g
P
f
b
<
=
» F -
ò
X
x
x x
where
X
is the vector of random variables,
g
(
x
) is the limit state
Table 5 – Random variables, their parameters and distributions
Random varaible
Distrib.
Média
Desvio-
padrão
C.V.
Fonte
70/70 model error
Normal
0.908
0.150
0.165
this work
80/40 model error
Normal
0.682
0.111
0.162
this work
f
c
Normal
fck + 1.65.
σ
4.00 MPa 0.150
MELCHERS [13]
Dead load
Normal
1.05 D
n
0.105 D
n
0.100
ELLINGWOOD et al.[11]
Arbitrary-point-in-time life load
Gamma
0.25 L
n
0.148 L
n
0.55
ELLINGWOOD et al.[11]
50-year extreme life load
Gumbel
1.00 L
n
0.250 L
n
0.25
ELLINGWOOD et al.[11]
Annual extreme wind load
Gumbel
0.33 W
n
0.155 W
n
0.47
BECK & SOUZA JR. [12]
50-year extreme wind load
Gumbel
0.90 W
n
0.306 W
n
0.34
BECK & SOUZA JR. [12]