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Reliability of material and geometrically non-linear reinforced and prestressed concrete structures

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Reliability of material and geometrically non-linearreinforced and prestressed concrete structures
Fabio Biondini
a,*
, Franco Bontempi
b
, Dan M. Frangopol
c
,Pier Giorgio Malerba
a
a
Department of Structural Engineering, Technical University of Milan, Piazza L. da Vinci, 32, Milan 20133, Italy
b
Department of Structural and Geotechnical Engineering, University of Rome, ‘‘La Sapienza’’, Via Eudossiana, 18-00184 Rome, Italy
c
Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, CO 80309-0428, USA
Accepted 5 March 2004Available online 9 April 2004
Abstract
A numerical approach to the reliability analysis of reinforced and prestressed concrete structures is presented. Theproblem is formulated in terms of the probabilistic safety factor and the structural reliability is evaluated by MonteCarlo simulation. The cumulative distribution of the safety factor associated with each limit state is derived and areliability index is evaluated. The proposed procedure is applied to reliability analysis of an existing prestressed concretearch bridge.
2004 Elsevier Ltd. All rights reserved.
Keywords:
Concrete structures; Non-linear analysis; Structural reliability; Bridges; Simulation
1. Introduction
This paper considers a direct and systematic ap-proach to the reliability analysis of reinforced and pre-stressed concrete structures subjected to static loads [4].The structural reliability is evaluated by Monte Carlosimulation. Therefore, repeated non-linear analyses arecarried out giving outcomes from a set of basic variableswhich deﬁne the structural problem (e.g. mechanical andgeometrical properties, dead and live loads, prestressingforces, etc.). The results of the analysis associated toeach singular realization are then statistically examinedand used to evaluate the reliability index associated witheach considered limit state. The proposed procedure isﬁnally applied to the reliability assessment of an existingarch bridge. The structure is modeled by using a com-posite reinforced/prestressed concrete beam element,whose formulation accounts for the mechanical non-linearity due to the constitutive properties of materials(i.e. cracking, softening and crushing of concrete;yielding, hardening and failure of steel; prestressing ac-tion), as well as for the geometrical non-linearity due tosecond order eﬀects.
2. Probability of failure and reliability index
A structure is safe if the applied actions
S
are lessthan its resistance
R
. The problem may also be formu-lated in terms of the probabilistic safety factor
H
¼
R
=
S
.Let
h
be a particular outcome of the random variable
H
.The probability of failure can be evaluated by the inte-gration of the density probability function
f
H
ð
h
Þ
withinthe failure domain
D
¼ f
h
j
h
<
1
g
:
P
F
¼
P
ð
H
<
1
Þ ¼
Z
D
f
H
ð
h
Þ
d
h
:
ð
1
Þ
The above equation is often approximated as
P
F
¼
U
ð
b
Þ
;
ð
2
Þ
*
Corresponding author. Tel.: +39-02-2399-4394; fax: +39-02-2399-4220.
E-mail address:
biondini@stru.polimi.it (F. Biondini).0045-7949/$ - see front matter
2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruc.2004.03.010Computers and Structures 82 (2004) 1021–1031www.elsevier.com/locate/compstruc
where
U
is the standard normal cumulative probabilityfunction and
b
¼
U
1
ð
P
F
Þ
is the
reliability index
whichrepresents, in the space of the standard normal variables(zero mean values and unit standard deviations), theshortest distance from the srcin to the surface whichdeﬁnes the limit state.
3. Reliability assessment by simulation methods
In practice the density function
f
H
ð
h
Þ
is not knownand at the most some information is available onlyabout a set of
n
basic random variables
X
¼½
X
1
X
2
X
n
T
which deﬁne the structural problem(e.g. mechanical and geometrical properties, dead andlive loads, prestressing actions, etc.).Moreover, in concrete design the limit states areusually formulated in terms of functions of randomvariables
Y
¼
Y
ð
X
Þ
which describe the structural re-sponse (e.g. stresses, strains, etc.), and such derivation isgenerally only available in an implicit form. A numericalapproach is then required and the reliability analysis canbe performed by Monte Carlo simulation [6], where re-peated analyses are carried out with random outcomesof the basic variables
X
generated in accordance to theirmarginal density functions
f
X
i
ð
x
i
Þ
,
i
¼
1
;
. . .
;
n
. Based onthe sample obtained through the simulation process, thedensity function
f
H
ð
h
Þ
or the cumulative function
F
H
ð
h
Þ
can be derived for each given limit state
h
ð
Y
Þ ¼
0, andthe corresponding probability of failure
P
F
¼
F
H
ð
1
Þ
, aswell as the reliability index
b
¼
U
1
½
F
H
ð
1
Þ
, can beevaluated.An analytical interpolation of the numerical resultscan also be attempted, for example in terms of cumu-lative function
F
H
ð
h
Þ
. To this aim, a fairly regular andnon-decreasing function
F
H
ð
h
Þ
withlim
h
!1
F
H
ð
h
Þ ¼
0
;
lim
h
!þ1
F
H
ð
h
Þ ¼
1
ð
3
Þ
can be chosen as described in Biondini et al. [1]:
F
H
ð
h
Þ ¼
12 1
"
þ
tanh
X
K k
¼
0
c
k
h
k
!#
:
ð
4
Þ
A good accuracy is usually achieved by assuming
K
¼
5and the coeﬃcients
c
k
are identiﬁed through a leastsquare minimization.
4. Failure criteria for concrete structures
4.1. Serviceability limit states
Splitting cracks and considerable creep eﬀects mayoccur if the compression stresses
r
c
in concrete are toohigh. Besides, excessive stresses either in reinforcing steel
r
s
or in prestressing steel
r
p
can lead to unacceptablecrack patterns. Excessive displacements
s
may also in-volve loss of serviceability and then have to be limitedwithin assigned bounds
s
and
s
þ
. Based on these con-siderations, the following constraints account for ade-quate durability at the serviceability stage:S1
:
r
c
6
a
c
f
ck
;
ð
5a
Þ
S2
:
j
r
s
j
6
a
s
f
syk
;
ð
5b
Þ
S3
:
j
r
p
j
6
a
p
f
pyk
;
ð
5c
Þ
S4
:
s
6
s
6
s
þ
;
ð
5d
Þ
where
a
c
,
a
s
and
a
p
are reduction factors of the charac-teristic values
f
ck
,
f
syk
, and
f
pyk
of the material strengths.
4.2. Ultimate limit states
When the strain in concrete
e
c
, or in the reinforcingsteel
e
s
, or in the prestressing steel
e
p
reaches a limit value
e
cu
,
e
su
or
e
pu
, respectively, the failure of the corre-sponding cross-section occurs. However, the failure of asingle cross-section does not necessarily lead to thefailure of the whole structure, since the latter is causedby the loss of equilibrium arising when the reactions
r
requested for the loads
f
can no longer be developed.Therefore, the following ultimate conditions have to beveriﬁed:U1
:
e
c
6
e
cu
;
ð
6
Þ
U2
:
j
e
s
j
6
e
su
;
ð
7
Þ
U3
:
j
e
p
j
e
pu
;
ð
8
Þ
U4
:
f
6
r
:
ð
9
Þ
4.3. Probabilistic safety factors and limit load multipliers
Since these limit states refer to internal quantities of the system, a check of the structural performancethrough a non-linear analysis needs to be carried out atthe load level. To this aim, it is useful to assume
f
¼
g
þ
H
q
, where
g
is a vector of dead loads andprestressing actions, and
q
is a vector of live loads whoseintensity varies proportionally to a unique multiplier
H
P
0. Using these vectors, the serviceability and ulti-mate limit states previously deﬁned can be directly de-scribed in terms of the corresponding limit loadmultipliers
H
, which assume the role of probabilisticsafety factors.It is worth noting that non-linear analysis plays afundamental role in the evaluation of the limit loadmultipliers. In fact, for reinforced and prestressed con-
1022
F. Biondini et al. / Computers and Structures 82 (2004) 1021–1031
crete structures the distribution of stresses and strains inthe materials (concrete, reinforcing and prestressingsteel), as well as the magnitude of the displacements andthe collapse loads, depend on non-linear phenomena ascracking and crushing of the concrete matrix, yielding of the reinforcement bars and/or of the prestressing cables,second order geometrical eﬀects, etc. As a consequence,the investigated ultimate limit states cannot be investi-gated in the linear ﬁeld and, in most cases, such kind of structures should be analyzed by taking material and,possibly, geometrical non-linearity into account if real-istic results under all load levels are needed.Nowadays, non-linear analysis is a tool that can beapplied more easily than in the past. In many reportsand normative codes this aspect is recognized and it ishighlighted that non-linear analysis can give moremeaningful results than linear analysis. For these rea-sons, a new trend in design is spreading, where the usualprocedure of non-linear veriﬁcation of cross-sections onthe basis of the results of linear analyses tends to bereplaced by a full non-linear analysis where the struc-tural safety is evaluated at the load level.
5. Application to an existing arch bridge
The proposed procedure is now applied to the reli-ability analysis of the existing three hinge arch bridgeshown in Fig. 1 [7]. The total length of the bridge is 158m, with a central span of 125 m, and the total width of the deck 8.10 m (Fig. 2). The box-girder cross-sectionhas the width 5.00 m and height varying from 7.00 m atthe abutments to 2.20 m at the crown (Fig. 3). Thelayout of the prestressing cables is shown in Fig. 4. Thenominal value of the prestressing stress is
r
p
;
nom
¼
1200MPa. The number of reinforcement bars varies from aminimum of 108
£
22 at the crown to a maximum164
£
22 at the abutments. The bridge was built by usingprestressed lightweight concrete with the followingmaterial properties:
Fig. 1. View of the arch bridge over the Rio Avelengo, Bolzano, Italy (reprinted with permission from
L’industria Italiana del Cemento
––[7]).Fig. 2. Schematic view and main geometrical dimensions of the bridge (reprinted with permission from
L’industria Italiana del Cemento
––[7]).
F. Biondini et al. / Computers and Structures 82 (2004) 1021–1031
1023
Fig. 3. Longitudinal, horizontal and transversal cross-sectional views of the bridge (reprinted with permission from
L’industria Italiana del Cemento
––[7]).
1 0 2 4
F .B i o n d i n i e t a l . / C o m p u t e r s a n d S t r u c t ur e s 8 2 ( 2 0 0 4 ) 1 0 2 1 –1 0 3 1
Fig. 4. Layout of the prestressing cables and some details about the distribution of the main reinforcement bars (reprinted with permission from
L’industria Italiana del Cemento
–– [7]).
F .B i o n d i n i e t a l . / C o m p u t e r s a n d S t r u c t ur e s 8 2 ( 2 0 0 4 ) 1 0 2 1 –1 0 3 1
1 0 2 5

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