The 11th International Conference on Mathematical Methods in Reliability, HONG KONG, CHINE, 03-/06/2019 - 07/06/2019Predicting reliability levels for critical details of bridges based on limited statistical traffic data is a relevant topic nowadays. That is why the comparison between results from various statistical approaches based on the recorded data for applied traffic actions is the main point of interest of this work. The object of the current study is the famous Millau viaduct, a cable-stayed bridge with the steel orthotropic deck located in Southern France. Values of load effects that are used in analysis are derived from a finite element model of a part of the deck. They are based on data from traffic monitoring that is provided from the bridge Weigh-In-Motion system covering several months of axle loads, distances and speeds of heavy trucks. The methodology is based on a definition of limit state functions based on several statistical distributions in order to assess and compare reliability indexes for the ultimate limit state. It includes a comparison between different approaches of extreme values theory, the methodology proposed in background works for European standards and the design load model. Moreover, this work covers the influence of applied loads of a high amplitude, as global effects, onto stresses from axle loads, as local effects

Fischer, Oliver

NESTEROVA, Mariia

NOWAK, Marcel

SCHMIDT, Franziska

English

Archive Ouverte en Sciences de l'Information et de la Communication

HAL Id: hal-02316778
https://hal.archives-ouvertes.fr/hal-02316778
Submitted on 15 Oct 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Reliability of a bridge with an orthotropic deck exposed
to extreme traffic events
Mariia Nesterova, Marcel Nowak, Franziska Schmidt, Oliver Fischer
To cite this version:
Mariia Nesterova, Marcel Nowak, Franziska Schmidt, Oliver Fischer. Reliability of a bridge with an or-
thotropic deck exposed to extreme traffic events. The 11th International Conference on Mathematical
Methods in Reliability, Jun 2019, HONG KONG, China. 8 p. ￿hal-02316778￿
RELIABILITY OF A BRIDGEWITH AN ORTHOTROPIC DECK
EXPOSED TO EXTREME TRAFFIC EVENTS
MARIIA NESTEROVA∗
Universite´ Paris-Est, MAST, EMGCU, IFSTTAR, F-77447 Marne-la-Valle´e, France,
email: mariia.nesterova@ifsttar.fr
MARCEL NOWAK
Technical University of Munich, Chair of Concrete and Masonry Structures, 80333
Mu¨nchen, Germany, email: marcel.nowak@tum.de
FRANZISKA SCHMIDT
Universite´ Paris-Est, MAST, EMGCU, IFSTTAR, F-77447 Marne-la-Valle´e, France,
email: franziska.schmidt@ifsttar.fr
OLIVER FISCHER
Technical University of Munich, Chair of Concrete and Masonry Structures, 80333
Mu¨nchen, Germany, email: oliver.fischer@tum.de
Predicting reliability levels for critical details of bridges based on limited statistical traffic
data is a relevant topic nowadays. That is why the comparison between results from various
statistical approaches based on the recorded data for applied traffic actions is the main point
of interest of this work. The object of the current study is the famous Millau viaduct, a
cable-stayed bridge with the steel orthotropic deck located in Southern France. Values of
load effects that are used in analysis are derived from a finite element model of a part of
the deck. They are based on data from traffic monitoring that is provided from the bridge
Weigh-In-Motion system covering several months of axle loads, distances and speeds of
heavy trucks. The methodology is based on a definition of limit state functions based on
several statistical distributions in order to assess and compare reliability indexes for the
ultimate limit state. It includes a comparison between different approaches of extreme
values theory, the methodology proposed in background works for European standards and
the design load model. Moreover, this work covers the influence of applied loads of a high
amplitude, as global effects, onto stresses from axle loads, as local effects.
Keywords: Millau viaduct, orthotropic deck, extreme load effects, reliability,
traffic actions
∗Corresponding Author
1. Introduction
The European road infrastructure includes a significant number of bridges with
orthotropic decks, which are exposed to heavy traffic loads and unfavorable en-
vironmental influence. In order to estimate if a structure operates safely, such
bridges need to be assessed on a regular basis to verify the original design life.
This can be done by reliability analysis (e.g. reliability index at the end of the
operating life) of the most critical detail of a structure.
Various methods for reliability analysis exist nowadays, each having advan-
tages and drawbacks. A correct choice of the most appropriate reliability method
depends on the studied structure, input data, initial conditions, and computational
sources [1]. In bridge engineering, the most frequently used are First and Sec-
ond Order Reliability Methods (FORM and SORM) described, for instance, by D.
Straub [2].
The main interest of the current study is in comparison of predicted reliability
levels at the end of design life based on a limit state function (LSF) defined with
various approaches of the extreme values theory (EVT) [3]. In a recent study
[4] of various LSF, one of the conclusions shows that not only fatigue but also
extreme values and serviceability have to be considered in the reliability analysis
of bridges.
Such as for bridges with steel orthotropic decks, critical details of a deck have
a shorter life than other structural elements of a bridge [5]. The current study is
focused on details of the orthotropic deck of the Millau viaduct obtained using the
extreme values approaches: Peaks Over Threshold (POT), Block Maximum (BM)
and level crossing counting (LCC). The stresses are derived from the FE model of
a part of the deck and based on 180 days of monitored by bridge Weigh-in-Motion
(BWIM) traffic data.
2. Methodology
To assess the reliability of a detail, first of all, the LSF G(x) is needed. It is based
on one of several approaches of the EVT and is expressed as:
G(x) = R− L(x) > 0, (1)
where L(x) represents the probabilistic model of a load effect – equivalent
stress in the critical welding between a rib and the deck plate – and R = Fu
is the probabilistic model of the material resistance. R is represented here by the
ultimate strength of steel, described by a log-normally distributed random variable
with coefficient of variance (CoV) CoV = 0.05.
For each described case, the reliability index β is found for the corresponding
probability of failure Pf , Eq. (2) with the probability density function fx(x) of the
load effect x and the cumulative distribution function (CDF) of the standardized
Normal distribution Φ−1 (3).
β = −Φ−1(Pf ) = −Φ−1
(∫
G(x)≤0
fx(x)dx
)
(2)
Φ(x) =
1√
2pi
∫ x
−∞
e−x
2/2dx (3)
2.1. POT approach
The POT approach has been recently proved to be a good solution for predic-
tions of extreme traffic actions [6]. As a time-series process, ”peak” values of
load effect (LE), that lay above a certain threshold, are fitted to the generalized
Pareto distribution (GPD). It was recently concluded [7] that the efficiency of this
method is high enough under the assumption of normal distribution of return level
estimators.
Let X be the sequence of n random variables Xi representing values of LE
with the distribution function F . Let Y be the sequence of m random variables Yj
representing exceedance of a threshold u and defined by Y = X − u, for every
threshold excess Xj ∈ X so that Xj > u. A few assumptions for the application
of the EVT include identical distribution and statistical independence of random
variables Xi.
According to the general principle of the POT approach [8], the CDF of thresh-
old exceedances tends to the upper tail of GPD (4), with shape and scale (σ > 0
and ξ). The approach is valid only if Yj = Xj − u ≥ 0, Xj ≥ u for ξ ≥ 0 and
u ≤ Yj ≤ u− σ/ξ for ξ < 0.
G(Yj ; ξ;σ;u) =
{
(1− [1 + ξ(Yjσ )]−1/ξ, ξ 6= 0
1− exp(−Yjσ ), ξ = 0
(4)
For a long period, estimates can be based on the CDF of extreme values from
a shorter period [9]. In other words, monitored values of a LE for a limited period
can be used for extrapolation of the LE in time. The value of the p-observation
return level Lr(p) for the probability P [X ≤ Xi|Xi > u] with the probability
of exceedance (PE) ζu = P{Xi > u}, is a quantile that exceeds once every
p observations. The procedure for selecting an optimized threshold is described
earlier by author [10]. Therefore, for a fixed period p, the LSF (1) based on the
GPD takes the following form:
R > Lr(u, σ, ξ, ζu) =
{
u+ σξ [(pζu)
−ξ − 1], ξ 6= 0
u+ σlog(pζu), ξ = 0
(5)
2.2. BM method
The BM method is based on a limiting theorem stating that the generalized Ex-
treme Values distribution (GEVD) is an approximation for describing sample
maxima for large sample sizes. This holds for the condition of the BM being
(approximately) independent observations with identical distributions [3]. A BM
series is generated by separating the time history for load effects X into intervals
of a chosen size and determining the maxima in each interval. Different modi-
fications of BM method are available to handle inhomogeneities in the data-set,
aiming at approximately obtaining idealized conditions of independent and iden-
tically distributed observations in a remaining subset for further model fit.
One such approach [11] proposes a composite distribution model that de-
scribes bridge load effects from a mixture of different loading event types, repre-
senting different number of trucks on the bridge contributing to a maximum value
of the LE. Such a model accounts for the actual physical nature of the bridge load-
ing process, and hence allows for a higher accuracy in extreme value estimates, as
also shown in a recent study [12]. Based on separate GEVD fitsGj(z) with model
parameters [µ, σ, ξ] to the corresponding BM series for each of the different load-
ing event types, the distribution of load effects GC(z) resulting from the mixture
of these loading event types asymptotically approaches the following composite
distribution:
GC (z) =
N∏
j=1
Gj (z) = exp
−
N∑
j=1
[
1 + ξj
(
z − µj
σj
)]−1/ξj (6)
Due to the small influence area of the considered load effect it was observed
that the main contribution for extreme values of LE results from a single vehicle.
Hence, there is no need for further classification for this study. The LSF (1) based
on the BM takes the following form (with Tref = 5d and Rt to be inserted in [d]):
R > Lr(µ, σ, ξ) =
{
µ− σξ
[
1− (Tref/Rt)−ξ
]
, ξ 6= 0
µ− σlog(Tref/Rt), ξ = 0
(7)
2.3. Rice formula
The LCC method, where Rices formula is fitted to the upper tail of an out-crossing
rate histogram (ORH) is based on the assumption of a stationary Gaussian process
describing the time variations of load effects on bridges [13]. The number of times
is counted at which positive values are crossed upwardly in a LE time history.
By normalizing the resulting level crossing histogram with respect to the time
history length, the ORH is obtained, representing for each level the mean rate of
its crossing during a reference period Tref . The Rices formula with its model
parameters [a0 = ln(ν0) − m2/2σ2, a1 = m/σ2, a2 = −1/2σ2] describes the
mean rate ν(x) of up-crossing for a certain level during Tref , and is fitted to the
significant tail regions to the ORH. The proper choice of the starting point for
the fit with Rices formula is crucial. It should be as low as possible, to ensure
sufficient representativeness for statistical extrapolation, but not too low to still
provide a reasonable approximation of the significant tail region. In this study,
optimal starting points are identified with evaluating the goodness of fit by means
of a modified Kolmogorov test [13]. The LSF (1) based on the LCC takes the
following form:
R > Lr(Rt, a0, a1, a2) = − a1
2a2
+
√
−a0
a2
+
(
a1
2a2
)2
− ln(Rt/Tref )
a2
(8)
3. Application to the Millau viaduct
3.1. Instrumentation
Figure 1. Cross section of the deck of the Millau Viaduct
Millau viaduct is a cable-stayed bridge that consists of 8 spans with a total
length of 2460 m. The stiff orthotropic steel deck is suspended by 11 steel cables
at each span. Each span of 342 m is composed of four or six meters long elements,
each of them acting almost as a simply supported beam. The cross-section of the
deck, Figure 1 [14], shows that the viaduct has two lanes in each direction: (i)
slow lane, where traffic is mainly composed of heavy vehicles, (ii) fast lane, that
is used by lorries only to overtake.
The data on traffic actions were provided from BWIM system that was lo-
cated in the middle of the first span of the viaduct. It includes axle weights [kN]
and spacing [m], vehicle speed [m/s], axles configuration, pavement temperature.
Recordings were made between October 2016 and June 2017 with a total 180 days
of recorded traffic.
3.2. Effects of traffic actions
For the current study, the critical detail is chosen to be the welded connection
between a longitudinal stiffener and the plate of the orthotropic deck that is located
right under a truck wheel assuming the vehicle is passing in the middle of the slow
lane.
The values of stresses that occurred in this connection were computed with
a detailed finite element model (FEM) of the part of the deck. The model takes
into account the self-weight of the bridge deck with its asphalt layer, the type of
passing vehicle and amplitudes of axle loads from each axle, as well it considers
the influence of vehicles queues and the wind [10].
The summary of random variables for each LSF based on described ap-
proaches is given in Table 1.
Table 1. Random Variables for reliability modeling
Case Random variable Distribution Mean CoV
Resistance steel strength Fu Log-Normal 400 MPa 0.05
POT
threshold u Normal 102 0.05
shape ξ Normal -0.3 0.12
scale σ Log-Normal 25.35 0.06
PE ζu Bi-nominal 8.2× 10−4 0.045
BM
location µ Normal 100.07 0.059
scale σ Normal 23.57 0.198
shape ξ Normal −1.36× 10−2 17.503
LCC
parameter a0 Normal 8.23 0.028
parameter a1 Normal −1.03× 10−2 0.798
parameter a2 Normal −8.06× 10−4 0.087
3.3. Comparison between various statistical approaches
In order to perform the reliability analysis, limit state functions are defined for
every method described in Section 2, and then used in the software UQLab to
obtain reliability indexes with SORM method, based on values of load effects
described in Section 3. Table 2 gives results for three time references: one year,
50 and 120 years. First three columns show comparison of three extreme values
approaches, the reliability index based on design load model is shown in the fifth
column, and the last column reflects minimum values of β according to European
standards [15].
Table 2. Reliability index β for different studied cases
β, reference POT BM LCC EN, LM1 [16] β, EN [15],
2017 9.8 19.7 9.6 - 4.7
50 years 7.6 12.5 8.0 8.0 3.8
120 years 7.4 11.4 7.7 - -
4. Discussion of results
This work has been carried out on the monitored traffic data from the Millau
viaduct. Three extreme value approaches have been performed in order to asses
the reliability of the connection between a longitudinal stiffener and the steel plate
of the orthotropic deck. The values of reliability indexes are found for the year of
monitoring, the 50 years reference period and at the end of the promised opera-
tional life of the viaduct.
The results show that during the operational life, the reliability of the most
critical detail of the deck is higher than the minimum value given in standards.
POT approach and Rice formula show similar results that are also close to results
obtained with the design load model (LM1) of standards. The reliability obtained
with BM method is much higher, which could be due to the choice of a block size
and high variance of distribution parameters, and should be further investigated.
The choice of an appropriate method depends on available tools and on the refer-
ence data-set. For robust results, reliability levels from several methods should be
compared.
Acknowledgments
This project has received funding from the European Union’s Horizon 2020 re-
search and innovation programme under the Marie Sklodowska-Curie grant agree-
ment No 676139. The grant is gratefully acknowledged.
The BWIM measurements for the Millau viaduct were obtained with the help
from CESTEL, Slovenia and EIFFAGE, France, which is highly appreciated.
References
[1] J. Morio, M. Balesdent, D. Jacquemart and C. Verge´. A survey of rare event sim-
ulation methods for static input-output models. Simulation Modelling Practice and
Theory 49, 287-304, 2014
[2] C. Klu¨ppelberg, D. Straub and I.M. Welpe. Risk - A Multidisciplinary Introduction.
Springer, 2014
[3] S. Coles, J. Bawa, L. Trenner and P. Dorazio. An introduction to statistical modeling
of extreme values. Springer, 2001
[4] S. H. Ghasemi and A. S. Nowak. Target reliability for bridges with consideration of
ultimate limit state. Engineering Structures, 152, 226237, 2017
[5] I. Lukacevic, B. Androic and D. Dujmovic. Assessment of reliable fatigue life of
orthotropic steel deck. Central European Journal of Engineering, 1, 306-315, 2011
[6] X.Y. Zhou, F. Schmidt, F. Toutlemonde and B. Jacob. A mixture peaks over threshold
approach for predicting extreme bridge traffic load effects. Probabilistic Engineering
Mechanics, 43, 121-131, 2016
[7] T. Schendel and R. Thongwichian. Confidence intervals for return levels for the
peaks-over-threshold approach. Advances in Water Resources, 99, 53-59, 2017
[8] J. Pickands. Statistical inference using extreme order statistics. The Annals of Statis-
tics, 3, 119-131, 1975
[9] C. Crespo-Minguillon and J. R. Casas. A comprehensive traffic load model for bridge
safety checking. Structural Safety, 19, 339-359, 1997
[10] M. Nesterova, F. Schmidt and C. Soize. Probabilistic analysis of the effect of the
combination of traffic and wind actions on a cable-stayed bridge. Bridge Structures,
submitted, 2019
[11] C. C. Caprani, E. J. OBrien and G. J. McLachlan. Characteristic traffic load effects
from a mixture of loading events on short to medium span bridges. Structural Safety,
30, 394-404, 2008
[12] M. Nowak, D. Straub and O. Fischer. Extended extrapolation methods for robust
estimates of extreme traffic load effects on bridges. Proceedings for the Sixth Inter-
national Symposium on Life-Cycle Engineering, 2018
[13] C. Cremona. Optimal extrapolation of traffic load effects. Structural Safety, 23, 31-
46, 2001
[14] M. Nesterova, F. Schmidt, E. Bru¨hwiler and C. Soize. Generalized Pareto Distribution
for reliability of bridges exposed to fatigue. Taylor & Francis Group, 2018
[15] EN-1990, Eurocode-1.3: Basis of structural design. European Committee For Stan-
dardization, Brussels, 2002
[16] EN-1991-2, Eurocode-1. Actions on structures - Part 2: Traffic loads on bridges.
European Committee For Standardization, Brussels, 2003


Reliability of a bridge with an orthotropic deck exposed to extreme traffic events

https://hal.archives-ouvertes.fr/hal-02316778/document

Reliability of a bridge with an orthotropic deck exposed to extreme traffic events

Abstract

Similar works

Full text

Available Versions

Archive Ouverte en Sciences de l'Information et de la Communication