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Reliability of Existing Bridge Structures
prof. Ing. Josef Vičan, CSc
University of Žilina
Faculty of Civil Engineering
Department of Structures and Bridges
Lecture Content Lecture Content
Reliability of Existing Bridge StructuresReliability of Existing Bridge Structures
1. General formulation of the reliability
assessment of building structures Reliability as a ability of structure to fulfil required
functions
Classification of reliability verification methods
Engineering methods for the reliability assessment of
building
structures and basic reliability condition
The partial safety factors method
2. Reliability of existing bridge structures Relationship between structural design and evaluation of existing
structures
Adjusted reliability level for existing bridge evaluation
Loading capacity as the basic parameter of existing bridge evaluation
3. Parameters entering process of reliability verification
Steel bridge actions and materials basic characteristics
Partial safety factors for existing bridge evaluation
Calibration of partial safety factors for action and
material
4. Conclusions
1. General formulation of the reliability assessment of building structures
1.1 1.1 Reliability of building structuresReliability of building structures
Structural reliability is an ability of a structure to meet required functions from the viewpoint of preserving real service indicators in actual conditions and limits over the required time period.
Partial reliability components are:
• safety – do not endanger human health and environment,
• serviceability – utilization of a structure for intended
purpose,
• durability – time period of reliable service.
The structure occurs in certain states during its lifetime:
• from the viewpoint of activity: - service - downtime
• from the viewpoint of failure: - failure-free state - state of failure.
Specific structural state - limit state
• state when performing required functions is stopped
• state when the structure does not meet the proposed requirements anymore
• the state of failure downtime
In the case of building structures, we can distinguish: • ultimate limit states – related to the safety and durability of structures• serviceability limit states – related to the serviceability of structures
Ultimate Limit State
Exceeding them leads to structural failure – structural collapse
• failures due to exceeding the material strength or due to excessive
deformation,
• lost of member or structural stability,
• fatigue or brittle failure,
• lost of structural equilibrium.
Serviceability Limit States
Due to their exceeding, service requirements of structures will not be fulfilled.
• excessive deformations affecting features or utilization of structure,
• unacceptable vibrations influencing psychics and convenience of people
as well as structural behaviour
• local failures (cracks) reducing structural durability.
To prevent attaining individual limit states, the reliability conditions shall be fulfilled. These reliability condition are defined in corresponding codes for structural design.
Actions
Building structure
Material and geometrical parameters
Transformation models of structural
response
Transformation models of structural
resistance
Material, member or structural resistance
Structural response
Reliability verification
Ultimate limit states
Serviceability limit states
Process of reliability verification of building structures
11.2 .2 Classification of reliability verification Classification of reliability verification methodsmethods
• Deterministic methods: - allowable stress design- safety factor design
• Probabilistic methods - 1. level – semi probabilistic methods
- 2. level – engineering methods- 3. level – mathematical methods Deterministic
methodsProbabilistic
methods
Methods of
2. level
Methods of
3. level
Methods of 1. level
Partial safety factors method
calibration calibration calibration
1.3 1.3 Engineering methods for reliability Engineering methods for reliability assessment of assessment of building structures building structures
The engineering probabilistic method (Ržanicyn, Cornell)
It is the most simple probabilistic method of structural reliability verification based on probabilistic evaluation of reliability margin G defined in the form:
R is the structural resistance as a function random variable enter parameters,E are action effects as a function of random variable enter parameters.
G R E 0 (1a)
G R / E 1 (1b)
Probability of failure :
f
f
P P(G 0) P(R E 0)
P P(G 1) P(R / E 1)
Assuming statistical independence of R and E, the probability of failure can be defined:
whereR(x) is the cumulative distribution function of structural resistance R
fE(x) expresses probability occurrence of action effects E in a neighbourhood of the point x
(x) dx(x) f = P REf
xRx)P(R
dxxf/dxxE/dxxP E 22
(2)
(3)
(x) dx(x) f = P REf
xRx)P(R
dxxf/dxxE/dxxP E 22
(3)
e,r,x
fE(e)fR(r)
mE mR
fE(e)fR(r)
fE(x)FR(x)
x
dx xdx
G
fG(G)
mG
Pr
Pf
fG(G)sG
b.sG
0
f G
r G
0
P f (G)dG
P f (G)dG
(4)
Reliability index (according to Cornell): G R E2 2 0,5
G R E( )
(5)
Reliability condition:
d
fdf PP
Design values of Pfd and d for planned structural lifetime of Td = 80 years
Limit states
ultimate serviceability Reliability
level Pfd
d Pfd d
decreased 5 . 10-4 3,30 1,6 . 10-1 1,00
basic 7 . 10-5 3,80 7 . 10-2 1,50
increased 8 . 10-6 4,30 2,3 . 10-2 2,00
5,01
297,0605,0198,0
5,1198,0/log605,0
102
ff
f
PP
P
(6)
Engineering probability method represents simplified approach using linear combination of two resultant random variable E and R.
• actually, E and R are linear and non–linear combinations of action effects, material and geometrical characteristics which can be statistically independent or dependent random variables.
• it is a system random variables Xi in n-dimensional space.
Reliability margin is a function of random variables X1, X2 ….. Xn
and reliability condition has a form:
Then probability of failure shall be written as follows:
f(x1, x2 … xn) is the compound probability density function of
random variables x1, x2 ….xn
021 )X, Xg(XXg n
1 2 nG g(X , X X )
2121 fD
nnf dX.dX.dX.)X,Xf(XP
(7)
(8)
(9)
Examples of the actual failure function
Methods how to solve the problem
• approximate methods - FORM, SORM
• simulation techniques - Monte Carlo, Importance
Sampling,Latin Hypercube Sampling, Response surface and others.
2 2n n
i i i i 3i 3 i 4i 1 i 1
G X X , G X X / X
(10)
Approximate methods
Enter values Xi are transformed on uncorrelated norm
random variables Yi
Position of design point D is found which lies on the
failure function g(y) and has the minimal distance from
centre of distribution C. The distance is reliability index
ß (Hasofer - Lind reliability index).
Failure function can be usually linearly distributed
(FORM) using Taylor progression or the quadratic
approximation form can be used (SORM).
fy1(y1)
fy2(y2)
D
g(y)=0C
Simulation techniques
Monte Carlo
- Repeated numerical simulation solving failure function g(X)
always
with another random generated vector of enter parameter Xi.
- Obtained set G (g1, g2 … gn) is statistically evaluated.
Probability of failure:
where Nf is the number of simulations with gj 0,
N is the global number of simulations.
To obtain correct results,large number of simulation is needed, so
that much computer time is necessary although powerful
computer is used. This disadvantage can be eliminate by means
of modern simulation techniques.
/NNP ff (11)
Importance Sampling
Concentration of simulations in the region of g(x) = 0 using
weight
function hy(x)
Where 1 [g(x)] = 1 for Xj from field of failure,
= 0 for other Xi
Concept of Importance Sampling can be applied also for another point, e.g. for surroundings of point corresponding with mean values.
(12) (x) dXh(x)h
(x)fg(x)P y
D y
xf
f
01
hy=fx(x)/pf1 hy=fx(x)/pf2
pf1 pf2
hyX(x)=0
1.4 The partial safety factor method
Reliability condition is defined in the partial safety factor method in a separate form as follows:
Ed are design action effects,
Rd is the design resistance of material, member or structure,
Cd is the nominal value of certain properties of structural
member or structure.
(13)dd
dd
CE
RE
- Separation of random variables E and R
EERR
ER
ER
ER
ER
ER
ER
ER
ER
22
225,022
5,022
5,022.
Separation is attained through the so-called separation
(sensitivity, linearised) function of action effects and structural
resistance.
(14)
From the reliability condition in the form:
the following equation can be derived:
is the separation function
of action effects
is the separation function
of structural resistance
Separation functions E and R are replaced in the method of
partial factors by constants:
expressing very well the real form of the functions E and R
within the range of E a R :
(15)d
5,022
5,022
/
/
ERRR
EREE
(16)
(17)
67160 ,/, RE
8,0
7,0
R
E
RdRREdEE
- Application of characteristic values
- design value of action
- design value of material property
- design value of geometrical property
- design value of action effects
- design value of material resistance
Assuming proportionality of loads effects E to the action F and model uncertainties, the following relations for partial safety
factors F and M can be derived:
- partial safety factor of action effects
- partial safety factor of material
d f kF F
mkd XX /
aaa nomd
,/ , /
d Ed f k nom
d k m nom Rd
E E F a aR R X a a
F f Ed nom
M m Rd nom
1 a / a
/ 1 a / a
f is a partial safety factor for action allowing for adverse
deviations
of loading from its representative values,
m is a partial safety factor for material properties considering
adverse deviations of material properties from their
characteristic
values,
Ed is a partial factor considering uncertainties of the model of
load
response,
Rd is a partial factor allowing for uncertainties of the resistance
model
a allows for effect of adverse deviations of geometrical
properties from
their nominal values anom.
Generally, the partial safety factors for action effects and materials can be derived as follows:
WhereE , R are the ratios of the mean values of action effects
or structural resistance respectively to the relevant characteristic values,so-called bias factor of action effects or structural resistance,E , R are the coefficients of variations of action effects or
structural resistance respectively.
1RdR
1R
RdRR
kdkM
EdEEk
EdEEkdF
11
RR/R
1E
1E/E
(18)
(19)
2. Reliability of existing bridge structures
2.12.1 Relationship between structural design and evaluation of existing structures
Reliability of existing bridge structure is the ability to meet
required functions within bridge remaining lifetime respecting
usual traffic condition and bridge maintenance.
Required functions:
• to carry all actions especially traffic action on bridges
• to preserve required operating efficiency of transport
communications
• to preserve required comfort and convenience for passengers
Number of bridges: 7423 persistent bridges 29 temporary bridges
Material: concrete - 93 %steel - 3 %others - 4 %
Global length of bridges: 106,521 km from this: 96,6 % massive bridges
3,4 % steel bridges
Evaluation : 22 % bridges do not meet required loading capacity
2.1 % is in accidental condition
Statistical data about road bridges in SlovakiaStatistical data about road bridges in Slovakia
Number of bridges: 2281 bridges
Material: 78 % massive bridges 22 % steel bridges
Global length of bridges: 78,030 km from this: 52,5 % massive bridges
47,5 % steel bridges
Evaluation: 23 % is more than 77 years 14 % is more than 100 years 2.4 % bridges do not meet required
loading capacity
Statistical data about railway bridges in SlovakiaStatistical data about railway bridges in Slovakia
Causes
• Bad concept and technology of the bridge erection
• Increasing of transport intensity
• Material degradation due to retrogressive
environment
• Insufficient and unqualified bridge maintenance
• Shortage of financial resources for bridge
maintenance
• Development of Slovak motorway network and
modernisation of European railway corridors
The main differences between existing bridges and newly designed ones:
• effect of the regular inspection as well as the results of the technical diagnostics, which reduce the uncertainties of input parameters of reliability verification,
• lengths of the bridge remaining lifetime; it means time,for which the results of evaluation are reliable,
• effect of the reliability level differentiation in dependence on function of the element in the whole system,
• actual bridge condition found by diagnostic investigation.
Therefore, the adjusted reliability level for existing bridge evaluation should be considered.
In assessing reliability of existing structures include bridges,it is necessary to consider differences they may have in comparison with newly designed structures.
2.2 2.2 Adjusted reliability level for existing bridge evaluation
Due to differences between design of new structures and evaluation
of the existing ones, the adjusted reliability level should be derived
using theoretical approach based on conditional probability
respecting basic information from regular bridge inspection.
Basic assumptions:
• The observed bridge member was designed for planned lifetime of
Td with
basic reliability level given by reliability index β(t):
µR(t),σR(t) are the mean value and the standard deviation of
normally distributed member resistance,
µE(t),σR(t) are the mean value and standard deviation of normally
distributed action effects.
0,52 2R E R Eβ(t) μ (t) μ (t) / σ (t) σ (t) 3.80 (20)
• An inspection carried out at time tinsp < Td has shown that the
verified
bridge member should not fail in the sense of exceeding any of
its limit
states. This state can be described by following equation:
Time dependent R and E enables considering changes of member
resistance and action effects in time to allow for e.g. effects of
material degradation.
The conditional probability that bridge member survives to planned
lifetime Td will be as follows:
i inspR(t) max[E (t)], for i 1, 2 . N (t ) (21)
i d
i insp
P R(t) max(E (t) i 1,..., N(T )
P R(t) max(E (t) i 1,....N(t )
(22)
i insp d
i insp
P R t max E t , for i N t 1 N T /
R(t) max E (t) , for i 1, . N t
The probability of failure for member remaining lifetime will be
then:
And corresponding reliability index:
Individual probability of failure Pf (tinsp) respectively Pf (Td) could be
calculated using formulae:
where
is the cumulative distribution function of random variable E(t)
f d f inspf dfu
f insp f insp
P (T ) P (t )1 P (T )P (t) 1
1 P (t ) 1 P (t )
23(
.)()( 1 tPt fuu (24)
t
RR
Rf dxdf
xxFtP
0
)()(
1
)(
)()(1)(
(25)
t
0 E
E d)(f)(
)(x1)t(L
e)x(F (26)
tpre
tpretLf
,0 , 0
,0 ,/
(27)
Action effects Ei(t) are considered as a set of load effects
repeating in time with frequency N(t), that is a random
variable having Poisson distribution of probability in form:
Parameter λ(t) represents intensity of action effects
occurrence within the requisite time and thus also intensity of
failures. It may be considered constant or in time linearly
dependent within observed time.
(28)
t
tLk
dtL
kketLktNP
0
...0for ,!/
Results of parametric studies show, that reliability index β(t) increases for member remaining lifetime due to positive information acquired upon performed inspection. If proper implementation of inspection can be assumed when designing a structural member, the member can be designed to a lower target reliability index βt. This can be determined by iteration, so that at the end of the member lifetime its value did not decrease below the basic design level βd= 3.80.
2,4
2,9
3,4
3,9
4,4
0,000 0,125 0,250 0,375 0,500 0,625 0,750 0,875 1,000
tinsp /T
u
=0,01250,0250,06250,1250,25
u
2,4
2,6
2,8
3,0
3,2
3,4
3,6
3,8
4,0
0,000 0,125 0,250 0,375 0,500 0,625 0,750 0,875 1,000tinsp/T
t
=0,01250,0250,06250,1250,25
As it has been shown in previous pictures, the reliability level is in all cases connected closely with a structural lifetime. In the case of new structures, it is their planned lifetime.
In the case of existing ones it should be the remaining lifetime for which the determined level of reliability is applicable. Due to difficulties in determining realistic remaining lifetime of existing bridge structural members, the reliability level for planned remaining lifetime has been derived.
The planned remaining lifetime is the difference between the design lifetime Td and the time during which the structure was in operation, provided all design requirements were respected – purpose of the structure, periodic inspections, current maintenance, etc.
Table 1 The reliability levels for existing bridge evaluationBridge evaluation after
20. years 40. years 60. years 70. yearsRemaining
lifetimeyears Pft
t Pft
t Pft
t Pft
t
3 5,60.10-4 3,26 1,08.10-3 3,07 1,58.10-3 2,95 1,82.10-3 2,915 3,73.10-4 3,37 6,70.10-4 3,21 9,65.10-4 3,10 1,11.10-3 3,06
10 2,23.10-4 3,51 3,71.10-4 3,37 5,18.10-4 3,28 5,91.10-4 3,2420 1,48.10-4 3,62 2,21.10-4 3,51 2,95.10-4 3,4430 1,23.10-4 3,67 1,73.10-4 3,5840 1,11.10-4 3,69 1,48.10-4 3,6250 1,05.10-4 3,7160 9,70.10-5 3,73
Table 1 The reliability levels for existing bridge evaluationBridge evaluation after
20. years 40. years 60. years 70. yearsRemaining
lifetimeyears Pft
t Pft
t Pft
t Pft
t
3 5,60.10-4 3,26 1,08.10-3 3,07 1,58.10-3 2,95 1,82.10-3 2,915 3,73.10-4 3,37 6,70.10-4 3,21 9,65.10-4 3,10 1,11.10-3 3,06
10 2,23.10-4 3,51 3,71.10-4 3,37 5,18.10-4 3,28 5,91.10-4 3,2420 1,48.10-4 3,62 2,21.10-4 3,51 2,95.10-4 3,4430 1,23.10-4 3,67 1,73.10-4 3,5840 1,11.10-4 3,69 1,48.10-4 3,6250 1,05.10-4 3,7160 9,70.10-5 3,73
The adjusted reliability levels shown in Table 1 are valid for primary bridge members.
The uniform reliability level, Pft= 2,3 . 10-3 or t = 2,80 was establishedfor secondary bridge members.
2.3 Loading capacity as the basic parameter
of
existing bridge evaluation
Process of reliability verification of existing bridge structures is
a crucial part of their overall evaluation, which is understood
as a complex assessment based on processing all available
information to reach the optimum most economic decision
concerning the bridge rehabilitation strategy.
Two approaches to existing bridge evaluation :
• classification approaches
• reliability – based approaches
Classification approaches involve assessment of an existing bridge structure based on results of periodic inspection, but without checking the reliability of existing bridge structure taking into account only current technical bridge condition.In classification approaches,this is expressed by various weighted coefficients which seek to take into account the influence of bridge member damage or failure upon its reliability and reliability of whole bridge structure.
Reliability-based approaches to the existing bridge evaluation look at direct influence of current bridge member technical condition on its behaviour and its reliability by means of its reliability verification.
The basic quantitative and qualitative parameter in reliability-based evaluation of existing bridge structure is its loading capacity expressed in the form of so-called Live load rating factor - LLRF.
Loading capacity of existing bridge structures
LLRF can be derived for separate bridge member from a marginal
condition of reliability of relevant limit state. In the case of ultimate limit
states:
where
where EQd is the design value of the variable short-term traffic load
effects are design values of others loads acting simultaneously with the traffic load ( permanent load, variable long-term load, climatic loads, brake forces,lateral strokes etc.)
(20)d dE R
n 1
d rs,di Qd di 1
n 1
d rs,di Qdi 1
E E LLRF E R
LLRF R - E E
(21)
1
1
n
irs,diE
Loading capacity is understood as the amount of bridge Loading capacity is understood as the amount of bridge
capacity used bcapacity used byy a variable short-term traffic action. It is a variable short-term traffic action. It is
expressed via the level of an appropriate variable load effects, expressed via the level of an appropriate variable load effects,
either road or railway trafficeither road or railway traffic,, which are which are considered by ideal considered by ideal
load models. load models.
In the case of railway bridges - dynamic load effects
of load model UIC-71 (EUIC,d) are considered as the appropriate
level of load effects.
In the case of road bridges – three types of loading
capacities shall be distinguish according to Slovak standards:
• normal loading capacity (n)
• exclusive loading capacity (r)
• exceptional loading capacity (e)
In accordance with appropriate type of road loading capacity,
the relevant traffic load models for normal, exclusive and
exceptional loads shall be applied.
where
Vj is the the loading capacity expressed by vehicle weight of
the
appropriate road traffic load model,
Vjk is the characteristic value of vehicle weight of the
appropriate
road traffic load model.
n 1
j d rs,di d jki 1
V R - E / Q .V , for j n, r,e
(22)
While the loading capacity of road bridges directly specified
the weight of vehicles passing the bridge structure without
any limitations, in the case of railway bridges the passage of
actual traffic load shall be specified additionally using
following formulae:
where
and λUIC is the actual railway traffic load efficiency,
ET is the characteristic value of actual railway traffic
load effects,
EUIC is the characteristic value of load model UIC-71
effects
δ is the dynamic factor of the load model UIC-71, δf is the dynamic factor of actual railway traffic load
effects.
UIC UIC
UIC T UIC
ψ LLRF λ
λ E / E
(23)
(24)fψ δ/δ
Load class Axle load[kN]
Equivalent uniformlydistributed load
[kNm-1]A 160 50
B 1 180 50B 2 180 64C 2 200 64C 3 200 72C 4 200 80D 2 225 64D 3 225 72D 4 225 80
The real railway traffic vehicles are simulated by means of represen-tative traffic load models included in nine classes. The axle forces and equivalent uniformly distributed load are defined for every representative load model.
In the approach described above, the UIC-71 load model
effects are used as a comparative level for determining
passage of actual traffic load over the observed bridge, so
that it means generalization but also simplification of the
approach presented above.
To avoid some shortages of the simplified standard practice
mentioned above, the concept of traffic loading capacity in
the form of Traffic Load Rating Factor (TLRF) was
developed having the following form
ETd is the design value of the dynamic effects of the actual
railway traffic load classified in nine classes presented on previous slide.
(25) T d rs,di TdLLRF R - E /E
3. Parameters entering process of existing bridge reliability verification
33.1 Load and material characteristics of steel bridge structures • Permanent and long-term actions
Self-weight of structural and non-structural bridge member
Characteristic values:
Ak is the nominal value of cross-section area,
ρ is the average material bulk density.
Design values:
- considering Gk as a nominal value
- considering Gk as a mean value
kk AG (26)
kGd GG (27)
GdEGGSdG 11
GdEGSdG 12
Table 2
Action G ωG G1 G2 G
Hot-rolled bars 1.050 0.062 1.209 1.198 1.223
Sheets 1.000 0.030 1.074 1.068 1.080
Compound cross-sections 1.008 0.067 1.173 1.162 1.188
Cast-in-factory members 1.030 0.080 1.232 1.217 1.250
Cast-in-place members 1.050 0.100 1.307 1.289 1.330
The values of partial safety factors for some types of permanent
actions γGi were calculated using adjusted reliability level derived
for existing bridge structure evaluation. In Table 2 are given the
adjusted values of γGi determined for adjusted reliability indexes
βt = 3.50(γG1)and βt = 3.25 (γG2)valid for bridge remaining lifetime
t=20 years or t=10 years respectively. Partial safety factor of
model uncertainties was taken into account by value γEd =1.0.
Calculated values are presented in Table 2 together with basic
value valid for βd=3.80(γG).
• Variable traffic actions
In case of railway bridges, the load model UIC-71 is used to
calculate LLRFUIC and λUIC.
Load class Axle load[kN]
Equivalent uniformlydistributed load
[kNm-1]A 160 50
B 1 180 50B 2 180 64C 2 200 64C 3 200 72C 4 200 80D 2 225 64D 3 225 72D 4 225 80
The real railway traffic vehicles are simulated by means of represen-tative traffic load models included in nine classes. The axle forces and equivalent uniformly distributed load are defined for every representative load model.
3.2 Partial safety factors for existing bridge evaluation
To determine adjusted values of partial safety factors for railway traffic action, the statistical characteristics of actual railway traffic action is necessary to know.
The basic information is possible to obtain:
- by means of information system of Slovak Railways IRIS-N, where data about freight trains are available,- in-situ measurements on real bridge structures,- numerical simulation of trains passing the bridge structures using appropriate transformation models of bridge structures and information acquired from IRIS-N,- combination of above mentioned approaches.
To obtain effects of the real railway traffic load, the in-situ measurements on the actual bridge structure across river Váh were carried out. The observed bridge structure is located near the railway station Turany and represents the three-spans steel railway bridge consisting of two truss girders and open bridge deck. The in-situ measurements were carried out in 43.4 m long span, which cross-section is presented in following Fig.
The passages of the 25 freight trains and 26 local trains were monitored within the 20-hour in-situ measurement. The measured values were statistically processed.
In-situ measurements and numerical simulation of traffic load effects
Results of in-situ measurements
The mean values (μs) and coefficients of variation (ωs) of the processed statistical data obtained by in-situ measurements
are as follows:μs,H4 = - 24.02 MPa, ωs,H4 = - 0.20, H4 = 0.34
for the
upper chordμs,S4 = 19.21 MPa, ωs,S4 = 0.20, S4 = 0.37
for the
bottom chord
Results of the numerical simulation
Using numerical simulation of train passages over the bridge, the
static load effects of the 205 freight train sets moving on the bridge within one week were obtained. Dynamic response of
the observed bridge chord members was allowed for using the
dynamic factor f in accordance with ENV 1991-3 (1995) that
is valid for actual railway traffic load taking into account the
actual train speed. The stress responses of the bridge chord
members obtained by numerical simulations were statistically
processed. and are presented.
The major statistical characteristics of the observed members
stress response are as follows
μs,H4 = - 32.84 MPa, ωs,H4 = - 0.20, H4 = 0.47
μs,S4 = 23.87 MPa, ωs,S4 = 0.19, S4 = 0.47
Bottom chord S4
0
5
10
15
2012
,00
14,4
4
16,8
8
19,3
2
21,7
6
24,2
0
26,6
4
29,0
8
31,5
2
33,9
6
36,4
0
38,8
4
Stress [MPa]
Fre
qu
en
cy [
%]
Upper chord H4
0
5
10
15
20
25
-56,
00
-52,
88
-49,
76
-46,
64
-43,
52
-40,
40
-37,
28
-34,
16
-31,
04
-27,
92
-24,
80
-21,
68
Stress [MPa]
Fre
qu
en
cy [
%]
• Partial safety factors for traffic load
The obtained statistical data was applied for determining values
of partial safety factor for railway traffic action respecting
adjusted reliability level valid for existing bridge evaluation.
Due to many quantities with very similar values, a simplification
has been performed and result values recommended for
practice are shown in Table 3.
Table 3 Bridge age Less than 60 year More than 60 year
Remaining bridge lifetime tr Remaining bridge lifetime tr
Bridge
component
10 tr 20
3 tr 10
tr 3
10 tr 20
3 tr 10
tr 3 Main bridge components
1.30 1.25 1.20 1.30 1.20 1.15
Secondary bridge
components
1.20 1.20 1.20 1.15 1.15 1.15
• Material characteristicsTo determine partial safety factors for structural steel, we have a large statistical sets of yield strength collected from 60-ties. Assuming gamma distribution of collected material properties we determine partial safety factor of structural steel for adjusted reliability level βt = 3.50.
where fyk is the characteristic value of steel yield strength,
fyd is the design value of steel yield strength,
aR is the non-symmetry coefficient of steel yield strength,
R is the variation coefficient of steel yield strength,
r is the mean value of cross-sectional characteristic,
r is the variation coefficient of cross-sectional
characteristic,
,d are constants allowing for adjustment of the reliability
index for gamma distribution.
5,022 ))(1(1
)1(64,11/
rRRdtRr
RRydykM a
aff
(28)
for 3 tr 20 years: M = 1.10 for S 235 M = 1.15 for S 355
for tr 3 years: M = 1.05 for S 235
M = 1.10 for S 355
Bridge age
60 years 60 years
Bridge remaining lifetime tr
comp tens comp tens
10 t r 20 years 1,25 1,40 1,20 1,30
3 t r 10 years 1,20 1,30 1,20 1,25
tr 3 years 1,15 1,25 1,15 1,20
3.3 Calibration of partial safety factor for action and material
The values of partial safety factors for actions and for material were determined for adjusted reliability level separately. Now we will verified proposed values of relevant partial safety factors using calibration.
Load model
As the representative load model of railway traffic load, the load model A according the UIC Kodex 700 V has been chosen.
• Resistance model
• Tension member S4 of bottom chord
Nt,4 = Rt . An = fy . φa . An (29)
fy is the steel yield strength,
φa = A/An is a ratio of real and nominal value of cross-sectional
area of the bottom chord.
Member resistance Rt has been determined using Monte – Carlo simulation.
λRt = mRt/mRk = 1,20 - bias factor of the member resistance,
vRt = 0,092 - coefficient of variation of the member resistance.
• Compression member of upper chord The buckling resistance of the bridge upper chord could be
determined according to formulae valid for pin-ended strut with initial out-of-straightness of sinus half wave derived by Šertler, Vičan and Slavík (1992) in form taking into account all the parameters as random variables.
RC = [ - (2 - 1)0,5] a
(30a) where = 0,5 [fy + 2 E ((b / L. n)2 + e.eoe,n.zn. b / L)]
1 = 2 Efy (b / L n)
(30b)
and a = A / An is the ratio of the actual and nominal values of the cross-section area
(fabrication factor)
fy is the actual steel yield strength.
The following symbols were used in relation (30a) and (30b):
L = Lcr / Ln, b = b / bn, e = eoe / eoe,n, zn = (z / L)n (31)
n is the nominal value of the member slenderness,
Lcr (Lcr,n) is the actual (nominal) buckling length of
chord member,b (bn) is the actual (nominal) width of the chord
cross-section,eoe (eoe,n) is the actual (nominal) equivalent value of the
relative initial out-of-straightness,
zn is the nominal distance of the extreme cross-
sectional fibres from the centroid of the chord cross-section.
The empirical distribution of steel yield strength and relative width b of the upper chord cross- section
0
2
4
6
8
10
12
14
16
23
8
24
8
25
8
26
8
27
8
28
8
29
8
30
8
31
8
32
8
fy [MPa]
Fre
qu
en
cy
[%]
0
1
2
3
4
5
6
7
0,9
6
0,9
7
0,9
8
0,9
9
1,0
0
1,0
1
1,0
2
1,0
3
1,0
4
1,0
5
1,0
6
1,0
7
1,0
8
1,0
9
1,1
0
1,1
1
1,1
2
f b
Fre
qu
en
cy
[%]
The remaining random variables have normal distribution with parameters
a – µa = 1.001, σa = 0.03e – µe = 0.963, σe = 0.022
Constants entering formulae (30) have following values:
E = 210 000 MPa, zn = 0.06912, n =23.42, eoe,n = 0.001
Histogram of the chord buckling resistance obtained by Monte-Carlo simulation
Statistical characteristics of the buckling resistance RC
µR = 266,101 MPa, σR = 19,603 MPa
λR = µRc/µRk = 1,15 bias factor of member
resistance ωRc = 0,070 variation coefficient of member
resistance
0
1
2
3
4
5
6
20
6,1
1
21
4,2
4
22
2,3
7
23
0,5
0
23
8,6
3
24
6,7
6
25
4,8
9
26
3,0
2
27
1,1
5
27
9,2
8
28
7,4
1
29
5,5
4
30
3,6
7
31
1,8
0
31
9,9
3
32
8,0
6
33
6,1
9
Rc [MPa]
Fre
qu
en
cy
[%]
Rackwitz–Fiessler method of design point
Design point D is laying on the failure border and has the maximum probability of failure.
G = R – E = 0 so that following relation is valid for point D (RD, ED)
RD = ED.
The approximation of cumulative distribution function of resistance FR
and load effects FE on cumulative distribution functions of normally
distributed random variables is the basic assumption of the above mentioned method.
Then standard deviation and mean value can be defined by following relations
)R(f/)R(F DRDRnR1
Similar relations are valid for standard deviation and mean value of load effects .
n is the probability density function of norm normal distribution,
is the cumulative distribution function of norm normal distribution.
)R(F.σRµR DRRD1
Now the reliability index β can be determined, which defines the distance of the design point D from the centre C of distribution.
Súradnice návrhového bodu určíme zo vzťahov
50222 ,
ERRRD /R 50222 ,
EREED /E
5022 ,
ERER σσ/µµ
In the analysed case, the normally distributed load effects has been considered, because they are a sum of effects of permanent , long-term variable and short-term variable actions. From this point of view, the approximation of cumulative distribution function has not be needed. The member resistance has been assumed as log-normally distributed random variable.
The calibration of partial safety factors has been performed for basic combination of permanent, long-term variable and short-term variable traffic actions represented by load model A.
MkTkfFTQkFQGkFG /REEE
γFi are partial safety factors of effects of permanent action (EGk), long-term variable action (EQk) and short-term variable traffic action (ETk), δf is the dynamic factor of real traffic load, γM is the partial safety factor of structural steel.
For set of partial safety factors determined separately for adjusted reliability level given by reliability index βt = 3,50
γFG = 1,10, γFQ = 1,20, γFT = 1,20, γM = 1,10
the characteristic value of Rk has been determined from marginal reliability condition and using bias factor of member resistance the mean value of resistance µR is calculated. By means of statistical characteristics of individual load effects, the mean value µE
and standard deviation σE of global load effects could then be determined. The standard deviation and mean value of the approximate normal distribution of the member resistance R in the point RD will be then as follows:
DRR .RRRDRDR
)/ln(lnRR )k(1R RD R
For proposed set of partial safety factors, the distance of design point from centre of distribution has been calculated by means of an iteration process.
β = 4,343 > βt = 3,50
β = 4,415 > βt = 3,50
In the case of compression resistance model of the chord H4 :
In the case of tension resistance model of chord S4:
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