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: