Application of the Direct Optimized Probabilistic Calculation Martin Krejsa Department of Structural Mechanics Faculty of Civil Engineering VSB - Technical

Application of the Direct Optimized Probabilistic

CalculationMartin Krejsa

Department of Structural MechanicsFaculty of Civil Engineering

VSB - Technical University OstravaCzech Republic

Synopsis of Presentation Direct Optimized Probabilistic Calculation – DOProC method

Background and introduction to the probabilistic method Program package ProbCalc

Probabilistic calculation approach to the propagationof a fatigue crack using DOProC method Propagation of cracks from the outer edge Propagation of cracks from the surface

Probabilistic calculation of fatigue cracks propagating using FCProbCalc program The choice of integration method Analysis of accuracy in relation to the time of calculation

Conclusions

Application of the Direct Optimized Probabilistic Calculation 1 / 26

Essential of DOProC Characteristics Should be effectively used for the assessment of structural

reliabilities and/or for other probabilistic calculations. Input random variables (load, geometry, material properties,

imperfections) are expressed by the empirical or parametric distributions in histograms.

Reliability function under analysis can be expressed analytically or using DLL library.

Error of calculation is done only by discretization of input and output variables and numerical error.

The number of intervals (classes) of each histogram is extremely important for the number of needed numerical operations and required computing time.

The number of numerical operations can be reduced using optimizing techniques of the probabilistic calculation.

Direct Optimized Probabilistic Calculation - DOProC Method 2 / 26

Principle of Numerical Calculation

B = f(A1, A2, …, Aj, … An)


Optimizing Techniques in DOProC

Grouping of input random variables, which can be expressed by the common histogram.

Interval optimizing - decreasing the number of intervals in input variable histograms.

Zonal optimizing - each histogram is divided into areas (zones) depending on their share in the result (failure).

Trend optimization – using correct or incorrect trend of input variable on the result.

Grouping of partial calculations results. Parallelization of the calculation – calculation is proceeded on

number of processors. Combination of the mentioned optimizing techniques.Direct Optimized Probabilistic Calculation - DOProC Method 4 / 26

Program Utility ProbCalc

5 / 26

Analytical definitionof calculation model

Calculator

Reliability function

Commandline

Grouping of input random variables

Table of input random variables

Direct Optimized Probabilistic Calculation - DOProC Method

Program Utility ProbCalc

Probability of failurepf = 1,28.10-6

meets requirements of EN1990 for consequences

class RC3/CC3with design probability

pd = 8,4.10-6

Histogram of reliability function RF

6 / 26Direct Optimized Probabilistic Calculation - DOProC Method

Usage of ProbCalc

Probabilistic assessment of load combinations, Probabilistic reliability assessment of cross-sections and

systems of statically (in)definite load-bearing constructions, Probabilistic aproach to assessment of mass concrete and

fibrous concrete mixtures, Reliability assessment of arch supports in underground and

mining workings, Reliability assessment of load-bearing constructions under

impact loads, Probabilistic calculation of fatigue crack progression in steel

structures and bridges.


Program FCProbCalc

Probabilistic calculation of fatigue cracks propagating using FCProbCalc program 8 / 26

FCProbCalc program desktop - entry of input quantities

Look to the reviewed road bridgePhoto: J.Odrobiňák

Bridge’s crosswise cut

Detail of Solved Steel Bridge’s Flange

Photo: J.Odrobiňák

Reliability Assessment of Steel Bridge’s Flange in Tension


Some of input values - no way to obtain using measurement, can be approximate only.

QuantityType of parametric

distribution

Parameters

Mean ValueStandard Deviation

Oscillation of stress peaks [MPa] Normal 30 3

Total number of oscillation of stress peaks per year N [-] Normal 106 105

Initial size of the crack a0 [mm] Lognormal 0,2 0,05

Smallest measurable size of the crack ad [mm] Normal 10 0,6

Yield stress of material fy [MPa] Lognormal 280 28

Nominal stress in flange [MPa] Normal 200 20

List of random input variables

Quantity Mean Value

Constant of material m 3

Constant of material C 2,2.10-13

Flange width bf [mm] 400

Flange thickness tf [mm] 25

List of constant input variables

Real values

Approximate values

Overview of Variable Input Quantities


Resistance of the structure R :

da

d

a

am

a

aFa

aR

0 ..

d

0..d..0

NNCNCS mN

N

m Cumulated effect of loads S :

N is total number oscillation of stress peak needed to increase the crack from a0 to ad or aac

N0 is number of stress range cycles in time of fatigue crack initialization

C , m are material constants

ac

ac

a

am

a

aFa

aR

0 ..

d

a0 is initial crack size ad is detectable crack size aac is acceptable crack size F(a) is calibration function, which corresponds to propagation

behavior of the crack

Probabilistic calculation approach to the propagation of a fatigue crack

where

where


All of these three events creates full space of event, which can come in time t, can be applied:

1 ttt FPDPUP

• Probability of crack undetection in time t :

where ad is minimal detectable crack size

• Probability of crack detection in time t, crack size a(t) is less than tolerable size aac:

• Probability of crack detection in time t, crack size a(t) is equal or greater than tolerable (acceptable) size aac:

actt aaPFP

dtt aaPUP

actdt aaaPDP

Probability of Defined Random Events


Crack’s propagations from the edge or from the surface are possible to monitor according to initial crack position.

Weakness of the same flange increased from the edge is quicker then from the surface one.

Places of Fatigue Damage Danger’s Concentration


432

39,3072,2155,10231,012,1

b

a

b

a

b

a

b

aF a

The propagation of the fatigue crack from the edge can be expressed by means of a calibration function F(a) :

where a is crack size, b is flange width (in this case 400 mm).

Fatigue Crack’s Propagations from the Edge


St

aM

t

aMMSMF a ..

4

3

2

21

c

aM 09,013,11

c

aM

2,0

89,054,02

24

3 11465,0

15,0

c

a

c

aMwhere

wffgS ..

22

sin135,01,01

t

agwhere 4 22

2

cossin.

c

af

t

a

b

cfw

sec

where is geometrical parameter. For sizeable width b is valid, that fw=1.

Coefficient S :

Fatigue Crack’s Propagations from the Surface

Calibration function F(a) in general definition:


Cumulated Effect of Loads S

Total number oscillation of stress peak per 111 years

Determined for each yearof the bridge operation using time step equals 1 year.


Total number oscillation of stress peak per 49 years

Structure Resistance Rad

Crack’s propagations from the edge

Crack’s propagations from the esurface

Yet possible to select 5 types of integration methods


Numerical integration

• Rectangular method

• Simpson method

• Romberg method

• Adaptive method

• Gauss quadrature

Gaussian quadrature integration points


Probability of Defined Random Events

The probability of the event U, D and F,

depending on years of operation of the bridge

The crack from the edge0 to 80 years of operation

The crack from the surface0 to 150 years of operation


Construction Inspection TimesFatigue crack from the edge

FCProbCalc program desktop


Construction Inspection TimesFatigue crack from the edge

The resulting probabilities and times of construction inspections

The dependence of the probability of failure Pf and

years of operation of the bridge

Adaptive numerical integration method chosen with a parameter tol0 = 1.10-4


Construction Inspection TimesFatigue crack from the surface

FCProbCalc program desktop


Construction Inspection TimesFatigue crack from the surface

The resulting probabilities and times of construction inspections

The dependence of the probability of failure Pf and

years of operation of the bridge


Adaptive numerical integration method chosen with a parameter tol0 = 1.10-4

Comparison of the calculated 1st time inspection

Calculated times of the 1st inspection of the bridge construction with a particular attention on used numerical integration method

Fatigue crack

from the edge


Time of the Calculation

Calculation time for each type of numerical integration and the specified number of intervals of input random variables

Fatigue crack

from the edge


Conclusion

Presentation approached the newly developed probabilistic method DOProC, applications and developed software tools (for details see eg http://www.fast.vsb.cz/popv/).

DOProC method appears to be an effective means for obtaining solutions for a number of probability problems.

26 / 26

DOProC method can solve such propagation of fatigue cracks in steel bridges and structures and to determine the inspection times using conditional probability.

Application of the Direct Optimized Probabilistic Calculation

Thank you for your attention !

Ostrava, Moravian-Silesian Region

Documents

Application of the Direct Optimized Probabilistic Calculation Martin Krejsa Department of Structural Mechanics Faculty of Civil Engineering VSB - Technical