Reliability analysis of statically Reliability analysis of statically

indeterminate steel frameindeterminate steel frame

(pilot study)(pilot study)

David PustkaDavid Pustka

VVŠB – Technical University of OstravaŠB – Technical University of Ostrava

Faculty of Civil EngineeringFaculty of Civil Engineering

Ludvíka PodéšLudvíka Podéštětě 1875 1875, 708 , 708 0000 Ostrava, Czech Republic Ostrava, Czech Republic

IntroductionIntroduction

In the last months was performed the In the last months was performed the pilot studypilot study demonstrating the demonstrating the applicapplicability of SBRAability of SBRA method method to to systems.systems.

This study is briefly explained in the following.This study is briefly explained in the following.

The The first order theoryfirst order theory and and staticstatic and and elasticelastic response response to the loading was assumedto the loading was assumed..

The The calculation of load efects wascalculation of load efects was based on based on simplified simplified deformation methoddeformation method..

All loads acting on structure All loads acting on structure werewere assumed as assumed as random random variablesvariables and and werewere described by described by load duration curvesload duration curves..

Example assignmentExample assignment

Statically indeterminate planar steel frameStatically indeterminate planar steel frame made of made of rolled-sections IPErolled-sections IPE::

g

jh

a b

ed f

ck

m

F1 F2 F2 F3

F4

F5

1

6

2

411

912

3

5

10

7 8

f1

F1 ... short-lasting loadF2 ... long-lasting & short-lasting loadF3 ... long-lasting & short-lasting load

F4 ... wind loadF5 ... wind loadf1 ... dead & snow load

Load effects calculation – normal Load effects calculation – normal forcesforces

Load effects calculation – shear Load effects calculation – shear forcesforces

Load effects calculation – bending Load effects calculation – bending momentsmoments

Load effects combination Load effects combination calculation, Resistance calculationcalculation, Resistance calculation

Load effects combinationLoad effects combination in any point of the structure (in in any point of the structure (in case of shear forces neglecting) case of shear forces neglecting) can becan be expresexpresssed ed byby normal stress normal stress xx : :

zI

M

A

Nx

The The resistanceresistance of the whole structure from the viewpoint of of the whole structure from the viewpoint of the safety was related to the safety was related to yield stressyield stress f fy y ::

var,9.0 yy ff

The The overstepoverstep of this of this reference reference valuevalue in any point of the in any point of the structure was considered asstructure was considered as failure failure..

Reliability assessmentReliability assessment

The The reliability assessmentreliability assessment in the in the critical cross-sectioncritical cross-section of of the structure the structure waswas based on the based on the reliability criterionreliability criterion::

df PP

where where PPdd is is target probabilitytarget probability given by codes and given by codes and PPff is is

probability of failureprobability of failure calculated from the calculated from the reliability functionreliability function RF RF of the critical cross-sectionof the critical cross-section ::

xyfRF

Reliability function calculationReliability function calculation

Resulting Resulting reliability function reliability function of critical cross-section:of critical cross-section:

Minimum : 4 Variance : 2538,769364

Maximum : 473 Skewnes : 0,067558461

Mean : 241,507527 Kurtosis : 1,852408027

StDeviation : 50,38620212 Median : 240,955801

Probability : 0,00007 Quantile : 20,57988281

Probability : 0,5 Quantile : 240,9809439

Probability : 0,99993 Quantile : 471,7175781

Quantile : 0 Probability : 00

100

200

300

400

500

600

700

6,748 80,029 153,311 226,592 299,873 373,154 446,436

SF -

Minimum : 21 Variance : 2580,789592

Maximum : 510 Skewnes : 0,075854437

Mean : 256,9382339 Kurtosis : 1,845579242

StDeviation : 50,80147234 Median : 255,6080107

Probability : 0,00007 Quantile : 30,64628906

Probability : 0,5 Quantile : 255,6314111

Probability : 0,99993 Quantile : 494,5277344

Quantile : 0 Probability : 00

100

200

300

400

500

600

700

23,865 100,271 176,678 253,084 329,490 405,896 482,303

SF +

Summary and conclusionSummary and conclusion

The demonstrated simulation based approach The demonstrated simulation based approach SBRASBRA can be used as a can be used as a tooltool for for reliability analysisreliability analysis of of systemssystems..

This approach allows to evaluate This approach allows to evaluate all possible all possible combinationscombinations of all input variables of all input variables in timein time due to due to direct direct Monte Carlo simulationMonte Carlo simulation technique. technique.

The calculation is The calculation is quickly, clearly quickly, clearly and leads to and leads to balanced reliability level.balanced reliability level.

For details, see For details, see dissertation work dissertation work (Pustka, V(Pustka, VŠB-TU ŠB-TU Ostrava, Ostrava, 20022002).).

ThankThank you you for attention … for attention …