Guendel_Influence of Material Strength Scattering on the Ductile Response of Steel Structures_FINAL

8/2/2019 Guendel_Influence of Material Strength Scattering on the Ductile Response of Steel Structures_FINAL

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1 INTRODUCTION

The exploitation of the capability of steel structuresto dissipate energy by means of plastic deformationsis a common and effective design strategy to achieve

a high resistance against exceptional loads like earth-quake. The crucial point in this design method is theprediction and control of the formation of plasticmechanisms with regard to their location and to theirultimate resistance combined with the prevention ofbrittle or other sudden failure modes. To this end socalled capacity design rules are applied, where brittleparts of the structure are designed with a sufficientoverstrength compared to the plastic limits of ductilemembers and thus enabling the development of theintended plastic mechanisms.

The overstrength used in the capacity design

needs to cover the differences between the nominalplastic resistances of members obtained with thenominal values of yield strength and member dimen-sions and the real resistances which include the in-fluence of strain hardening, cross-sectional dimen-sions and the actual yield strength of the material. Inparticular the scattering of the yield strength, whichin reality is usually significantly higher than thenominal value, is of high importance. Current Euro-pean material standards however provide no re-quirements with regard to a limitation of the upper

yield strength values.The contribution presents results from the re-

cently finished research project OPUS founded byRFCS. In this project actual material properties ofstructural steel were obtained directly from the

monitoring of European steel producers. The datawere evaluated statistically and used to assess theexpected distribution of yield strength in four refer-ence steel buildings. The structures were used to in-vestigate the influence of scattered material strength

on their response to seismic actions and on failuremodes.

2 MATERIAL STRENGTH

2.1 Production standards

The limits for mechanical material properties ofstructural steel as yield stress, tensile strength and ul-timate elongation are defined in the European pro-duction standard series EN10025. These standardsdefine depending on the material thickness

minimum values for the yield stress and ultimateelongation as well as lower and upper limits for thetensile strength. An upper limit for the yield stress,as given by ISO-DIS24314 (steel grades for seismicapplication), is not implemented in EN10025. How-ever, the European seismic standard EN1998-1 rec-ommend to consider an factor of 1.25 for materialoverstrength and a factor of 1.1 for strain hardeningin the capacity design.

2.2 Measured material properties

In the research project OPUS two European steelproducers provided more than 13000 material datasets from three plants including yield stress, tensilestrength, ultimate elongation and nominal thickness.

Influence of material strength scattering on the ductile response of steelstructures

M. Gndel, B. Hoffmeister & M. FeldmannInstitute for Steel Structures, RWTH Aachen University, Germany

ABSTRACT: In this paper investigations on the influence of material strength scattering on the seismic per-formance of steel structures are presented. Real material properties obtained directly from the monitoring ofEuropean steel producers has shown a significant overstrength for low steel grades, which is not always cap-tured by provisions in EN1998-1. The real material scattering is considered in non-linear time step analyses on

reference steel structures either braced by moment resisting or concentrically braced frames. The evaluation offailure criteria shows that local and global deformation behaviour is less depending on material scattering thanon random artificial accelerograms. However, connection and foundation forces correlate strongly with theyield stress of adjacent dissipative elements, but are nearly independent to different accelerograms.


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The data were obtained for HE-, IPE- and UPN-sections in steel grades S235M, S275M, S355M andS460M produced according to EN10025. The datawere grouped according to steel grade and flangethicknesses (3 to 16 mm and 16 to 40 mm). Thenumber of samples in each group was between 60and 8200.

The statistical evaluation of the yield strength led

to coefficient of variations (COV) mainly between0.05 and 0.07. These results are in good agreementwith data from literature (Faber et al 2001). The ratiofrom mean values to nominal value dependedstrongly on the steel grade: for S235 it was 1.40, forS275 between 1.20 and 1.32, for S355 between 1.12and 1.28, for S460 between 1.08 and 1.13 (Braconiet al 2009, unpub.). Therefore, for low steel gradesalready the ratio from mean value to nominal valuewas significantly higher than the overstrength factorof 1.25 proposed in EN1998-1. The statistical distri-bution of the yield strength samples can be described

sufficiently well by a 2-parameter lognormal distri-bution.

3 REFERENCE STEEL STRUCTURES

3.1 Procedure

The numerical investigations were carried out onfour typical multi-storey steel structures with differ-ent bracing systems and type of use. In the first stepthe buildings were designed for ordinary loads ac-

cording to EN1991 and E1993 using a 3D-modelconsidering dead load, imposed load, wind and snowloads. In the next step the initial design was ex-tended by adopting requirements for moderate seis-mic loads according to EN1998-1. This design wasdone by the lateral force method (ag = 0.1, Soil typeC, 3 % damping). Finally, non-linear time-stepanalyses were carried out on 2-D models for prob-abilistic investigations.

3.2 Office building braced by moment resisting

frames (MRF)The first structure was a 5 storey office building withdimensions of 21 x 36 m in plane and a height of17.5 m (equal storey heights). The concrete floors(without composite action) were designed to providesufficient diaphragm action. It was braced by mo-ment resisting frames in X-direction and by concen-tric bracings in Y-direction. The moment resistingframes consisted of HEB400-columns and IPE400-beams, both in steel grade S235. The dissipativeelements were the horizontal beams and column

bases, which were rigidly connected to the founda-tion. The 2-D non-linear analyses were focused onthe moment resisting frames in X-direction. Thefundamental period in this direction was T = 1.27 s(Fig. 1).

3.3 Office building braced by concentrically bracedframes (CBF)

The second structure was of the same geometricaldimensions and the same type of use as the first one,but it was braced by concentric bracings also in X-direction. The columns were made of HEB340 andthe beams IPE400 (hinged), both in steel gradeS235. The concentric bracings the dissipative ele-

ments - were made of CHS, d = 139.7 mm in S235with thickness decreasing from t = 12.5 mm in thefirst storey to t = 4 mm in the fifth storey. The crosssection dimensions were selected to fulfil the homo-geneous as well as the slenderness criteria accordingto EN1998-1. The 2-D non-linear analysis was car-ried out only for the concentrically braced frames inX-direction, where the fundamental period was T =1.12 s (Fig. 2).

3.4 Industrial building braced by moment resisting

frames (MRF)

The next structure was a four storey industrial build-ing with dimensions of 22.5 x 30 m in plane and aheight of 20 m. The unequally spaced storey heights(first to fourth storey 4 m, 4 m, 5 m and 7 m) andhigh masses also in upper storeys are typically forindustrial buildings. The resistance to lateral loadswas provided by moment resisting frames in X-direction and concentrically braced frames in Y-direction. The moment resisting frame consisted ofHEB700-columns and IPE500-beams with exception

of the first storey, where IPE550s were used to pre-vent soft storey failure. The steel grade of all ele-ments was S355. The column bases were designed ashinged connected to the foundation, as transferringthe full plastic moment of the large column sectionsto the foundation was judged as uneconomic. Thestructure was relatively soft in X-direction resultingin a rather high fundamental period (T = 1.81 s, seeFig. 3).

3.5 Industrial building braced by concentricallybraced frames (CBF)

The last structure was the same industrial building asin the previous section, but in this example theanalysis was focused on the concentrically bracedframes in Y-direction. The beams were made ofHEA700 in S355 and the columns were stillHEB700-sections (bending around the weak axis).The concentric bracings first to fourth storey -were made of CHS 244.5x8, 244.5x6 (both in S355),193.7x10 and 193.7x4 (both in S235). These sec-tions were used to fulfil the slenderness as well asuniformity criteria according to EN1998-1. The fun-damental period of the structure in Y-direction wasT = 1.01 s (Fig. 4).


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Figure 1. Reference structure 1 - Office building MRF: funda-mental period = 1.27 s.

Figure 2. Reference structure 2 - Office building CBF: funda-mental period = 1.12 s.

Figure 3. Reference structure 3 - Industrial building MRF: fun-damental period = 1.81 s.

Figure 4. Reference structure 4 - Industrial building CBF: fun-damental period = 1.01 s.

4 NONLINEAR DYNAMIC ANALYSIS

4.1 Model and FE-program

The influence of material scattering on the seismicperformance was investigated by non-linear dynamicanalyses. The analyses were carried out using theFE-program DYNACS developed at the Institute forSteel Structures at RWTH Aachen University (Kuck

1993, unpubl.). The structures were modelled in 2-Dby fibre beam elements, with increasing elementdensity in dissipative regions of the moment resist-ing frames (e.g. column feet, beam-column connec-tions). The non-linear material behaviour was con-sidered by a bi-linear model with kinematichardening described by yield stress, tensile stress andultimate elongation (Fig. 5). Braces were describedby special developed non-linear springs elements,representing the cyclic behaviour including plastifi-cation under tension, global buckling under com-pression and cyclic degradation. The analyses in-

cluded large deformations to consider the influenceof the P--effect.

Figure 5. Real stress strain curve of steel and simplified mate-rial law used in the non-linear analyses.

4.2 Ground motion histories

In the time step analyses artificial ground motionhistories with a p.g.a. of 0.1 g were used, which ful-filled the target spectrum for low seismicity (type 2)and soil type C according to EN 1998-1 (5 % damp-ing). The filter function was defined by a trapezoidalshape, where the time intervals for the initial ramp,strong motion duration and ending ramp are 5 s. Theground motion histories were generated with thesoftware SIMQUE (Gelfi 2006), whereas a baselinecorrection of the accelerograms was carried out af-terwards (see Figs. 6-7).


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-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

t [s]

a[m/s]

Akz 1

Akz 2

Akz 3

Akz 4

Akz 5

Akz 6

Akz 7

0 5 10 15 Figure 6. 7 artificial accelerograms fulfilling the demand spec-tra acc. to EN1998-1 for low seismicity and soil type C (5%damping).

0.0

0.2

0.4

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T [s]

Sa

[g]

target spectrum

accel. 1

accel. 2

accel. 3

accel. 4

accel. 5

accel. 6

accel. 7

Figure 7. Elastic response spectra of 7 artificial accelerogramsfor low seismicity and soil type C (5% damping vs. target spec-

tra acc. to EN1998-1.

4.3 Failure criteria

Seismic demand levels are usually defined in rela-tion to performance levels as Damage Limitation,Severe Damage and Near Collapse. The investiga-tions hereafter were carried out for the performancelevel Severe Damage acc. to EN1998-3, whichcorresponds to an earthquake hazard level with amedium return period of 475 years. A crucial pointin assessing structures by non-linear time-step analy-sis is the definition of limit states, as they are partlynot exactly defined in European seismic standards.The seismic performance of structures can be evalu-ated by general deformation criteria like roof driftand storey drift or local ductility criteria. Further-more, non seismic-specific verifications as shear ca-pacity, global buckling, etc. have to be carried out. Inthis study following limit states were defined as fail-ure criteria (Table 1):

- ultimate rotation acc. to EN1998-3 including the

limiting effect of axial forces (only MRF)- ultimate deformation in compression acc. to

EN1998-3 (only CBF)- ultimate deformation in tension acc. to EN1998-3

(only CBF)

- dynamic instability- shear capacity acc. to EN1993-1-1; for ratios

higher than 0.5 the moment capacity is reducedcorrespondingly

- global buckling acc. to EN1993-1-1- lateral torsional buckling of columns acc. to

EN1993-1-1

Furthermore, the verification of sections subjected tocombined axial and bending forces was considereddirectly in each time step by using fibre elementswith non-linear material behaviour. Global deforma-tion criteria as roof and storey drift according toFEMA356 were only used as indicative criteria. Ad-ditionally, maximum connection forces and founda-tion forces were recorded for further investigations.All verifications were carried out for each structuralelement with regard to the maximum value during atime history automatically by user-defined Matlabsubroutines (Matlab 2010). Only global buckling and

lateral torsional buckling were checked separately inthe relevant time step of each accelerogram.

Table 1. Seismic failure criteria.______________________________________________Criteria limit reference______________________________________________Roof drift 2.5% (ind.) FEMA356Storey drift 2.5% (ind.) FEMA356Ultimate rotation 6y (limit) EN1998-3Ultimate def. in compression 4c (limit) EN1998-3Ultimate def. in tension 7t (limit) EN1998-3_____________________________________________y = chord rotation at yielding, for 0.3 < N/Npl < 0.5:

y*= y (1 N/Npl) acc. to FEMA350c = axial deformation of the brace at buckling loadt = axial deformation of the brace at tensile yielding load.

4.4 Incremental Dynamic Analysis (IDA)

The seismic performance of all reference structuresas well as evaluation of their limit states were inves-tigated for nominal yield stress values. The originalaccelerograms were multiplied with a gradually in-creased factor until the dynamic instability of thestructure was reached. All structures resisted signifi-

cantly higher p.g.a. levels than considered in the ini-tial design by the lateral force method. The availableq-factors determined on the basis of the IDA are 6.7for the office building MRF, 5.4 for the office build-ing CBF, 7.0 for the industrial building MRF and 9.2for the industrial building CBF; all q-factors are re-lated to the acceleration corresponding to the firstplastic hinge in the structure. The high resistance canbe explained, as many seismic design requirementslead to an overstrength of the structure compared tothe resistance required for the applied seismic designload. Such effects are more considerable for struc-

tures designed for moderate seismic loads, as theseseismic design requirements have to ensure a suffi-cient performance of structures not only for lowseismicity but also for high seismicity with longerstrong motion periods.


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0

0.5

1

1.5

2

0 5 10 15

multiplier factor [-]

ultimaterotationratio[-]

(1)

(6)

(3)

(7)(5)

(2)

(4)

(...) accelerogram

Figure 8. Reference structure 1 - Office building MRF: maxi-mum ultimate rotation ratio in the IDAs .

roof drift storey drift beam rot. column rot. shear force0%

25%

50%

75%

100%

125%

150%

175%

200%

ra

tio[-]

Figure 9. Reference structure 1 - Office building MRF: capacityratio of failure criteria at load factor 10 (mean, maximum andminimum).

0

0.5

1

1.5

2

0 5 10 15


ultimatedef.rati

o[-]

(1)

(6)

(3)

(7)

(5)

(2)

(4)

(...) accelerogram

Figure 10. Reference structure 2 - Office building CBF: maxi-mum ultimate tension deformation ratio in the IDAs.

roof drift storey drift tension def. shear force0%

25%

50%

75%

100%

125%

150%

175%

200%

ratio[-]

compr. def.

~650 %

Figure 11. Reference structure 2 - Office building CBF: capac-ity ratio of failure criteria at load factor 7 (mean, maximum andminimum).

0

0.5

1

1.5

2

0 5 10 15


ultimaterotationratio[-]

(1)

(6)

(3)

(7)

(5)

(2)(4)

(...) accelerogram

Figure 12. Reference structure 3 - Industrial building MRF:maximum ultimate rotation ratio in the IDAs.

roof drift storey drift beam rot. column rot. shear force0%

25%

50%

75%

100%

125%

150%

175%

200%

ra

tio[-]

Figure 13. Reference structure 3 - Industrial building MRF: ca-pacity ratio of failure criteria at load factor 8 (mean, maximumand minimum).

0

0.5

1

1.5

2

0 5 10 15


ultimatedef.rati

o[-]

(1)

(6)

(3)(7)

(5)

(2)

(4)(...) accelerogram

Figure 14. Reference structure 4 - Industrial building CBF:maximum ultimate tension deformation ratio in the IDAs.

roof drift storey drift def. tension shear force0%

25%

50%

75%

100%

125%

150%

175%

200%

ratio[-]

compr. def.

~700 %

Figure 15. Reference structure 4 - Industrial building CBF: ca-pacity ratio of failure criteria at load factor 8 (mean, maximumand minimum).


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In structures braced by moment resisting framesthe ultimate rotation ratio was the controlling failurecriterion (Figs. 8-9 and 12-13). In the office as wellas in the industrial building the columns were thecritical elements, which was also related to the re-duction of ultimate rotation capacity due to axialloads. The scattering of the ultimate rotation ratiobetween accelerograms was considerably high (80

140 % and 80 130 %). The other failure criteriawere not dominant excepting the indicative criterionstorey drift.

In the buildings braced by concentric bracings theultimate deformation of the bracings in compressionwas the governing failure criterion (Figs. 11 and 15).However, the ultimate deformation ratio of braces incompression according to EN1998-3 is questionable.The deformation of braces in tension and compres-sion is approximately identical, but the ultimate de-formation capacity in compression is always lowerthan in tension. The first one is defined as 4 times

the axial deformation at buckling load and the sec-ond one is defined as 7 times the axial deformationat tensile yielding load. Strict application of this fail-ure criterion may prevent plastifications of slenderbraces in tension, which are still in the bandwidth ofthe slenderness criterion of EN1998-1. Therefore, inthe following the investigations were focused on thedeformation in tension criterion. The capacity ratiosof the other failure criteria were rather low. The scat-tering of the results between different accelerogramswas lower than for the MRF, especially for the in-

dustrial building.

5 RESULTS OF CRUDE MONTE-CARLO-SIMLUATION

5.1 Procedure

Following the IDA, the influence of material scatter-ing on the seismic performance was investigated.The analyses were carried out with accelerogramsmultiplied by the load factor, at which the first fail-ure criterion was reached in the IDA applying nomi-

nal material properties (see previous section). Thedeformation limits (ultimate rotation and ultimatedeformation in tension and compression) were as-sessed with the nominal yield stress and were keptunchanged.

The samples of the material properties (yieldstress, tensile stress and ultimate elongation) weregenerated on the basis of the monitored data in sec-tion 2.2. The material properties between differentstructural elements were assumed as uncorrelatedexcepting members representing columns as they

were assumed as continues structural elements fromstorey 1 to 3. For each structure 5000 samples withdifferent material properties were generated and in-vestigated in non-linear time-step analyses with 7different accelerograms.

5.2 Buildings braced by moment resisting frames

In the analyses of MRF-structures the scattering ofdeformation parameters as roof drift and element ro-tation for different material samples was moderate(COV = 0.03 to 0.07). The scattering between differ-ent accelerograms were obviously predominant(COV = 0.16 and 0.17, Fig. 17 and 21), which hasalready been stated by other authors (e.g. Kook

1994). As ultimate rotation was the dominant failurecriterion (see previous section), the failure probabil-ity of the structures seems to be nearly independentfrom material scattering. Furthermore, local rotationsof beams and columns did not correlate with the roofdrift (Fig. 16), which is an indication for a signifi-cant influence of higher modes.

In contrast to this observation connection andfoundation forces were highly correlated with theyield stress of the adjacent dissipative element (Fig.18). The differences between particular accelero-

grams were however very small.

5.3 Buildings braced by concentrically bracedframes

Similar results were also obtained for concentricallybraced frames. The scattering of global and local de-formation parameters for different material sampleswithin one accelerogram was significant lower thanthe scattering between different accelerograms. Fur-thermore, higher mode effects were clearly visible,as deformation of the braces correlates less with the

roof drift. Again the deformation of the braces wasless dependent to the material scattering; therefore,also the failure probability of CBF structures seemsto be less dependent to material scattering.

The evaluation of connection and foundationforces adjacent to bracings (dissipative elements)show similar tendencies as for the MRF: correlationto the actual yield strength and low scattering be-tween different accelerograms. However, especiallyfor the office building the scattering was higher.

0.010

0.011

0.012

0.013

0.014

0.015

0.22 0.24 0.26 0.28

roof drift [m]

rotation[rad]

Figure 16. Reference structure 1 - Office building MRF: col-umn base rotation over roof drift for accelerogram 1.


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Figure 17. Structure 1 - Office building MRF: Box plot of col-umn foot rotation 1

ststorey (multiplier 10).

300

350

400

450

500

250 300 350 400

yield stress [MPa]

moment[kNm]

Figure 18. Structure 1 - Office building MRF: maximum mo-ment at joint vs. yield stress (beams 2nd storey, multiplier 10).

Figure 19. Structure 2 - Office building CBF: Box plot of ten-sion deformation braces 3rd storey (multiplier 7).

1300

1500

1700

1900

2100

2300

250 300 350 400

yield stress [MPa]

axialforce[kN]

Figure 20. Reference structure 2 - Office building CBF: maxi-mum tension force at joint vs. yield stress, (braces 3 rd storey,multiplier 7).

Figure 21. Structure 3 - Industrial building MRF: Box plot ofcolumn head rotation 1st storey (multiplier 8).

900

1000

1100

1200

1300

1400

1500

350 400 450 500 550

yield stress [MPa]

moment[kNm]

Figure 22. Structure 3 - Industrial building MRF: maximummoment at joint vs. yield stress (beams 2

ndstorey, multip. 8).

Figure 23. Structure 4 - Industrial building CBF: Box plot oftension deformation braces 4th storey (multiplier 8).

2300

2500

2700

2900

3100

3300

3500

350 400 450 500 550yield stress [MPa]

axialforce[kN]

Figure 24. Structure 4 - Industrial building CBF: maximum ten-sion force at joint vs. yield stress (braces 3

rdstorey, multiplier

8)


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6 CONCLUSIONS AND FURTHERINVESTIGATIONS

6.1 Results

In this study the influence of material scattering onthe seismic performance of steel structures was in-vestigated via non-linear time step analyses. Basedon the results of four reference structures - office and

industrial building braced by MRF and CBF -following conclusions are made:

- The global and local deformation behaviour of theinvestigated steel structures is less dependent onmaterial scattering.

- The scattering of global and local deformation be-haviour of the structure due to different accelero-grams is predominant.

- Connection and foundation forces correlatestrongly with the yield stress of adjacent dissipa-tive elements, but they are nearly independent to

the scattering of the seismic action.

6.2 Outlook

In the performed investigations the ultimate rotationcapacity and the ultimate deformation capacity areassumed as constant. However, there are experimen-tal and analytical results, which show a negative cor-relation between deformation capacity and materialstrength (e.g. Feldmann 1994). Even if the materialproperties correlate less with the deformation behav-

iour, the material strength may influence the defor-mation capacity. If the deformation capacity is nega-tive correlated with the material strength, realmaterial properties increase the failure probability inrelation to nominal material strength.

The statistical evaluation of real material data insection 2.2 has clearly shown that the mean values ofsteel strength are significant higher than the nominalvalues. Under consideration of the strong correlationbetween the material strength of dissipative elementsand forces in adjacent connections and foundations,non-ductile joints have to be designed with an enor-

mous overstrength to fulfil capacity design rules.However, in the current model the connection forcesare only evaluated as demand measure and the inher-ent existing overstrength of the joint itself is not con-sidered. To determine a realistic and economic over-strength factor for connections also the realscattering of joint capacities has to be determinedand considered.

On the basis of 5000 non-linear time step analy-ses for each structure with different material samplessome general conclusions can be made on the failure

probability. However, to determine the influence ofdifferent parameters on the absolute failure probabil-ity, the number of samples shall be increased. Asnon-linear time step analysis even with efficientsoftware and strong computers are still time consum-

ing (5 to 10 min for each sample), crude MonteCarlo simulations are still not applicable. Hence, fur-ther investigations will be carried out with more ef-ficient probabilistic methods. Investigations in(Whaarts 2002) have shown that combining direc-tional sampling with adaptive response surfacemethods (Directional Adaptive Response surfaceSampling, DARS) provide a robust and efficient

approach. Therefore, DARS will be implemented inthe open source reliability platform FERUM 4.0 tocarry out further investigations (Bourinet 2009).

ACKNOWLEDGEMENT

The research leading to these results has receivedfunding from the Research Program of the ResearchFund for Coal and Steel RFSR-CT-2007-00039.

REFERENCE

Bourinet, J.-M. 2009. FERUM 4.0 Users GuideBraconi, A. et al 2009. Optimizing the seismic performance of

steel and steelconcrete structures by standardizing materialquality control, Midterm report (unpublished). Grant agreenumber RFSR-CT-2007-00039

Faber, H.M. et al 2001. JCSS Probabilistic Model Code. JointCommittee on Structural Safety. www.jcss.ethz.ch.

Feldmann, M. 1994. Zur Rotationskapazitt von I-Profilen sta-tisch und dynamisch belasteter Trger. PhD-thesis. Series atInstitute for Steel Structures at RWTH Aachen UniversityVolume 30

Feldmann, M., Schfer, D. and Eichler, B. 2009. Vorhersageduktilen Festigkeitsversagens von Stahlbauteilen mit Hilfeschdigungsmechanischer Methoden. Stahlbau, 78: 784794. doi: 10.1002/stab.200910094

Gelfi, P. 2006. User Manual - SIMQKE_GR Version 1.2.Kook, S. K. 1994. Beitrag zur Definition der Bauwerksregulari-

tt und zur Bestimmung der Verhaltensbeiwerte fr dieErdbebenbemessung von Stahlbauten. PhD-thesis. Series atInstitute for Steel Structures at RWTH Aachen UniversityVolume 27.

Kuck, J., Hoffmeister, B. 1993. User manual for DYNACS - AProgram for DYnamic Nonlinear Analysis of Compositeand Steel structures (unpublished).

Matlab 2010. User Manual Matlab 2010a. Version7.10.0.499Whaarts, P. H. 2000. Structural reliability using Finite Element

Analysis An appraisal of DARS: Directional AdaptiveResponse surface Sampling. PhD-thesis. Delft UniversityPress, Delft.

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