Optimal design of tapered steel portal frame structures

Optimal design of tapered steel

portal frame structures exposed to

extreme effects

PhD dissertation

Tamás Balogh

Supervisor László Gergely Vigh, PhD

Associate Professor

Budapest University of Technology and Economics Faculty of Civil Engineering, Department of Structural Engineering

Budapest 2017

Tamás Balogh – PhD Dissertation

Declaration of Authenticity

I, the undersigned, Tamás Balogh, declare that this dissertation is my original work, gathered

and utilized especially to fulfil the purposes and objectives of this study. The work and results of

other researchers, which are referred squarely in order to separate from the original work, are

specifically acknowledged.

Budapest, 6th June 2017

Tamás Balogh


Acknowledgements

First of all, I would like to express my gratitude to my supervisor Dr. László Gergely Vigh,

who has been helping me as a mentor since my MSc studies and who encouraged me to start my

research. I would like to thank in particular his advices and help during the consultations, this

research work could not have been done without his guidance.

I also would like to thank to the members and colleagues of Department of Structural

Engineering, in particular to Dr. József Simon, Eduardo Charters, Bettina Badari, Dr. Árpád

Rózsás, Lili Laczák, Péter Hegyi, Dr. László Horváth, Dr. László Dunai, Kitti Gidófalvy, Dr. Ádám

Zsarnóczay, Dr. Viktor Budaházy, Dr. Attila László Joó, Dr. Mansour Kachichian. I am very

grateful for their friendly support, valuable advices and comments that they gave during personal

and seminar consultations.

I highly appreciate the help and guidance of Mario D’Aniello and Professor Raffaele Landolfo

during my short study program in Naples, Italy. They treated me not only as a student, but as a

colleague and as a friend.

I would like to express my deepest gratitude to my beloved fiancée and my family. Their endless

encouragement and kindness helped me to get through the difficulties to continue my work.

Without their support I could not have finished my thesis.

The research work is completed under the support of the following projects and programs:

• Development of quality-oriented and harmonized R+D+I strategy and functional model at

BME project by the grant TÁMOP-4.2.1/B- 09/1/KMR-2010-0002,

• Talent care and cultivation in the scientific workshops of BME project by the grant TÁMOP-

4.2.2.B-10/1--2010-0009,

• Campus Hungary program supported by the grant TÁMOP-4.2.4.B/1-11/1-2012-0001,

• HighPerFrame R&D project GOP-1.1.1-11-2012-0568, supported by the Új Széchenyi Terv,

• János Bolyai Research Scholarship of the Hungarian Academy of Sciences.


Abstract

In the last decade seismic and fire design became everyday practice in Hungary since the

harmonized European Norms require the designers verifying the structures in extreme design

situations more precisely. Economical configurations may hard to be found because high degree of

nonlinearity characterizes these extreme effects and the structural response to them. Optimal design

of tapered steel portal frame structures exposed to extreme effects has not been extensively studied

earlier. However, steel is very heat conductive material and this kind of structure is very sensitive

to stability failure modes, thus fire design situation may easily become the leading design situation.

Furthermore, my calculations showed that optimal design to seismic effects may be also important

(even in a seismically moderate area) because seismic action can be dominant comparing to wind

action in case of high seismicity or high vertical loads. The aims of this thesis are: a) developing

appropriate and effective tools to obtain optimal structural configurations considering extreme

effects based on structural reliability; b) analysing optimal safety levels that lead economical

solutions; c) obtaining optimal solutions for a number of design situations within the framework of

a parametric study; d) deriving conceptual design concepts related to the design of tapered portal

frames to extreme effects.

In this study, a new and effective reliability assessment framework is developed and applied to

structural reliability calculation of portal frames under seismic and fire exposure. Reliability

analysis is based on first order reliability method; state-of-the-art analysis and evaluation tools are

incorporated. This framework is used for estimation of possible target reliability indices for seismic

and fire effects. The calculated and recommended values in Eurocode differ, it seems that lower

target values would be more appropriate in case of extreme effects.

A reference structure, namely a tapered storage hall, is optimized in more than 60 cases

considering different initial design conditions. The optimization is performed using a genetic

algorithm based heuristic structural optimization algorithm. The developed reliability assessment

framework is invoked within the objective function evaluation during the structural optimization.

The objective functions express the life cycle cost of the structures. After the analysis of the results,

valuable conclusions can be drawn related to the optimal safety level and conceptual design of steel

tapered portal frame structures.


Összefoglalás

Az elmúlt évtizedben szerkezetek megbízhatóságának tűz- és földrengési hatásokkal szembeni

igazolása mindennapos gyakorlattá vált Magyarországon is a harmonizált európai szabványok

(Eurocode szabványok) hazai bevezetésével. Ezen rendkívüli hatásokkal és a méretezéssel

foglalkozó szabványfejezetek egyrészt a korábbinál több és modernebb szabványosított méretezési

módszert/előírást tartalmaznak, másrészt azonban a korábbinál szigorúbb feltételeknek kell

megfeleltetni szerkezeteinket.

Gazdaságos szerkezetek tervezése sok esetben nehézkes és iteratív folyamattá válik a rendkívüli

hatásokban, a szerkezet viselkedésében és a méretezési módszerekben található nagyfokú

nemlinearitás miatt. Változó keresztmetszetű acél keretszerkezetek extrém hatásokra való

optimális tervezése a viszonylag kevéssé kutatott területek közé tartozik, annak ellenére, hogy

például tűz esetén az acél anyag merevségi és szilárdsági tulajdonságainak drasztikus változása a

szerkezetet rendkívül érzékennyé teszi különböző stabilitásvesztési tönkremenetelekre és a

tűzhatást is figyelembe vevő tervezési szituáció mértékadóvá válhat. Továbbá számításaim szerint

a földrengési hatások sem hanyagolhatók el, ugyanis könnyen előfordulhat olyan tervezési helyzet,

még moderált szeizmicitású övezetben is, hogy a szeizmikus hatásból származó terhek

meghaladják a szélhatásból számított terhek intenzitását és a földrengési hatásokat is magába

foglaló tervezési helyzet válik mértékadóvá.

Az kutatásom céljai a következők: a) hatékony eszközök kifejlesztése melyekkel változó

keresztmetszetű acél keretszerkezetek esetében optimális szerkezetkialakítások meghatározhatók

extrém terhekre a szerkezeti megbízhatóság figyelembe vételével; b) optimális biztonsági szint

vizsgálata mellyel gazdaságos szerkezetkialakítások érhetők el; c) paraméteres vizsgálat keretén

belül optimális szerkezetkialakítások meghatározása számos lehetséges tervezési helyzetben; d)

koncepcionális tervezési javaslatok kidolgozása a paraméteres vizsgálatok eredményei alapján.

Ebben a dolgozatban bemutatok egy új és hatékony, extrém hatások figyelembevételével

szerkezeti megbízhatóság számítására alkalmas keretrendszert. Ezen keretrendszer segítségével

határozom meg a szerkezeti megbízhatóságokat szeizmikus- és tűzterhekre. A keretrendszer

elsőrendű megbízhatósági analízist alkalmaz, komplex nemlineáris analízis és kiértékelő

módszerek a határállapot függvénybe kerültek beépítésre. A számított és az Eurocode 0 által

megkövetelt megbízhatósági szintek eltérnek, az eredmények alapján az előírtnál alacsonyabb


biztonsági szint jövőbeni alkalmazása gazdaságosabb tervezést eredményezhet extrém terhek

esetében.

Egy raktár funkciójú, változó keresztmetszetű példaszerkezet szerkezetoptimálását összesen 60

különböző tervezési helyzetben végeztem el. A felírt szerkezetoptimálási feladatot genetikus

algoritmus segítségével oldottam meg. A kifejlesztett megbízhatósági analízis keretrendszer

beépítésre került az optimáló algoritmusba, a bemutatott célfüggvények a szerkezet életciklus

költségeit fejezik ki, amely magába foglalja a létesítési költségeket és tönkremenetelkor keletkező

károk kockázatát. Az eredmények alapján értékes következtetéseket tudtam levonni változó

keresztmetszetű acél keretszerkezetek extrém terhekre való optimális tervezésével kapcsolatban.


6

Table of contents

Table of contents ............................................................................................................................. 6

List of symbols ................................................................................................................................ 8

Abbreviations .................................................................................................................................. 8

1. Introduction ............................................................................................................................... 9

1.1. Literature review ............................................................................................................... 9

1.2. Aims and outline of this research ................................................................................... 15

1.3. Details of the investigated structure ............................................................................... 17

1.4. Preliminary research and results ..................................................................................... 19

2. Optimization algorithm development .................................................................................... 21

2.1. Basic description of the optimality problem .................................................................. 21

2.2. Objective function in case of fire optimization ............................................................. 23

2.3. Objective function in case of seismic optimization....................................................... 25

2.4. Genetic algorithm based optimization framework ........................................................ 26

2.5. Global optimum ............................................................................................................... 28

3. Reliability analysis.................................................................................................................. 29

4. Reliability evaluation of frames under fire exposure ........................................................... 33

4.1. Overview from the methodology.................................................................................... 33

4.2. Event tree and Bayesian probabilistic network ............................................................. 36

4.3. Limit state function ......................................................................................................... 38

4.4. System reliability ............................................................................................................. 39

4.5. Fire effects ....................................................................................................................... 41

4.6. Random variables ............................................................................................................ 42

4.7. Structural analysis ........................................................................................................... 45

4.8. Verification of the elements ............................................................................................ 48

5. Reliability evaluation of frames under seismic excitation ................................................... 50

5.1. Overview from the methodology and limit state function ............................................ 50

5.2. Seismic effects and hazard curves .................................................................................. 53

5.3. Random variables ............................................................................................................ 55

5.4. Structural model and analysis ......................................................................................... 57

5.5. Verification of structural model ..................................................................................... 58

5.6. Calculation of target displacements ............................................................................... 59

5.7. Verification of the elements ............................................................................................ 60

6. Target reliability in case of extreme effects .......................................................................... 62

6.1. Minor, moderate and large consequences of failure...................................................... 62

6.2. Target reliability estimation in case of fire design situation ......................................... 64

6.3. Target reliability estimation in case of seismic design situation .................................. 68

7. Reliability based structural optimization results ................................................................... 75

7.1. Optimized variables ......................................................................................................... 75

7.2. Convergence and performance of the optimization algorithm ..................................... 76

7.2.1. Settings of the algorithm ..................................................................................... 76

7.2.2. Shape of the objective function .......................................................................... 78

7.3. Optimal solutions in fire design situation ...................................................................... 80


7

7.3.1. Parametric study .................................................................................................. 80

7.3.2. Parametric study results ...................................................................................... 83

7.4. Optimal solutions in seismic design situation ............................................................... 89

7.4.1. Parametric study .................................................................................................. 89

7.4.2. Parametric study results ...................................................................................... 90

8. Summary and conclusions ...................................................................................................... 95

9. Future research ........................................................................................................................ 96

10. New scientific results.............................................................................................................. 98

10.1. Thesis I. ............................................................................................................................ 98

10.2. Thesis II............................................................................................................................ 98

10.3. Thesis III. ......................................................................................................................... 99

10.4. Thesis IV. ....................................................................................................................... 100

References.................................................................................................................................... 102

Publications of the author related to the theses ..................................................................... 102

Other references ...................................................................................................................... 103

Appendix A – Parametric study tables and results.................................................................... 109

Appendix B – EC design of steel frames for extreme effects .................................................. 114

Appendix C – Evaluation of limit state function in case of fire design ................................... 120

Appendix D – Evaluation of limit state function in case of seismic design ............................ 124

Appendix E – Fire optimization framework .............................................................................. 128

Appendix F – Seismic optimization framework ....................................................................... 129

Appendix G – New scientific results in Hungarian .................................................................. 130

G 1. I. Tézis ............................................................................................................................ 130

G 2. II. Tézis .......................................................................................................................... 130

G 3. III. Tézis ......................................................................................................................... 131

G 4. IV. Tézis ......................................................................................................................... 132


8

List of symbols

Greek Roman

αu load amplifier of the design loads to reach the characteristic resistance ag ground acceleration

β reliability index b flange width of an I section γov material overstrength factor db diameter of the tension-only braces γI importance factor f(X) joint distribution function Δ deformation g gravity acceleration, 9.81 m/s2 μ mean g(x) inequality design constraint η utilization (demand-to-capacity ratio) h height of an I section ρ correlation coefficient h(x) equality design constraint σ standard deviation p(x) penalty function θ deformation tf flange thickness of an I section

Φ cumulative distribution function of standard normal distribution

tw web thickness of an I section

tp thickness of fire protection

q behaviour factor

x vector of design variables

C cost

C(x) initial cost function

CLC(x) life cycle cost function

D damage

G(X) limit state function

P probability

Pf failure probability

R(x) risk function

Sa spectral acceleration

Sd spectral displacement

W(x) structural weight function

X vector of random variables

Abbreviations

AISC American Institute of Steel Construction LTB lateral torsional buckling CoV coefficient of variation MCS Monte Carlo simulation CP collapse prevention MPP maximum probability point DL damage limitation MRF moment resisting frame EFEHR European Facility for Earthquake Hazard and

Risk MRSA modal response spectrum analysis

EN European Norm PA pushover analysis FE finite element PEER Pacific Earthquake Engineering Research

Center FB flexural buckling PGA peak ground acceleration FEMA Federal Emergency Management Agency PSHA probabilistic seismic hazard analysis FORM first order reliability method SLS serviceability limit state GA genetic algorithm SORM second order reliability method IDA incremental dynamical analysis SRSS square root of the sum- of the squares IDR interstorey drift THA time history analysis JCSS Joint Committee on Structural Safety ULS ultimate limit state LFM lateral force method


9

1. Introduction

In Hungary, the reliability of structures has to be verified against stricter requirements since the

introduction of European Standards (ENs). The verification against extreme effects (e.g. fire and

seismic effects) has started to play a significant role in the structural design practice. In lot of cases,

the complexity and nonlinearity of these extreme effects makes the design procedure to a time-

consuming iterative process. For this reason, the conceptual design of structures has become more

important. This is especially true when the aim is to find an economical or optimal solution. The

problem is even more difficult and complex if the structural behaviour is highly nonlinear, e.g.

when the structure is sensitive for stability failure modes, if the dominant failure mode may be an

interaction between different stability failure modes and if the structural configuration is non-

conventional. These statements are particularly true in case of tapered portal frames. Structural

optimization tool may be used effectively in order to find optimal and economical solutions in these

cases.

1.1. Literature review

Nowadays, many researchers deal with structural optimization since it is still a developing area.

Due to the large number of publications in this field, presenting and listing all of the connected

papers and books is definitely hopeless. In this thesis, only the relevant research works and studies

are referred. In Hungary, the researchers of Structural Engineering Department of Budapest

University of Technology and Economics (BME) presented solutions for optimal design of cold-

formed steel elements in [1], for new generation steel wind-turbine tower in [2] and for steel

stiffened plates related to optimal stiffener geometry in [3]. These papers directly focused on

practical problems, similarly to the work of József Farkas and Károly Jármai from University of

Miskolc, who gave solutions for various problems, e.g. in [4] and in [5], from the field of optimal

design of steel structures. In [6], they presented a detailed cost calculation method for steel

structures and optimized a multi-storey steel frame based on structural costs, considering seismic

effects. They also gave solution for cost optimization of a welded box beam and a stiffened plate

in [7]. From the Department of Structural Mechanics of BME, János Lógó [8] gave a

comprehensive overview about relevant literature from the field of structural optimization and

mathematical programming, including early researches and applications. For this reason, for


10

inquiring readers the author suggests to read and study the work of János Lógó as a good and

comprehensive introduction to this field. In [8], he presented a solution for optimization of a

haunched steel frame, which had been experimentally tested in the laboratory of Structural

Engineering Department. In recent publications, with his colleagues, he dealt with optimal design

considering uncertain loading positions, e.g. in [9] and in [10], and optimal design of curved folded

plates, e.g. in [11] and in [12]. György Rozványi, also from the Department of Structural

Mechanics, earned wide international reputation with his oeuvre in the field of topology

optimization [13]. Anikó Csébfalvi, from Department of Structural Engineering, University of

Pécs, deals with optimization of frame and truss structures. In [14], she presented genetic algorithm

based heuristic weight optimization of frames with semi-rigid joints. She proposed and applied on

three dimensional truss structures the so-called ANGEL algorithm, a metaheuristic optimization

algorithm, that combines ant colony optimization, genetic algorithm and gradient-based local

search, in [15] and in [16].

This literature review focuses on summarizing the most relevant literatures related to the

optimal design of steel multi-storey and portal frames considering various loading conditions

(considering “conventional”, seismic and fire effects). Because of the discrete and highly nonlinear

nature of the optimality problem, the researchers mostly apply evolutionary or other heuristic

strategies to find the optimum of the objective function that measures the fulfilment of design and

performance criteria. A number of studies, e.g. [17], [18], [20], [21], [22], [19], [23], [24], [25] and

[26], exist related to the optimization of regular or tapered portal frames considering

“conventional” loading conditions (dead load, snow load, etc.) in order to achieve a more economic

design usually by minimizing the weight or the cost of the structure. Nowadays cost optimization

is becoming the most widespread; however, in lot of cases the calculation of the structural costs

may be difficult and controversial.

As regards to the optimal seismic design of frames, Kaliszky and Lógó in [27] presented a

method for elasto-plastic optimal weight design of frame structures subjected to seismic excitation

according to the design rules of Eurocode 8 Part 1 (EC8-1) [28] standard. Due to the high

nonlinearity of the design problem they proposed an iterative design procedure where a

mathematical programming problem and a pushover analysis had to be carried out in each iteration

step. Salajegheh, Gholizadheh and Khatibinia, in [29], analysed the optimal design of a multi-

storey steel frame and a spatial truss structure considering seismic effects using time history


11

analysis. Oskouei, Fard and Aksogan, in [30], presented the optimal design of multi-storey frames

exposed to seismic effects using both linear and nonlinear static analyses. The aim of optimization

was to find a structure with minimum structural weight considering constraints for allowable

stresses, deformations and the position of plastic hinges. They could achieve lighter solutions by

using semi-rigid connections and nonlinear static analysis method.

There is lack of studies dealing with optimal fire design of steel structures in the literature, only

few studies are available in this topic. Jármai in [31] presented optimal solutions for a simple one-

storey frame constructed using square hollow sections. Particle swarm optimization technique [32]

was applied in order to minimize the objective function which expressed the initial cost of the

structure. Internal forces in the elements were calculated using first order theory and the gas

temperature was calculated according to ISO standard fire curve [33]. The author concluded that

by using passive fire protection significant cost savings can be achieved.

Conventional prescriptive design criteria may be not able to describe well the structural

performance for highly nonlinear loading conditions, such as seismic and fire effects. For this

reason, more and more researcher and designer apply the so-called Performance Based Design

(PBD) concept [34]. The performance of the structure is often characterized by the reliability or

failure probability through the seismic or fire risk. The main advantages of PBD comparing to

prescriptive design are the following: 1) PBD gives the opportunity to take into account the

uncertainties and the consequences in the design; 2) PBD makes the comparison easier among

structures with similar initial costs and with similar demand-to-capacity (D/C) ratios.

Nowadays, performance based optimization of structures is a rapidly evolving, state-of-the-art

topic; the risk or the structural reliability provides a good measure for structural optimization.

Kaveh et al., in [35], published a paper on performance based seismic design of steel frames using

ant colony optimization algorithm [32]. The performance of the structures was evaluated with

nonlinear static structural analysis; they pointed out that the presented method was able to obtain

lighter frames having less damage. Saadat, Camp and Pezeshk, in [36], also presented performance

based seismic design optimization of steel frames, but considering direct economic and social

losses. The optimization objective, optimized with genetic algorithm (GA) [37] [38], was selected

as the lifetime cost considering initial costs and possible losses due to seismic effects in the future.

In [39], Rojas, Foley and Pezeshk presented a GA based optimization method in order to minimize

both the structural weight and the expected annual losses considering constraints related to


12

performance objectives and to element resistance. Fragility functions of HAZUS [40] were used

for performance evaluation according to PEER framework [41].

Liu, Wen and Burns presented a life cycle cost oriented seismic design optimization in [42].

The authors minimized life cycle cost of steel MRF (moment resisting frames) structures using a

detailed failure cost function considering AISC steel design specifications and seismic design

provisions. The applied search engine was a GA based procedure. In [43], the authors presented a

seismic design optimization with reliability constraints. The response of MRF structures was

assessed with the help of pushover analysis (PA) within a FEMA-356 [44] conforming

performance evaluation procedure. The objective function which expressed the initial cost had been

minimized with GA. They solved the reliability problem with both Monte Carlo simulation (MCS)

[45] [46] and first order reliability method (FORM) [47].

No study may be found in the literature related to performance based optimization of structures

exposed to fire effects, although such optimization results would provide useful information related

to optimal design considering both initial costs and possible failure risks. This is especially true in

case of steel structures, because their structural response and load bearing capacity is highly

sensitive to elevated temperatures. Results would also help to provide more information about

possible target reliability indices. The target safety level is an important issue from the point of

view of economic design because it ensures a balance between the initial costs and failure

consequences (Fig. 2-1a).

Related to structural reliability, a Hungarian researcher and engineer Gábor Kazinczy may have

been the first who proposed the application of probability theory (accounting the variability in

manufacturing and the quality of construction materials) for assessing the safety of structures [48]

[49]. In Hungary, among others Endre Mistéth did a remarkable research work related to structural

reliability theory, he summarized his oeuvre in a book [50] that is well-known among scientists

and engineers working on this field.

Structural failure reliability is the probability that the structure will retain its safety over the

design period (service life) under specified conditions [51]. Reliability index is frequently used in

the literature as the measure of structural reliability. In case of normally distributed and not

correlated joint density functions, structural reliability (R), reliability index (β) and the probability

of failure (Pf) are in the following relation:

( ) ( )ββ ΦΦ =−−=−= 11 fPR , (1)


13

where Φ(•) is the standard normal cumulative distribution function. Probability of failure may

be calculated with integration (Eq. (15)) the joint density function over the domain (failure region)

where the value of limit state function (that separates feasible and non-feasible alternatives) is

violated [51], Eurocode 0 (EC0) [52]. The density function shows the relative frequency of

realizations calculated using random variables (e.g. strength, geometry, load intensity).

Suggestions related to distribution, mean value and standard deviation of random variables can be

found in the literature, e.g. in [53] or in studies connected to reliability assessment of structures

[54].

To the best of the author’s knowledge, there is a lack of studies in the literature on

comprehensive reliability calculation of complex structural systems exposed to fire; the available

studies mainly deal with the reliability calculation of simple, separated elements. For example, in

an earlier study Holickỳ et al. [55] analysed the reliability of unprotected simple supported steel

beams with SORM, which had been verified according to Eurocode 3 Part 1-2 (EC3-1-2) [56].

Jeffers et al. in [57] analysed protected simple supported steel beams as well using both ISO

standard fire curve (equivalent fire effect that is commonly used for fire design given in

temperature vs. time format) [33] and Eurocode 1 Part 1-2 (EC1-1-2) [58] conforming parametric

fire curves (Eurocode conforming curves that were obtained on the basis of properties of the

compartment and the combustible material) to model the temperature in the compartment. The

reliability of the beam was assessed using MCS with Latin Hypercube Sampling (LHS). They

pointed out that probability calculation is needed to ensure consistent reliability level in the fire

resistant design and further discussion is necessary in order to decide the acceptable level of risk

in structural fire engineering. Guo and Jeffers [59] presented a detailed discussion on the reliability

calculation theory extended for calculation of the failure probability of a protected steel column

under fire exposure. They calculated the reliability of a simple pinned column with FORM, SORM

and MCS. Based on the resulted probabilities they showed that there could be significant difference

between MCS and FORM, where FORM resulted more conservative failure probabilities. Li et al.

[60] investigated the reliability of steel column elements protected by intumescent coating. They

were able to assess the aging effect of the intumescent coating on the structural reliability.

Reliability analysis of complex structures exposed to fire can be found in [61] and in [62]. In

the first study Boko et al. analysed an unprotected steel roof structure with SORM and FORM.

They pointed out that using rules and recommendations from EC1-1-2 and EC3-1-2 appropriate


14

safety level can be ensured. The analysed truss structure was not taken into account as structural

system, the presented reliability indices are related single elements. In the second study, Boko et

al. presented the analysis of a steel portal warehouse without fire protection. The reliability related

to the failure of the beam was obtained with different parametric fire curves (EC1-1-2) calculated

using different values for fuel load, fire area and opening factor. They pointed out that the usage

of ISO standard fire curve leads for conservative structural reliability value.

It can be concluded that structural reliability calculation under fire exposure is still a developing

area and (because of some shortcomings in the available studies: simplification of fire curve,

simplification related the consideration of failure within the reliability analysis, simplified analysis

and analysis of an isolated element) it does not ensure strong and consistent basis related to

reliability level of different structures that were designed using prescriptive rules of modern codes

(EC1-1-2, EC3-1-2).

As it can be seen in the presented table (Table 1-1), different standards and recommendations

give different values for target reliability in terms of reliability index.

50 years service life: EC0

Low consequence

(CC1)

Medium consequence

(CC2)

High consequence

(CC3) 3.3 3.8 4.3

50 years service life: Probabilistic Model Code of JCSS [53] Relative cost of safety

measure Minor

consequences Moderate

consequences Large

consequences High (A) 1.67 1.98 2.55

Moderate (B) 2.55 3.21 3.46 Low (C) 3.21 3.46 3.83

Service life: ISO 2394 [63] Relative cost of safety

measure Some


consequences Great

consequences High (A) 1.5 2.3 3.1

Moderate (B) 2.3 3.1 3.8 Low (C) 3.1 3.8 4.3

Table 1-1 – Target reliability index values from standards and recommendations

Holickỳ pointed out in his study [64] that the suggested safety level is inconsistent among the

codes. Further studies, e.g. [65], [66], [67], [BT5], [BT9] and [BT12] showed and pointed out that

the structural reliability against seismic effects does not achieve EC0 required level, the achievable

reliability index for conventional structures is β≈2.0 – 3.0, due to the high uncertainties in the

seismic effects. In case of fire effects, in [BT8] and in [BT10] the authors showed that the reliability

level was inconsistent and lower than EC0 suggested levels in the case of structures that had been

designed according to the prescriptive rules of EC1-1-2 and EC3-1-2 standards. Further issue is

that the EC0 does not differentiate groups according to the relative cost of safety measures, in this


15

way, it recommends the same target reliability for persistent, seismic and fire design situations.

This method does not seem to be able to provide solutions with consistent reliability that is one of

the bases of safe and economic design. It shows that further research work is necessary in order to

extend our knowledge on optimal design considering fire or seismic effects to refine or check the

available target reliability indices for extreme design situations.

1.2. Aims and outline of this research

Extend the existing information in the literature related to the optimal design of steel tapered

portal frames motivated this research because to the best of author’s knowledge there is no study

focusing on structural fire or seismic optimization of tapered portal frame structures (Fig. 1-1).

When seismic and fire effects are taken into account, highly nonlinear, discrete and non-convex

nature of the design problem (Appendix B) makes the design of tapered steel frames a time-

consuming and iterative process. This is caused by not only the design procedure but also the nature

of extreme effects, the structural response and the fact that the structure typically consists of slender

elements and it is sensitive to stability failure modes.

Fig. 1-1 – Investigated structural configuration – steel tapered portal frame (Rutin Ltd.) [68]

By fire design, it is not evident whether protected or unprotected strengthened structure is

characterized by better performance; passive or active safety measures provide more economical

structures. In case of seismic design, it is also not clear what is the influence of the sheeting system

rigidity on the performance of tapered portal frame structures. It is not obvious which target

reliability indices would be appropriate for fire or seismic design that makes the evaluation of

E

A

A

B

C

D

E

1

2

3

4

5

6

7

8

VH/1

O/2

M/6

ZKL/2

ZKK/3

KM/3

KM/3

VJM/4

M/7 S/1

VBM/12

VJM/4

M/6V

BM/12

HG/5

VBM/16

VJM/8

VJM/8

M/1

M/2

VBM/17

F

T/1

HG/7

VBM/12

T/1

D

BV/7

T/1

HO/6

VH/1

VJM/4

VBM/4

T/5

VBM/16

M/2

VJM/8

VJM/8

G/1

VBM/17

M/1

O/3

VJM/4

VJM/8

VJM/8

VBM/1

VBM/17

T/3

KK/6

KK/2

KK/2

HO/7

HO/7

BV/2

BV/4

BV/4

BV/5

FO/3

HG/4

VBM/12

VJM/4

VJM/4

M/6

VJM/4

VBM/13

C

B

T/1

VJM/11

T/1 KK/2

KK/6

KK/1

HO/1

HO/2

BV/7BV/3

T/7

T/5

VJM/11

T/8

T/8

T/7

T/7

T/7T/8

T/8

ZS/2

G/6

G/5

O/4

VBM/9

VBM/14

VJM/4

VJM/11

VJM/11

M/5

M/1

VBM/15

VBM/19

T/5

HG/7

ZKL/2

ZKK/3

VJM/4

BV/4

BV/6BV/4

T/3

HG/7

ZK/1

ZKK/4

FO/10

G

VJM/4

HG/1

VBM/7

ZK/1

ZK/3

ZK/3

M/7 ZKK/4

KM/3

M/7

KM/3KM/3

KM/3

ZKK/3

ZKK/4

VBM/12

VJM/8

VJM/4

VJM/4

O/6

VBM/4

VBM/1

K

VH/1

VJM/4

G/8

VJM/4M/6

M/3

S/1

VBM/11

VBM/11

VBM/12

VH/1

M/6

VBM/13

S/1

VBM/10VBM/10

S/1VH/1

VJM/4

VJM/4

BV/5

BV/2

T/1 VH/1

VBM/2

VBM/5

VBM/3

VJM/11

I

K/6

K/6

K/6

K/4

RT/1

K/11

K/2

K/9

VJM/1ZS/3

G/9

LK/1

T/6T/5

T/2

G/10

T/2

K/1T/2

VJM/1

FO/6

VBM/4

K/8

VH/1

ZS/1

FO/8

FO/5

FO/1

G/7

K/6

K/6

K/7

H

LK/3

T/2T/6

T/4

FO/6

VBM/4

LK/2

K/3 K/6

K/6

K/7

T/1

VH/1

FO/2

G/2

FO/7

KK/2

K/6

K/7

K/6

K/5

K/6

K/6 K/6

J

VBM/8

S/1M/7

VJM/4

VBM/6

VBM/5

VJM/4

T/1

KK/2

T/1

VH/1

M/3

S/1

S/1 VBM/6

HO/2

KK/2

BV/7

ZKK/4

HG/7

HG/7

KK/6

HO/5

VBM/17

VJM/8

S/1

ZKK/3BV/3

BV/6

M/6

VBM/7

S/1

S/1

VJM/4

M/8

HO/2

BV/4

VBM/8VJM/4

HG/7

G/4VH/1

HO/7

BV/4

BV/4

BV/4

VJM/2

T/4

VJM/2

VJM/2

VJM/2

VBM/9

VJM/11

VJM/11

VJM/11

M/5H

O/4

KK/2

VBM/15

T/1

FO/4

T/5

VH/1

VBM/2

O/5

T/5

VBM/3

O/1

BV/7KK/6

G/3

VH/1

VBM/14

VJM/4

VJM/4

HG/3

M/1

VBM/19

9.0 m

5.9 m

0.0 m

Type Valuedead load of the

framecalculated

dead load of theroof system

0.2 kN/m2

weight of theequipment

0.2 kN/m2

snow 1.25 kN/m2

velocity pressureof wind

0.58 kN/m2


16

performance based design results difficult. Assessment of target reliability indices for extreme

effects, development of comprehensive and effective reliability analysis tools, investigation the

influence of different design variables on optimal solutions and derivation of new design concepts

and recommendations would be very helpful for researchers and also for practicing engineers to

find more economical solutions with better performance. The primary objectives of this study are

the following:

a) to develop a structural optimization framework,

b) to develop effective reliability analysis tools,

c) to define target reliability indices considering fire or seismic effects,

d) to perform a parametric study in order to identify the most influencing parameters,

e) to derive new and valuable design concepts for practicing engineers.

In the course of this research work, a complex structural optimization algorithm framework

(Appendix E – Fire optimization framework; Appendix F – Seismic optimization framework) has

been developed that adopts state-of-the-art design and analysis tools (Section 4 and 5) with respect

to performance assessment and optimization methods. In this context, the appellation framework

means that many different analysis and evaluation tools are incorporated and connected together

with structural optimization algorithm and it does not intend to refer to the universality of the

algorithm. The research and this thesis focus on the investigation of optimal design of tapered steel

frame structures subjected to extreme loading conditions in general instead of giving slightly more

optimal solution (e.g. a solution with 0.5% less structural weight, etc.) for a specific frame in a

specific design situation. The developed optimization framework is a tool, wherewith economic

and well performing structural solutions are determined in the course of parametric studies (Section

7.3 and 7.4). The algorithm evaluates the objective function (Section 2.2 and 2.3) for thousands of

possible design alternatives in each optimization process, thus it provides a comprehensive and

strong basis for the definition of new design concepts and recommendations and the determined

structural solutions can be considered optimal from practical point-of-view.

Wide range of possible design situations is covered in the parametric studies, namely fire effects

with different intensity and duration; seismic effects in seismically less and more intensive areas;

structures with low and high gravity loads; structural failure with low, moderate and high

economical consequences. By the analysis of optimization results, emphasis is laid on the amount

of passive fire protection (fire design), on the influence of sheeting system rigidity (seismic design)


17

and on the optimal configurations. The considered sites for seismic design structural optimization

are characterized with moderate and high seismicity as it is described in Section 5.2. The severity

of fire effect is based on the quality and quantity of the stored materials. Different fire curves (for

calculation of fire curves see Section 4.5) related to different fire design situations are presented in

Section 7.3.1. In this study, intumescent coating fire protection is applied due to the facts that

painting is practical, aesthetic and easy to use. The properties of a specific product, namely

Polylack A paint of Dunamenti Tűzvédelem Hungary Ltd. [69], are considered in the calculations

and applied as passive fire protection in case of the protected structures. However, the calculated

paint thicknesses can be converted if a different product is used; the only criterion is that the

prescribed thicknesses in the design sheet need to be given according to MSZ EN 13381-8 [70]

standard. According to Hungarian regulations [71] [72] a minimum active safety measure, namely

automatic smoke detection system, is selected for the reference structure.

Details of the investigation on possible target reliability indices for the investigated structure in

both fire and seismic design situations is presented in Section 6. The results can be used to get more

information about the achievable reliability of complex structural systems subjected to extreme

effects and may be used later by the refinement of partial factors, prescriptive requirements in the

codes. The investigation covers a wide parametric range (consequence classes, load severity), thus

the result and recommendations may be generalized for other structural configuration types.

In order to reduce the complexity of the investigated problem, the reliability of purlins, sheeting

and its connections is not incorporated in this study. Furthermore, the failure of thin walled

elements and their connections under fire or seismic loading conditions is highly uncertain. The

considered failure modes are focusing only on the failure of columns, beams, connections and

bracing elements.

1.3. Details of the investigated structure

The basic configuration of the portal frame that is investigated in this research is shown in Fig.

1-1 with dead and meteorological loads acting on the structure (the presented structure was

investigated from different perspectives in the framework of HighPerFrame RDI project). The

structure has altogether 8 main frames and it is divided into two fire compartments; the first one is

considered to be a small office, while the second part with 36 m total length (7 frames, Fig. 1-1)

has storage hall function.


18

The primary elements of tapered frames are welded and they are connected with bolted

connections; the steel grade is selected for S355. Secondary elements (e.g. wind bracing) are

constructed from S235 steel grade using prefabricated, rolled sections. Based on the outcomes of

a refined numerical study [73], base connections can be considered as pinned connections while

the beam-to-beam and beam-to-column connections are clearly rigid connections according to the

guidelines of Eurocode 3 Part 1-8 (EC3-1-8) [74] standard. Actual properties of the connections

are taken into consideration within the nonlinear structural analysis with the help of nonlinear

spring elements (Section 4.7 and 5.4). Columns are restrained against torsion at the middle of the

eave height, while there are altogether six brace element equally distributed in the roof level in

order to prevent the lateral torsional buckling of compressed flange of beam elements. At high

temperatures, the sheeting and the purlins cannot be considered as supports for the flanges, because

they lose their stiffness very quickly due to their high section factor and thin walls.

The safety of the structures is verified in persistent design situation considering load

combinations, combination factors and partial safety factors from EC0. The utilization (demand-

to-capacity ratio) of the elements is calculated using geometrically nonlinear analysis on imperfect

structural model considering out-of-plane buckling with the help of reduction factor method of

Eurocode 3 Part 1-1 (EC3-1-1) [75], similarly to fire and seismic design situations. The

serviceability of the frame is checked in quasi-permanent load combination [52] in order to prevent

the aesthetically disturbing deflection of the main frame. The considered failure modes in persistent

design situation are:

1. shear buckling of column web,

2. strength and stability failure of tapered columns,

3. shear buckling of beam web,

4. strength and stability failure of non-tapered and tapered beam parts,

5. failure of connections,

6. failure of side and wind bracings.

The optimized structural configurations satisfy the above mentioned criteria thus they represent

adequate solutions in persistent design situation considering conventional loading conditions.

Components Cost rate Cost of main frame elements 2.25 €/kg Cost of bracing system 2.25 €/kg Cost of sheeting system (purlins included) 25 €/m2 Cost of passive fire protection 24 €/mm∙m2 Cost of automatic smoke detection system 40 €/m2

Table 1-2 – Cost components and rates considered in this study


19

The structural initial cost together with the failure consequences are used by the derivation of

the objective functions in case of both fire and seismic design optimization (Section 2.2 and Section

2.3). Cost components and rates (Table 1-2) have been determined based on consultation with

Hungarian industrial representatives; in order to represent Hungarian circumstances. Some

parameters are varied in the parametric study (Section 7.3.1 and Section 7.4.1) characterizing the

sensitivity of the optimum solutions and giving a strong basis for suggested design concepts for a

wide range of possible cases.

1.4. Preliminary research and results

The numerical framework’s development started in 2012 [BT1] with investigation of

prescriptive optimal seismic design of CBF (concentrically braced frame) (Fig. 1-2) structures

based on the suggestions and rules of EC8-1. With lot of improvements a more settled and

comprehensive application was published later in [BT3].

Fig. 1-2 – Axonometric view and ground plan of the investigated CBF [BT3]

Based on the results in [BT3], it could be declared that the developed algorithm was numerically

stable and suitable for cross-section and bracing system layout optimization of steel multi-storey

CBF buildings. This earlier application proved the applicability of the presented algorithm and the

steps of deriving design concepts based on the results of a parametric optimization study. It could

be also concluded that from practical point of view there may not be difference between the

achieved solutions and the global optima, since the fact that one or two elements could be slightly

different would not change the global concepts which had been derived from the obtained results.

This framework and objective function was applied later by the structural optimization of BRB

(buckling restrained brace) structures in [BT5] and in [BT7]. The probability of failure of the

structures was evaluated with the help of performance evaluation framework of Zsarnóczay [67].

The results of the research confirmed the good performance of designed frames thus the proposed


20

design procedure was appropriate for the design of concentrically braced BRB frames with

chevron-type brace topology.

From the point of view of providing well performing solutions, the introduction of Performance

Based Design concept for multi-storey steel frames was an important milestone in [BT4] instead

of the application of prescriptive rules. The optimization framework has not been changed

significantly since the early applications; only the objective function, the penalization and the

objective function’s evaluation were different by each application. The last step in the development

was the reliability based structural optimization of steel tapered portal frame structures.


21

2. Optimization algorithm development

2.1. Basic description of the optimality problem

In most of the cases in the available literature, the aim of studies dealing with structural

optimization of steel portal frame structures is to find structural configurations with minimum

structural weight or minimum initial cost, e.g. in [23], in [21] and in [24], among others. In case of

minimum initial cost (C(x)) or minimum structural weight (W(x)) design, the aim is to solve one

of the following problems:

( )xCmin or ( )xWmin , (2)

where x is a vector which contains the design variables. Typically, these solutions are considered

the possible cheapest solutions. During a structural optimization procedure, aiming to find a

solution with minimum structural cost or weight, structural reliability is ensured and risk of

possible failure is limited by the application of prescribed design rules and partial safety factors of

the selected code.

However, many recent publications (see Section 1.1) have pointed that the structural reliability

against seismic and fire effects did not achieve EC0 required level in every cases if the structure

was designed according to Eurocode conforming prescriptive design rules. Thus a solution with

the minimum weight or initial cost may not necessarily be the optimal solution when the whole life

cycle of the structure is considered. The optimal configuration may be the one that gives the

minimum cost considering the life cycle of the structure, the risk of different damage states and the

amount of total losses. This aspect motivated my research to develop reliability based optimization

framework instead of cost optimization with prescriptive design constraints.

Fig. 2-1 – Optimal design concept: a) interpretation of life cycle cost; b) life cycle optimum

In some cases the structural reliability may be significantly increased and the expected losses

may be significantly decreased by slightly increasing in the initial cost. This is illustrated in Fig.

safety

Cost,Risk

safetya) b)

Cost,Risk

( )xR( )xC

( ) ( ) ( )xxx RCCLC += ( )xLCC

( )xR( )xC

unfeasible area


22

2-1b, where the red point indicates the optimal configuration having the sum of cost and risk (R(x))

minimum, the green one shows feasible optimum having minimum initial cost and maximum

acceptable risk according to the standard, e.g. according to [52]. The risk of a failure in CLC(x)

function means the risk of a failure in seismic or fire design situation. The dashed line (CLC(x)) is

the so-called life cycle cost (Fig. 2-1a). In life cycle optimization the primary aim is finding a

solution with minimal life cycle cost:

( ) ( ) ( )[ ]xxx RCminCmin LC += . (3)

Throughout the optimization process, in this study the global optimum (minimum in this case)

of the objective function shall be found which expresses the life-cycle cost (Fig. 2-1a) of the

investigated structure. The infeasible solutions are eliminated in the process with the help of

equality (g(x)) and inequality (h(x)) constraints:

( ) ,k,...,,i;gi 210 =≤ x , (4)

( ) m,..,kj;h j 10 +== x . (5)

Equality constraints may express the equilibrium conditions, so stable solutions are only accepted.

Inequality constraints express other design constraints, such as strength and stability checks of the

main frame elements in persistent design situation (Section 1.3). Solutions that violate the design

constraints are also unfeasible and are shown with grey colour in Fig. 2-1b.

By the optimization of steel frame structures the typical design variables are steel profile sizes

or cross section dimensions of column, beam and bracing elements. The fact that we are interested

in practically acceptable solutions makes the optimality problem discrete since available

dimensions of steel plates and steel profiles are discrete on the market. It is also conceivable that

large number of local optima may exist since cross sections with different types and sizes can have

the same load bearing capacity (e.g. higher and more slender, shorter and less slender cross sections

have the same moment resisting capacity). Furthermore, the highly nonlinear nature of extreme

effects makes the existence of local optima more likely.

Life cycle cost consists of a number of different cost components. These cost components may

be roughly differentiated into the following groups in case of seismic or fire design of structures:

• design and construction costs (design fees, infrastructure, construction, non-structural

components, management, etc.);

• maintenance costs (service, repairs, downtime cost, etc.);


23

• operation costs (insurance, management, energy costs, cleaning, etc.);

• risk of total losses due to a possible failure (missing income or malfunction in

production, repair costs, replacement costs, etc.).

Many of these cost components do not depend on the value of selected design variables (cross-

section dimensions of beams, columns and bracings, thickness of passive fire protection), thus these

components can be considered approximately constant (e.g. energy costs are clearly not dependent

on the selected variables). For sake of simplicity, components independent on the selected variables

are neglected in this research similarly to [64]. In this study, the life cycle cost function is a function

that consists of the initial cost of steel superstructure, the initial cost of passive and active fire safety

measures (in case of fire design) and the risk of failure, as described in the following sections. It is

assumed that the cost factor given for the passive fire protection (Table 1-2) consists of not only

the cost of intumescent coating but the cost of primer and the cost of the finish coat, as well. Aging

effect of the paint and the cost of repainting is not considered in this study.

2.2. Objective function in case of fire optimization

Life cycle cost function of the investigated structure (CLC(x)) may be formulated on the

following way [BT11] similarly to [64]:

( ) ( ) ( )( )

( )( )

4444444444 34444444444 21

444 3444 21

x

x

x

xxx

R

erventionintignitionfignitionfff

C

LC

PC.PC.PC

...CCCC

+⋅+⋅+⋅+

+++=

050010

210

. (6)

In Eq. (6), C0(x), C1(x) and C2 are the initial cost, the cost of passive and the cost of active safety

measures, respectively, while Cf and Pf (x) refer to total losses and the failure probability (calculated

with reliability analysis, Section 3 and 4.1) related to the service life which equals to 50 years. The

last two terms express the damage cost which is caused by moderate fire (quenched before

flashover) and by intervention (e.g. damage caused by sprinkler system and/or firefighting). Cf

contains direct (e.g. value of stored material or the construction of a new storage hall) and indirect

cost components (e.g. missing income or malfunction in production).


24

Fig. 2-2 – The shape of proposed objective function with two decisive variables [BT11]

The optimal solution is associated with a structure that results the lowest CLC(x) objective function

value (Fig. 2-2). Almost every component of Eq. (6) depends on the value of design variables, for

example, if the thickness of the flanges or passive fire protection is increased, this increment will

directly change the C0(x) or C1(x) cost components. Furthermore, in case of a stronger or a better

protected frame the failure probability is lower compared to a less protected one and the risk of the

structural failure in fire design situation is decreased.

The initial cost is proportional to the weight of the structure:

( ) i,d,l,t,bCcld

cltbnC iiiish

n

isi

i

n

isiiif

bp

∀∈+⋅⋅⋅

+⋅⋅⋅⋅= ∑∑==

xx1

2

10 4

ρπ

ρ . (7)

This approach is clearly an approximation; however, it is often used by industrial representatives

in cost calculations and bids. In Eq. (7), nf, np, cs and Csh are the number of frames, the number of

steel plates of a frame, cost rate in €/kg unit and the cost of the sheeting and bracing system,

respectively. The weight of the ith plate is calculated by multiplying bi (width), ti (thickness), li

(length) and ρ (density). The nb and di are the number of bracing elements and the diameter of ith

steel bar, respectively. The cost of the passive fire protection is considered to be proportional to

the protected surface, thus it can be formulated as follows:

( ) j,tctlAnC j,p

n

jpj,pjjf

e

∀∈⋅⋅⋅= ∑=

xx1

1 , (8)

where ne, Aj, lj, tp,j and cp are the number of protected elements, the protected surface, the protected

length of jth element, the protection thickness in case of jth element and the cost rate in €/(mm·m2)

unit, respectively. Due to the fact that column base connections are pinned, the dimensions of

foundation are not design variables and the cost of foundation is not considered in this study.

1.5

2.5

3.5

4.5

Sta

ndar

dize

d co

st fu

nctio

n, R

elia

bilit

y in

dex

01

2 01

22

4

6

8

10

12

x 104

Active safety measurePassive safety measure

Cos

t fun

ctio

n [E

uro]

1.5

2.5

3.5

4.5

Sta

ndar

dize

d co

st fu

nctio

n, R

elia

bilit

y in

dex

Active safety measure


25

2.3. Objective function in case of seismic optimization

Similarly to fire design situation in Section 2.2, the objective function expresses the life cycle

cost of the structure. In this case, CLC(x) may be formulated on the following way:

( ) ( ) ( )( )

( )( )434214434421

xx

xxxx

R

ff

C

LC PCCCC ⋅++= 10 . (9)

In Eq. (9), C0(x) and C1(x) are the initial cost (considering the sheeting system and purlins as

well) and the cost of the bracing system, respectively, while Cf and Pf (x) refer to total losses (Cf

contains direct and indirect cost components) and the failure probability (calculated with reliability

analysis as it is described in Section 3 and 5.1) related to the service life which equals to 50 years.

The optimal solution is associated with a structure that results lowest CLC(x). Similarly to fire

design situation, in seismic design situation only one performance objective is considered, namely

the failure of the structure. The performance objective is investigated with the consideration of 475

years return period seismic action, thus it corresponds to significant damage performance level of

EC8-1 [28].

In case of calculation of C0(x) the basic idea is that the initial cost is proportional to the weight

of the structure:

( ) i,l,t,bCcltbnC iiish

n

isiiif

p

∀∈+⋅⋅⋅⋅= ∑=

xx1

0 ρ . (10)

This approach is clearly an approximation; however, it is often used by industrial representatives

in cost calculations and bids. In Eq. (10), nf, np, cs and Csh are the number of frames, the number of

steel plates of a frame, cost rate in €/kg unit and the cost of the sheeting and bracing system,

respectively. The weight of the ith plate is calculated by multiplying bi (width), ti (thickness), li

(length) and ρ (density). The cost of the bracing system can be formulated on the following way in

case of slender steel round bars:

( ) i,l,dcld

C ii

n

ibi

ib

∀∈⋅⋅⋅⋅

=∑=

xx1

2

1 4ρ

π. (11)

In Eq. (11), nb, di and cb are the number of bracing elements, the diameter of ith steel bar and the

cost rate in €/kg unit.

Eq. (10) and (11) express the cost of steel superstructure. Due to the fact that column base

connections are pinned, the dimensions of foundation are not design variables and the cost of


26

foundation is not considered in this study. The considered cost factors are further discussed at the

description of the parametric study and it has to be noted that the cost factors have been selected

on the basis of consultation with Hungarian industrial representatives (Table 1-2).

2.4. Genetic algorithm based optimization framework

Due to the above mentioned issues, an advanced and universally applicable optimization

algorithm needed to be invoked that is able to handle the special characteristics and difficulties of

the investigated problem. Due to the fact that the problem is non-convex (large number of local

optima exist), simple gradient based optimization algorithm is clearly not applicable. A feasible

and favourable solution is the application of heuristic algorithms because their applicability is

confirmed by numerous examples from the literature, such as in e.g. [42], [19], [14], [43], [36] and

[5], among others.

In this study, genetic algorithm (GA) [37] [38] is invoked to find the optimal solutions due to

the fact that GA is able to handle highly nonlinear problems, different optimal solutions in parallel

and discrete objective functions, it can scan a very large search space during its operation and its

operation is stable. Its applicability to similar and other nonlinear structural optimization problems

is shown e.g. in [23], in [39], in [24], in [BT3] and in [BT4]. GA literally imitates the biological

evolution; the best individuals survive and transmit their genes for the newer generations; for this

reason, the technical terms often have biological origin (Fig. 2-3). The design variables are stored

in chromosome-like data structures, namely in a series of vectors considering the symmetry of the

frame. Example: n is the number of individuals which is commonly referred as the population size;

ne is the number of elements; hi, bi, tw,i and tf,i are the cross section dimensions of the ith element:

=

n

...

x

x

x

X 2

1

(12)

[ ]eeee n,fn,wnni,fi,wii,f,w ttbh...ttbh...ttbh 1111=x (13)


27

Fig. 2-3 – Illustrative flowchart from the operation of GA

GA starts seeking optimum from a randomly generated initial set with the help of improvement

of individuals and the search space during its operation. Uniform crossover (Fig. 2-3) is invoked

in the optimization algorithm where the genes of parental individuals are selected randomly with

even chance. Crossover ratio controls the percentage of best individuals participating in the

crossover. After the crossover the chromosomes are varied further within the mutation procedure

(Fig. 2-3). The elite individuals are responsible to preserve the best genomes, thus they are not

allowed to be mutated. Mutation helps to avoid the local optimum in the optimization process.

Mutation ratio gives the number of mutated individuals which are selected randomly excluding the

elites, thus one individual may be mutated more than once.

GA can handle the constraints only with the help of so-called penalty functions [76]. Using

penalty functions the problem can be transformed into unconstrained format:


28

( ) ( ) ( ) ( )( )

( )( )x

x

xxxxx

iilim,

ilim,i

i

iULSSLSLC g;ggCmin!ηηηη

η <

≤

=⋅⋅ 2

1, (14)

where gULS(x) and gSLS(x) are the penalty functions related to ultimate and serviceability limit states

related conventional design situation (Section 1.3). The ηi and ηlim,i are the calculated and

acceptable D/C ratio (1.0, i.e. 100%) in the investigated limit states.

2.5. Global optimum

Heuristic algorithms (e.g. genetic algorithm) seek the optimum with the development of the

search space during their operation and they evaluate large number of possible solutions (thousands

of possible solutions in this study). With good settings heuristic algorithms can find solutions

situated very close to the global optimum with no difference compared to the global optimum from

practical point-of-view. There may be neighbouring solutions with very slight difference in the

objective function value but both solutions are acceptable for an engineer. In this way, within the

framework of parametric studies the application of heuristic algorithms provides a comprehensive

basis for the definition of new design concepts and recommendations.

The finding of global optimum cannot be guaranteed in this research due to the fact that the

problem is discrete, non-convex and highly nonlinear. Brute-force search technique (evaluate all

of the possible solutions and select the best) could guarantee the global optimum, however,

considering e.g. fire design optimization with 15 variables (Section 7.1) the number of possible

variations is incredibly high (billions) even with some restrictions. For this reason, brute-search

technique can be used only if the complexity of the problem is reduced significantly compared to

the optimization problem presented in this thesis.

Further details about the settings and convergence of the structural optimization framework can

be found in Section 7.2.


29

3. Reliability analysis

The aim of reliability analysis is mainly to obtain the probability of failure (Pf) of the

investigated structure. The structural failure is modelled with the so-called limit state function,

G(X)=0 in Fig. 3-1 that separates the safe (G(X)>0) and unsafe (G(X)<0) potential solutions.

Fig. 3-1 – Joint distribution function of reliability problem with two random variables

Basically, the ratio of safe solutions to the all possible solutions gives the reliability of the structure

in a given design situation (R=1-Pf). This is theoretically equal with the value of the following

integral if the distribution of the random variables can be approximated with continuous

distribution functions.

( ){ } ( )( )

XXXX

dfGPPG

f ∫<

=<=0

0 , (15)

where the f(X) is the joint density function, which already contains the random variables from the

effect and resistance side. In case of normally distributed and not correlated joint density function,

the integral can be evaluated in the following way:

( )βσµ

−=

−= ΦΦfP , (16)

where β, μ and σ are the so-called reliability index (frequently used in literature as the measure of

structural reliability), mean value and standard deviation, respectively.

The integral is typically calculated numerically, because of the fact that the limit state function

can be often discrete or highly complex in real design cases. For this reason, the exact failure

probability can be rarely calculated; the main difference between the numerical technics is the

degree of approximation. The Monte Carlo simulation (MCS) [45] [46] may give the most accurate

approximation from the available methods, where a series of realizations are simulated and the

9.510

10.511

11.512

12.5

2

2.5

3

3.5

40

0.5

1

1.5

2X1X

( ) 0=XG

( ) 0<XG

( ) 0>XG


30

unfeasible solutions are simply counted. The high nonlinearity in the structural response makes

MCS method computationally expensive because it can happen that evaluation of 105 - 106 of

analyses are necessary to get accurate solution especially when the failure probability is low. In

order to eliminate the lack of computational capacity, there are other approximated methods, e.g.

FORM and SORM (second order reliability method), where the limit state function is approximated

with linear and second order terms of the Taylor series [47] in the so-called maximum probability

point (MPP; the closest point of failure surface to the centre in standard normal space). FORM

algorithm, which is based on Hasofer – Lind – Rackwitz – Fiessler (HLRF) iteration method (Fig.

3-2) [47] [77] [78] is adopted in this work. The implied and extended HLRF algorithm can handle

correlated and non-normally distributed random variables with:

• transformation of non-normally distributed variables into normally distributed ones with

normal tail approximation,

• transformation of correlated variables into non-correlated space.

Fig. 3-2 – Adopted Hasofer – Lind – Rackwitz – Fiessler (HLRF) iteration method

The optimized objective function is presented in Section 2.1 in general and more detailed form

in Section 2.2 and 2.3. The objective function expresses the initial costs and the possible risk of

Definition of discrete and random variables

Distribution, parameters, correlation

Design point

i

iii

Xu

σµ−

=

Normal tail approximation

'UTU =

( ) ( )UGUG ∇

( ) ( )

( )2

1

1

∂∂

∂∂

−

=

∑

∑

=

=

n

i

ii

n

i

iii

x

UG

ux

UGUG

σ

σ

β

( )

( )2

1

∂∂

∂∂

=

∑=

n

i

ii

ii

i

x

UG

x

UG

σ

σα

iii*iX σβαµ +=

Convergence?yes

no

failureP

( )( ) 0

21

=

=

UG

UUmin! Tβ

HLRF iteration


31

losses caused by a failure due to fire or seismic effects. The risk is calculated as the product of

financial losses and the probability of failure, thus the result of adopted FORM analysis is directly

connected to the optimized function. The following uncertain variables are taken into account as

random variables (Section 4.6 and 5.3): strength of the steel material; stiffness; permanent loads;

meteorological loads (wind and snow); seismic effects; fire effects; effect- and resistance

uncertainty factors. Based on the results of a sensitivity analysis, some uncertain variables have

been merged into uncertainty factors [79].

The coordinates of MPP are calculated on the basis of sensitivity of limit state function that is

approximated with partial derivatives in every substep. Due to the complexity and discrete nature

of the design problem and variables (e.g. thickness of the flange plate, thickness of insulation),

partial derivatives can be calculated e.g. with central difference method, that means that the limit

state function shall be evaluated a lot of times during the iteration. The accuracy and feasibility of

the reliability analysis have been checked with the help of comparison to MCS results for simple

nonlinear problems from [51] and for complex and comprehensive reliability assessment of a

tapered steel frame exposed to fire effects [BT13]. It is found that the adopted reliability algorithm

is characterized by good accuracy.

The failure of a steel frame structure can be caused by several failure mechanisms. In case of

stability failure modes, it can be said that the investigated frame is failed if stability failure occurs,

so failure components compose a series reliability system, where the following simple

approximation can be given for the boundaries of the system failure probability [80]:

{ } ( )∏==

−−≤≤n

ii,fS,fi,f

n..i

PPPmax11

11 , (17)

where Pf,i, n and Pf,S are the failure probability related to the ith failure mode, the number of failure

modes and the failure probability of the system, respectively. Plastic sway mechanism is a failure

mode that composes a parallel reliability system since formation of several plastic hinges is needed

for failure. For a parallel system, simple lower and upper bound can be given as follows:

{ }i,fn..i

S,f

n

ii,f PminPP

11 ==

≤≤∏ . (18)

If the value of the limit state function can be calculated with respect to the all relevant components

(failure modes) within reliability analysis simultaneously then the correlation between failure

modes is already taken into account (as it is the case in this study, see Section 4.3 and 5.1). In case


32

of the whole structure, that consists of several frames itself, the reliability system is also series

because in the case of failure of one frame the structure is considered to be failed. If the whole

structure is modelled in the limit state function as a three-dimensional structure with respect for the

random variables (and their correlation) of every individual frame, the correlation between the

frames is already taken into consideration. However, in some cases the three-dimensional

modelling of the system incredibly increases the complexity of the reliability calculation. A

possible approximation is the application of e.g. simple bounds [80], Ditlevsen bounds [81] or

multivariate normal cumulative distribution function to approximate the system reliability with

respect to the correlation between the failures of separated frames:

( ) ( )( )ρβ,P mf,Sf,S ΦΦΦβ −−=−≅ −− 111 , (19)

In Eq. (19), the Φ, Φm, βS,f and PS,f, are the single- and multivariate standard normal cumulative

distribution functions, the reliability index (P = Φ(-β)) of the system and the probability of failure

related to the system, respectively. The β and ρ reliability index vectors containing the reliability

indices of individual frames and correlation matrix are in the following form, where the n is the

number of frames:

jiij

nnn

n

n

n

f,n

f,

f,

......

...

...

;...

ρρ

ρρρρ

ρρρρ

β

ββ

=

=

=

1

1

1

1

321

3

221

112

2

1

ρβ . (20)

This approximation allows the researcher to consider different reliability indices and different

correlation coefficients among the frames representing e.g. different scenarios.

In this study, the system’s reliability index is calculated for seismic effects with the help of

application nonlinear three-dimensional analysis on a spatial model within the evaluation of limit

state function (Section 5.1). In the course of reliability analysis for fire effects, two-dimensional

structural model is applied and multivariate normal cumulative distribution function is used to

calculate system reliability index and consider correlation among the frames’ failure (Section 4.4).


33

4. Reliability evaluation of frames under fire

exposure

4.1. Overview from the methodology

The methodology (Fig. 4-1) connects a complex structural reliability analysis to a Bayesian

probabilistic network [80]. Full system failure probability over the reference period (that is selected

equal to the structure’s design lifetime) is calculated as a product of conditional failure probability

given flashover occurrence (PS,f|flashover(X)) and probability of severe fire (Pflashover):

( ) flashoverflashoverf,Sfailure PPP ⋅= X , (21)

While calculation of failure probability and Bayesian networks are straightforward and well

known, the suggestions of the author are concentrated mainly in the composition of the limit state

function and the whole framework [BT13].

Fig. 4-1 – Overview from the proposed methodology and the limit state function.

It is recommended to model fire effects as accurate as possible in the design process. For

example, ISO standard fire curve can be considered only as a non-realistic, comparable effect

which can barely be the basis of realistic reliability calculation, notwithstanding the majority of

verifications against fire are based directly or indirectly on the ISO curve. CFD (computational

fluid dynamics) fire modelling softwares, e.g. FDS v6.0.1 [82], are the best tools for consideration

different fires and scenarios, local fires, spatial distribution of temperature; however, the number

of possible scenarios is considerably high. For this reason, the applicability of complex models in

structural reliability calculation seems difficult because they overcomplicate the investigated

problem and the results depend on the decisions made related to the considered fire scenarios. In

order to reduce the complexity but keep the accuracy, one or two zone model calculations, e.g.


34

[83], seem to be the best alternatives. They provide the opportunity for consideration important

influencing parameters, such as quantity or quality of combustible material, ventilation. Parametric

fire curves are also able to capture the effect of these influencing parameters, nevertheless, their

applicability is limited (EC1-1-2) since the equations are based on a limited number of

experimental fires considered by their development.

Further issue is that the structural reliability in a lot of cases is ensured through verification of

isolated structural elements without consideration of realistic boundary conditions, interaction

among the elements and the nonlinear system response on elevated temperature; it makes the

calculated safety level more uncertain. In case of complex structural systems, the reliability of a

structure is not equal to the reliability of one of its elements. More accurate information can be

achieved by modelling the whole structure and follow the geometrically and materially nonlinear

structural response in fire.

The following sections provide more information on components of the methodology: event

tree analysis and Bayesian network (Section 4.2); limit state function (Section 4.3); consideration

of fire effects (Section 4.5); structural analysis (Section 4.7). This methodology offers a more

complex and comprehensive basis for the calculation of structural reliability than earlier studies in

the literature: a) the reliability calculation does not focus on one single element but the whole

structure; b) the presented methodology is able to consider any type of fire curve; c) reliability

analysis includes the nonlinear analysis of the whole structure; d) the structural reliability is

assessed on time basis. While these features may be found separately in earlier studies, the main

novelty of this methodology is that it offers the aforementioned features together for complex

structural systems.

The feasibility, accuracy, fast convergence and sensitivity of the algorithm (Fig. 4-2) (Fig. 4-3)

is proven with the help of reliability analysis of example structures protected by intumescent

painting; further details may be found in [BT13] and in [79]. It is found that the FORM

approximation underestimates the failure probability in the investigated case, the difference

between the results of FORM and MCS is observed from -1% to -34%, nevertheless, linear

approximation resulted good approximation of failure probability in the range of interest. The

sensitivity factors (negative normalized gradients of the limit state function at MPP are presented

in Table 4-1. For further details see [BT8] and [BT13].


35

Fig. 4-2 – Conditional probabilities related to structural failure from FORM and MCS [BT13]

Random variable Sensitivity factors - α

Value at MPP point

Yield stress [MPa] -0.091 386.81

Equipment load [kN/m2] 0.101 0.201

Wind load [kN/m2] 0.118 0.238

Snow load [kN/m2] 0.563 0.481 Resistance factor for the column-base

connection [-] 0 1.241

Resistance factor for the column-beam connection [-]

0 1.241

Resistance factor for the beam-beam connection [-]

0 1.241

Right column section modulus factor [-] 0 1

Left beam section modulus factor [-] 0 1

Right beam section modulus factor [-] -0.185 0.995

Effect model uncertainty factor [-] 0.429 1.029

Resistance model uncertainty factor [-] -0.431 0.930

Steel temperature uncertainty factor [-] 0.497 1.048

Table 4-1 – The sensitivity factors related to random

variables in fire design problem [BT13]

Fig. 4-3 – The convergence of the reliability analysis

algorithm [BT13]

Based on the literature review (Section 1.1), more research work is needed on developing

consistent basis in order to obtain target reliability levels for different structural systems designed

considering different fire scenarios and design situations. Despite of the fact that the framework is

used for reliability calculation of tapered portal frames the proposed methodology is inherently

applicable for any kind of structure analysed as a system by modelling its nonlinear behaviour and

not only as a totality of separated elements. The methodology gives the opportunity to apply more

realistic and problem dependent fire curves. For the time being, similarly complex and

comprehensive framework (Fig. 4-1) is not available in the literature, for further details see [BT13].

One of its novel features is the complex application of state-of-the-art analysis and evaluation tools

within the evaluation of limit state function.

15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

Time demand [min]

Con

diti

onal

fai

lure

pro

babi

lity

MCS simulation results (1 exposed frame)MCS simulation results (5 exposed frames)FORM results (1 exposed frame)FORM results (5 exposed frames)Fitted lognormal distribution (1 exposed frame)Fitted lognormal distribution (5 exposed frames)

1 2 3 4 5 6 7 80.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

Iteration

flashoverfailβ

( )0021

2070

.

.G

=

=

βX

2983

9182

5940

.

.

.

fail

flashover

flashoverfail

=

=

=

β

β

β

( )60350

0370

.

.G

=

=

βX ( )

59970

0060

.

.G

=

=

βX

( )59380

0000

.

.G

=

=

βX


36

4.2. Event tree and Bayesian probabilistic network

Beside the calculation of the conditional failure probability given flashover, the probability of

occurrence of flashover (simultaneous ignition of most of the flammable material in a

compartment) also has to be taken into account. Bayesian probabilistic network [80] is

implemented in the proposed framework (shown in Fig. 4-4 where F and T refer to false and true

events, respectively) in order to take into account the ignition occurrence, the effect of active safety

measures in the calculation of flashover probability. For more refined net please refer to [84].

Fig. 4-4 – Event tree and Bayesian network

The ignition probability can be assessed using statistical data [85], while the probability of fire

growth can be approximated using event tree analysis. With respect to the effectiveness of active

safety measures (e.g. sprinkler system, detection by heat, etc.) and the possibility of fire growth

from ignition, relevant data can be found in the literatures in [86] [87]. For the current study, the

probability of ignition is assumed equal to P(I)=10·10-6/(m2·year) as [86] recommends for an

IgnitionActivesafety

measure

Flashover

1,flashoverP

Active p.

Ignition T F

F 0 1

T 0.99 0.01

Ignition

T F

Pignition 1-Pignition

Flashover

Active p. Ignition T F

F F 0 1

F T 1 0

T F 0 1

T T PFL|A 1 - PFL|A

Ignition

1.0·10-5

fire/m2/year

4.5·10-6

6.5·10-6

Fire stoppedby occupants

yes - 0.45

no - 0.65

5.53·10-6

0.98·10-6

yes - 0.85

no - 0.15

Fire stopped byfire brigade

50,flashoverP

fire/m2/year

fire/year


37

industrial building with no standard public fire brigade. The probability that occupants or

professional fire brigade stop the fire is taken into account in the event tree analysis as illustrated

in Fig. 4-4. The calculation of flashover probability for the whole service life of 50 years is based

on the assumption that the ignitions are independent. A considerably low 0.01 probability is

assigned to the failure of active safety measures.

PFL|A is the probability of fire growth from moderate fire to severe fire if active safety measure

is applied. According to [86], PFL|A is equal to 0.02, 0.0625 and 1.0 in case of fire extinguish system

(sprinkler), smoke detection system and no applied safety measure, respectively. Considering

smoke detection system the calculated flashover probability is then equal to Pflashover≈3.51∙10-6/m2

for 50 years in case of the investigated industrial portal frame. The author emphasize the fact that

structural reliability actually depends on the function of the building as different ignition fire

development parameters are associated to different function. Thus, the usage of equivalent fire

effects (e.g. ISO fire curve) in design may not lead homogeneous reliability level (code conforming

designed buildings, facilities should have approximately the same reliability against failure in fire).

Application of active safety measures (e.g. sprinkler system) may drastically increase the reliability

level and may unnecessitates passive fire protection. The optimal fire protection method can be

selected only on the basis of cost and risk analysis.


38

4.3. Limit state function

Fig. 4-5 – Limit state function for fire design

The composition of the limit state function (Fig. 4-5) involves the following steps in one

iteration step of reliability analysis: 1) fire effect modelling; 2) calculation of the steel temperature;

3) nonlinear structural analysis; 4) failure mode verification; 5) evaluation of the limit state function

G(X, t) (Annex B).

( )( )X

XRt

tt,G −=1 , (22)

where the t, tR(X) and X are the time (where the time demand shall be substituted), the time capacity

(resistance time) and the vector of discrete design variables, respectively. The calculation of tR(X)

is based on the loss of load bearing capacity of the structure. The point of failure is equal with the

time step where the D/C ratio of the frame exceeds 1.0 or where plastic sway mechanism/global

stability failure occurs. This point gives the fire resistance capacity of the structure in time unit. As

it is shown in Fig. 4-5, the limit state function is formulated on time basis contrary to other studies,

e.g. [59], [60], [61] and [62], where steel or maximum gas temperature, internal force or

displacement is used for this purpose. First of all, it is practical to use the time as a measure of

capacity and demand due to the fact that the interest of the designer is focused on the reliability


39

according to the stability of the structure within required time of evacuation that is given by national

or international standards. Another advantage is that it gives the opportunity for consideration

possible failure during the decay phase of the fire in the analysis. Internal forces to calculate the

value of the limit state function can be used solely in case of separated elements not in the case of

a complex structure thus the use of internal forces would overcomplicate the reliability calculation

and separated reliability analysis would be necessary for every failure components with the

consideration of the possible correlation. Further details can be found in [BT13] and in [BT14].

As it can be seen in Fig. 4-5, General Method of EC3-1-1 standard is invoked to evaluate the

resistance the tapered members due to the fact that the EC3-1-2 conforming method for verification

of e.g. columns subjected to compression and bending moment (Appendix B.2) is developed for

uniform elements. Furthermore, a part of the structure is analysed within the framework of GMNI

numerical analysis instead of linear analysis of separated elements. In case of an analysis of part

of the structure, EC3-1-2 recommends the use of advanced analysis methods that are based on

acknowledged principles and methods of the theory of structural mechanics considering the

changes of material properties caused by elevated temperatures (EC3-1-2). The applicability of

General Method for fire design of tapered elements is confirmed in [88].

In this thesis, the failure of main frames (consist of columns, beams and connections) is

considered only by the evaluation of structural reliability of the investigated tapered steel portal

frame structure. The failure of longitudinal elements and the behaviour in longitudinal direction

are not covered due to the computational demand of time step thermal and GMNI analyses of a

three-dimensional model.

4.4. System reliability

The reliability calculation of the structure using HLRF iteration is presented in Section 3.

System reliability index is evaluated with the help of multivariate normal cumulative distribution

function. In order to characterize the sensitivity of the investigated problem for the issue of spatial

location of fire and different correlation among the components, a short example is presented

covering different correlation among the frames and different fire sizes. Let us assume an industrial

portal frame structure that consists of 7 individual frames and assume that internal frames have

βf|flashover=1.0 conditional structural reliability in fire design situation, respectively. A conditional

reliability index equal to 1.0 well represents an average case based on the experiences of this


40

research. As regards to the outer frames, their reliability index is much higher due to the fact that

they may be exposed less severe effects because of their spatial location; thus n may be set to 5.

The sensitivity of conditional system reliability index on the correlation coefficient and on the

number of exposed frames can be seen in Fig. 4-6 for the example frame. The correlation

coefficient refers to the correlation considered among the reliability of exposed inner frames. It has

to be noted that when only 1 or 2 frames are exposed, the fire curve becomes less severe. It can be

accounted in reliability calculation with the usage of different fire curves for different cases.

Fig. 4-6 – The effect of correlation and number of exposed frames on system reliability index(example)

The results of preliminary calculations showed that correlation and spatial distribution of fire

play an important role and have a great influence on the system reliability index (in this case

βS,f|flashover≈0.1-1.0). Some of the random variables are supposed to be highly correlated, such as

strength, section dimensions of the frames and the intensity of meteorological loads. However, due

to the following reasons the correlation among the frames is supposed to be low (namely ρ=0–0.6)

in case of fire design: I) there is a spatial variation in the location of the combustible material; II)

all of the frames may not be exposed to fire at the same time; III) it is very likely that the

temperature varies spatially; IV) there is a certain spatial variation in the equipment load.

In order to cover a wide range of possible outcomes, in this study the system reliability is

calculated by use of Eq. (19) considering a low ρ=0.4 and a considerably high ρ=0.9 correlation

among the failure of the frames. Different fire scenarios are not covered, it is assumed that all of

the frames are exposed to fire.


41

4.5. Fire effects

Realistic modelling of the fire effects is important in order get realistic structural reliability

from the calculation. The most widely used representation of fire effects in fire resistant design is

the fire curve, which gives the temperature as a function of time as it is illustrated in Fig. 4-7a.

Fig. 4-7 – a) The shape of different fire curves; b) Design gas temperatures and zone interface elevation calculated

in OZone V2.2.6

Different fire curves are used in design practice, among them some represents only a

comparable effect, e.g. ISO standard fire curve in [33], and do not intend to express real and

physical effects. Earlier studies have shown, e.g. in [61] and in [BT10], that the usage of ISO fire

curve for fire resistant design can hardly ensure consistent reliability level, since the structural

reliability depends on the function and properties of the fire compartment and on the amount and

properties of the fire load. Other fire curves that are obtained with advanced methods and models,

e.g. one- and two-zone models [83] can represent fire severity and temperatures closer to the reality.

The fire effects in this study are modelled with fire curves obtained with the help of two-zone

model in OZone V2.2.6 software [83] in order to represent more realistic fire. Two-zone models

may be criticised by some experts from the field of fire engineering because of their shortcomings

(one dimensional calculation, constant temperature within a zone, etc.), however, based on

comparisons in [83] it can be concluded that OZone is characterized by relatively high accuracy in

calculation of compartment or steel element maximum temperature.

OZone is able to consider these influencing parameters, such as the fire load, combustion heat,

fire growth rate, ventilation, geometry of the compartment, etc. An example design fire curve

obtained by OZone can be seen in Fig. 4-7b. It is called design curve (it is assumed that the curve

represents 95th percentile of the effect) because it is calculated using the design value of parameters

of compartment and stored material [58]. 95th percentile is selected because in [87] it is suggested

that the uncertainty in fire load should be followed with the help of Gumbel distribution having 0.3

0 20 40 60 80 100 1200

500

1000

1500

2000

t [min]

ISO standard fire curve

OzoneV2 fire curve

Parametric fire curve

20 40 60 80 100 1200

500

1000

1500

2000

t [min]

T [

°C]

0

5

10Heat zone gas temperature

Cold zone gas temperature

Zone interface elevation

Flashover

0

Zone i

nte

rface e

levati

on [

m]a) b)

Design curve ~95th fractile

mean

5th percentile120

1

1

2

T [

°C]


42

CoV and characteristic value equal to 80th percentile. According to the further instructions in [87]

the design value could be calculated close to 95th percentile. In order to avoid the numerical

instabilities within the reliability analysis, the decaying period of the curves (Fig. 4-7) is neglected

and substituted with the maximum gas temperature.

4.6. Random variables

The random variables, considered in the reliability analysis, are listed in Table 4-2. Due to the

small variation and the fact that their effect on the global behaviour is small, the uncertainty in the

Young’s modulus and global geometry is neglected. Among the loads, the weight of equipment (as

permanent load) and the meteorological loads, namely wind and snow loads, are considered as

random variables. Because of the accuracy in manufacturing and assembly the uncertainty in dead

loads are negligible. The uncertainty of yield strength, section moduli and connection parameters

has been selected according to the Probabilistic Model Code of Joint Committee on Structural

Safety [53]; the CoV (coefficient of variation) values related to section factor moduli are slightly

higher in Table 4-2 than in the JCSS because of the tapered elements. ρ=0.7 correlation is

considered among the section modulus factors.

Random Variable μ CoV Distribution References Yield stress [MPa] 388 0.07 Lognormal [53] Equipment [kN/m2] 0.2/0.5 0.2 Normal

Wind load [kN/m2] 0.06 1.963 Lognormal Calculation, [52],

[53], [94]

Snow load [kN/m2] 0.205 1.03 Weibull Calculation, [52],

[53], [93] Resistance factor for the column-base connection

[-] 1.25 0.15 Lognormal [53]

Resistance factor for the column-beam connection [-]

1.25 0.15 Lognormal [53]

Resistance factor of ridge beam-beam connection [-]

1.25 0.15 Lognormal [53]

Right column section modulus factor [-] 1 0.05 Normal [53] Left beam section modulus factor [-] 1 0.05 Normal [53]

Right beam section modulus factor [-] 1 0.05 Normal [53] Effect model uncertainty factor [-] 1 0.15 Lognormal

Resistance model uncertainty factor [-] 1 0.2 Lognormal Model uncertainty in LTB reduction factor -

LTχ 1.15 0.1 Normal [54]

Model uncertainty in FB reduction factor - Zχ 1.15 0.1 Normal [54]

Steel temperature uncertainty factor [-] 1 0.3 Lognormal [BT13], [61] Table 4-2 – Random variables

The uncertainty in the fire effects is considered through the introduction of a global uncertainty

factor that has been calculated with Monte Carlo simulation. Since OZone is not appropriate for

evaluating numerous simulations, the uncertainty in the gas temperatures is obtained using


43

parametric fire curves of EC1-1-2 [58] with a little modification according to [89] in order to

eliminate the discontinuity in the calculation (limits suggested in EC1-1-2) for some parameters to

separate fuel and ventilation controlled fires cause an unreasonable discontinuity in the

compartment temperatures). The investigated compartment can be found in the literature [90],

6.4x3.2x2.6m compartment with 4.4m2 openings; mean fire load 420MJ/m2 with CoV=0.3

according to EC1-1-2. The maximum gas temperature is used as the output parameter of 100,000

simulations and it is found that the uncertainty in the maximum temperature can be approximated

with a lognormal distribution (because the effect of uncertain parameters may be multiplied with

each other) that has a mean equal to 1.0 and a CoV equal to 0.25. For further details, see [BT13].

This uncertainty is considered in the reliability analysis through the uncertainty of the steel

temperatures as it is described in [BT13]. For this reason the fire curve, as input parameter of the

analysis, has to be given as a representation of mean gas temperatures. In order to calculate the

mean fire curve, it is assumed that every point of the design curve is 95th percentile of the above

obtained lognormal distribution (Fig. 4-7b).

The reliability problem is time-variant because the meteorological loads vary in time. In order

to reduce the complexity of reliability analysis, the problem is transformed into a time-invariant

problem with the help of the so-called Turkstra’s rule [53]. Its application is presented for similar

problem in [91]. The leading action, namely the fire effect, is considered with its lifetime (50 years)

maximum, while snow and wind loads are accounted with the distributions of daily maximums

derived from meteorological data (wind speeds and snow water equivalents) which have been

downloaded from CARPATCLIM database [92]. In CARPATCLIM, different meteorological data

sets of Carpathian basin are given for 50 years in 10 km by 10 km grid. Due to the fact that not a

specific frame with a specific location is analysed in this paper, the aim is to obtain distributions

that represent the standardized characteristic load intensities for Hungary according to the notes

and instructions of the EN standards, namely EC0, Eurocode 1 Part 1-3 (EC1-1-3) [93] and

Eurocode 1 Part 1-4 (EC1-1-4) [94].

The determination of distribution function of the wind loads based on statistical data can be

seen in Fig. 4-8. The characteristic value of variable actions on buildings is defined as a value that

has 0.02 exceedance probability within 1 year reference period [52]. Firstly, the yearly maximum

wind velocities are selected for each grid (the data set contained data from 50 years). Using annual

maximums, extreme distribution is fitted on the data in order to find the wind speed which has


44

exactly 0.02 annual exceedance probability. The basic wind velocity in Hungary is vb0=23.6 m/s,

so a node is selected which results the same velocity as characteristic value (Fig. 4-8a). Daily

maximum wind velocities of 50 years related to the selected node (Fig. 4-8b) are used in calculation

of the distribution. As the best fit, lognormal distribution is selected to describe the variability in

the daily maximum wind velocities (Fig. 4-8c).

Fig. 4-8 – Evaluation of the distribution of daily maximum wind speeds: a) EN conforming characteristic wind

speeds in Hungary; b) daily maximum wind speeds for 50 years at the selected coordinate; c) fitted distribution.

According to the recommendations of JCSS Probabilistic Model Code [53], uncertainties are

considered in gust (cg), pressure (cp) and roughness coefficients (cr). Further details can be found

in [BT14] and in EC1-1-4.

In case of the snow loads, similar procedure is carried out in order to obtain the distribution of

daily maximum values that fitted to the standardized characteristic load according to EC1-1-3. It

has to be noted that the daily maximums are not independent, however, the application of yearly

maximum’s distribution is clearly too conservative. The representation of meteorological loads as

stochastic processes would be the most accurate solution, but it would overcomplicate the

reliability analysis. Calculations showed that application of daily maximums served internal forces

in better agreement with internal forces calculated using the load combination of EC0 standard for

extreme design situations. For this reason, this method assumed to lead EN conforming design.

The problem should to be further divided into two fundamental cases, since in Hungary there

is no snow in a significant part of the year. Two independent reliability analyses have to be carried

out with and without the consideration of snow load in the analysis. The calculated reliabilities can

be summed easily if we assume that the ignition and the meteorological loads are independent:

( ) ( )

( ) flashoverswflashoverfsw

flashoverwflashoverfwf

PPP

...PPPP

⋅⋅+

+⋅⋅=

++ x

xx, (23)

0 4 8 12 16 20 24Daily maximum wind velocities [m/s]

17.518

18.519

19.520

20.521

21.522

22.523

46

46.5

47

47.5

48

48.5

5

10

15

20

25

30

Days

a)

b)

c)


45

In Eq. (23), Pw is the probability that only wind load acts on the frame and there is no snow load

because the temperature is too high, while Pw+s is the probability that wind and snow loads act on

the frame at the same time. Pw and Pw+s can be derived from the meteorological data sets.

The given description of derivation and consideration of meteorological loads and their

distribution within the reliability analysis is applied in order to consider representative

meteorological loads which are consistent with standardized reliability level. No correlation is

considered between the snow and wind loads since the data are related to different sites and snow

water equivalents in CARPATCLIM [92] are predicted with numerical models and not measured.

This section does not discuss all of the aspects of selection and consideration of random

variables. More detailed discussion can be found in [79], in [BT13] and in [BT14].

4.7. Structural analysis

Type of structural analysis and numerical model is an important issue from the point-of-view

of reliability analysis of structures subjected to fire effects mainly because of the high nonlinearity

in the thermal effects and in the structural response. The determination of the internal forces using

one simple static analysis and a linear elastic structural model may not be reliable enough.

In parallel with the increasing the steel temperature, the strength and stiffness of steel are

decreasing (EC3-1-2), which causes a complex, nonlinear structural response under fire exposure.

The heating of elements is clearly nonlinear, furthermore due to the differences in heating intensity

and stiffness of different elements the internal forces redistribute during the heating. Time-step

analysis is required to follow load-history-dependent response using a structural model that can

represent the change in the stiffness during the analysis in case of elements, plates and fibres, as

well.

In this research, an open source finite element code, namely OpenSees [95], is used with its

OpenSeesThermal extension [96] analysing the nonlinear structural behaviour on elevated

temperatures (by considering the temperature-dependent properties of steel material and the

connections).


46

Fig. 4-9 – Validation of OpenSees model using experimental data of portal frames [98]

A two-dimensional structural model (Fig. 4-10a) is developed in OpenSees using

nonlinearBeamColumn beam elements with fiber sections and Steel01 bilinear material model

[97], considering equivalent geometrical imperfections (that involve the effect of the geometrical

imperfections, structural imperfections, residual stresses, variation of the yield strength) according

to EC3-1-1. Based on the results of an earlier comparison with experimental data from [98], it can

be stated that the developed model is able represent realistically the nonlinear response of a frame

exposed to elevated temperatures (Fig. 4-9). The second frame from the system is modelled only

with its loading conditions assuming that purlins are constructed as continuous beams. In order to

represent well the response of a tapered frame, the column and beam elements are divided in the

analysis into 10 and 20 smaller elements, respectively. Geometrically and Materially Nonlinear

Imperfect (GMNI) analysis is carried out on the developed model. Using the calculated steel

temperature for every sub-element, the analysis can follow the change in the properties of the steel

material according to EC3-1-2 and the development of additional internal forces from the

constrained thermal expansion.

Fig. 4-10 – a) The imperfect 2D structural model in OpenSees; b) Bilinear material models for connections

0 0.02 0.04 0.06 0.08 0.1 0.120

0.5

1

1.5

2

2.5

3

3.5

4x 10

8

φ [rad]

My

[Nm

m]

Column base conn. T=20°C

Column base conn. T=500°C

Column-beam conn. T=20°C

Column-beam conn. T=500°C

Ridge connection T=20°C

Ridge connection T=500°C

b)

2D stuctural model in OpenSees: 2 x 10 + 2 x 20 = 60 ForceBeamColumn Elements

ec eb

ϕ

mm.

.h.

hec 92200

5900525900003090

200

52≅

⋅+⋅=+⋅=φ

mm.L

eb 5472002

19000

2002=

⋅=

⋅=

h

a)


47

The properties of the connections [73] are not design variables in the analysis, their stiffness,

strength and nonlinear behaviour (Fig. 4-10b) is modelled with nonlinear spring elements.

Connection properties are changed dynamically in the finite element analysis according to the

calculated temperature of the connections.

The temperature of the elements is initial parameter for the finite element calculation. Since the

sheeting system is supported by thin-walled purlins, it is assumed that the elements of the frame

are heated from four sides. While an iterative algorithm is given in EC3-1-2 to calculate the steel

temperatures in case of unprotected and protected steel sections, the use of standardized closed

formulae may be difficult because physical properties of intumescent paint is not known. The

everyday practice selects the appropriate thickness from the design sheets only based on the critical

temperature and the section modulus (the data are given based on furnace tests according to [70]).

Thus, no closed formula exists to calculate the temperatures of a steel plate when it is protected by

intumescent coating. In this study, the iterative algorithm of EC3-1-2 is adopted in the algorithm

[BT13] and the necessary so-called equivalent thermal resistant [99] is calculated based on ECCS

(European Conventions for Constructional Steelwork) recommendations [100] and on data given

in the design sheet [69], that gives the required thickness wherewith after 30, 45 or 60 minutes the

steel temperature reaches exactly the given critical temperature. The intumescent coating starts to

expand on cca. Tgas=200C° compartment temperature [60], thus the equivalent thermal resistance

of the coating is negligible until the temperature reaches Tgas=200C°. This issue has been accounted

by the development of the algorithm. Using the suggestion from the ECCS, the following formula

can be derived for equivalent thermal conductivity:

V

A.d..d,

V

A015702121434629 −+=

λ

mK

W, (24)

where λ, d and A/V are the equivalent thermal conductivity, the thickness of intumescent coating

and the section factor of the cross section, respectively. The thickness of the intumescent coating

and the section factor has to be substituted in mm and 1/m unit. For further details, see [BT13] and

[BT10]. In case of the investigated protection material, the applicability of linear approximation,

for the equivalent conductivity is confirmed by [BT13].


48

4.8. Verification of the elements

The temperature effects are considered in the analysis and in the verification procedure and

summed with the gravity and meteorological forces. The considered failure modes are the

following according to the regulations of EC3-1-1, EC3-1-2 and EC3-1-5 [101]:

a) strength and stability failure of beam and column elements;

b) shear buckling of the web plates;

c) strength failure of the connections [73];

d) plastic sway mechanism.

In conformity with the fact that the listed failure components composes a series system, failure

mechanism associated with the lowest resistance time is selected for the basis of the evaluation of

the limit state function (Annex B).

The stability verification of beam and column elements is carried using the so-called General

Method (GM) from EC3-1-1 (Fig. 4-5), where the in-plane stability failure is considered using

imperfect structural model (Fig. 4-10), while the out-of-plane stability failure is taken into

consideration with reduction factors.

y,yy

y

y,y fkW

M

fkA

N

⋅⋅+

⋅⋅=

θθ

η , (25)

ηα θ

1=,k,ult , (26)

111

−

+=

θθθ αα

α,LTB,cr,FB,cr

,op,cr , (27)

θ

θθ

αα

λ,op,cr

,k,ult,op = , (28)

2

1

2

1 22θθθ

θθθθ

θ

λλαΦ

λλαΦ ,op,op

,LTB

,op,op

,FB ;++

=++

= , (29)

2222

11

θθθθθθ λΦΦχ

λΦΦχ

,op,LTB,LTB

fi,LTB

,op,FB,FB

fi,FB;

−+=

−+= , (30)

θ

θθ

α

αλ

,op,cr

,k,ult,op = , (31)


49

( )fi,LTBfi,FBfi,op ;min χχχ = , (32)

01.fi,op

≤χη

, (33)

where ky,θ is the reduction factors related to yield strength, χop,fi and αult,k,θ are the reduction factor

taking into account the out-of-plane stability failure and the minimum load multiplier in order to

reach the characteristic resistance of the critical cross section, respectively. FB and LTB

abbreviations refer to flexural and lateral torsional buckling failure modes. αcr,op,θ is the critical load

amplifier in order to reach the critical intensity of internal forces causing out-of-plane flexural

buckling or lateral torsional buckling, respectively.

The buckling length according to in-plane buckling based on which the equivalent imperfection

has been selected for nonlinear structural model 2.5 times column height (Fig. 4-10) due to the

sway frame behaviour, finite stiffness of the connected beam and the pinned column base

connection. The columns are restrained against torsion and buckling at the middle of the eave

height, thus the reduction factor for out-of-plane lateral buckling and lateral torsional buckling is

selected based on 0.5 times column height buckling length. The critical load amplifiers for different

stability failures are combined with Dunkerley theorem [117], that can be used as an approximate

superposition technic of critical load amplifiers related to different loading conditions. The

decrease of the stiffness and strength has a significant unfavourable effect on resistance of the

elements that has to be considered by the classification of the sections and by the strength and

stability verifications (Appendix B.2) in every evaluation of the cross section resistances.

The failure of the purlins and sheeting is not considered in this study. Although they lose their

load bearing capacity very early because of the thin wall and high section factor, but due to the

relatively low loading and possible catenary action [102], similarly to composite floor slabs [103],

they may not fall down. The serious deformation and damage of sheeting seems unavoidable in

fire design situation.


50

5. Reliability evaluation of frames under seismic

excitation

5.1. Overview from the methodology and limit state function

In both moderate and high seismicity areas, the seismic resistant design is an important issue

not just because of the life safety requirement, but also in order to avoid significant losses caused

by a seismic excitation. This part of the thesis adopts the earlier introduced concepts and focuses

on seismic reliability calculation of low-rise industrial halls constructed with steel portal frames

(Fig. 1-1). The applied reliability analysis method is FORM similarly to the fire design situation.

Seismic reliability calculation of tapered steel portal frames is not extensively studied. In most

cases the researchers analyse multi-storey MRFs, e.g. in [104] and in [43]. However, it is possible

that the seismic effects become the leading action in higher seismicity areas even in case of a light-

weight structure. The original layout of the model frame (Fig. 1-1) is investigated within the

framework of a parametric study where a comparison had been made between the resultant seismic

and wind forces (Fig. 5-1, where qe is the equipment load and qb is the wind pressure). The results

showed that in case of higher gravitational loads, the seismic forces may become leading action in

some regions of Hungary. This is even true in the surrounding areas (Fig. 5-1) or in case of high

concentrated loading. It does not necessarily mean that seismic design situation becomes the

leading design situation in transversal direction (it depends on the size, the geometry and the

loading conditions of the facility), because significant bending moments may occur in persistent

design situation, as well. Red points in Fig. 5-1 only indicate the possible sites where it is very

likely that seismic action becomes the leading effect. This is more likely when significant

concentrated loads (reaction forces of a slab, crane, etc.) act on the column because these loads

considerably increase the seismic mass and the base shear forces.

The optimal design of tapered portal frame structures subjected to seismic effects has not been

studied earlier, however as it is shown in Fig. 5-1, the seismic effects may become leading effects

compared with wind effects in moderate and high seismicity areas when the seismic mass is high.


51

Fig. 5-1 – Major results of a parametric study on the dominancy of seismic loads on wind loads in Hungary (seismic

force is dominant: green – in longitudinal direction; red – in transversal direction) based on EFEHR PGA data,

Type I response spectrum of EC8-1 and soil type C

In seismic reliability analysis, similarly to the fire design, the structural reliability is calculated

considering the structure as a system and it is not calculated for separated elements. The structural

response is calculated on 3D model in OpenSees [95] FE software. In this section only the main

concept of the reliability assessment methodology is presented and demonstrated on tapered portal

frames, further problem specific details are discussed in the further sections. The advantage of the

presented formulation and methodology compared to the earlier method in [BT4] is that this

method does not apply unreliable methodology to account the uncertainties (e.g. rotation of

fragility curves that is a common technic among many researchers nowadays); different

uncertainties are directly taken into account within the reliability analysis. The known and

important issue is that the solution is very sensitive for the treatment of uncertainties is concluded

and showed in [BT4] as well.

As a first step, seismic hazard analysis (SHA) is carried out based on EFEHR site specific

hazard curves in order to find the distribution of peak ground acceleration and the uncertainties in

seismicity (Fig. 5-2). Based on the fact that the structure mainly vibrates in the first mode, the

14 16 18 20 22 2445

46

47

48

49

50a)

14 16 18 20 22 2445

46

47

48

49

50b)

14 16 18 20 22 2445

46

47

48

49

50c)

14 16 18 20 22 2445

46

47

48

49

50d)

14 16 18 20 22 2445

46

47

48

49

50e)

14 16 18 20 22 2445

46

47

48

49

50f)

Longitude

Lat

itu

de

c) qe=1.5 kN/m2, qb=0.5 kN/m

2

e) qe=1.0 kN/m2, qb=1.0 kN/m

2

b) qe=1.0 kN/m2, qb=0.5 kN/m

2

d) qe=0.2 kN/m2, qb=1.0 kN/m

2

f) qe=1.5 kN/m2, qb=1.0 kN/m

2

a) qe=0.2 kN/m2, qb=0.5 kN/m2


52

numerical analysis is carried out in both principal directions with a load distribution according to

the first vibration mode and the distribution of seismic mass. The mass is coming from the weight

of the structure, the sheeting system, the equipment and from the additional reaction forces on the

columns. Due to the relatively precise construction of steel structures in Europe and the type of the

structure, the 5% accidental eccentricity according to EC8-1 is neglected in the analysis.

Fig. 5-2 – Overview from the methodology and the formulation of limit state function

Preliminary results showed (Section 6.3) that significant differences may be observed between

seismic reliability index of elastic and dissipative structural solutions, thus proper selection

between elastic and dissipative models is an important issue. In transversal direction the structure


53

is very sensitive to stability failure modes, thus significant energy dissipation may not be realized

even in the case of a site characterized with high seismicity, as it is shown in [79] through analysis

results of the investigated portal frame. However, in order to get realistic reliability indices the fact

that the structure is able to absorb seismic energy through the failure of tension-only braces is need

to be considered. It has to be noted that this assumption can be made only if tension failure of the

braces is ductile and it is the leading failure mode (i.e. connections are full strength). The

connections of the investigated frames in this study are assumed to be full strength and the

structural behaviour in longitudinal direction is assumed to be ductile.

Structure’s regularity in plan (EC8-1) (except that the in-plane stiffness of the roof may not be

sufficiently large in transversal direction) and the fact that the primary seismic elements are

different in transversal- and in longitudinal directions make the behaviour in perpendicular

directions separable. This differentiation can be made because the longitudinal behaviour has

negligible effect on the transversal response and negligible internal forces are developed from

longitudinal excitation in the main frame elements. As it is shown in Fig. 5-2, in the limit state

function (Annex C) linear elastic- and nonlinear static analyses are performed in transversal- and

in longitudinal directions, respectively. The analyses are completed on a 3D imperfect numerical

model (Section 5.4). Internal forces, displacements and deformations calculated in different

directions are combined with square root of the sum- of the squares (SRSS) combination rule of

EC8-1. Limit state function related to tension-only braces is formulated on deformation basis where

the deformation limits are selected on the basis of the provisions of [44] (Section 5.7).

5.2. Seismic effects and hazard curves

Hazard curve (Fig. 5-3) is used to characterize the seismicity of a site, it gives the probability

of exceedance of different PGAs for a given reference period. According to the guidelines of EC8-

1, PGA is selected for ULS design satisfying life safety criterion having 0.1 exceedance probability

for 50 years (Fig. 5-3). Hazard curve is a discrete cumulative distribution function that is calculated

with the help of probabilistic seismic hazard analysis (PSHA) [105] considering data of historical

earthquakes from area, line and point sources situated around the investigated site.

The estimation of hazard curve is uncertain due the fact that there are many possible attenuation

and ground motion prediction models in the literature. In EFEHR database [106] one can select

hazard curves representing different percentiles (5%, mean, 95%, etc.) (Fig. 5-3). This type of


54

uncertainty is not covered in the present design procedure, since engineers typically use mean

hazard curve (e.g. in Hungarian National Annex of EC8-1 offers peak ground acceleration values

read from mean curves) or hazard curves that are obtained using a given set of logic tree weights

within PSHA. This uncertainty is considered in the present research (Fig. 5-2) as it is described in

Section 5.3.

Fig. 5-3 – Different hazard curves [106]

The surroundings of Komárom city is characterized by the highest seismicity in our country

(Fig. 5-4), thus this site is one of the two investigated sites in this thesis. The second site is the city

of Râmnicu Sărat (situated in Romania, east from Transylvania) characterized by the highest

seismicity in our neighbourhood (the middle of the red zone in Romania). In this research, Type I

response spectrum of EC8-1 with soil type C is considered.

Fig. 5-4 – Mean seismic hazard

map of Europe [106]


55

5.3. Random variables

Table 5-1 summarizes the random variables considered in the reliability calculation. Most of

the random variables are associated with distribution types and distribution parameters derived

from literature review. The global geometry is not considered uncertain since it has only negligible

effect on the global seismic response and the internal forces. Due to the fact that seismic action is

very rare and the intensive part of the excitation is short, according to the regulations of EC0 by

combination of loads the meteorological loads are neglected in the reliability analysis in seismic

design situation. The uncertainties in the global stiffness are summarized into three random

variables. In transverse direction the uncertainty in the stiffness of bracing system and sheeting and

the uncertainty caused by connections, frames and foundations are separated. The effect of the

spatial variation in stiffness of bracing elements, sheeting and frames has been analysed. The results

confirmed that the applied uncertainty factors covered this issue [79]. ρ=0.7 correlation is

considered among the section modulus factors.

The sensitivity of the reliability problem on different random variables is investigated for

Komárom site with the help of three different example structures, in order to have the sensitivity

factors for a problem where the beam failure, the side bracing failure and the column failure is the

leading failure mode, respectively. The sensitivity factors of the most influential random variables

are marked with thick black frame in Table 5-1. Not surprisingly, the uncertainty in seismic effects

governs the reliability problem.

Uncertainties in the seismic action are given by two random variables. In order to determine

the distribution parameters, seismic hazard analysis is completed for the given sites, using the

database of European Facility for Earthquake Hazard and Risk (EFEHR). The uncertainty in peak

ground acceleration is obtained by fitting a continuous cumulative distribution function on discrete

points of mean EFEHR hazard curve. Lognormal distribution seemed to be the best fit [79], its

parameters can be found in Table 5-1 for both sites.

It can be seen from the calculated parameters that the seismic action is characterized by high

degree of uncertainty. In case of the Hungarian site the uncertainty is higher due to the fact that

historical seismic data are available for many but less dominant sources, while in case areas with

high seismicity typically there are dominant sources. This high uncertainty becomes dominant in

the reliability analysis and increases the deviation of the joint probability distribution, for this


56

reason, it is difficult to design a structure with high reliability against structural failure in seismic

design situation [53].

Random variable μ CoV Distribution Reference Sens. fact. (α)†

Sens. fact. (α)††

Sens. fact.

(α)††† Yield strength [MPa] 388 0.07 Lognormal [53] 0.025 0 0.047

ag [g] Komárom 0.092 1.333

Lognormal Calculation, [106] -0.813 -0.823 -0.808 Râmnicu Sărat

0.168 0.682

Equipment [kN/m2] 0.2/1.0 0.2 Normal -0.038 -0.045 -0.058 Beam section modulus factor 1 0.05 Normal [53] 0.043 0 0

Column section modulus factor 1 0.05 Normal [53] 0 0 0.046 Model uncertainty in LTB reduction fact. -

LTχ 1.15 0.1 Normal [54] 0.076 0 0.080

Model uncertainty in FB reduction factor - Zχ 1.15 0.1 Normal [54] 0 0 0

Global stiffness factor in transverse direction – sheeting and bracing

1 0.2 Normal [79] 0 0 0

Global stiffness factor in transverse direction – other

1 0.1 Normal [79] -0.032 0 -0.02

Global stiffness factor in longitudinal direction 1 0.25 Normal [79] 0 -0.063 0 Effect model uncertainty 1 0.1 Lognormal [79] [53] -0.090 0 -0.095

Resistance model uncertainty 1 0.2 Lognormal 0.161 0 0.169 Model uncertainty – connection resistance 1.25 0.15 Lognormal [53] 0 0 0 Model uncertainty – wind (roof) bracing

resistance 1.2 0.1 Lognormal [54] 0 0 0

Model uncertainty – side (wall) bracing resistance 1.2 0.1 Lognormal [54] 0 0.079 0

Intensity and record-to-record uncertainty

Komárom 1 0.762 Lognormal

Calculation, [106], [107]

-0.542 -0.552 -0.542 Râmnicu Sărat

1 0.492

Deformation capacity factor for side braces 1 0.1 Lognormal 0 0.074 0

Sensitivity factor: negative normalized gradients of the limit state function at MPP; † Beam failure is the leading failure mode; †† Side bracing failure is the leading failure mode; ††† Column failure is the leading failure mode.

Table 5-1 – Random variables in seismic design situation

The variation among the hazard curves and the so-called record-to-record uncertainty have to

be also considered in the reliability analysis. Regarding to record-to-record uncertainty, its source

is the high variation among the dynamical actions which have different amplitudes, frequency

contents and dominant periods. FEMA-P695 [107] gives recommendations to consider this type of

uncertainty as values for lognormal standard deviation parameter (βRTR=0.2...0.4 based on the

ductility). Regarding to the calculation of variation in the intensity, hazard curves with different

percentile (5%, 15%, mean, 85% and 95%) are obtained from EFEHR for the same site and

lognormal distribution is fitted on the data. The record-to-record uncertainty and the intensity

uncertainty are summed analytically since both are followed by lognormal distributions, the

resulted uncertainty factor can be found in Table 5-1.

Based on the regulations of EC3-1-8 the beam-to-column connections of the investigated

structure (Fig. 1-1) may be considered as rigid and the column base connection can be considered

as pinned, thus uncertainty is considered only according to the capacity of the connections.


57

Despite the fact that 3D structural model is applied, in order to reduce the complexity of

reliability analysis the correlation among the frames’ self-weight, the yield strength and stiffness

of frames’ material is neglected. It means, that a full correlation is supposed to be among the

properties of frames. There is no doubt that the correlation among the frames’ response is much

higher in case of seismic excitation than in case of fire exposure. There is no spatial variation in

the effect, the frames are subjected to the load at the same time with almost the same intensity.

However, this restriction is clearly an approximation which cannot be examined accurately on the

present level of this research because it would extremely overcomplicate the reliability problem.

5.4. Structural model and analysis

For the calculation of seismic response and internal forces a 3D frame model (Fig. 5-5) is built

in OpenSees [95] FE environment using 12 DOF nonlinear beam-column finite elements

(nonlinearBeamColumn elements), 12 DOF contact spring elements with zero length (connections

– ZeroLength elements), 12 DOF contact spring elements (sheeting – twoNodeLink elements) and

6 DOF linear and nonlinear truss elements (end-wall, solid circular tension-only wind and side

braces) [97].

Fig. 5-5 – 3D structural model in OpenSees

The internal forces in the primary and secondary elements are calculated on imperfect model

with Geometrically Nonlinear Imperfect Analysis (GNIA) in case of elastic and Geometrically and

Y

Z

X

end wall (linear truss)

sheeting (linear twoNodeLink) bracing (nonlinear truss)

columns & beams (nonlinearBeamColumn)

connections (nonlinear ZeroLength)

x

y

z

Kx, Ky


58

Materially Nonlinear Imperfect Analysis (GMNIA) in case of dissipative structure, respectively.

The imperfections are obtained according to EC3-1-1. The analysis is carried out in both principal

directions, the internal forces are combined with SRSS according to EC8-1. Detailed explanation

can be found in [79].

In order to represent well the stiffness of tapered frames, the column and beam elements are

divided into 8 and 4 smaller elements, respectively, based on average of second moment of area.

The properties of the connections are selected on the basis of [79]. In case of the tension-only

elements, three different material models are connected in parallel in order to ensure the

consideration of prestressing and buckling when the bars become compressed elements. The

stiffness of the end wall is represented in the model with the help of truss elements. The shear

stiffness of the wall is calculated using Stressed Skin Design of EC3-1-3 [108].

Side braces are modelled with nonlinear material models with 0.1% post yielding hardening,

while non-dissipative parts that are checked in terms of load bearing capacity are modelled with

materially linear elements. The local z vector of 12 DOF contact spring elements representing the

sheeting system is selected parallel to the normal vector of sheeting panels. Only Kx and Ky elastic

stiffness are taken into consideration in the analysis (Fig. 5-5), Kx is calculated on the basis of the

sum of purlins’ and panels’ cross section area. Nonlinear behaviour is not considered due to the

high uncertainties in the nonlinear cyclic response of thin walled purlins and trapezoidal panels and

due to the lack of reliable and appropriate models related to nonlinear cyclic response.

The investigated frame structure mainly vibrates in the first vibration mode in both principal

directions. For this reason, LFM can be used in transversal direction, where the design spectral

acceleration of EC8-1 [28] related to the first vibration period and the whole mass are used to

calculate the seismic forces. These forces are distributed among the nodes proportionally to the

first vibration shape and to the distribution of seismic masses. The first vibration period and

vibration shape are obtained in transversal and longitudinal directions with eigenvalue analyses

before every seismic analysis within the objective function (Fig. 5-2). In longitudinal direction PA

is appropriate for nonlinear seismic analysis, since the first vibration mode is dominant.

5.5. Verification of structural model

There is lack of available experimental results on test of whole frame structures in the literature.

For this reason, the verification of the tapered frame model is completed using a full scale


59

experimental test of two tapered steel frames from the US [109]. As it is shown in Fig. 5-6, the

developed FE model is able to represent adequately the initial stiffness of a tapered steel portal

frame and the developed plastic hinge at the beginning of the tapered element. The lateral torsional

buckling phenomena is not considered in the structural analysis, since 12 DOF beam elements are

used instead of 14 DOF elements. For this reason, the effect of occurrence of inelastic LTB at cca.

25 cm top displacement cannot be seen in the presented results. LTB is checked with the help of

reduction factor method according to EC3-1-1.

Fig. 5-6 – Site photo from the specimen [109] and frame model’s verification results

In order to check the capability of 3D OpenSees model to represent the stiffness of the global

structural system including sheeting system, a comparison has been made using full scale test

results from HighPerFrame RDI project [79]. The full scale test was completed in the depot of

Rutin Ltd. using two tapered frames [110] with different layouts related to the purlins and sheeting

system. From the large number of test results, three layouts (F-RW: pure frames without sheeting

and bracing; PE-R: frames with sheeting system in roof level with LTP20 and LTP45 trapezoidal

sheeting, respectively) have been selected for comparison and validation purposes. After the

calibration of zeroLength element stiffness parameters, which represent the stiffness of the sheeting

system, 0.5 – 2.5% difference is found between the simulated and measured/calculated global

structural stiffness [79]. The results of this investigation and calibration showed that the developed

model is able to represent accurately the global structural response and stiffness with or without

trapezoidal sheeting system.

5.6. Calculation of target displacements

Tension-only bracing elements are able to dissipate seismic energy, thus in longitudinal

direction nonlinear static analysis (Pushover, PA) is carried out in order to evaluate the nonlinear

structural response. EC8-1 proposes the so-called N2 method [111], evaluating the acting seismic


60

forces based on the capacity curve (it characterizes the connection between the control

displacement and base shear force; Fig. 5-6), elastic spectra and equal displacement rule. The result

of the calculation is the target displacement related to the control node. In this research, the eave

node of the middle frame is selected for control node.

The implemented N2 method is slightly modified and improved comparing to the original

method from the point-of-view of calculation equivalent bilinear representation on the basis of

equal energy concept. In the modified N2 method, the equivalent bilinear approximation is defined

by the modification of yielding plateau and not by the modification of initial stiffness. Recent

research work on Department of Structural Engineering [BT6] pointed out that this approximation

gives better results comparing to nonlinear time-history analysis results.

5.7. Verification of the elements

The following failure modes are considered and checked by the calculation of the objective

function within the reliability analysis:

a) strength and stability failure of beam and column elements;

b) shear buckling of the web plates;

c) strength failure of the connections [73];

d) strength check of tension-only wind bracing elements;

e) deformation check of side braces.

The verification procedure related to the listed failure modes is completed with respect to the

requirements of EC3-1-1 and EC8-1 standards (Annex C). The algorithm is able to calculate the

D/C ratio of the members based on the class of the cross sections [75] considering the effective

width of the plate elements of a Class 4 cross section. Strength and stability verification of column

and beam elements are completed in accordance to General Method of EC3-1-1 [75]. According

to the method, in-plane stability failures are considered via geometrically nonlinear analysis on

imperfect model considering local and global imperfections according to EC3-1-1, while out-of-

plane stability modes are handled by the reduction factor method (Fig. 5-2a):

( )( )

( )( ) yy

y

y fW

M

fA

N

⋅+

⋅=

x

x

x

xη , (34)

ηα

1=k,ult ;

111

−

+=

LTB,crFB,cr

op,cr ααα , (35)


61

op,cr

k,ultop

αα

λ = , (36)

( ) ( )2

201

2

201 22opopLTB

LTB

opopFBFB

.;

. λλαΦ

λλαΦ

+−+=

+−+= , (37)

2222

11

opLTBLTB

LTB

opFBFB

FB ;λΦΦ

χλΦΦ

χ−+

=−+

= , (38)

( )LTBFBop ;min χχχ = , (39)

01.op

≤χη

, (40)

where χop and αult,k are the reduction factor taking into account the out-of-plane stability failure and

the minimum load multiplier in order to reach the characteristic resistance of the critical cross

section, respectively. αcr,op is the critical load amplifier in order to reach the critical intensity of

internal forces causing out-of-plane flexural buckling or lateral torsional buckling, respectively.

The buckling length of the column is selected equal to the half of the column length related to

FB (flexural buckling) for minor axis and related to LTB (lateral torsional buckling) due to the

longitudinal bracing member at the half of the eave height (Fig. 1-1) that restrains the displacement

and torsion of the cross section. The buckling length related to FB for major axis is set 2.5 times

of the column length because of the sway frame construction and the pinned column base

connections. The shape and amplitude of local and global imperfections are selected on the basis

of the discussed buckling lengths. According to the beams, their buckling length is equal to the

distance between the purlins supposing that the sheeting system has enough stiffness to restrain the

out-of-plane displacement of the cross sections connected to the purlins.

The dissipative zones are verified in terms of deformation capacity (ductility). Plastic hinges

are considered in the side bracing elements. The deformation capacity of plastic hinges is selected

on the basis of the provisions of [44]. According to FEMA-356, the deformation limit for brace in

compression is 4Δc (Δc – axial deformation at expected buckling load), while for brace in tension

is 7Δy (Δy – axial deformation at expected tensile yielding load).


62

6. Target reliability in case of extreme effects

6.1. Minor, moderate and large consequences of failure

The consequence of possible failure is the key issue by making a decision on target reliability

level. Different safety levels lead economical design e.g. in case of an agricultural silo, in case of

an office building or in case of a power plant. The referred standards and recommendations (Table

1-1) differentiate the acceptable reliability level based on the consequence that is typically divided

into 3 main classes, namely minor, moderate and large according to JCSS. Related to low

consequence class, quite low reliability indices may be found in Table 1-1 because some buildings

do not worth to be strengthened rather to be rebuilt. The Probabilistic Model Code of Joint

Committee on Structural Safety provides some information (Table 6-1) in order to help the designer

by the selection among consequence classes.

Class ρ† Examples

Class 1 – Minor Consequences

<2 Low or negligible risk; economic consequences are low or

negligible (e.g. agricultural structures, silos, masts)

Class 2 – Moderate Consequences

2 – 5 Medium risk; economic consequences are considerable (e.g.

office buildings, industrial buildings, apartment buildings)

Class 3 – Large Consequences

5 – 10 High risk; economic consequences are significant (e.g. bridges,

theatres, hospitals, high rise buildings)

† the ratio ρ is defined as the ratio between total costs (construction costs including non-structural costs + failure costs) and construction costs

Table 6-1 – Consequence classes [53]

Despite the fact that EC0 and JCSS mainly differentiate based on the function of the building

(which has significant effect on the possible consequences) the investigated structure may be

classified in each classes based on the value of the stored material and the direct/indirect economic

consequences. The classification can be made after the evaluation the possible ratio ρ related to

analysed frame structure (because ρ also depends on the initial cost of the building that may be

significantly higher than the cost of a lightweight frame).

A cost evaluation is made in order to calculate the total cost of the investigated structure based

on the recommendations and cost breakdowns (Fig. 6-1) of (www.steelconstruction.info) and

[112]. The structural and fire protection costs are calculated with the help of the presented formulae

in Section 2.2. The estimated initial cost of the superstructure may vary between Ctot=75,000 € and

Ctot=100,000 € using cost components in Table 1-2. Considering the cost breakdown in Fig. 6-1


63

and other structural but not included cost components, such as foundation, concrete floor, podiums

and platforms, the total constructional cost may reach 150,000 €. Furthermore, without including

contents the cost of non-structural elements may govern the total cost of the facility (Fig. 6-1), thus

the total cost of the structure can certainly reach 150,000 – 300,000 €.

Fig. 6-1 – Cost breakdown of structural steelwork (www.steelconstruction.info) and cost breakdown of office

buildings, hotels and hospitals [112]

Assuming that multi-storey office buildings (Class 2) and bridges (Class 3) are more expensive

(may be 5 – 10 and 10 – 100 times more expensive, respectively) than a lightweight storage

building, the following ρ ratios (Table 6-2) may be evaluated for the investigated structure based

on Table 6-1.

Class ρ† Failure cost range

Class 1 – Minor Consequences 1 – 4 Cf = 0 – 300,000 €

Class 2 – Moderate Consequences 10 – 100 Cf = 1,500,000 – 15,000,000 €

Class 3 – Large Consequences 100 – Cf = 15,000,000 – † the ratio ρ is defined as the ratio between total costs (Ctot + Cf) and construction cost

Table 6-2 – Consequence classes for the investigated structure based on failure cost value

It is important to note that selection of boundaries of consequence classes is clearly based on

assumptions decided by the author and it may be valid only for industrial steel portal frame

structures with similar size and with similar function to the investigated frame. Further

investigation is necessary for better understanding the possible components (and their weights) of

failure cost function and deriving the boundaries more precisely. For the parametric study in case

of both optimal fire and optimal seismic design (Section 7.3 and Section 7.4), optimal solutions are

derived for three different Cf failure cost values, representing large, moderate and minor

consequences, namely for Cf,1= 30 m€, Cf,2 = 3 m€ and Cf,3= 0.3 m€, respectively.


64

6.2. Target reliability estimation in case of fire design situation

The appropriate reliability level of structures exposed to fire can be ensured on different ways,

using active and/or passive fire safety measures. First of all, as the most obvious case, the structure

can be strengthened in order to avoid the failure on high temperature when the strength and stiffness

of the material is considered on a reduced value. Passive protection measures, such as intumescent

coating, can reduce the temperature of the protected elements. The active protection tools (e.g.

alarm and water extinguish system) result safer solutions by decreasing the possibility of ignition

and flashover.

Holickỳ in [64] showed a method for the calculation of optimum reliability on a general

example with a few random variables. In [84] a Bayes belief network is presented related to fire

design, a simple example and the effectiveness of different safety measures are presented. Within

the framework of this research, an investigation defining the target/optimum reliability levels of a

tapered steel frame structure is presented where the cost function is formulated similarly to Eq. (6),

based on [64]:

( ) ( ) fignitionfffLC C.PC.yCxCCy,xPCy,xC 050010210 +⋅+⋅+⋅++⋅= . (41)

Cf, C1 and C2 vary, however reference values are assumed to be as 3 million €, 4,200 € and 27,400

€ (Fig. 6-2, Table 1-2), respectively, for the reference structure. Cf contains direct (e.g. value of

stored material or the construction of a new storage hall) and indirect cost components (e.g. missing

income or malfunction in production). The typical shape of the function is presented in Fig. 2-2.

The analysed structure is the earlier presented (Fig. 1-1) tapered portal frame structure which

is used as a warehouse. The dimensions are selected to be equal to a structure that has been designed

by practicing engineers considering 0.2 kN/m2 equipment load. The cost of the superstructure

(including the sheeting, purlins and bracing elements) is approximated as C0≈57,000 € (using Table

1-2). The frame is protected by intumescent coating; the appropriate thickness of this passive safety

measure is designed on the basis of the section factor and the critical temperature of the element

(Fig. 6-2). The intumescent paint of a Hungarian producer [69] is considered in the design process,

the thicknesses in Fig. 6-2 are calculated for 30 minutes fire resistance.


65

Fig. 6-2 – Calculation details related to

prescriptive fire design [BT11]

The considered three fire design

cases are the same as the considered

design cases within the parametric study

(Fig. 7-6): 1) extreme (the combustible

material is rubber tire); 2) severe (the

combustible material is rubber tire and

wood); 3) moderate (the combustible

material is wood). With different time

demands, namely R30, R45 and R60,

altogether 9 cases are investigated. The probability of failure under fire exposure contains the

probability of occurrence of two independent events, namely the probability of severe fire and the

conditional probability of failure if the severe fire is occurred. Thus the effect of passive and active

safety measures can be easily separated in the calculation since the first probability is dependent

on the active safety measures (Fig. 6-3), while the second probability depends on the amount of

applied passive protection.

Demand\Severity Extreme fire Severe fire Moderate fire

R30 case #1 case #2 case #3



Table 6-3 – Investigated cases

In this investigation the thicknesses of intumescent coating are varied with one amplifier from

zero to tenfold value. The conditional probabilities of structural failure have been calculated using

developed reliability analysis framework (FORM, Fig. 4-1), a small chance of malfunction of

active measures is considered in the network (Fig. 4-4). It has to be noted that in this section,

according to Hungarian regulations [71] [72] a minimum active safety measure, namely automatic

smoke detection system, is selected.

1

b)


66

Fig. 6-3 – Cost and efficiency of different active safety measures including the installation, construction and

maintenance for the service life [BT11]

I investigated in an earlier study [BT11] the problem more deeply, regarding to different active

safety measures using different assumptions regarding to the consideration of meteorological loads

in the reliability analysis, see details in [BT8]. The main conclusions of these studies are the

following: a) active safety measures and passive protection have to be applied in order to satisfy

the criteria of EC0 (Table 1-1) (Fig. 6-4); b) passive protection without active safety measures is

not effective enough when the target reliability is greater than 3.2 – 3.3 (Fig. 6-4); c) the suggested

target reliability indices of JCSS seemed to be more appropriate for practically expectable cases.

Fig. 6-4 – Optimal reliability indices for various consequences [BT11]

The resulted target reliability indices of this investigation are presented in Fig. 6-5 for 9

considered cases and for different Cf/Ctot ratios. This ratio for the reference structure is around 30

considering normal design conditions. Because R60 and R45 time demands are more demanding

than R30; R60, R45 and R30 demand levels are associated with high, moderate and low relative

cost of safety measures, respectively. Using these differentiations, the reliability indices are

0 0.5 1 1.5 210

−3

10−2

10−1

100

Active safety measures [−]

Con

ditio

nal p

roba

bilit

y of

flas

hove

r [−

]

Active safety measure a b c d b+d c+d

Cost [€/m2] 0 25 40 50 75 90Relative cost of safety

measure0 0.5 0.8 1.0 1.5 1.8

1 0.25 0.0625 0.02 0.005 0.00125

ignitionignitionflashover

ignitionflashover

PP

P

⋅

=∩

ignitionflashoverP

Scenario:• a: None of active safety measure is applied• b: Automatic fire detection and alarm by heat• c:Automatic fire detection and alarm by smoke• d: Sprinkler system• b+d: Sprinkler system + detection by heat• c+d: Sprinkler system + detection by smoke

( ) y.IFL e.yP 755317651 −=

100

101

102

103

104

105

106

2.5

3

3.5

4

4.5

Cf/(C1+C2)

Tar

get r

elia

bilit

y in

dex

EN 0 low consequences

JCSS 0 moderate consequences

JCSS 0 minor consequences

JCSS 0 large consequences

EN 0 high consequences

EN 0 medium consequences

Only passive protection Passive protection + active safety measures


67

presented in Table 6-4 (without considering case #3 where the fire is not the leading action)

similarly to Table 1-1 using the boundaries of Table 6-2 to differentiate the consequence classes.

However, more accurate values can be read from Fig. 6-5 for a specific design situation. The

calculated reliability indices are lower than the suggested values in EC0, they are closer to the

recommendations of Joint Committee on Structural Safety [53]. Despite the fact that this

conclusion is derived based on analysis of a single storey portal frame structure, the results may be

also valid for other type of steel structures because wide range of correlation coefficient and wide

range of possible failure consequences are covered. The higher correlation (Fig. 6-5) among the

frames may represent the case of a smaller structure with fewer frames or a structure with smaller

compartments. In case of higher correlation among the frames the failure probability of the system

is lower and higher target indices can be calculated.

Fig. 6-5 – Target reliability indices for various consequences for fire design of example portal frame

The reliability depends on the severity of the fire effect, thus it depends on the function of the

building, the quality and the quantity of the combustible material. It has to be noted that β=2.82

reliability index implies that the structure has almost 1.0 conditional failure probability in fire. In

these cases the fire effect is too severe and the protection and strengthening of the structure may

not be economical. β=2.82 reliability index is a lower bound because the occurrence of flashover

is quite rare in the investigated cases [BT11]. For high Cf/Ctot ratios, the curves are flatter than in

the case of lower Cf/Ctot ratios because the effectiveness of the intumescent painting is not in linear

connection with the layer thickness [BT11]. In Fig. 6-5, from the point where the curves change to

horizontal lines, the curves only indicate the values because the optimum thickness amplifiers are

out of the investigated range in these cases. The proper target reliability index value should be

selected considering the severity of a possible fire event. The application of comparable design

2,8

3

3,2

3,4

3,6

1 10 100 1000

β-

targ

et r

elia

bil

ity

ind

ex

(ρ=

0.4)

Cf /Ctot

1 2 34 5 67 8 9

2,8

3

3,2

3,4

3,6

3,8

1 10 100 1000

β-

targ

et r

elia

bil

ity

ind

ex

(ρ=

0.9)

Cf /Ctot

1 2 34 5 67 8 9


68

curves (e.g. ISO standard fire curve) may lead inconsistent structural reliability. In case of the

investigated example considering both ρ=0.4 and 0.9 correlation coefficients, reliability index

β=2.9 – 3.3 may be achieved based on the time demand R45, R30 and R15, respectively, using

EC3-1-2 conforming prescriptive design and ISO standard fire curve [BT10]. It means that design

based on ISO standard curve is too conservative in case of structures with low failure consequences

and it may be unsafe when a possible failure has considerable consequences. The target reliability

index may be selected between 2.8 and 3.7 based on the possible failure consequences for industrial

steel tapered portal frames with storage function. The presented values are calculated on the basis

of Hungarian circumstances, considering the regulations of OTSZ 5.0 and TvMI 5.1.

The presented values may be used later in performance based design, however, further research

work is needed in order to extend and validate the suggested numbers and in order to understand

better the components of failure costs (Cf). The target reliability indices may be also influenced by

the acceptance ability of the society and global economy of the country, so in some cases minimum

limits may be used in order to ensure the minimum desired safety.

50 years service life: calculated

Relative cost of safety measure Minor


consequences Large

consequences High – Severe fire 2.8 (2.8 – 2.9) 2.8 – 3.0 (2.8– 3.1) 2.8 – 3.2 (2.8 – 3.4)

Moderate – Medium fire 2.8 (2.8 – 3.0) 2.8 – 3.2 (2.8 – 3.3) 2.8 – 3.5 (3.0 – 3.6) Low – Minor fire 2.8 (2.8 – 3.0) 2.8 – 3.3 (2.8 – 3.4) 3.2 – 3.5 (3.2 – 3.7)

Table 6-4 – Calculated target reliability indices for industrial steel tapered portal frame considering ρ=0.4 and

ρ=0.9 correlation among the frames

6.3. Target reliability estimation in case of seismic design situation

Since earlier and recent studies showed that the reliability level is much lower in seismic design

situation than in conventional design situation the question arose that what would be proper target

reliability index in seismic design situation. This thesis provides information about possible target

reliability indices for tapered portal frame structures that are calculated using the presented

reliability analysis algorithm for three reference structures (Table 6-5). C0 and C1 cost values have

been calculated based on values from Table 1-2. The presented structures are verified in persistent

and seismic design situations without considering the stiffness of sheeting system in the calculation.

It has to be noted that ~0.2g peak ground acceleration is considered for Komárom site with C soil

type from EFEHR hazard curve related to 10% exceedance probability in 50 years instead of 0.15g

that is given in the National Annex of EC8-1.


69

# hc1 - hc2 x tw,c + bc x tf,c hb1 - hb2 x tw,b + bb x tf,b db Equipment

load C0 C1

1 300-700x6+200x11 380-700x6+180x8 16 mm 0.2 kN/m2 47,750 € 860 € 2 300-750x6+220x14 380-770x6+180x12 24 mm 1.0 kN/m2 51,370 € 1,940 € 3 400-750x6+260x14 400-770x6+220x14 28 mm 2.0 kN/m2 55,260 € 2,640 €

Table 6-5 – Reference structural configurations

For sake of simplicity, the reliability indices are calculated only in longitudinal direction related

to the failure of side diagonal braces without considering the sheeting system’s stiffness in the

analysis. For dissipative design, the limit state function expresses a deformation check of tension

braces where the allowable deformation is set 7Δy (Δy – axial deformation at expected tensile

yielding load). The nonlinear structural response is obtained with pushover analysis considering

0.01Einit post yielding stiffness and considering second order (also known as P-Δ) effects of the

gravity loads on the columns. N2 method [111] is invoked in order to find the target displacement.

In elastic design case, the limit state function expresses tension verification of braces in tension

supposing full strength connection at the ends.

The calculated probabilities are not conditional values, they contain the occurrence of seismic

action via the hazard curve. The diameter of side bracing is varied from 4 to 84 mm altogether in

six investigations, since the reliability is calculated using both elastic and dissipative limit state

functions (Fig. 6-6). Due the design criteria related to persistent design, tension-only braces with

diameter db<16mm cannot be used since 16mm is minimum that is able to resist the resultant wind

force.

Fig. 6-6 – Failure probabilities and reliability indices of investigated structures for Komárom site as a function of

the bracings’ diameter

Regarding to the tension-only braces, db=16mm, 24mm and 28 mm diameters would be

sufficient according to elastic design rules of EC8-1 in the first, second and third case, respectively.

The presented reliability indices in Fig. 6-6 imply that the EC8-1 conforming elastic design rules

4 12 20 28 36 44 52 60 68 76 840

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Diameter [mm]

β −

rel

iabi

lity

inde

x

4 12 20 28 36 44 52 60 68 76 840

0.1

0.2

0.3

0.4

0.5

0.6

Diameter [mm]

P f − fa

ilure

pro

babi

lity

#1 dissipative#2 dissipative#3 dissipative#1 elastic#2 elastic#3 elastic


70

provided β ≈ 1.5 reliability index for ~1.0 D/C ratio in the investigated cases due to the high

uncertainty in the seismic effects (Table 5-1 and Fig. 5-3). It does not necessarily mean that all of

the structures designed according to the prescriptive design rules of EC8-1 standard have similarly

low reliability. The presented values may be not generalized easily due to the following issues: a)

the indices are valid for the investigated structural configuration considering that the dead loads

are dominant; b) the structure is investigated in longitudinal direction thus its behaviour can be

represented well with a simple SDOF system; c) the failure probabilities are calculated for an

elastic structure neglecting the possibility of energy dissipation by plastic deformation except the

fact that q=1.5 behaviour factor (also known as seismic response modification factor, R, in ASCE7-

10 [115]) is used for the calculation of EC8-1 design spectra. In case of structures where higher

vibration modes become dominant, higher reliability index may be calculated, however, the results

imply the fact that the desired reliability level may not be achieved in some cases using the

prescriptive rules of EC8-1. In case of brittle failure modes, dashed lines (Fig. 6-6) may describe

better the achievable reliability level for seismic design situation. Reinforced concrete structures

are not incorporated in this study, other researchers published low reliability values for shear failure

of reinforced concrete sections, e.g. [113], thus this issue also needs to be further investigated.

In case of dissipative structural configurations where significant plastic deformations are

allowed and the failure mechanism of the structure is controlled and designed, significantly higher

reliability indices may be achieved using EC8-1 conforming design. For example, Zsarnóczay in

[67] presented reliability indices between 2.9 and 3.6 for a large set of BRB frames (designed with

q=7 behaviour factor). The investigated structure may also be capable to absorb energy during

seismic excitation, tension failure of bracing elements with full strength connections can be

considered as a dissipative failure mode. For this reason, the considerable structural reliability

achieved with the help of prescriptive rules of EC8-1 is higher than β ≈ 1.5. Considering tension-

only braces with db=16mm, 24mm and 28 mm diameters, respectively, the resulted reliability

indices vary between 2.3 and 2.5. These reliability indices satisfy the criteria of JCSS (Table 1-1)

suggested for seismic design (high relative cost of safety measure) of a structure having moderate

consequences in case of failure, however, the target indices of EC0 (Table 1-1) may be hardly

achieved.

The above discussed reliability index values (Fig. 6-6) and the following life cycle cost

function, Eq. (42), are applied in order to investigate the optimal safety level in seismic design


71

situation for Komárom site considering structures loaded with low, moderate and high gravitational

and seismic loads (Table 6-5). The cost function is formulated similarly to Eq. (9) but here with

only one variable that refers to a diameter for the braces:

( ) ( ) 210 xCCxPCxC ffLC ⋅++⋅= , (42)

where C1 is the cost factor related to the braces. Different sites are not incorporated in this

investigation, thus for sites having hazard curve characterized with different mean or different

variation the calculated optimal reliability indices may be slightly different than the presented

values in Section 7.4.2. The minimum of the presented life cycle function is found using direct

search algorithm. Some increase or decrease in the seismicity have similar effect on failure

probability than some increase or decrease in the gravitational loads. For this reason, the presented

results may be extended for wider set of possible cases.

The calculated target reliability indices for seismic design are presented in Fig. 6-7 for 6

considered cases (Table 6-5) and with different Cf/Ctot ratios (this ratio for the reference structure

is around 50 considering normal design conditions). It has to be noted that the presented values

may be only valid for steel portal frame structures with similar configuration (tension-only braces

for wind- and side bracing) and size analysed in longitudinal direction.

Fig. 6-7 – Target reliability indices for various consequences for seismic design of example portal frames

The results are summarized in Table 6-6; however, more accurate values can be read from Fig.

6-7 in a specific design situation. It can be concluded that the possible target reliability indices

depend not only on the consequence but also on the seismicity or the severity of the seismic effect

and on the required strengthening cost. In Fig. 6-5, from the point where the curves change to

horizontal lines, they only indicate the values because the optimal diameter amplifiers are out of

1

1,5

2

2,5

3

3,5

4

4,5

1 10 100 1000 10000

β-

targ

et r

elia

bil

ity

ind

ex

Cf /Ctot

elastic 0.2kN/m2

elastic 1.0kN/m2

elastic 2.0kN/m2

dissipative 0.2kN/m2



JCSS (high relative cost)

JCSS (moderate relative cost)

JCSS (low relative cost)

EC0 recommended limits


72

the investigated range (4-84! mm). The presented indices in this section will be compared and

evaluated later together with results of Section 7.4.

Table 6-6 – Calculated target reliability indices in longitudinal direction for industrial steel tapered portal frame

situated next to Komárom - brittle and dissipative failure mechanism

The calculated reliability indices are lower than the suggested values in EC0 and they are closer

to the recommendations of JCSS. Recommended values of Joint Committee on Structural Safety

for low and moderate cost of safety measure seem to be appropriate for low and medium seismic

mass, respectively. If significant concentrated loads act on the columns, higher seismic mass have

to be considered during the design. In these cases target indices may be lower than the calculated

ones and JCSS recommended values for high relative cost of safety measure may be used. Due to

the fact that a wide range of possible consequences, different intensity of gravitational loads, elastic

and dissipative structural configurations have been taken into account by the derivation of

reliability index values; these conclusions may be extended to other type of steel structures, as well.

The EC8-1 standard defines importance classes in order to differentiate structures having

different failure consequences (similarly to the consequence classes in EC0). EC8-1 uses the so-

called importance factor (γI) multiplying the considered PGA in the analysis (it can be interpreted

as the return period of considered seismic event changes). The examples given in the code are

mainly related to structures with different functions, however, higher or lower consequence class

may be selected during the design if the failure causes more or less significant economic losses,

respectively. This differentiation (in better agreement with the discussion in the code) is made and

possible target indices are presented in Table 6-6 based on assumptions made by the author related

to Cf/Ctot boundaries. Importance classes III and IV represent extraordinary cases, importance

classes I and II cover most of the cases in practice for industrial steel portal frames.


73

The investigated configurations having tension-only braces with db=16mm, 24mm and 28 mm

diameters, respectively, have been designed considered importance class II (ordinary structure,

γI=1.0) and elastic design rules of EC8-1. In case of brittle failure modes, where no plastic

behaviour may be developed, the resulted reliability indices varied between 1.2 and 1.8 considering

all the importance classes for each configuration (marked as light red zone in Fig. 6-7). These

values are rather low (lower than the lowest limit of JCSS) and may be not acceptable. In case of

dissipative failure mode the indices varied from 2.3 to 2.7 (marked as light blue zone in Fig. 6-7).

Higher values (β>3) may be reached with design of DCH [28] structures (e.g. BRBF) [BT7]. These

results point another advantage of dissipative design (it is well known that using dissipative design

the cost of bracing system can be significantly reduced), namely that seismic structural reliability

can be remarkably increased. Even in the case of a simple industrial portal frame designed

elastically against seismic effects, it is favourable to give the opportunity to development of plastic

deformations in order to avoid brittle failure modes (e.g. design full strength brace’s connections

in order to let yield tension failure being the leading failure mode and avoid shear failure of bolts

with some overstrength).

It seems that the target indices suggested by EC0 [52] are too high and not appropriate in

seismic design situation in case of industrial portal frames having similar function and geometry

to the investigated frame. Although this conclusion has been drawn based on results in longitudinal

direction (for Komárom site); similar or lower eliability indices may be calculated in transversal

direction since the cost of improving structural capacity is higher. Further investigation is necessary

concentrating on target reliability indices for seismic design in order to extend the validity of

conclusions for different kinds of structures, failure modes and sites. Parametric study results will

help to extend the conclusions for wider range in Section 7.4.2. For the time being, β=2.0-3.5 seems

to be economical and optimal for most of the practical cases. This target value may be satisfied

without any problems if the designer makes an effort on overstrength brittle failure components

and ensure dissipative failure modes.

The resulted reliability index considering elastic limit state (brittle) are too low for a structure

designed elastically according to the recent regulation, namely EC8-1. It seems that the rules need

to be revised in the future. At the moment, from the point of view of the seismic effects EC8-1

introduces a safety factor in the design procedure since the considerable peak ground acceleration

has 10% probability of exceedance in 50 years (0.9 fractile of 50 years mean hazard curve, as it


74

can be seen in Fig. 5-3). However, this 10% exceedance probability results only β ≈ 1.3 reliability

index for 50 years and the fact that resistance is accounted on design value by the verification

cannot increase the structural reliability significantly in case of the investigated steel structure since

the limit state function is dominantly sensitive for seismic uncertainties (Table 5-1). In spite of the

fact that the review of the rules in the code is not one of the scopes of this study, one possible issue

may be the neglecting the uncertainty of hazard level (Fig. 6-8) in code conforming design (the

input parameter related to seismic effect intensity is typically the 0.9 fractile of 50 years mean

hazard curve). It is found that if the desired safety level were β ≈ 2.0 reliability index considering

elastic limit state function, the 0.95 fractile hazard curve should be used for design purposes in

order to cover the uncertainty in the intensity because there is a considerable uncertainty related to

the usage of different attenuation and ground motion prediction models.

Fig. 6-8 –Hazard curves with different fractiles for Komárom site [106]


75

7. Reliability based structural optimization results

The aim of this research is to define new and valuable concepts for seismic and fire design of

tapered portal frame structures based on the results of structural optimization procedure. In order

to give comprehensive and useful concepts, it is important to characterize the sensitivity of the

design problem and the optimum on different design parameters, conditions and cost components.

For this reason, the achievable optimal solutions are derived in several cases, within the framework

of parametric studies.

7.1. Optimized variables

Similarly to a design problem in practice, the selected optimization variables are discrete thus

the value of the presented objective functions (Section 2.2 and 2.3) are evaluated only in discrete

points of the search domain. In case of fire design optimization, the variables are the dimensions

of the main frame elements and the thicknesses of intumescent coating (Table 7-1), while the

dimensions of the main frame elements and the diameter of tension-only bracing elements (Table

7-2) are selected as optimized variables in seismic design optimization.

Column Beam tw,c column web thickness tw,b beam web thickness tf,c column flange thickness tf,b beam flange thickness bc column flange width bb beam flange width hc1 column height at the base hb1 the height of non-tapered beam hc2 column height at eave hb2 beam height at the end of the tapered part tp,c1 intumescent coating thickness on the lower part tp,b1 on the non-tapered part of the beam tp,c2 intumescent coating thickness on the upper part tp,b2 on the tapered part of the beam tp,c intumescent coating thickness in the connection zone

Table 7-1 – Optimization variables in fire design situation

Column Beam tw,c column web thickness tw,b beam web thickness tf,c column flange thickness tf,b beam flange thickness bc column flange width bb beam flange width hc1 column height at the base hb1 the height of non-tapered beam hc2 column height at the eave hb2 beam height at the end of the tapered part Bracing db diameter of tension-only side and wind bracing

elements

Table 7-2 – Optimization variables in seismic design situation


76

7.2. Convergence and performance of the optimization algorithm

7.2.1. Settings of the algorithm

The convergence and capability to find the optimal solution of heuristic algorithms is highly

influenced by proper selection of their parameters. In case of GA these parameters are the mutation

ratio, crossover ratio, number of mutated genes and elite ratio. An investigation on adjustment of

GA parameters for building structure optimization is presented in [BT2] where various

combinations of parameters are evaluated with respect to convergence rate, stability and

computational demands. The parametric study was completed on an example four-storey building,

with fixed bracing layout.

In order to find the best settings with reasonable resource needs (an evaluation of the objective

function takes cca. 50-60 seconds by fire- and 90-100 seconds by seismic design optimization that

makes the optimization procedure very demanding), the developed algorithm is further tested

within the framework of a sensitivity analysis. Due to the similarity in degree of

nonlinearity/complexity, in the objective functions and in the number and type of design variables,

the sensitivity analysis is carried out only using structural fire design optimization algorithm. The

best settings is used in case of seismic design optimization, as well.

The convergence of the best set is shown in Fig. 7-1 and in Table 7-3, where the results of

altogether 11 optimization processes are presented. The algorithm serves consistent results from

engineering point-of-view, little scatter appears in the results due to the fact that the problem is

extremely nonlinear and the applied population size need to be limited (with the last settings within

an optimization process altogether 4900 structures are investigated that increases the computation

time to 70-80 hours). In case of hc2, hb1, hb2, bc, and bb the observed standard deviation is 2.8-6.7%,

which is acceptable from practical reasons and does not mean any difference from designer point

of view between the solutions. The column base connection can be considered pinned, thus in case

of hc1 12.5% standard deviation is obtained because this parameter does not have significant

influence on the internal forces and stiffness.


77

Fig. 7-1 – Convergence of the developed reliability based optimization algorithm

As the final setting, crossover ratio, mutation ratio, number of mutated genes and elite ratio are

set to 0.4, 1.0, 2 and 0.2, respectively. In order to reduce the computational time, the population

size is changed dynamically where this parameter set to 200, 40, 20 and 10 in 0-20, 21-30, 31-40

and 41-50 iteration steps, respectively. When the population size is reduced, the best 40, 20 or 10

candidates are kept for further analysis.

[mm] hc1 hc2 hb1 hb2 bc bb tf,c tf,b tw,c tw,b tp,c1 tp,c2 tp,b2 tp,b1 tp,c

ME

AN

=979

1933

ST

D=

0.48

5%

235 665 230 680 195 180 10 9 6 6 0.4 0.4 0.6 0.3 0.2 195 670 230 670 195 185 10 9 6 6 0.4 0.4 0.6 0.3 0 235 640 230 700 200 190 10 8 6 6 0.4 0.4 0.6 0.4 0 240 700 250 725 185 170 11 9 6 6 0.4 0.5 0.6 0.3 0 300 595 245 715 200 175 11 9 6 6 0.4 0.5 0.6 0.4 0 245 620 225 640 195 175 11 10 6 6 0.5 0.5 0.6 0.3 0 195 625 225 665 190 180 11 9 6 6 0.4 0.5 0.6 0.4 0 215 590 225 705 190 175 12 9 6 6 0.4 0.5 0.6 0.4 0 220 605 240 675 205 180 10 9 6 6 0.5 0.5 0.7 0.4 0.1 255 665 215 740 195 170 10 9 6 6 0.5 0.5 0.7 0.4 0 245 725 200 695 195 165 9 10 6 6 0.5 0.6 0.6 0.4 0.1

Table 7-3 – Results of the sensitivity analysis

The performance and convergence of the developed optimization framework is compared to

the performance of Particle Swarm Optimization (PSO) heuristic algorithm [32]. The optimization

is performed considering only conventional effects (dead loads and meteorological loads). This

optimization problem with the same number of variables is also characterized discrete and highly

nonlinear nature. Matlab [119] PSO algorithm is selected for comparison with default settings, no

preliminary sensitivity analysis has been made.

0 5 10 15 20 25 30 35 40 45 500.9

1

1.1

1.2x 10

7

Iteration

Life

cyc

le c

ost [

HU

F]


78

Fig. 7-2 – Comparison of performance with MATLAB PSO algorithm

The results of the comparative study are shown in Fig. 7-2 where objective function is the

penalized (Section 2.4) cost of steel superstructure without the cost of passive or active fire

protection measures. It can be seen that both optimization algorithm is characterized with good

convergence rate, the difference in objective function value is lower than 0.5%, GA based

optimization framework achieved slightly better solution, however, from practical point-of-view,

there is no difference between the structural configurations obtained with GA (critical column

section: 710x6+185x10; beam section at the middle: 210x6+185x8) and PSO (critical column

section: 715x7+200x8; beam section at the middle: 200x6+185x8).

7.2.2. Shape of the objective function

GA seeks the optimum with the development of the search space during its operation and it

evaluates large number of possible solutions. The operation of GA and the presented objective

function is illustratively shown in cost-risk coordinate system in Fig. 7-3.

Fig. 7-3 – Illustration of operation of GA and the shape of objective function


79

Fig. 7-4 – Objective function values for different column flange thickness, protection thickness and column flange

width

In fact, the objective function can be hardly illustrated due to the number of optimization

variables. Only sections of the objective function can be illustrated for one or two selected

variables. Fig. 7-4 shows the shape of the objective function close to the optimal solution for fire

design situation. Case #10 (column: 130-720x6+195x11; beam: 285-730x6+170x10; tp,c1 = 0.5

mm; tp,c2 = 0.6 mm; tp,b1 = 0.8 mm; tp,b2 = 0.4 mm; tp,c = 0.2 mm) is selected as reference case from

Table A-2; the thickness of column flange (tf,c = 8 – 13 mm), the width of column flange (bc = 180

– 240 mm) and the thickness of intumescent coating (tp,c2 = 0.3 – 1.2 mm) are varied due to the fact

that stability failure of column governs the failure of the frame in the reliability analysis.


80

Fig. 7-5 – Objective function values as a function of column flange thickness (with constant protection thickness

tp,c2= 0.6 mm) and protection thickness (with constant column flange thickness tf,c= 11mm)

In Fig. 7-5, the objective function values are illustrated for different column flange thicknesses

and protection thicknesses. It can be seen that the original optimized configuration results the

lowest objective function value with ~92889 EUR. The figures show well the shape of objective

function and the fact that the developed optimization framework is able to find optimal solution

with regard to many parameters. Despite the fact that the global optimum cannot be proven due to

the huge search space, with good settings the developed optimization framework can find solutions

with no difference compared to the global optimum from practical point-of-view.

7.3. Optimal solutions in fire design situation

7.3.1. Parametric study

Table A-1 (in Appendix A) summarizes the investigated cases. Altogether, the optimal

solutions have been successfully obtained in 33 different cases from 36 which cover a wide range

of possible design cases. The cost components are essential issue of the solution, the listed costs

have been obtained with the consideration of Hungarian circumstances based on consultations with

practicing engineers. The cost of sheeting and bracings Csh (Eq. (6)) is generally set to 25 €/m2

(except group F where Csh is set to 50 €/m2). The time demand, the value of cost components, the

application of active fire protection, the severity of fire effect and equipment load are varied in this

study. Some other parameters like the meteorological loads, the type and the weight of sheeting

system and the main geometry remained to be unchanged. It has to be noted that in case of industrial

8 9 10 11 12 13flange thickness [mm]

0.9

0.95

1

1.05

1.1

1.15

1.2105

bc = 180mm

bc = 195mm

bc = 210mm

bc = 225mm

bc = 240mm


81

frame structures, according to Hungarian regulations [71], a minimum active safety measure,

namely automatic smoke detection system, is selected for most of the cases.

Fig. 7-6 – Ozone fire curves on design and on mean value

Within the framework of the presented parametric study the optimal solutions are investigated:

I) in case of different demand levels (A case group – reference cases); II) in case of different

constructional costs and losses (see D, E and F case groups); III) with different active safety

measures (see G and H case groups); IV) without passive fire protection (see C case group); V)

with different gravity load intensities (see B case group). As the most common fire protection in

Hungary, smoke detection device is assumed in most cases as active safety measure. However, the

optimal solutions with only passive protection are also analysed.

From the point-of-view of the severity of fire effect, altogether three different cases are

considered. The fire effect is represented by fire curves (Fig. 7-6) that have been obtained with the

help of two-zone fire model in Ozone software [83]. In Fig. 7-6 the design and mean fire curves

are presented and the ISO standard fire curve is also shown as a reference. The uncertainties in the

steel temperature are considered in the analysis with the help of a global uncertainty factor (Table

4-2) whose parameters and distribution type is obtained in [BT13]. For this reason, the temperature

input shall be the mean fire curve in case of which it was assumed that every point of the design

curve is the 95th percentile of a lognormal distribution with CoV=0.25.

The considered three fire design cases (Fig. 7-6 and Table A-1): 1) extreme (the combustible

material is rubber tyre with qf,d≈470 MJ/m2 design fire load, with 30 MJ/kg combustion heat from

EC1-1-2 and tα=150s fast fire growth rate from EC1-1-2); 2) severe (the combustible material is

rubber tyre and wood with qf,d≈670 MJ/m2 design fire load, with ~24 MJ/kg combustion heat on

average and tα=200 fast fire growth rate); 3) moderate (the combustible material is wood with

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

t [min]

T [

C°]

fire curve #1 (design)

fire curve #1 (mean)





ISO curve (design)


82

qf,d≈1070 MJ/m2 design fire load, with 17.5 MJ/kg combustion heat and tα=300 fast fire growth

rate).

The gas temperatures calculated with Ozone seem very high, however, there is a physical limit

for maximum compartment gas temperature e.g. because of fracture of openings or sheeting. It has

to be noted that these temperatures have been obtained considering design values of influencing

parameters thus these curve can be considered as design curves (Section 4.5) that represent a highly

unlikely event. The maximum measured gas temperatures in compartment fires in the literature are

between 1300 and 1400C°, for example, similar values are measured [114] during Cardington test

where the combustible materials are mainly wood or combination of wood and plastic. The

assumed combustible material in my example is rubber tyre that has considerably higher

combustible heat than wood has. Thus, the possible peak temperature may be higher than 1400C°

and the applied distribution related to the fire curve should be truncated close to that value. In order

to avoid numerical instabilities during HLRF iteration, the truncation is neglected. However, due

to the fact that conditional probabilities in fire calculated with FORM are considerably higher than

failure probabilities e.g. for persistent design situation, the value of random variables (coordinates

in design space) at MPP are much closer to the mean value than in case of reliability calculation in

persistent design situation. For this reason, neglect of truncation does not affect the calculated

reliabilities.

The considered failure costs (Cf = 0.3, 3 and 30 m€) are selected based on the investigation

presented in Section 6.1. The main aim of the standards is the protection of human life. The

consideration of human life in the failure cost is inherently questionable, the target safety level is

highly influenced by the acceptance ability of the society and global economy of the country, thus

in some cases minimum limits may be used in order to ensure the minimum desired safety. In this

investigation, due to the industrial storage function and due to the limited size of the building, the

possible loss of human life is not accounted by the estimation of failure costs, thus the applied Cf

values may express only economic losses.

In order to characterize the importance of proper fire design, the initial cost of optimal solutions

for the above listed 36 cases are also obtained without considering fire design situation during the

design. The life cycle cost and fire risk related to these solutions are obtained thereafter with passive

fire protection selected for Tcr=500-550 °C critical temperature according to the common

Hungarian fire protection design practice.


83

7.3.2. Parametric study results

The optimization results are summarized in Table A-2 (in Appendix A) according to cases of

the parametric study (Table A-1). In the first and second columns, the dimensions of cross sections

can be seen for both column and beam elements, while in the following columns the thicknesses

of intumescent coating fire protection have been presented. The D/C column shows the demand-

to-capacity ratio in ULS of persistent design situation related to the critical failure mode. βopt is the

reliability index related to the optimum safety level that results minimum life cycle cost value (CLC

in Fig. 2-1). Structural cost C0 (Eq. (6)) contains the cost of the purlins, the sheeting system and the

bracing system, as well. In order to take into account the whole frame’s cost in the calculation the

outer frames have been considered with the following dimensions: 300-300x6+200x8 (columns)

and 300-300x6+200x8 (beams).

The calculated reliability indices, initial and life cycle cost components of cases #1 - #9 show

that fire curve #1 is the most demanding from the considered cases, while fire curve #3 represents

much less severe fire. Not surprisingly, in case of 45 and 60 minutes time demand levels, the

optimized solutions requires more investments regarding to the initial cost of the steel structure and

the initial cost of passive protection. Furthermore for R45 and R60, passive protection with thick

layers has to be used for good performance and safety. However in case of fire curve #3, when the

design aim is to satisfy R30 criterion, there is no need for passive fire protection (see the results of

case #7 and #16). Nevertheless, this is not the case for fire curves #1 and #2. By case #19 and #20,

because the fire effect is too demanding the algorithm cannot find good and stable solution. For

this reason, it is not safe and economical to ensure the fire safety without passive fire protection,

as it is also shown in [BT8], because the steel plates are very heat conductive and they loss their

stiffness and strength very quickly in severe fire.

Generally, the D/C ratio of the frames in persistent design situation is high (Table A-2, cases

#1 - #36), thus the presented solutions are possible design alternatives also for conventional loading

conditions.


84

Fig. 7-7 – Differences in the width-to-thickness ratio (slenderness) of the plate elements comparing to the optimized

reference cases (Table A-2, Table A-3)

The optimized solutions are compared with solutions designed by practicing engineers with

C0≈57,000€ (column: 300-700x6+180x10, beam: 380-700x6+165x8) considering 0.2 kN/m2

equipment load, and optimized by the developed algorithm with C0≈55,700€ (column: 185-

665x6+205x9, beam: 215-700x6+185x8) and with C0≈56,260€ (column: 130-855x6+210x8,

beam: 230-815x6+190x8) considering only serviceability and ULS constraints in persistent design

situation for 0.2 kN/m2 and 0.5 kN/m2 equipment load, respectively. The solutions provided in

Table A-2 have larger C0 cost in most of the cases, however, they have lower C1 cost and they have

lower CLC cost in fire design situation (Table A-3 in Appendix A).

It can be seen from the results that the flanges and webs are less slender (Fig. 7-7) compared to

the width-to-thickness ratio of plates of the optimized reference frames. Probably, the most

economical solution cannot be achieved only with protection elements with more slender sections

(with higher plate width-to-thickness ratio), which may be optimal and adequate in persistent

design situation, using thick passive protection. From the point-of-view of conceptual design

stockier sections combined with less passive protection ensure better performance during fire. Less

slender sections also give lower A/V value, thus the heating of these sections are slower comparing

to sections which have higher A/V section ratio. Due to the fact that structural fire design is

generally new for the structural designer society in Hungary, the issue of structural fire design is

often assigned to fire safety engineers, who may be not well educated from the point-of-view of

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18−40

−30

−20

−10

0

10Column web slenderness (ref. h/t=110.8 and h/t=142.5)

Cases

Diff

eren

ce [%

]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18−60

−40

−20

0

20Beam web slenderness (ref. h/t=116.7 and h/t=135.8)

Cases

Diff

eren

ce [%

]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18−50

−40

−30

−20

−10

0Beam flange slenderness (ref. b/2t=11.6 and b/2t=11.9)

Cases

Diff

eren

ce [%

]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18−60

−50

−40

−30

−20

−10

0Column flange slenderness (ref. b/2t=11.4 and b/2t=13.1)

Cases

Diff

eren

ce [%

]


85

structural engineering, and select the amount of fire protection based only the section factor (A/V

– ratio of perimeter and surface) supposing that the critical temperature of the element is e.g. 550

°C. It will be shown later in this section that this method is not reliable and not safe in some cases

and the most economical solution cannot be achieved only with protection of slender elements that

would be optimal in persistent design situation using thick passive protection. It is important to

consider the fire design situation during structural design (and not after it) in order to achieve

economical and well performing solutions.

The calculated optimum/target reliability indices are listed in Table A-2. Comparing to the

standardized target indices (in Table 1-1), it can be seen that the calculated values for cases #1 - #9

(β=2.82-3.45) are lower than the suggested values of EC0. It has to be noted that β=2.82 reliability

index implies that the structure has almost 1.0 conditional failure probability in fire. In these cases

the fire effect is too severe and the protection and strengthening of the structure may not be

economical. β=2.82 reliability index is a lower bound because the occurrence of flashover is quite

rare in the investigated case (Fig. 4-1). Due to the highly nonlinear, uncertain and extreme nature

of the fire effect (especially when this nature is combined with extreme intensity, e.g. see fire curve

#1 and #2 for R45 and R60 demand levels), ensuring of high reliability is too expensive (relative

cost of safety measure is moderate or high), thus, the resulted reliability indices are low comparing

to other cases. It has to be noted that some conservative assumptions have been made by the

formulation of reliability analysis due to the lack of knowledge. By reducing this uncertainty and

conservative assumptions, the calculated target reliability indices may be increased. The

optimization procedures have been performed considering ρ=0.4 correlation coefficient (as a more

likely value for the investigated structure) in Eq. (20), however, the reliability indices are presented

for ρ=0.9 as well in Table A-2, in order to characterize the effect of low and high correlation. With

the consideration of higher correlation among the frames, higher reliability indices are calculated

(β=2.84-3.51). These values better characterize smaller structures with smaller fire compartment.

The difference between the probabilities of failure varied from 0% to -50%, thus the correlation

has a significant effect on the reliability of the structure.

As it can be seen by comparing Table 1-1 and Table A-2 and as it is pointed out in [BT11], the

target values of JCSS Probabilistic Model Code and ISO 2394 standard are more applicable for fire

design of industrial steel tapered portal frames. Further issue is that the EC0 does not give different

groups according to the relative cost of safety measures, in this way, it recommends the same target


86

reliability for persistent, seismic and fire design situation. This method may does not seem proper

for providing solutions with consistent reliability which is one of the bases of safe and economic

design.

It is a very important conclusion that the reliability indices related to optimum solutions vary

in a wide range when different fire curves and different time demands are considered during the

design. This observation implies that the optimum safety level depends on the heating rate and the

maximum temperature in the compartment. Furthermore, the safety level significantly depends on

the occurrence of severe fire and flashover, thus it is dependent on the function of the building and

the amount of active safety measures. For this reason, the safety of two identical frame structures

is different when the function of the buildings is different. These conclusions predict the fact that

comparable effects, such as ISO standard fire, cannot be the basis for consistent and reliable

structural fire design. In order to achieve consistent reliability level, safe and economical solutions,

it is important to model the fire effect as accurately as possible.

Fig. 7-8 – Optimal safety levels as a function of additional costs comparing to the configurations #1 - #18 in Table

8: a) ρ=0.4; b) ρ=0.9 correlation coefficient.

Based on the results of 36 optimized cases, a table with possible values for target reliability

indices is constructed (Table 7-4), similarly to Table 1-1. The presented target indices may be also

valid for other type of steel structures and not only for industrial steel portal frame structures;

because of the consideration of low and high correlation (smaller structure/compartment) among

the frames, because of various failure consequences and various initial design conditions the

presented results cover a wide range of possible cases. Further investigation is necessary in order

to define target indices for different type of structural configurations. Optimized cases with high

initial cost components (Table A-1, Fig. 7-8) or demanding fire curve are categorized in high

relative cost of safety measure row, while cases optimized considering fire curve #3 resulted low

additional costs (Fig. 7-8) are categorized in the last row. In Fig. 7-8 the initial costs of the

0 2 4 6 8 10 12 14 16 182.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

(C0,opt

+ C1,opt

)/C0,ref

- C0,ref

[%]

β

0 2 4 6 8 10 12 14 16 182.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

(C0,opt

+ C1,opt

)/C0,ref

- C0,ref

[%]

β


87

optimized cases are compared with reference structures (Table A-3), optimized in persistent design

situation for 0.2 kN/m2 (column: 185-665x6+205x9, beam: 215-700x6+185x8) and 0.5 kN/m2

(column: 130-855x6+210x8, beam: 230-815x6+190x8) equipment load using the developed

algorithm. Solutions with β=2.82 are not accounted because the fire effect and time demand are

too severe in these cases. It can be seen that this table is in better agreement with the

recommendations of JCSS and ISO 2394 than with EC0. Due to the limited number investigated

cases, there is no defined range in columns related to minor and large consequences, thus further

investigation is needed later in order to extend and validate the suggested numbers. Further

investigation is necessary for better understanding the possible components (and their weights) of

failure cost function. The target values are also influenced by the acceptance ability of the society

and global economy of the country, so in some cases minimum limits may be used in order to

ensure the minimum desired safety.

50 years service life: calculated target reliability indices Relative cost of safety measure

Fire effect severity

Minor consequences

Moderate consequences

Large consequences

High High 2.8 (2.8)* 2.8 – 3.2 (2.8 – 3.3) 3.6 (3.7)* Moderate Medium 2.8 (2.9)*† 2.9 – 3.4 (3.0 – 3.5) 3.6 (3.8)*

Low Low 2.9 (3.0)* 3.1 – 3.5 (3.3 – 3.6) 3.7 (3.8)* * based on limited number of cases, further investigation is necessary; † interpolated

Table 7-4 – Calculated target reliability indices for tapered portal frames with storage function (with ρ=0.4 and

ρ=0.9 correlation coefficient)

Comparing the results of cases #2, #5 and #8 to results of cases #34, #35 and #36, it can be seen

that the application of more active safety measures can result cheaper structure in terms of initial

cost of steel superstructure and passive fire protection, however, active safety measures are

generally expensive. It can be also concluded that life cycle cost values are lower with only alarm

system, thus in the investigated case the application of both alarm and extinguish systems may not

lead to economical design. Comparing to the results of cases #2, #5 and #8 to results of cases #31,

#32 and #33, it can be concluded that the initial costs are much higher, nevertheless, they result the

lowest life cycle costs (considering cases where the equipment load is 0.2 kN/m2 and where the

cost components are the same). In case of the investigated and similar structural configurations

with storage function, optimal solution may be achieved with less active safety measure (if the

presented safety level meets the allowable minimum safety limit), but with more passive fire

protection and stronger structure. This conclusion is in good agreement with the results of an earlier

study [BT11].


88

In order to investigate the achievable performance using common practice in structural fire

engineering, the passive protection of the above mentioned reference frames (which have been

optimized considering only constraints related to persistent design situation) is selected based only

on the section factor of the sections and according to the producer’s manual [69], assuming that the

critical temperature is 550 °C. The A/V factor in case of the columns is between 250 and 303 1/m,

while in case of the beams it vary from 280 to 305 1/m. The calculated reliability indices and life

cycle costs can be seen in Table A-3 (Appendix A).

The calculated reliability indices vary in a wide range and they rarely achieve the EC0

recommended β target indices because of several reasons: a) the structural fire design is

characterized by high degree of uncertainty, the EC0 recommended target indices may not refer

well to extreme situations; b) the design of intumescent coating is based and generally the fire

design is often based on ISO standard fire curve which is not able to represent real fire thus cannot

be used as the basis for consistent, safe and economical structural fire design; the reliability depends

on the quantity and quality of the combustible materials and depends on the function of the

building; c) the reliability of a structural system is generally lower than the reliability of separated

elements (structural reliability is often calculated for separated elements in the literature, e.g. in

[87], in [55] and in [54]); d) the structural fire design should be completed by the structural designer

and should be included in the design process from the beginning of searching possible economic

solutions; e) the persistent design situation and fire design situation may be contradictory objectives

in some cases, the cross section (see Table A-2 and Table A-3; compare e.g. cases #1 - #3 or cases

#10 - #12) which is close to optimum for conventional loads is not optimum for fire design; f) the

common practice that the passive protection is selected after the persistent design assuming the

critical temperature of the element may be unreliable (Fig. 7-9) and unsafe.

Fig. 7-9 – The life cycle costs of optimized (blue) and reference cases (red)

The life cycle cost values are higher than values in Table A-2 for optimized cases (Fig. 7-9);

the achievable saving for life cycle with the presented method varies from -0.1 to +76% comparing

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 366080

100

120140

160180200

Case No.

CL

C [

1000

EU

R]


89

to the common design practice. The difference in probability of failure varies between -85 and

+1290%, the highest negative values have been calculated for fire curve #3. In most of the cases

the difference is positive and significant positive differences can be observed for R45 and R60 time

demands, especially when the fire effect is severe or extreme (fire curves #1 and #2 , respectively).

It shows that the common practice and the application of ISO standard curve are unsafe in lot of

cases.

Related to the protection of connections, it is observed that the protection thicknesses at the

beam-to-column connections are much lower in case of the optimized cases than in cases presented

in Table A-3, where the thicknesses are selected based on the thicknesses of connected elements

which would be a reasonable engineering decision if it was a real design situation. Due to the

generally slender structural configuration and due to the fact that the Young’s modulus decreases

at high temperature, the leading failure mode in fire design situation is loss of stability of main

elements. Furthermore the heating of connection zones is slower than the heating of connected

elements. Thus, the beam-to-column connections are not fully utilized in fire design situation and

there is no need for thick protection in the connection zones. However, the heating of the

connections is generally more uncertain and thicker protection does not mean significant additive

cost, for this reason, an engineering practice according to which the connection is protected as the

connected elements can be considered safe and good in the case of the investigated structure and

structural configuration.

7.4. Optimal solutions in seismic design situation

7.4.1. Parametric study

Similarly to the case of fire design, optimal solutions are obtained for several design cases

within the framework of a parametric study. The investigated cases (Table A-4) cover a wide range

of possible design situations. The effect of different loading conditions, different load intensities,

different sites with higher and lower seismic intensities and sheeting system rigidity on the optimal

solutions are investigated.

As regards to the cost components in Eq. (9), (10) and (11), the initial cost rate of the main

frame elements (cs), the initial cost rate of the bracing elements (cb) and the cost rate of the sheeting

system (Csh) are set to 2.25 €/kg, 2.25 €/kg and 25 €/m2, respectively. The failure cost is varied in

order to obtain results related to design cases where the risk is higher and lower. The selected


90

failure costs have been obtained based on Section 6.1 and may represent minor, moderate and large

consequences.

The consideration of different loading conditions means that optimal configurations are derived

for frames both with and without additional concentrated forces on the columns at 2/3 of the eave

height (Fig. 7-10a). These concentrated forces may represent the reaction force of an internal slab

or a crane; their intensity is set to 100kN. These forces increase the seismic vibration mass of the

building and they cause significant torsional moment if the pattern is asymmetrical (Fig. 7-10c).

S1, S2 and S3 load cases (Table A-4) represent cases where there are no additional concentrate

forces on the columns, there are concentrated forces on half of the columns (Fig. 7-10c) and there

are concentrate forces on all of the columns (Fig. 7-10b).

Fig. 7-10 – a) Additional concentrated forces on columns; b) symmetrical pattern; c) asymmetrical pattern

7.4.2. Parametric study results

Optimization results are summarized in Table A-5. In the first and second columns, the

dimensions of cross sections can be found for both column and beam elements, respectively, while

the D/C column shows the demand-to-capacity ratio in ULS of persistent design situation related

to the critical failure mode. βopt is the reliability index related to the optimum safety level that

results minimum life cycle cost value (CLC in Fig. 2-1). C0 contains the cost of purlins, sheeting-

and bracing system, as well. In order to take into account the whole frame’s cost in the calculation

the outer frames have been considered with the following dimensions: 300-300x6+200x8

(columns) and 300-300x6+200x8 (beams).

First of all, general conclusions can be drawn related to data presented in Table A-5. Not

surprisingly, in case of design situations where the considerable gravity load is high the resulted

optimal solutions have thicker elements and they are more expensive because the seismic forces

are higher (e.g. compare case #1 to #3 or case #4 to #6, etc.). Due to the increase in the level of

a) b) c)


91

seismic forces, the resulted optimal reliability indices are lower because the strengthening of the

structure becomes more expensive and it may become not economical.

It can be seen from the parameters of the fitted lognormal distributions (Table 5-1) that the

seismic effect is characterized by high degree of uncertainty irrespectively to the seismicity of the

investigated site. During the consideration of historical earthquakes in PSHA high uncertainty is

partly coming from the evaluation of statistical data set containing from a number of sites. In case

of low and moderate seismicity (e.g. Hungary) typically there is no dominant site in investigated

set and large number of different sources influence the site’s hazard curve contrary to sites

characterized by high seismicity where few sources may affect dominantly the uncertainty in the

hazard curve. For this reason, significant difference may be realized between hazard curves of

Komárom and Râmnicu Sărat; the hazard curve related to Komárom site is much more uncertain

(Table 5-1) than the curve that characterizes the seismicity of Râmnicu Sărat. This uncertainty

highly influences the results of reliability analysis. It might be surprising and interesting for the

reader that a certain configuration is characterized by similar reliability indices for the both sites

(only slight differences can be observed). The 90th percentiles are very close to each other,

nonetheless, the mean of PGA distributions are significantly different.

Contrary to the optimized solutions for fire effects, the results (Table A-5) of seismic design

optimization show that the D/C ratio in persistent design situation is very low by a number of cases.

It shows that seismic design situation may be the leading design situation when the seismic mass

or/and the target reliability index is high regardless to the moderate seismicity of the site. Regarding

to cross section dimensions, strong conclusions cannot be drawn compared to the original structural

configuration; both stockier but smaller and thinner but higher sections may be appropriate in order

to find optimal configurations.

According to the opinion of many designers, sheeting system decreases the structure’s safety

because its stiffness increases the global structural stiffness; therefor it increases the seismic forces

acting on the building. Nevertheless, based on the results of this study sheeting system has

beneficial effect on the structural safety and the cost of resulted configurations by the industrial

frames with similar function, size and configuration to the investigated frame. This statement is

confirmed (Table A-5, Fig. 7-11) for high and moderate seismicity, for low and high vertical forces,

as well. The tendency is clear in the results; the same structural safety can be achieved beside 0-

15% less superstructure’s cost with the consideration of sheeting system in the structural analysis


92

depending on the consequence class, on the severity of seismic action and on the intensity of

vertical loading; the saving is higher in case of higher intensity of horizontal actions. It is important

to note that sheeting system has to be designed and checked against seismic effects in order to

ensure its contribution to the global behaviour during the earthquake.

Fig. 7-11 – The cost of the superstructure and the resulted reliability indices for high seismicity,

In fact, the contribution of sheeting system decreases the vibration period of the building thus

it increases the base shear force (as the results of HighPerFrame RDI project showed) (Table 7-5).

It seems that the contribution of sheeting system is not beneficial and it may decrease the structural

safety through increasing the seismic load on the building. Nonetheless, different beneficial effects

may be developed during seismic excitation due to the sheeting system. These beneficial effects

decrease the level of internal forces in the main elements; resulting cheaper and safer solutions.

Considered sheeting LTP45+Z200 LTP20+Z200 none Transversal period 0,5932 s 0,6371 s 0,7462 s

Longitudinal period 0,2912 s 0,3102 s 0,3660 s Design acceleration in transversal direction 5,6407 m/s2 5,3119 m/s2 4,5356 m/s2

Design acceleration in longitudinal direction 5,6407 m/s2 5,6407 m/s2 5,6407 m/s2 Transversal base shear force 225,89 kN 212,72 kN 181,63 kN

Longitudinal base shear force 225,89 kN 225,89 kN 225,89 kN Table 7-5 – The effect of sheeting system’s stiffness on the base shear force; case RLTP4502 for ~ 0.3g PGA [BT9]

According to previous results [79] [BT9] [BT12], the sheeting system decreases the value of

internal forces in the main frame elements (Fig. 7-12) through developing a spatial contribution

together with the wind and side braces. These parts compose a box-like structure; a part of the

horizontal forces is transferred to the ground through the braces at the end of structure similarly to

a simple supported beam (Fig. 7-13). Furthermore, it helps a global response being developed

where adjacent frames help to each other in resisting the seismic forces. Another advantage is that

the sheeting behaves as a chord of main frame elements thus it increases their bending stiffness and

it contributes in resisting the bending moments. As the results of the example in [BT9] showed the

bending moments in the structure may be decreased by cca. 5-25% (Fig. 7-12).

40

60

80

100

2 2,5 3 3,5

C0

+ C

1[1

000€

]

β

without sheetingwith sheeting


93

Fig. 7-12 – Bending moment diagram of the main frames after SRSS sum (0.3g PGA; continuous blue – outer frames,

continuous red – inner frames, dashed blue – second and fifth frames): a) LTP45+Z200; b) no sheeting system is considered [BT9]

Fig. 7-13 – Base shear forces [N] of the frames after SRSS sum (0.3g PGA): a) LTP45+Z200; b) no sheeting system

is considered [BT9]

In order to consider the contribution of sheeting system in the global behaviour, spatial 3D

structural model need to be applied together with realistic modelling of the stiffness and the

properties of connections of trapezoidal sheeting and purlins in the analysis. This model need to

able to describe the internal forces and stresses in the main elements, in the braces and in the

sheeting system as well. The beneficial effect of the sheeting system can be utilized only if the

sheeting’s and connections’ failure is avoided. By many of the investigated cases the base shear

forces are higher than possible resultant horizontal forces coming from wind actions. If the D/C

ratio related to the failure of sheeting, purlins or connections is not significant due to wind forces,


94

the contribution of sheeting system in seismic design situation may be utilized without additional

costs.

Panels between frames that subjected to significantly different horizontal actions due to their

different loading conditions may be designed against higher internal forces. In case of the

investigated structure, panels that are subjected to high seismic forces are the second and the fourth

panels. By longer industrial halls that consist of large number of frames the contribution of sheeting

system may be lower because the braced bays are situated far from the middle and the panels are

not capable to transfer the loads to the braces.

Based on the results of 28 optimized cases, a table with possible values for target reliability

indices is presented in Table 7-6, similarly to Table 7-4 that shows possible target reliability indices

for fire design. It can be seen that this table is in better agreement with the recommendations of

JCSS and ISO 2394 than with EC0. The target values are also influenced by the acceptance ability

of the society and global economy of the country.

50 years service life: calculated target reliability indices

Seismic mass Seismic effect

severity Minor


consequences Large consequences

High High 2.2* 2.5 – 2.7 2.8 – 3.0* Moderate Medium 2.3* 2.6 – 2.8 2.8 – 3.0†*

Low Low 2.6* 2.8 – 3.0 3.0 – 3.5* * based on limited number of cases, further investigation is necessary; † assumed value, there is problem with the convergence

Table 7-6 – Calculated possible target reliability indices in seismic design situation for tapered portal frames with

storage function


95

8. Summary and conclusions

Seismic and fire effects are considered extreme due to their highly nonlinear and severe nature

and also due to their generally extreme consequences. Protection against these severe loading

conditions often require expensive solutions, protection and strengthening compared to the design

against conventional effects such as dead and live loads. For this reason, conceptual design of

optimal structural configurations and solutions came to the fore, however, to find these solutions

by trial-and-error method for a designer in everyday practice can be difficult and highly time-

consuming. New and valuable design concepts may help designers in conceptual design phase in

order to develop economical structural solutions characterized by lower initial costs or considerably

better performance and safety.

The global aims of this study are: a) finding optimal solutions for tapered portal frame structures

subjected to seismic or fire effects; b) analyzing optimal structural safety under extreme loading

conditions that leads cost effective structures; c) introducing new design concepts based the results

of a parametric study covering a wide range of possible design situations. The applicability,

efficiency and convergence of the proposed methodology and numerical algorithm are confirmed

by optimization results presented in this thesis. The results are discussed regarding structural initial

costs, strength, stiffness and possible/acceptable risks of significant losses under severe fire or

seismic loading conditions. Wide range of possible design situations is covered in this research,

namely fire effects with different intensity and duration; seismic effects in seismically less and

more intensive areas; structures with low and high gravity loads. Main observations and outcomes

are summarized in Section 10; the new design concepts and suggestions connected to design of

tapered steel frames subjected to extreme effects.


96

9. Future research

In this thesis, a comprehensive reliability calculation methodology is presented for steel frame

structures subjected to fire effects. The limit state function is formulated on time basis as a practical

measure that shows how much time the structure is able to sustain under fire exposure. The required

time values for evacuation, namely 15, 30, 45, etc. minutes, are prescribed in national regulations

[71], however, no study is available in the literature that investigates many issues that would be

important to evaluate more accurately fire safety of structures in Hungarian circumstances:

• statistical data related to elapsed time from fire ignition till fire detection;

• statistical data related to elapsed time from fire detection till extinguishment the fire;

• statistical data related to required time for fire growth into fully developed fire;

• statistical data related to the required time to extinguish different fires.

For this reason, the uncertainties of required time level are not taken into account in the reliability

analysis. R values should be considered in the reliability based structural optimization procedure

as a variable based on firefighting statistics and based on acceptable risk of society and economy

in order to evaluate values for optimal fire safety related to different structural configurations and

consequence classes. Furthermore, based on firefighting and ignition statistics flashover

occurrence could be taken into consideration with probability distributions resulting more

comprehensive results related to optimal fire reliability of different structures.

Related to the consideration of fire effects and system reliability in reliability analysis for fire,

it would be necessary to consider different fire scenarios where the size of exposed part of the

structure is different and further research should be made on evaluation system reliability index.

In this study, column base-, beam-to-beam and beam-to-column connections were considered

with fixed parameters as boundary conditions in each design cases. In the course of a future

research work, it would be interesting to consider semi-rigid connections to analyse how the

connection stiffness affect the target reliability index and optimal solutions.

This research was part of the HighPerFrame RDI project where the aim was study of optimal

fire and seismic design of steel tapered portal frame structures in cooperation with Rutin Ltd. The

reference structure investigated in this research (Fig. 1-1) is a storage building and it is situated in

Hungary. The applied cost factors are partly provided by Rutin Ltd. As regards to target reliability

indices and optimal configurations, the effect of different cost factors is partly covered in the


97

presented parametric studies, nevertheless further research is needed to extend the conclusions with

the consideration of different cost factors for steel superstructure and for the fire protection.

Optimal structural configurations and possible target reliability indices are presented for fire

and seismic effects, respectively, based on comprehensive analyses of steel tapered portal frames.

Further research is important to strengthen these conclusions, to develop more accurate target

reliability index values and to analyse different structural configurations.


98

10. New scientific results

This section presents a summary of new scientific results achieved within the framework of this

research and presented in this dissertation.

10.1. Thesis I.

I solved two highly nonlinear and non-convex reliability based optimization problem of steel

tapered portal frame structures exposed to seismic or fire effects. I developed and verified a

reliability based heuristic optimization framework in order to perform the structural optimization.

I/a. I connected a genetic algorithm based optimization algorithm to an advanced reliability

assessment framework that is based on first order reliability method. The application of first

order reliability method reduces the computational demand, thus it increases the

performance of the optimization framework. I reduced the level of approximation compared

to earlier studies by adoption of state-of-the-art analysis and evaluation tools, analysis of

the whole structural system, nonlinear numerical analysis on imperfect model to evaluate

the nonlinear structural response and General Method of Eurocode 3 for stability

verification of tapered steel elements subjected to compression and bending.

I/b. I analysed the convergence and stability of the developed optimization framework in a

sensitivity analysis. I showed a proper settings for the algorithm, wherewith the

optimization is stable and characterized by good accuracy from engineering point-of-view.

I proved that the developed optimization framework is applicable to find optimal solutions

in the course of reliability based optimization of tapered portal frames exposed to extreme

effects.

Connected publications: [BT1], [BT2], [BT3], [BT4]

10.2. Thesis II.

In this thesis, I presented the development and essential parts of a complex and effective

methodology for reliability evaluation of complex structural systems subjected to fire. Its

application is illustrated for tapered portal frame structures.

II/a. I developed a comprehensive limit state function evaluation methodology for complex

structural systems with the consideration of complex structural response through nonlinear


99

analysis on imperfect model. The methodology is able to consider any type of fire curve

and a limit state function formulated on time basis. While these features may be found

separately in earlier studies, the main novelty of this methodology is that it offers the

aforementioned features together for complex structural systems.

II/b. I verified the calculated reliability indices and convergence of the framework through a

comparison with Monte Carlo simulation results of a tapered steel storage hall

prescriptively designed according to Eurocode. I found that the first order approximation

underestimates the failure probability of the investigated problem. The difference between

the resulted failure probabilities of first order reliability method and Monte Carlo simulation

is observed from -1% to -34% in case of low and high probabilities, nevertheless, first order

approximation resulted good approximation of failure probability in the range of interest.

Connected publications: [BT8], [BT13]

10.3. Thesis III.

In this thesis, I investigated the optimal fire design concept of steel tapered portal frame

structures. I calculated target reliability indices using the presented reliability analysis framework

and I performed a parametric study for a reference tapered storage hall structure with the help of

developed structural optimization framework in order to derive new and valuable design concepts.

III/a. I confirmed that the optimum reliability and the optimal configuration depends on the

quantity and quality of the combustible materials and it depends on the function of the

building. It is stated, that the application of comparable effects, such as ISO standard fire,

often leads to either conservative or unsafe solutions. β=2.9 – 3.5 reliability index may be

achieved using Eurocode 3 conforming prescriptive design and ISO standard fire curve in

case of the reference structure. In order to achieve consistent reliability level as well as safe

and economical solutions, it is important to model the fire effect as accurately as possible.

III/b. I presented target reliability indices for the reference structure considering three different

consequence classes and three different fire curves representing fires with different

intensities. I showed that in case of the reference structure active safety measures and

passive protection have to be applied together in order to satisfy the criteria of Eurocode 0,

passive protection without active safety measures is not effective enough when the target

reliability is greater than 3.2 – 3.3. I pointed out that the suggested target reliability indices


100

proposed by Joint Committee of Structural Safety seem to be more appropriate and the

target reliability index may be selected between 2.8 and 3.7 for industrial steel tapered portal

frames.

III/c. I pointed out the problem that Eurocode 0 standard does not differentiate groups according

to the relative cost of safety measures, in this way, it recommends the same target reliability

for persistent and fire design situations. However, the calculated reliability indices for fire

effects are lower than the suggested values in Eurocode 0; they are closer to the

recommendations of Joint Committee of Structural Safety and ISO 2394 standard. Different

target safety levels need to be developed for fire effects in Eurocode 0.

III/d. Through optimization results I showed for steel tapered portal frame structures similar to

the reference configuration that less slender, stockier sections (with lower plate width-to-

thickness ratio: -10% to -20% for webs and -20% to -40% for flanges on average) are more

appropriate in case of severe fire effect (especially with high time demand) due to the fact

that the structure is sensitive for stability failure. These sections, combined with less passive

fire protection ensure better performance and safety in fire. I pointed out that the

conventional fire design practice that fire design is not incorporated directly in conceptual

design phase and dimensioning and that structural fire safety is ensured by selection of

passive fire protection for slender elements adequate in persistent design situation is often

unreliable and not economical.

Connected publications: [BT8], [BT10], [BT11], [BT14]

10.4. Thesis IV.

In this research, I investigated the optimal seismic design of steel tapered portal frame

structures. I calculated target reliability indices using the presented reliability analysis framework

and I performed a parametric study for a reference tapered storage hall structure with the help of

developed structural optimization framework in order to derive new and valuable design concepts.

IV/a. I presented target reliability indices for the reference structure considering two sites

characterized with moderate and high seismicity (Komárom and Râmnicu Sărat,

respectively), three different consequence classes and both elastic and dissipative design. I

showed that Eurocode 8 conforming dissipative design leads significantly higher structural

reliability compared to elastic design even in the case of a single storey steel tapered portal


101

frame structure. For structures similar to the reference building β=2.0 – 3.5 seems to be

economical and optimal for most of the practical cases notwithstanding that it is lower than

the required level in Eurocode 0.

IV/b. I pointed out the problem that Eurocode 0 does not differentiate groups according to the

relative cost of safety measures, in this way, it recommends the same target reliability for

persistent and seismic design situations. However, the calculated reliability indices for

seismic effects are lower than the suggested values in Eurocode 0; they are closer to the

recommendations of Joint Committee of Structural Safety and ISO 2394 standard. Different

target safety levels need to be developed for seismic effects in Eurocode 0.

IV/c. I investigated the effect of sheeting system on the global structural safety of reference

structure. I showed that although the sheeting system increases the structural rigidity and

the intensity of seismic forces acting on the structure but it helps to distribute the horizontal

forces among the frames and to transfer the forces to the ground in contribution with the

bracing elements (the bending moments in the structure may be decreased by cca. 5-25%).

On the whole it has a beneficial effect on the structural safety and the cost of industrial

frames with similar function size and configuration to the investigated frame (3% to 16%

savings in superstructure’s cost and 1% to 13% in life cycle costs considering the results of

optimized structures).

IV/d. Regarding to optimal seismic design of single storey steel tapered portal frame structures

I showed that even though elastic design is applied it is favourable to provide ductile

behaviour for the structure because plastic failure modes lead significantly higher structural

reliability under seismic excitation. It may be also favourable and result cheaper and more

reliable structure to model the sheeting system and to consider its rigidity in seismic design

situation if the sheeting panels and the connections of sheeting system are verified against

seismic forces.

Connected publications: [BT9], [BT10], [BT12]


102

References

Publications of the author related to the theses

[BT1] Balogh T. and Vigh L. G. (2012) Genetic Algorithm based optimization of regular steel

building structures subjected to seismic effects. In Proc. 15th World Conference on Earthquake Engineering, 4975, Lisbon, Portugal.

[BT2] Balogh T. (2013) Adjustment of genetic algorithm parameters for building structure

optimization. In Proc. Second Conference of Junior Researchers in Civil Engineering. June 17-18, 2013, Budapest, Hungary.

[BT3] Balogh T. and Vigh L. G. (2013) Cost Optimization of Concentric Braced Steel Building

Structures. WORLD ACADEMY OF SCIENCE, ENGINEERING AND TECHNOLOGY: INTERNATIONAL SCIENCE INDEX 78, International Journal of Civil, Environmental, Structural, Construction and Architectural Engineering, 7(6), 469 - 478.

[BT4] Balogh T., D’Aniello M., Vigh L. G. and Landolfo R. (2014) Performance Based Design

Optimization of Steel Concentric Braced Structures. In Proc. the 7th European Conference on Steel and Composite Structures, Eurosteel 2014. September 10-12, 2014, Naples, Italy.

[BT5] Zsarnóczay Á., Balogh T. and Vigh L. G. (2014) Design of frames with Buckling Restrained

Braces – FEMA P695 based evaluation of an Eurocode 8 conforming design procedure. In Proc. the 7th European Conference on Steel and Composite Structures, Eurosteel 2014. September 10-12, 2014, Naples, Italy.

[BT6] Vigh L. G., Balogh T., Zsarnóczay Á. and Castro J. M. (2015) Eurocode 8 Part 1 – research report. Prepared for ECCS Technical Committee TC13 – Seismic Design and CEN SC8/WG2 Committee.

[BT7] Zsarnóczay Á., Balogh T. and Vigh L. G. (2015) On the European Norms of design of

Buckling Restrained Braced Frames – European Norms of design of BRBF. The Open Civil Engineering Journal, Bentham Science Publisher. 11, pp. 3-15. ISSN 1874-1495

[BT8] Balogh T. and Vigh L. G. (2015a) Optimal design of tapered steel portal frame structures subjected to fire effects (in Hungarian). MAGESZ Steel Structures, 1st special issue, 84-94.

[BT9] Balogh T. and Vigh L. G. (2015b) Optimal design of tapered steel portal frame structures subjected to seismic effects (in Hungarian). MAGESZ Steel Structures, 1st special issue, 95-106.

[BT10] Balogh T. and Vigh L. G. (2015c) The calculation of structural reliability of portal frame

structures with First Order Reliability Method. In Proc. of 12th Hungarian Conference on Mechanics (XII. MAMEK), Miskolc, Hungary.

[BT11] Balogh T. and Vigh L. G. (2015d) Optimum reliability of a steel tapered portal frame

structure exposed to fire. In Proc. ASFE 2015 International Conference in Dubrovnik, 15-16 October 2015, Dubrovnik, Croatia.

[BT12] Balogh T. and Vigh L. G. (2015e) Seismic reliability based optimization of steel portal frame structures. In Proc. SECED 2015: Earthquake Risk and Engineering towards a Resilient World Conference. July 9-10, 2015, Cambridge, UK.

[BT13] Balogh T. and Vigh L. G. (2016a) Complex and comprehensive method for reliability

calculation of structures under fire exposure. Fire Safety Journal. 86. p. 41-52. [BT14] Balogh T. and Vigh L. G. (2016b) Optimal fire design of steel tapered portal frames.

Periodica Polytechnica Civil Engineering. Accepted, Online First, p. 8985.


103

Other references

[1] Honfi D. and Dunai L. (2005) Optimization of cold-formed steel compression members using

genetic algorithm. In proc. Int. Symposium on Neural Networks and Soft Computing in Structural Engineering (NNSC 2005), June 30 – July 2, 2005, Krakow, Poland.

[2] Vigh L. G. and Pap Cs. (2011) Design of new generation steel wind-turbine tower: Advanced analysis and optimal design of conical segments. In Proc. the 6th European Conference on Steel and Composite Structures, Eurosteel 2011. August 31 – September 2, 2011, Budapest, Hungary.

[3] Simon J., Kemenczés A. and Vigh L. G. (2014) Optimal stiffener geometry of stiffened plates: Non-linear analysis of plates stiffened with flat bars under uniaxial compression. In Proc. the 7th European Conference on Steel and Composite Structures, Eurosteel 2014. September 10-12, 2014, Naples, Italy.

[4] Farkas J. and Jármai K. (2008) Design and optimization of metal structures. Horwood Publishing, West Sussex, England.

[5] Farkas J. and Jármai K. (2013) Optimum design of steel structures. Heidelberg, New York: Springer Verlag.

[6] Jármai K., Farkas J. and Kurobane Y. (2006) Optimum seismic design of a steel frame.

Engineering Structures. 28. p. 1038-1048. [7] Jármai K. and Farkas J. (1999) Cost calculation and optimization of welded steel structures.

Journal of Constructional Steel Research, 50. p. 115-135. [8] Lógó J. (1988) Design of frame structures using multiples objective functions and

mathematical programming (in Hungarian: Rúdszerkezetek méretezése több célfüggvényes

matematikai programozással). PhD Dissertation. Budapest University of Technology and Economics. Budapest.

[9] Lógó J. (2013) On the optimal layout of structures subjected to probabilistic or multiply loading. Structural and Multidisciplinary Optimization. 48(6). p. 1207-1212.

[10] Lógó J., Pintér E. and Vásárhelyi A. (2013) On the optimal topologies considering uncertain loading positions. In proc. the 10th World Congress of Structural and Multidisciplinary Optimization. May 19-24, 2013, Orlando, USA.

[11] Balogh B. and Lógó J. (2014a) Optimal design of curved folded plates by optimality criteria

method. In proc. Engineering Optimization IV. September 8 – 11, 2014, Lisbon, Portugal. [12] Balogh B. and Lógó J. (2014b) Optimal design of curved folded plates. Periodica Polytechnica

– Civil Engineering. 58(4). p. 423-430. [13] Rozvany G. I. N., Zhou M. and Birker T. (1992) Generalized shape optimization without

homogenization. Structural Optimization. 4. p. 250-254. [14] Csébfalvi A. (2007a) Optimal design of frame structures with semi-rigid joints. Periodica

Polytechnica – Civil Engineering. 51(1). p. 9-15. [15] Csébfalvi A. (2007b) ANGEL method for discrete optimization. Periodica Polytechnica – Civil

Engineering. 51(2). p. 37-46. [16] Csébfalvi A. (2011) Multiple constrained sizing-shaping truss-optimization using ANGEL.

Periodica Polytechnica – Civil Engineering. 55(1). p. 81-86. [17] Thomas K., Tam H. and Jennings A. (1988) Optimal plastic design of frames with tapered

members. Computers & Structures. 30. p. 537-544. [18] Hayalioglu M. S. and Saka M. P. (1992) Optimum design of Geometrically nonlinear elastic-

plastic steel frames with tapered members. Computers & Structures. 44. p. 915-924.


104

[19] Hayalioglu M. S. and Degertekin S. O. (2005) Minimum cost design of steel frames with semi-

rigid connections and column bases via genetic optimization. Computers & Structures. 83. p. 1849-1863.

[20] Saka M. P. (1997) Optimum design of steel frames with tapered members. Computers & Structures. 63. p. 797-811.

[21] Kravanja S. and Zula T. (2001) Cost optimization of industrial building structures. Advances in Engineering Software. 41. p. 442-450.

[22] Kravanja S., Turkalj G., Silih S. and Zula T. (2013) Optimal design of single-storey steel

building structures based on parametric MINLP optimization. Journal of Constructional Steel Research. 81. p. 86-103.

[23] Hadril P., Mielonen M. and Fülöp L. (2010) Advanced design and optimization of steel portal

frames. Journal of Structural Mechanics. 43. p. 44-60. [24] Phan D. T., Lim J. B. P., Tanyimboh T. T., Lawson R. M., Xu Y., Martin S. and Sha W. (2013)

Effect on serviceability limits on optimal design of steel portal frame. Journal of Constructional Steel Research. 86. p. 74-84.

[25] Kazemzadeh S. A. and Hasancebi O. (2015) Computationally efficient discrete sizing of steel frames via guided stochastic search heuristic. Computers & Structures. 156. p. 12-28.

[26] McKinstray R., Lim J. B. O., Tanyimboh T. T., Phan D. T. and Sha W. (2015) Optimal design of long-span steel portal frames using fabricated beams. Journal of Constructional Steel Research. 104. p. 105-114.

[27] Kaliszky S. and Lógó J. (2006) Optimal design of elasto-plastic structures subjected to normal

and extreme loads. Computers & Structures. 84. p. 1770-1779. [28] MSZ EN 1998-1:2008 (2008) Eurocode 8, Design of structures for earthquake resistance –

Part 1: General rules, seismic actions and rules for buildings. Hungarian Standards Institution. Budapest, Hungary.

[29] Salajegheh E., Mohammadi A. and Ghaderi S. S. (2008) Optimum Performance Based Design of Concentric Steel Braced Frames. In Proc. the 14th World Conference on Earthquake Engineering. October 12-17, 2008, Beijing, China.

[30] Oskouei A. V., Fard S. S. and Askogan O. (2012) Using genetic algorithm for the optimization

os seismic behavior of steel planar frames with semi-rigid connections. Structural and Multidisciplinary Optimization. 45. p. 287-302.

[31] Jármai K. (2008) Optimization of a steel frame for fire resistance with and without protection. In proc. Design, Fabrication and Economy of Welded Structures International Conference. p. 79-89., Miskolc, Hungary.

[32] Barbazon A., O’Neill M. and McGarraghy S (2015) Natural Computing Algorithms. Springer-Verlag. London.

[33] ISO 834 – 10 (2014) Fire resistance tests – Elements of building constructions. [34] Federal Emergency Management Agency (2006) FEMA-445 Next-Generation Performance-

Based Seismic Design Guidelines. USA [35] Kaveh A., Azar B. F., Rezazadeh F. S. and Talatahari S. (2010) Performance-based seismic

design of steel frames using ant colony optimization. Journal of Constructional Steel Research. 66. p. 566-574.

[36] Saadat S., Camp C. V. and Pezeshk S. (2014) Seismic performance-based design optimization considering direct economic loss and direct social loss. Engineering Structures. 76. p. 193-201.

[37] Holland J. H. (1975) Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, 1975.


105

[38] Goldberg D. E. (1989) Genetic Algorithm in Search, Optimization and Machine Learning. Kluwer Academic Publishers. Boston, MA.

[39] Rojas H. A., Foley C. and Pezeshk S. (2011) Risk-Based Seismic Design for Optimal Structural

and Nonstructural System Performance. Earthquake Spectra: August 2011, Vol. 27, No. 3, pp. 857-880.

[40] DHS, Department of Homeland Security (2003) HAZUS-MH MR1: Advanced Engineering

Module, Technical User’s Manual. Washington D.C. [41] Deierlein G.G. (2004) Overview of a comprehensive framework for earthquake performance

assessment. In proc. International Workshop “Performance-Based Seismic Design Concepts and Implementation. p. 15-25. 28 June – 1 July, 2004. Bled, Slovenia.

[42] Liu M., Wen Y.K. and Burns S.A. (2004) Life cycle cost oriented seismic design optimization

of steel moment frame structures with risk-taking preference. Engineering Structures. 26. p. 1407-1421.

[43] Lagaros N.D, Garavelas A. Th. and Papadrakis M. (2008) Innovative seismic design

optimization with reliability constraints. Computer methods in applied mechanics and engineering. 198. p. 28-41.

[44] Federal Emergency Management Agency (2000b) FEMA-356 Prestandard and Commentary

for the Seismic Rehabilitation of Buildings. USA. [45] Neumann J. and Richtmyer M. D. (1947) Statistical methods in neutron diffusion. Los Alamos

Scientific Laboratory report LAMS–551. [46] Metropolis N. and Ulam S. (1949) The Monte Carlo Method. Journal of the American

Statistical Association. 44(247). p. 335-341. [47] Hasofer A. M. and Lind N. C. (1974) An exact and invariant second-moment code format. J.

Engrg. Mech. Div., ASCE, 100(1), p. 111-121. [48] Kazinczy G. (1921) On the degree of safety (in Hungarian). Építőipar - Építőművészet, 44,

78-79, 86-87. [49] Elishakoff I. (2004) Safety Factors and Reliability: Friends and Foes? Kluver Academic

Publishers. Dordrecht. [50] Mistéth, E. (2001). Structural Reliability (in Hungarian). Akadémiai Kiadó, Budapest. [51] Choi S. K., Grandhim R. V. and Canfield R. A. (2007) Reliability-based Structural Design.

Springer-Verlag. London. [52] MSZ EN 1990:2011 (2011) Eurocode 0, Basis of structural design. Hungarian Standards

Institution. Budapest, Hungary. [53] Joint Committee on Structural Safety (JCSS) (2000) Probabilistic Model Code. [54] Nadolski V. and Sykora M. (2015) Model uncertainties of steel members. In Proc. the ESREL

2015 European Safety and Reliability Conference, 7-10 September, 2015, Zürich, Switzerland. [55] Gulvanessian H. M., Holickỳ M., Cayot L. G. and Schleich J. B. (1999) Reliability analysis of

a steel beam under fire design situation. In Proc. the 2nd European Conference on Steel and Composite Structures, Eurosteel 1999. May 26 – 29, 1999, Prague, Czech Republic.

[56] MSZ EN 1993-1-2:2013 (2013) Eurocode 3, Design of steel structures – Part 1-2: General

rules – Structural fire design. Hungarian Standards Institution. Budapest, Hungary. [57] Jeffers A. E., Jia Z., Shi K., Guo Q. (2013) Probabilistic Evaluation of Structural Fire

Resistance, Journal of Fire Technology, Springer. 49. p. 793-811. [58] MSZ EN 1991-1-2:2005 (2005) Eurocode 1, Action on structures – Part 1-2: General actions

– actions on structures exposed to fires. Hungarian Standard Institution. Budapest, Hungary. [59] Guo Q. and Jeffers A. E. (2015) Finite Element Reliability Analysis of Structures Subjected to

Fire. Journal of Structural Engineering, ASCE. 141-4.


106

[60] Li Q. G., Zhang C. and Wang Y. C. (2014) Probabilistic Analysis of steel columns protected by intumescent coatings subjected to natural fires. Structural Safety. 50. p. 16-26.

[61] Boko I., Torić N. and Peroš B. (2010) Reliability of Steel Structures under Fire Conditions. In. proc. 6th International Seminar on Fire and Explosion Hazards. 2010, Leeds, United Kingdom.

[62] Boko I., Torić N. and Peroš B. (2013) Probabilistic analysis of the fire resistance of a steel

roof structure exposed to fire. In: Deodatis G., Ellingwood B. R.. Frangopol D. M. (eds.), Safety, Reliability, Risk and Lyfe-Cycle Performance of Structures and Infrastructures, Taylor & Francis Group. p. 4361-4366. London.

[63] ISO 2394 (2015) General principles on reliability for structures [64] Holickỳ M. (2011) The target reliability and working life. In Safety and Security IV.

Southempton, Boston, USA. WIT Press. [65] Ellingwood B. and Galambos T. V. (1982) Probability-Based criteria for structural design.

Structural Safety. 1. p. 15-26. [66] Dymiotis C. (2002) Reliability Based Code Calibration for Earthquake-Resistant Design. In

Proc. JCSS Workshop on Reliability Based Code Calibration, Zürich, Switzerland. [67] Zsarnóczay Á. (2013) Experimental and Numerical Investigation of Buckling Restrained

Braced Frames for Eurocode Conform Design Procedure Development. PhD Dissertation. Budapest University of Technology and Economics. Budapest.

[68] Dunai L. and Papp Z. (2015) High Performance Steel Structural System – HighPerFrame Research and Development Project. MAGÉSZ Steel Structures, 1st special issue, 3-5.

[69] Polylack A manual (2015) Technical data sheet of Polylack A intumescent painting (in Hungarian). Dunamenti Tűzvédelem Hungary Ltd. (http://www.dunamenti.hu/).

[70] MSZ EN 13381-8:2013 (2013) Test methods for determining the contribution to the fire resistance of structural members. Part 8: Applied reactive protection to steel members. Hungarian Standards Institution. Budapest, Hungary.

[71] OTSZ 5.0 (2014) 54/2014. (XII. 5.) Edict of Hungarian Home Secretary. [72] TvMI 5.1:2015.03.05. (2015) Fire Protection Technical Guideline – Planning, design and

installation of fire alarm systems (in Hungarian). Edict of Hungarian Home Secretary. [73] Horváth L. and Kövesdi B. (2015) Innovative design methods and solutions in fire design of

steel frames (in Hungarian). MAGÉSZ Steel Structures, 1st special issue, 13-20. [74] MSZ EN 1993-1-8:2012 (2012) Eurocode 3, Design of steel structures – Part 1-8: Design of

joints. Hungarian Standards Institution. Budapest, Hungary. [75] MSZ EN 1993-1-1:2009 (2009) Eurocode 3, Design of steel structures – Part 1-1: General

rules and rules for buildings. Hungarian Standards Institution, Budapest, Hungary [76] Lin C. Y. and Wu W. H. (2004) Self-organizing adaptive penalty strategy in constrained

genetic search. Structural and Multidisciplinary Optimization. 26. p. 417-428. [77] Racwitz R. and Fiessler B. (1977) An algorithm for calculation of structural reliability under

combined loading. Berichte zur Sicherheitstheorie der Bauwerke. Lab. f. Konst. Ingb., München.

[78] Hohenbichler M. and Rackwitz R. (1981) Non-normal dependent vectors in structural safety. J. Engrg. Mech. Div. ASCE, 107(6). p. 1227-1238.

[79] Dunai L., Horváth L., Joó A., Kövesdi B., Vigh L. G. et al. (2015) HighPerFrame – Research

and development of high performance steel structure and sheeting system (in Hungarian). Final research report. BME Department of Structural Engineering, Budapest, Hungary.

[80] Faber M. H. (2009) Risk and Safety in Engineering – Lecture notes. ETH Zürich. 2009


107

[81] Ditlevsen O. V. (1979) Narrow reliability bounds for structural systems. Journal of Structural Mechanics. 4. p. 453-472.

[82] McGrattan K., Hostikka S., McDermott R., Floyd J., Weinschenk C. and Overholt K. (2013) Fire Dynamics Simulator v6.0.1 User’s Guide, NIST Special Publication, USA.

[83] Cadorin J. F., Pintea D. and Franssen J. M. (2001) The Design Fire Tool OZone V2.0 –

Theoritical Description and Validation on Experimental Fire Tests, 1st draft. University of Liege, Belgium.

[84] Holickỳ M., Schleich J. B. (2001) Modelling of a Structure under Permanent and Fire Design

Situation, in Safety, Risk, Reliability – Trends in Engineering, 21-23 March 2001, Malta, p. 1001-1006.

[85] Tillander K. (2004) Utilisation of statistics to assess fire risks in buildings. PhD dissertation. Helsinki University, Finland.

[86] Vassart O., Cajot L. G. and Brasseur M. (2008) DIFISEK Program: Dissemination of Fire

Safety Engineering Knowledge documents - WP1 Thermal and mechanical actions. [87] Handbook 5 – Leonardo da Vinci Pilot Project (2005) Implementation of Eurocodes – Design

of Buildings for the Fire Situation, Luxemburg. [88] Couto C., Vila Real P., Ferreira J. and Lopes N. (2014) Numerical validation of the general

method for structural fire design of web-tapered beams. In Proc. the 7th European Conference on Steel and Composite Structures, Eurosteel 2014. September 10-12, 2014, Naples, Italy.

[89] Hamilton S. R. (2011) Performance-based fire engineering for steel framed structures: A probabilistic methodology. PhD dissertation, Stanford University, USA.

[90] Zhang C., Li Q. ans Wang C. Y. (2014) Analysis of steel columns protected by intumescent coatings subjected to natural fires. Structural Safety. 50. p. 16-16.

[91] Sỳkora M. (2002) Reliability analysis of a steel frame. Acta Polytechnica. 42. pp. 27-34. [92] Szalai S., Antofie T., Barbose P., Bihari Z., Lakatos M., Spinoni J., Szentimrey T. and Vogt J.

(2012) The CARPATCLIM project: creation a gridded Climate Atlas of the Carpathian Region for 1961-2010 and its use in the European Drought Observatory of JRC. In Proc. 12th EMS Annual Meeting and 9th European Conference on Applied Climatology (ECAC). 10 – 14 September, 2012, Lodz, Poland.

[93] MSZ EN 1991-1-3:2005 (2005) Eurocode 1, Actions on structures – Part 1-3: General actions – Snow loads. Hungarian Standards Institution. Budapest, Hungary.

[94] MSZ EN 1991-1-4:2007 (2007) Eurocode 1, Actions on structures – Part 1-4: General actions

– Wind loads. Hungarian Standards Institution. Budapest, Hungary. [95] McKenna F. and Feneves G. L. (2012) Open system for earthquake engineering simulation.

Pacific Earthquake Engineering Research Center, USA. [96] Jiang J. and Usnami A. (2013) Modeling of steel frame structures in fire using OpenSees.

Computers and Structures. 118. pp. 90-99. [97] OpenSeesWiki (2015) (http://opensees.berkeley.edu/) [98] Rubert A., Schaumann P. (1985) Tragverhalten stählerner Rahmensysteme bei

Brandbeanspruchung (in German). Stahlbau. 54, No.9, pp. 280-287. [99] Li Q. G., Lou G. B., Zhang C. and Wang Y. C. (2012) Assess the fire resistance of intumescent

coatings by equivalent constant thermal resistance. Fire Technology. 48. pp. 529-546. [100] European Commission for Constructional Steelwork (ECCS) (1985) Design manual on the

European recommendation for the fire safety of steel structures (Report). Brussels, Belgium. [101] MSZ EN 1993-1-5:2012 (2012) Eurocode 3, Design of steel structures – Part 1-8: Plated

structural elements. Hungarian Standards Institution. Budapest, Hungary.


108

[102] Moss P. J., Dhakal R. P:, Bong W. A. and Buchanan A. H. (2009) Design of steel portal

frame buildings for fire safety. Journal of Constructional Steel Research. 65. pp. 1216-1224. [103] Cameron N. J. K. and Usnami A. S. (2005) New design method to determine the membrane

capacity of laterally restrained composite floor slabs in fire Part 1: theory and method. The Structural Engineer. 83. pp. 28-33.

[104] Liu Z., Atamturkur S. and Juang H. C. (2014) Reliability based multi-objective robust

optimization of steel moment resisting frame considering spatial variation of connection

parameters. Engineering Structures. 76. pp. 393-403. [105] Baker J. W. (2008) An introduction to Probabilistic Seismic Hazard Analysis (PSHA).

Stanford University. [106] EFEHR (2015) European Facility for Earthquake Hazard and Risk

(http://www.efehr.org:8080/jetspeed/portal/) [107] Federal Emergency Management Agency (2009) FEMA-P695 Quantification of Building

Seismic Performance Factors. USA. [108] MSZ EN 1993-1-3:2007 (2007) Eurocode 3, Design of steel structures – Part 1-3:

Supplementary rules for cold formed members and sheeting. Hungarian Standards Institution. Budapest, Hungary.

[109] Uang C. M., Smith M. D. and Shoemaker W. L. (2011) Earthquake Simulator Testing of Metal Building Systems. In Proc. Structures Congress 2011 ASCE. 14 – 16 April, 2011, Las Vegas, Nevada, USA.

[110] Joó A. and Mayer R. (2015) Cladding stiffness of industrial steel buildings – Part 2: Full

scale test (in Hungarian). MAGÉSZ Steel Structures, 1st special issue, 66-77. [111] Fajfar P. (2000) A Nonlinear Analysis Method for Performance Based Seismic Design.

Earthquake Spectra. 16(3). pp. 573-592. [112] Miranda E. and Taghavi S. (2003) Response Assessment of Nonstructural Building

Elements. PEER Report 2003/05, Pacific Earthquake Engineering Research Center, Berkeley, CA, USA.

[113] Simon J. (2016) Seismic Performance and Damage Assessment of Hungarian Road Bridges. PhD Dissertation. Budapest University of Technology and Economics. Budapest.

[114] Gottfried J. S., Rein G., Busby L. A. and Torero J. L. (2010) Experimental review of the homogenous temperature assumption in post-flashover compartment fires. Fire Safety Journal, 45 pp. 249-261.

[115] ASCE/SEI 7-10 (2013) Minimum design loads for buildings and other structures. American Society of Civil Engineers.

[116] Andrade A., Camotim D. and Providencia P. (2005) Critical moment formulae for doubly

symmetric web-tapered I-section steel beams acted by ending moments. In Proc. 4th European Conference on Steel and Composite Structures, Eurosteel 2005. Maastricht, Netherlands.

[117] Kollár L. (2008) Structural Stability in Engineering Practice. Taylor & Francis. London, UK.

[118] European Commission for Constructional Steelwork (ECCS) Technical Committee 8 (2006) Structural Stability, Resolution of ECCS/TC8 with respect to the general method in EN

1993-1-1 (Report). [119] MathWorks MATLAB 2016a User’s manual, 2016.


109

Appendix A – Parametric study tables and results

#

Demand (resistance)

cs [€/kg]

see Eq. (4)

cp

[€/ (mm·m2)] see Eq. (5)

C2 [€/m2]

see Eq.(3)

Active safety measure

Cf

[m €] see Eq. (3)

Fire curve

Equipment [kN/m2]

A –

ref

eren

ce c

ase

gro

up

1 R30 2.25 24 40 smoke detection 3.0 1 0.2









B










C

19 R30 2.25 - 40 smoke detection 3.0 1 0.2



D




E




F†




G

31 R45 2.25 24 - - 3.0 1 0.2

32 R45 2.25 24 - - 3.0 2 0.2

33 R45 2.25 24 - - 3.0 3 0.2

H

34 R45 2.25 24 75 sprinkler system 3.0 1 0.2

35 R45 2.25 24 75 sprinkler system 3.0 2 0.2

36 R45 2.25 24 75 sprinkler system 3.0 3 0.2 R30, R45 and R60 refer to 30, 45 and 60 minutes time demand, respectively; m EUR refers to million euros. †A fix cost component, namely the cost of sheeting and bracings, Csh, is generally set to 25 €/m2, however, in case of group F Csh is set to 50 €/m2

Table A-1 – Investigated fire design cases within parametric study


110

# hc1 - hc2 x tw,c* +

bc x tf,c

hb1 - hb2 x tw,b* +

bb x tf,b t p,c

1

t p,c

2

t p,b

2

t p,b

1

t p,c D/C

[%] βopt

0.4 β

0.9 C0 C1 C2

CLC

††

1 215-630x6+190x11 210-765x6+155x10 0.6 0.6 0.8 0.4 0.2 100 3.15 3.32 56.4 4.2 27.4 90.5

2 260-555x8+185x12 245-625x9+150x11 0.7 0.9 1.3 0.6 0.2 99 2.97 3.12 59.3 5.9 27.4 97.1

3 235-645x6+165x14 225-560x8+165x12 0.1 0.3 0.3 0.0 0.5 99 2.82 2.84 59.1 1.3 27.4 95.0

4 225-690x6+190x10 220-725x6+170x9 0.4 0.4 0.5 0.3 0.0 99 3.27 3.44 56.3 2.8 27.4 88.1

5 205-635x6+180x12 215-645x6+175x10 0.6 0.8 1.2 0.5 0.3 99 3.02 3.19 57.1 5.5 27.4 93.8

6†

7 220-685x6+200x9 225-760x6+180x8 0.0 0.0 0.0 0.0 0.0 100 3.45 3.51 55.8 0.0 27.4 84.1

8 190-650x6+195x10 235-775x6+175x8 0.4 0.4 0.6 0.3 0.0 100 3.21 3.38 55.9 3.1 27.4 88.4

9 165-575x6+195x12 230-635x6+175x10 0.5 0.5 1.3 0.4 0.0 100 3.11 3.28 57.3 4.9 27.4 92.4

10 130-720x6+195x11 285-730x6+170x10 0.5 0.6 0.8 0.4 0.2 100 3.04 3.21 57.7 4.3 27.4 93.0

11 130-750x6+180x12 330-550x8+165x12 0.5 0.8 0.9 0.3 0.2 100 2.85 2.94 60.1 4.5 27.4 98.5

12 150-755x6+165x14 300-560x8+155x13 0.1 0.2 1.0 0.1 0.4 100 2.82 2.83 60.2 2.9 27.4 97.7

13 180-735x6+180x12 340-685x6+175x10 0.3 0.4 0.6 0.3 0.0 100 3.28 3.38 58.5 3.1 27.4 90.6

14 145-820x6+185x11 235-710x6+165x11 0.5 0.7 1.0 0.5 0.2 100 2.92 3.06 57.8 5.0 27.4 95.4

15 205-760x6+190x11 340-490x7+185x12 0.0 0.0 0.3 0.0 0.0 100 2.82 2.82 60.4 0.7 27.4 95.7

16 160-805x6+200x9 250-800x6+185x8 0.0 0.0 0.0 0.0 0.0 100 3.41 3.57 56.5 0.0 27.4 84.8

17 115-745x6+200x10 270-735x6+170x10 0.5 0.5 0.6 0.3 0.0 100 3.16 3.33 57.3 3.4 27.4 90.5

18 120-740x6+185x12 315-640x6+175x11 0.4 0.6 0.9 0.3 0.1 99 2.95 3.11 58.8 4.2 27.4 95.1

19†

20 175-705x6+185x10 210-715x6+185x8 0.0 0.0 0.0 0.0 0.0 100 2.82 2.87 55.8 0.0 27.4 90.4

21 195-650x6+215x9 215-710x6+185x8 0.0 0.0 0.0 0.0 0.0 100 3.43 3.60 55.9 0.0 27.4 84.2

22 170-535x9+170x22 200-595x12+135x20 1.4 1.7 2.3 1.3 0.6 98 3.59 3.73 66.1 10.1 27.4 104.1

23 250-540x8+200x15 215-595x10+170x13 1.1 1.5 1.9 1.2 0.4 99 3.61 3.76 62.7 9.7 27.4 100.2

24 155-745x6+210x13 235-765x8+175x11 0.6 0.7 1.0 0.5 0.4 89 3.70 3.82 60.5 5.4 27.4 93.6

25 225-555x6+195x12 240-630x6+190x9 0.0 0.1 0.0 0.0 0.0 100 2.82 2.82 57.4 0.1 27.4 92.1

26†

27 170-665x6+205x9 215-700x6+185x8 0.0 0.0 0.0 0.0 0.0 100 2.85 2.95 55.7 0.0 27.4 89.6

28 250-605x6+185x12 230-600x6+200x8 0.0 0.0 0.0 0.2 0.1 100 2.82 2.82 115.1 0.8 54.7 177.8

29 210-550x6+190x13 245-595x6+185x10 0.0 0.0 0.1 0.0 0.2 99 2.82 2.85 116.1 0.6 54.7 178.6

30 190-635x6+210x9 225-740x6+180x8 0.3 0.3 0.4 0.2 0.0 100 3.08 3.25 111.5 4.5 54.7 173.8

31 190-525x10+175x20 195-600x12+145x19 1.6 1.9 2.5 1.5 0.5 99 2.89 3.06 66.2 11.5 0.0 83.5

32 230-585x10+250x15 230-525x8+200x16 1.0 1.6 1.7 1.1 0.0 90 2.95 3.11 67.7 9.8 0.0 82.2

33 230-580x7+230x13 245-490x7+180x13 0.7 0.8 1.3 0.6 0.4 97 3.07 3.24 61.3 6.1 0.0 70.7

34 215-630x6+180x12 225-685x6+190x8 0.0 0.1 0.1 0.1 0.1 100 3.09 3.09 56.5 0.6 51.3 111.4

35 225-635x6+190x11 215-700x6+165x10 0.0 0.0 0.0 0.0 0.0 99 3.09 3.10 56.8 0.0 51.3 111.1

36 205-655x6+195x10 230-765x6+175x8 0.0 0.0 0.0 0.0 0.0 100 3.12 3.22 55.9 0.0 51.3 109.9

The dimensions (hc1, bc, tf,c, etc.) are given in mm unit; C0, C1 and CLC are given in 1000€ unit. C2 is equal to 27400, 54700, 0 and 51300 € for cases #1 - #27, cases #28 - #30, cases #31 - #33 and cases #34 - #36, respectively * There is an additional constraint related to the minimum thickness of the web; the minimum considered plate thickness is 6mm in order to avoid problems related to corrosion and welding † There were numerical problems during the optimization procedure; the algorithm did not find stable solutions. †† The presented CLC values are calculated with ρ=0.4 correlation coefficients but without damage cost by moderate fire and damage cost by intervention.

Table A-2 – Optimized structural configurations in considered fire design cases


111

# hc1 - hc2 x tw,c +

bc x tf,c hb1 - hb2 x tw,b +

bb x tf,b tp,c1 tp,c2 tp,b2 tp,b1 tp,c

β

0.4 β

0.9 C0 C1 C2

CLC

†† ΔCLC [%]†

1 185-665x6+205x9 215-700x6+185x8 0.54 0.58 0.61 0.58 0.6 2.97 3.12 55.7 4.2 27.4 91.8 1.4

2 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 2.89 3.02 55.7 10.8 27.4 99.7 2.7

3 185-665x6+205x9 215-700x6+185x8 2.15 2.30 2.47 2.30 2.4 2.84 2.92 55.7 16.9 27.4 106.8 12.3

4 185-665x6+205x9 215-700x6+185x8 0.54 0.58 0.61 0.58 0.6 3.13 3.24 55.7 4.2 27.4 89.9 2.1

5 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.01 3.18 55.7 10.8 27.4 97.8 4.3

6 185-665x6+205x9 215-700x6+185x8 2.15 2.30 2.47 2.30 2.4 2.89 3.02 55.7 16.9 27.4 78.4

7 185-665x6+205x9 215-700x6+185x8 0.54 0.58 0.61 0.58 0.6 3.52 3.67 55.7 4.2 27.4 88.0 4.6

8 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.40 3.56 55.7 10.8 27.4 94.9 7.4

9 185-665x6+205x9 215-700x6+185x8 2.15 2.30 2.47 2.30 2.4 3.16 3.33 55.7 16.9 27.4 102.4 10.8

10 130-855x6+210x8 230-815x6+190x8 0.68 0.72 0.72 0.67 0.7 2.95 3.10 56.3 5.5 27.4 93.9 1.0

11 130-855x6+210x8 230-815x6+190x8 1.80 1.90 1.90 1.77 1.9 2.86 2.97 56.3 14.5 27.4 104.5 6.1

12 130-855x6+210x8 230-815x6+190x8 2.35 2.50 2.50 2.30 2.5 2.83 2.88 56.3 18.9 27.4 109.5 12.1

13 130-855x6+210x8 230-815x6+190x8 0.68 0.72 0.72 0.67 0.7 3.24 3.41 56.3 5.5 27.4 91.0 0.4

14 130-855x6+210x8 230-815x6+190x8 1.80 1.90 1.90 1.77 1.9 2.96 3.11 56.3 14.5 27.4 102.8 7.7

15 130-855x6+210x8 230-815x6+190x8 2.35 2.50 2.50 2.30 2.5 2.85 2.95 56.3 18.9 27.4 109.1 14.0

16 130-855x6+210x8 230-815x6+190x8 0.68 0.72 0.72 0.67 0.7 3.46 3.62 56.3 5.5 27.4 90.0 6.1

17 130-855x6+210x8 230-815x6+190x8 1.80 1.90 1.90 1.77 1.9 3.35 3.51 56.3 14.5 27.4 99.4 9.8

18 130-855x6+210x8 230-815x6+190x8 2.35 2.50 2.50 2.30 2.5 3.08 3.25 56.3 18.9 27.4 105.7 11.1

19 185-665x6+205x9 215-700x6+185x8 - - - - - 2.82 2.83 55.7 0.0 27.4 90.3

20 185-665x6+205x9 215-700x6+185x8 - - - - - 2.82 2.87 55.7 0.0 27.4 90.3 -0.1

21 185-665x6+205x9 215-700x6+185x8 - - - - - 3.34 3.50 55.7 0.0 27.4 84.4 0.2

22 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 2.89 3.02 55.7 10.8 27.4 151.7 39.7

23 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.01 3.18 55.7 10.8 27.4 133.1 27.5

24 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.40 3.56 55.7 10.8 27.4 104.0 7.8

25 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 2.89 3.02 55.7 10.8 27.4 94.5 10.4

26 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.01 3.18 55.7 10.8 27.4 94.3

27 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.40 3.56 55.7 10.8 27.4 94.0 12.3

28 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 2.89 3.02 111.4 22.5 54.7 194.4 9.3

29 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.01 3.18 111.4 22.5 54.7 192.5 7.8

30 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.40 3.56 111.4 22.5 54.7 189.6 9.1

31 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 1.93 2.11 55.7 10.8 0.0 146.9 76.0

32 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 2.10 2.32 55.7 10.8 0.0 120.1 46.0

33 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 2.61 2.80 55.7 10.8 0.0 80.1 8.2

34 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.15 3.27 55.7 10.8 51.3 120.3 7.9

35 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.27 3.42 55.7 10.8 51.3 119.4 7.5

36 185-665x6+205x9 215-700x6+185x8 1.38 1.48 1.57 1.48 1.5 3.64 3.79 55.7 10.8 51.3 118.2 7.5

The dimensions (hc1, bc, tf,c, etc.) are given in mm unit; C0, C1, C2 and CLC are given in 1000€ unit. The last column shows the difference in CLC. The structures have been optimized considering only dead, equipment and meteorological loads, thus the D/C ratio of every configuration is 100% in persistent design situation. † Compared to Table A-2; positive value means that cases optimized in persistent design situation resulted higher life cycle cost. †† The presented CLC values are calculated with ρ=0.4 correlation coefficients but without damage cost by moderate fire and damage cost by intervention.

Table A-3 – Persistent design situation optimized structural configurations in fire design situation


112

# Site

Soil type (EC8-1)

Sheeting Cf

[m €] Load case

Equipment [kN/m2]

A –

ref

eren

ce c

ase

g

rou

p

1 Komárom C - 3.0 S1 0.2

2 Komárom C - 3.0 S2 0.2

3 Komárom C - 3.0 S3 0.2

4 Râmnicu S. C - 3.0 S1 0.2

5 Râmnicu S. C - 3.0 S2 0.2

6 Râmnicu S. C - 3.0 S3 0.2

B

7 Komárom C LTP45+Z200 3.0 S1 0.2

8 Komárom C LTP45+Z200 3.0 S2 0.2

9 Komárom C LTP45+Z200 3.0 S3 0.2

10 Râmnicu S. C LTP45+Z200 3.0 S1 0.2

11 Râmnicu S. C LTP45+Z200 3.0 S2 0.2

12 Râmnicu S. C LTP45+Z200 3.0 S3 0.2

C

13 Komárom C - 3.0 S1 1.0

14 Komárom C - 3.0 S2 1.0

15 Komárom C - 3.0 S3 1.0

D

16 Komárom C LTP45+Z200 3.0 S1 1.0

17 Komárom C LTP45+Z200 3.0 S2 1.0

18 Komárom C LTP45+Z200 3.0 S3 1.0

E

19 Komárom C - 30.0 S1 0.2

20 Komárom C - 30.0 S2 0.2

21 Komárom C - 30.0 S3 0.2

22 Komárom C LTP45+Z200 30.0 S1 0.2

23 Komárom C LTP45+Z200 30.0 S2 0.2

24 Komárom C LTP45+Z200 30.0 S3 0.2

F

25 Komárom C - 0.3 S1 0.2

26 Komárom C - 0.3 S2 0.2

27 Komárom C - 0.3 S3 0.2

28 Komárom C LTP45+Z200 0.3 S1 0.2

29 Komárom C LTP45+Z200 0.3 S2 0.2

30 Komárom C LTP45+Z200 0.3 S3 0.2 m EUR refers to million euros.

Table A-4 – Investigated seismic design cases within parametric study


113

# hc1 - hc2 x tw,c+bc x

tf,c [mm] hb1 - hb2 x tw,b+bb x

tf,b [mm] db

[mm] βopt

D/C [%]

C0

[1000 €] C1

[1000 €] CLC

[1000 €]

1 245-570x6+245x11 180-790x6+210x14 24 2.86 75.2 50.9 1.9 59.2

2 250-535x6+280x14 215-785x6+250x18 28 2.52 62.1 57.4 2.6 77.7

3 300-800x6+320x17 430-715x7+295x19 36 2.53 41.3 65.9 4.4 87.4

4 215-465x6+250x12 260-410x6+215x15 24 2.86 81.0 51.9 1.9 60.2

5 205-640x6+310x15 215-785x8+300x19 30 2.74 48.3 62.9 3.0 75.2

6 295-750x6+295x14 210-795x7+315x20 36 2.68 47.0 63.9 4.4 79.3

7 215-450x7+195x13 305-500x6+190x13 24 2.82 90.3 49.7 1.9 58.9

8 260-400x6+265x13 255-460x6+220x14 30 2.67 76.2 52.3 3.0 66.7

9 245-495x6+280x15 260-445x7+270x17 38 2.59 57.1 58.3 4.9 77.6

10 210-415x7+185x14 270-440x6+185x14 22 3.03 96.9 49.7 1.6 55.0

11 220-400x6+210x17 235-420x6+220x16 30 2.77 73.4 53.4 3.0 64.8

12 215-580x6+285x12 185-775x6+240x16 38 2.72 57.0 54.6 4.9 69.3

13 235-425x7+280x18 195-820x6+205x19 28 2.76 75.8 57.4 2.6 68.7

14 255-620x6+320x21 360-600x6+335x22 30 2.40 49.3 71.1 3.0 98.7

15 345-750x6+350x17 335x560x6+360x23 40 2.42 47.0 72.7 5.4 101.4

16 215-450x7+230x16 190-810x6+215x19 30 2.71 87.4 55.7 3.0 68.8

17 220-490x6+265x17 275-490x6+230x20 32 2.43 75.6 58.5 3.5 84.6

18 285-505x6+315x16 280-525x6+270x19 38 2.47 67.7 61.2 4.9 86.3

19 255-770x6+335x20 410-815x7+300x20 38 3.48 35.1 68.9 4.9 81.3

20†

21 430-770x6+405x25 440-810x8+390x25 58 2.99 26.8 86.3 11.3 139.5

22†

23 240-690x7+310x20 260-800x6+345x22 44 2.84 36.6 70.9 6.5 145.1

24 310-640x6+370x19 260-775x6+375x24 52 2.84 34.2 75.9 9.1 152.7

25 240-575x6+210x9 210-480x6+190x12 20 2.58 99.6 47.6 1.3 50.4

26 265-440x6+225x11 185-720x6+165x12 20 2.22 99.6 47.6 1.3 52.9

27 265-505x6+230x11 220-590x6+200x13 26 2.19 95.9 49.6 2.3 56.2

28 265-610x6+200x9 240-570x6+170x11 20 2.60 93.5 46.8 1.3 49.5

29 250-435x6+225x10 220-400x6+220x10 22 2.27 99.7 47.6 1.6 52.7

30 280-465x6+225x10 190-610x6+215x10 26 2.18 99.3 47.7 2.3 54.4

The dimensions (hc1, bc, tf,c, etc.) are given in mm unit; C0, C1 and CLC are given in 1000€ unit. † There were numerical problems during the optimization procedure; the algorithm did not find stable solutions.

Table A-5 – Optimized structural configurations in considered seismic design cases


114

Appendix B – EC design of steel frames for extreme

effects

B.1. Prescriptive seismic design

The seismic resistant design of EC8-1 is based on the structural response of structure subjected

to seismic excitation. The response and the seismic forces are mainly dependent on the ground

acceleration, the structural stiffness, the seismic mass and the structural ductility. This seismic mass

may be calculated using combination of actions for seismic design situation according to EC0 and

EC8-1 standards:

∑∑ ⋅⋅++ ikiEdjk QAG ,,2, """" ϕψ ,

where Gk, Qk and AEd represent characteristic value of dead loads, characteristic value of variable

effects and design value of seismic action, respectively. ψ2 is the so-called quasi-permanent

combination coefficient and φ considers the correlation among the storeys’ occupancy.

Fig. B-1 – Determination of internal forces

The seismic forces may be obtained with the help of different methods, namely with lateral

force method (LFM; linear static analysis), modal response spectrum analysis (MRSA; linear

dynamic analysis), pushover analysis (PA; nonlinear static analysis) or time history analysis (THA;

nonlinear dynamic analysis); depending on the regularity, the importance and the behaviour of the

investigated structure. In this section, the seismic design of the frame is illustrated with the help of


115

MRSA since it may be the most common method applied by practicing designers. The steps of

determination of internal forces is broadly illustrated in Fig. B-1.

After calculation of seismic masses, the next step is the vibration analysis where the vibration

periods of the structure are calculated. Finite element method (FEA) is the most common way to

solve the following system of differential equations (damping is neglected) for multiple-degree-of-

freedom (MDOF) systems in order to find the vibration periods:

( ) ( ) 0KUUM =+ tt&& ,

where M, K and U are the mass matrix, the stiffness matrix and the displacement vector

respectively. The number of rows is equal to the sum of masses’ degree of freedoms. The vibration

periods are nonlinearly dependent on the structural stiffness (k) and the seismic mass (m), e.g. in

case of a single-degree-of-freedom (SDOF) structure:

k

mT π2= .

The design seismic forces related to the nth vibration period are the product of modal masses

(EC8-1) of nth vibration shape and design spectral acceleration read from the so-called response

spectrum EC8-1 (Fig. B-1). Based on the Hungarian National Annex of EC8-1, I considered Type

I response spectrum in this research given by the following formulae:

( )

−+=

3

252

3

2

q

,

T

TSaTS

B

gd if 0 ≤ T ≤TB,

( )q

,SaTS gd

52= if TB ≤ T ≤TC,

( )

= gC

gd a,;T

T

q

,SamaxTS 20

52 if TC ≤ T ≤TD,

( )

= gDC

gd a,;T

TT

q

,SamaxTS 20

522

if TD ≤ T.

In the equations above ag, q, S and T are the peak ground acceleration (PGA) on rock, the

behaviour factor (EC8-1), the soil factor (EC8-1) and the vibration period, respectively. The

response spectrum is linearly dependent on the peak ground acceleration that is selected with the

help of hazard curve (Section 5.2.) for a specific site.


116

The next step is to perform linear static analyses in order to calculate the displacements, internal

forces, etc. from seismic forces. Calculated effects in each vibration shape may be combined with

square root of the sum- of the squares (SRSS) combination rule of EC8-1 (equation below).

Resultant seismic effects calculated from seismic excitation related to different directions may be

combined with SRSS or 30% (EC8-1) combination rules.

∑= 2i,EE EE .

In case of elastic elements (no energy dissipation is allowed via plasticity), seismic verification

may be performed according to the rules of EC3-1-1 while the adequacy of dissipative elements

may be checked according to material specific rules of EC8-1. The stability verification of beam

and column elements may be carried using the so-called General Method (GM) (EC3-1-1), where

the in-plane stability failure is considered using imperfect structural model (Fig. 4-10), while the

out-of-plane stability failure is taken into consideration with reduction factors:

( )( )

( )( ) yy

y

y fW

M

fA

N

⋅+

⋅=

x

x

x

xη ,

ηα

1=k,ult ,

111

−

+=

LTB,crFB,cr

op,cr ααα ,

op,cr

k,ultop

α

αλ = ,

( ) ( )2

201

2

201 22opopLTB

LTB

opopFB

FB

.;

. λλαΦ

λλαΦ

+−+=

+−+= ,

2222

11

opLTBLTB

LTB

opFBFB

FB ;λΦΦ

χλΦΦ

χ−+

=−+

= ,

( )LTBFBop ;min χχχ = ,

011 .op

M ≤⋅

χηγ

.

The steps of checking stability failure are presented above, where χop and αult,k are the reduction

factor taking into account the out-of-plane stability failure and the minimum load multiplier in


117

order to reach the characteristic resistance of the critical cross section, respectively. FB and LTB

abbreviations refer to flexural and lateral torsional buckling failure modes. αcr,op is the critical load

amplifier in order to reach the critical intensity of internal forces causing out-of-plane flexural

buckling or lateral torsional buckling, respectively. In this study, the critical bending moment for

tapered steel members is calculated using an approximation from the literature [116], the

calculation is based on an equivalent non-tapered element. The critical load amplifiers for different

stability failures are combined with Dunkerley theorem [117], that can be used as an approximate

superposition technic of critical load amplifiers related to different loading conditions. χop is

calculated as the minimum of reduction factors related both of the flexural and lateral torsional

buckling failure modes, because the interpolation is considered unsafe [118]. Both of these

reduction factors are obtained using the global relative slenderness.

As can be seen in this section, prescriptive seismic verification of steel structures according to

ENs is highly nonlinear and non-convex problem. In order to find adequate configuration that is

safe in both persistent and seismic design situations, the above described steps may be carried out

at least several times. Optimal design is even more difficult and time-consuming.

B.2. Prescriptive fire design

According to EC0, fire design of structures is performed considering accidental combination of

effects in the following form:

∑∑ ⋅+⋅++≥ 1

21111 >i

i,ki,,k,dj

j,k Q""Q""A""G ψψ ,

where ψ1 is the combination factor for variable action on frequent value (EC0), while Ad represents

the design value of indirect effects of thermal action in fire. The verification may be carried out via

member analysis, analysis of a part of the structure or analysis the whole structure. Due to the fact

that analysis of a part of the structure or analysis of the whole structure require complex imperfect

models, nonlinear analysis and better characterization of fire effect, member analysis based on

prescriptive rules is the most common fire design method among structural engineers. For this

reason, main steps (Fig. B-2a) of prescriptive fire design according to ENs is presented based on

member analysis.


118

a)

b) Fig. B-2 – a) Main steps of prescriptive fire design based on member analysis; b) reduction factors related to

strength and stiffness of steel material (EC3-1-2)

As a first step and as the basis of the calculation of structural temperatures, the gas temperatures

in the compartment need to be calculated. EC1-1-2 standard gives nominal temperature curves to

represent the temperatures in the fire compartment. Mostly the so-called standard temperature-time

curve (Fig. B-2a) is used:

( )1834520 10 ++= tloggθ ,

where t is the time in minutes. The steel temperature for protected and unprotected section can be

calculated with incremental, time step formulae from EC3-1-2 standard. In case of protected

sections, the steel temperature may be evaluated as follows:

( )( ) ( ), , 1 10

, ,11 3

p p g t s t

s t g t

p s s

A Vt e

d c

ϕλ θ θ

θ θρ ϕ

−−∆ = ∆ − − ∆

+ p p

p p

s s

cd A V

c

ρϕ

ρ= ,

where t, λp, dp, cp, ρp, cs, ρs and Ap/V are the time, thermal conductivity, thickness, specific heat,

unit mass of insulation material, specific heat, unit mass of protected steel and section factor of the

protected section, respectively. Δθg,t and Δθs,t stand for the gas and steel temperature at time step t,

respectively. According to Hungarian fire safety regulation [71], the safety of the structure need to

be confirmed within a time interval, thus steel temperature need to be evaluated considering 15,

30, 60, etc. minutes exposition in fire. Knowing the maximum steel temperature within the

examined time interval is essential since significant reduction in strength and stiffness of steel

material need to be considered according to EC3-1-2 (Fig. B-2b) on elevated temperatures.


119

For example, the stability verification of uniform beam or column elements subjected to

bending and compression may be carried using member analysis based on the rules of EC3-1-2. In

case of member analysis, the internal forces in the element may be determined for t=0 time without

considering thermal elongation during the linear structural analysis. The boundary conditions may

also be considered unchanged (EC3-1-2).

1≤++

fi,M

y

,yz

Ed,fi,zz

fi,M

y

,yy

Ed,fi,yy

fi,M

y

,yfimin,

Ed,fi

fkW

Mk

fkW

Mk

fAk

N

γγγχ θθθ

,

1≤++

fi,M

y

,yz

Ed,fi,zz

fi,M

y

,yyfi,LT

Ed,fi,yLT

fi,M

y

,yfi,z

Ed,fi

fkW

Mk

fkW

Mk

fAk

N

γγχ

γχ θθθ

,

θ

θθ λλ

,E

,y

k

k= ,

2

1 2θθ

θ

λλαΦ

++= ,

22

1

θθθ λΦΦχ

−+=fi

,

where ky,θ and kE,θ are the reduction factors related to yield strength and Young’s modulus, while

ky, kz and kLT are interaction coefficients considering the interaction between flexural buckling and

lateral torsional buckling failure modes (EC3-1-2). The calculation of relative slenderness is based

on the value of relative slenderness on normal temperature, however, different buckling length may

be considered.

The fact that the steel temperature is dependent on steel section dimensions and material

properties are sensitive to the steel temperature makes the design highly iterative and nonlinear.

This complexity is increased by nonlinearity of verification procedure. Similarly to seismic design,

fire design of steel structures is iterative, non-convex and time-consuming.


120

Appendix C – Evaluation of limit state function in

case of fire design

Determination of gas temperatures in the compartment (fire curve)

Fig. C-1 – Fire curve (compartment gas temperature as a function of the time)

Determination of steel temperatures in every time step

Fig. C-2 – Calculation of steel temperatures

The steel temperatures related to each cross section are determined with taking into

consideration of steel temperature uncertainty factor (Section 4.6).

( )( ) ( ), , 1 10

, ,11 3

p p g t s t

s t g t

p s s

A Vt e

d c

ϕλ θ θ

θ θρ ϕ

−−∆ = ∆ − − ∆

+

p p

p p

s s

cd A V

c

ρϕ

ρ=


121

Determination of internal forces in every time step

Fig. C-3 – Calculation of internal forces

The internal forces are calculated with OpenSeesThermal FE software in every time step. Since

the analysis considers the steel temperatures, thermal extension and the degradation of steel

material’s properties, the calculated internal forces vary in time.

Resistance time related to stability failure of column and beam elements

y,yy

y

y,y fkW

M

fkA

N

⋅⋅+

⋅⋅=

θθ

η

ηα θ

1=,k,ult

111

−

+=

θθθ αα

α,LTB,cr,FB,cr

,op,cr

θ

θθ

αα

λ,op,cr

,k,ult,op =

2

1

2

1 22θθθ

θθθθ

θ

λλαΦ

λλαΦ ,op,op

,LTB

,op,op

,FB ;++

=++

=

2222

11

θθθθθθ λΦΦχ

λΦΦχ

,op,LTB,LTB

fi,LTB

,op,FB,FB

fi,FB ;−+

=−+

=


122

( )fi,LTBfi,FBfi,op

;min χχχ =

01.fi,op

=χη

→ [ ]mint ,R 1

The D/C ratio related to the stability failure of beams and columns is evaluated in every time

step. Where D/C ratio exceeds 1.0, the current time step is considered the resistance time. Every

beam and column is characterized with different resistance time, the minimum of these values is

selected.

Resistance time related to shear web buckling failure of columns and beams

3

thfkV

wy,yfi,w

fi,R,bw,z

⋅⋅⋅⋅= θχ

τ

θε

λkt.

hw,w

⋅⋅⋅=

437 →

fi,wχ

yf.

235850=ε

01.V

V

fi,R,bw,z

= → [ ]mint ,R 2

The D/C ratio related to the shear web buckling failure of beams and columns is evaluated in

every time step. Where D/C ratio exceeds 1.0, the current time step is considered the resistance

time. Every beam and column is characterized with different resistance time, the minimum of these

values is selected.


123

Resistance time related to strength failure of connections

Fig. C-4 – The properties of the connections [73]

0.1,,

=fiRy

y

M

M → [ ]mint ,R 3

The D/C ratio related to the strength failure of connections is evaluated in every time step.

Where D/C ratio exceeds 1.0, the current time step is considered the resistance time. Every

connection is characterized with different resistance time, the minimum of these values is selected.

Resistance time related to plastic sway mechanism

Fig. C-5 – Plastic sway mechanism

( )( ) ∞→tTu s → [ ]mint ,R 4

Value of limit state function

[ ] ( )4321 ,R,R,R,RR t;t;t;tminmint =

RtG

301−= is the value of limit state function in case of R30 demand level.

0 0.�� 0.�� 0.�� 0.�� 0.� 0.��0

0.�

1

1.�

2

2.�

3

3.�

4x 10

8

φ [rad]

M� N��

]

Colmn a�� onn. T��0°C

Colmn a�� onn. T��00°C

Colmn-�a� onn. T��0°C

Colmn-�a� onn. T��00°C

Ridg� onn� e�on T��0°C

Ridg� onn� e�on T��00°C


124

Appendix D – Evaluation of limit state function in

case of seismic design

Vibration analysis

Fig. D-1 – Vibration analysis

Vibration analysis considering both principal directions is performed in OpenSess FE

environment. Seismic masses are determined according to the regulations of EC0 standard in

seismic load combination.

Seismic analysis in transversal direction

Fig. D-2 – Determination of design spectral accelerations

In transversal direction, the structure is very sensitive to stability failure modes thus

considerably seismic energy cannot be absorbed. For this reason, linear structural analysis is

performed in transversal direction. The seismic forces are determined as the product of design

spectral acceleration (Fig. D-2) and the seismic mass. According to EC8-1, q=1.5 behaviour factor

is applied considering some energy dissipation due to e.g. stir of the connections. Internal forces

and deformations are calculated in linear static analysis using OpenSees FE environment.

Sa

T


125

Fig. D-3 – Calculation of internal forces and displacements based on linear static analysis in OpenSees

Seismic analysis in longitudinal direction

Fig. D-4 – Calculation of target displacement

In case of dissipative design, the global response of the structure is calculated by geometrical

and material non-linear static analysis (pushover analysis) in OpenSeses and the seismic

verification is completed at the target displacement level (Fig. D-4). Seismic forces are distributed

among the nodes proportionally to the first vibration shape and to the distribution of seismic

masses.

Combination of seismic analysis results

Internal forces, displacements and deformations calculated in different directions are combined

with square root of the sum- of the squares (SRSS) combination rule of EC8-1:

22y,Ex,EE EEE +=

Check stability failure of column and beam elements

( )( )

( )( ) yy

y

y fW

M

fA

N

⋅+

⋅=

x

x

x

xη ,

ηα

1=k,ult

,

u [mm]

F [kN]


126

111

−

+=

LTB,crFB,cr

op,cr ααα ,

op,cr

k,ultop

α

αλ = ,

( ) ( )2

201

2

201 22opopLTB

LTB

opopFB

FB

.;

. λλαΦ

λλαΦ

+−+=

+−+= ,

2222

11

opLTBLTB

LTB

opFBFB

FB ;λΦΦ

χλΦΦ

χ−+

=−+

= ,

( )LTBFBop ;min χχχ = ,

op

CDχη

=1

The D/C ratio is evaluated related to the stability failure of every beam and column of the

structure. The highest value is selected and compared to the D/C ratios related to other failure

modes.

Check shear web buckling failure of column and beam elements

3

thfV

wyw

R,bw,z

⋅⋅⋅=χ

τελ

kt.

hww

⋅⋅⋅=

437 → wχ

R,bw,zV

VCD =2

The D/C ratio is evaluated related to the shear web buckling failure of every beam and column

of the structure. The highest value is selected and compared to the D/C ratios related to other failure

modes.

Check strength failure of connections

R,y

y

M

MCD =3


127

The resistance of connections is selected based on [73]. The highest D/C ratio is selected and

compared to the D/C ratios related to other failure modes.

Check strength failure of tension-only wind bracing

yfA

NCD

⋅=4

Tension-only wind bracing elements are checked in terms of tension load bearing capacity

assuming full strength connections. Every bracing element is subjected to different tension force;

the highest D/C ratio is selected and compared to the D/C ratios related to other failure modes.

Deformation check of plastic hinges in the side bracings

According to FEMA-356, the deformation limit for brace in compression is 4Δc (Δc – axial

deformation at expected buckling load), while for brace in tension is 7Δy (Δy – axial deformation at

expected tensile yielding load). D/C5 ratio is calculated as the ratio of calculated deformations in

the elements and the deformation limits; the highest D/C ratio is selected and compared to the D/C

ratios related to other failure modes.

Value of limit state function

( )54321 CD;CD;CD;CD;CDmaxCD =

CDG −= 1 is the value of limit state function.


128

Appendix E – Fire optimization framework

Start

Initialization

yes

noi < niter

Crossover

Mutation

i=i+1

Selection

Replacement

Maximum number of iteration reached

Print the results

Objective function evaluation, penalization

Ranking


Ranking

( ) ( ) ( )( )

( )( )

444444444 3444444444 21444 3444 21xx

xxxx

R

erventionintignitionfignitionfff

C

LCPC.PC.PCCCCC +⋅+⋅+⋅+++= 050010210

( ) ( ) ( ) ( ) ( )( )

( )x

x

xxxxx

ii

ii

i

iULSSLSLCgggC

ηη

ηη

η <

≤

=⋅⋅lim,

lim,

2

1;min!


( ) j,tctlAnC j,p

n

j

pj,pjjf

e

∀∈⋅⋅⋅= ∑=

xx1

1

( )2

01 1

, , , , ,4

p bn n

if i i i s i s sh i i i i i

i i

dC n b t l c l c C b t l d l i

πρ ρ

= =

⋅= ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ∈ ∀∑ ∑x x

Reliability analysis (FORM)

Bayesian probabilisticnetwork

Limit state function

9.510

10.511

11.512

12.5

2

2.5

3

3.5

40

0.5

1

1.5

2X1X

( ) 0=XG

( ) 0<XG

( ) 0>XG

( )failureP X

ignitionP

Event tree analysis

fla sh overP

( ),S fai lure flashoverP X

Thermal and mechanical analysis of complex model

T [C°]

t [min]

( )( )

01, <−=X

XRt

ttG

Evaluation of failure probability

( ) flashoverflashoverf,Sfailure PPP ⋅= X



Design point

i

iii

Xu

σµ−

=


'UTU =

( ) ( )UGUG ∇

( ) ( )

( )2

1

1

∂∂

∂∂−

=

∑

∑

=

=

n

i

ii

n

i

iii

x

UG

ux

UGUG

σ

σ

β

( )

( )2

1

∂∂

∂∂

=

∑=

n

i

ii

ii

i

x

UG

x

UG

σ

σα

iii*iX σβαµ +=

Convergence?yes

no

failureP

( )( ) 0

21

=

=

UG

UUmin!Tβ

HLRF iteration

IgnitionActivesafety

measure

Flashover

1,flashoverP

Active p.

Ignition T F

F 0 1

T 0.99 0.01

Ignition

T F

P ignition 1-Pignition

Flashover

Active p. Ignition T F

F F 0 1

F T 1 0

T F 0 1

T T PFL|A 1 - PFL|A

Ignition

1.0·10-5

fire/m2/year

4.5·10-6

6.5·10-6

Fire stoppedby occupants

yes - 0.45

no - 0.65

5.53·10-6

0.98·10-6

yes - 0.85

no - 0.15

Fire stopped byfire brigade

50,flashoverP

fire/m2/year

fire/year

Event tree analysis and Bayesian probabilistic network

T [C°]

t [min]

u [mm]

T [C°]

( )xcrM

( )xz,crN

op,crαop,cr

k,ultop

αα

λ = ( )LTzop ,min χχχ =

( ) ( )( ) ( )

( )( ) ( )

yyy

y

yyfTkW

M

fTkA

N,T

⋅⋅+

⋅⋅=

θθ

η,,

x

x

x

xx opχ

η

k,ultα

t [min]tR

01.

Fire resistance

( )( )X

XRt

ttG −=1,

1

5 frames are exposed to fire

( ) ( ) ( )flashoverswflashoverfswflashoverwflashoverfwf

PPPPPPP ⋅⋅+⋅⋅=++ xxx

( ) ( )( )ρβ,P mf,Sf,S ΦΦΦβ −−=−≅ −− 111

=

=

1

1

1

1

321

3

221

112

2

1

nnn

n

n

n

f,n

f,

f,

......

...

...

;...

ρρρ

ρρρ

ρρ

β

β

β

ρβ

Reliability analysis – failure probability of single frame

System reliability

Evaluation of the limit state function

Appendix E –Fire optimization framework


129

Appendix F – Seismic optimization framework

Start

Initialization

yes

noi < niter

Crossover

Mutation

i=i+1

Selection

Replacement

Maximum number of iteration reached

Print the results


Ranking


Ranking

( ) ( ) ( ) ( ) ( )( )

( )x

x

xxxxx

ii

ii

i

iULSSLSLCgggC

ηη

ηη

η <

≤

=⋅⋅lim,

lim,

2

1;min!


Evaluation of the failure probability

Appendix F – Seismic optimization framework

( ) ( ) ( )( )

( )( )434214434421

xx

xxxx

R

ff

C

LC PCCCC ⋅++= 10

( ) iltbCcltbnC iiish

n

i

siiif

p

∀∈+⋅⋅⋅⋅= ∑=

,,,1

0 xx ρ

( ) ildcld

C ii

n

i

bi

ib

∀∈⋅⋅⋅⋅

=∑=

,,41

2

1 xx ρπ



Design point

i

iii

Xu

σµ−

=


'UTU =

( ) ( )UGUG ∇

( ) ( )

( )2

1

1

∂

∂

∂∂

−

=

∑

∑

=

=

n

i

ii

n

i

iii

x

UG

ux

UGUG

σ

σ

β

( )

( )2

1

∂

∂

∂∂

=

∑=

n

i

ii

ii

i

x

UG

x

UG

σ

σα

iii*iX σβαµ +=

Convergence?yes

no

failureP

( )( ) 0

21

=

=

UG

UUmin! Tβ

HLRF iteration

Reliability Analysis

9.510

10.511

11.512

12.5

2

2.5

3

3.5

40

0.5

1

1.5

2X1X

( ) 0=XG

( ) 0<XG

( ) 0>XG

Seismic Hazard Analysis

Limit state function

Sa·m, V[kN]

Sd, d[mm]

Sa

T

d [mm]

F [kN]

1. Linear analysis in transversal direction 2. Nonlinear analysis in longitudinal direction

( )xcr

M

( )xz,cr

N

op,crαop,cr

k,ultop

α

αλ = ( )

LTzop ,min χχχ =

k,ultα

( ) 01 <−=op

gχη

x

( )( )

( )( ) yy

y

y fW

M

fA

N

⋅+

⋅=

x

x

x

xη3. Combination of seismic analyses’ results

4. Calculation of limit state function value

εσ ,,,,,2

,

2

, dMVNEEE yExEE ⇒+=( ) 01 <−=

op

Gχη

x


130

Appendix G – New scientific results in Hungarian

G 1. I. Tézis

Dolgozatomban megoldottam két nagymértékben nemlineáris és nem konvex

szerkezetoptimálási problémát, változó keresztmetszetű acél keretszerkezetek megbízhatósági

alapú optimálását tűz-, illetve szeizmikus hatásokra. Kidolgoztam és verifikáltam egy

megbízhatóság alapú heurisztikus optimáló keretrendszert a szerkezetoptimálás végrehajtására.

I/a. Összekötöttem egy genetikus algoritmus alapú optimáló algoritmust egy fejlett

megbízhatósági analízis keretrendszerrel, amely elsőrendű megbízhatósági analízisen

alapul. Az elsőrendű megbízhatósági analízis alkalmazása csökkenti a számítási igényt és

így növeli az optimáló keretrendszer hatékonyságát. Korábbi kutatásokkal összehasonlítva

a szerkezet megbízhatóságát pontosabban tudtam meghatározni a fejlett analízis és

kiértékelő módszereknek, összetett szerkezeti rendszer analízisének (nem különálló elemek

analízise), nemlineáris analízisnek imperfekt modellen és az Eurocode 3 szabvány általános

stabilitásvizsgálati eljárásának köszönhetően.

I/b. Vizsgáltam az általam kifejlesztett szerkezetoptimáló keretrendszer konvergenciáját és

stabilitását érzékenységi vizsgálat keretein belül. Megmutattam, hogy a kifejlesztett

algoritmus stabil és mérnöki szempontból megfelelő pontossággal rendelkezik.

Bemutattam, hogy az általam kifejlesztett optimáló keretrendszer alkalmas optimális

megoldások meghatározására tűz-, illetve szeizmikus hatásoknak kitett acél portálkereteket

esetében.

Kapcsolódó publikációk: [BT1], [BT2], [BT3], [BT4]

G 2. II. Tézis

A disszertációmban bemutattam egy átfogó és hatékony módszertant komplex szerkezetek

tűzhatás alatti megbízhatóságának kiértékelésére. Alkalmazását változó keresztmetszetű keretek

megbízhatóságának kiértékelésére mutattam be.

II/a. Kidolgoztam egy átfogó módszertant a határállapot függvény kiértékelésére tűzhatás

esetén, amely az egész szerkezet viselkedését figyelembe veszi a szerkezet nemlineáris

analízisének beépítésén keresztül, továbbá lehetővé teszi bármely típusú és lefutású


131

tűzgörbe alkalmazását, illetve amelyben a határállapot függvény idő alapon van felírva. A

módszertan újszerűségét az adja korábbi vizsgálatokhoz képest, hogy a felsorolt előnyöket

egyszerre nyújtja komplex szerkezetek vizsgálata során.

II/b. A számított tönkremeneteli valószínűségeket és az algoritmus stabilitását Monte Carlo

megbízhatósági analízis eredményekkel ellenőriztem egy acél keretszerkezet esetén. Az

eredmények alapján az elsőrendű megbízhatósági analízis segítségével alacsonyabb

tönkremeneteli valószínűség adódott. A határállapot függvény lineáris közelítése

jelentősebb hibát (maximum 34%) csak alacsony és magas feltételes tönkremeneteli

valószínűségek esetén szolgáltatott, a vizsgálat szempontjából releváns tartományban a

közelítés pontosabb.

Kapcsolódó publikációk: [BT8], [BT13]

G 3. III. Tézis

Dolgozatomban vizsgáltam változó keresztmetszetű acél keretszerkezetek optimális tervezését

tűzhatásra. A bemutatott megbízhatósági analízis keretrendszer segítségével meghatároztam

lehetséges megbízhatósági index célértékeket egy vizsgált példaszerkezetre és új tervezési

koncepciók kidolgozása céljából végrehajtottam egy paraméteres vizsgálatot egy mintaszerkezetre

a kifejlesztett optimáló keretrendszer segítségével.

III/a. Igazoltam, hogy az optimális megbízhatósági szint és az optimális konfiguráció függ az

éghető anyagok mennyiségétől és minőségétől, illetve a létesítmény funkciójától. Ennek

megfelelően összehasonlító tűzhatások (pl. ISO szabványos tűzgörbe) használata gyakran

vezethet túl konzervatív, vagy alulméretezett szerkezeti kialakításokhoz. A vizsgálatok

szerint β=2,9 – 3,5 megbízhatósági index érhető el az Eurocode 3 szabvány előíró jellegű

méretezési szabályaival és ISO szabványos tűzgörbe alkalmazásával. A problémára

megoldást jelent a tűzhatás minél pontosabb figyelembevétele a méretezés során.

III/b. Megbízhatósági index célértékeket meghatároztam három különböző intenzitású

tűzgörbére és három különböző következmény-osztályra. Megmutattam, hogy a

referenciaszerkezet esetén aktív és passzív védelmi eszközök együttes alkalmazásával

érhető el β>3,2-3,3 megbízhatósági index célérték, amely már az Eurocode 0 szabványban

megadott határértékeket is teljesíti. Rámutattam, hogy az optimális szerkezeti

megbízhatósági szintek közelebb állnak a Joint Committee of Structural Safety által javasolt


132

követelményszintekhez és a megbízhatósági index célértékére 2,8 és 3,7 közötti érték

veendő fel raktár jellegű acél portálkeret szerkezetek esetén a kockázat függvényében.

III/c. Rámutattam, hogy az Eurocode 0 nem különbözteti meg a hagyományos-, illetve a

tűzhatásokkal szembeni méretezés során alkalmazandó megbízhatósági célértékeket.

Tűzhatás esetében alacsonyabb célértékek adódnak, amelyek közelebb helyezkednek el a

Joint Committee of Structural Safety ajánlásban és ISO 2394 szabványban javasolt

célértékekhez. Rámutattam, hogy különböző megbízhatósági szintek definiálására van

szükség az Eurocode 0 szabványban tűzhatással szembeni tervezés esetében.

III/d. A példaszerkezet esetén rámutattam, hogy zömökebb (kisebb b/t aránnyal rendelkező:

átlagosan -10% - -20% gerinc és -20% - -40% öv esetén) szelvények hatékonyabbak

tűzhatás esetén, mivel a szerkezet rendkívül érzékeny különböző stabilitásvesztési

tönkremenetelekre. Zömök szelvények, kisebb mértékű passzív tűzvédelem alkalmazásával

jobb teljesítményt nyújtanak, mint a karcsúbb szelvények vastagabb tűzvédelemmel.

Rámutattam, hogy gyakran nem gazdaságos és nem megbízható az az elterjedt tűzhatással

szembeni méretezési gyakorlat, mely során a tűzhatást nem vesszük figyelembe a

koncepcionális tervezés és a szerkezeti elemek méreteinek megválasztása során, illetve

amikor a tűzhatással szembeni megbízhatóságot tartós tervezési helyzetre méretezett karcsú

szelvények passzív tűzvédelmével biztosítjuk.

Kapcsolódó publikációk: [BT8], [BT10], [BT11], [BT14]

G 4. IV. Tézis

Kutatásom során vizsgáltam változó keresztmetszetű acél keretszerkezetek optimális tervezését

szeizmikus hatásra. Bemutatott megbízhatósági analízis keretrendszer segítségével meghatároztam

lehetséges megbízhatósági index célértékeket egy vizsgált példaszerkezetre és új tervezési

koncepciók kidolgozása céljából végrehajtottam egy paraméteres vizsgálatot egy mintaszerkezetre

a kifejlesztett optimáló keretrendszer segítségével.

IV/a. Lehetséges megbízhatósági index célértékeket mutattam be moderált és magas

szeizmicitású területekre (Komárom és Râmnicu Sărat), három különböző következmény-

osztályra, mind rugalmas-, mind disszipatív tervezés esetén. Megmutattam, hogy acél

portálkeretek esetében is az Eurocode 8 szerinti disszipatív tervezéssel jelentősen magasabb

megbízhatósági szint érhető el, mint rugalmas tervezéssel. Rámutattam, hogy az Eurocode


133

8 szabályai szerint elérhető és az optimális szerkezeti megbízhatósági szintek is elmaradnak

az Eurocode 0 szabványban megkívánt szinttől és gazdaságos szerkezetkialakítás

érdekében a megbízhatósági index célértékére 2,0 és 3,5 közötti érték veendő fel egyszintes

acél portálszerkezetek esetén.

IV/b. Rámutattam, hogy az Eurocode 0 nem különbözteti meg a hagyományos-, illetve a

szeizmikus hatásokkal szembeni méretezés során alkalmazandó megbízhatósági

célértékeket. Szeizmikus hatások esetében alacsonyabb célértékek adódnak, amelyek

közelebb helyezkednek el a Joint Committee of Structural Safety ajánlásban és ISO 2394

szabványban javasolt célértékekhez. Rámutattam, hogy különböző megbízhatósági szintek

definiálására van szükség az Eurocode 0 szabványban szeizmikus hatások esetében.

IV/c. Megvizsgáltam a burkolati rendszer hatását a globális szerkezeti biztonság szempontjából.

Megmutattam, hogy a burkolati rendszer növeli a szerkezet globális merevségét és segít

elosztani a terheket a keretek között, illetve továbbítani a vízszintes erőket az alapozásra a

merevítőrendszerrel együttdolgozva. Ezáltal a burkolati rendszernek a vizsgált szerkezet

esetében kedvező hatása van a szerkezeti megbízhatóságra és a szerkezet költségére nézve

(3% - 16% megtakarítás az acélszerkezet költségében és 1% - 13% életciklus költségben az

optimált szerkezetkialakítások eredményei alapján).

IV/d. Földrengési terhekkel szembeni optimális méretezéssel kapcsolatban rámutattam, hogy

rugalmasan méretezett egyszintes acél keretszerkezetek esetén is érdemes a szerkezetet úgy

konstruálni és megtervezni, hogy lehetőséget adjunk a földrengési energia elnyelésére

koncentrált képlékenyedés révén, ugyanis ezáltal jelentősen magasabb szerkezeti

megbízhatóság érhető el. Szintén kedvező hatása lehet a szerkezet költségére és

megbízhatóságára nézve, ha modellezzük és figyelembe vesszük a burkolati rendszer

merevségét a fölrengési terhekkel szembeni méretezés során, amennyiben a burkolati

paneleket és a burkolati rendszer kapcsolatait szeizmikus hatásokból származó erőkre is

méretezzük.

Kapcsolódó publikációk: [BT9], [BT10], [BT12]

Documents

Optimal design of tapered steel portal frame structures