Optimal design of tapered steel portal frame structures exposed to

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Optimal design of tapered steel portal frame structures exposed to

extreme effects

PhD disse rta tion

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

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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, 6^th June 2017

Tamás Balogh

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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.

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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.

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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

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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.

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6

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

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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

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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

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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

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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:

(

β

)

Φ

( )

β Φ− =

−

=

−

=1 P_f 1

R , (1)

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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

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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

Moderate 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

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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/7S/1

VBM/12

VJM/4

M/6 VBM/12

HG/5

VBM/16 VJM/8

VJM/8 M/1

M/2 VBM

/1 7

F

T/1 HG/7 VBM/12

T/1

D BV/7

T/1

HO/6

VH/1

VJM/4 V BM /4

T/5 VBM/16 M/2

V JM/8 VJM/8

G/1

VBM/17 M/1

O/3

VJM/4

V JM/8 VJM/8 VBM/1 VBM

/17

KK/6 T/3

KK/2

KK/2 HO/7 HO/7

BV/2

BV/4 BV/4 BV/5

FO/3

HG/4 VBM/12

VJM/4

M/6

VJM/4VBM/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 V JM/1 1

T/8 T/8 T/7 T/7

T/7 T/8

T/8 ZS/2

G/6

G/5

O/4

VBM/9 VBM/14 VJM/4

V JM/1 1 VJM/11

M/5 M/1

V BM /15 VB M/1 9

T/5

HG/7 ZKL/2 ZKK/3

VJM/4

BV/4

BV/6 BV/4

T/3 HG/7

ZK/1 ZKK/4

FO/10

G

VJM/4

HG/1 VB

M/7 ZK/1 ZK /3 ZK/3

M/7ZKK/4 KM/3 M/7

KM/3KM/3 KM/3 ZKK/3

ZKK/4

VB

M/12

VJM/8

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

VB

M/10 VBM/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/1 ZS/3

G/9

LK/1

T/6 T/5

T/2 G/10

T/2 T/2 K/1

VJM/1

FO/6

VBM/4 K/8 VH/1

ZS/1FO/8

FO/5 FO/1

G/7

K/6 K/6 K/7

H

LK/3

T/6 T/2 T/4

FO/6 VBM/4

LK/2 K/3K/6 K/6 K/7

T/1 VH/1

FO/2

G/2

FO/7KK/2

K/6 K/7

K/6 K/5

K/6

K/6 K/6

J

VBM/8 M/7S/1

VJM/4 VBM/6

VBM/5 VJM/4 T/1

KK/2

T/1 VH/1

M/3 S/1 S/1VBM/6

HO/2

KK/2

BV/7

ZKK/4 HG/7

HG/7 KK/6 HO/5

VB M/17 VJM

/8

S/1 ZKK/3

BV/3 BV/6

M/6 VBM/7

S/1 S/1

VJM/4

M/8

HO/2

BV/4

VBVJM/4M/8 HG/7

G/4 VH/1

HO/7

BV/4 BV/4 BV/4

V JM/2

T/4 VJM

/2 VJM

/2

VJM/2 VBM/9 VJM /11

VJM /11 VJM/11

M/5

HO/4

KK/2V BM /15

T/1

FO/4

T/5 VH/1 V BM/2

O/5

T/5 VBM/3

O/1

BV/7

KK/6 G/3 VH/1

VBM/14 VJM/4

VJM/4 HG/3

M/1 V BM/1 9

9.0 m

5.9 m

0.0 m

Type Value

dead load of the

frame calculated dead load of the

roof system 0.2 kN/m² weight of the

equipment 0.2 kN/m² snow 1.25 kN/m² velocity pressure

of wind 0.58 kN/m²

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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)

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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.

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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

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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

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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.

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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:

( )

x C

min or minW

( )

x , (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

safety

a) b)

Cost, Risk

( )x ( )x R

C

( )x C( )x R( )x

C_LC = + C_LC( )x

( )x ( )x R

C

unfeasible area

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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:

( )

x min

[

C

( )

x R

( )

x

]

C

min _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:

( )

; i , ,...,k,

g_i 12

0≤ x = , (4)

( )

; j k ,..,m

h_j 1

0= 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.);

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• 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]:

( ) ( ) ( )

( )

( )44444444443 4

4 4 4 4 4 4 4 4

4 2

1

4 4 4 3 4

4 4 2 1

x x

x

x x

x

R

ervention int ignition f

ignition f

f f

C LC

P C . P

C . P

C

...

C C

⋅ +

+

⋅ +

+ + +

=

05 0 01

0

2 1

0

. (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).

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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:

( )

ⁿ ^b ^t ^l ^c ^d ^l ^c ^C ^b^,^t ^,^l ^,^d ^, ⁱ

C _sh _i _i _i _i

n

i

s i i n

i

s i i i f

p b

∀

∈ +

⋅

⋅ ⋅ +

⋅

=

∑ ∑

=

x x

1 2

1

0 ρ 4π 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 i^th 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 i^th 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:

( )

n A l t c t , j

C _p_,_j

n

j

p j , p j j f

e ⋅ ⋅ ⋅ ∈ ∀

=

∑

=

x x

1

1 , (8)

where ne, Aj, lj, tp,j and cp are the number of protected elements, the protected surface, the protected length of j^th element, the protection thickness in case of j^th 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

Standardized cost function, Reliability index

0 1

2 0 1

2 2

4 6 8 10 12

x 10⁴

Active safety measure Passive safety measure

Cost function [Euro]

1.5 2.5 3.5 4.5

Standardized cost function, Reliability index

Active safety measure

Optimal design of tapered steel portal frame structures exposed to