Helsinki University of Technology Laboratory of Steel Structures Publications 29 Teknillisen korkeakoulun teräsrakennetekniikan laboratorion julkaisuja 29

Espoo 2003 TKK-TER-29

ADVANCED STEEL STRUCTURES

1. STRUCTURAL FIRE DESIGN 2. FATIGUE DESIGN

Wei Lu Pentti Mäkeläinen

AB TEKNILLINEN KORKEAKOULUTEKNISKA HÖGSKOLANHELSINKI UNIVERSITY OF TECHNOLOGYTECHNISCHE UNIVERSITÄT HELSINKIUNIVERSITE DE TECHNOLOGIE D’HELSINKI

Helsinki University of Technology Laboratory of Steel Structures Publications 29 Teknillisen korkeakoulun teräsrakennetekniikan laboratorion julkaisuja 29

Espoo 2003 TKK-TER-29

ADVANCED STEEL STRUCTURES

1. Structural Fire Design

2. Fatigue Design

Wei Lu Pentti Mäkeläinen Helsinki University of Technology Department of Civil and Environmental Engineering Laboratory of Steel Structures Teknillinen korkeakoulu Rakennus- ja ympäristötekniikan osasto Teräsrakennetekniikan laboratorio

Distribution: Helsinki University of Technology Laboratory of Steel Structures P.O. Box 2100 FIN-02015 HUT Tel. +358-9-451 3701 Fax. +358-9-451 3826 E-mail: srt-sihteerit@hut.fi Teknillinen korkeakoulu ISBN 951-22-6732-2 ISSN 1456-4327 Yleisjäljennös - Painopörssi Espoo 2003

PREFACE This report is prepared in the Laboratory of Steel Structures at Helsinki University of Technology (HUT) in 2003. This work is a part of the project “Teräsrakennetekniikan opetusmateriaalin ajanmukaistaminen” (Teaching material updating for advanced steel structures). This project is financially supported by TKK/Opintotoimikunta (joint student-faculty committee in HUT), which is gratefully acknowledged. This report was developed to use as a part of teaching materials either for graduate courses Rak-83.122 Advanced Steel Structures or for postgraduate studies Rak-83.J. Two topics are included in this report: structural fire design and fatigue design. The structure of each topic is basically composed of three parts: theoretical backgrounds, design rules and worked examples. The design rules that are presented in this report are based on ENV 1991-1 (1994): Eurocode 1-Basis of design and actions on structures-Part 1: Basis of design; EN 1991-1-2 (2002): Eurocode 1: Actions on structures –Part 1-2: General actions-Actions on structures exposed to fire; ENV 1993-1-1 (1992): Eurocode 3: Design of Steel Structures-Part 1.1: General rules and rules for buildings; and ENV 1993-1-2 (1995): Eurocode 3: Design of Steel Structures-Part 1.1: General rules-structural fire design. The materials used in the part of structural fire design are based on the books and papers that are collected, and researches that have been carried out in the Laboratory of Steel Structures. The materials used in the part of fatigue design are based on materials available in the Laboratory of Steel Structures and the materials distributed in the short course Fatigue of Materials and Structures organized by Laboratory for Mechanics of Materials at HUT. I would like to express my thanks to these authors and organizers. The authors wish to express gratitude for Lic.Sc. (Tech.) Olli Kaitila, Lic.Sc. (Tech.) Jyri Outinen, Mr. Olavi Tenhunen and D.Sc. (Tech.) Zhongcheng Ma for providing extra materials, nice discussions and useful comments. Many thanks go to secretary Mrs. Elsa Nissinen for her kind assistance. Wei Lu, D.Sc. (Tech.) Espoo, August 2003

CONTENTS

PREFACE ........................................................................................................................................... 3

CONTENTS........................................................................................................................................ 4

1 STRUCTURAL FIRE DESIGN................................................................................................ 6 1.1 INTRODUCTION...................................................................................................................... 6

1.1.1 Development of fire in buildings .................................................................................. 6 1.1.2 Fire safety..................................................................................................................... 7 1.1.3 Fire protection.............................................................................................................. 7 1.1.4 Structural fire safety design methods ........................................................................... 8

1.2 DESIGN CURVES AND FIRE MODELS...................................................................................... 9 1.2.1 Nominal temperature-time curves ................................................................................ 9 1.2.2 Natural fire models: compartment fires or parametric fires...................................... 10 1.2.3 Natural fire models: localized fire models ................................................................. 15 1.2.4 Natural fire models: advanced fire models ................................................................ 15

1.3 MATERIAL PROPERTIES OF STEEL AT ELEVATED TEMPERATURE........................................ 16 1.3.1 Mechanical properties of materials ........................................................................... 16 1.3.2 Thermal properties ..................................................................................................... 23

1.4 PASSIVE PROTECTION FOR STEELWORK .............................................................................. 26 1.4.1 Fire protection systems .............................................................................................. 26 1.4.2 Thermal properties of fire protection systems............................................................ 28

1.5 HEAT TRANSFER IN STEEL................................................................................................... 29 1.5.1 Type of heat transfer................................................................................................... 29 1.5.2 Heat transfer equation for steel.................................................................................. 30

1.6 MECHANICAL ANALYSIS OF STRUCTURAL ELEMENT .......................................................... 37 1.6.1 Required fire resistance time...................................................................................... 37 1.6.2 Mechanical actions..................................................................................................... 41 1.6.3 Design value of material temperature........................................................................ 42 1.6.4 Design value of fire resistance time ........................................................................... 43 1.6.5 Critical temperature ................................................................................................... 44 1.6.6 Load bearing capacity................................................................................................ 45

1.7 DESIGN OF STEEL MEMBERS EXPOSED TO FIRE .................................................................. 47 1.7.1 Design methods .......................................................................................................... 47 1.7.2 Classification of cross-sections .................................................................................. 47 1.7.3 Tension members........................................................................................................ 47 1.7.4 Moment resistance of beams ...................................................................................... 48 1.7.5 Lateral-torsional buckling.......................................................................................... 49

1.7.6 Compression members with Class 1, Class 2 or Class 3 cross-section ..................... 49 1.8 USE OF ADVANCED CALCULATION MODELS ....................................................................... 50 1.9 GLOBAL FIRE SAFETY DESIGN ............................................................................................ 51 1.10 DESIGN EXAMPLE ACCORDING TO EUROCODE 3.................................................................. 52

1.10.1 Introduction ................................................................................................................ 52 1.10.2 Design loads and load distribution in the frame ........................................................ 54 1.10.3 Fire resistance and protection of a tension member BE ............................................ 54 1.10.4 Fire resistance and protection of steel beam AB........................................................ 58

1.11 REFERENCES........................................................................................................................ 62

2 FATIGUE DESIGN.................................................................................................................. 64 2.1 INTRODUCTION .................................................................................................................... 64

2.1.1 Different approaches for fatigue analysis .................................................................. 64 2.1.2 A short history to fatigue ............................................................................................ 65

2.2 FATIGUE LOADING .............................................................................................................. 66 2.3 STRESS METHODS................................................................................................................ 67

2.3.1 Standard fatigue tests ................................................................................................. 68 2.3.2 S-N curves................................................................................................................... 69 2.3.3 One dimensional analysis for fatigue assessment ...................................................... 76

2.4 STRAIN METHODS ............................................................................................................... 77 2.4.1 Cyclic material law..................................................................................................... 77 2.4.2 Fatigue life.................................................................................................................. 79

2.5 CRACK PROPAGATION METHODS ........................................................................................ 82 2.5.1 Characteristic of fatigue surfaces............................................................................... 82 2.5.2 Fatigue mechanism..................................................................................................... 83 2.5.3 Linear elastic fracture mechanics .............................................................................. 84 2.5.4 Crack propagation under fatigue load ....................................................................... 86 2.5.5 Short crack behavior .................................................................................................. 88

2.6 FATIGUE ANALYSIS UNDER VARIABLE LOADS.................................................................... 88 2.6.1 Fatigue testing under variable loading ...................................................................... 88 2.6.2 Palmgren-Miner rule.................................................................................................. 89 2.6.3 Cycle counting ............................................................................................................ 90 2.6.4 Crack propagation under variable loading................................................................ 92

2.7 FATIGUE ANALYSIS OF WELDED COMPONENTS................................................................... 93 2.7.1 Factors affecting the fatigue life................................................................................. 94 2.7.2 S-N methods for evaluating fatigue life ...................................................................... 96 2.7.3 Crack propagation method....................................................................................... 106

2.8 CALCULATION EXAMPLES ACCORDING TO EUROCODE 3................................................... 107 2.8.1 Introduction .............................................................................................................. 107 2.8.2 Given values ............................................................................................................. 108 2.8.3 Stress calculations .................................................................................................... 108 2.8.4 Assessment for the trolley carrying the full load of 150 kN ..................................... 111 2.8.5 Assessment for the trolley returning empty .............................................................. 113 2.8.6 Assessment for the trolley returning carrying load of 70 kN ................................... 113 2.8.7 Assemblage of the calculated damage and determination of the fatigue life ........... 114

2.9 REFERENCES...................................................................................................................... 115

1 STRUCTURAL FIRE DESIGN

1.1 Introduction

1.1.1 Development of fire in buildings A real fire in a building grows and decays in accordance with the mass and energy balance within the compartment in which it occurs. The energy released depends upon the quantity and type of fuel available and upon the ventilation conditions. Figure 1.1 illustrates that the fire in a building can be divided into three phases: the growth or pre-flashover period, the fully developed or post-flashover fire and the decay period [2]. The most rapid temperature rise occurs in the period following flashover, a point at which all organic materials in a compartment spontaneously combust. Anyone who has not escaped from a compartment before flashover is unlikely to survive.

Figure 1.1 Typical temperature development in a compartment [2]

In the pre-flashover phase, the room temperature is low and the fire is local in the compartment. This period is important for evacuation and fire fighting. Usually, it has not significant influence on the structures. After flashover, the fire enters into the fully developed phase, in which the temperature of the compartment increase rapidly and the overall compartment is engulfed in fire. The highest temperature, the highest rate of heating and the largest flame occur during this phase, which gives rise to the most structural damage and much of the fire spread in buildings. In the decaying period, which is formally identified as a stage after the temperature falling to 80 percent of its peak value, the temperature decreases gradually. It is worth to point out that this period is also important to the structural fire engineering because for the insulated steel structures and unprotected

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steel structures of low section factor, the internal temperature of cross-section will still increase significantly even though in the decaying periods [12].

1.1.2 Fire safety Fire safety design is an important aspect of building design. A properly designed building system greatly reduces the loss of the life and the property of finical losses in, or in the neighborhood of, building fires. Current fire safety concepts are defined as optimal packages of integrated structural, technical and organizational fire precaution measures, which allow well defined objectives agreed by the owner, the fire authority and the designer, to be fulfilled [8]. The essential requirements for the limitation of fire risks have to be fulfilled in the following ways:

o The load-bearing capacity of the construction can be assumed for a specific period of time; o The generation and spread of fire and smoke within the works are limited; o The occupants can leave the works or can be rescued by other means o The safety of rescue teams is taken into consideration.

The central objective of fire safety in the current Fire Codes is to confine the fire within the compartment in which it started. These consist of a collection of requirements, only or mostly related to the structural fire resistance of load-bearing elements and to walls and slabs necessary to guarantee the compartmentation. It should be noticed that the objectives of fire safety are a historical concept, in which the contents can be changed with the development of fire science. Besides, the additional objective can also be implemented if the client or authorities require a particular building or a project.

1.1.3 Fire protection Structural fire protection is only one part of the package of fire safety measure used in a building. There are two broad groups of measures [1]:

o Fire prevention, designed to reduce the chance of a fire occurring; o Fire protection, designed to mitigate the effects of a fire should it nevertheless occur.

Fire prevention includes eliminating or protecting possible ignition sources in order to prevent a fire occurring. Fire protection measures may be passive or active, which are used according to the phase of fire development as shown in Figure 1.2 [5]. Active measures [1] include detection and alarm, fire extinction, and smoke control, all of which may be operated manually or automatically. Early detection and extinction lead to early fire fighting and decease the risk of a large fire. For instance, the combination of automatic sprinklers and a designed smoke-control system has been used to protect people escaping from fire in large buildings. Passive measures include structural fire protection, layout of escape routes, fire brigade access routes, and control of combustible materials of construction [1]. Normally for pre-flashover fires,

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passive protection includes selection of suitable materials for building contents and interior linings that do not support rapid flame spread in the growth period. In post-flashover fires, passive protection is provided by structures and assemblies, which have sufficient fire resistance to prevent both spread of fire and structural collapse. The controls of fire spread include controlling fire spread within the room of origin, to adjacent room, to other storeys and to other buildings. The most important component of passive fire protection is fire resistance of the structure.

Figure 1.2 Fire evolution and fire protection [5]

Often a combination of the above measures is applied. Ideally, the fire safety design concept should allow for a certain between the various measures i.e. emphasis on one or two of the possible measures should lead to relaxation of the remaining one(s). For instance, a sprinkler installation would lead to reduce overall requirements for the fire resistance. Such a trade off is not generally accepted at present but needs to be pursued with the appropriate authorities [5].

1.1.4 Structural fire safety design methods Currently, the design methods may be classified into two classes, i.e. [12]

o Methods related to fire resistance only; o Methods related to global fire safety.

The first category of methods concerns the verification methods of fire resistance. The Eurocodes are for the time being strictly limited to this category. The method related to fire resistance is governed by two basic models: a heat model and a structural model. The heat model defines the evolution of air temperature, the convective and radiative boundary conditions and the spreading of fire in a fire affected room if possible. The structural model defines elements or parts of the structures, thus allowing the prediction of the temperature increase in the structure or in elements ensuring compartmentation, of the collapse temperature or the collapse time for a given load. To

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date, the use of a conventional fire scenario based on the ISO standard fire curve is common practice in Europe and elsewhere. Safety level in buildings referring to fully developed mainly. The second category is based on the fire risk assessment technology, which is being developed for particular buildings, important structures or individual projects. The purpose of developing the global fire safety concept is to establish the basis for realistic and credible assumptions to be used in fire situation for thermal actions, active measures and structural response.

1.2 Design Curves and Fire Models When dealing with fire resistance, the ignition stage is generally neglected, although this stage is generally the most critical for human life since it is during this stage that toxic gases are produced and the temperature can reach 100 °C and more. To select the relevant fire model, the fire scenarios need to be defined. It is a selection of the possible worst cases as far as the location and the amount of fire load are concerned [5]. For instance:

o In a small room, it is assumed a fully developed fire, using the maximum fire load which can be in the compartment;

o In a large room, at least two assumptions can be made, either a uniformly distributed fire load leading to a fully developed fire in the compartment or localized fires depending on the possible location of the fire load;

o For element located outside the facade of the building, flames coming through windows and doors will be considered.

A design fire shall be expressed as a relationship between temperature, time and space location, which may be [5]:

o A nominal temperature-time curve uniform in the space; o A ‘real fire’ either specified in terms of parametric fire exposure, or given by an analytical

formula for localized fire, or obtained by computer modeling.

1.2.1 Nominal temperature-time curves The nominal temperature-time curves are a set of curves, in which no physical parameters are taken into account. The main purpose of the prescription of the nominal curves was to make the fire resistance tests reproducible. The ability of fire resistance of building elements can be evaluated under the same heating curve [18]. Fire resistance times specified in most national building regulations relate to test performance when heated according to an internationally agreed time-temperature curve defined in ISO834 (or Eurocode 1 Part 2-2), which does not represent any type of natural building fire. This standard temperature-time curve involves an ever-increasing air temperature inside the considered compartment, even when later on all consumable materials have been destroyed. This has become the standard design curve, which is used in furnace testing of components. The quoted value of fire

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resistance time does not therefore indicate the actual time for which a component will survive in a building fire, but is a like-against-like comparison indicating the severity of a fire that the component will survive [17]. Where the structure for which the fire resistance is being considered is external, and the atmosphere temperatures are therefore likely to be lower at any given time (which means that the temperatures of the building materials will be closer to the corresponding fire temperatures), a similar “External Fire” curve may be used. In cases where storage of hydrocarbon materials makes fires extremely severe a “Hydrocarbon Fire” curve is also given. The formula for describing these curves are given as follows [3]: for standard temperature-time curve θθθθg = 20+345log10(8t+1) ( 1.1 )

for external fire curve θθθθg = 660(1-0.687e-0.32t-0.313e-3.8t)+20 ( 1.2 )

for hydrocarbon curve θθθθg = 1080(1-0.325e-0.167t-0.675e-2.5t)+20 ( 1.3 )

These three nominal temperature-time curves according to these formulas are shown in Figure 1.3.

Figure 1.3 Nominal temperature-time curves

1.2.2 Natural fire models: compartment fires or parametric fires Before get into the details of this model, the following definitions are clarified [2]:

o Heat of combustion or the calorific value of material is defined as the amount of heat in calories evolved by the combustion of one-gram weight of a substance [MJ/kg].

0

200

400

600

800

1000

1200

0 30 60 90 120 150

Time (min)

Tem

pera

ture

( C

)

Standard Fire

External Fire

Hydrocarbon Fire

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o Mass loss rate is defined as the mass of fuel that is vaporized from the solid or liquid fuel per

unit time [kg/s].

o Burning rate is the amount of fuel that is burned within the compartment in terms of airflow per unit time [kg/s]. This is distinct from the mass loss rate and is dependent on the available oxygen.

o Heat release rate is defined as the rate at which the heat is released [J/s] and can be measured

experimentally or obtained by calculation. It is the source of the gas temperature rise and the driving force behind the spreading of gas and smoke.

Background Parametric fire models provide a simple means to take into account the most important physical phenomenon that may influence the development of a fire in a particular building. Like nominal fires, they consist of time temperature relationships, but these relationships contain some parameters represent particular aspects of the reality. Normally, three parameters are included in these models, namely, the fire load present in the compartment, the openings in the walls and/or in the roofs, and the type and nature of the different walls of the compartment. These models assume that the temperature is uniform in the compartment, which limits the application to post-flashover fires in compartment of moderated dimensions. These models require the following data: fire load density, rate of heat release and heat losses. a. Fire load density Fire load density is defined as the total amount of combustion energy per unit of floor area and is the source of the fire development. The fire load is composed of the building components such as wall and ceiling linings, and building contents such as furniture. The characteristic value of fire load density is provided by: qf.k = [ΣΣΣΣMk.iHuiΨΨΨΨl]/A ( 1.4 )

where

Mk.i is the combustible materials [kg] Hui is the net calorific value [MJ/kg]; Mk.i Hui is the total amount of energy contained in material and released assuming

complete combustion. Ψl is the optional factor to accessing protected fire load. For instance by putting

it into a cabinet. A is the floor area.

b. Rate of heat release ( RHR ) The calculation of RHR is different from ventilation controlled fire to fuel controlled fire. The fuel controlled fire refers to the case that there is always enough oxygen to sustain combustion. While for

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the ventilation controlled fire, the size of openings in the compartment enclosure is factor to control the amount of the air to enter the compartment. When fire is ventilation controlled, according to Kawagoe (1958), the burning rate m� [kg/s] can be calculated as [2]

m� = 0.092 Av√√√√Hv ( 1.5 )

Where Av is the area of the openings (m2) and Hv is the height of the openings (m). This equation is derived from the experiments for a room with a single opening. Despite the findings showing that the burning rate depends on the shape of the room and the width of the window proportion to the wall in which it is located, this equation is formed the basis of most post-flashover fire calculation. The corresponding ventilation controlled heat released rate (MW) for steady burning is calculated as [1]:

Qvent = m� Hui ( 1.6 )

The duration of fire can be calculated as: tb = E/Qvent ( 1.7 )

where E is the energy content of fuel available for combustion (MJ). In addition, the amount of ventilation in a fire compartment is described by the opening factor O (m0.5) given by O = Av√√√√Heq/At ( 1.8 )

where Heq is weighted average window heights on all walls (m) and At is total area of enclosures (walls, ceiling and floor, m2). If this formula is multiplied by gravity g, then the product is related to the velocity of gas flow through openings. Researches show that if the ventilation openings were enlarged, a condition would be reached beyond which the burning rate would be independent on the size of the opening and would be determined instead by the surface and burning characteristics of the fuel. For the fuel controlled fire, the duration of the fire can be assumed as 25 min for slow fire growth rate, 20 min for medium growth rate and 15 min for fast growth rate. The RHR can be calculated as Qfuel = E / tlim ( 1.9 )

When the duration is not known, the RHR is estimated from the information about the fuel and the temperatures in the fire compartment. In the current Eurocode[3], the RHR is implicitly calculated using Av√Heq. c. Heat losses Heat losses suffered by the combustion gases are important factors to the temperature development of a compartment fire. Heat losses occur to the compartment boundaries by convection and radiation and by the ventilation flow [3]. The most popular way to model the heat losses to the compartment boundaries is through the concept of the ‘thermal inertia”, b, of the wall material, i.e. b = √√√√ρρρρcλλλλ ( 1.10 )

where λ is heat conductivity (W/mK); c is the heat capacity (J/kgK) and ρ is mass density (kg/m3).

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Parametric temperature-time curves Current Eurocode[3] gives an equation for parametric temperature-time curves for any combination of fuel load, ventilation openings and wall lining materials. The equation of temperature Θg (ºC) for heating phase is provided by Θg = 1325(1-0.324e-0.2t*-0.204e-1.7t*-0.472e-19t*)+20 ( 1.11 )

where t* is fictitious time (hours) given by t* = t·Γ ( 1.12 )

where t is the time (hours) and Γ = (O/b)2/(0.04/1160)2 ( 1.13 )

In the case of compartment with O = 0.04 m0.5 and b = 1160 J/m2s0.5K, the parameter curve is almost exactly the ISO curve. The maximum temperature occurred at t* = t*

max where t*

max = tmax·Γ ( 1.14 )

with tmax = max [(0.2·10-3·qt.d/O); tlim] ( 1.15 )

The time tmax corresponding to the maximum temperature is given by tlim in case the fire is fuel controlled. If tmax is given by (0.2·10-3·qt.d/O), the fire is ventilation controlled. When tmax = tlim, t* is the temperature formula is replaced by t* = t·Γlim ( 1.16 )

with Γlim = (Olim/b)2/(0.04/1160)2 ( 1.17 )

where Olim = 0.1·10-3·qt.d/tlim ( 1.18 )

If O > 0.04 and qt.d < 75 and b < 1160, Γlim has to be multiplied by k given by k = 1+ [(O-0.04)/0.04]·[(qt.d-75)/75]·[(1160-b)/1160] ( 1.19 )

This is due to the fact that the influence of the openings is still present when the fire is fuel controlled. [3] uses a reference decay rate equal to 625 ºC per hour for fires with duration less than half an hour, decreasing to 250 ºC for fires with duration greater than two hours. The temperature curves in the cooling period are given by Θg = Θmax –625 (t*-t*

max·x) ( 1.20 )

for t* ≤ 0.5 Θg = Θmax –250(3-t*

max) (t*-t*max·x) ( 1.21 )

for 0.5 < t* ≤ 2 Θg = Θmax –250 (t*-t*

max·x) ( 1.22 )

for t* > 2, where t* = t·Γ; tmax = (0.2·10-3·qt.d/O)·Γ and x = 1.0 if tmax > tlim , or x = tlim·Γ / t*max if tmax

= tlim .

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Examples The effects of the fire load density and the ventilation of the fire compartment on the gas temperature are shown in Figure 1.4 and Figure 1.5. These calculations are based on the formula given above with the parameters given in the figures, which are based on the seminar materials [16]. These curves are suitable for using as alternatives of nominal curve of internal members of a compartment.

Figure 1.4 Parametric temperature-time curves considering the effects of openings

Figure 1.5 Parametric temperature-time curves considering the effects of fire loads

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1.2.3 Natural fire models: localized fire models The models mentioned above have assumed a fully developed fire occurs and the same temperature conditions throughout the fire compartment. However, in some circumstances, possibly in a large space where there are no nearly combustibles, or in a fire partially controlled by sprinklers, there could be a localized fire which has much less impact on the building structure than a fully developed fire. The thermal actions of a localized fire can be assessed using the analytical formula that takes into account the relative height of the flame to the ceilings. These formulas are given in [3] and [5].

1.2.4 Natural fire models: advanced fire models Two kinds of numerical models are available to model the real fires: multi zone models and field models. The multi zone models are used when the fire is localized, e.g. in the growth phase of a fire. The fire compartment is divided into a hot zone, with a uniform temperature, above a fresh air zone and a fire plume that feeds the hot zone just above the fire. A two-zone model is shown in Figure 1.6. For each of the zones, the heat and mass balance is solved. (Semi) empirical relations govern plume entrainment, irradiative heat exchange between zones and mass flow through openings to adjoining compartments. Particularly, the (growth of the) fire size should be taken as an input besides the parameters mentioned in the one zone model [18]. The application of this model is mainly in pre-flashover conditions, in order to know the smoke propagation in buildings, and estimate the life safety in function of toxic gas concentration temperature, radiative flux and optical density.

Figure 1.6 Zone model [10]

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Field models are also called Computational Fluid Dynamics (CFD) models. These models are based on two or three dimensional heat and mass transport, solving the equations of conservation of mass, momentum and energy for discrete points in the enclosed compartment. In this model, material properties and boundary conditions may be defined as the function of temperatures. The fire simulation problem represents one of the most difficult areas in computational fluid dynamics: the numerical solution of re-circulating, three dimensional turbulent, generating eddies or vortices of many sizes. The energy contained in large vortices cascades down to smaller and smaller vortices until it diffuses into heat. Eddies exist down to the size where the viscous forces dominate over inertial forces and energy is dissipated into heat. Field models will provide accurate information about temperatures from the pieces of the fire room [18]. The complexity and the CPU time needed with field models allow few applications of such model in respect to fire resistance particularly for fully developed fire. In fire domain the use of field model is often reduced to the application of smoke movement.

1.3 Material Properties of Steel At Elevated Temperature

1.3.1 Mechanical properties of materials When structural components are exposed to fire, they experience temperature gradients and stress gradients, which are both varied with time. Mechanical properties of materials for fire design purpose must be consistent with the anticipated fire exposure.

1.3.1.1 Components of strains The deformation of steel at elevated temperature is described by assuming that the change in strain consists of three components: mechanical or stress-related strain, thermal strain and creep strain [1]. Stress-related strain Figure 1.7 shows the stress-strain curves at various temperatures for S275 steel in Eurocode 3. It can be seen that the steel suffers a progressive loss of strength and stiffness at temperature increases. The change can be seen at temperatures as low as 300°C. Although melting does not happen until about 1500°C, only 23% of the ambient-temperature strength remains at 700°C. At 800°C this has reduced to 11% and at 900°C to 6% [17]. A value of yield strength is required at elevated temperature. Most normal construction steel has well-defined yield strength at normal temperatures, but this disappears at elevated temperatures as shown in Figure 1.7. In Eurocode 3 [7], the 2% proof strength is used as the effective yield strength. However, Kaitila [9] summarizes the possible values for yield strength at elevated temperature from the literatures: Ala-Outinen and Myllymäki, and Ranby suggested the use of the 0.2% proof stress for the effective yield strength at elevated temperature; In the Steel Construction Institute (SCI) recommendation, the use of 0.5% proof stress is suggested for members failing by buckling in

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compression (mainly columns) and the 1.5% proof stress fro members failing in bending (mainly beams); and Kirby and Preston recommend using 1% proof stress as the effective yield strength.

Figure 1.7 Reduction of stress-strain properties with temperature for S275 steel (EC3 curves) [17]

The modulus of elasticity is needed for buckling calculations and for elastic deflection calculation, but these are rarely attempted under fire conditions because elevated temperatures lead rapidly to plastic deformations. For the fire design of individual structural members such as simply supported beams that are free to expand during heating, the stress-related strain is the only component that needs to be considered. If the reduction of strength with temperature is known, member strength at elevated temperature can easily be calculated using simple formulae. The stress-related strains in fire-exposed structures may be well above yield levels, resulting in extensive plastification, especially in buildings with redundancy or restraint to thermal expansion. Computer modeling of fire-exposed structures requires knowledge of stress-strain relationships not only in loading, but also in unloading, as members deform and as structural members cool in real fires [1]. Thermal strain Thermal strain is the thermal expansion (∆L/L) that occurs when most materials are heated, with expansion being related to the increase in temperature. Thermal strains is not important for fire design of simply supported members, but must be considered for frames and complex structural systems, especially where members are restrained by other parts of structure since thermal strains can induce large internal forces [1].

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Creep strain Creep is the term that describes long-term deformation of materials under constant load. Under most conditions, creep is only a problem for members with very high permanent loads. Creep is relatively insignificant in structural steel at normal temperatures. However, it becomes very significant at temperatures over 400 or 500°C and highly depends on the stress level. At higher temperatures, the creep deformations in steel can accelerate rapidly, leading to plastic behavior. Creep strain is not usually included explicitly in fire engineering calculations because of the added complexity. This applies to both hand and computer methods. The effects of creep are usually allowed for implicitly by using stress-strain relationships that include an allowance for the amount of creep that might be expected in a fire-exposed member [1].

1.3.1.2 Testing regimes Constant temperature tests of material can be carried out in the following four regimes [1]:

1) The most common test procedure to determine stress-strain relationship is to impose a constant rate of increase of strain and to measure the load, from which the stress can be derived;

2) A similar regime is to control the rate of increase of load and measure the deformation; 3) A creep test is one in which the load is kept constant and deformation over time is measured; 4) A relaxation test is one in which a constant initial deformation is imposed and the reduction

in load over time is measured. Two other possible test regimes are available when the effects of changing temperature are added.

5) A transient creep test is that the specimen is subjected to initial load, then the temperature is increased at a constant rate while the load is maintained at a constant level, and the deformations are measured.

6) An alternative is that the applied load is varied throughout the test in order to maintain a constant level of strain as temperature is increased at a constant rate.

The most common of these are regimes (1) and (5). The regime (1) tests depend on the rate of loading because of the influence of the creep. The region (5) tests depend on the rate of temperature increase. The stress-strain relationships at elevated temperature can be obtained directly from steady-state tests at certain elevated temperatures (Regime 1) or they can be derived from the results of transient tests. This procedure can be demonstrated from the following example, i.e. the small-scale tensile tests of steel at high temperature [15]. This research has been carried out in the Laboratory of Steel Structures at Helsinki University of Technology from the years 1994-2001 in order to investigate mechanical properties of several structural steels at elevated temperatures by using mainly the transient state tensile test method.

19

The testing device is illustrated in Figure 1.8. The oven, in which test specimen is situated during the tests, was heated using three separate temperature controlled resistor elements. The air temperature is measured with three separate temperature-detecting elements. The steel temperature was measured accurately from the test specimen using temperature-detecting element that was fastened to the specimen during the heating.

Figure 1.8 High temperature tensile testing device [15]

During the transient test, the specimen is under a level of constant load and a constant rise of temperature. The temperature and the strain are measured; the temperature and strain curve are recorded. The results are then converted into stress-strain relations using the scheme shown in Figure 1.9. Thermal elongation, which has been measured separately [14], is subtracted from the total strain.

Figure 1.9 Converting the stress-strain curves from the transient state test results [15]

In the steady-state tests, the test specimen was heated up to a specific temperature, and then a normal tensile test was carried out. The mechanical properties can be determined directly from the

20

recorded stress-strain curve. The comparisons of Comparison of the steady state and transient state test results of structural steel S350GD+Z at temperature 800°C are shown in Figure 1.10. It can be seen that the results using these two testing methods are different.

Figure 1.10 Comparison of the steady state and transient state test results of structural steel S350GD+Z at temperature 800°°°°C [15]

1.3.1.3 Mechanical properties provided in Eurocode 3 The steel grades in Eurocode 3 [7] are based on EN 10025 (S235, S275, S355) and EN 10113 (S420, S460). The mechanical properties of steel at 20 °C is taken as those given in Eurocode 3, Part 1.1 for normal design. The stress-strain relationship at elevated temperature is given in Figure 1.11 and can be used to determine the resistance to tension, compression, moment or shear. This is suitable for the heating rate from 2 to 50 K/min.

Strain range Stress Tangent modulus ε ≤ εp.θ ε⋅Ea.θ Ea.θ

εp.θ < ε < εy.θ fp.θ - c + (b / a)[a2 - (εy.θ - ε)2]0.5 b(εy.θ - ε) / {a[a2 - (εy.θ - ε)2]0.5} εy.θ ≤ ε ≤ εt.θ fy.θ 0 εt.θ < ε < εu.θ fy.θ[1 - (ε - εt.θ) / (εu.θ - εt.θ)] -

ε = εu.θ 0.00 - Parameters εp.θ = fp.θ / Ea.θ , εy.θ = 0.02, εt.θ = 0.15 , εu.θ = 0.20 Functions a2 = (εy.θ - εp.θ)⋅(εy.θ - εp.θ + c / Ea.θ)

b2 = c(εy.θ - εp.θ) Ea.θ + c2 c = (fy.θ - fp.θ)2 / [(εy.θ - εp.θ) Ea.θ - 2(fy.θ - fp.θ)]

21

fy.θ is the effective yield strength; fp.θ is the proportional limit; Ea.θ is the slope of the linear elastic range; εp.θ is the strain at proportional limit

εy.θ is the yield strain; εt.θ is the limiting strain for yield strength εu.θ is the ultimate strain

Figure 1.11 Stress-strain relationship for steel at elevated temperature (Eurocode 3) [7]

The variations of the reduction factor for effective yield strength, for proportional limit and for the slope of the linear elastic range are shown in Figure 1.12 [17]. The reduction factors, relative to the appropriate value at 20 °C, are given in Table 1.1.

Figure 1.12 Reduction factors for stress-strain relationship of steel at elevated temperature [17]

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000 1200

Temperature (C)

Red

uctio

n fa

ctor Effective yield strength

Slope of linear elastic range

Proportional limit

22

Table 1.1 Reduction factors for stress-strain relationship of steel at elevated temperatures [7]

Reduction factors at temperature relative to the value at 20 °°°°C Steel temperature

θθθθa Effective yield strength ky.θθθθ = fy.θθθθ / fy

Proportional limit kp.θθθθ = fp.θθθθ / fy

Slope of the linear elastic range kE.θθθθ = Ea.θθθθ / Ea

20 1.000 1.000 1.000 100 1.000 1.000 1.000 200 1.000 0.807 0.900 300 1.000 0.613 0.800 400 1.000 0.420 0.700 500 0.780 0.360 0.600 600 0.470 0.180 0.310 700 0.230 0.075 0.130 800 0.110 0.050 0.090 900 0.060 0.0375 0.0675

1000 0.040 0.0250 0.0450 1100 0.020 0.0125 0.0225 1200 0.000 0.0000 0.0000

Note: For intermediate values of the steel temperature, linear interpolation may be used As an example, Figure 1.13 illustrates the stress-strain relationship for steel grade S 355 at elevated temperature using the values given above. For steel grade S355, the yield strength is 355 MPa and the elastic modulus is 210 000 MPa. In Figure 1.13, no strain hardening is included. However, for temperature below 400 °C, the alternative strain hardening option can be used according to Annex B in Eurocode 3, Part 1.2 [7].

Figure 1.13 Stress-strain relationship for S355 at elevated temperature

23

Hot-rolled reinforcing bars are treated in Eurocode 4 in similar fashion to structural steels, but cold-worked reinforcing steel, whose standard grade is S500, deteriorates more rapidly at elevated temperatures than do the standard grades. Its strength reduction factors for effective yield and elastic modulus are shown in Figure 1.14. It is unlikely that reinforcing bars or mesh will reach very high temperatures in a fire, given the insulation provided by the concrete if normal cover specifications are maintained. The very low ductility of S500 steel (it is only guaranteed at 5%) may be of more significance, in which high strains of mesh in slabs are caused by the progressive weakening of supporting steel sections [17].

Figure 1.14 EC3 Strength reduction for structural steel (SS) and cold-worked reinforcement (Rft) at high temperatures [17]

1.3.2 Thermal properties Such material properties as density, specific heat and thermal conductivity are needed for heat transfer calculation in solid materials. Density, ρ, is the mass of the material per unit volume in kg/m3. Specific heat, cp, is the amount of heat required to heat a unit mass of material by one degree with unit of J/kgK. Thermal conductivity, λ, represents the rate of heat transferred through a unit thickness material per unit temperature difference with unit of W/mK. Two other derived properties which are often needed, i.e. the thermal diffusivity given by λ/ρc with unit of m2/s and thermal inertia given by α=λρc with unit of W2s/m4K2. When materials with low thermal inertia are exposed to heating, surface temperature increase rapidly, so that these materials ignite more readily. In the following sections, the values of some thermal properties provided in Eurocode 3 [7] are described.

0 300 600 900 1200

100

80

60

40

20

% of normal value

Temperature (°C)

Effective yield strength(at 2% strain)

Elastic modulus

SSRft

SS

Rft

24

1.3.2.1 Specific heat In Eurocode 3, Part 1.2 [7], the specific heat of steel (J/kgK) may be determined as follows:

ca = 425 + 7.73·10-1θθθθa - 1.69·10-3·θθθθa2 - 2.22·10-6·θθθθa

3 (20°°°°C ≤≤≤≤ θθθθa < 600°°°°C) ca = 666 + 13002 / (738 - θθθθa) (600°°°°C ≤≤≤≤ θθθθa < 735°°°°C) ca = 545 + 17820 / (θθθθa - s731) (735°°°°C ≤≤≤≤ θθθθa < 900°°°°C) ca = 650 (900°°°°C ≤≤≤≤ θθθθa ≤≤≤≤ 1200°°°°C)

The variation of specific heat with temperature is illustrated in Figure 1.15. The value of specific heat undergoes a very dramatic change in the range 700-800°C. The apparent sharp rise to an "infinite" value at about 735°C is actually an indication of the latent heat input needed to allow the crystal-structure phase change to take place. When simple calculation models are being used a single value of 600J/kgK is allowed, which is quite accurate for most of the temperature range but does not allow for the endothermic nature of the phase change.

Figure 1.15 Variation of the specific heat of steel with temperature [17]

1.3.2.2 Thermal conductivity The thermal conductivity of steel may be defined as follows [7] λλλλa = 54-3.33·10-2·θθθθa (20°°°°C ≤≤≤≤ θθθθa < 800°°°°C) λλλλa = 27.3 (800°°°°C ≤≤≤≤ θθθθa ≤≤≤≤ 1200°°°°C) The variation of thermal conductivity with temperature is shown in Figure 1.16. For simple design calculations the constant conservative value of 45W/m°C is allowed.

5000

0 200 400 600 800 1000 1200

Temperature (°C)

Specific Heat(J/kg°K)

4000

3000

2000

1000

ca=600 J/kg°K(EC3 simple calculationmodels)

25

Figure 1.16 Eurocode 3 representations of the variation of thermal conductivity of steel with temperature [17]

1.3.2.3 Thermal elongation In most simple fire engineering calculations thermal expansion of materials is neglected, but for steel members which support a concrete slab on the upper flange the differential thermal expansion caused by shielding of the top flange and the heat-sink function of the concrete slab causes a “thermal bowing” towards the fire in the lower range of temperatures. In Eurocode 3, Part 1.2 [7], the thermal elongation is defined as the function of temperature and may be determined as follows: ∆∆∆∆l /l = 1.2××××10-5θθθθa+0.4××××10-8θθθθa

2-2.416××××10-4 (20°°°°C ≤≤≤≤ θθθθa < 750°°°°C) ∆∆∆∆l /l = 1.1××××10-2 (750°°°°C ≤≤≤≤ θθθθa ≤≤≤≤ 860°°°°C) ∆∆∆∆l /l = 2××××10-5θθθθa-6.2××××10-3 (860°°°°C < θθθθa ≤≤≤≤ 1200°°°°C) where, l is the length at 20°C; ∆l is the temperature induced expansion; and θa is the steel temperature. The variation of thermal elongation with temperature is illustrated in Figure 1.17. When the exposed steel sections reach a certain temperature range within which a crystal-structure change takes place and the thermal expansion temporarily stops. In simple calculation models, the relationship between thermal elongation and steel temperature may be considered to be constant. In this case the elongation may be determined from ∆∆∆∆l /l = 14××××10-6(θθθθa-20) ( 1.23 )

10

20

30

40

50

60

0 200 400 600 800 1000 1200

Thermalconductivity(W/m°K)

Temperature (°C)

λa=45 W/m°K (EC3 simple calculation models)

26

Figure 1.17 Thermal elongation of steel as a function of the temperature ( Eurocode 3, Part 1.2) [7]

1.4 Passive Protection for Steelwork

1.4.1 Fire protection systems The traditional approach to fire resistance of steel structures has been to clad the members with insulating material. This may be in alternative forms [17]:

o Boarding (plasterboard or more specialized systems based on mineral fiber or vermiculite) fixed around the exposed parts of the steel members. This is fairly easy to apply and creates an external profile that is aesthetically acceptable, but is inflexible in use around complex details such as connections. Ceramic fiber blanket may be used as a more flexible insulating barrier in some cases.

o Sprays that build up a coating of prescribed thickness around the members. These tend to use vermiculite or mineral fiber in a cement or gypsum binder. Application on site is fairly rapid, and does not suffer the problems of rigid boarding around complex structural details. Since the finish produced tends to be unacceptable in public areas of buildings these systems tend to be used in areas that are normally hidden from view, such as beams and connections above suspended ceilings.

o Intumescent paints, which provide a decorative finish under normal conditions, but which foam and swell when heated, producing an insulating char layer which is up to 50 times as thick as the original paint film. They are applied by brush, spray or roller, and must achieve a specified thickness that may require several coats of paint and measurement of the film thickness.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 200 400 600 800 1000 1200

Temperature (C)

Elon

gatio

n

27

All of these methods are normally applied as a site operation after the main structural elements are erected. This can introduce a significant delay into the construction process, which increases the cost of construction to the client. The only exception to this is that some systems have recently been developed in which intumescents are applied to steelwork at the fabrication stage, so that much of the site-work is avoided. However, in such systems there is clearly a need for a much higher degree than usual of resistance to impact or abrasion. These methods can provide any required degree of protection against fire heating of steelwork, and can be used as part of a fire engineering approach. However traditionally thicknesses of the protection layers have been based on manufacturers’ data aimed at the relatively simplistic criterion of limiting the steel temperature to less than 550°C at the required time of fire resistance in the ISO834 standard fire. Fire protection materials are routinely tested for insulation, integrity and load-carrying capacity in ISO834 furnace test. Material properties for design are determined from the results by semi-empirical means. Full or partial encasement of open steel sections in concrete is occasionally used as a method of fire protection, particularly in the case of columns for which the strength of the concrete, either reinforced or plain, can contribute to the ambient-temperature strength. In the case of hollow steel sections concrete may be used to fill the section, again either with or without reinforcing bars. In fire this concrete acts to some extent as a heat sink, which slows the heating process in the steel section. In a few buildings hollow-section columns have been linked together as a system and filled with water fed from a gravity reservoir. This can clearly dissipate huge amounts of heat, but at rather high cost, both in construction and maintenance. The most recent design codes are explicit about the fact that the structural fire resistance of a member is dependent to a large extent on its loading level in fire, and also that loading in the fire situation has a very high probability of being considerably less than the factored loads for which strength design is performed. This presents designers with another option that may be used alone or in combination with other measures. A reduction in load level by selecting steel members that are stronger individually than are needed for ambient temperature strength, possibly as part of a strategy of standardizing sections, can enhance the fire resistance times, particularly for beams. This can allow unprotected or partially protected beams to be used. The effect of loading level reduction is particularly useful when combined with a reduction in exposed perimeter by making use of the shielding and heat sink effects of the supported concrete slab. The traditional down stand beam (Figure 1.18) gains some advantage over complete exposure by having its top flange upper face totally shielded by the slab; supporting the slab on shelf angles welded to the beam web keeps the upper part of the beam web and the whole top flange cool, which provides a greater enhancement. The recent innovation of “Slimflor” beams, in which an unusually shallow beam section is used and the slab is supported on the lower flange, either by pre-welding a plate across this flange or by using an asymmetric steel section, leaves only the lower face of the bottom flange exposed. Alternative fire engineering strategies are beyond the scope of this lecture, but there is an active encouragement to designers in the Eurocodes to use agreed and validated advanced calculation models for the behavior of the whole structure or sub-assemblies. The clear implication of this is

28

that designs which can be shown to gain fire resistance overall by providing alternative load paths when members in a fire compartment have individually lost all effective load resistance are perfectly valid under the provisions of these codes. This is a major departure from the traditional approach based on the fire resistance in standard tests of each component. In its preamble Eurocode 3 Part 1-2 also encourages the use of integrated fire strategies, including the use of combinations of active (sprinklers) and passive protection, although it is acknowledged that allowances for sprinkler systems in fire resistant design are at present a matter for national Building Regulations.

Figure 1.18 Inherent fire protection to steel beams [17]

1.4.2 Thermal properties of fire protection systems Typical values of thermal properties of insulating materials are given in Table 1.2, from ECCS (1995) [1].

Table 1.2 Thermal properties of insulation materials

Materials Density ρρρρi

(kg/m3)

Thermal conductivity λλλλi

(W/mK)

Specific heat ci

(J/kgK)

Equilibrium moisture content

% Sprays Sprayed mineral fiber 300 0.12 1200 1 Perlite or vermiculite plaster

350 0.12 1200 15

High density perlite or vermiculite plaster

550 0.12 1200 15

Boards: Fiber-silicate or fiber-calcium silicate

600 0.15 1200 3

Gypsum plaster 800 0.20 1700 20 Compressed fiber boards Mineral wool, fiber silicate 150 0.20 1200 2

Downstand beam

Shelf-angle beam

Slimflor beam

29

1.5 Heat Transfer in Steel

1.5.1 Type of heat transfer Heat transfer involves the following three processes: conduction, convection and radiation, which can occur separately or together depending on the circumstances. Conduction Conduction is the mechanism for heat transfer in solid materials. In materials that are good conductors of heat, the heat is transferred by interaction involving free electrons. In materials that are poor conductors, heat is conducted via mechanical vibrations of molecular lattice. Conduction of heat is an important factor in the ignition of solid surfaces, and in fire resistance of barriers and structural members. In the steady state, the heat transfer by conduction is directly proportional to the temperature gradient between two points and the thermal conductivity, λ, i.e. [1]

hD =λλλλ dθθθθ/dx ( 1.24 )

where hD is the heat flow per unit area (W/m2), λ is the thermal conductivity (W/mK), θ is the temperature (°C), and x is the distance in the direction of heat flow (m). In the transit state, i.e. the temperatures are changing with time, the amount of heat required to change the temperature of the materials must be included. For one dimension heat transfer by conduction with no internal heat being released, the governing equation is [1] δδδδ 2θθθθ /δδδδ 2x=(1/αααα)/(δδδδ θθθθ/δδδδ t) ( 1.25 )

where t is time (s) and α = λ/ρc is thermal diffusivity (m2/s). These equations can be solved using analytical, graphical or numerical methods. Convection Convection is heat transfer by the movement of fluids, either gases or liquids. Convective heat transfer is an important factor in flame spread and in the upward transport of smoke and hot gases to the ceiling or out of window from a compartment fire. For given conditions, the heating transfer is proportional to the temperature difference between to materials, so that the heat flow per unit area can be calculated using [1]

hD =ααααc ∆∆∆∆θθθθ ( 1.26 )

where αc is the convective heat transfer coefficient (W/m2K) and ∆θ is the temperature difference between the surface of the solid and the fluid (°C). In Eurocode 3, Part 1.2 [7], the coefficient of heat transfer by convection is given as follows:

30

Table 1.3 Coefficient of heat transfer by convection

ααααc (W/m2K) Exposed sides the standard temperature-time curve is used 25 the external fire curve is used 25 the hydrocarbon temperature-time is used 50 the simplified fire models are used 35 the advanced fire models are used 35 Unexposed side of separating members the radiation effects are not included 4 the radiation effects are included 9

Radiation Radiation is transfer of energy by electromagnetic waves that can be travel through a vacuum or through a transparent solid or liquid. Radiation is the main mechanism for heat transfer from flames to fuel surfaces, from hot smoke to building objects and from a burning building to an adjacent building. The heat flow per unit area can be calculated as [1]:

hD =ΦΦΦΦ εεεε σσσσ [(θθθθe + 273) 4 – (θθθθr +273 ) 4] ( 1.27 )

where Φ is the configuration factor that is a measure of how much of the emitter is ‘seen’ by the receiving surface. ε is the resultant emissivity of two surface and can be calculated as ε=1/(1/εr + 1/εe - 1). σ is the Stefan-Boltzmann constant and its value is σ = 5.67 × 10-8 W/m2K4. θe is the temperature of emitting surface (°C) and θr is the temperature of receiving surface (°C).

1.5.2 Heat transfer equation for steel The rise of temperature in a structural steel member depends on the heat transfer between any two elements that are at different temperature. Conduction, radiation and convection are the modes by which thermal energy flows from regions of high temperature to those of low temperature. On the external surfaces of the elements, all three mechanisms are present. Inside the element, heat is transferred from point to point only by conduction. Calculation of heat transfer requires knowledge of the geometry of element, thermal properties of the materials and heat transfer coefficient at boundaries. Practical difficulties are that some of thermal properties are temperature dependent as shown in 1.3.2.

1.5.2.1 General equation The general approach to study the increase of the temperature in structural elements exposed to fire is based on the integration of the Fourier-differential equation for transient conduction inside the member. This equation is given as [16]:

31

dtdch

zzyyxx netθθθθθθθθρρρρ

δδδδθθθθδδδδθθθθλλλλ

δδδδδδδδ

δδδδθθθθδδδδθθθθλλλλ

δδδδδδδδ

δδδδθθθθδδδδθθθθλλλλ

δδδδδδδδ )()()()(

=+

+

+

D ( 1.28 )

where x, y, and z is the Cartesian coordinates inside the structural element; λ is thermal conductivity; ρ is the density; c is the specific heat and hD is the net heat flux that is due to convection and radiation, i.e.

neth� = cneth .� + rneth .

� ( 1.29 )

where, the net convective heat flux can be determined by

cneth .� = ααααc (θθθθg - θθθθm) ( 1.30 )

in which αc is the coefficient of heat transfer by convection, θg is the gas temperature in the vicinity of the fire exposed member (°C) and θm is the surface temperature of the member (°C). The net radiative heat flux component per unit of surface area is determined by:

rneth .� =ΦΦΦΦ εεεε σσσσ [(θθθθr + 273) 4 – (θθθθm +273 ) 4] ( 1.31 )

in which θr is the effective radiation temperature of the fire environment (°C). The solution of Fourier-differential equation can be obtained when the initial and boundary conditions are known. For fire, the initial conditions consist of the temperature distribution at the beginning of the analysis (usually the room temperature before fire); boundary conditions must be defined on every surface of the structure, for instance, boundaries exposed to fire and boundaries unexposed to fire. Usually fire simulations are based on the temperature history of the fire, for instance the standard fire curve. However, any other any fire conditions can be assumed, using other type of temperature-time curves. Numerical methods are necessary to solve this equation. Many computer programs are available and it is possible to carry out thermal analysis for very complex structural elements. For instance, Ma and Mäkeläinen [11] in the Laboratory of Steel Structures at HUT has developed a computer program to perform temperature analysis of steel-concrete composite slim floor structures exposed to fire based on this heat transfer equation. As an example, Figure 1.19 shows the section shape of a new slim floor beam, which is composed of a three-plate-welded beam, a profiled steel deck and a concrete slab over the steel deck. Figure 1.20 shows the temperature distribution of this slim floor beam under standard temperature-time curve when the fire exposure is 60 minutes. In many cases, the general form of the equation can be greatly simplified. For instance, thermal conductivity, density and specific can be assumed to be independent of temperature; internal heat generation is absent or can be neglected; and three-dimensional problems can be studied as two-dimensional or one dimensional idealizations.

32

Figure 1.19 Section shape of new slim floor beam [11]

Figure 1.20 Temperature distribution of the new slim floor beam under ISO fire (60 minutes) [11]

1.5.2.2 Temperature calculation for unprotected steel members Since the thermal conductivity is high enough to allow the difference of temperature in the cross-section to be neglected. This assumption means that thermal resistance to heat flow is negligible. Any heat supplied to the steel section is instantly distributed to give a uniform steel temperature. With this assumption, the energy balance can be made based on the principle that the heat entering the steel over the exposed surface area in a small time step ∆t (s) is equal to the heat required to raise temperature of the steel by ∆θ (°C), i.e. [1]

60

117

183

Concrete

Rannila 120

Steel Deck

Reinforcement Mesh

Asymmetric

Steel Beam

10

20

400

200

18

fillet weld

fillet weld

33

heat entering = heat to raise temperature

neth� Am ∆∆∆∆t = ρρρρa ca V ∆∆∆∆ θθθθa ( 1.32 )

and the temperature increase of steel can be calculated as

∆∆∆∆ θθθθa.t = (Am/V)(1/ρρρρa ca) neth� ∆∆∆∆t ( 1.33 )

where neth� is the heat flow per unit area (W/m2) and is given by:

neth� = ααααc (θθθθg - θθθθm) + ΦΦΦΦ εεεε σσσσ [(θθθθr + 273) 4 – (θθθθm +273 ) 4] ( 1.34 )

The meanings and the values of other symbols are given in Table 1.4.

Table 1.4 Parameter values for exterminating temperature increase

Symbols Meanings Values according to Eurocode Unit ρa Density of steel 7850 kg/m3 ca Specific heat of steel see 1.3.2.1 or 600 for a simple calculation model J/kgK αc Coefficient of heat transfer

by convection see Table 1.3 W/m2K

Φ Configuration factor can be taken as 1. A lower value may be chosen to consider position and shadow effect

----

ε Resultant emissivity of two surface

can be calculated as ε=εm·εf with εm = 0.8 and εf = 1.0, ε = 0.8

-----

σ Stefan-Boltzmann constant 5.67 × 10-8 W/m2K4 θg Temperature of gas nominal temperature-time curve or parametric

temperature-time curves °C

θr Effective radiation temperature of the fire environment

θr = θg °C

Am/V Section factor see Table 1.6 1/m ∆θa.t Temperature change of steel Calculation results °C

Solving the increasemental equation step by step gives the temperature development of the steel element during the fire. A spreadsheet for calculating steel temperatures is shown in Table 1.5. In order to assure the numerical convergence of the solution, some upper limit must be taken for the time increasement ∆t. In Eurocode 3, Part 1.2 [7], it suggested that the value of ∆t should not be taken as more than 5 seconds.

Table 1.5 Spreadsheet calculation for temperatures of unprotected steel section [1]

Time Steel temperature θθθθa

Fire temperature θθθθg

Temperature change in steel ∆∆∆∆θθθθa

t1 = ∆t Initial steel temperature θa0

Fire temperature half way through time step (at ∆t / 2)

Calculating from increasemental equation with θa and θg from this row

t2 = t1 + ∆t θa + ∆θa θa Temperature for previous row

Fire temperature half way through time step (at t1+∆t / 2)

Calculating from increasemental equation with θa and θg from this row

34

An important parameter in determining the rise of temperature of the steel section is section factor, Am/V (sometimes given as F/V or A/V or Hp/V in different countries). The section factors for some of the unprotected steel members in Eurocode 3, Part 1.2[7] are shown in Table 1.6.

Table 1.6 Section factor for unprotected steel members [7]

Open section exposed to fire Am/V: Perimeter / Section area

Open section exposed to fire on three sides Am/V: Surface exposed to fire / Section area

Hollow section or welded box section with uniform thickness exposed to fire on all sides if t << b, Am / V = 1 / t

Tube exposed to fire on all sides Am / V = 1 / t

When the profile is in contact with a concrete slab, which has a thermal conductivity greatly lower than that of steel, the effective exposed perimeter Am must be calculated using directed exposed part. This requires an assumption of an adiabatic condition at the contact surface. The result is safe. In fact some thermal energy passes through the colder body and, if it is neglected, the increase of the temperature in the steel element is higher. It is very important to understand this point, because it gives the key to deciding if the simplified solution of the thermal problem is appropriate or if it is necessary to solve the complete the heat transfer equation.

1.5.2.3 Temperature calculation for protected steel members For members with passive protection the basic mechanisms of heat transfer are identical to those for unprotected steelwork, but the surface covering of material of very low conductivity induces a considerable reduction in the heating rate of the steel section. Also, the insulating layer itself has the capacity to store a certain, if small, amount of heat. It is acceptable to assume that the exposed insulation surface is at the fire atmosphere temperature (since the conduction away from the surface is low and very little of the incident heat is used in raising the temperature of the surface layer of

35

insulation material). The calculation of steel temperature rise ∆θa.t in a time increment ∆t is now concerned with balancing the heat conduction from the exposed surface with the heat stored in the insulation layer and the steel section:

( ) ( ) tgtatgp

aa

ppta et

VA

cd

.10/

... 13/1

1/θθθθθθθθθθθθ

φφφφρρρρλλλλ

θθθθ φφφφ ∆−−∆−

+

=∆ but 0. ≥∆ taθθθθ ( 1.35 )

in which the relative heat storage in the protection material is given by the term

VA

dcc p

paa

pp

ρρρρρρρρ

φφφφ = ( 1.36 )

in which Ap/V section factor for protected steel member, where Ap is generally the inner perimeter of the protection material and the values are shown in Table 1.7.

Table 1.7 Section factors of steel members insulated by fire protection materials[7]

Sketch Description Section factor (Ap/V)

Contour encasement of uniform thickness

Steel perimeter / Steel section area

Hollow encasement of uniform thickness

2(b+h) / Steel section area

Contour encasement of uniform thickness to fire on three sides

(Steel perimeter-b) / Steel section area

Hollow encasement of uniform thickness exposed to fire on three sides

(2h+b) / Steel section area

Normally, the section factors represent the ratio of the effective surface exposed to fire to the volume of the element. When there is a protective coating, the surface to be taken into account is not

36

the external surface of the profile but the inner steel surface. cp is the specific heat of protection material; λp is thermal conductivity of the fire protection material; ρp is the density of fire protection material. These values are given in Table 1.2. dp is the thickness of fire protection material. The value of ∆t should not be taken as more than 30 seconds. Fire protection materials often contain a certain percentage of moisture that evaporates at about 100°C, with considerable absorption of latent heat. This causes a “dwell” in the heating curve for a protected steel member at about this temperature while the water content is expelled from the protection layer. The incremental time-temperature relationship above does not model this effect, but this is at least a conservative approach. A method of calculating the dwell time is given, if required, in the European pre-standard for fire testing.

1.5.2.4 Example: temperature analysis for both unprotected and protected steel members The following example shows the temperature analysis of steel beam with three-side exposure to fire and box protection with gypsum board under standard fire. The cross-section and the required parameter of the gypsum board are given in Figure 1.21. The results of temperature-time curves for unprotected steel beam and protected beam together with the standard fire curve are shown in Figure 1.22. The thickness of 12.5 mm gypsum board is used and it can be seen that with this thickness, the temperature of steel beam drops dramatically at 30 minutes.

Figure 1.21 Cross-section of steel beam and properties of protection material

Figure 1.22 Temperature-time curves of unprotected and protected steel beam together with standard fire

37

1.6 Mechanical Analysis of Structural Element Fire resistance is a measure of the ability of building element to resist a fire. Fire resistance is most often quantified as the time to which the element can meet certain criteria during an exposure to a standard fire test. Structural fire resistance can also be quantified using temperature or load capacity of a structural element exposed to a fire. Verification of fire resistance should be in one of the following domain [3]:

o Time domain: tfi.d ≥ tfi.requ o Strength domain Rfi.d.t ≥ Efi.d.t o Temperature domain Θd ≤ Θcr.d

where tfi.d is the design value of fire resistance; tfi.requ is the required fire resistance time; Rfi.d.t is the design value of the resistance of the member in the fire situation at time t; Efi.d.t is the design value of the relevant effects of actions in the fire situation at time t; Θd is the design value of material temperature; Θcr.d is the design value of critical material temperature. tfi.requ, Efi.d.t, Θd are the variables to describe fire severity. Fire safety is a measure of the destructive impact of a fire, or measure of the forces or temperatures that could cause collapse or other failure as a result of the fire. tfi.d, Rfi.d.t, and Θcr.d are used to describe the fire resistance.

1.6.1 Required fire resistance time The required fire resistance time is usually a time of standard fire exposure specified by a building code, or the equivalent time of standard fire exposure calculated for a real fire in building.

1.6.1.1 Standard fire exposure Required fire resistance time are specified in National Codes, for instance, in Finland, the required fire resistance time is prescribed in E1 National Building Code of Finland, Structural Fire Safety, Regulation, Helsinki, Ministry of the Environment (cited with abbreviation: RakMK E1), and E2 National Building Code of Finland, Fire Safety in Industrial and Warehouse buildings, Helsinki, Ministry of the Environment (cited with abbreviation: RakMK E2). Required fire resistance time normally depends on factors such as: type of occupancy, height and size of the building, effectiveness of fire brigade action, and active measures such as vents and

38

sprinklers [5]. An overview of fire resistance requirements in various European countries as a function of above factors is given in Table 1.8 [5]. From this table it shows that

o For one storey buildings, no or low requirements are needed and ISO-fire class is possibly up to R30;

o For 2 to 3 storeys buildings, no up to medium requirements are needed and ISO-fire class is possibly up to R60;

o For more than 3 storey buildings, medium requirements are needed and ISO-fire class is R60 to R120;

o For tall buildings, high requirements are needed and ISO-fire class is R90 and more. Although quite large variations exist, the required fire resistance time is not beyond 90 to 120 minutes. If requirements are set, the minimum values are 30 minutes (some countries have minimum requirements of 15 or 20 minutes). Intermediate values are usually given in steps of 30 minutes, leading to a schema of 30, 60, 90, 120, ... minutes.

1.6.1.2 Equivalent time of fire exposure Equivalent time of fire exposure is a quantity which relates a non-standard or natural fire exposure to the standard fire, i.e. an equivalent time of exposure to standard fire is supposed to have the same severity as a real fire in the compartment. This equivalent can be determined based on equal area concept, maximum temperature concept and minimum load capacity concept [1]. The key difference lies in the definition of ‘severity’. Equal area concept Figure 1.23 illustrates the concept first proposed by Ingberg (1928), by which two fires are considered to have equivalent severity if the areas under each curves are equal, above a certain temperature (150 or 300 °C). This has little theoretical significance because the product of temperature and time is not heat as expected. However, his work formed the starting points of current regulations of fire class [1].

Figure 1.23 Equivalent fire severity on equal area concept [1]

39

Table 1.8 Minimum Periods (minutes) for elements of structure [5]

In the following building types According to the regulations of Building

type n h H X L b x(*) S B CH D F I L NL FIN SP UK

Y 0 0 0 30*2 0/60 (7)

0 0 0 - 0*1 Industrial hall

1 0 10 20 100 50 2

N 0 (1)*3 (1) 30*2 30/90 (7)

0-60 0 0 - 0*1

Y 0 0 0 0 H 60/90 (7)

30 0 0 90 0*1 Commercial center and shop

1 0 4 500 80 80 4

N (1) (1)*3 (1) 30 V 90/120 (7)

(3) 0 30 90 0*1

Y 0 0 (2) 60 (8)(9) 30 0 60(4) 90 30 Dancing 2 5 9 1000 60 30 4 N 0 30 90 60 60 30 0 60(5) 90 60

Y 60(6) 0 30*3

(2) 60 (8) (10)

90 60 60(4) 60 60 School 4 12 16 300 60 20 4

N 60(6) 60 90 60 60 90 60 60(5) 60 60

Y 60(6) 0 30*3

(2) 60 (8) (9)

90 60 60(4) 60 30 Small rise office building

4 10 13 50 50 30 2

N 60(6) (1)*3 90 60 60 90 60 60(5) 60 60

Y 60(6) 30 60*3

(2) 60 (8) (11)

90 60 60(4) 90 60 Hotel 6 16 20 60 50 30 2

N 60(6) 60 90 60 60 90 60 60(5) 90 60

Y 120 60 (2) 60 (8)(12) 90/120 120 60(4) 120 90 Hospital 8 24.5 28 60 70 30 2 N 120 90 90 60 120 120 120 60(5) 120 90

Y 120 60 90*3

(2) 120 (8) (9)

90 60 120 (4)

120 120 Medium rise office building

11 33 37 50 50 30 2

N 120 90 90 120 90 120 90 120 (5)

120 (3)

Y 120 90 90 120 (8) (9)

120 90 120 (4)

120 120 High rise office building

31 90 93 100 50 50 2

N 120 90(3) (3) 120 120 (3) 90 120 (5)

120 (3)

40

Maximum temperature concept Maximum temperature concept developed by Law, Pettersson et al. and others is to define the equivalent fire severity as the time of exposure to the standard fire that would result in the same maximum temperature in a steel member as would occur in a complete burnout of the fire compartment as shown in Figure 1.24 [1]. This concept is widely used and current Eurocode is based on this method.

Figure 1.24 Equivalent fire severity based on maximum temperature concept [1]

Minimum load capacity concept In this concept, the equivalent fire severity is the time of exposure to the standard fire that would result in the same load bearing capacity as the minimum which would occur in a complete burnout of the fire compartment as shown in Figure 1.25 [1]. The load bearing capacity of a structural member exposed to the standard fire decreases continuously, but the strength of the same member exposed to a natural fire increases after the fire enters the decay period and the steel temperature decreases.

Figure 1.25 Equivalent fire severity based on minimum load capacity concept [1]

41

Equivalent time of fire exposure in Eurocode The equivalent time of ISO fire exposure is defined by [3] te.d = (qf.d ·kb ·wf) kc or te.d = (qf.d ·kb ·wt) kc ( 1.37 )

where qf.d is the design fire load density; kb is the conversion factor; wf is the ventilation factor and kc is the correction factor function of the material composing structural cross-sections.

1.6.2 Mechanical actions Mechanical actions include actions from normal conditions of use and indirect fire actions. Indirect actions may occur as result of restrained thermal expansion and depend on the temperature development in the structural system and different in stiffness. A typical example of indirect action due to fire is temperature-induced stress due to non-uniform temperature distribution over the cross-section. In normal condition of use, the load combination for ultimate limit state verification in Eurocode is defined as [6]: E d = γγγγ G·G k + γγγγ Q1·Q k1 + ΣΣΣΣγγγγ Qi·Q ki ( 1.38 )

The actions during fire exposure is in accordance with the accidental design situation and the load combination is defined as [6]: E fi.d = γγγγ GA·G k + ψψψψ 1.1·Q k1 + ΣΣΣΣψψψψ 2.i·Q ki + Ad ( 1.39 )

where

γG = 1.35 Partial factor for permanent loads: strength design γQ = 1.5 Partial factor for variable loads: strength design

γGA = 1.0 Partial factor for permanent loads: accidental design situations ψ1.1 Table 1.9 Combination factor: variable loads ψ 2.i Table 1.9 Combination factor: variable loads E d Design value of effects of actions from normal design

E fi.d Constant design value in fire exposure G k Characteristic value of permanent action

Q k1 Characteristic value of dominant variable action Q ki Characteristic value of other variable actions Ad Design value of accidental action: indirect action in fire

Due to the low probability that both fire and extreme severity of external actions occur at the same time and indirect actions not being considered for standard fire exposure, the above two formulas can be simplified as: E d = γγγγ G·G k + γγγγ Q1·Q k1 ( 1.40 )

in normal condition, and E fi.d = γγγγ GA·G k + ψψψψ 1.1·Q k1 ( 1.41 )

in fire situation. The values of combination factors are given in Table 1.9.

42

Table 1.9 Values of combination factor [6]

Actions ψ0 ψ1 ψ2 Imposed loads in buildings, category (EN 1991-1-1) Category A: domestic, residential areas Category B: office areas Category C: congregation areas Category D: shopping areas Category E: storage areas Category F: traffic area, vehicle weight = 30 kN Category G: traffic area, 30 kN < vehicle weight = 160 kN Category H: roofs

0.7 0.7 0.7 0.7 1.0 0.7 0.7 0

0.5 0.5 0.7 0.7 0.9 0.7 0.5 0

0.3 0.3 0.6 0.6 0.8 0.6 0.3 0

Snow loads on building (see EN 1991-1-3)* Finland, Iceland, Norway, Sweden Remainder of CEN member States, for sites located at altitude H > 1000 m a.s.l H = 1000 m a.s.

0.7

0.7 0.5

0.5

0.5 0.2

0.2

0.2 0

Wind loads on buildings (see EN 1991-1-4) 0.6 0.2 0 Temperature (non-fire) in building (see EN 1991-1-5) 0.6 0.5 0 Note: value of ψ may be set by national annex * for countries not mentioned above, see relevant local conditions

The reduction factor can be defined either as [3] ηηηη fi = E fi.d.t / R d ( 1.42 )

in which the loading in fire is taken as a proportion of ambient-temperature design resistance when global structural analysis is used, or [3] ηηηη fi = E fi.d.t / E d ( 1.43 )

in which loading in fire is taken as a proportion of ambient-temperature factored design load when simplified design of individual members is used and only the principal variable action is used together with the permanent action. This may be expressed in terms of the characteristic loads and their factors as

1.1.

1.1.1

kQkG

kkGAfi QG

QGγγγγγγγγψψψψγγγγ

ηηηη++

= ( 1.44 )

1.6.3 Design value of material temperature The design value of material temperature, Θd, is the maximum temperature reached in fire or temperature at the time specified by code. This temperature can be determined using heat transfer analysis.

43

1.6.4 Design value of fire resistance time Fire resistance time can be described using fire resistance class (grade), or fire resistance level, or fire resistance rating. Fire resistance rating is normally assigned starting with 15 and 30 minutes, and continuing in whole numbers of hours or parts of hours, for instances, 30/60/90 minutes. Fire resistance rating can be obtained using full-scale fire resistance test, calculation or expert opinions. These ratings are listed in various documents maintained by testing authorities, code authorities or manufacturers and can be classified into three categories, i.e. generic ratings, which apply to typical materials, proprietary ratings, which are linked to particular manufacturers, and approved calculation methods. Full-scale testing is the most common method of obtaining fire resistance ratings [1]. Fire resistance tests are carried out on representative specimens of building elements. For example, if a representative sample of a floor system has been exposed to the standard fire for at least two hours while meeting the specified failure criteria, a similar assembly can be assigned a two hour fire resistance rating for use in a real building. For fire resistance testing, most European countries have standards similar to ISO 834 and in the United States, Canada and some other countries is ASTM E119. The relevant British Standards are BS 476 Parts 20-23 (BSI, 1987) [1]. The test is mainly carried out in a furnace that is composed of a large steel box lined with firebricks or ceramic fiber blanket. The furnace will have a number of burners, most often fuelled with gas but sometimes with fuel oil. There exist an exhaust chimney, several thermocouples for measuring gas temperatures and usually a small observation. The most common apparatus for full-scale fire resistance testing is the vertical wall furnace. The minimum size specified by most testing standards is 3.0×3.0 m2 (ISO 834 or ASTM E119). Some furnaces are 4.0 m tall [1]. Three failure criteria for fire resistance testing are stability, integrity and insulation [1]. To meet the stability criteria, a structural element must perform its loading bearing function and carry the applied loads for the duration of the test without structural collapse. The integrity and insulation criteria are intended to test the ability of a barrier to contain a fire, to prevent fire spreading from the room of origin. To meet the integrity criterion, the test specimen must not develop any cracks or fissures that allow smoke or hot gases to pass through the assembly. To meet the insulation criterion, the temperature of the cold side of the test specimen must not exceed a specified limit, usually an average increase of 140 °C and a maximum increase of 180 °C at a single point. Fire resistance of building elements, such as walls, beams, columns and doors etc., depends on many factors including the severity of the fire test, the material, the geometry and support conditions of the element, restraint from the surrounding structure and the applied loads at the time of the fire. Furnace testing using the standard time-temperature atmosphere curve is the traditional means of assessing the behavior of frame elements in fire, but the difficulties of conducting furnace tests of representative full-scale structural members under load are obvious. The size of furnaces limits the size of the members tested, usually to less than 5m, and if a range of load levels is required then a separate specimen is required for each of these. Tests on small members may be unrepresentative of the behavior of larger members.

44

A further serious problem with the use of furnace tests in relation to the behavior of similar elements in structural frames is that the only reliable support condition for a member in a furnace test is simply supported, with the member free to expand axially. When a member forms part of a fire compartment surrounded by adjacent structure which is unaffected by the fire its thermal expansion is resisted by restraint from this surrounding structure. This is a problem that is unique to the fire state, because at ambient temperatures structural deflections are so small that axial restraint is very rarely an issue of significance. Axial restraint can in fact work in different ways at different stages of a fire; in the early stages the restrained thermal expansion dominates, and very high compressive stresses are generated, but in the later stages when the weakening of the material is very high the restraint may begin to support the member by resisting pull-in. Furnace tests that allow axial movement cannot reproduce these restraint conditions at all; in particular, in the later stages a complete collapse would be observed unless a safety cut-off criterion is applied. In fact a beam furnace test is always terminated at a deflection of not more than span/20 for exactly this reason. Only recently has any significant number of fire tests been performed on fire compartments within whole structures [13]. Some years may pass before these full-scale tests are seen to have a real impact on design codes. In fact full-scale testing is so expensive that there will probably never be a large volume of documented results from such tests, and those that exist will have the major function of being used to validate numerical models on which future developments of design rules will be based. At present, Eurocodes 3 and 4 allow for the use of advanced calculation models, but their basic design procedures for use in routine fire engineering design are still in terms of isolated members and fire resistance is considered mainly in terms of a real or simulated furnace test [1].

1.6.5 Critical temperature For a separating member, the critical temperature is the temperature on the unexposed surface allowed fire to spread to other room. For a load bearing member, the critical temperature Θcr.d of a member is the temperature at which a member is calculated to fail under its given loading. This temperature can be calculated from knowledge of loads, load capacity at normal temperature and effects of elevated temperature on materials. This can be determined for all types of member when using Eurocode 3 from the degree of utilization, µ0, of the member in the fire design situation. The following equation, plotted in Figure 1.26, defines the critical temperature [7]:

482196740

11939 83330

+

−= ,,

ln.µµµµ

θθθθcr ( 1.45 )

This is used for all except the very slender Class 4 sections, for which a single conservative critical temperature of 350°C is specified. The degree of utilization, µ0, is a proportion of the design loading in fire to the design resistance, where the latter is calculated at ambient temperature (or at time t = 0) using the material partial safety factors that apply in fire design rather than in normal strength design [17]:

45

0dfi

dfi0 R

E

..

.=µµµµ ( 1.46 )

A simple conservative version, which can be used for tension members and restrained beams, where lateral-torsional buckling is not a possibility, is [17]

=

1M

fiMfi0 γγγγ

γγγγηηηηµµµµ . ( 1.47 )

in which the reduction factor ηfi may already be conservative.

Figure 1.26 Critical temperature, related to degree of utilization [17]

1.6.6 Load bearing capacity The process of calculating structural behavior is shown in Figure 1.27 and consists of three essential component models: a fire model, a heat transfer model and a structural model. The fire model can be nominal temperature curves, parametric temperature curves or a real fire as discussed in Section 1.2. The heat transfer model has been discussed in Section 1.5. For non-load bearing elements designed to contain fires, the output from a heat transfer model can be used directly to assess whether the time to critical temperature rise on the unexposed face is acceptable. For simple structural elements with a single limiting temperature, the output from a heat transfer model can be used directly to assess whether the critical temperature is exceeded. For more complicated structural elements or assemblies, the output from the heat transfer model is essential input to a structural model for calculating load-bearing capacity. For instance, for materials with low thermal conductivities like concrete, it becomes very important to know the thermal gradients during the fire because these affect the temperature of the reinforcing steel [1].

46

Figure 1.27 Flow chart for calculating the load capacity of a structure exposed to fire [1]

The structural model can be divided into an isolated element, a sub-system and a total structure. The method for determining the load-bearing capacity can be a testing method, a simple calculation method or a complicated computer model. Table 1.10 [8] shows the possible method can be used for various model. In Section 1.7, the simple calculation method for isolated method based on Eurocode 3, Part 1.2 [7] are described.

Table 1.10 Design combination for calculating structural behavior in fire [8]

Structural models Fire exposure models Isolated member Sub-system Total structure

Standard fire Nominal temperature-time curves

Testing and calculations

Parametric fire Equivalent fire exposure Testing and calculations

Testing and calculations

Homogenous temperature distribution

Calculation Calculation For research only Natural fire

Zone, field models Calculation Calculation For research only

Assessment of methods

No interaction between neighboring elements is considered

A reasonable interaction between neighboring elements is considered

All interaction of the global structure system are considered

47

1.7 Design of Steel Members Exposed to Fire This section gives the rules for steelwork that can be unprotected, insulated by fire protection materials or protected by heat screens. Fire resistance is calculated using simple calculation models that are simplified design methods for individual members, which are based on conservative assumptions. This simplified method follows the ultimate strength design method as for normal temperature, except that there are reduced loads for the fire condition and reduced values of modulus of elasticity and yield strength of steel at elevated temperature. The effects of restraint caused by thermal deformation are not included. Structural design at normal temperature requires prevention of collapse (the strength limit state) and preventing excessive deformation (serviceability limit state). Design for fire resistance is mainly concerned to prevention of collapse. Large deformations are expected under severe fire exposure, so they are not normally calculated unless they are going to affect structural performance.

1.7.1 Design methods The load-bearing capacity of a steel member shall be assumed to be maintained after a time t in a given fire if Efi.d.t ≤≤≤≤ Rfi.d.t ( 1.48 )

where Efi.d.t is the design effect of actions for fire design situation and Rfi.d.t is the corresponding design resistance of the steel member for fire design situation at time t. As an alternative, the verification may be carried out in the temperature domain for the member with uniform temperature distribution as shown in Section 1.5.2. Applied loads have been described in Section 1.6.2. The design resistance Rfi.d.t at time t shall be determined in the hypothesis of a uniform temperature in the cross-section, by modifying the design resistance for normal temperature design to take into account of the mechanical properties at elevated temperatures. The design resistance of the steel member may be axial force, bending moment or shear force individually or in the combinations.

1.7.2 Classification of cross-sections In a fire design situation, the classification of cross-section should be made as for normal temperature design without any change [7].

1.7.3 Tension members The design resistance Nfi.θ.Rd of a tension member with a uniform temperature θ should be determined from [7]: Nfi.θθθθ.Rd = A·(fy·ky.θθθθ)/γγγγ M.fi = (A·fy/γγγγ M.1)·ky.θθθθ·(γγγγM.1/γγγγ M.fi) = ky.θθθθ·NRd·(γγγγM.1/γγγγ M.fi) ( 1.49 )

48

where ky.θ is the reduction factor for yield strength of steel ate temperature θ reached at time t as shown in Section 1.3.1.3 and NRd is the design resistance of the cross-section Npl.Rd for normal temperature design. The design resistance Nfi.θ.Rd of a tension member with a non-uniform temperature θ should be determined from [7]: Nfi.θθθθ.Rd = ΣΣΣΣAi·(fy·ky.θθθθi)/γγγγ M.fi ( 1.50 )

where Ai is an element area of the cross-section with a temperature θi; ky.θ.i is the reduction factor for yield strength of steel ate temperature θi; and θi is the temperature in the elemental area Ai. The design resistance Nfi.θ.Rd of a tension member with a non-uniform temperature θ may conservatively be taken as equal to the design resistance of a tension member with uniform maximum temperature.

1.7.4 Moment resistance of beams Moment resistance in fire for Class 1 or 2 sections with uniform cross-sectional temperature θ is calculated from the normal plastic resistance moment of strength design, with the reduction factor ky.θ for yield strength at elevated temperature, and with an adjustment for the relative material safety factors in normal design and fire design [7]: Mfi.θθθθ.Rd = ky.θθθθ·MRd·(γγγγM.1/γγγγM.fi) ( 1.51 )

For a Class 3 section the same expression can be used, but with the elastic moment of resistance used for MRd. In the case of beams supporting concrete slabs on the top flange, the non-uniform temperature distributions may be accounted for analytically in calculating the design moment resistance by dividing the cross-section into uniform-temperature elements, reducing the strength of each according to its temperature, and finding the resistance moment by summation across the section. Alternatively it may be dealt with conservatively using two empirical adaptation factors κ1 and κ2 to define the moment resistance at time t as [7]: Mfi.t.Rd = Mfi.θθθθ.Rd / (κκκκ1·κκκκ2) ( 1.52 )

where κ1 is the factor for non-uniform cross-sectional temperature and κ2 is the factor for temperature reduction towards the supports of a statically indeterminate beam. The values of κ1 and κ2 are specified in Eurocode 3, Part 1.2 as follows [7]:

o for a beam exposed on all four sides: κ1 = 1.0; o for a beam exposed on three sides, with a composite or concrete slab on side four: κ1 = 0.70 o at the supports of a statically indeterminate beam: κ2 = 0.85 o in all other cases: κ2 = 1.0

49

Shear resistance is determined using the same general process as for bending and tension resistance, with the same adaptation factors as those above in cases with non-uniform temperature distribution. The general expression, covering uniform and non-uniform temperature cases, is [7]: Vfi.t.Rd = ky.θθθθ.max·VRd·(γγγγM1/γγγγM.fi)·(1/(κκκκ1·κκκκ2)) ( 1.53 )

1.7.5 Lateral-torsional buckling In cases where the compression flange is not continuously restrained, the design resistance moment against lateral-torsional buckling is calculated for Class 1 or 2 sections using the formula from Eurocode 3 Part 1.1, with minor amendment for the fire state [7]: Mb.fi.t.Rd = (χχχχLT.fi/1.2)·Wpl.y·ky.θθθθ.com·fy/γγγγM.fi ( 1.54 )

where

χLT.fi = reduction factor for lateral-torsional buckling in fire design situation ky.θ.com = reduction factor for yield strength of steel at the maximum temperature of the

compression flange θcom at time t θcom = conservatively can be assumed to be equal to the maximum temperature

The 1.2 is an empirical correction factor for a number of effects. The lateral-torsional buckling reduction factor χLT.fi is determined as in ambient-temperature design, except that the normalized slenderness used is adapted to the high-temperature steel properties [7]:

comE

comyLTcomLT k

k

..

....

θθθθ

θθθθθθθθ λλλλλλλλ = ( 1.55 )

where kE.θ.com = elastic modulus reduction factor at the maximum compression

flange temperature at time t Lateral-torsional buckling resistance should be considered when the non-dimensional slenderness, com..LT θλ , greater than 0.4. For lower slenderness only the consideration of the bending resistance is necessary.

1.7.6 Compression members with Class 1, Class 2 or Class 3 cross-section If there is a temperature gradient over the cross-section, it is not possible to consider accurately the variation of strength segment by segment without a computer program because of the domination of thermal bowing and instability consideration. An approximate design method is used in Eurocode 3, Part 1.2, i.e. it is assumed that the whole cross-section is at the maximum temperature θmax [7]. The design buckling resistance of columns of Class 1, 2 or 3 is calculated as follows, allowing for a reduction in strength and an increase in normalized slenderness at high temperatures. The number

50

1.2 in this formula is an empirical correction factor for a number of effects, i.e. 17% reduction (1/1.2) in strength to allow for other effects [7]: Nb.fi.t.Rd = A·ky.θθθθ.max·fy·(χχχχfi/1.2)·(1/γγγγM.fi) ( 1.56 )

The flexural buckling reduction factor χfi is the lower of its values about the yy and zz axes determined as in ambient-temperature design, except that the normalized slenderness used is adapted to the fire situation as follows:

o Buckling curve c is always used, i.e. α = 0.49 o The buckling length lfi is determined as shown in Figure 1.28 [17], provided that each storey

of the building comprises a separate fire compartment, and that the fire resistance of the compartment boundaries is not less than that of the column. Because the continuing columns are much stiffer than the column in the fire compartment it is assumed that they cause the end(s) of the heated column to be restrained in direction, so the effective length factor is taken as 0,5 for intermediate storeys and 0,7 for the top storey.

o The normalized slenderness of the column for the maximum temperature is given by:

max..

max..max.

θθθθ

θθθθθθθθ λλλλλλλλ

E

y

kk

= ( 1.57 )

o Figure 1.28 Buckling lengths of columns in fire [17]

1.8 Use of Advanced Calculation Models Both Eurocodes 3 and 4 also permit the use of advanced calculation models, which give a realistic analysis of the behavior of the structure in fire [17]. All computational methods are to some extent approximate, are based on different assumptions, and are not all capable of predicting all possible

Bracingsystem

lfi=0,7L

lfi=0,5L

51

types of behavior. It is therefore stipulated that the validity of any such model used in design analysis must be agreed by the client, the designer and the competent building control authority. Computational models may cover the thermal response of the structure to any defined fire, either nominal or parametric, and should not only be based on the established physical principles of heat transfer but should also on known variation of thermal material properties with temperature. The more advanced models may consider non-uniform thermal exposure, and heat transfer to adjacent structure. Since the influence of moisture content in protection materials is inevitably an additional safety feature it is permitted to neglect this in analysis. When modeling the mechanical response of structures, the analysis must be based on acknowledged principles of structural mechanics, given the change of material properties with temperature. Thermally induced strains and their effects due to temperature increase and differentials must be included. Geometric non-linearity is essential when modeling in a domain of very high structural deflections, as is material non-linearity when stress-strain curves are highly curvilinear. It is, however, acknowledged that within the time-scale of accidental fires transient thermal creep does not need to be explicitly included provided that the elevated-temperature stress-strain curves given in the Code are used.

1.9 Global Fire Safety Design The building’s response to fire is highly dependent on the prevailing state of fire: either pre- or post-flashover conditions. In a prescriptive approach, the assessment is based on standard fire conditions and refers to building components only. Traditionally, the only option was to carry out standard fire tests. During last decade, however, various calculation models have been developed. The first generation of these models to avoid significant costs, associated with standard fire testing. In a performance-based method, more recent developments allow to analyze the structural response under natural fire conditions, not only of building components but also of the entire building or major subsystems thereof. Worldwide research and development have contributed to establish the basis for realistic and credible assumptions to be used in the fire situation for thermal actions, active measures and structural response. Hence global fire safety design (see Figure 1.29) consists first in a realistic fire resistance design in order [5]:

o to proceed a global structural fire analysis in the fire situation; o to consider a realistic i.e. accidental combination rule for actions during fire exposure and o to design according to natural fire conditions.

The global fire safety design considers the active fire safety and fire fighting measures in view of their impact on the probable evolution of the natural fire. In this respect the danger of fire activation has to be taken into account. This finally leads to the design of a natural fire resistance of the structure. This design natural fire resistance shall exceed the required fire resistance period that shall depend on both objectives to avoid any human fatalities and to reduce consequences of structural

52

failure. Fire safety should include the safety for occupants and firemen and may take into account the protection of properties and environment.

Figure 1.29 Global fire safety concept [5]

1.10 Design Example according to Eurocode 3

1.10.1 Introduction A simple 4-storey frame as shown in Figure 1.30 is braced against horizontal sway deflection. Identical frames at 6 m spacing. The 60 minutes of fire protection for structural members are required, i.e. tR=60 min. Other given parameters are:

Materials

Steel grade: S275Lightweight concrete C40 (slab)

53

Figure 1.30 A simple framed braced against horizontal sway

Characteristic floor loadings

Permanent: Gk 1.9kN

m2⋅:= Primary variable: Qk.1 3.8

kN

m2⋅:=

Dimensions of the frame

Frame spacing: L 6 m⋅:=Length of beam AB: LAB 5 m⋅:=

Height of the storey: LGH 3.5 m⋅:=

Material properties and partial safety factors

Norminal value for the yield strength fy 275 MPa⋅:=

Modulus of elasticity E 210000MPa⋅:=

Poisson's ratio ν 0.3:=

Density of steel: ρ a 7850kg

m3⋅:=

Partial safety factor --resistance of Class 1, 2 or 3 cross-section: γM0 1.1:=

--resistance of Class 4 cross-section: γM1 1.1:=

--resistance of member buckling: γM1 1.1:=

54

1.10.2 Design loads and load distribution in the frame

1.10.3 Fire resistance and protection of a tension member BE

Load distribution on member BE and member AB

The loads at the supports of the frame can be calculated using the models shown in the following figure. From minor beam AB,

RA12

Gd Qd+( )⋅ LAB⋅:= RA 123.975kN=

RB RA:= RB 123.975kN=

NBE RA RB+:= NBE 247.95kN=

RDGd Qd+( ) 2⋅ LAB⋅ NBE+

2:=

RD 371.925kN=

RF RD:= RF 371.925kN=

Loading on column GH

RGH 2 RD⋅ 2 RB⋅+:= RGH 991.8kN=

Design loads on beams

Partial safety factor for permanent actions under unfavorable effects γG 1.35:=

Partial safety factor for leading variable actions under unfavorable effects γQ.1 1.5:=

Permanent actions: Gd γG Gk⋅ L⋅:=

Variable actions: Qd γQ.1 Qk.1⋅ L⋅:=

Gd 15.39kNm

=

Qd 34.2kNm

=

Ambient temperature design

NSd 247.95kN⋅:=Design loading NSd NBE:=

Partial safety factor for the fire situation shall be taken as:-- for thermal properties of steel: γM.fi 1.0:=

-- for mechanical properties of steel: γM.fi 1.0:=

55

According to equation (1.44) According to equation (1.49), the design resistance is: (from Table 1.1)

Try IPE 100. The dimension of IPE 100 are as follows:

hBE 100 mm⋅:= bBE 55 mm⋅:= tfBE 5.7 mm⋅:= twBE 4.1 mm⋅:= ABE 1032 mm2⋅:= RBE 7 mm⋅:=

According to EC 3 Part 1-1 5.4.3, design plastic resistance of the gross cross-section is defined

as Npl.Rd Afy

γM0⋅

A A BE:=

The design plastic resistance is Npl.Rd Afy

γM0⋅:=

Design "OK!!" Npl.Rd NSd≥if

"NOT OK!!!" otherwise

:= Design "OK!!"=

Fire resistance of tension member

Design loading in fire

γGA 1.0:= ( Permanent loads: accidental design situations ) ψ1.1 0.5:= ( Combination factor: variable loads, office buildings)

The reduction factor

ηfiγGA Gk⋅ ψ1.1 Qk.1⋅+

γG Gk⋅ γQ.1 Qk.1⋅+:= ηfi 459.7701 10 3−×=

The design load in fire is

Nfi.d ηfi NSd⋅:= Nfi.d 114kN=

Design resistance in fire at ambient temperature

NRd Npl.Rd:= NRd 258kN=

At ambient temperature, θ 20:=

ky.20 1.0:=

The design resistance at ambient tempartaure

Nfi.20.Rd ky.20 NRd⋅γM1γM.fi

⋅:= Nfi.20.Rd 283.8kN=

56

The degree of utilization at t = 0, i.e. θ = 20 °C is: According to equation (1.45), the critical temperature is: For an equivalent uniform temperature distribution in a cross-section, the increase of temperature in an unprotected steel member during a time interval ∆t may be determined using equation (1.33). For IPE 100: if In a simple calculation model, the specific heat may be considered as independence of the steel temperature. According to section 1.3.2.1 According to Table 1.3, the convective heat transfer coefficient for standard or external fire curves is: When calculating net radiation heat flux per unit of surface area, the following parameters are given:

Critical temperature

µ0Nfi.d

Nfi.20.Rd:= µ0 401.6913 10 3−×=

θcr.t 39.19 ln1

0.9674µ03.833

⋅1−

⋅ 482+:= θcr.t 619.1392=

Fire resistance time

b bBE:= b 55mm= tf tfBE:= tf 5.7mm= tw twBE:= tw 4.1mm=

h hBE:= h 100mm= R RBE:= R 7mm= A ABE:= A 1.032 103× mm2=

steel_perimeter 2 b 2 t f⋅+ b 2 R⋅− t w−( ) h2

t f− R−

2⋅++ π R⋅+

⋅:= steel_perimeter 399.7823mm=

fire_exposure "all-round":=

The perimeter of the cross section is

Am steel_perimeter b− fire_exposure "3-sided"if

steel_perimeter fire_exposure "all-round"if

:=Am 399.7823mm=

The section factor is V A:=AmV

387.38591m

=

c a 600J

kg K⋅⋅:=

α c 25W

m2 K⋅⋅:=

57

Stephan-Boltzmann constant configuration factor Φ = 1.0 emissivity of member surface εm = 0.625 emissivity of fire compartment εf = 0.8 Using spreadsheet calculation, the time for steel to reach its critical temperature is 9 minutes 40 seconds. For a uniform temperature distribution in a cross-section, the temperature increase of an insulated steel member during a time interval ∆t may be calculated using equation (1.35) and equation (1.36). Based on Table 1.7, when protection with gypsum boards: With thickness of dp = 20 mm, after 60 minutes, the temperature of steel is 613.995 °C and is less than the critical temperature 619.139 °C. Thus, 20 mm gypsum board provides more than 60 minutes fire protection.

σ 5.67 10 8−⋅W

m2 K4⋅⋅:=

Fire protection of tension member

fire_exposure "all-round":= protection_type "box":=

steel_perimeter 2 b 2 tf⋅+ b 2 R⋅− tw−( ) h2

tf− R−

2⋅++ π R⋅+

⋅:= steel_perimeter 399.7823mm=

The section factor can be calculated as:

Ap_gypsteel_perimeter b− protection_type "contour"if

2 h⋅ b+ protection_type "box"if

fire_exposure "3-sided"if

steel_perimeter protection_type "contour"if

2 h⋅ 2 b⋅+ protection_type "box"if

fire_exposure "all-round"if

:=

Ap Ap_gyp:=

Appropriate area of fire protection material per unit length is Ap 310mm=

Section factor: V A:=ApV

300.38761m

=

Try the following gypsum boarding:

Density ρ p 800kg

m3⋅:= ,

Specific heat cp 1700J

kg K⋅⋅:= ,

Thermal conductivity λp 0.2W

m K⋅⋅:=

58

1.10.4 Fire resistance and protection of steel beam AB

Ambient temperature design

Applying bending moment: MSd18

Gd Qd+( )⋅ LAB2⋅:=

Try IPE 300. The dimensions of the cross-section of IPE 300 are as follows:

hAB 300 mm⋅:= bAB 150 mm⋅:= tfAB 10.7 mm⋅:= twAB 7.1 mm⋅:=

RAB 15 mm⋅:= AAB 5381 mm2⋅:= WAB 557 103⋅ mm3⋅:= Wpl.AB 628 103⋅ mm3⋅:=

Section classification

According to EC3 Part 1.1, Table 5.3.1, for rolled section, when web is subjected to bending:

sectType_web "Class 1"dtw

72 ε⋅≤if

"Class 2" 72 ε⋅dtw

< 83 ε⋅≤if

"Class 3" 83 ε⋅dtw

< 124 ε⋅≤if

"Class 4" otherwise

where ε235 MPa⋅

fy:= , d h 2 tf R+( )⋅−

when flange is subjected to compression:

sectType_flange "Class 1"ctf

10 ε⋅≤if

"Class 2" 10 ε⋅ctf

< 11 ε⋅≤if

"Class 3" 11 ε⋅ctf

< 15 ε⋅≤if

"Class 4" otherwise

where cb2

59

Wpl.yy Wpl.AB:=Wyy 557 103× mm3=Wyy WAB:=A 5.381 103× mm2=A AAB:=

R 15mm=R RAB:=tw 7.1mm=tw twAB:=tf 10.7mm=tf tfAB:=

b 150mm=b bAB:=h 300mm=h hAB:=ε 924.4163 10 3−×=For beam AB

Therefore, the Class of the cross-section is

sectType_beam sectType_web sectType_web sectType_flange≥if

sectType_flange otherwise

:=

sectType_beam "Class 1"=

The Class of the web d h 2 tf R+( )⋅−:= d 248.6mm=

sectType_web "Class 1"dtw

72 ε⋅≤if

"Class 2" 72 ε⋅dtw

< 83 ε⋅≤if

"Class 3" 83 ε⋅dtw

< 124 ε⋅≤if

"Class 4" otherwise

:=

sectType_web "Class 1"=

The Class of the flange is cb2

:= c 75mm=

sectType_flange "Class 1"ctf

10 ε⋅≤if

"Class 2" 10 ε⋅ctf

< 11 ε⋅≤if

"Class 3" 11 ε⋅ctf

< 15 ε⋅≤if

"Class 4" otherwise

:=

sectType_flange "Class 1"=

Moment resistance

The concrete floor slab provides full lateral resistant to the top flange, hence no need to consider lateral-torsional buckling.

According to EC3, Part 1.1, 5.4.5.2, bending about one axis, the design moment resistance of a cross-section without holes for fasteners may be determined as:

Mc.RdWpl.yy fy⋅

γM0sectType_beam "Class 1"( ) sectType_beam "Class 2"( )∨if

Wyy fy⋅

γM0sectType_beam "Class 3"( )if

60

Given key1 sectType_beam "Class 1":=

key2 sectType_beam "Class 2":=

key3 sectType_beam "Class 3":=

Therefore

Mc.RdWpl.yy fy⋅

γM0key1 1 key2 1∨if

Wyy fy⋅

γM0key3 1if

:= Mc.Rd 157kN m⋅=

moment_Resistance "OK!!" MSd Mc.Rd≤if

"NOT OK!!!" otherwise

:= moment_Resistance "OK!!"=

Shear resistance

According to EC3, Part 1.1, 5.4.6, the design value of shear force VSd shall satisfy:

VSd Vpl.Rd≤

where Vpl.Rd is the design plastic shear resistance given by:

Vpl.Rd

Avfy

3

⋅

γM0

where Av is the shear area and may be taken as follows:

Av A 2 b⋅ tf⋅− tw 2 R⋅+( ) tf⋅+:=

Applied shear: VSd12

Gd Qd+( )⋅ LAB⋅:= VSd 123.975kN=

The shear area: Av A 2 b⋅ tf⋅− tw 2 R⋅+( ) tf⋅+:= Av 2.568 103× mm2=

Vpl.Rd 370.6545kN=The design plastic shear resistance: Vpl.Rd

Avfy

3

⋅

γM0:=

shear_Resistance "OK!!" VSd Vpl.Rd≤if

"NOT OK!!!" otherwise

:= shear_Resistance "OK!!"=

61

Design resistance in fire The design resistance of Class 1 or Class 2 cross-section with a non-uniform temperature distribution may conservatively be determined using equation (1.51). The design moment resistance at ambient temperature of Class 1 or Class 2 cross-section with a uniform temperature may be determined from:

Fire resistance of minor beamDesign loading in fire

The design loading in fire: Mfi.d ηfi MSd⋅:= Mfi.d 71.25kN m⋅=

Critical temperature

ultilizaiton factor µ0Mfi.d

Mfi.0.Rd:=

The critical temperature is

θcr.t 39.19 ln1

0.9674µ03.833

⋅1−

⋅ 482+:= θcr.t 669.5458=

Fire resistance time

steel_perimeter 2 b 2 tf⋅+ b 2 R⋅− tw−( ) h2

tf− R−

2⋅++ π R⋅+

⋅:= steel_perimeter 1.16 103× mm=

fire_exposure "3-sided":=

The perimeter of the cross section is

Am steel_perimeter b− fire_exposure "3-sided"if

steel_perimeter fire_exposure "all-round"if

:=Am 1.01 103× mm=

The section factor is V A:=AmV

187.706=

Mfi.20.Rd ky.20 Mc.Rd⋅γM1γM.fi

⋅:= Mfi.20.Rd 172.7kN m⋅=

For beam supporting concrete slab: κ1 0.7:= κ2 1.0:=

Therefore, the fire resistance at ambinent temperature

Mfi.0.RdMfi.20.Rd

κ1 κ2⋅:= Mfi.0.Rd 246.7143kN m⋅=

62

Using spreadsheet calculation, the time for steel to reach its critical temperature is 15 minutes 23 seconds. With thickness of dp = 15 mm, after 60 minutes, the temperature of steel is 570.545 °C and is less than the critical temperature 669.546 °C. Thus, 15 mm gypsum board provides more than 60 minutes fire protection.

1.11 References 1. Buchanan, A. H. (2001). Structural Design for Safety. John Wiley & Sons. 2. Drysdale, D. (1999). An Introduction to Fire Dynamics. John Wiley & Sons. 3. EN 1991-1-2 (2002). Eurocode 1: Actions and Structures, Part 1-2: General Actions-Actions on

Structures Exposed to Fire. 4. ECCS (1983). European Recommendations for the Fire Safety of Steel Structures: Calculation of

the Fire Resistance of Loading Bearing Elements and Structural Assemblies Exposed to the Standard Fire. Elsevier.

5. ECCS (2001). Model Code on Fire Engineering, First Edition. 6. ENV-1991-1-1 (1994). Eurocode 1: Basis of Design and Actions on Structures, Part 1: Basis of

design.

Fire protection of steel beam

When protection with gypsum boards:

fire_exposure "3-sided":= protection_type "box":=

steel_perimeter 2 b 2 tf⋅+ b 2 R⋅− tw−( ) h2

tf− R−

2⋅++ π R⋅+

⋅:= steel_perimeter 1.16 103× mm=

The section factor can be calculated as:

Ap_gypsteel_perimeter b− protection_type "contour"if

2 h⋅ b+ protection_type "box"if

fire_exposure "3-sided"if

steel_perimeter protection_type "contour"if

2 h⋅ 2 b⋅+ protection_type "box"if

fire_exposure "all-round"if

:=

Ap Ap_gyp:=

Section factor: V A:=ApV

139.37931m

=

63

7. ENV-1993-1-2 (1995). Eurocode 3: Design of Steel Structures, Part 1-2: General Rules-Structural Fire Design.

8. ESDEP Working Group 4B, Protection: Fire. Teräsrakenneyhdistys 9. Kaitila, O. (2002). Finite Element Modeling of Cold-Formed Steel Members at High

Temperatures. TKK-TER-24, Laboratory of Steel Structures, Helsinki University of Finland, Espoo, Finland.

10. Korhonen, E. (1999). Natural Fire Modeling of Large Space. Master’s Thesis. 11. Ma, Z. and Mäkeläinen, P. (1999). Temperature Analysis of Steel-Concrete Composite Slim

Floor Structures Exposed to Fire. TKK-TER-10, Laboratory of Steel Structures, Helsinki University of Technology, Espoo, Finland.

12. Ma, Z. (2000). Fire Safety Design of Composite Slim Floor Structures. TKK-TER-18, Helsinki

University of Technology. 13. Newman, G. M., Robinson, J. T. and Bailey, C. G. (2000). Fire Safety Design: A New Approach

to Multi-Storey Steel-Framed Buildings. SCI publication P288. 14. Outinen, J. and Mäkeläinen, P. (1997). Mechanical Properties of Austenitic Stainless Steel

Polarit 725 (EN 1.4301) at Elevated Temperatures. TKK-TER-1, Laboratory of Steel Structures, Helsinki University of Technology, Espoo, Finland.

15. Outinen, J., Kaitila, O. and Mäkeläinen, P. (2001). High-Temperature Testing of Structural Steel

and Modeling of Structures at Fire Temperatures. TKK-TER-24, Laboratory of Steel Structures, Helsinki University of Technology, Espoo, Finland.

16. Schaumann, P. (2002). Fire Design of Steel and Composite Structures. Seminar materials,

Laboratory of Steel Structures, Helsinki University of Technology. 17. Structural Steelworks Eurocodes, Development of a Trans-National Approach. 18. Twilt, L., Van De Leur, P., Cajot, L.G., Schleich, J.B., Joyeux, D. and Kruppa, J. (1996). Input

Data for the Natural Fire Design of Building Structures. IABSE Report: Basis of Design and Actions on Structures, Background and Application of Eurocode 1.

2 FATIGUE DESIGN

2.1 Introduction Fatigue is a terminology to describe the damage and failure of materials under cyclic loads in engineering applications. Fatigue failures generally take place at a stress much lower than the ultimate strength of the material. The failure is due primarily to repeated stress from a maximum to a minimum. Fatigue failure may occur in many different forms such as mechanical fatigue when the components are under only fluctuating stress or strain; creep-fatigue when the components under cyclic loading at high temperature; thermo mechanical fatigue when both mechanical loading and temperature are cyclic; corrosion fatigue when the components under cyclic loading impose in the presence of a chemically aggressive environment. Fatigue is the mechanism that cracks grow in a structure under fluctuating stress. The progression of fatigue damage can be classified into the following stages [15]:

o Substructural and microstructural changes which cause nucleation of permanent damage; o The creation of microscopic cracks; o The growth and coalescence of microscopic flaws to form dominant cracks, which may

eventually lead to catastrophic failure (This stage of fatigue is the mark between crack initiation and propagation);

o Stable propagation of the dominant macrocrack; o Structural instability or complete fracture.

Final failure generally occurs in regions of tensile stress when the reduced cross-section becomes insufficient to carry the peak load without rupture. Fatigue damage of structures subjected to elastic stress fluctuations occurs at regions where the localized stress exceeds the yield stress of material. After a certain number of load fluctuations, the accumulated damage causes the initiation and subsequent propagation of a crack or cracks, in the plastically damaged regions. This process can cause the fracture of components. Many structures, such as building frames, do not experience sufficient fluctuating stress to give rise to fatigue problems. Others do, such as bridges, cranes, and offshore structures.

2.1.1 Different approaches for fatigue analysis The total fatigue life, Nt, is defined as the sum of the number of cycles to initiate a fatigue crack, Ni, and the number of cycles to propagate a fatigue crack to a critical size, Np, i.e. Nt = Ni + Np ( 2.1 )

65

No simple or clear boundary between fatigue crack initiation and propagation. Furthermore, a pre-existing crack in a structural component can reduce or eliminate the fatigue crack initiation life and, thus, reduce the total fatigue life. According to the definition of the fatigue life, three approaches for fatigue analysis can be classified: stress method, strain method and crack propagation method. Stress method and strain method characterize the total fatigue life in terms of cyclic stress range or strain range. In these methods, the number of stress or strain cycles to induce fatigue failure in initially uncracked or smooth surfaced laboratory specimen is estimated under controlled cyclic stress or strain. The resulting fatigue life includes the fatigue crack initiation life to initiate a dominant crack and a propagation of this crack until catastrophic failure. Normally, the fatigue initiation life is about 90% of total life due to the smooth surface of the specimen [15]. Under high cycle (> 102 to 104), low stress fatigue situation, the material deforms primarily elastically and the failure time has traditionally been described in terms of stress range. However, stresses associated with low cycle fatigue (< 102 to 104) are generally high enough to cause plastic deformation prior to failure. Under these circumstances, the fatigue life is described in terms of strain range. The low cycle approach to fatigue design has found particularly widespread use in ground vehicle industries [15]. The basic premise of crack propagation method is that all engineering components are inherently flawed. The size of a pre-existing flaw is generally determined from nondestructive flaw detection techniques, such as visual, dye-penetrant or X-ray techniques, or the ultrasonic, magnetic or acoustic emission methods. The fatigue life is then defined as the number of cycles to propagate the initial crack size to a critical size. The choice of critical size of cracks may be based on the fracture toughness of the material, the limit load for the particular structural part, the allowable strain or the permissible change in the compliance of the component. The prediction of fatigue life is mainly base on linear elastic fracture mechanics. The crack propagation method, which is a conservative approach to fatigue, has been widely used in fatigue critical applications where catastrophic failures will results in the loss of human lives such as aerospace and nuclear industries [15].

2.1.2 A short history to fatigue The history of fatigue design goes back to the middle of the nineteenth century, marked by the beginning of industrial revolution and, in particular, the advent of railroads in central Europe. The first known investigators concerned with fatigue phenomena were designers of axles for locomotives. Wöhler’s experiments with axles in 1852 were the first known laboratory tests with the objective to derive and quantitatively describe the limits of fatigue. This was followed by more elaborate analyses of stresses and their effect on fatigue by Berber, Goodman and others. Continuous efforts of researches in the twentieth century have given a new impetus to the development of theories, such as the effects of plastic deformation on fatigue-resulting in the strain method discovered by Manson and Coffin. In parallel, Pairs and Others continued the theory of crack propagation started by Griffith. Research accomplishment of Morrow Socie and their followers brought the state of fatigue analysis to the present day level [17]. Fatigue was incorporated into design criteria near the end of the nineteenth century and has been studied since. However, the most significant developments have occurred since the 1950s. At present, fatigue is part of design specification for many engineering structures [1].

66

2.2 Fatigue Loading Structural components are subjected to two kinds of load history in fatigue design. The simplest one is the constant-amplitude cyclic loading fluctuation. Figure 2.1 illustrates a constant-amplitude cyclic stress fluctuation and this kind of loading normally occurs in machinery parts such as shafts and rods during periods of steady-state rotation.

Figure 2.1 Terminology used in constant-amplitude loading

Constant-amplitude loading can be described using the following parameters (see Figure 2.1):

o stress range, ∆σ, which is the algebraic difference between the maximum stress, σmax, and the minimum stress, σmin, in the cycle, i.e.

∆∆∆∆σσσσ = σσσσmax - σσσσmin ( 2.2 )

o mean stress, which is the algebraic mean of σmax and σmin in the cycle, i.e. σσσσm = (σσσσmax + σσσσmin) / 2 ( 2.3 )

o stress amplitude, which is half the stress range in a cycle, i.e. σσσσa = (σσσσmax - σσσσmin) / 2 ( 2.4 )

o stress ratio, which represents the relative magnitude of the minimum and maximum stress in a cycle, i.e.

R = σσσσmin / σσσσmax ( 2.5 )

The values of R corresponding to various loading case are shown in Figure 2.2. The complete reversal load is changing from a minimum compressive stress to an equal maximum tensile load (R = -1). The stress fluctuation from a given minimum tensile load to a maximum tensile load is characterized by a positive value between 0 and 1 (0 < R < 1). Comparing to the constant-amplitude loadings, the variable-amplitude loadings are more complex. In variable-amplitude loading history, the probability of the same sequence and magnitude of stress ranges recurring during a particular time interval is very small and cannot be represented by an analytical function (see Figure 2.3). This type of loading is experienced by many structures, such as wind loading on aircraft, wave loading on ships and offshore platforms, and truck loading on bridges.

67

Figure 2.2 Comparison of R-rations for various loadings [ 1]

Figure 2.3 Variable-amplitude loading history

Either constant-amplitude loadings or variable-amplitude loadings can cause unidirectional stresses in the structural components, such as pure axial tension and compression, pure bending, or pure torsion. For the components with complicated geometries, these loadings may cause the stresses acting on the components simultaneously in different directions. In this course, only the unidirectional cases are discussed. Further reading about multi-axial loading can be found in Socie and Marquis [14].

2.3 Stress Methods In the case of static loading, the yield strength or ultimate strength of material is obtained from the tensile testing. The structural components are designed according to these values. Likely, under the fluctuating stress, the significant strength is fatigue strength or fatigue limit. The fatigue strength is defined as the intensity of the reverse stress causing the failure after a given number of cycles. The fatigue limit (or endurance limit) is defined as the maximum value of fully reverse stress that can be repeated an infinite number of times on a test specimen without causing a failure.

68

In stress methods, it is necessary to determine fatigue strength and/or fatigue limit (analogous to yield strength) for the material so that cyclic stresses can be kept below that level avoiding fatigue failure for the required number of cycles. The structural components are designed that the maximum stress never exceeds the materials fatigue strength or fatigue limit. The stresses and strains remain in the elastic region such that no local yielding occurs to initiate a crack.

2.3.1 Standard fatigue tests Types of testing The history of standard fatigue tests goes back to Wöhler who designed and built the first rotating-beam test machine that produced fluctuating stresses of constant amplitude in test specimens. R.R Moore later adopted this technique to a simply supported rotating beam in fully reversed, pure bending. The scheme of Moore rotating beam fatigue test machine is shown in Figure 2.4 (a). While the specimen rotates, the two bearings near each end of the test specimen permit the load to be applied and two bearings outside of these provide support. The constant force is provided by the hanging weight. This testing has the advantage of providing a constant bending moment and a zero shear over the length of the specimen. When rotated one-half revolution, the stress below the neutral axis are reversed from tension to compression and vice versa. Upon completing the revolution, the stresses are again reversed. The counter registers the number of revolution. The testing stops when the specimen breaks. Another testing type is called axial loading testing. Figure 2.4 (b) shows a test machine operating by hydraulic forces and controlled by electrical signals. The machine loads the specimen through hydraulic actuator. The direction of the axial force is changed when the flow of the oil is reversed. The servo-valve is in charge of reversing the flow direction of oil. This testing system allows the combination of a cyclic and a steady load applied at the same time.

Figure 2.4 Fatigue testing machines [17]

69

Besides aforementioned two testing types, there are some other fatigue testing. Similar to the rotating beam fatigue testing, the reversed bending test is that one end of specimen is fixed and the other end is pushed alternative up and down. These differ from the stresses caused by rotating bending in that the maximum stress are limited to the top and bottom instead of producing the maximum stress all round the circumference. Torsional fatigue tests are performed on a cylindrical specimen subjected to fully reversed, torsional loading. Test specimens Two types of specimen are used in the fatigue test. The simplest test specimen is called unnotched or smooth specimen. No stress raiser in the region where failure occurs. Another test specimen is called the notched specimen and contains stress raisers in the section where failure is expected to occur. Figure 2.5 shows some examples of flat specimens for fatigue tests.

Figure 2.5 Test specimens for standard fatigue tests (schematic)

2.3.2 S-N curves The most common way to describe the fatigue testing data is using S-N curves that show the relationship between the number of cycles, N, for fracture, and the maximum (or mean or amplitude or range) value of the applied cyclic stress. Generally, the abscissa is the logarithm of N and the ordinate may be the stress or the logarithm of stress. In stress method, the stress is designated using S, while in other method the stress is expressed as σ. The reason for this discrepancy is due to a tradition of the stress method throughout its long history [17].

2.3.2.1 Linear part of S-N curves A typical standard S-N curve is shown in Figure 2.6 (a). Most of the fatigue tests are performed in the high-cycle fatigue domain, where a linear relationship between stress range and fatigue life exists in log-log diagram. This linear relationship can be expressed as: N = NR·( σσσσ / σσσσR ) -m ( 2.6 )

70

based on the stress level or N = NR·( ∆∆∆∆σσσσ / ∆∆∆∆σσσσR ) -m ( 2.7 )

based on the stress range. In the formula, σ ( or ∆σ) is the fatigue strength at loading cycle, N; σR ( or ∆σR) is the characteristic value at loading cycle NR = 2 x 106 and m is the slope exponent. There are two strategies to establish the S-N curves in the high-cycle regime [8]:

o Testing at different load levels, so that the slope exponent and the characteristic value of the S-N curve can be determined (Figure 2.6 (b));

o Testing at one load level and assuming a fixed slope exponent of the S-N curve that is almost the same for similar types of structures.

Figure 2.6 P-S-N curves and scatter of test results at two stress levels

In order to obtain the meaningful engineering data, a large amount of testing should be carried out. However, even though the same specimen is used in the fatigue tests, the results show a wide range of dispersion. This is due to the different geometrical micro irregularities of surfaces for the same type of specimen. These different local concentrations cause different fatigue life. Therefore, it is necessary to carry out the statistical analysis of fatigue data. This in turn brings the necessity to consider the effect of failure probability. The curves formed by integrating the failure probability into S-N curve are called P-S-N curves. The standard S-N curve corresponds to a 50 percent of probability of failure (P = 0.5). The S-N curves corresponding to other failure probabilities are shown in Figure 2.6 (a).

2.3.2.2 Fatigue limit The main consideration of fatigue analysis is to design the structural components for an infinite life or for a limited life. The stress values corresponding to these two lives are fatigue limit that forms a mechanical property specific for each material, and fatigue strength at given number of cycles (normally located in the sloping part) as shown in Figure 2.7. The objective for the infinite life design is to ensure the working stress due to loading is under the fatigue limit. While the objective for the limit life design is to predict number of cycles available within the fatigue life based on the stress level, or conversely to determine the stresses based on a given number of cycles [17].

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Figure 2.7 S-N curves (schematic): with and without fatigue limit [7]

Figure 2.7 shows that for ferrous alloys, there is a clearly defined value for fatigue limit, under which failure does not occur. The “knee” point of fatigue limit is normally in the range of 105 to 107 cycles [7]. Many high strength steels, aluminum alloys and other materials do not generally exhibit a “knee” point of fatigue limit. For these materials, the fatigue limit is defined at the stress level corresponding to 107 cycles [15]. To determine a fatigue limit experimentally, the test results are evaluated statistically using either the data of specimens that survived (run-outs) or of those that failed. A common procedure is staircase method, which can be described as follows (see Figure 2.8):

o Estimate mean value, ∆σm, and standard deviation, d, of the fatigue limit based on the preliminary knowledge;

o Perform the first fatigue test at the stress level ∆σm+d; o If the specimen fails, decrease the stress level by d. If the specimen survives (run-out),

increase the stress level by d; o Continue until 15 to 30 specimens have been tested; o A statistical evaluation of all tests yields the mean value of ∆σD,50 and standard deviation of

the fatigue limit.

Figure 2.8 Determination of the fatigue limit with staircase method [8]

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However, in a preliminary design work, it is necessary to approximate the S-N curve without actually running a fatigue test. For steel it has been found that a good approximation of the S-N curve can be drawn using the following rules [11]:

o Obtain the ultimate tensile strength, σu, of the specimen from a simple tensile test; o Plot the S-N curves with the following point: (a) 0.9·σu at 103 cycles, (b) 0.5·σu at 106 cycles.

2.3.2.3 Factors affecting S-N curves Many factors have influences on S-N curves. Generally any change on the static mechanical properties or microstructure is likely to affect the S-N curve. Other factors to be considered are chemical environment, cyclic frequency, temperature, residual stresses and surface effects. Factors affecting fatigue limit The specimens in the aforementioned testing are free of stress concentrations and residual stresses. In order to use the fatigue limit of standard specimen in the rotating bending test in the design of the real structural components, the standard fatigue limit should be multiplied by the following factors: loading mode factor, size effect factor, surface roughness factor and reliability factor [17]. Figure 2.9 indicates the effects of the loading mode on the value of fatigue limit. Figure 2.9 (a) shows that S-N curves obtained from axial loading test are lower than those from rotating bending test. A principal difference between axial and the rotate bending test is that the entire section is uniformly stressed in axial loading rather than linear stress distribution, i.e. maximum at far end and zero at center as in rotating bending testing (see Figure 2.9 (b)). When the same specimen subjected to torsional loading, the equivalent stress can be calculated using von Mises criterion, i.e. σσσσeq = ( 0 + 3ττττ2 )0.5 ( 2.8 )

Then the fatigue limit for torsion, τf, can be calculated assuming that σeq is equal to the standard fatigue limit, σf

′ ττττ f = ( 1 / √√√√3 )· σσσσf

′′′′ ( 2.9 )

Figure 2.9 Effects of loading modes

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The surface roughness or the local irregularities are the high stress concentration points where a fatigue failure generally originates. The fatigue limit for a polished specimen has a higher value than those with rough surfaces. There is an inverse relationship between the fatigue limit and the magnitude of the irregularity [17]. The size of the specimen must be considered when determining the fatigue limit. Experiments show that from rotating beam tests and from torsion tests, the values of the fatigue limit change inversely to diameters of specimens, while from axial loading tests the size has no influences on the fatigue limit [17]. The values of fatigue limit from standard rotating bending tests are based on 50 percent probability of reliability. Thus, a reliability factor must be multiplied to the standard fatigue limit so as to consider the probability of the fatigue test data. In addition, for steel, there is an empirical relation between the rotating beam fatigue limit and tensile strength, i.e. the fatigue limit from standard test is about a half of the tensile strength. Therefore, the fatigue limit for steel can be defined as: σσσσ f = kl ·ks · kd · kr · σσσσf

′′′′ = kl ·ks · kd · kr · ( σσσσu / 2 ) ( 2.10 )

where kl is the loading factor; ks is the surface roughness factor; kd is the size factor; kr is the reliability factor; and σu is the tensile strength of steel. The values of aforementioned factors are empirical factors based on testing. Stress concentration caused by notches and holes Stress raisers such as notches, holes or sharp corners can cause large rise in stress above the nominal stress. Under static loading and beyond elastic limit of ductile material, plastic deformation can cause stress redistribution, i.e. the high peak stress caused by the stress raisers is redistributed to an almost uniform stress across the cross-section. However, the stress raiser will reduce the fatigue life of the structural component. Figure 2.10 illustrates the fatigue limit of notched specimen comparing to un-notched specimen. The stress increase related to the normal stress is described by the stress concentration factor, Kt, i.e. Kt = σσσσmax / σσσσn ( 2.11 )

where σmax is the maximum stress at notches that can be determined using either experimental stress analysis or numerical methods such as Finite Element Analysis (FEA). σn is the nominal stress that can be calculated, for instance, for tension member shown in Figure 2.11, as N / A, in which N is the tension loading and A is the cross-section area without notch. The value of Kt can be checked from manuals. Figure 2.12 provides some examples of values of Kt. Stress concentration factor Kt aforementioned is based on elasticity theory. A discrepancy found between the theoretical and experimental data demands using a fatigue notch factor instead of this stress concentration factor. The fatigue notch factor is defined as: Kf = fatigue limit of smooth specimen / fatigue limit of notched specimen ( 2.12 )

A notch sensitivity factor, which relates the fatigue notch factor and the stress concentration factor, is defined as the ratio of effective stress increase in fatigue due to the notch to the theoretical stress increase given by the elastic stress concentration factor. The notch sensitivity factor can be expressed as:

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Figure 2.10 Illustration of S-N curves for notched and un-notched fatigue tests

Figure 2.11 Definition of stress-concentration

factor

( a ) SCFs for cut-outs in infinite, uni-axially stressed plates

(b) SCF for a rounded transition between two shaft diameters

Figure 2.12 Examples of values of stress concentration factors [8]

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q = (Kf – 1) / (Kt -1) ( 2.13 )

In addition, two relations have been developed to relate notch root radius and material behavior to the notch sensitivity factor, q. One is based on Peterson, i.e. q = 1 / ( 1 + √√√√(a / r) ) ( 2.14 )

in which r is the notch radius and a is the material property constant. The other is based on Neuber, i.e. Kf = 1 + (Kt – 1) / (1 + √√√√(ρρρρ / r )) ( 2.15 )

in which ρ is material constant related to grain size. Therefore, the notch sensitivity factor can be expressed as: q = 1 / ( 1 + √√√√(ρρρρ / r) ) ( 2.16 )

Generally, Kf << Kt for ductile materials and sharp notches but Kf ≅ Kt for brittle materials. The fatigue limit of the notched specimen can be related to that of un-notched specimen by: σσσσf-notched = σσσσf-unnotched / Kf ( 2.17 )

Mean stresses As mentioned above, the S-N curves are generated with fully reverse load (R = -1) and zero mean stresses. However, non-zero mean stresses can also play an important role in resulting fatigue data. The effects of mean stresses on the fatigue limit corresponding to the infinite fatigue life are illustrated in the limit stress diagram that is severed as a practical design tool as shown in Figure 2.13. The abscissa in the limit stress diagram is the mean stress of applied loading and the ordinate is the allowed stress amplitude. The line in the diagram is fatigue limit corresponding to the infinite life. These two lines are based on Goodman rule and Söderberg rule, whose mathematical expressions are given in the figure. Note that when mean stress is σm = 0, the allowed stress amplitude, σa, is the fatigue limit measured from fully reversed loading. If σa = 0, the allowable mean stress in either yield or ultimate strength from a monotonic test since the stress is not fluctuating when the stress amplitude is zero.

Figure 2.13 Effects of mean stress on allowable stress amplitude [Mek]

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2.3.3 One dimensional analysis for fatigue assessment Two approaches are described in this section to perform the fatigue assessment for the structural components: nominal stress method and notch stress method.

2.3.3.1 Nominal stress method The nominal stress can be determined from the applied loading such as forces and moments, and the cross-sectional properties of a component or structure in accordance with the basic theory of strength of materials. For instance, the nominal stress for beam-like components is composed of the normal stress σn in the longitudinal direction and the mean shear stress,τn, in the web, which can be calculated as: σσσσn = N/A + (M/I)·z, ττττn = Q/As ( 2.18 )

where, N, Q and M are axial force, shear force and bending moment, respectively; A, As and I are cross-sectional area, effective shear area and moment of inertia; z is the distance from the neutral axis as shown in Figure 2.14.

Figure 2.14 Nominal stress in a beam-like component [8]

Any stress increase resulting from discontinuities is considered by S-N curve, i.e. an S-N curve is generally valid only for a specific geometric in addition to material type, surface and manufacturing condition [8]. It should be kept in mind that the results cannot be transferred to other geometries or component sizes.

2.3.3.2 Notch stress method In the notch stress method, the local maximum stress due to stress risers can simply be calculated using the notch factor and nominal stress, i.e. σσσσmax = Kf·σσσσn ( 2.19 )

In addition to methods mentioned above, the notch factor might be determined using other methods, for instance, Siebel and Stieler, Sonsino, and Taylor and Wang [8]. The advantage of notch stress method is that the local geometric is taken into account and the disadvantage of this approach is that it can only be used if the stress concentration factor is known. Besides, as mentioned before, the local maximum stress can be directly calculated using method such as FEM.

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The examples of using nominal stress method and notch stress method for performing fatigue analyses are provided in Section 2.7 Fatigue analysis of welded components.

2.4 Strain Methods From a design point of view, the easy answer to fatigue is to use low stress so as to keep both static and cyclic analysis in elastic range. However, the stress raisers such as notches create stress concentrations and elevate the stress into plastic range. The solution to this phenomenon is strain method. A strain method is used to predict the fatigue life of the structural component based on the fluctuating strain. This method is also known as low cycle fatigue where the cyclic stresses are high enough to cause yielding, thus, leading to the life span ranging from 1 to 10,000 cycles. Since the crack initiation involves local yielding and therefore strain life method gives a reasonable estimation about the crack-initiation stage.

2.4.1 Cyclic material law The material law may differ from static loading and cyclic loading. The stress-strain curve for high carbon steel under monotonic (static) stress is shown in Figure 2.15. The engineering stress-strain curve is drawn with the stress calculated using initial cross-section. This is the stress-strain curve that we have used as the material law from testing. However, when the specimen is under tension or compression, the cross-section is changing. The stress-strain curve that obtained with stress calculated from real cross-section is called true stress-strain curve. The cyclic stress-strain curve is using true stress-strain definition. Assuming a metal has hypothetical properties that stay constant under load cycling. The load history begins from point O. At first the deformation is elastic and is represented by straight line OA. Beyond point A, the deformation is plastic represented by line AB. Beyond point B, it is unloading. Since it is assumed no changes in metal properties, the subsequent reverse loading has an equal but symmetrically opposite pattern. From O′ to A′ the deformation again elastic and from A′ to B′ it is plastic. Unloading from B′ brings us back to point O. This procedure is shown in Figure 2.16 (a). However, in real life, due to the Bauschinger’s effect, the yield point A′ is less than A. The cycling process produces strain hardening thus changing the position of B, B′, B′′ and B′′′. The hysteresis loop is drifted (see Figure 2.16 (b)). With the cyclic hardening prevails, the Bauschinger’s effect diminishes. After a few cycles, the material exhibits a stabilized behavior, i.e. the hysteresis loop is stabilized (see Figure 2.17). The phenomena aforementioned can be observed using either stress-controlled tests or strain-controlled tests. The stress-controlled tests are carried out with prescribed stress fluctuation and the strain-controlled tests with prescribed strain fluctuation. Because of the fact that the stress control at large load is cumbersome, the strain control tests are more convenient even though both tests give similar results. In both tests, the material may show either cyclic hardening or cyclic softening (see Figure 2.17). From the testing data, it is shown that low-strength steel tends to soften in the range of smaller stress cycles and to harden for greater stress cycles, high-tensile strength steel exhibits softening in every aspect [8].

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Figure 2.15 Stress strain curves under static loading [17]

Figure 2.16 Cyclic stress-strain curves [17]

Figure 2.17 Cyclic hardening and cyclic softening

The purpose of material tests is to produce the stabilized hysteresis loop as shown in Figure 2.18 (a). However, if the hysteresis loop is changing over the whole life until crack initiation, the hysteresis loop at the half number of cycles should be taken [8]. The stress-strain relationship in the stable state can be obtained using two methods: a multi-specimen testing program and a multi-step testing program. In the multi-specimen program, a number of specimens are tested, each one at a different strain amplitude (Figure 2.19 (a)) until a corresponding stable hysteresis loop is obtained. A series of resulted hysteresis loop is plotted in a common σ-ε diagram (Figure 2.18 (b)). Connecting the hysteresis tips, a stress-strain curve is obtained, which represents a relation of cyclic stress and strain. A more economic way to obtain the cyclic stress-strain curves is using multi-step testing program, in which the periodical increasing and decreasing load cycles are applied (Figure 2.19 (c)). After stabilization, connecting the reverse points of the corresponding loops yields the cyclic material law, i.e. a similar cyclic stress-strain curve to that shown in Figure 2.18 (b). This cyclic material law might differ slightly from that obtained from multi-specimen testing program. Figure 2.18 (b) also shows that the stress-strain curve under monotonic test is different from that under cyclic loading. Therefore, using monotonic curve for fatigue design may lead to incorrect safe limits.

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Figure 2.18 Cyclic Stress-strain relationship

Figure 2.19 Load history for multi-specimen and multi-step testing program

The cyclic material law is usually approximated by the Ramberg-Osgood equation, separating the total strain amplitude εa into an elastic and plastic part: εεεεa = σσσσa / E + (σσσσa / K′′′′)1/n′′′′ ( 2.20 )

where K′ and n′ are material dependant constants (cyclic hardening coefficient and cyclic hardening exponent). Since the equation is non-linear over the entire range, a linear curve is often assumed up to a fictitious yield point σy, at which the plastic strain assumes a value that can no longer be neglected, e.g. 0.001% [8].

2.4.2 Fatigue life In strain-controlled constant-amplitude tests, the crack initiation behavior of the material is investigated. The crack initiation is usually found by a drop of the stabilized stress σa by 5 %, which corresponds to a crack depth of approximately 0.5 mm in a small-scale specimen [8]. The crack initiation life, Nf, versus the strain amplitude, εa, is called strain-life or strain S-N curve (see Figure 2.20). Two parts are included in the strain-life curve part: Elastic part based on Basquin relation:

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εεεεe = σσσσa / E = σσσσf′′′′ (2Nf)b / E ( 2.21 )

where εe is the elastic strain amplitude; σa is the true stress amplitude; 2Nf is the reversals to failure ( 1 reversal = 0.5 cycles); σf′ is the fatigue strength coefficient and b is the fatigue strength exponent (see Figure 2.20). Plastic part based on Coffin’s and Manson’s separately developed relations: εεεεp = εεεεf′′′′ (2Nf)c ( 2.22 )

where εp is the plastic strain amplitude; εf′ is the fatigue ductility coefficient and c is the fatigue ductility exponent (see Figure 2.20). Therefore, the strain-life curve can be expressed as: εεεεa = εεεεe + εεεεp = σσσσf′′′′ (2Nf)b / E + εεεεf′′′′ (2Nf)c ( 2.23 )

For practical application for steel, the following approximation can be used for determining the parameters in strain S-N curves [17]. The fatigue strength coefficient with the hardness less than 500 BHN can be approximated using σσσσf′′′′ = σσσσu + 50 ksi ( 2.24 )

where σu is the tensile strength. The fatigue ductility coefficient can be approximated by: εεεεf′′′′ = εεεεf = ln (100/(100-%RA)) ( 2.25 )

where εf is true fracture ductility; and %RA is the percentage of reduction in cross-sectional area at fracture, defined as: %RA = 100 (A0 – Af) / A0 ( 2.26 )

For most metals the cyclic strain hardening exponent n′ is 0.1 ≤≤≤≤ n′′′′ ≤≤≤≤ 20 ( 2.27 )

In addition, the life at which the elastic and plastic strains are equal is called transition life represented by 2Nt (see Figure 2.20), and can be expressed as: 2Nt = εεεεf′′′′ (Eεεεεf′′′′/σσσσf′′′′)1/(b-c) ( 2.28 )

Further, the cyclic hardening coefficient can be expressed in terms of σf′ and εf′ K′′′′ = σσσσf′′′′ / (εεεεf′′′′)n′′′′ ( 2.29 )

and the cyclic hardening exponent can be expressed in term of b and c as: n′′′′ = b / c ( 2.30 )

The strain S-N curve defined above is established according to a smooth specimen. If the specimen is notched, the stress concentration effect must be taken into account. When all the stresses are in the elastic range, the peak stress σ can be expressed as: σσσσ = Kt σσσσn ( 2.31 )

where Kt is the stress concentration factor and σn is the nominal stress. The maximum strain can be expressed as: εεεε = Kt εεεεn ( 2.32 )

However, when peak stress is higher than the yield strength, a local plastic deformation results a nonlinear stress-strain relationship. Thus, the concentration factors for maximum stress and strain are different, i.e. σσσσ = Kσσσσ σσσσn ( 2.33 )

where Kσ is the stress concentration factor and

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εεεε = Kεεεε εεεεn ( 2.34 )

where Kε is the strain concentration factor. These two factors are interdependent and can be related using Neuber rule [8] as follows: Kt

2 = Kσσσσ·Kεεεε ( 2.35 )

The Neuber rule can be rewritten according to the stress, strain and concentration factors, i.e. σσσσ·εεεε = (Kt·σσσσn)2/E ( 2.36 )

The maximum stress and strain should also satisfy the cyclic material law, i.e. equation 2.20. Thus, the value of maximum strain can be determined with these two equations. This process is schematically shown in Figure 2.21. The fatigue life of the notched specimen then can be determined from the strain S-N curve with this maximum strain.

Figure 2.20 Strain S-N curve and their parameters

Figure 2.21 Determination of maximum strain

based on Neuber’s rules

It has been shown that Neuber’s rule is superior to other approximation formulae for diverse materials and load conditions since the calculation results lie a little on the conservative side in many cases. However, in order to account for the yielding of the entire cross-section, the equation of 2.36 must be extended by additional parameters [8]. When a specimen is subjected to a fully reversed load with superimposed steady stress, the effects of the mean stress should be taken into account. The effect of tensile mean stress is most critical and the compressive mean stress would somehow improve the fatigue behavior. When taking the tensile mean stress into account, the strain S-N curve can be modified according to Manson as [17]: εεεεa = (σσσσf′′′′ - σσσσm) (2Nf)b / E + εεεεf′′′′ [(σσσσf′′′′ - σσσσm) / σσσσf′′′′]1/n′′′′ (2Nf)c ( 2.37 )

according to Morrow as [17]: εεεεa = (σσσσf′′′′ - σσσσm) (2Nf)b / E + εεεεf′′′′ (2Nf)c ( 2.38 )

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2.5 Crack Propagation Methods Fracture mechanics is firstly related to the problems of unstable fracture; however, the fracture mechanics theory was found to be the best model of the crack propagation in fatigue. This method is mainly applied to low cyclic fatigue and finite life problems in predicting the remaining life of cracked components. Unlike the analysis of unstable fracture, plastic zones are relatively small so that the Linear Elastic Fracture Mechanics (LEFM) already offers suitable solutions for crack propagation problem. The application of fracture mechanics generally presupposes an existing cracks, which may be a defect or a flaw, or a small crack initiated by cyclic loads. In engineering design, crack lengths can range from 0.1 mm to several meters. Below this range is the special field of short cracks, which behave quite differently from usual cracks [8].

2.5.1 Characteristic of fatigue surfaces Figure 2.22 shows a macroscopic view of a typical crack surface of a round specimen. The cracks initiate at point A. The origin of the fatigue crack may be more or less distinct. In some cases a defect may be identified as the origin of the crack, in other cases there is no apparent reason why the crack should start at a particular point. If the critical section is at a high stress concentration fatigue initiation may occur at many points, in contrast to the case of un-notched parts where the crack usually grows from a point only [7]. The crack propagates in a slow and stable mode, D, that exhibits beach marks (also called clamshell marks). These beach marks are concentric rings that point toward the areas of the initiation. Beach marks are formed when the crack grows intermittently and at different rates during random variations in the loading pattern under the influence of a corrosive environment. Therefore, under the constant load, the beach marks cannot be observed. In order to create this kind of beach mark, in fatigue tests two levels of load are applied. In addition, on the crack surface, striations are formed which is a clear indication for a fatigue crack (see Figure 2.23) [8]. Although somewhat similar in appearance, striations are not beach marks as one beach mark may contain thousands of striations.

Figure 2.22 Typical fracture surface with initiation (A), stable crack propagation (D) and final

fracture (D) [8]

Figure 2.23 Typical striations around an inclusions [8]

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The rough region G is the final fracture area. A large final fracture area for a given material indicates a high maximum load, whereas a small area indicates that the load was lower at fracture [7].

2.5.2 Fatigue mechanism Fatigue damage is characterized by the nucleation, coalescence and stable growth of cracks leading to ultimately to net section yielding or brittle facture. Cyclic plastic shear strains eventually cause the nucleation of the slip band as illustrated in Figure 2.24, in which the applied tensile load is vertical and the resulting shear stress is at 45°. The slip band will be first formed in those grains whose crystallographic slip planes and directions are favorably oriented with respect to the applied cyclic shear stress. Each grain will have different preferred slip plane. At low stresses and strains, only a few grains have favorable orientations and only a few slip bands form. At high stresses and strains, a large number of slip bands form. During repeated cyclic loading, these slip bands grow and coalescence into a single dominant fatigue crack [14]. The nucleation process can be described using intrusion and extrusion model illustrated in Figure 2.25. This figure shows a cross-section view of a deforming grain in the material. Slip bands are formed due to the dislocation movement within individual grains. Cyclic shear stresses cause the dislocation to move, particularly the plastic deformation results in some slip bands coming out of the surface of the material (extrusion) and some bands going into the surface of the material (intrusion). Figure 2.24 Crack nucleation within grains [14]

Figure 2.25 Slip band formation [14]

The cohesion between the layers in slip band is weakened by oxidation of fresh surfaces and hardening of the strained material. At some points in this process small cracks develop in the intrusions. These micro cracks grow along slip planes, i.e. a shear stress driven process. Growth in a shear mode, which is called stage I crack growth, extends over a few grains (see Figure 2.26). During continued cycles, the micro cracks in different grains coalesce resulting in one or a few dominating growth under the primary action of maximum principal stress and this is called stage II growth (see Figure 2.26). The crack path is now essentially perpendicular to the tensile stress. However, the crack advancement is still influenced by the crystallographic orientation of the grains and the crack grows in a zigzag path along the slip planes.

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Figure 2.26 Stage I and Stage II growth process

Figure 2.27 Crack opening modes

Once a crack has been initiated, subsequent crack propagation may occur in several ways. Three basic modes of crack surface displacement can be classified (see Figure 2.27) [9]:

o Mode I. Opening or tensile mode, where the crack surfaces move directly apart; o Mode II. Sliding or in-plane shear mode, where the crack surface slide over one another in a

direction perpendicular to the leading edge of the crack; o Mode III. Tearing or out-of-plane shear mode, where the surfaces move relative to one

another and parallel to the leading edge of the crack. In isotropic materials, brittle fracture usually occurs in Mode I. Although fractures induced by sliding (Mode II) and tearing (Mode III) do occur, their frequency is much less than the opening mode fracture [2].

2.5.3 Linear elastic fracture mechanics In linear elastic fracture mechanics, the stress and displacement fields in the vicinity of crack tips subjected to three modes of deformation are given in Figure 2.28. The symbols used in Figure 2.28 are defined as shown in Figure 2.29. In addition, K is stress intensity factor that presents a relationship of a loading mode, geometry of the stressed part and the length of crack and is defined as: K = f·σσσσ·(ππππ·a)1/2 ( 2.39 )

where σ represents the loading, a is the length of the initial crack and f is the compliance function that describes the geometry of the part. Mode I covers the most common form of cracks caused by fatigue and the compliance function corresponding to four standard specimens with a different geometry shown in Figure 2.30 can be defined as [17]: Center crack loaded in tension fI = (sec(ππππa/2b))1/2 ( 2.40 )

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Edge crack loaded in tension fI = 1.12 – 0.231·a/b + 10.55 (a/b)2 – 21.72 (a/b)3 + 30.39 (a/b)4 ( 2.41 )

Double edge cracks loaded in tension fI = 1.12 + 0.203·a/b – 1.197 (a/b)2 + 1.93 (a/b)3 ( 2.42 )

Edge crack loaded in bending fI = 1.122 – 1.40·a/b + 7.33 (a/b)2 – 13.08 (a/b)3 + 14.0 (a/b)4 ( 2.43 )

From the formula shown in Figure 2.28, it can be seen that the stress intensity factor corresponding to three opening mode uniquely defines the stress state. The unit of the stress intensity factor is MPa(m)1/2.

Figure 2.28 Stresses and deformations in vicinity of crack tip for three modes of deformation

Figure 2.29 Coordinate system in the vicinity of

a crack

Figure 2.30 Concerning mode I: (a) center cracked plate in tension (b) edge cracked plate in

tension (c) double-edge cracked plate in tension (d) cracked beam in bending

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From tests it is known that instable fracture (final fracture) occurs when the crack length reaches a critical value ac or the stress intensity factor, KI, reaches the critical value, KIC, which is called fracture toughness. The fracture toughness can be determined experimentally from tests with predetermined crack size a. KIC assumes a fracture without plastic deformation. Under certain conditions (e.g. relatively thin plates), larger plastic deformations occur and the critical intensity factor is then defined as KC, which is higher than KIC [8]. Notched plates under loading with existing cracks at notches are shown in Figure 2.31 (a) and (b). Since the geometry of the opening mode has changed, the expression of the stress intensity factor must be changed [17]. This can be done through an empirical method where the stress gradients in the vicinity of the notch are taken into account. As indicated in Figure 2.31 (c), the area in the vicinity of the notch is divided into the areas of high stress gradients and of low stress gradients. Within the domain of d = 0.13 (Dr)1/2 ( 2.44 )

the stress intensity factor is computed based on a stress concentration factor, Kt, and the stress intensity factor is calculated as: K = fKtσσσσ(ππππa)1/2 ( 2.45 )

In the remaining domain, the stress intensity factor is equal to K = fσσσσ[ππππ(D+a)]1/2 ( 2.46 )

Figure 2.31 Effect of notches on the stress intensity factor [17]

2.5.4 Crack propagation under fatigue load Consider a fatigue load that fluctuates at constant amplitude where the stresses vary between constant limit σmax and σmin. The range of the stress intensity factor can be expressed as: ∆∆∆∆K = Kmax – Kmin = f (σσσσmax - σσσσmin) (ππππa)1/2 ( 2.47 )

From the test of measuring the crack growth rate, it has been found that three regions can be divided on the crack propagation curve (see Figure 2.32):

o Region I: crack formation o Region II: moderate crack propagation o Region III: accelerated crack growth and fracture

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Figure 2.32 Three regions of crack growth rate [8]

According to Paris and Erdogan [8], the crack propagation rate da/dN (increase of crack length per cycle) and the range of the stress intensity factor can be expressed as: da/dN = C(∆∆∆∆K)m ( 2.48 )

C and m are material parameters. In logarithmic scale, the crack propagation law is a straight line, which describes the major part of the crack propagation domain (region II). Below a threshold value ∆Kth, the crack propagation is zero due to the existing fatigue limit, while in the upper part the crack propagation rate increases rapidly and the ∆Kc reflects the failure point. The number of cycles N between an initial crack length ai and final crack length af can obtained from integration of equation, i.e.

−

⋅⋅−⋅

=−− 1

21

22

11

12

1m

f

m

i

mm

f

aafmC

N

ππππσσσσ∆∆∆∆ )(

( 2.49 )

The initial crack length can be computed from the equation a = 1/ππππ [∆∆∆∆K/(a(σσσσmax-σσσσmin))]2 ( 2.50 )

using ∆∆∆∆Kth = Kth - Kmin ( 2.51 )

and crack at failure can be calculated using the same equation with ∆∆∆∆Kc = Kc - Kmin ( 2.52 )

The values of both the stress intensity factor at threshold point ∆Kth and the fracture toughness ∆Kc are provided through testing for practical application in design for fatigue [17]. In case of a geometry function depending on the crack length, an incremental solution is possible for step-wise increased crack length ∆a, i.e.

88

+

−

⋅⋅−⋅

=−− 1

21

22

11

12

1mmm

m aaafmC

N

)()( ∆∆∆∆ππππσσσσ∆∆∆∆

∆∆∆∆ ( 2.53 )

The total cycles to failure can be calculated as Nf = ΣΣΣΣ (∆∆∆∆Ni) ( 2.54 )

When the value of lower limit is less than zero, σmin < 0, the cracks stops growing due to compression at lower limit since the propagation occurs only at tensile stresses. However, when σmin > 0, the propagation law has to be amended with stress ratio based on testing data. One relation developed by Forman, Kearney and Engle has the form [17] da/dN = A (∆∆∆∆K)n / [(1-R)Kc - ∆∆∆∆K] ( 2.55 )

where A and n are material properties.

2.5.5 Short crack behavior Due to improved measurement techniques, very small cracks can be detected (smaller than grain size). Normally they start from the material surface along the slip bands under mode II, i.e. shear mode and can stop or propagate to larger size. At short cracks in ductile material, the plastic zone is comparably large. The crack propagation is determined by such effects as grain boundaries, material phases, inclusions and pores. The behavior can be non-normal: the crack propagation rate may decrease with increasing crack length. The transition from a short crack to a long crack can occur if the cyclic load becomes so large that the threshold value of the stress intensity factor range is exceeded. Models for considering above-mentioned effects have been developed by Newman et al. and Hou and Chang [8].

2.6 Fatigue Analysis Under Variable Loads Until now we have described fatigue properties of structural component under constant amplitude. In this section, analysis methods concerning to variable loading will be discussed.

2.6.1 Fatigue testing under variable loading In stress method, the fatigue test under variable load can be performed in the following scenarios [8]:

o Prototype testing under realistic conditions (cars under real life condition); o Application of the original loading to components or structures in a laboratory; o Application of synthetic load histories to components and structures.

The load histories can be created from load spectra either in the form of block-program loading or random loading. Block-program loading is a simplified representation of the load process, where load amplitudes of the same size are gathered in blocks as shown in Figure 2.33. It has been found

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that the block sequence in Figure 2.33 (low-high-low) is a good compromise between different kinds of variable amplitude loading. The typical random loading is shown in Figure 2.34. The sequence of load amplitudes during a random loading history is significantly different from that in a block-program test. The type of load history strongly affects the fatigue life. Under random loading, the shorter fatigue life is expected. This is due to the effect of sequence of amplitude, i.e. frequently changing amplitudes and mean stresses are more damaging than similar load cycles following each other. In addition, due to the high costs and long testing time of variable amplitude tests, the small amplitudes are frequently omitted. The omissions of the small amplitude have influences on the fatigue life. Roughly it can be said amplitudes below half of the fatigue limit are non-damaging [8]. Figure 2.33 Block-program loading [8]

Figure 2.34 Random loading [8]

2.6.2 Palmgren-Miner rule The fatigue life of a component under variable loading can be calculated using the Palmgren-Miner rule, which is a linear damage rule assuming that:

o The variable load that takes place irregularly can be replaced using an sequence of blocks of uniform cycles (see Figure 2.35 (a) and (b)).

o The number of stress cycles imposed on a component, expressed as a percentage of the total number of stress cycles of the same amplitude necessary to cause failure, gives the fraction of damage.

o The order in which the stress blocks of different amplitudes are imposed does not affect the fatigue life.

o Failure occurs when the linear sum of the damage from each load level reaches a critical value.

If ni is the number of cycles corresponding to the stress amplitude, σi, in a sequence of m blocks, and if Ni is the number of cycles to failure at σi, then the Palmgren-Miner’s rule states that the failure would occur when

1Nnm

1i i

i =∑=

( 2.56 )

90

and Di = 1 / Ni ( 2.57 )

is called the damage of a single cycle at stress level σi. The scheme of Palmgren-Miner’s rule is shown in Figure 2.35 (c). The rule is first introduced by Palmgren in analysis of ball bearings and adapted by Miner for aircraft structure [17]. It should be noted that, when variable amplitude loading is applied, the stresses less than the fatigue limit still cause damage due to the fact that larger amplitude cycles may start to propagate the crack. However, linear Palmgren-Miner’s rule assumes independence of damage accumulation. This can be overcome in practical design using a slope line after fatigue life instead of using a horizontal line as shown in Figure 2.35 (c).

Figure 2.35 Scheme of Palmgren-Miner’s rule

Empirically, tests have shown that differences between low-high sequences and high-low sequence. Thus, there are two main shortcomings of the linear damage rule: assuming sequence independence and assuming independence of damage accumulation. These two shortcomings might be overcome by non-linear damage rules.

2.6.3 Cycle counting When using linear Palmgren-Miner’s rule to estimate the fatigue life, the variable amplitude loading has to be transformed into a series of constant amplitude loadings. Several methods are available to do cycle counting, for instance, level crossing counting, peak counting, simple range counting and rainflow counting. In this section, we only provide the details of rainflow counting. Rainflow is a generic term to describe any cycle counting method that identifies closed hysteresis loops in stress-strain response of material subjected to cyclic loading. Several algorithms are available to perform the counting, however, they all require that the entire load history be known before the counting process starts [4]. The basic rule of rainflow counting is defined as follows:

o In order to eliminate the counting of half cycles, the load history has to be drawn as starting and ending at the greatest magnitude;

o A flow of rain has to be stopped when

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a. The rain begins at a local maximum and falls opposite a local maximum that is greater than that where it came from

b. The rain encounters a previous flow Figure 2.36 illustrates the procedure of cycle counting using rainflow method. Figure 2.36 (a) is the initial loading history. The counting is firstly started from the tension peaks. The details of counting based on above-mentioned rule are described as follows:

o Route 1 starts from A and falls down at B. Since the value of C is less than that of A, the rain can continually fall down to line CD. Similarly, the value of A is larger than that of E, C, I, K, and M, it will stop at the position shown in Figure 2.36 (b). This procedure is carried out based on rule (a);

o Route 2 starts from C and stops as shown in Figure 2.36 (b) due to it encounter the previous rain flow (Route 1). This is the rule (b);

o Route 3 starts from E and stops due to the value of G being larger than that of E (rule (b)); o Route 4 is based on rule (b); o Route 5 is based on rule (a); o Route 6 is based on rule (b); o Route 7 is based on rule (a).

Similarly, the rainflow counting from compression peaks are shown in Figure 2.36 (c). Figure 2.36 (d) shows the cycles from both tension side and compression side. This can be done as follows:

Figure 2.36 Scheme of rainflow counting

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o Start from Route 1 of tension side and find the ending point of Route 1. Then check the route of compression side that starts from this same point. In this case it is Route 6′. This is one cycle of loading. Similarly, other cycles in loading history are obtained as shown in Figure 2.36 (d).

After this counting, the stress range and number of cycles corresponding to the stress range are obtained and the damage can be estimated according to Palmgren-Miner’s rule under this variable history. Rainflow counting is easy to do manually for relatively simple loading history, however, for more complex loading history numerical methods are used [4]. Figure 2.37 shows the rainflow counting procedure for a strain history. The similar procedures aforementioned for getting the stress range are applied to obtain the cycles of the strain ranges (Figure 2.37 (b)). In addition, when combined with the stress-strain relationship of material law, the hysteresis loops together with the mean stress effects are provided from rainflow counting (see Figure 2.37 (c)). Using equation (2.23), the fatigue life corresponding to each strain range level can be calculated. The total damage under this strain history can be computed using Palmgren-Miner’s rule with each fatigue life calculated above.

Figure 2.37 Rainflow counting for a strain history

2.6.4 Crack propagation under variable loading In previous sections we have paid our attentions to predict fatigue life under variable loading using stress method and strain method. In this section, we will investigate the crack propagation behavior under variable loading. The crack propagation behavior under constant loading can differ considerably from variable amplitude loading. Under variable amplitude loading, the crack increment, ∆a, is dependent not only on the present crack size, but also on the load history, i.e. load interaction or load sequence effects. A tensile overload induces compressive residual stresses, which are beneficial for the following stress cycles. Figure 2.38 shows that a single overload can considerably decrease crack growth rate, i.e. crack retardation. On the other hand, a compressive overload creates tensile residual stresses, which have acceleration effects. Besides, the crack closure behavior is very complex particularly under variable amplitude loading [8].

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Figure 2.38 Retardation effect of an overload on crack growth [8]

2.7 Fatigue Analysis of Welded Components In welded steel structures, most of the fatigue cracks start to grow from welds, rather than from other details, because [7]:

o Most welding processes leave minute metallurgical discontinuities from which cracks may grow. As a result, the initiation period, which is normally needed to start a crack in plain wrought material, is either very short or no existent. Cracks therefore spend most of their life propagating, i.e. getting longer.

o Most structural welds have a rough profile. Sharp changes of direction generally occur at the

toes of butt welds, and at the toes and roots of fillet welds (see Figure 2.39). These points cause local stress concentrations (see Figure 2.40). Small discontinuities close to these points will therefore react as though they are in a more highly stressed member and grow faster.

Figure 2.39 Local stress concentrations at welds

Figure 2.40 Typical stress distribution at weld

toe

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2.7.1 Factors affecting the fatigue life Welded components can be regarded as manufacturing-related notches that reduce the fatigue strength. The effects of the following parameters on the fatigue behavior are investigated. Influence of mean stress and material strength From a large amount of test specimens, it has been found that the stress ratio R has little influence on fatigue behaviors of welded components [8]. This is because at the critical crack initiation points of the welded structures, tensile residual stresses up to yielding are expected. The stress cycles are remained in tensile, irrespective of the R-values of the external load. Therefore, the influence of stress ratio is only taken into account very cautiously or not at all in the codes or regulations. Similarly, the influence of the material strength is not considered for welded components due to the strong notch effects. With regard to the crack propagation behavior, crack closure does not occur if high-tensile residual stresses are presented in the area of the crack tip. Influence of imperfections Imperfections can reduce the fatigue strength of the welded components considerably. These imperfections include volumetric imperfections (blowholes and pores, and slug inclusions), planer imperfections (cracks and lack of fusion), imperfections of the weld geometry (weld reinforcement and undercut) and imperfections of the weld geometry (angular and axial misalignment) [8]. Some typical imperfections are shown in Figure 2.41.

Figure 2.41 Imperfections in welded joints [1]

Normally, these imperfections can cause stress concentration that lowers the fatigue strength. These imperfections may be caused by: (1) improper design that restricts accessibility for welding; (2) incorrect selection of a welding process or welding parameters; (3) improper care of electrode or flux, or both and (4) other causes including welder performance [1]. The severity of a discontinuity, which is due to the imperfections, is governed by its size, shape, and orientation, and by the magnitude and direction of the design and fabrication stresses. Generally, the severity of discontinuity increases as the size increases, and as the geometry becomes more planar and the orientation more perpendicular to the direction of tensile stresses. Thus, volumetric discontinuities are usually less injurious than planar, crack-like discontinuities. Also crack-like discontinuities whose orientation is perpendicular to the tensile stress can be injurious than those

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parallel to the tensile stress. Furthermore, a surface discontinuity whose plane is perpendicular to the tensile stress is more severe than if it were embedded [1]. Influence of residual stresses Residual stresses are those exist in a component that is free from externally applied loads. They are caused by non-uniform plastic deformations in neighboring regions. Furthermore, residual stresses are always balanced so that the stress field is static equilibrium. Because fatigue life is governed by the stress range instead of stresses, tensile residual stresses usually have only a secondary effect on fatigue behavior of components. On the other hand, excessive tensile residual stress can also initiate unstable fracture in materials with low-fracture toughness. In welded components, residual stresses are caused by the inability of the deposited molten weld metal to shrink freely as it cools and solidifies. The magnitude of the residual stresses depends on such factors as the deposited weld beads, weld sequence, total volume of deposited weld metal, weld geometry, and strength of the deposited weld metal and of the adjoining base metal as well as other factors. Often, the magnitude of these stresses exceeds the elastic limit of the lowest strength region in the weldment [1]. Influence of plate thickness The thickness of plate has an adverse effect on the fatigue strength due to the following reasons [8]:

o Stress gradient effect: the tensile region of the stress field (including residual stresses) around the weld toe is larger in thicker plates so that an initial defect will experience a larger stress during crack initiation and early crack propagation, thus, resulting in a shorter fatigue life.

o Technological size effect: this effect is mainly attributed to material size and surface effects.

In particular, for welded joints, the ratio between plate thickness and weld toe radius is larger for thicker plates, thus, resulting in a higher stress concentration and, hence, in a reduced crack initiation period.

o Statistical size effect: the likelihood of finding a significant defect in a larger volume is

increased compared to a small one. Influence of post-weld treatment Using post-weld treatment of the weld, it is possible to improve the fatigue strength of welded joints considerably, especially the fatigue limit. The improvement mainly involves an extension of the crack initiation life and can be achieved by [8]:

o A reduction of the stress peak related to the weld shape; o Removal of crack-like weld imperfections at the weld toe; o Removal of detrimental tensile residual stresses, up to the formation of favorable

compressive residual stresses in the area susceptible to crack initiation.

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Post-weld treatment is of particular interest in connection with the repair of fatigue cracks. However, it must be guaranteed that the fatigue strength of the area, which is not subjected to post-weld treatment, is high enough.

2.7.2 S-N methods for evaluating fatigue life Several S-N methods are available for estimating the fatigue life of welded components: nominal stress method, structural hot spot stress method, notch stress method, notch stress intensity method and notch strain method [8]. Fatigue assessment according to nominal stress method uses standard S-N curves together with detail classes of basic joints that can be found in several standards and guidelines. Notch strain method is not widely used for welded components for two reasons. Firstly, several materials are involved in welded components: base metal, heat affected zone and weld metal. Reliable cycle data for different type of materials are rare and the numerical efforts to analyze the local stress and strain are high. Secondly, the local material in the critical area is far from smooth and homogeneous. The early crack propagation phase may form the major part of the fatigue life. In principle, the notch stress and notch stress intensity method are closely related [8]. Thus, in this section, the nominal stress method, the hot spot stress method and notch stress method are discussed.

2.7.2.1 Definitions of stresses Before calculating the fatigue life using three methods mentioned above, the concepts of nominal stress, hot spot stress and notch stress in welded joints are defined firstly (see Figure 2.42). Nominal stresses are those derived from simple beam models or from coarse mesh FEM models. Stress concentrations resulting from gross shape of the structure are included in the nominal stress.

Figure 2.42 Stress distribution at welded joints

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Structural hot spot stresses, also called geometric stresses, include nominal stresses and stresses from structural discontinuities. The latter are not the stresses due to the presence of welds. Instead, they are extrapolated using various methods from the points at certain distance away from weld toe. Notch stresses are the total stress at the weld toe and include the structural stresses and the stresses due to presence of the weld. FEM can be used to calculate the notch stress. However, due to the small notch radius and steep gradient, a very fine mesh is necessary.

2.7.2.2 Nominal stress method The simplest and most common method for estimating fatigue life is nominal stress method. Eurocode 3, Part 1.1 is mainly based on this method. In this section, after shortly introducing the determination of nominal stress of welded joints, the design procedure based on Eurocode 3, Part 1.1 are described in details. Calculation of nominal stress Usually, the nominal stress is related to the section in which the crack is to be expected. This is in most cases the section in front of weld toe, if a crack is expected to initiate from there (see Figure 2.14). If a crack is expected to propagate through weld from an unwelded root face, the relevant nominal stress is referred to the section through the weld throat (see Figure 2.43 (b)). In case of bi-axial stress states, the largest principal stress σ1 is taken (see Figure 2.43 (a)).

Figure 2.43 Example of cracks at welded joints with relevant principal stress σσσσ1 [8]

S-N curves in Eurocode 3, Part 1.1 The fatigue strength in Eurocode 3, Part 1.1 is defined by a series of log ∆σ - log N or log ∆τ - log N curves (see Figure 2.44), each applying to a typical detail category. Each category is designated by a number which represents the reference value ∆σC of the fatigue strength at 2 million cycles, i.e. NC = 2 x 106. The values are rounded values. Some common detail types and their fatigue categories are shown in Figure 2.45 and more details types are provided in Table 9.8.1 to Table 9.8.7 in Eurocode 3, Part 1.1 [6].

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Figure 2.44 Family of design curves [6]

Figure 2.45 Some common detail type and their fatigue categories [7]

In addition, two other concepts are defined in Figure 2.44 (a). One is the constant amplitude fatigue limit, ∆σD, which is the limiting stress range value above which a fatigue assessment is necessary. The number of cycles corresponding to constant amplitude fatigue limit is 5 million cycles, i.e. ND =

99

5 x 106. The other is cut-off limit, ∆σL, which is a limit below which stress ranges of the design spectrum do not contribute to the calculated cumulative damage. The number of cycles corresponding to this value is 108 cycles, i.e. NL = 108. The cut-off limit is put forward because when variable amplitude loading is applied, the stresses less than the fatigue limit still cause damage due to the fact that larger amplitude cycles may start to propagate the crack. The fatigue strength curves for nominal stresses are defined by log N = log a – m log ∆∆∆∆σσσσR ( 2.58 )

where ∆σR is the fatigue strength; N is the number of stress range cycles; m is the slope constant of the fatigue strength curves with value of 3 and/or 5; and loga is a constant that depends on the related part of the slope and their values are given in Table 2.1. Similar fatigue strength curves are used for shear stresses (Figure 2.44 (b)) and only one slope value is taken, i.e. m = 5. These curves are based on representative experimental investigations and thus include the effects of local stress concentrations due to the weld geometry, size and shape of acceptable discontinuities, the stress direction, residual stresses, metallurgical conditions, and in some cases, the welding process and post-weld improvement procedures.

Table 2.1 Numerical values for fatigue strength curves [6]

Detail category log a for N < 108 Stress range at constant amplitude fatigue limit

Stress range at cut-off limit

Normal stress range ∆∆∆∆σσσσC

(N/mm2) N ≤≤≤≤ 105 (m = 3 )

N > 105 (m = 5 )

∆∆∆∆σσσσD (N/mm2)

∆∆∆∆σσσσL (N/mm2)

160 12.901 17.036 117 64 140 17.751 16.786 104 57 125 12.601 16.536 93 51 112 12.451 16.286 83 45 100 12.301 16.036 74 40 90 12.151 15.786 66 36 80 12.001 15.536 59 32 71 11.851 15.286 52 29 63 11.701 15.036 46 26 56 11.551 14.786 41 23 50 11.401 14.536 37 20 45 11.251 14.286 33 18 40 11.101 14.036 29 16 36 10.951 13.786 26 14

Shear stress range ∆∆∆∆ττττC

(N/mm2) N < 108 (m = 5 )

------ ∆∆∆∆ττττL (N/mm2)

100 16.301 ------- 46 80 15.801 ------- 36

Fatigue analysis based on Eurocode 3, Part 1.1 No fatigue assessment is required when any of the following condition is satisfied according to Eurocode 3, Part 1.1:

o The largest nominal stress range ∆σ satisfies:

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γγγγFf ∆∆∆∆σσσσ ≤≤≤≤ 26 / γγγγMf N/mm2 ( 2.59 )

o The total number of stress cycles, N, satisfies: N ≤≤≤≤ 2 x 106 [(36/γγγγMf) / (γγγγFf ∆∆∆∆σσσσE.2)]3 ( 2.60 )

o For a detail for which a constant amplitude fatigue limit ∆σD is specified, the largest stress range ∆σ satisfies:

γγγγFf ∆∆∆∆σσσσ ≤≤≤≤ ∆∆∆∆σσσσ / γγγγMf ( 2.61 )

In these conditions, ∆σE.2 is the equivalent constant amplitude stress range (N/mm2), which is defined as the constant amplitude stress range that would result in the same fatigue life as for the spectrum of variable amplitude stress ranges, when the comparison is based on a Miner’s summation [6]. γFf is the partial safety factor for fatigue loading and its value are provided in Eurocode 1 [5]. A value of γFf = 1.0 may be applied in the design calculation. γMf is the partial safety factor for fatigue strength and its value are provided in Table 2.2, which is Table 9.3.1 in Eurocode 3, Part 1.1 [6].

Table 2.2 Partial safety factor for fatigue strength γγγγMf [6]

Inspection and access “Fail-safe” components

Non “fail-safe” components

Periodic inspection and maintenance. Accessible joint details.

1.00 1.25

Periodic inspection and maintenance. Poor accessibility.

1.15 1.35

Otherwise, the fatigue assessment criterion for constant amplitude loading is: γγγγFf ∆∆∆∆σσσσ = ∆∆∆∆σσσσR / γγγγMf ( 2.62 )

where ∆σ is the nominal stress range and ∆σR is the fatigue strength for the relevant detail category for the total number of stress cycles N during the required design life. For variable amplitude loading, the fatigue assessment shall be based on Palgren-Miner rule of cumulative damage. If the maximum stress range due to variable loading is higher than the constant amplitude fatigue limit, a cumulative damage assessment may be made using: Dd ≤≤≤≤ 1 ( 2.63 )

where Dd = ΣΣΣΣ (ni / Ni ) ( 2.64 )

in which ni is the number of cycles of stress range ∆σi during the required design life; and Ni is the number of cycles of stress range γFf·γMf·∆σi to cause failure for the relevant detail category. Cumulative damage calculations shall be based on one of the following:

a) a fatigue strength curve with a single slope constant m = 3; b) a fatigue strength curve with double slope constants (m =3 and m = 5), changing at the

constant amplitude fatigue limit; c) a fatigue strength curve with double slope constants (m =3 and m =5), and a cut-off limit at

N = 1000 million cycles;

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d) a fatigue strength curve with a single slope constant m = 5 and a cut-off limit at N = 100 cycles.

Case (c) is most general. When using case (c) and with a constant amplitude fatigue limit ∆σD at 5 million cycles, Ni may be calculated as follows:

o if γFf ∆σi ≥ ∆σD / γMf Ni = 5 x 106 [[(∆∆∆∆σσσσD / γγγγMf) / (γγγγFf ∆∆∆∆σσσσi)]3 ( 2.65 )

o if ∆σD / γMf > γFf ∆σi ≥ ∆σL / γMf Ni = 5 x 106 [[(∆∆∆∆σσσσD / γγγγMf) / (γγγγFf ∆∆∆∆σσσσi)]5 ( 2.66 )

o if γFf ∆σi < ∆σL / γMf Ni = ∝∝∝∝ ( 2.67 )

Nominal shear stress ranges, ∆τ, should be treated similarly to nominal normal stress ranges, but using a single slope constant m = 5. Ni may be calculated as:

o if γFf ∆τi ≥ ∆τL / γMf, Ni = 2 x 106 [[(∆∆∆∆ττττC / γγγγMf) / (γγγγFf ∆∆∆∆ττττi)]5 ( 2.68 )

o if γFf ∆τi < ∆τL / γMf, Ni = ∝∝∝∝ ( 2.69 )

Fatigue assessment of hollow sections The fatigue strength curves to be used in conjunction with the hollow details shown in Table 9.8.6 in Eurocode 3, Part 1.1, are those shown in Figure 2.44. They have double slope constant of m = 3 and m =5. The fatigue strength curves to be used in conjunction with the hollow section joint details for lattice girders shown in Table 9.8.7 in Eurocode 3, Part 1.1, are given in Figure 2.46. They have a single slope constant of m = 5. The corresponding values for numerical calculations of the fatigue strength are given in Figure 2.3. In these calculations, the throat thickness of a fillet weld shall not be less than the wall thickness of the hollow section member that it connects. The member force for hollow sections according to Eurocode 3, Part 1.1 may be analyzed neglecting the effect of eccentricities and joint stiffness, assuming hinged connections, provided that the effects of secondary bending moments on stress range are considered. In the absence of rigorous stress analysis and modeling of the joint, the effects of secondary bending moment may be taken into account by multiplying the stress range due to axial member forces by appropriate coefficients as follows:

o for joints in lattice girders made from circular hollow sections, see Table 2.4. o for joints in lattice girders made from rectangular hollow sections, see Table 2.5.

The values in these two tables are approximate empirical values or values based on testing.

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Figure 2.46 Fatigue strength curves for joints

in lattice girders

Table 2.3 Numerical values for fatigue strength curves for hollow sections

Detail category

∆∆∆∆σσσσC (N/mm2)

log a for N<108

Stress range at cut-off

limit (N=108)∆∆∆∆σσσσL

(N/mm2) 90 16.051 41 71 15.551 32 56 15.051 26 50 14.801 23 45 14.551 20 36 14.051 16

Table 2.4 Coefficients to account for secondary bending moments in joints of lattice girders made from

circular hollow sections [6]

Type of joint Chords Verticals Diagonals K type 1.5 1.0 1.3 Gap joints N type 1.5 1.8 1.4 K type 1.5 1.0 1.2 Overlap joints N type 1.5 1.65 1.25

Table 2.5 Coefficients to account for secondary bending moments in joints of lattice girders made from

rectangular hollow sections [6]

Type of joint Chords Verticals Diagonals K type 1.5 1.0 1.5 Gap joints N type 1.5 2.2 1.6 K type 1.5 1.0 1.3 Overlap joints N type 1.5 2.0 1.4

Fatigue strength modifications For the construction details not listed in Eurocode 3, Part 1.1, all hollow section members and tubular joints with wall thickness greater than 12.5 mm, fatigue assessment shall be carried out using the procedure based on geometric stress ranges, i.e. hot spot stresses method whose calculation procedure are described in next section [6]. In addition, for non-welded details or stress relieved welded details; the effective stress to be used shall be determined by adding the tensile portion of the stress range and 60% of the compressive portion of the stress range.

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The influence of the thickness of the parent metal in which a potential crack may initiate and propagate are taken into account in Eurocode 3, Part 1.1 in the following way: the variation of fatigue strength with thickness shall be taken into account, when material thickness is greater than 25 mm, by reducing the fatigue strength using: ∆∆∆∆σσσσR.t = ∆∆∆∆σσσσR (25 / t)0.25 ( 2.70 )

with t > 25 mm. When the material thickness of the constructional detail is less than 25 mm, the fatigue strength shall be taken as that for a thickness of 25 mm. This reduction shall be only applied only to structural details with welds transverse to the direction of the normal stresses. Where the detail category in the classification tables already varies with thickness, the above correction for thickness shall not applied. “Safe-life” and “fail-safe” concepts of structural design “Fail-safe” and “safe-life” are the two concepts of structural design [7]. In the “safe-life” method, the designer starts by making an estimation of the load spectrum to which the critical structural components are likely to be subjected in service. These components are then analyzed or tested under that load spectrum so as to obtain its expected life. Finally a factor of safety is applied in order to give a safe life during which the possibility of fatigue failure is considered to be sufficiently remote. It is clear that via making the safety factor sufficient large, the designer can govern the probability of failure associated with his design. On the other hand, if a fatigue crack does occur, it may well be catastrophic, and safety depends on achieving a specified life without a fatigue crack developing. With this method, the emphasis is on prevention of crack initiation [7]. With the “fail-safe” concept, the basis of design is that, even if failure of part of the main structure does occur, there will always be sufficient strength and stiffness in the remaining part to enable the structure to be used safely until the crack is discovered. This concept implies that periodic in-service inspection is a necessity, and that the methods used must be such as to ensure that cracked members will be discovered so that repairs or replacements can be made. It is clear that with this method of design the probability of partial failure is much greater than with the “safe-life” design. In developing a “fail-safe” structure, the “safe-life” should also be evaluated, in order to make sure that it is of the right order of magnitude. However, the emphasis, instead of being on the prevention of crack initiation, is on producing a structure in which a crack will propagate slowly, and which is capable of supporting the full design load after partial failure. The basic principle of “fail-safe” design is therefore to produce a multiple load-path structure, and preferably a structure containing crack arresters. In addition, the structural elements must be arranged so as to make inspections as easy as possible. In areas where that is not possible, the elements must be oversized so that either fatigue cracking does not occur in them, or fatigue crack growth is so slow that there is no risk of failure [7].

2.7.2.3 Hot-spot stress method The concept of a hot spot stress is originated in the design of offshore structures. The problem of local stresses in the vicinity of weld toes in tubular joints is one of the most difficult stress distributions in steel structures. The nominal stresses at these connections are often impossible to

104

define. The geometrical hot spot stress was introduced with the definition of reference points for stress evaluation and extrapolation at certain distance away from the weld [3]. Niemi [10] has listed several cases where the hot spot stress approach is more suitable than the nominal stress weld classification approach:

o there is no clearly defined nominal stress due to complicated geometric effects; o the structural discontinuity is not comparable with any classified details in the design rules; o for the above-mentioned reasons, the finite element method is used; o field testing of a prototype structure is performed using hot spot strain gauge measurements; o offset or angular misalignments exceeds the fabrication tolerances, thus invalidating some of

the basic conditions for using nominal stress approach. In Eurocode 3, Part 1.1, the hot spot stress is defined as the maximum principal stress in the parent material adjacent to the weld toe taking into account only the overall geometry of the joint, excluding local stress concentration effects due to the weld geometry and discontinuities at the weld toe. The maximum value of geometric stress range or hot spot stress range shall be found by investigating various locations at the weld toe around welded joint or the stress concentration area. The hot spot stresses may be determined using stress concentration factors obtained from parametric formulae within their domains of validity, a finite element analysis or an experimental model. Since the local stress concentration due to weld geometry and irregularities at the weld toe cannot easily be determined, the influence of the local weld notch stresses can be excluded by carrying out an extrapolation procedure of the geometric stresses from outside this region. The hot spot stresses or strains arrived at in this manner are divided by the nominal stress or strain to arrive at the stress concentration factor (SCF) or strain concentration factor (SNCF) [16]. As an example, two extrapolation methods based on ECSC are shown in Figure 2.47, i.e. linear extrapolation and quadratic extrapolation [16]. In linear extrapolation method, two points on the curve determined from all data points are used for the extrapolation: the first is 0.4t from the weld toe with a minimum of 4 mm. The second point is taken to be 0.6t further. In quadratic extrapolation, the first point is 0.4t from the weld toe with a minimum of 4 mm. The second point on the curve is taken 1.0t further. The quadratic extrapolation is carried out through these two points and other points between these two points and thereby obtaining the quadratic SCF. The stresses at interpolation points can be obtained either from a FEM calculation or from an experimental measurement. The determination of hot spot stress is only a step in determining the fatigue life of a specific connection and load case. A S-N curve relates the hot spot stress range to the expected fatigue life of a welded joint. It has been shown that for seam-welded structures the fatigue resistances are similar. Thus, one S-N can be used to describe the fatigue behavior [8]. In Eurocode 3, Part 1.1, the fatigue strength curves to be used for fatigue assessments based on hot spot stress range, shall be [6] (see Figure 2.48):

a) For full penetration butt welds: • Category 90, when both weld profile and permitted weld defects acceptance criteria

are satisfied. • Category 71, when only permitted weld defects acceptance criteria are satisfied.

105

Figure 2.47 Method of extrapolation to the weld toe

Figure 2.48 Fatigue strength curves for hot spot stress method [6]

b) For load carrying partial penetration butt welds and fillet welds: • Category 36 or alternatively a fatigue strength curve obtained from adequate fatigue

test results. As for the hot spot stress method, the following things might be concerned [16]: Stress or strain based definition Although in most design recommendations, the hot spot stress and stress concentration factors are used, in many cases these are really based on strains. This is due to the fact that the strain can be measured easily by individual strain gauges, whereas stresses would require strain gauge rosettes to measure various strain components. Another reason is that stresses cannot significantly exceed the yield stress and in lower cycle fatigue, the failure mechanism is strain based rather than stress based. The nominal stress and strain can be easily converted using σ = E ε. Type of stress to be used In Eurocode 3, Part 1.1, the hot spot stress are principal stresses. However, the use of stresses perpendicular to the weld might be possible, this because:

o Principal stresses can be significantly higher than stresses perpendicular to the weld toe, yet closer to the weld the stresses are diverted to the weld by the stiffening influence of weld and attached wall. Therefore, the difference between principal stresses and stresses perpendicular to the weld toe decreases closer to the weld.

o Only stress components perpendicular to the weld are enlarged by stress concentration caused by the global weld shape and the wall of the adjacent member. This is also the reason why the direction of crack growth is usually mainly along the toe of the weld, especially at the initial stage of the crack.

106

o Strains perpendicular to the weld toe can be measured by simple strain gauges instead of strain gauge rosettes.

o Extrapolation of principal strains or stresses would require extrapolation of all components, which is rather cumbersome.

o The direction of the principal stress would be different for different load cases, prohibiting superposition of load cases.

Factors not covered by hot spot stress method The following factors are not covered in the hot stress method

o The stress fields around the hot spot, e.g., the stress gradient; o Global geometry of the weld, especially the leg length; o The condition of the weld toe, e.g., the toe radius or the influence of weld toe improvement

techniques. Finally, one thing should mention that the hot spot stress method is not suitable for the analysis of fatigue cracks from embedded weld defects or weld roots. In those cases fracture mechanics is often a suitable assessment tool [10].

2.7.2.4 Notch stress method Notch stress method requires knowledge of the stress distribution in the vicinity of the weld, which is usually obtained by means of a FE analysis. The influence of the notch and the notch stress can be obtained from a FE analysis of a small region in the vicinity of the weld using a fine 2D (shell) or 3D (solid) mesh. As a result, additional stress concentration factors can be established, which is to be multiplied with the SCFs of the hot spot stress method [16]. The main advantage compared to nominal and structural hot spot stress method is that the local geometry of the weld seam can be considered, e.g. the effect of the throat thickness, the flank angle and the actual weld toe radius [8]. However, a number of disadvantages to the notch stress method exist [16]:

o The determination of the effect of local stress raisers in a uniform way for inclusion in design guidelines is still a problem;

o The weld shape, especially the leg length, affects not only the local notch stresses but also the hot spot stress, since the weld toe is moved away from the highest stress range;

o To take full advantage of this method, the weld profile must be controlled. Usually, this is very difficult and hence expensive, to the extent that other techniques might be preferred to enhance the fatigue behavior.

2.7.3 Crack propagation method The crack propagation approach has found much application in the fatigue assessment of welded joints mainly due to the following reasons:

107

o The crack initiation phase is normally shorter than the crack propagation phase due to the

relatively sharp notches and weld imperfections; o Unwelded root gaps and weld defects, if present, act as crack starter with a short crack

initiation phase. The calculations of propagation are distinguished from the positions of the cracks, i.e. cracks initiating from weld toes and cracks from unwelded root faces. The factors that affect the propagation are considered in the stress intensity factor. The detail calculation can be found in corresponding literatures, for instance, Radaj [12, 13].

2.8 Calculation Examples According to Eurocode 3

2.8.1 Introduction This example is a fatigue analysis of an existing design to check the fatigue life of critical weld details. Details of the crane are shown Figure 2.49. The crane trolley runs on rails supported by two box girders. The box girders have diaphragms at intervals along this length, and the critical welded details have been identified in the inset sketch and numbered 1 to 5.

Figure 2.49 Details of the crane

108

The crane travels the length of the girders 20 times/day carrying a load of 15 tons (150 kN) including dynamic effects, the dead weight of the trolley being 1 ton (10kN). The analysis is carried out for the case when the trolley returns empty, and then for the case when the trolley returns carrying a load of 7 tons (70kN). The crane operates 200 days per year, i.e. the following cycles are accumulated each year:

o 20 x 200 times a load of 150 kN o 10 x 200 times trolley returns empty o 10 x 200 times trolley returns with a load of 70 kN

The weld descriptions and their categorization for fatigue purpose by Eurocode 3, Part 1.1 are as follows:

Weld EC3 Category Description 1 EC 100 Longitudinal web to bottom flange manual fillet weld, closing welds

of the box section, 4 mm throat 2 EC 80 Transverse manual fillet at bottom edge of diaphragm to web weld 3 EC 80 Transverse manual fillet at top edge of diaphragm to top flange weld 4 EC 112/EC 71 Web to top flange longitudinal manual T-butt weld under crane rail 5 EC 80 Welded stud bolt for fastening rail.

2.8.2 Given values

2.8.3 Stress calculations

2.8.3.1 Calculation of moment inertia and section modulus

bt_flange 500 mm⋅:= tt_flange 10 mm⋅:= hweb 500 mm⋅:= tweb 10 mm⋅:=

bb_flange 500 mm⋅:= tb_flange 10 mm⋅:=

kN 103 N⋅:= W1 150 kN⋅:= W2 70 kN⋅:= Wdead 10 kN⋅:=

Ls 15 m⋅:=

Area of each element

At_f bt_flange tt_flange⋅:= At_f 5 103× mm2=top flange

bottom flange Ab_f bb_flange tb_flange⋅:= Ab_f 5 103× mm2=

web Aw hweb tweb⋅:= Aw 5 103× mm2=

109

The values of the calculation are shown in the following table:

The position of neutral axis can be calculated as: ycgΣAyΣA

, i.e.

The section modulus can be calculated as:

Distance from centroid of each element to bottom flange

yt_f hweb tb_flange+tt_flange

2+:= yt_f 515mm=top flange

bottom flange yb_ftb_flange

2:= yb_f 5 mm=

web yw

hweb2

tb_flange+:= yw 260mm=

Area A y* Ay Ay^26500 515 3347500 17239625005000 5 25000 12500010000 260 2600000 67600000021500 5972500 2400087500

* Distance from bottom flange

Element

WebsTotal

top flangebottom flange

ycg5972500mm3⋅

21500 mm2⋅:= ycg 277.791mm=

The moment of inertia can be calculated as:

Total1 ΣAy2

Total1 2400087500mm4⋅:=

Iweb112

5003⋅ 10⋅ mm4⋅:= Iweb 1.042 108× mm4=

Itop_f112

650⋅ 103⋅ mm4⋅:= Itop_f 5.417 104× mm4=

Ibottom_f112

500⋅ 103⋅ mm4⋅:= Ibottom_f 4.167 104× mm4=

Total2 ΣA( ) ycg2⋅

Total2 21500mm2⋅ ycg2⋅:= Total2 1.659 109× mm4=

I Total1 2 Iweb⋅+ Ibottom_f+ Itop_f+ Total2−:= I 9.494 108× mm4=

ZtopI

500 mm⋅ 10 mm⋅+ 10 mm⋅+ ycg−( ):= Ztop 3.92 106× mm3=

ZbottomI

ycg:= Zbottom 3.418 106× mm3=

110

2.8.3.2 Calculation of moment inertia and section modulus Participation of the crane rail is ignored. The highest bending stresses will be at mid-span when the trolley is also at mid-span. As the trolley passes from one end to the other end of the girders, the bending moment due to living loading will go from zero to maximum and back to zero. The load is assumed to be carried equally between the two girders. Maximum bending moment range per girder:

∆MW1 Wdead+( ) Ls⋅

2 4⋅:= ∆M 300kN m⋅=

The stresses can be calculated using simple bending theory, i.e.

Bending σM y⋅

Ior σ

MZ

This calculation leads to the following results for the stress ranges at different weld details under the full load conditions.

∆σ y5( ) 76.535MPa=y5 h ycg−:=Point 5

∆σ y4( ) 73.375MPa=y4 h tt_flange− ycg−:=Point 4

∆σ y3( ) 73.375MPa=y3 h tt_flange− ycg−:=Point 3

∆σ y2( ) 53.019MPa=y2 ycg tb_flange− 100 mm⋅−:=Point 2

∆σ y1( ) 78.996MPa=y1hweb

2:=Point 1

∆σ y( )∆M y⋅

I:=

h 520mm=h hweb tt_flange+ tb_flange+:=MPa 106 Pa⋅:=

111

2.8.4 Assessment for the trolley carrying the full load of 150 kN

According to Eurocode 3, Ni may be calculated as follows:

Ni ∆σi( ) 5 106⋅

∆σDγMf

γFf ∆σi⋅

3

⋅ γFf ∆σi⋅∆σDγMf

≥if

5 106⋅

∆σDγMf

γFf ∆σi⋅

5

⋅∆σDγMf

γFf ∆σi⋅>∆σLγMf

≥if

∞ γFf ∆σi⋅∆σLγMf

<if

in which,

γFf 1.0:= is the partial safety factor for fatigue loading;

γMf 1.35:= is the partial safety factor for fatigue strength according to Table 9.3.1;

∆σD is the stress range at constant amplitude fatigue limit;

∆σL is the stress range at cut-off limit;

The bending stress ranges are summarized in the following table:

Weld Bending stress range

1 ∆σ y1( ) 78.996MPa=

2 ∆σ y2( ) 53.019MPa=

3 ∆σ y3( ) 73.375MPa=

4 ∆σ y4( ) 73.375MPa=

5 ∆σ y5( ) 76.535MPa=

According to the welded details, the following categories and characteristic values for each welded details are provided according to Eurocode 3

∆σC

100

80

80

112

80

MPa⋅:= ∆σD

74

59

59

83

59

MPa⋅:= ∆σL

40

32

32

45

32

MPa⋅:= ∆σi

∆σ y1( )∆σ y2( )∆σ y3( )∆σ y4( )∆σ y5( )

:=

112

Ni ∆σi ∆σD, ∆σL,( ) 5 106⋅

∆σDγMf

γFf ∆σi⋅

3

⋅ γFf ∆σi⋅∆σDγMf

≥if

5 106⋅

∆σDγMf

γFf ∆σi⋅

5

⋅∆σDγMf

γFf ∆σi⋅>∆σLγMf

≥if

∞ γFf ∆σi⋅∆σLγMf

<if

:=∆σi

78.996

53.019

73.375

73.375

76.535

MPa=

N1 Ni ∆σi0 0,∆σD0 0,

, ∆σL0 0,,

:= 1.67 106×=

N2 Ni ∆σi1 0,∆σD1 0,

, ∆σL1 0,,

:= 2.8 106×=

N3 Ni ∆σi2 0,∆σD2 0,

, ∆σL2 0,,

:= 1.057 106×=

N4 Ni ∆σi3 0,∆σD3 0,

, ∆σL3 0,,

:= 2.941 106×=

N5 Ni ∆σi4 0,∆σD4 0,

, ∆σL4 0,,

:= 9.31 105×=

let

N150

N1

N2

N3

N4

N5

:= N150

1.67 106×

2.8 106×

1.057 106×

2.941 106×

9.31 105×

=

Since the crane travels the length of girders 20 times per day and the crane operates 200 days a year. Therefore,

n150

20 200⋅

20 200⋅

20 200⋅

20 200⋅

20 200⋅

:= n150

4 103×

4 103×

4 103×

4 103×

4 103×

=

113

2.8.5 Assessment for the trolley returning empty The weight of the empty trolley is 10 kN compared to 160 kN for the fully loaded trolley. The bending stress ranges due to the passage of the empty trolley will be 1/16 of those for the full trolley. These ranges are all less than 10 MPa. The cut-off limits for all categories for direct stress ranges, ∆σL, have a minimum value of 14 MPa for category EC 36. Adopting this value the applied stress ranges due to empty return trolley are all less than ∆σL / γMf and can be ignored.

2.8.6 Assessment for the trolley returning carrying load of 70 kN In this case, each detail experiences half the number of cycles of stress ranges at a level of (80/160), i.e. half the full stress ranges calculated above. These cycles have to be assessed separately to find their damage sum n/N per year.

The damage per year for each welded detail can be calculated using niNi

, i.e.

D150n150N150

→

:=

D150

2.395 10 3−×

1.428 10 3−×

3.786 10 3−×

1.36 10 3−×

4.296 10 3−×

=

∆σi∆σi2

:= ∆σi

39.498

26.51

36.687

36.687

38.267

MPa=

N1 Ni ∆σi0 0,∆σD0 0,

, ∆σL0 0,,

:= 2.574 107×=

N2 Ni ∆σi1 0,∆σD1 0,

, ∆σL1 0,,

:= 6.089 107×=

N3 Ni ∆σi2 0,∆σD2 0,

, ∆σL2 0,,

:= 1.199 107×=

N4 Ni ∆σi3 0,∆σD3 0,

, ∆σL3 0,,

:= 6.609 107×=

N5 Ni ∆σi4 0,∆σD4 0,

, ∆σL4 0,,

:= 9.715 106×=

114

2.8.7 Assemblage of the calculated damage and determination of the fatigue life

The contributions of the damage due to the different loading cases for the same detail are added. The sum of the n/N contributions is used in Palmgren-Miner’s Rule, and for design purpose Σ

nN

1

The fatigue life in years is then the reciprocal of the sum Σn/N per year.

The damage per year for each welded detail can be calculated using niNi

, i.e.

D70n70N70

→

:=

D70

7.771 10 5−×

3.285 10 5−×

1.667 10 4−×

3.026 10 5−×

2.059 10 4−×

=

D150

2.395 10 3−×

1.428 10 3−×

3.786 10 3−×

1.36 10 3−×

4.296 10 3−×

= D70

7.771 10 5−×

3.285 10 5−×

1.667 10 4−×

3.026 10 5−×

2.059 10 4−×

=

let

N70

N1

N2

N3

N4

N5

:= N70

2.574 107×

6.089 107×

1.199 107×

6.609 107×

9.715 106×

=

Since the crane travels the length of girders 10 times per day and the crane operates 200 days a year. Therefore,

n70

10 200⋅

10 200⋅

10 200⋅

10 200⋅

10 200⋅

:= n70

2 103×

2 103×

2 103×

2 103×

2 103×

=

115

2.9 References 1. Barsom, J. M. and Rolfe, S. T. (1999). Fatigue and Fracture Control in Structures: Application

of Facture Mechanics. American Society for Testing and Materials (ASTM), U.S.A. 2. Boresi, A. P., Schmidt, R. J. and Sidebottom, O. M. (1993). Advanced Mechanics of Materials,

Fifth Edition, John Wiley and Sons, Inc.. 3. Doerk, O., Fricke, W. and Weissenborn, C. (2003). Comparison of different calculation method

for structural stresses at welded joints. International journal of fatigue, Vol. 25, pp.359-369. 4. Downing, S. D. and Socie, D. F. (1982). Simple Rainflow Counting Algorithms. International

Journal of Fatigue, Vol. 4, No. 1, pp. 31-40. 5. ENV-1991-1-1 (1994). Eurocode 1: Basis of Design and Actions on Structures, Part 1: Basis of

design. 6. ENV 1993-1-1 (1992). Eurocode 3:Design of Steel Structures: Part 1.1, General Rules and Rules

for Buildings. 7. European Steel Design Educational Program (ESDEP). Working Group 12, Fatigue.

Teräsrakenneyhdistys. 8. Fricke, W. (2003). Fatigue of Materials and Structures. Seminar organized by Laboratory for

Mechanics of Materials. http://www.tu-harburg.de/skf/fatiguecourse/.

The total damage can be calculated as:

D D150 D70+:=

D

2.472 10 3−×

1.461 10 3−×

3.953 10 3−×

1.39 10 3−×

4.502 10 3−×

=

The fatigue life for each welded detail can be calculated as:

Fatigue_life1D

→

:=Fatigue_life

404.496

684.364

252.992

719.358

222.108

=

116

9. Hertzberg, R. W. (1996). Deformation and Fracture Mechanics of Engineering Materials, John Wiley and Sons, Inc..

10. Marquis, G. and Kähönen, A. (1995). Fatigue Testing and Analysis using the Hot Spot Method.

VTT publications, No. 239, Espoo, Finland. 11. Mechanics of Materials Laboratory, Course Notes (2003).

http://courses.washington.edu/me354a/notes.html 12. Radaj, D. (1990). Design and Analysis of Fatigue Resistant Welded Structures. Abington

Publishing, Woodhead Publishing Ltd. in Association with The Welding Institute, Cambridge England.

13. Radaj, D. and Sonsino, C. M. (1998). Fatigue Assessment of Welded Joints by Local

Approaches. Abington Publishing, Woodhead Publishing Ltd. in Association with The Welding Institute, Cambridge England.

14. Socie, F. and Marquis, G. B. (2000). Multiaxial Fatigue. Society of Automotive Engineers

(SAE), Inc., Warrendale, Pa.. 15. Suresh, S. (1998). Fatigue of Materials, Second Edition. Cambridge University Press. 16. Von Wingerde, A.M., Packer, J.A. and Wardenier, J. (1995). Criteria for the Fatigue Assessment

of Hollow Structural Section Connections. Journal of Constructional Steel Research, Vol. 35, No.1, pp.71-115.

17. Zahavi, E. and Torbilo, V. (1996). Fatigue Design Life Expectancy of Machine Parts. CRC

Press, A Solomon Press Book.

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