Fire and fatigue calcualtion

Document Sample
Fire and fatigue calcualtion Powered By Docstoc
					Helsinki University of Technology Laboratory of Steel Structures Publications 29
Teknillisen korkeakoulun teräsrakennetekniikan laboratorion julkaisuja 29
Espoo 2003                                                                  TKK-TER-29



Wei Lu     Pentti Mäkeläinen

Helsinki University of Technology Laboratory of Steel Structures Publications 29
Teknillisen korkeakoulun teräsrakennetekniikan laboratorion julkaisuja 29
Espoo 2003                                                                  TKK-TER-29


    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
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

 Teknillinen korkeakoulu

ISBN 951-22-6732-2
ISSN 1456-4327
Yleisjäljennös - Painopörssi
Espoo 2003

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

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

Wei Lu, D.Sc. (Tech.)
Espoo, August 2003

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

1.1    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

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,

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

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

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
       θg = 20+345log10(8t+1)                                                                            ( 1.1 )
for external fire curve
       θg = 660(1-0.687e-0.32t-0.313e-3.8t)+20                                                           ( 1.2 )
for hydrocarbon curve
       θg = 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.

                                                                              Hydrocarbon Fire
                                                                              Standard Fire
                        Temperature ( C )

                                             600                                  External Fire



                                                   0        30      60        90           120     150
                                                                     Time (min)

                                                   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].

   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

   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.


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 = [ΣMk.iHuiΨl]/A                                                                         ( 1.4 )

        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

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 (m ) 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
      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 = √ ρ cλ                                                                                   ( 1.10 )

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

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 )
                                                       0.5            2 0.5
In the case of compartment with O = 0.04 m and b = 1160 J/m s K, the parameter curve is almost
exactly the ISO curve. The maximum temperature occurred at t* = t*max where
        t*max = tmax·Γ                                                                                 ( 1.14 )
        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 )
        Γlim = (Olim/b)2/(0.04/1160)2                                                                  ( 1.17 )
        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 )
    *                    *                       -3
for t > 2, where t = t·Γ; tmax = (0.2·10 ·qt.d/O)·Γ and x = 1.0 if tmax > tlim , or x = tlim·Γ /   t*max   if tmax
= tlim .


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

           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

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

                                    Figure 1.6     Zone model [10]

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. 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

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].

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]. 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
     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.

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

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] 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
                                                 2              2 0.5
     εp.θ < ε < εy.θ     fp.θ - c + (b / a)[a - (εy.θ - ε) ]             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)]

               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.


                                                                                         Effective yield strength
                       Reduction factor


                                                                                              Slope of linear
                                                                                              elastic range

                                                         Proportional limit

                                                     0         200       400       600        800       1000        1200

                                                                               Temperature (C)

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

       Table 1.1        Reduction factors for stress-strain relationship of steel at elevated temperatures [7]
        Steel         Reduction factors at temperature relative to the value at 20 °C
         θa           Effective yield strength        Proportional limit      Slope of the linear elastic range
                           ky.θ = fy.θ / fy
                              θ      θ                  kp.θ = fp.θ / fy
                                                           θ      θ                   kE.θ = Ea.θ / Ea
                                                                                         θ      θ
           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

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].
                      % of normal value

                     100                              Effective yield strength
                                                             (at 2% strain)



                                  Elastic modulus

                         0                300               600                  900   1200
                                                           Temperature (°C)

 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.
24 Specific heat

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

       ca = 425 + 7.73·10-1θa - 1.69·10-3·θa2 - 2.22·10-6·θa3             (20°C ≤ θa < 600°C)
       ca = 666 + 13002 / (738 - θa)                                       (600°C ≤ θa < 735°C)
       ca = 545 + 17820 / (θa - s731)                                     (735°C ≤ θa < 900°C)
       ca = 650                                                           (900°C ≤ θa ≤ 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.
                         Specific Heat



                                    ca=600 J/kg°K
                                    (EC3 simple calculation


                                0         200      400        600       800    1000      1200
                                                           Temperature (°C)

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

The thermal conductivity of steel may be defined as follows [7]

       λa = 54-3.33·10-2·θa                     (20°C ≤ θa < 800°C)
       λa = 27.3                                (800°C ≤ θa ≤ 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.



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





                           0        200   400         600       800       1000       1200
                                                   Temperature (°C)

 Figure 1.16     Eurocode 3 representations of the variation of thermal conductivity of steel with temperature
                                                     [17] 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 = 1.2×10-5θa+0.4×10-8θa2-2.416×10-4            (20°C ≤ θa < 750°C)
       ∆l /l = 1.1×10-2                                     (750°C ≤ θa ≤ 860°C)
       ∆l /l = 2×10-5θa-6.2×10-3                            (860°C < θa ≤ 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(θa-20)                                                                             ( 1.23 )








                                           0   200   400       600    800    1000   1200

                                                           Temperature (C)

      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

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

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.

                                   Downstand beam                                        Slimflor beam

                                                     Shelf-angle beam

                                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    Thermal conductivity      Specific heat      Equilibrium moisture
                                      ρi               λi                    ci                   content
                                   (kg/m3)         (W/mK)                 (J/kgK)                   %
 Sprayed mineral fiber               300              0.12                  1200                          1
 Perlite or vermiculite              350              0.12                  1200                         15
 High density perlite or             550              0.12                  1200                         15
 vermiculite plaster
 Fiber-silicate or fiber-            600              0.15                  1200                         3
 calcium silicate
 Gypsum plaster                      800              0.20                  1700                         20
 Compressed fiber boards
 Mineral wool, fiber silicate        150              0.20                  1200                         2

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 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]
      h =λ dθ/dx                                                                               ( 1.24 )
where h 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]
      δ 2θ /δ 2x=(1/α)/(δ θ/δ t)                                                               ( 1.25 )
where t is time (s) and α = λ/ρc is thermal diffusivity (m /s). These equations can be solved using
analytical, graphical or numerical methods.


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]
      h =αc ∆θ                                                                                 ( 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:

                            Table 1.3          Coefficient of heat transfer by convection

                                                                                       αc (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 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]:
      h =Φ ε σ [(θe + 273) 4 – (θr +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. 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]:

       δ         δθ δ            δθ δ            δθ D                   dθ
          λ (θ )    +     λ (θ )
                                      +
                                        δ z  λ (θ ) δ z  + hnet = ρ c (θ ) dt                 ( 1.28 )
      δ x        δ x δ y         δ y                 

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 h is the net heat flux that is due to
convection and radiation, i.e.
      hnet = hnet .c + hnet .r                                                                   ( 1.29 )

where, the net convective heat flux can be determined by
      hnet .c = αc (θg - θm)                                                                     ( 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:
      hnet .r =Φ ε σ [(θr + 273) 4 – (θm +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.


                                                                                  Reinforcement Mesh

                                                               fillet weld

                                            Steel Beam                                    Rannila 120
                                                                                          Steel Deck


                                                                    fillet weld



                         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] 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]

  heat entering = heat to raise temperature
        hnet Am ∆t = ρa ca V ∆ θa                                                                                   ( 1.32 )

and the temperature increase of steel can be calculated as
       ∆ θa.t = (Am/V)(1/ρa ca) hnet ∆t                                                                             ( 1.33 )

where hnet is the heat flow per unit area (W/m2) and is given by:
        hnet = αc (θg - θm) + Φ ε σ [(θr + 273) 4 – (θm +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 or 600 for a simple calculation model          J/kgK
  αc             Coefficient of heat transfer    see Table 1.3                                              W/m2K
                 by convection
  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     can be calculated as ε=εm·εf                               -----
                 surface                         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               °C
                                                 temperature-time curves
  θr             Effective         radiation     θr = θg                                                    °C
                 temperature of the fire
  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               Fire temperature                 Temperature change in steel
                               θa                           θg                                     ∆θa
  t1 = ∆t         Initial steel temperature   Fire temperature half way          Calculating    from      increasemental
                  θa0                         through time step (at ∆t / 2)      equation with θa and θg from this row

  t2 = t1 + ∆t    θa + ∆θa                    Fire temperature half way          Calculating    from      increasemental
                  θa Temperature        for   through time step (at t1+∆t / 2)   equation with θa and θg from this row
                  previous row

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

                                                                               Tube exposed to fire on
                                                                               all sides

                                                                               Am / V = 1 / t

 Hollow section or welded box section with
 uniform thickness exposed to fire on all sides
 if t << b, 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. 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

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:
                 λ p / d p Ap  1 
      ∆θ a.t =                                             (          )
                                       ( g.t − θ a.t )∆t − e φ / 10 − 1 ∆θ g.t but ∆θ a .t ≥ 0
                  ca ρ a V  1 + φ / 3 
                                         θ                                                                                ( 1.35 )
                                      

in which the relative heat storage in the protection material is given by the term
           cpρ p         Ap
      φ=            dp                                                                                                    ( 1.36 )
           ca ρ a        V

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

                                                                                   Contour encasement       Steel perimeter /
                                                                                   of uniform thickness     Steel section area

                                                                                   Hollow encasement of     2(b+h) / Steel
                                                                                   uniform thickness        section area

                                                                                   Contour encasement
                                                                                                            (Steel perimeter-b)
                                                                                   of uniform thickness
                                                                                                            / Steel section area
                                                                                   to fire on three sides

                                                                                   Hollow encasement of
                                                                                   uniform thickness        (2h+b) / Steel
                                                                                   exposed to fire on       section area
                                                                                   three sides

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

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. 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

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


        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. 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

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. 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]

                          Table 1.8        Minimum Periods (minutes) for elements of structure [5]
                In the following building types                                        According to the regulations of
 Building        n       h   H        X   L       b   x(*)   S       B   CH      D        F           I       L    NL    FIN     SP    UK
Industrial      1    0       10   20      100   50    2      Y   0       0       0       30*2   0/60      0        0     0       -     0*1
hall                                                                                            (7)
                                                             N   0       (1)*3   (1)     30*2   30/90     0-60     0     0       -     0*1
Commercial      1    0       4    500     80    80    4      Y   0       0       0       0H     60/90     30       0     0       90    0*1
center and                                                                                      (7)
shop                                                         N   (1)     (1)*3   (1)     30 V   90/120    (3)      0     30      90    0*1
Dancing         2    5       9    1000    60    30    4      Y   0       0       (2)     60     (8)(9)    30       0     60(4)   90    30
                                                             N   0       30      90      60     60        30       0     60(5)   90    60

School          4    12      16   300     60    20    4      Y   60(6)   0       (2)     60     (8)       90       60    60(4)   60    60
                                                                         30*3                   (10)
                                                             N   60(6)   60      90      60     60        90       60    60(5)   60    60

Small rise      4    10      13   50      50    30    2      Y   60(6)   0       (2)     60     (8)       90       60    60(4)   60    30
office                                                                   30*3                   (9)
building                                                     N   60(6)   (1)*3   90      60     60        90       60    60(5)   60    60

Hotel           6    16      20   60      50    30    2      Y   60(6)   30      (2)     60     (8)       90       60    60(4)   90    60
                                                                         60*3                   (11)
                                                             N   60(6)   60      90      60     60        90       60    60(5)   90    60

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

Medium          11   33      37   50      50    30    2      Y   120     60      (2)     120    (8)       90       60    120     120   120
rise office                                                              90*3                   (9)                      (4)
building                                                     N   120     90      90      120    90        120      90    120     120   (3)
High     rise   31   90      93   100     50    50    2      Y   120     90      90      120    (8)       120      90    120     120   120
office                                                                                          (9)                      (4)
building                                                     N   120     90(3)   (3)     120    120       (3)      90    120     120   (3)

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

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

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 = γ G·G k + γ Q1·Q k1 + Σγ 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 = γ GA·G k + ψ 1.1·Q k1 + Σψ 2.i·Q ki + Ad                                            ( 1.39 )

        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
        Ed                      Design value of effects of actions from normal design
      E fi.d                    Constant design value in fire exposure
       Gk                       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 = γ G·G k + γ Q1·Q k1                                                                    ( 1.40 )
in normal condition, and
      E fi.d = γ GA·G k + ψ 1.1·Q k1                                                               ( 1.41 )
in fire situation. The values of combination factors are given in Table 1.9.

                                        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                              0.7   0.5   0.3
               Category B: office areas                                             0.7   0.5   0.3
               Category C: congregation areas                                       0.7   0.7   0.6
               Category D: shopping areas                                           0.7   0.7   0.6
               Category E: storage areas                                            1.0   0.9   0.8
               Category F: traffic area, vehicle weight = 30 kN                     0.7   0.7   0.6
               Category G: traffic area, 30 kN < vehicle weight = 160 kN            0.7   0.5   0.3
               Category H: roofs                                                     0     0     0
               Snow loads on building (see EN 1991-1-3)*
               Finland, Iceland, Norway, Sweden                                     0.7   0.5   0.2
               Remainder of CEN member States, for sites located at altitude
               H > 1000 m a.s.l                                                     0.7   0.5   0.2
               H = 1000 m a.s.                                                      0.5   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]
      η 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]
      η 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
               γ GA G k + ψ 1.1Q k .1
      η fi =                                                                                          ( 1.44 )
                γ G G k + γ Q.1Q k .1

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

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.

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]:
                            1             
      θcr = 39.19 ln           3 , 833
                                        − 1 + 482                                                 ( 1.45 )
                      0 ,9674 µ0
                                          

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]:

             E fi.d
      µ0 =                                                                                          ( 1.46 )
             R fi.d.0

A simple conservative version, which can be used for tension members and restrained beams, where
lateral-torsional buckling is not a possibility, is [17]
                 γ       
      µ 0 = η fi 
                 γ       
                                                                                                   ( 1.47 )
                  M1     

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].

          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-    Testing and
                    time curves                  calculations
 Parametric fire    Equivalent fire exposure     Testing and              Testing and
                                                 calculations             calculations
 Natural fire       Homogenous                   Calculation              Calculation             For research only
                    temperature distribution
                    Zone, field models           Calculation              Calculation             For research only
                                                 No interaction           A reasonable            All interaction of the
                                                 between neighboring      interaction between     global structure
           Assessment of methods
                                                 elements is              neighboring elements    system are considered
                                                 considered               is considered

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.θ.Rd = A·(fy·ky.θ)/γ = (A·fy/γ M.1)·ky.θ·(γM.1/γ = ky.θ·NRd·(γM.1/γ
          θ               θ                           θ                    θ                     ( 1.49 )

where ky.θ is the reduction factor for yield strength of steel ate temperature θ reached at time t as
shown in Section 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.θ.Rd = ΣAi·(fy·ky.θi)/γ
             θ                 θ                                                                  ( 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.θ.Rd = ky.θ·MRd·(γM.1/γ
             θ         θ                                                                          ( 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.θ.Rd / (κ1·κ2)
                        θ                                                                         ( 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

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.θ.max·VRd·(γM1/γ·(1/(κ1·κ2))
                    θ                                                                            ( 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]: = (χ·Wpl.y·ky.θ.com·fy/γ
                                         θ                                                       ( 1.54 )

   χ = 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 χ is determined as in ambient-temperature design, except that the normalized
slenderness used is adapted to the high-temperature steel properties [7]:
                            k y.θ.com
        λ LT.θ.com = λ LT                                                                        ( 1.55 )
                            k E.θ.com


 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, λLT .θ .com , 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

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]: = A·ky.θ.max·fy·(χfi/1.2)·(1/γ
                         θ                                                                         ( 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:
                             k y.θ. max
           λ θ. max = λ                                                                            ( 1.57 )
                             k E.θ. max

                                                        l fi=0,7L

                   system                                                   l fi=0,5L

                                    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

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

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:

     Steel grade:           S275
     Lightweight concrete   C40 (slab)

                         Figure 1.30       A simple framed braced against horizontal sway

Characteristic floor loadings
                                         kN                                               kN
  Permanent:                  Gk := 1.9⋅           Primary variable:         Qk.1 := 3.8⋅
                                          2                                                2
                                         m                                                m

 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 := 7850⋅
  Partial safety factor
  --resistance of Class 1, 2 or 3 cross-section:                        γ M0 := 1.1
  --resistance of Class 4 cross-section:                                γ M1 := 1.1
  --resistance of member buckling:                                      γ M1 := 1.1

Partial safety factor for the fire situation shall be taken as:
-- for thermal properties of steel:                                       γ := 1.0
-- for mechanical properties of steel:                                    γ := 1.0

1.10.2 Design loads and load distribution in the frame

Design loads on beams
 Partial safety factor for permanent actions under unfavorable effects              γ G := 1.35

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

 Permanent actions:          Gd := γ G⋅ Gk⋅ L                Gd = 15.39
 Variable actions:           Qd := γ Q.1⋅ Qk.1⋅ L            Qd = 34.2

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,
            (        )
  RA := ⋅ Gd + Qd ⋅ LAB
                               RA = 123.975kN

  RB := RA                     RB = 123.975kN

  NBE := RA + RB               NBE = 247.95kN

          (Gd + Qd)⋅2⋅ LAB + NBE
  RD :=

                             RD = 371.925kN

  RF := RD                   RF = 371.925kN

Loading on column GH

     RGH := 2⋅ RD + 2⋅ RB      RGH = 991.8kN

1.10.3 Fire resistance and protection of a tension member BE

Ambient temperature design
Design loading              NSd := NBE                NSd := 247.95⋅ kN

Try IPE 100. The dimension of IPE 100 are as follows:
   h BE := 100⋅ mm        b BE := 55⋅ mm          t fBE := 5.7⋅ mm     t wBE := 4.1⋅ mm A BE := 1032⋅ mm    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 A ⋅
               γ M0

      A := A BE
 The design plastic resistance is               Npl.Rd := A ⋅
                                                                γ M0

  Design :=       "OK!!" if Npl.Rd ≥ NSd
                                                                                      Design = "OK!!"
                  "NOT OK!!!" otherwise

Fire resistance of tension member

Design loading in fire

According to equation (1.44)
   γ GA := 1.0   ( Permanent loads: accidental design situations )
   ψ 1.1 := 0.5 ( Combination factor: variable loads, office buildings)

The reduction factor

              γ GA⋅ Gk + ψ 1.1⋅ Qk.1                                        −3
    η fi :=                                            η fi = 459.7701× 10
               γ G⋅ Gk + γ Q.1⋅ Qk.1

 The design load in fire is

     Nfi.d := η fi⋅ NSd                                   Nfi.d = 114kN

 Design resistance in fire at ambient temperature

According to equation (1.49), the design resistance is:
  NRd := Npl.Rd                                        NRd = 258kN

 At ambient temperature,                θ := 20
       ky.20 := 1.0           (from Table 1.1)
 The design resistance at ambient tempartaure

                                       γ M1
       Nfi.20.Rd := ky.20⋅ NRd⋅                              Nfi.20.Rd = 283.8kN

Critical temperature

The degree of utilization at t = 0, i.e. θ = 20 °C is:
                Nfi.d                                              −3
     µ 0 :=                                µ 0 = 401.6913× 10

According to equation (1.45), the critical temperature is:

     θ cr.t := 39.19⋅ ln                        − 1 + 482
                                                                          θ cr.t = 619.1392
                             0.9674µ 0
                                                    

Fire resistance time

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:
     b := b BE              b = 55mm              t f := t fBE          t f = 5.7mm    t w := t wBE   t w = 4.1mm
                                                                                                                    3   2
     h := h BE              h = 100mm             R := RBE              R = 7 mm       A := A BE      A = 1.032 × 10 mm

      steel_perimeter := 2⋅ b + 2⋅ t f + b − 2⋅ R − t w +
                                                (                )      h − t − R  ⋅ 2 + π⋅ R
                                                                              f                     steel_perimeter = 399.7823mm
                                                                       2                     

      fire_exposure := "all-round"

 The perimeter of the cross section is

       A m :=      steel_perimeter − b if fire_exposure               "3-sided"
                   steel_perimeter if fire_exposure              "all-round"                    A m = 399.7823mm

                                                          Am                  1
 The section factor is               V := A                      = 387.3859
                                                            V                 m

In a simple calculation model, the specific heat may be considered as independence of the steel
temperature. According to section
      c a := 600⋅
                    kg⋅ K
According to Table 1.3, the convective heat transfer coefficient for standard or external fire curves
      α c := 25⋅
                   m ⋅K
When calculating net radiation heat flux per unit of surface area, the following parameters are given:

                                                                    −8        W
  Stephan-Boltzmann constant                  σ := 5.67⋅ 10              ⋅
                                                                              2    4
                                                                             m ⋅K
  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

Fire protection of tension member

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

    steel_perimeter := 2⋅ b + 2⋅ tf + b − 2⋅ R − t w +
                                     (                  )          h − t − R  ⋅ 2 + π⋅ R
                                                                                               steel_perimeter = 399.7823mm
                                                                  2 f                   
The section factor can be calculated as:

    A p_gyp :=     if fire_exposure   "3-sided"
                      steel_perimeter − b if protection_type                      "contour"
                      2⋅ h + b if protection_type             "box"
                   if fire_exposure   "all-round"
                      steel_perimeter if protection_type                     "contour"
                                                                                                     A p := A p_gyp
                      2⋅ h + 2⋅ b if protection_type           "box"

 Appropriate area of fire protection material per unit length is                           A p = 310mm
                                                               Ap                      1
 Section factor:     V := A                                           = 300.3876
                                                               V                       m
 Try the following gypsum boarding:

     Density                           ρ p := 800⋅        ,

     Specific heat                     cp := 1700⋅             ,
                                                     kg⋅ K

     Thermal conductivity                  λp := 0.2⋅
                                                      m⋅ K

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.

1.10.4 Fire resistance and protection of steel beam AB

Ambient temperature design

Applying bending moment:                M Sd :=
                                                             (           )
                                                             ⋅ Gd + Qd ⋅ LAB

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

     h AB := 300⋅ mm       b AB := 150⋅ mm                            tfAB := 10.7⋅ mm            twAB := 7.1⋅ mm
                                                    2                                     3   3
     RAB := 15⋅ mm         A AB := 5381⋅ mm                           W AB := 557⋅ 10 ⋅ mm                          3   3
                                                                                                  W pl.AB := 628⋅ 10 ⋅ mm

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" if             ≤ 72⋅ ε

                       "Class 2" if 72⋅ ε <                 ≤ 83⋅ ε

                       "Class 3" if 83⋅ ε <                 ≤ 124⋅ ε

                       "Class 4" otherwise

              235⋅ MPa
where ε :=
                    ,                                            d            (
                                                                        h − 2⋅ tf + R     )
when flange is subjected to compression:

                                            c                                         b
 sectType_flange         "Class 1" if            ≤ 10⋅ ε                where c
                                            tf                                        2

                         "Class 2" if 10⋅ ε <                ≤ 11⋅ ε

                         "Class 3" if 11⋅ ε <                ≤ 15⋅ ε

                         "Class 4" otherwise

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

      t f := t fAB     tf = 10.7mm             t w := t wAB                t w = 7.1mm   R := RAB           R = 15mm

                                           3         2                                            3     3
      A := A AB             A = 5.381 × 10 mm                W yy := W AB W yy = 557 × 10 mm                  W pl.yy := W pl.AB

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

      sectType_web :=           "Class 1" if               ≤ 72⋅ ε

                                                                  d                         sectType_web = "Class 1"
                                "Class 2" if 72⋅ ε <                   ≤ 83⋅ ε

                                "Class 3" if 83⋅ ε <                   ≤ 124⋅ ε

                                "Class 4" otherwise

The Class of the flange is          c :=                  c = 75mm
    sectType_flange :=          "Class 1" if               ≤ 10⋅ ε

                                                                  c                       sectType_flange = "Class 1"
                                "Class 2" if 10⋅ ε <                   ≤ 11⋅ ε

                                "Class 3" if 11⋅ ε <                   ≤ 15⋅ ε

                                "Class 4" otherwise
 Therefore, the Class of the cross-section is

    sectType_beam :=           sectType_web if sectType_web ≥ sectType_flange
                               sectType_flange otherwise
                                                                                              sectType_beam = "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,, bending about one axis, the design moment resistance of a
cross-section without holes for fasteners may be determined as:

              W pl.yy ⋅ fy
 M c.Rd                         if ( sectType_beam                "Class 1" ) ∨ ( sectType_beam       "Class 2" )
                     γ M0

              W yy ⋅ fy
                             if ( sectType_beam             "Class 3" )
                 γ M0

Given     key1 := sectType_beam               "Class 1"
          key2 := sectType_beam               "Class 2"
          key3 := sectType_beam               "Class 3"


                      W pl.yy ⋅ fy
      M c.Rd :=                        if key1       1 ∨ key2     1               M c.Rd = 157kN⋅ m
                         γ M0

                      W yy ⋅ fy
                                   if key3       1
                       γ M0

     moment_Resistance :=            "OK!!" if M Sd ≤ M c.Rd                          moment_Resistance = "OK!!"

                                     "NOT OK!!!" otherwise

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:

                   fy 
             A v⋅ 
                   3
                  γ M0

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

  A v := A − 2⋅ b ⋅ tf + tw + 2⋅ R ⋅ tf   )
Applied shear:
                       VSd := ⋅ Gd + Qd ⋅ LAB    )                           VSd = 123.975kN

The shear area:                                             (            )
                                     A v := A − 2⋅ b ⋅ t f + t w + 2⋅ R ⋅ t f
                                                                                              A v = 2.568 × 10 mm

                                                                              fy 
                                                                      Av⋅ 
The design plastic shear resistance:                      Vpl.Rd :=
                                                                              3
                                                                         γ M0                  Vpl.Rd = 370.6545kN

     shear_Resistance :=          "OK!!" if VSd ≤ Vpl.Rd
                                                                                          shear_Resistance = "OK!!"
                                  "NOT OK!!!" otherwise

Fire resistance of minor beam
Design loading in fire

The design loading in fire:                  M fi.d := η fi⋅ M Sd             M fi.d = 71.25kN⋅ m

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:
                                      γ M1
    M fi.20.Rd:= ky.20⋅ M c.Rd⋅                             M fi.20.Rd = 172.7kN⋅ m

For beam supporting concrete slab:                              κ 1 := 0.7       κ 2 := 1.0

Therefore, the fire resistance at ambinent temperature

                    M fi.20.Rd
     M fi.0.Rd :=                                           M fi.0.Rd = 246.7143kN⋅ m
                        κ 1⋅ κ 2

  Critical temperature
                                               M fi.d
  ultilizaiton factor               µ 0 :=
                                             M fi.0.Rd

  The critical temperature is

       θ cr.t := 39.19⋅ ln                       − 1  + 482
                                                                             θ cr.t = 669.5458
                            0.9674µ 0
                                                       
  Fire resistance time

  steel_perimeter := 2⋅ b + 2⋅ t f + b − 2⋅ R − t w +
                                        (                  )     h − t − R  ⋅ 2 + π⋅ R
                                                                 2 f                                                  3
                                                                                                steel_perimeter = 1.16 × 10 mm
                                                                                      

 fire_exposure := "3-sided"

The perimeter of the cross section is

     A m :=    steel_perimeter − b if fire_exposure                  "3-sided"                             3
                                                                                              A m = 1.01 × 10 mm
               steel_perimeter if fire_exposure                 "all-round"

The section factor is              V := A                                                              = 187.706

Using spreadsheet calculation, the time for steel to reach its critical temperature is 15 minutes 23

 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 +
                                     (              )     h − t − R  ⋅ 2 + π⋅ R
                                                          2 f                     steel_perimeter = 1.16 × 103 mm
                                                                               
 The section factor can be calculated as:

     A p_gyp :=   if fire_exposure     "3-sided"
                       steel_perimeter − b if protection_type        "contour"
                       2⋅ h + b if protection_type       "box"
                  if fire_exposure     "all-round"
                       steel_perimeter if protection_type         "contour"
                                                                                            A p := A p_gyp
                       2⋅ h + 2⋅ b if protection_type     "box"

                                                                                       Ap                1
     Section factor:              V := A                                                    = 139.3793
                                                                                       V                 m

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

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.

2.1    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 + N p                                                                                 ( 2.1 )

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

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].

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.
             ∆σ = σmax - σmin                                                                  ( 2.2 )

     o mean stress, which is the algebraic mean of σmax and σmin in the cycle, i.e.
             σm = (σmax + σmin) / 2                                                            ( 2.3 )
     o stress amplitude, which is half the stress range in a cycle, i.e.
             σa = (σmax - σmin) / 2                                                            ( 2.4 )
     o stress ratio, which represents the relative magnitude of the minimum and maximum stress in
       a cycle, i.e.
             R = σmin / σmax                                                                   ( 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

                      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.

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]

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]. 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·( σ / σR ) -m                                                                       ( 2.6 )

based on the stress level or
       N = NR·( ∆σ / ∆σR ) -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). 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].

                 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]

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. 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.
       σeq = ( 0 + 3τ2 )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′
       τ f = ( 1 / √3 )· σf′                                                                      ( 2.9 )

                                  Figure 2.9     Effects of loading modes

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:
      σ f = kl ·ks · kd · kr · σf′ = kl ·ks · kd · kr · ( σu / 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 = σmax / σn                                                                               ( 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:

     Figure 2.10    Illustration of S-N curves for                 Figure 2.11       Definition of stress-concentration
         notched and un-notched fatigue tests                                             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]

      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,
      Kf = 1 + (Kt – 1) / (1 + √(ρ / 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 + √(ρ / 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:
      σf-notched = σf-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

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]

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. 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:
     σn = N/A + (M/I)·z,       τn = 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. 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.
     σmax = Kf·σn                                                                              ( 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.

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].

 Figure 2.15    Stress strain curves under static         Figure 2.16      Cyclic stress-strain curves [17]
                 loading [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.

                                     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:
      εa = σa / E + (σa / 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:

      εe = σa / E = σf′ (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:
      εp = εf′ (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:
      εa = εe + εp = σf′ (2Nf)b / E + εf′ (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
      σf′ = σu + 50 ksi                                                                             ( 2.24 )

where σu is the tensile strength. The fatigue ductility coefficient can be approximated by:
      εf′ = εf = 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 = εf′ (Eεf′/σf′)1/(b-c)                                                                   ( 2.28 )

Further, the cyclic hardening coefficient can be expressed in terms of σf′ and εf′
      K′ = σf′ / (εf′)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:
      σ = Kt σ n                                                                                    ( 2.31 )

where Kt is the stress concentration factor and σn is the nominal stress. The maximum strain can be
expressed as:
      ε = Kt ε n                                                                                    ( 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.
      σ = Kσ σn                                                                                     ( 2.33 )

where Kσ is the stress concentration factor and

      ε = Kε ε n                                                                                                 ( 2.34 )

where Kε is the strain concentration factor. These two factors are interdependent and can be related
using Neuber rule [8] as follows:
      Kt2 = Kσ·Kε                                                                                                ( 2.35 )
The Neuber rule can be rewritten according to the stress, strain and concentration factors, i.e.
      σ·ε = (Kt·σn)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                      Figure 2.21      Determination of maximum strain
                       parameters                                                     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]:
      εa = (σf′ - σm) (2Nf)b / E + εf′ [(σf′ - σm) / σf′]1/n′′ (2Nf)c                                            ( 2.37 )
according to Morrow as [17]:
      εa = (σf′ - σm) (2Nf)b / E + εf′ (2Nf)c                                                                    ( 2.38 )

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         Figure 2.23     Typical striations around an
initiation (A), stable crack propagation (D) and final                     inclusions [8]
                    fracture (D) [8]

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

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.

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
       K = f·σ·(π·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(πa/2b))1/2                                                                    ( 2.40 )

  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

       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               Figure 2.30      Concerning mode I: (a) center
                     a crack                                       cracked plate in tension (b) edge cracked plate in
                                                                  tension (c) double-edge cracked plate in tension (d)
                                                                               cracked beam in bending

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 = fKtσ(πa)1/2                                                                             ( 2.45 )
In the remaining domain, the stress intensity factor is equal to
       K = fσ[π(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 (σmax - σmin) (πa)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

                               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              1           1      
        Nf =                    m        m − m                                                 ( 2.49 )
                       m                      −1
                                                  a f 2 −1   
               C ⋅ ∆σ ( − 1) ⋅ π 2   ⋅f  ai 2               
The initial crack length can be computed from the equation
        a = 1/π [∆K/(a(σmax-σmin))]2                                                             ( 2.50 )
        ∆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.

                                                                 
                         1                1            1         
       ∆N =                                    −                                              ( 2.53 )
                        m                 m −1            m
                                                             −1   
              C ⋅ ∆σ m ( − 1) ⋅ π 2      a 2
                                      ⋅f         (a + ∆a) 2      
The total cycles to failure can be calculated as
       Nf = Σ (∆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

     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

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

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
       i =1    i
                   =1                                                                           ( 2.56 )

        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

       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

     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].

                  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           Figure 2.40     Typical stress distribution at weld
                    welds                                                       toe

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

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

   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.

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. 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

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. 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 σ1 [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].

                               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 =

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 ∆σR                                                                         ( 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     Stress range at
                                                         amplitude fatigue limit       cut-off limit
        Normal stress range
             ∆σC            N ≤ 105        N > 105                 ∆σD                     ∆σL
           (N/mm )          (m = 3 )       (m = 5 )              (N/mm2)                 (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
             ∆τC                   N < 108                         ------                  ∆τL
           (N/mm )                 (m = 5 )                                              (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:

                 γFf ∆σ ≤ 26 / γMf N/mm2                                                                        ( 2.59 )
   o The total number of stress cycles, N, satisfies:
                 N ≤ 2 x 106 [(36/γMf) / (γFf ∆σE.2)]3                                                          ( 2.60 )

   o For a detail for which a constant amplitude fatigue limit ∆σD is specified, the largest stress
     range ∆σ satisfies:
                 γFf ∆σ ≤ ∆σ / γMf                                                                              ( 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 γMf [6]

                 Inspection and access                               “Fail-safe”                  Non “fail-safe”
                                                                    components                     components
 Periodic inspection and maintenance.                                   1.00                           1.25
 Accessible joint details.
 Periodic inspection and maintenance. Poor                               1.15                          1.35

Otherwise, the fatigue assessment criterion for constant amplitude loading is:
        γFf ∆σ = ∆σR / γMf                                                                                      ( 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 )
        Dd = Σ (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;

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

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 [[(∆σD / γMf) / (γFf ∆σi)]3                                             ( 2.65 )

   o if ∆σD / γMf > γFf ∆σi ≥ ∆σL / γMf
            Ni = 5 x 106 [[(∆σD / γMf) / (γFf ∆σi)]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 [[(∆τC / γMf) / (γFf ∆τi)]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

   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.

                                                              Table 2.3       Numerical values for fatigue
                                                                   strength curves for hollow sections

                                                               Detail         log a for       Stress range
                                                              category         N<108           at cut-off
                                                                ∆σC                          limit (N=108)
                                                              (N/mm2)                             ∆σL
                                                                90             16.051              41
                                                                71             15.551              32
                                                                56             15.051              26
                                                                50             14.801              23
                                                                45             14.551              20
                                                                36             14.051              16

 Figure 2.46    Fatigue strength curves for joints
                in lattice girders

 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
    Gap joints            K type                      1.5                  1.0                 1.3
                          N type                      1.5                  1.8                 1.4
   Overlap joints         K type                      1.5                  1.0                 1.2
                          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
    Gap joints            K type                      1.5                  1.0                 1.5
                          N type                      1.5                  2.2                 1.6
   Overlap joints         K type                      1.5                  1.0                 1.3
                          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.

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:
      ∆σR.t = ∆σR (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]. 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

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

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.

  Figure 2.47    Method of extrapolation to the          Figure 2.48       Fatigue strength curves for hot
                  weld toe                                             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.

   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

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]. 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:

   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

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

  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

 b t_flange := 500⋅ mm      t t_flange := 10⋅ mm          h web := 500⋅ mm   tweb := 10⋅ mm

 b b_flange := 500⋅ mm      tb_flange := 10⋅ mm

 kN := 10 ⋅ N               W 1 := 150⋅ kN            W 2 := 70⋅ kN     W dead := 10⋅ kN

 Ls := 15⋅ m

2.8.3 Stress calculations Calculation of moment inertia and section modulus

 Area of each element
                                                                                     3       2
 top flange       A t_f := bt_flange ⋅ tt_flange                      A t_f = 5 × 10 mm

 bottom flange                                                                       3       2
                    A b_f := b b_flange ⋅ tb_flange                   A b_f = 5 × 10 mm

 web                                                                             3       2
                    A w := h web⋅ tweb                                A w = 5 × 10 mm

Distance from centroid of each element to bottom flange
top flange        y t_f := h web + tb_flange +                               y t_f = 515mm
bottom flange      y b_f :=                                                  y b_f = 5 mm

web                          hweb
                   y w :=             + tb_flange                            y w = 260mm

The values of the calculation are shown in the following table:
       Element        Area A                  y*      Ay       Ay^2
      top flange       6500                   515   3347500 1723962500
    bottom flange      5000                    5     25000    125000
        Webs           10000                  260   2600000 676000000
        Total          21500                        5972500 2400087500
 * Distance from bottom flange

The position of neutral axis can be calculated as: ycg                           , i.e.
        y cg :=
                              2                        y cg = 277.791mm
                  21500⋅ mm
The moment of inertia can be calculated as:

                        2                                                                    4
       Total 1    ΣAy                                                            ⋅
                                                             Total 1 := 2400087500mm

              1     3        4                                               8           4
       Iweb := ⋅ 500 ⋅ 10⋅ mm                                Iweb = 1.042 × 10 mm
                1         3    4                                                 4           4
       Itop_f := ⋅ 650⋅ 10 ⋅ mm                              Itop_f = 5.417 × 10 mm
                   1         3    4                                                      4           4
       Ibottom_f := ⋅ 500⋅ 10 ⋅ mm                           Ibottom_f = 4.167 × 10 mm

       Total 2    (ΣA )⋅ y cg 2
                              2           2                                          9           4
       Total 2 := 21500⋅ mm ⋅ y cg                            Total 2 = 1.659 × 10 mm

       I := Total 1 + 2⋅ Iweb + Ibottom_f + Itop_f − Total 2                                             8       4
                                                                                         I = 9.494 × 10 mm

The section modulus can be calculated as:

        Ztop :=                                                                                              6       3
                  (500⋅ mm + 10⋅ mm + 10⋅ mm − ycg )                                     Ztop = 3.92 × 10 mm

        Zbottom :=                                                                                                   6   3
                      y cg                                                               Zbottom = 3.418 × 10 mm
110 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:

               (W 1 + W dead )⋅ Ls
      ∆M :=                        ∆M = 300kN⋅ m
                  2⋅ 4
The stresses can be calculated using simple bending theory, i.e.
                                 M⋅ y                           M
      Bending             σ                    or          σ
                                    I                           Z

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

 MPa := 10 ⋅ Pa                         h := h web + tt_flange + tb_flange
                                                                               h = 520mm

              ∆M ⋅ y
 ∆σ( y ) :=

                               h web
 Point 1              y 1 :=
                                                                       ( )
                                                                    ∆σ y 1 = 78.996MPa

 Point 2              y 2 := y cg − tb_flange − 100⋅ mm                ( )
                                                                    ∆σ y 2 = 53.019MPa

 Point 3              y 3 := h − tt_flange − y cg                      ( )
                                                                    ∆σ y 3 = 73.375MPa

 Point 4              y 4 := h − tt_flange − y cg                      ( )
                                                                    ∆σ y 4 = 73.375MPa

 Point 5              y 5 := h − y cg                                  ( )
                                                                    ∆σ y 5 = 76.535MPa

The bending stress ranges are summarized in the following table:
      Weld             Bending stress range
        1                 ( )
                        ∆σ y 1 = 78.996MPa

        2               ∆σ(y 2) = 53.019MPa

        3               ∆σ(y 3) = 73.375MPa

        4               ∆σ(y 4) = 73.375MPa

        5               ∆σ(y 5) = 76.535MPa

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

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

                              ∆σD  
                                      
                         6   Mf  
            ( )
       Ni ∆σi       5⋅ 10 ⋅             if γ Ff ⋅ ∆σi ≥
                             γ Ff⋅ ∆σi                  γ Mf

                              ∆σD  
                                      
                         6   Mf  
                                             ∆σD                 ∆σL
                    5⋅ 10 ⋅             if      > γ Ff ⋅ ∆σi ≥
                             γ Ff⋅ ∆σi     γ Mf                γ Mf

                    ∞ if γ Ff ⋅ ∆σi <
                                        γ Mf

in which,

        γ Ff := 1.0 is the partial safety factor for fatigue loading;
        γ Mf := 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;

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

                                                                                       ∆σ(y 1) 
         100                     74                       40                    
         80                       59                        32                      ∆σ(y 2) 
                                                                                         
 ∆σC :=  80  ⋅ MPa       ∆σD :=  59  ⋅ MPa        ∆σL :=  32  ⋅ MPa      ∆σi :=  ∆σ(y 3) 
         112                     83                       45                     ∆σ y 
                                                                                    ( 4) 
         80                      59                       32                     ∆σ(y )
                                                                                            5 

        78.996                                            ∆σD  
                                                                   
         53.019                                             γ
                                                       6   Mf  
                          Ni(∆σi, ∆σD , ∆σL) := 5⋅ 10 ⋅              if γ Ff ⋅ ∆σi ≥
 ∆σi =  73.375 MPa                                                  
        73.375                                           γ Ff⋅ ∆σi                  γ Mf

                                                                          5
        76.535                                                 ∆σD  
                                                                         
                                                            6   γ Mf       ∆σD                 ∆σL
                                                       5⋅ 10 ⋅             if γ    > γ Ff ⋅ ∆σi ≥
                                                                γ Ff⋅ ∆σi       Mf                γ Mf

                                                       ∞ if γ Ff ⋅ ∆σi <
                                                                           γ Mf

 N1 := Ni ∆σi , ∆σD , ∆σL  = 1.67 × 10
             0, 0  0, 0  0, 0       
 N2 := Ni ∆σi , ∆σD , ∆σL  = 2.8 × 10
             1, 0  1, 0  1, 0       
 N3 := Ni ∆σi , ∆σD , ∆σL  = 1.057 × 10
             2, 0  2, 0  2, 0       
 N4 := Ni ∆σi , ∆σD , ∆σL  = 2.941 × 10
             3, 0  3, 0  3, 0       
 N5 := Ni ∆σi , ∆σD , ∆σL  = 9.31 × 10
             4, 0  4, 0  4, 0       

let                                                                     1.67 × 106 
                      N1 
                                                                                 6 
                      N2 
                                                                        2.8 × 10 
                      
             N150 :=  N3                                      N150 =  1.057 × 106 
                                                                                    
                     N 
                                                                                  6
                      4                                                2.941 × 10
                                                                                    
                      N5
                                                                      9.31 × 105
                                                                                    
Since the crane travels the length of girders 20 times per day and the crane operates 200 days a
year. Therefore,

                                                     4 × 103 
               20⋅ 200                                   3
               20⋅ 200                              4 × 10 
                       
      n150 :=  20⋅ 200                    n 150 =  4 × 103 
               20⋅ 200                                     
                                                    4 × 103 
               20⋅ 200                                     
                                                     4 × 103 

The damage per year for each welded detail can be calculated using        , i.e.

                   →                         2.395 × 10− 3 
                 n150                        
         D150 :=                                         −3
                                              1.428 × 10 
                                        D150 =  3.786 × 10− 3 
                                                              
                                                1.36 × 10− 3 
                                                              
                                                4.296 × 10− 3 

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 / γ Mf 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.

                                           39.498
         ∆σi                                     
∆σi :=                              ∆σi =  36.687 MPa
                                           36.687
                                           38.267
 N1 := Ni ∆σi , ∆σD , ∆σL  = 2.574 × 10
             0, 0  0, 0  0, 0  
 N2 := Ni ∆σi , ∆σD , ∆σL  = 6.089 × 10
             1, 0  1, 0  1, 0  
 N3 := Ni ∆σi , ∆σD , ∆σL  = 1.199 × 10
             2, 0  2, 0  2, 0  

 N4 := Ni ∆σi , ∆σD , ∆σL  = 6.609 × 10
             3, 0  3, 0  3, 0  
 N5 := Ni ∆σi , ∆σD , ∆σL  = 9.715 × 10
             4, 0  4, 0  4, 0  


                                                       2.574 ×
                        N1 
                                                                          7
                        N2 
                                                       6.089 ×       10 
                        
                N70 :=  N3                    N70 =  1.199 ×         7
                                                                      10 
                       N 
                                                                       7
                        4                             6.609 ×       10
                                                                        
                        N5
                                                     9.715 ×            6
                                                                     10       
Since the crane travels the length of girders 10 times per day and the crane operates 200 days a
year. Therefore,
                                                           10
                 10⋅ 200                                   3
                 10⋅ 200                           2×     10 
                         
         n70 :=  10⋅ 200                    n70 =  2 ×     3
                                                            10 
                 10⋅ 200                          
                                                   2×     10
                 10⋅ 200                                    
                                                           10
 The damage per year for each welded detail can be calculated using                     , i.e.
                    →                           7.771 ×       −5
                  n 70                                     10
        D70 :=                                                  −5
                                                 3.285 ×   10       
                                            D70 =  1.667 ×   10
                                                                      
                                                   3.026 ×   10
                                                                      
                                                   2.059 ×
                                                             10
2.8.7 Assemblage of the calculated damage and determination of the fatigue

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         1
The fatigue life in years is then the reciprocal of the sum Σn/N per year.
                         2.395 × 10− 3 
                                                   7.771 ×
                                                                10

                                   −3                              −5
                         1.428 × 10               3.285 ×     10        
                 D150 =  3.786 × 10− 3     D70 =  1.667 ×     10
                                                                        
                         1.36 × 10− 3             3.026 ×     10
                                                                        
                         4.296 × 10− 3             2.059 ×          −4
                                                              10        

The total damage can be calculated as:

                                 2.472 × 10− 3 
                                           −3
      D := D150 + D70            1.461 × 10 
                            D =  3.953 × 10− 3 
                                               
                                 1.39 × 10− 3 
                                               
                                 4.502 × 10− 3 
 The fatigue life for each welded detail can be calculated as:

                      →                            404.496
                     1                           684.364
      Fatigue_life :=                                    
                      D          Fatigue_life =  252.992
                                                   719.358
                                                   222.108

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

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.

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

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).

12. Radaj, D. (1990). Design and Analysis of Fatigue Resistant Welded Structures. Abington
    Publishing, Woodhead Publishing Ltd. in Association with The Welding Institute, Cambridge

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.

TKK-TER-15   Hara, R., Kaitila, O., Kupari, K., Outinen, J., Perttola, H.
             Seminar on Steel Structures: Design of Cold-Formed Steel Structures, 2000.
TKK-TER-16   Lu, W.
             Neural Network Model for Distortional Buckling Behaviour of Cold-Formed Steel Compression
             Members ,2000.
TKK-TER-17   Kaitila, O., Kesti, J., Mäkeläinen, P.
             Rosette-Joints and Steel Trusses, Research Report and Design Recommendations, 2001.
TKK-TER-18   Ma, Z.
             Fire Safety Design of Composite Slim Floor Structures, 2000.
TKK-TER-19   Kesti, J.
             Local and Distortional buckling of Perforated Steel Wall Studs, 2000.
TKK-TER-20   Malaska, M.
             Behaviour of a Semi-Continuous Beam-Column Connection for Composite Slim Floors, 2000.
TKK-TER-21   Tenhunen, O., Lehtinen, T., Lintula, K., Lehtovaara, J., Vuolio, A., Uuttu, S., Alinikula, T., Kesti, J.,
             Viljanen, M., Söderlund, J., Halonen, L., Mäkeläinen, P.
             Metalli-lasirakenteet kaksoisjulkisivuissa, Esitutkimus, 2001.
TKK-TER-22   Vuolio, A.
             Kaksoisjulkisivujärjestelmien rakennetekniikka, 2001.
TKK-TER-23   Outinen, J., Kaitila, O., Mäkeläinen, P.
             High-Temperature Testing of Structural Steel and Modelling of Structures at High Temperatures,
TKK-TER-24   Kaitila, O.
             Finite Element Modelling of Cold-Formed Steel Members at High Temperatures, 2002.
TKK-TER-25   Lu, W.
             Optimum Design of Cold-Formed Steel Purlins using Genetic Algorithms, 2003
TKK-TER-26   Mäkeläinen, P., Tenhunen, O., Vuolio, A., Lintula, K., Viljanen, M., Bergman, J.,
             Hänninen, J., Alinikula, T., Palmi, P.
             Kaksoisjulkisivun suunnitteluohjeet, 2003
TKK-TER-27   Vuolio, A.
             Structural Behaviour of Glass Structures in Facades, 2003
TKK-TER-28   Tenhunen, O.
             Metalli-lasirakenteisen kaksoisjulkisivun materiaalien soveltamiskriteerit, 2003

ISBN 951-22-6732-2
ISSN 1456-4327

Shared By: