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General principles of the use of safety factors indesign and assessment

F.M. Burdekin *

Emeritus Professor, UMIST, Sackville Street, Manchester M60 1QD, United Kingdom

Received 30 August 2005; accepted 30 August 2005Available online 21 August 2006

Abstract

Any structure or component can be made to fail if it is subjected to loadings in excess of its strength. Structural integrityis achieved by ensuring that there is an adequate safety margin or reserve factor between strength and loading effects. Thebasic principles of allowable stress and limit state design methods to avoid failure in structural and pressure vessel com-ponents are summarised. The use of risk as a means of defining adequate safety is introduced where risk is defined as theproduct of probability of failure multiplied by consequences of failure. The concept of acceptable target levels of risk isdiscussed. The use of structural reliability theory to determine estimates of probability of failure and the use of the reli-ability index b are described. The need to consider the effects of uncertainties in loading information, calculation of stres-ses, input data and material properties is emphasised. The way in which the effect of different levels of uncertainty can be

dealt with by use of partial safety factors in limit state design is explained. The need to consider all potential modes offailure, including the unexpected, is emphasised and an outline given of safety factor treatments for crack tip dependentand time dependent modes. The relationship between safety factors appropriate for the design stage and for assessment ofstructural integrity at a later stage is considered. The effects of redundancy and system behaviour on appropriate levels ofsafety factors are discussed. 2006 Elsevier Ltd. All rights reserved.

Keywords: Structural reliability; Limit state design; Safety margins and safety factors

1. Definitions and general considerations

1.1. Safety margins and safety factors

For structural integrity applications safety is assured by ensuring that the resistance to failure is greaterthan the combined effects of the various types of loading which may occur. It is necessary to consider sepa-rately all modes of failure which may occur. For present purposes the resistance effects will be defined bythe term R and the loading effects by the term L. Thus for safety, R L > 0.

1350-6307/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfailanal.2005.08.007

* Address: Formerly School of Mechanical, Aeronautical and Civil Engineering, Department of Civil and Structural Engineering,University of Manchester, P.O. Box 88, Manchester M60 1QD, United Kingdom. Tel.: +44 161 200 4600; fax: +44 161 200 4601.

E-mail address: [email protected].

www.elsevier.com/locate/engfailanal

Engineering Failure Analysis 14 (2007) 420433
mailto:[email protected]:[email protected]


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The safety margin for any particular mode of failure, Z is given by:

Z R L

The overall safety factor for any failure mode, c is given by:

c R=L

In practice both the resistance effects R and the loading effects L will involve a number of variables or mate-rial properties, each of which may be subject to uncertainty or scatter. In addition, in order to compare theload and resistance effects, it is necessary to have equations giving the relationship between them for each po-tential mode of failure which predicts failure when R = L. There will also be uncertainties in this modellingequation.

The margin of safety, or alternatively the safety factor, which is appropriate for a particular applicationmust take into account the following:

The scatter or uncertainty in the variables which form the input data for load and resistance effects. Any uncertainty in the equation used to model failure. The consequences of failure.

The possibility of unknown loadings or mechanisms of failure occurring. The possibility of human error causing unforeseen events.

1.2. Modes of failure

The potential modes of failure can be divided into those which cause failure on the net cross section andthose which cause failure by progressive growth of a crack as follows:

Net section failure Crack tip failure

Plastic collapse FractureBending Fatigue

Buckling Stress corrosionOverall CreepLocalLateral torsional

TorsionShear

Note: the net cross section may be reduced by crack growth.

1.3. Types of loading

The types of loading which may have to be considered include the following:

Dead permanent effects self-weight Live imposed service

Office/factory floor loading Human movement Traffic Pressure Thermal Temperature difference

Environmental wind, wave, tide, snow

Extreme/accident earthquake, impact, failure of other members or component.

F.M. Burdekin / Engineering Failure Analysis 14 (2007) 420433 421


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Note: Probability of occurrence/uncertainty is different for each case and the overall probability must beobtained by combining the probabilities for each mode as independent events.

2. Codes and standards: system failure and effects of redundancy on safety margins

2.1. Allowable stress and limit state design codes

It is necessary to define exactly what constitutes failure. Codes and standards are generally based on one oftwo alternatives in this respect, namely allowable stress or limit state design. In allowable stress codes, theintention is that the stress under the maximum loading conditions should nowhere exceed the material yield orultimate strength divided by an appropriate safety factor (typically 1.5 for yield strength or 2.53.0 for ulti-mate strength). In limit state design, the structure is designed to reach a defined limit state under loading con-ditions derived from the maximum expected multiplied up by a load factor. The usual limit states are eitherthe ultimate state in which the structure actually fails or a serviceability limit state in which the performance ofthe structure is impaired to an unacceptable extent. With limit state design, it is common practice to use par-tial safety factors, where separate factors cL and cR are applied to the load and resistance parts of the failure

equation, respectively. These factors cL and cR may then be broken down further to partial safety factors cL1,cL2, cL3, cL4 and cR1, cR2, cR3, cR4, etc. applied to the individual variables for loading and resistance terms inthe failure equation, respectively. This is discussed further below.

Limit state design codes have been in use in the UK and some European countries for some years. Thesewill be superseded in due course by EuroCodes although individual member states have the right to place theirown values for certain requirements where guidance is given in the EuroCode by boxed numbers. This sit-uation applies to guidance on partial safety factors in EuroCodes. The most relevant of the EuroCodes isEuroCode 3 which is for steel structures and was published in 1993 although it is not yet in widespread use[1]. EuroCode 3 gives conventional guidance on design of steel structures to avoid failure by plastic collapseand by buckling. As far as fracture is concerned the guidance is given in the form of material selection require-ments based on the Charpy V notch impact test for different grades of steel and thicknesses at different min-

imum temperatures. A full presentation on the approach to safety in EuroCodes is given in a separate paper atthis symposium [2].

2.2. Effects of redundancy on safety margins

It is important to distinguish between local failure of a component and failure of a complete system. Whilstfailure of a critical component in a non-redundant structure may cause complete failure of the whole structure,in a redundant structure alternative load paths may be available such that there is a reserve capacity after fail-ure of a single component. Even within a single member there may be redundancies as shown by the exampleof comparing a fixed ended beam with a simply supported beam under uniformly distributed loadingas shownin Fig. 1. In Fig. 1a for the simply supported case, as the magnitude of the uniform load increases, yielding firstoccurs at the mid length position and a further increase of load causes the spread of yield across the completecross section at mid length until collapse occurs by the formation of a plastic hinge there. The load capacitydepends on the material yield strength ry, the span L, and the section moduli, Ze for elastic behaviour and Zpfor plastic behaviour. For the fixed ended case in Fig. 1b, however, yielding first occurs at the ends of the beamwhere the bending moment is highest. This starts to spread across the cross section at the ends as the load isincreased further. The bending moment at the mid length position for fully elastic conditions is only half thatat the ends so that a significant increase in loading is required before yielding starts to occur at midlength.Even when a full plastic hinge has developed at each end of the beam, collapse does not occur until a mech-anism develops with a plastic hinge at mid span as well. In general, overall collapse will not occur until thereare sufficient plastic hinges for a mechanism, although there will be some rotation at some hinges before thefull mechanism develops. For the fixed ended beam, rotation will develop at the ends at a fixed resistancemoment equal to the fully plastic moment of the beam and material (Zp ry) and the shape will change

towards the shape of a simply supported beam. The results for load capacities for these different conditions

422 F.M. Burdekin / Engineering Failure Analysis 14 (2007) 420433


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are summarised in Table 1, and the ratios of loads for first hinge and for collapse to those for first yield are

given in Table 2.The ratio of the fully plastic modulus to the elastic modulus Zp/Ze is a property of the cross section and is

known as the shape factor. For a solid rectangular bar, the shape factor has a value of 1.5, whilst for a typicalstructural I-beam the shape factor is about 1.1. Thus considering the results set out in Tables 1 and 2, it can beseen that a simply supported structural I-beam can carry about 10% additional load after first yield occursbefore failure occurs by plastic collapse. The corresponding fixed ended case, however, can carry nearly

w / unitlength

wL/2

M

L

Plastic hinge at collapse

Elastic

Collapsew =8 Zp y/L2

w =8 Ze y/L2

MP

M0 M

wL/2 wL/2

M

w / unitlength

L

Plastic hinges at collapse

1

23

3. w = 16 Zp y/L2

2. w =12 Zp y/L2

1. w =12 Ze /L2

1. w =12 Ze y/L2

2. w =12 Z /L2

MP

MP

b

a

wL/2

y

yp

0

Fig. 1. Beams with different end conditions under uniformly distributed loads. (a) Simply supported and (b) fixed ends.

Table 1Uniformly distributed load levels for different limiting conditions in simply supported and fixed ended beams

w for first yield w for first hinge w for collapse

Simply supported 8Zery/L2 8Zpry/L

2 8Zpry/L2

Fixed ended 12Zery/L2

12Zpry/L2

16Zpry/L2



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50% additional load after first yield occurs before it actually fails by plastic collapse. The fixed ended beamcase is typical of a redundant situation, where local exceedance of normal limiting stresses does not mean fail-ure of the structure or component.

Allowable stress design would limit permissible loads to those at which the yield strength is first reached.Limit state design, however, is based on designing for complete failure but then applying appropriate safetymargins to the input variables to ensure that the failure condition is not reached in practice. Similar consid-erations apply to the difference between local and global collapse in determining the plastic collapse parameterLr in the fracture assessment diagram treatments using the R6 [3] or BS 7910 [4] approaches. However, in thiscase, the relationship between the amount of plasticity and the increase of crack tip driving force is very

important and specific cases should be assessed by elastic plastic finite element analysis.In principle, failure of a redundant structure should be considered as a system in which the probability offailure of individual elements is assessed sequentially after load redistribution following each failure and theoverall failure probability obtained by combining the results.

3. General background to reliability analysis and partial factors

3.1. Reliability analysis

There are a number of general texts describing the general principles of reliability analysis; amongst themreferences [57]. For general structural assessment purposes it is standard practice to assess safety by a com-parison of load and resistance effects using an established design relationship able to predict failure. When

there are uncertainties in the input variables, or scatter in the materials data, reliability analysis methodscan be employed to determine the probability of failure, i.e. the probability that the load effects will exceedthe resistance effects. This is shown in Fig. 2 where the failure region is in the overlap zone between the loadand resistance distributions. The failure equation is written in terms of load and resistance effects with theinput variables grouped together appropriately. For normally distributed load and resistance parameters, withmeans lL and lR, and standard deviations sL and sR, respectively, the reliability index b is given by:

b lR lLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis

2R s

2L

p : 1

Table 2Ratios of loads at which different limiting conditions are reached for simply supported and fixed ended beams

w for first hinge

w for first yield

w for collapse

w for first yield

Simply supported Zp/Ze Zp/ZeFixed ended Zp/Ze 1.33Zp/Ze

pdf

Load Resistance

SR1

L

Load / strength

R

SL1

R L

2 2R L1 1s s

=

+

Fig. 2. Basic definition of reliability index b.



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One convenient method to estimate probability of failure is the first-order second moment method (FOSM)where the reliability index b is estimated by an iterative numerical procedure. In a multi-dimensional graphinvolving all the variables, the failure equation can be represented by a failure surface as shown in Fig. 3,which represents a two-dimensional cross section of the failure surface in the plane of two of the variables,plotted on a normalised basis. The reliability index b is the shortest distance from the origin to the failure sur-

face and can be determined using this approach by an iterative method, provided the failure surface is con-tinuous with no sharp changes in slope. When all the variables have a normal distribution, there is aunique relationship between the reliability index b and the probability of failure as shown in Fig. 4. Fornon-normal distributions, methods are available to transform them into equivalent normal distributions,although there may be some loss of accuracy in estimating the probability of failure the more the distributionsdeviate from normal.

Fig. 5 shows a case where the load and resistance distributions have the same mean values as shown inFig. 2, but the standard deviations for the distributions in Fig. 5 are much lower than in Fig. 2. Considerationof Eq. (1) shows that if the standard deviations of the load and resistance distributions, sL and sR, are reduced,the value of the reliability index will be increased. It can be seen from Fig. 4 that an increase ofb correspondsto a reduction in the probability of failure and in Fig. 5, it can be seen that this is represented by the overlapregion of the distributions becoming vanishingly small.

0

2

4

6

0 4

X1

X2

Failure surface

Design point

2

Fig. 3. Reliability index b in terms of normalised failure surface.

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 2 5

Reliability Index Beta

Probability

ofFailure

1 3 4 6

Fig. 4. Relationship between b and probability of failure.



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The possible effects of time on the probability of failure are shown schematically in Fig. 6 where it isassumed that the load effects distribution can increase with time whilst the resistance distribution can decrease

with time. The increase in severity of load effects with time might be due for example to crack growth, whilstthe decrease in resistance might be due to deterioration of fracture toughness for example by radiation effects.In the example shown in Fig. 6, the variability has been assumed to remain constant with time, although thisneed not necessarily be the case. In practice, time may well affect the standard deviations of the distributions aswell as the mean values and hence it is essential to have realistic data or modelling to predict variations ofproperties or other effects with time. It can be seen that the reduction in difference between the mean valuesleads directly to a reduction in the reliability index b and hence an increase in probability of failure. To allowfor these effects at the design stage, it is necessary to predict the occurrence of crack growth with time and thedegradation of properties with dose and time, but in principle this can be done.

Characteristic values are often taken to represent upper bounds for distributions of load effects and lowerbounds for distributions of resistance effects as follows:

CL lL

nL

sL;

CR lR

nR

sR; 2

where nL and nR are the number of standard deviations above or below the relevant mean values of the dis-tributions chosen to represent characteristic values. This is shown in Fig. 7.

The design point, which is where the probability of failure is greatest, is given by the following expressions,where Ld and Rd represent the design points on the load and resistance distribution curves, respectively, andare at the same position:

Ld lL sLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2R s

2L

p b sL; Rd lR sRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis

2R s

2L

p b sR: 3

pdf

Load Resistance

SR2

L

Load / strength

R

SL2

R L

2 2

R 2 L 2s s

=

+

Fig. 5. Reliability index for lower standard deviations of load and resistance.

pdf

Load Resistance

SR

L

Load / strength

R

SL

=

+

R L

Rs2 2

SL

SR

sL

Fig. 6. Change in reliability index with time.



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In the case of failure by fracture or plastic collapse the failure equation can be written as follows:

KI Kr Kmat q 0; 8

where KI is the applied stress intensity factor (and represents loading effects), Kr is the permitted value of thefracture ratio in the R6/BS 7910 assessment diagram approach given by the expression below, Kmat is thematerial fracture toughness and q is the plasticity interaction factor for primary and secondary stresses(Kr.Kmat q represents resistance effects).

Kr 1 0:14L2r 0:3 0:7exp 0:66L

6r

; 9

where Lr is the ratio of applied load to yield collapse load for the cracked structure. Note that plastic collapseis one of the failure mechanisms identified in Section 1.2.

It should be emphasised that these general explanations are presented to assist understanding of the generalprinciples of partial safety factors and their relationship to target reliability and variability/uncertainty ofdata. The situation is more complicated if the data are not normally distributed and where the load and resis-tance expressions themselves are functions of multiple variables. In these cases it is much more convenient tomake use of specially written computer software, such as the UMIST or TWI programs.

Calibration studies to give recommended values for partial factors for use with the fracture clauses in BS

7910 and also for the SINTAP programme were carried out at UMIST and TWI and reported in Ref. [8].The general basis for a FOSM fracture mechanics analysis with characteristic values and partial safety factorsis shown in Fig. 10. The calibration studies were based on assuming that the failure equations for a level 3 frac-ture analysis using the mean values of distributions did predict failure, and determining the combinations ofpartial factors necessary to give a required target probability of failure. It should be noted that in these analyses,a decision was made to make the partial factors on stress and on yield strength consistent with those for thestructural design code EuroCode 3. It should also be noted that there is no unique relationship between partialfactors and target reliability across the full range of values of input variables and hence it is necessary to

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4

Coefficient of variation on load effects

Partialfactor Beta0.739

Beta3.09

Beta3.8

Beta4.27

Fig. 8. Relationship between partial factor and COV on load effects with different values of target reliability index b.

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4

Coefficient of variation for resistance effects

Partialfactor

Beta 0.739

Beta 3.09

Beta 3.8

Beta 4.27

Fig. 9. Relationship between partial factor and COV on resistance effects with different values of target reliability index b.



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compromise with values of partial factors, which are sometimes conservative for some input values. Checks onthis aspect showed that the recommended values of partial factors in BS 7910 corresponded to notional prob-abilities of failure, which lay between the target value and up to one order of magnitude safer. Consideration ofthe effects of modelling uncertainties was also reported in Ref. [8] by comparisons of predicted results with those

from series of wide plate tests. It was found that removal of the modelling uncertainty of the failure equationgenerally amounted to a reduction in the generally recommended partial factors of the order of 0.050.1 onstress, and 0.21.0 on fracture toughness but these effects were not included in the recommendations for BS7910. Further studies on these matters are reported in other papers at this symposium [9,10].

It is important to note that all potential failure modes have to be considered and combined. Furthermore,the possibility of unforeseen modes of failure and of human error should be considered. These are difficultareas and are best treated as independent probabilistic events.

4. Risk assessment and acceptability

Two standard dictionary definitions of risk are as follows:

(i) The chance of loss or injury (Chambers Dictionary).(ii) The chance of bad consequences (Oxford Dictionary).

The general public has a basic perception of risk in connection with every day activities. This is usuallymanifest as a perception of injury or death.

In the engineering and scientific fields risk has a more precise definition as follows:

Risk = frequency of occurrence of an adverse event consequences of the event.

This can also be interpreted as follows:

Risk = probability of occurrence of adverse event consequences of the event.

There have been two authoritative reports on risk published by the Royal Society in 1983 [11] and 1992 [12].Fig. 11 shows information from the 1983 Royal Society report where the frequency of events per year, whichcause more deaths than N, is plotted against N itself for a number of common activities. The results for air,sea, and rail travel and for failure of dams are taken from actuarial figures. The figures for accidents involvingnuclear reactors and public assemblies are based on modelling calculations. The difference between the groupof the top four items in Fig. 11 and the two at the bottom is extremely significant. Clearly the general public isprepared to live with the level of risk involved in every day activities such as travel although any major acci-dent leading to a significant number of deaths does cause great concern. Such accidents are usually investi-gated by a public enquiry, which in turn leads to recommendations to try to ensure that similar events are

avoided in future. Engineering activities are expected to work to a completely different level of risk than that

pdf

LoadKI

ResistanceKr . Kmat

SR

L R

SL

Designpoint

CL CRL R

Load / strength

Fig. 10. Partial safety factors for fracture mechanics application.



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which members of the public may be prepared to accept when they have a free choice. This raises the conceptof perception of risk which has been addressed by the Health and Safety Executive (HSE) in reports anddiscussion documents [1315] and the Standing Committee on Structural Safety [16].

It should be noted that on a plot of frequency/probability of occurrence versus consequences using loga-rithmic scales, constant risk is represented by a straight line, so that each category in the figure is a line ofconstant risk. This can be compared with the figures put forward by the HSE as a basis for the ALARP prin-ciple (as low as reasonably practicable) shown in Fig. 12 [13]. This suggests three main regions on a risk dia-gram as follows:

1. Frequency (F) consequences (N) > 0.1 per year, risks unacceptable.2. 101 > F N> 104, ALARP region.3. 104 > F N, risks negligible and acceptable.

In the ALARP region it is required that control measures be taken to drive the residual risk towards theacceptable region. If society expects risk reductions, the residual risk in this region is only tolerable if suchreductions are impracticable or require action grossly disproportionate to the reduction in risk achieved.

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

10 100 1000

DEATHS, N

FREQ.EVE

NTS>N/YR

AIR

PASSENGERS

SHIPPING

RAILPASSENGERS

DAMS

NUCLEAR

REACTORS

PUBLICASSEMBLIES

Fig. 11. Event frequency versus consequences for various types of incident [11].

1E-06

1E-05

1E-04

1E-03

1E-02

1E-01

1 10 100 1000 10000

DEATHS, N

FREQ.EVENTS>

N/YR

LOCAL

TOLERABILITYLOCAL

SCRUTINY

NEGLIGIBLE

Intolerable

ALARP

Region

Negligible

Fig. 12. HSE guidance on tolerability of societal risk.



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The general value adopted in EuroCodes for the target reliability index b is 3.8 for ultimate limit state con-ditions in structures for which failure would have major consequences, corresponding to a failure probabilityof about 7 105 (see Fig. 4). To account for the fact that there is more uncertainty about variable (live) loadsthan for permanent (fixed or dead) loads, partial factors for variable loads are given as 1.5, those for perma-nent loads as 1.35 applied to best estimates (mean values) of loading. Because the probability of accidental

loading is much less than that for normal design loadings, the partial factors for accidental loads are givenas 1.05 for UK applications of EC3 (general EC3 values 1.0). Since these factors have been derived to dealwith the appropriate uncertainties in loading for plastic collapse failure with a target reliability index of 3.8it would seem sensible to adopt the same partial factors for fracture/plastic collapse failure to ensure consis-tency with existing procedures.

The resistance partial factors on material yield strength cM are given as 1.05 for UK applications of EC3 (gen-eral EC3 values 1.1), applied to characteristic values of material strength, i.e. mean minus 2 standard deviations.

It should be noted that account must be taken of both the expected lifetime at risk and the number of sim-ilar structures at risk in deciding an acceptable probability of failure per year.

5. Examples

5.1. Limiting thermal stress

It is required to assess safety margins and safety factors such that thermal stresses must not exceed a pre-scribed limit. For target reliability it is decided that a reliability index of 3.0 is required (i.e. the probability ofapplied thermal stress exceeding the limiting value is 103). Assume that the maximum value of residual stressis not permitted to exceed 355 MPa, with no uncertainty. This gives lR = 355 and sR = 0. The uncertainty inestimating the applied thermal stress is assumed to have a standard deviation of 50 MPa (i.e. a high probabil-ity of being able to estimate the thermal stress occurring within 100 MPa). In both cases these figures areassumed to apply throughout the lifetime of the structure so that adjustments for time considerations arenot required. Noting that the safety margin is defined as lR lL and the safety factor as lR/lL, Eq. (1)can be re-written as:

lR lL bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis

2R s

2L

q: 10

Thus, using the numbers assumed in this example, the safety margin can be calculated as:

Z R L 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi02 502

q 150 MPa: 11

Hence to meet the required level of safety in terms of the safety margin, the best estimate of thermal stressesshould not exceed 205 MPa (355 150).

Alternatively, using the safety factor concept and the same assumed input figures

c R=L 355=205 1:73: 12

The position remains the same in that the best estimate of thermal stresses should not exceed 205 MPa, andthis gives rise to a safety factor on best estimates (mean values) of 1.73.

Note that if characteristic values were used, based on mean + 2sL for load effects and mean 2sR for resis-tance effects, the value of CL would be 305 MPa (205 + 2 50) whilst the value of CR would be 355(355 2 0). The safety factor on characteristic values would then be 1.16 (355/305). This could be separatedinto partial factors of 1.16 on load effects and 1.0 on resistance effects. (This does not follow the arbitrary divi-sion ofaL as 0.7 and aR as 0.8 assumed in Section 3.1 because of the fixed value of limiting maximum stressassumed here.)

5.2. Limiting temperature difference

Avoidance of brittle fracture is sometimes sought by maintaining a safe temperature margin between the

minimum operating temperature and a nominal transition temperature. The uncertainties in these estimates



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are rarely taken into account explicitly (perhaps because of lack of firm data). For illustration purposes, it willbe assumed that for avoidance of fracture in parts of an offshore structure, a probability of failure of 10 3/yearor reliability index value, b, of 3 on a per year basis and the example will be based on determining the safetymargin for a Charpy test energy of 27 J not to lie below minimum service temperature. The minimum ambienttemperature per year for the under water regions of an offshore structure will be assumed to be 5 C with a

standard deviation of 2

C (lL = 5,sL = 2). The variability of the Charpy test results gives a standard devia-tion on the 27 J temperature of say 5 C (estimate assumed to lie within say 10 C of the mean).

Using Eq. (10), the safety margin can be calculated as:

Z R L 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi22 52

q 16:15 C: 13

This means that to achieve the required target safety level, the mean 27 J temperature should not be higherthan 11.15 C (5 16.15). A grade of steel having a toughness of 27 J at 15 C would suit.

It should be noted that safety factors in terms of transition temperatures are meaningless because of theintervention of zero on the temperature scale in C.

6. Conclusions

The principles of formal methods to determine safety margins or safety factors to meet a given target safetyrequirement have been explained demonstrating that safety margins and safety factors depend on the follow-ing factors:

Target reliability requirements which in turn depend on the consequences of failure. Variability or uncertainty in the input data or assumptions. Modelling uncertainties.

In addition the following factors have to be taken into account, but it is more difficult to make quantitativeallowance for them:

The possibility of unknown loadings or mechanisms of failure occurring. The possibility of human error causing unforeseen events.

The way in which safety factors are used in structural codes has been explained. The basis of dividing over-all safety factors into partial factors on the input data for the load and resistance parts of the failure equationhas been described. Two simple examples have been given of determining safety margins/factors for limitingthermal stresses and for Charpy test requirements.

References

[1] prEN 1993 1.1, EuroCode 3, Design of Steel Structures; 1993.

[2] Sedlacek G. Use of safety factors for the design of steel structures according to the Eurocodes, Paper No. 2 TAGSI Symposium; 2003.[3] British Energy Generation Report R/H/R6-Rev 4, Assessment of the integrity of structures containing defects; 2000.[4] BS 7910 British Standards Institution, Guidance on the determination of the significance of defects (Incorporating Amendment 1);

October 2000.[5] Baker MJ. Reliability considerations in structural design a state of the art report, CIRIA Report No. 73, London; 1978.[6] Melchers RE. Structural reliability analysis and prediction. 2nd ed. Chichester: Ellis Horwood; 1999, ISBN 0471983241.[7] Ang AH-S, Tang WH. Probability concepts in engineering planning and design: basic principles, vol. 1. New York: Wiley; 1975.[8] Burdekin FM, Hamour W, Pisarski HG, Muhammed A. Derivation of partial safety factors for BS 7910:1999, I Mech E Conference;

1999.[9] Muhammed A. Background to the derivation of partial safety factors for BS 7910 and API 579, Paper No. 6, TAGSI Symposium;

2003.[10] Wilson R. A comparison of the simplified probabilistic method in R5 with the partial safety factor approach, Paper No. 7, TAGSI

Symposium; 2003.[11] Risk Assessment A Study Group Report, The Royal Society, ISBN 0 85403 208 8; 1983.

[12] Risk Analysis, Perception and Management, Report of a Royal Society Study Group, The Royal Society, ISBN 0 85403 467 6; 1992.



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[13] Health and Safety Executive, Reducing Risks, Protecting People HSEs decision making process, HSE Books, ISBN 0 7176 21510;2001.

[14] Health and Safety Executive, Advisory Committee on Major Hazards Second Report, HMSO, London; 1979.[15] Health and Safety Executive, The tolerability of risk from nuclear power stations, HMSO, London; 1992.[16] Standing Committee on Structural Safety Reports, published on a periodic basis by the Institution of Structural Engineers, London.


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