The failure probabilities obtained from a probabilistic analysis should not exceed a specified target failure probability (pf). Alternatively, this desired failure probability may be expressed in
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terms of the target reliability index (βt), as a minimum value, specified for a given reference
period, such that
pf = Ф (- βt) (3.5)
where Ф is the cumulative normal distribution function. Good quality management practices are assumed in the selection of the level of reliability such that gross errors are avoided. The
probability of failure and the associated target reliability, as they are time-dependent, are related to a given reference period, defined as “a certain a priori specified period of time” by
SANS 2394: 2016 (Cl 8.1 page 38). The standard reference period, βt,1 is taken as one year.
The target reliability βt,n for other reference periods, n, may be determined using that for a one-
year reference period using the expression: Φ (βt,n) = [Φ(βt,1)]n
where Φ is the distribution function of a standardized normal distribution.
SANS 2394: 2016 (Cl 8.4 page 39) describes how target reliability should be chosen by taking into account the consequence and the nature of failure, the economic losses, the social inconvenience, effects to the environment, sustainable use of natural resources, and the amount of expense and effort required to reduce the probability of failure. The choice of target reliability is also dependent on the limit state considered as this relates to the severity of the consequence of failure, and thus the related costs of failure, or conversely, of safety. The ultimate limit state is associated with structural collapse and therefore has a higher target reliability, as the costs of human safety as well as other costs such as extensive loss of function and repair costs and replacement of the structure are considered. Serviceability limit states are generally related to performance and some loss of use, inconvenience or repair costs, rather than loss of human life or injury, and therefore tend to be associated with a lower target reliability. SANS 2394 (2016) describes the choice of a suitable target reliability for SLS such that the loss of functionality is limited, and/or the occurrence of damage is within an acceptable economic level, as there is generally no risk of loss of life.
As described by sources such as Rackwitz (2000), Holicky (2009) and SANS 2394 (2016), the target reliability may be determined from a reliability cost optimisation. The aim of such a probabilistic optimisation is to determine a reliability level (the target reliability) at which the total cost of the structure is a minimum. The cost objective function, Ctot, describes the total cost as:
Ctot= Co + C1d +Σ Cf pf (d) (3.6)
where d is the decision parameter, Co is the initial costs independent of d, C1 is the costs of
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probability of failure as a function of d. The decision parameter is generally a vector of multiple decision parameters such as section geometry, material properties and reinforcement area that influence, for instance, the resistance of the structure. The costs Co + C1d are the construction
costs, while Cf pf (d) are the expected failure costs related to each decision parameter.
Considering the ULS LSF of Equation 3.1, Equation 3.5 may be rewritten as: pf (d) = P[g = R – E < 0] = Ф (- βt)
Thus, the target reliability is related to the decision parameter through the LSF. The minimum total cost is the derivative of Equation 3.6. The optimum d may then be determined from the condition:
δ
pf (d) /δd = -
C1/ CfAs defining all costs may be difficult, a cost optimization is performed for a range of C1/ Cf
values to find cost optimal target reliability levels. For SLS, pf (d) may be determined using
Equation 3.2. The SLS cost optimisation model derived by Van Nierop (2017) is an example of this.
Target reliabilities may be differentiated according to the type of structure and the
consequences of failure. The type of structure is described in terms of its design working life, which Holický (2009) defines as “the period for which a structure or part thereof is to be used for its intended purpose with anticipated maintenance but without major repair being necessary”. Design working life as differentiated by types of structure by SANS 10160-1 (2011) is given in Table 3.2
Table 3.2: Notional design working life to SANS 10160-1 (2011)
Design Working Life Category Indicative Design Working Life (Years) Description Of Structures 1 10 Temporary structures.a b
2 25 Replaceable structural parts, for example bearings, agricultural structures and similar structures with low consequences of failure. 3 50 Building structures and other common structures.c
4 100
Building structures designated as essential facilities such as having post-disaster functions (hospitals and communication centres, fire and rescue centres), having high consequences of failured or having
another reason for an extended design working life.
a Structures or parts of structures that can be dismantled with a view to being re-used should not be
considered as temporary.
b Refer to SANS 10160-8 for the assessment of temporary structures during execution.
c The design working life category applies to the reference reliability class referred to in 4.5.2.3. d Consequences of structural failure could be determined in accordance with annex A.
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EN1990-1, MC 2010 and SANS 2394 (2016) define similar categories for the design life of a structure. A design working life of 50 years would be chosen for liquid retaining structures (category 3), which is also the general design life for most common structures.
Consequence classes relate to the cost of failure of, for example, loss of human life, and economic, social and environmental costs. The Joint Committee on Structural Safety (JCSS, 2008) defines consequence classes by considering the risk to life cost as a ratio of the total costs of failure to construction costs. EN1990 (2002) defines reliability classes (or consequence classes) RC1 to RC3 for the ultimate limit state. RC1 has a low consequence, RC2 has a medium consequence and RC3 has a high consequence. RC2 is taken as the reference class. SANS 10160-1 (2011) defines similar reliability classes (RC1 to RC4) and gives a
recommended target reliability for each consequence class. A more detailed description is given by SANS 2394: 2016 whereby five consequence classes are defined. The consequences related to each class include the expected number of fatalities. Examples of typical structures for each class are also given, but suitable target reliabilities relating to the consequence classes are not. As serviceability limit states are not usually associated with loss of human life, with the relative cost depending on performance and use of the structure, a low-order consequence class would be typically applicable. However, the consequence classes are generally derived with the ULS in mind. SANS 2394: 2016 links a quality level to the consequence classes, defining 3 levels of quality. Control organisms are recommended for each quality level. In LRS, the consequence of failure would not generally include loss of life, rather a loss of function. However, this loss of function has more serious societal consequences than usual for SLS as access to clean water may be interrupted for an extended period.
A review of the literature on target reliability confirmed that serviceability has been treated nominally by design standards. Target reliabilities for ULS have been derived through proper optimisation and calibration processes but this is not the case for SLS. The Joint Committee on Structural Safety (JCSS, 2008) made recommendations for target reliability index values (βt) for
irreversible serviceability limit states (such as concrete cracking) based on economic optimisation, given here as Table 3.3, for a one year reference period, along with the corresponding failure probabilities. The reference level of relative cost of safety is Normal.
Table 3.3: Target Reliability Indices for Irreversible SLS – Source: JCSS (2008).
Relative Cost of Safety Measure SLS Target Index, β Probability of Failure, pf
High 1,3 10-1
Normal 1,7 5.10-2
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The JCSS (2008) states that values chosen for target values may vary by about 0,3 from the βt
values of Table 3.3.
Retief and Dunaiski (2009) summarisedtarget reliability levels recommended by ISO 2394: 1998 (SANS 2394: 2004) and EN1990: Eurocode 1: Basis of structural design as shown in Table 3.4, for both the ultimate and the serviceability limit states.
Table 3.4: Target reliability levels (β) according to ISO 2394 (1998) and EN 1990 (Source: Retief and Dunaiski, 2009)
Relative cost of safety measures
ISO 2394 Minimum values for
Consequences of failure
Small Some Moderate Great
High 0 1,5 (A) 2,3 3,1 (B)
Moderate 1,3 2,3 3,1(C) 3,8 (D)
Low 2,3 3,1 3,8(D) 4,3(E)
A for serviceability limit states = 0 for reversible and = 1,5 for irreversible states
B for fatigue limit states = 2,3 to 3,1 depending on the possibility of inspection
For ultimate limit states the safety classes: C = 3,1 D = 3,8 E = 4,3
Reliability Class
EN 1990 Minimum values for
Ultimate LS Fatigue Serviceability LS
Reference
period 1 year 50 years 1 year 50 years 1 year 50 years
RC1 4,2 3,3
RC2 4,7 3,8(F) 1,5 to 3,8 2,9 1,5
RC3 5,2 4,3(G)
F With ISO 2394 clause 4.2(b) moderate safety costs & RC2 consequences , but EN 1990 is more conservative; EN1990 value agrees with ISO 2394 for either low safety cost or great consequences
G The EN1990 value for RC3 agrees with ISO 2394 for low safety cost and great consequences ISO:
2,3 – 3,1
EN: 1,5 – 3,8
Fatigue: ISO 2394 – restricted range;
EN1990 – range from serviceability LS equivalent to ultimate LS
The fib MC 2010 gives guidance on target reliabilities for serviceability as well as for ULS, as reported by Bigaj van Vliet and Vrouwenvelder (2013) and summarised here in Table 3.5. These SLS target reliabilities are common across the design standards of Eurocode, MC 2010 and SANS 10160.
Table 3.5: MC 2010 SLS Target reliabilities (Source: Bigaj van Vliet & Vrouwenvelder, 2013)
Limit State Target Reliability Index, βt Reference Period tR (years)
Serviceability, reversible 0 tSLS Serviceability, irreversible 0,7 200 1,1 100 1,5 50 2,1 15 3,0 1
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SANS 2394 (2016) recommends similar target reliabilities to the earlier SANS 2394 (2004) version for the ULS. It is also stated that for ULS, the target probability of failure may be
increased by a factor of 5 for a higher coefficient variation of the basic variables. Conversely, for variables with a low variability, the target probability may be reduced by a factor of 2. Bigaj van Vliet and Vrouwenvelder (2013) noted that the MC 2010 SLS target reliability indexes
correspond approximately to those of ISO 2394 (SANS 2394) for small consequences of failure and moderate costs of safety measures.
Summarising, a reliability index of 1,5 for a 50-year reference period is generally recommended for irreversible serviceability states such as cracking in buildings, as highlighted in Tables 3.4 and 3.5, although the derivation of this value is unclear. It should be noted that full probabilistic analyses have not been performed to obtain this value, unlike the ULS values given in the standards. In the design of LRS, where serviceability cracking is the limiting condition and has a greater importance due to the higher consequences of a loss of function if cracking results in water leakage, a higher target reliability may be required. Retief & Dunaiski (2009) (Section 5.9, page 51) stated that target reliabilities for serviceability are an indication of appropriate levels of reliability but that ‘further refinement of the scheme of target reliabilities may be feasible’. Van Nierop (2018) performed a cost optimisation to estimate an appropriate target reliability applicable for a typical LRS, the value of which was found to be greater than 2,0 (Reference period of 50 years). This is higher than the general target reliability of 1,5 for a 50 year reference period recommended by design standards such as EN 1990 and ISO 2394. The decision parameter used by Van Nierop (2018) in the optimisation of βt the amount of
reinforcement required. The resulting βt,SLS was compared to that found using the generic
decision parameter used by Rackwitz (2000) for estimating ULS target reliabilities. The
reinforcement area as a decision parameter was found to more cost-efficient than the generic in reducing failure probabilities, and therefore resulted in a higher assessment of the βt,SLS
applicable to LRS. Some simplifications were made in this study but the results suggests that current standards are underestimating the required SLS level of reliability in at least some SLS situations.