Civil Engineering July 2022 | Vol 30 No 6

Civil Engineering July 2022 29 random variables which can each be de- scribed by some probability distribution. Let M be R – E so that failure occurs when E exceeds R and M becomes nega- tive. M is called the safety margin and is shown to the right on Figure 2. If R and E both follow a normal distribution, then M will also be normally distributed. f(r), f(e) and f(m) are the prob- ability density functions of the load, re- sistance and safety margin distributions. The probability of failure, P f , is P f = P ( E > R ) = P ( M < 0) = Φ M (0) (1) From Figure 2 it is clear that the prob- ability of failure, P f , is the area under f(m) below zero on the horizontal axis. The cumulative distribution function (CDF) of M , evaluated at zero, equals the prob- ability of failure P f . M can be transformed to the stan- dardised normal distribution, U , so that u 0 = 0 – μ M σ M = – μ M σ M (2) The term – u 0 is known as the reliability index β and the probability of failure can hence be described by Equation 3 as P f = Φ U ( u 0 ) = Φ U (– β ) (3) In the case of independent normally distributed random variables, β can be calculated explicitly as β = μ M σ M = μ R – μ E √ σ 2 R + σ 2 E (4) β (or – u 0 ) can be interpreted as the number of standard deviations from the mean to zero for the limit state function M . A higher β value implies a smaller probability of failure P f . Without discussing FORM in depth, the direction of the vector indicates the contribution of the load and resistance respectively to the overall uncertainty. The direction of the vector is described by sensitivity factors for the load, α E , and resistance, α R . ISO 2394 allows ap- proximations of the sensitivity factors of α E = –0.7 and α R = 0.8 which were intro- duced by Konig and Hosser (1982). REFERENCE PERIOD, DESIGN LIFE AND TARGET RELIABILITY The design working life is an assumed period of time for which a structure is to be used for its intended purpose without any major repair being necessary. The concept of a reference period is therefore fundamen- tally different from the concept of design working life. This is especially relevant to bridges where the reference period for β and the design life are typically not equal. Design life for bridges ISO 2394, EN 1990, TMH7 and Holicky (2009), specify a design working life of 100 years for large or major bridges. SANS 10160 does not cover bridges, but speci- fies a design working life of 100 years for structures described as: “Building structures designated as essential facilities such as having post-disaster functions (hospitals and communication centres, fire and rescue centres), having high consequences of failure or having another reason for an extended design working life.” Recent failures and loss of life, including the Morandi Bridge collapse and the Florida International University bridge collapse, make it clear that bridge failures can have great consequences for human life and economic activities. A design life of 100 years for South Africa is therefore reasonable. Target reliability and reference period Dunaiski and Retief (2009) motivate a 50 year ULS β T value for South Africa of 3.0 which is implemented in SANS 10160-1 (SABS 2018) for Reliability Class 2 (RC2). RC2 specifies moderate for loss of human life, economic, social or considerable envi- ronmental consequences. RC2 corresponds to CC2 in EN 1990, although EN 1990 specifies a higher value for β T . The South African β T of 3.0 is a significant deviation from the Eurocode value of 3.8, but: Q Q There is agreement with ASCE-7 pro- cedures indicating that the suggested value of the reliability index is similar to international practice Q Q There is rationale for the difference with the Eurocode in that the struc- tures in more developed countries are potentially used for longer and require a longer design life and higher reliability Q Q An upwards adjustment of the reliability level, for example in the Eurocode, is indicative of increasing conservatism Q Q There is no reason to believe that the current reliability implemented in SABS 0160-1989 is no longer sufficient. It was previously motivated that bridge failures may have high consequence for loss of human life and economic activi- ties. It can therefore be concluded that 20 10 0 –10 40 30 20 10 0 0.05 0 0.10 f ( m ) 0.1 0 0.2 f ( r ) f ( e ) μ E μ R E R P f M – u 0 σ M μ M = μ R – μ E Figure 2 Failure zone of the limit state function