Civil Engineering July 2022 | Vol 30 No 6

30 July 2022 Civil Engineering the bridges considered in this work fall in the RC3 category in SANS 10160 which carries a β T of 3.5 for a 50-year reference period. A β T of 1.5 is specified for the irreversible SLS, similar to ISO 2394 and EN 1990. Disconnect between reference period and design working life There is a disconnect between the refer- ence period of 50 years in SANS 10160 and the design working life for bridges of 100 years. This is in contrast with ISO 2394 which provides lifetime values. Holicky (2011) shows that the optimum β opt from a cost optimisation exercise is dependent on the cost ratio between the malfunctioning cost C f and the cost per unit of the decision param- eter C 1 , the discount rate q and the design working life n . The discount rate is used to determine the present value of future cash flow. However, it is shown that n has an insignificant effect on β opt between n = 50 and n = 100. The discount rate q has an insignificant influence on β opt over the typical range of 0.01 to 0.05. Target reliability The β T can be set to β opt if the cost ratio is unknown. A conservative value for a lower bound of the design working life, for example 50 years, can be used in these cases for longer design working lives, in this case 100 years. It was argued in the previous section that a β = 3.5 should be used for bridges in South Africa, and according to the arguments presented in this section this value is applied for a design life of 100 years. It is therefore taken as a lifetime value. The β = 1.5 for SLS is adopted for the same reasons for 100 years. DESIGN VALUES AND PROBABILITY DISTRIBUTION The load and resistance distributions are often not normal and a more generic description of the design point is given by Ditlevsen and Madsen (2007) as X d = F x –1 [Φ( αβ )] (5) where X represents either E or R where F x –1 is the inverse CDF of the effect or the resistance. Van der Spuy (2020) showed that traffic load effects for South African data follow a generalised extreme value (GEV) distribution, in specific a Weibull (Type 3) distribution. For the derivation of the load model, daily maxima values of the hogging, sagging and shear were used for span lengths ranging from 5 m to 50 m in 5 m increments. Seven years of data from the Roosboom station on National Route 3 was used in this analysis yielding approximately 2 500 daily maxima for each load effect and each span length. As there are many sub populations of different vehicle types in the WIM data, only the upper tail, consisting of the heaviest vehicles, of a parent distribution contributes significantly to the extrapo- lated value at the return period (Bailey 1996; Zhou, Schmidt & Jacob 2012; Zhou 2013). To determine F x in Equation 6, a censored GEV distribution was fitted to the upper 2√ n of points where n is the number of daily maxima (Castillo 1988). FORMULATION OF THE DESIGN FORMAT FOR PARTIAL FACTORS Partial factors ensure that the charac- teristic load effects are away from the characteristic resistance of a structure by a sufficient safety margin, determined by β . The partial factor format for transient loads is described in fib Bulletin 80 (fib 2016) as γ E = γ Ed , M γ e (6) where γ E is the PF for loading, in this case traffic loading. γ Ed , M is the PF accounting for model uncertainty in the estimation of the load effect from the load model. Model uncertainty is the uncertainty related to imperfect knowledge or idealisations of the mathematical models used or uncertainty related to the choice of probability dis- tribution types for the stochastic variables. γ e is the reliability based PF accounting for variability of the traffic loads and uncertainties relating to the model of variable action. It is custom to model the loading with an extreme value (EV) distribution. The design value of the load effect, also de- noted as E d , by extrapolating to the return period which corresponds to the chosen β value. E c denotes the characteristic value for the same load effect. The reli- ability based partial factor, γ e , is given by Equation 7 (Holicky 2009; fib 2016). Table 1 Reliability based partial factors for hogging Hogging (kNm) Span length (m) E D , SLS E C E D , ULS γ e , SLS γ e , ULS 5 248 250 253 0.99 1.01 10 822 841 877 0.98 1.04 15 1 743 1 779 1 845 0.98 1.04 20 2 436 2 490 2 589 0.98 1.04 25 3 052 3 160 3 379 0.97 1.07 30 3 117 3 178 3 301 0.98 1.04 35 3 840 3 907 4 043 0.98 1.04 40 4 482 4 547 4 680 0.99 1.03 45 5 458 5 557 5 758 0.98 1.04 50 6 602 6 749 7 049 0.98 1.04 Table 2 Reliability based partial factors for sagging Sagging (kNm) Span length (m) M D , SLS M C E D , ULS γ e , SLS γ e , ULS 5 395 401 411 0.99 1.03 10 1 248 1 269 1 307 0.98 1.03 15 2 011 2 034 2 073 0.99 1.02 20 3 233 3 315 3 472 0.98 1.05 25 4 568 4 729 5 055 0.97 1.07 30 5 922 6 121 6 527 0.97 1.07 35 7 563 7 808 8 301 0.97 1.06 40 9 180 9 461 10 021 0.97 1.06 45 11 094 11 459 12 189 0.97 1.06 50 12 665 13 061 13 851 0.97 1.06 Table 3 Reliability based partial factors for shear Shear (kN) Span length (m) V D , SLS V C V D , ULS γ e , SLS γ e , ULS 5 336 336 337 1.00 1.00 10 482 485 489 1.00 1.01 15 561 566 575 0.99 1.02 20 708 722 747 0.98 1.04 25 805 819 844 0.98 1.03 30 875 890 917 0.98 1.03 35 957 976 1013 0.98 1.04 40 1 022 1 045 1 090 0.98 1.04 45 1 098 1 130 1 194 0.97 1.06 50 1 119 1 151 1 211 0.97 1.05