## 3 The Robustness Quantification Framework for Precast Concrete Frames under Seismic Loading
### Uncertainty Characterization and Failure Probability  
Based on the limit state function of each structural element, the failure probability of each element can be obtained using widely adopted methods, e.g., the first-order reliability method or simulation techniques. During the calculation of the failure probability of each element, properties of concrete, steel reinforcement as well as external loads are probabilistically modelled as random variables, as shown in Table 1.  
According to JCSS [36], the permanent load is modelled as a normally distributed variable with its mean value equal to its specified value and the coefficient of variation (c.o.v) equal to 0.05. For the variable load, a reference year of 50 years is adopted here. Consequently, the variable load is modelled by a Gumbel distribution with a mean value of 0.6 times the specified value and a c.o.v value equal to 0.35. The seismic load can be simplified to be the equivalent static load imposed on the centerline of the beams. Based on the findings in [PERSON] and [PERSON] [37], the seismic load can be modelled as a Gumbel variable with the mean equal to the standard value and a coefficient value of 0.30. According to [PERSON] and [PERSON] [38], the concrete strength might have an influence on the reliability of concrete structures and can be modelled as a lognormally distributed variable with a mean value equal to two times the standard deviation beyond the specified strength, and the c.o.v ranging from 0.10 to 0.18. It is well known that the concrete modulus is related to its strength and a relationship between them can be adopted from Eurocode 2 [39]. The concrete modulus is assumed to be lognormally distributed with its mean value related to its specified strength and its c.o.v equal to 0.15. The ultimate concrete compressive strain is assumed to be a lognormal variable with a specified value equal to its mean and the c.o.v equal to 0.15. Regarding the properties of steel reinforcement, the yield strength is assumed to be lognormally distributed. According to JCSS [36], the mean value of the yield strength of steel is the addition of its characteristic value and two times its standard deviation and the standard deviation is fixed as 30 MPa. Note that generally the geometrical properties including the width, depth and length of structural elements are treated as deterministic variables as their variability is small compared to the corresponding specified values. However, the variation of the concrete cover of reinforcing bars is found to be significant [38]. In this case, a bounded beta distribution seems to be the most suitable model to model the uncertainty in this case. More specifically, a lower bound of zero, a mean equal to the nominal value of the concrete cover and an upper bound of three times the mean can be adopted. With statistics of all basic variables at hand, the failure probability of each element can be calculated. It is worth noting that during the process of robustness quantification the accidental situation is adopted for the combination of load actions. The partial factor for actions due to permanent loads and seismic loads is 1.0; the partial factor for actions due to variable loads is 0.5. In the calculation of the equivalent static shear forces due to seismic loads, the severe earthquake situation should be adopted. Note that the variation of the modulus of elasticity of concrete leads to the variation of the stiffness of the frame. However, in case the same value of the modulus of elasticity of concrete is applied to all structural members, the load effect coefficients (\(a_{ij}\) and \(b_{ij}\) in Equation (4)) can be guaranteed to be constant. With the failure probabilities of all structural elements, the failure probability of the corresponding structural system can be calculated according to the beta-unzipping method at level 1. According to the beta-unzipping method at level 1, the critical elements with