In practice, many structures among which quay walls are designed according the Dutch national guidelines (CUR). Dutch national practical guidelines (NPR), with guidelines for renovation, is still in development for the building industry. These guidelines follow the Dutch Norms (NEN) and Annexes. The guidelines propose procedures in which newly-built or existing quay walls are designed. This study investigates the effects of past performance on the semi-probabilistic level I method for the purpose of design and evaluation of quay walls. This research is performed with a case study considering a CUR class III quay wall from. The objective of this research is gathering insight into the different aspects of past performance, among which degradation and information about survived years, on the reliability level and corresponding influence factors. Firstly, prior analyses including deterministic validation are performed. The output resulting from characteristic values of the cross-section is validated by means of Blum and analyses with the subgrade reaction method. The computed deterministic output appears to be in accordance with the results from the reference study. Afterwards, prior probabilistic analyses were performed in which the reliability, weight factors and corresponding partial safety factors are reconsidered. Failure mechanism ’yielding of front wall’ is a frequent phenomenon and is assessed in this research. Level II FORM is used for the calculation and level III Importance sampling for the validation. The cross-section is adjusted according the reference case and the results are reasonably in compliance with the results found by GeoDelft for CUR class III. The computed 50 year reliability index β = 4.53 and corresponds well to the target reliability level of β_t = 4.5. Additionally, the situation in which random input variables are correlated and model uncertainty is included, is considered as well. These correlations and model uncertainty are determined based on previous researches among which. The cohesion of clay, internal friction angles, wall friction angles and water levels are correlated. The model uncertainty factor is lognormally distributed and applies as multiplication factor on the maximum bending moment. Explicitly, the latter results in a significant influence on the limit state. Effects given the reference period are considered as well. Large numbers of the dominant load variable q are simulated. The (extreme value) distribution converges to a Gumbel distribution with σ = 0.61. The stochastic distribution of the dominant load is transformed from and to different reference periods: 1, 5, 10, 25, 50 and 100 years. Eventually, for posterior analyses the 50 year reference case is translated to an annual situation in which the annual reliability index and sensitivity factors are derived. Including model uncertainty and cross correlated random variables, one finds a reliability index β = 2.33 as assumption for the posterior analyses. The Equivalent Planes method (EPM) has already been applied in the field of flood defences for reliability updating. This method formulates an failure plane equivalent to two or more combined limit states. In this research, the method has been applied in the temporal context. This means that the Equivalent Planes method is considered in the derivation of the reliability given effect(s) of past performance. The annual reliability index and sensitivity factors are used in this reliability updating method. The autocorrelation represents the correlation of the concerned variable in time. Time-dependent variables including uniform load and water levels assume an auto-correlation of 0, other parameters initially assume an auto-correlation equal to 1. The annual reliability index and sensitivity values are iteratively applied in this method for combining limit states. The equivalent failure plane Z uses a simplified expression in the standard normal space. Eventually a time-dependent reliability curve, as presented by the green line in figure 1, is found. Without model uncertainty, the blue curve is obtained. A higher initial annual reliability index results in a relatively smaller increase of the conditional reliability index. Hence, the effect of past performance decreases when a higher start value of the reliability index is used. Due to the reduced cross-correlation between the water levels on both sides, the time-dependent reliability significantly increases. At last, the time-related effects of quay walls are considered. These effects include:

• The irreducible time-dependent uncertainty related to the model uncertainty factor. Randomness or natural variation is included in the model uncertainty factor. This is performed by considering situations with a reduced autocorrelation ρ(Z_i, Z_j) (0.25, 0.50 and 0.75). The reducible time-dependent uncertainty of load variables including q, w_a and w_p. Knowledge uncertainties (epistemic uncertainties) are reducible in time, meaning autocorrelation approaching 1. The autocorrelations of the considered variables distributions are derived by using transformed random distributions.

• Degradation by corrosion of the stiffest elements in the steel front wall. Corrosion is studied by considering the effects of a log normally distributed wall thickness loss according to corrosion curve 3. This corrosion rate affects the primary element characteristics of the equivalent combined wall. The correlation between the water levels on the active and passive is reconsidered and changed from 0.75 to 0.25.


As follows, the below figures show the annual development of the annual reliability as a function of time t. Notice that the annual reliability index increases as the extent to which the uncertainty is epistemic increases. Further, the reliability converges less rapid to larger value(s) in case of auto-correlation approaching 0. In this research, corrosion is considered as an epistemic uncertainty. Two modelling approaches have been considered: an engineering approach, a second order approach. The engineering approach solely considers a reducing section modulus W, whereas the second order approach is additionally including the second moment of inertia I. Corrosion curve 3 results in both approaches to a flattening of the conditional reliability index as time progresses. In addition, the speed in which the influence of time-independent epistemic uncertainties decreases, is less in case of corrosion. Hence, the involvement of stochastic degradation negatively affects the extent to which the uncertainties are reducible. The updated reliability index is also calculated per reference period. This is performed with probabilistic calculation rules. The corresponding sensitivity values, given survival of previous years (see figure(s) 5), can be used in the semi-probabilistic level I method for derivation of the updated partial safety factors. These factors can be applied in the derivation of the design values per random variable considering a service life time t. Hence, the reliability of a quay wall and the transformed sensitivity coefficients can be updated with the Equivalent Planes method. Incorporation of degradation and other time-related effects is seemingly possible. However, further research with finite element modelling is recommended for verification purposes.