Reliability Analysis Of Vertical Breakwaters Based On Copulas

Breakwaters can effectively reduce the wave effects and prevent the sands for harbors, once damaged, it will produce huge damage to human safety and property. For vertical breakwaters may fail because of sliding and overturning, designers usually calculate its reliability using Approximate Probability Methods such as First Order Second Moment Method and Monte Carlo Method to ensure the safety, however their accuracy is lower than the accuracy of Full Probability Method. With some software such as Matlab, the integration procedure of Full Probability Method can be easily realized, while the most difficult point is how to make the joint density function.Copulas are functions which can be used to make bivariate and multivariate joint density functions. For marginal distribution, Copulas has no strict requirements, the distributions frequently used in ocean engineering such as normal distribution, log-normal distribution and Gumbel distribution can be well adopted. As one kind of Copulas, symmetrical Archimedean Copula has been widely used in financial and hydrological field because of its simple structure and fewer parameters. Symmetrical Archimedean Copula mainly used for bivariate joint distribution while the asymmetric type mainly used for multivariate distributions.Among the variables influenced the reliability of vertical breakwater, horizontal and uplifting wave forces are main loads while the weight of the breakwater and friction are main resistance. Both Gumbel and lognormal distributions are used to fit two wave forces. Results indicate that the lognormal distribution is a better marginal distribution for horizontal force, while the Gumbel distribution fits better for the uplifting force. Three methods are used to make the joint density function of two wave forces: ①three bivariate Archimedean copulas functions including Clayton, Gumbel-Hougaard and Frank Copula constructed with best-fitted marginal distributions, goodness fitting tests are carried out to determine the best bivariate copula model; ②the bivariate logistic Gumbel distribution with Gumbel marginal distributions; ③the bivariate lognormal distribution with lognormal marginal distributions. This paper take three methods to get the failure probability of the breakwater:① think that the weight of the caisson and the friction factor are fixed value, integral the joint density function of two wave forces; ②think the weight and the friction factor are distributed normally, using the fully nested Archimedean Copula to make a four dimensional joint density function and integral it to get the failure probability; ③use the product of the bivariate density function of two wave forces, the density function of weight, and the density function of friction factor as the four dimensional joint density function and integral it.This paper take the vertical breakwater in Qinhuangdao as an example, constructing bivariate and multivariate joint density function with Copula function, calculating the failure probability with direct integration methods. The results are compared with those of bivariate logistic Gumbel distribution, bivariate lognormal distribution and JC methods. The results show that failure probability calculated by direct integration methods are comparable to those of Joint Committee on Structural Safety methods, and they are a little small than the latter. Results with Copula function are smaller than other direct integration methods.