## Methods of Reliability Analysis for Marine Structures

##### Master thesis

##### Date

2015##### Metadata

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- Institutt for marin teknikk [2279]

##### Abstract

Analytical solutions of structural reliability problems are often tedious or impossibleto obtain. The task is further complicated when the safety margin,i.e. the relation between variables and response, is implicit. Such is the casefor many practical problems, where structural response is obtained from finiteelement models and/or by semi-analytical equations. This thesis describes apractical approach to solution of such problems by response surface methods,i.e. ways of approximating the analytical safety margin by sampling at discretepoints. If each such sample is computationally demanding, it is necessary tolimit the number of sampling points without introducing unacceptable lackof-fit. Theoretically, once the response surface is given, an accurate approximationof failure probability can then be found by the Crude Monte Carlomethod. However, with low probabilities of failure and/or high dimensionality,this method becomes computationally unfeasible.Two response surface methods are tested for a stiffened panel, where the effectsof distribution types are investigated by comparing between more realisticmodels and corresponding gaussian approximations. The evaluations are performedfor a stiffened panel based on three different limit states; von-Misesstress in the plate along the midspan, axial capacity and a check according relevantclassification guidelines. For von-Mises stress and ultimate capacity limitstates, finite element software ABAQUS is used to sample the safety margin.The third is modelled from Det Norske Veritas recommended practice for bucklingof stiffened panels, corresponding to a check for plate side at midspan. Apurely quadratic response surface as suggested by Bucher and Buorgund [4],along with a hyperplane based on vector projection as suggested by Kim andNa [15] are employed. From the quadratic response surface, probability offailure is evaluated by Crude Monte Carlo, Importance Sampling and a FirstOrder Reliability method (FORM).The response surface obtained by vector projection yields similar results as thequadratic response surface in combination with simulation methods, but withsome deviations. These differences are generally larger for the non-gaussiancase than for the gaussian distributions. From the results, it can not be concludedwhether the differences are method-specific or caused by underlyingcalculations, e.g. variable transformations.
Effects of probability distributions are important, and the results with all variablestaken from the gaussian distribution is highly conservative compared tousing more relevant probability densities.
It is shown how the structural reliability problem can be solved for implicitlimit states in a sensible manner. The procedures shown are efficient from acomputational perspective, and the results from both approaches are equivalent.A difference between the two methods in terms of applicability is noted.The purely quadratic, "Bucher-Buorgund", response surface samples the safetymargin using two iterations with enough sampling points in each to uniquelydetermine the polynomial description, and simulations are used to find themost accurate probability of failure measure. The Vector Projection approachsamples the safety margin by continuously establishing a hyperplane approximationand shifting the sampling points until a convergence criteria is met. Theprobability of failure is evaluated simultaneously by FORM, which is highly efficientcompared to Monte Carlo. This leads to an unknown, potentially fairlylarge, number of safety margin samples but swift probability of failure calculations.If the results are considered equivalent, it can then be recommendedto use the Bucher-Buorgund approach for problems where the safety marginsamples are computationally demanding but the failure probabilities are moderate,whereas the Vector Projection approach is feasible for any probability offailure when the safety margin sampling is fast.The results in terms of failure probabilities are not thought of as directly applicableto design but are deemed valid in the sence of highlighting some importantconsiderations and show the essence of solving similar problems. A validstarting point for further analysis and design purposes would be to extend themodel with respect to boundary conditions, imperfections and an increasednumber of basic variables along with correlation effects.