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418                                                   Part IY Structural Reliability
























                             Figure 23.2   j3 Index Method


                 Example 23.1 is given in Section 23.11 to demonstrate the j3 index method.
                 Analytical Approach
                 As an approximate method to compute the failure probability, the first order reliability method
                 (FORM) is the most widely accepted one. FORM also provides the sensitivity of the failure
                 probability with  respect  to  different  input  parameters, which  is  essential to  optimize the
                 reliability of the structure in design, construction and maintenance.
                 The second order reliability method (SORM) is to  approximate the limit-state surface by a
                 second order  surface fitted at  the  design point.  The most  common approximation is the
                 parabolic surface.
                 FORM and SORM usually give good approximation for small probabilities, but their accuracy
                 and feasibility decrease with increasing non-linearity for the limit state and number of non-
                 normal random variables. In such cases, the failure probability may be estimated by simulation
                 methods.
                 Simulation Approach
                 Instead of using analytical solution, Monte Carlo Simulation (MCS) is a numerical technique
                 based on experiments on a digital computer. The failure probability is interpreted as relative
                 frequency.  MCS  involves randomly sampling a large number of realization of the  failure
                 function, gp), and observe the results, i.e, whether the failure function is less or equal to zero.
                 If the  experiment is repeated N  times and  failure occurs n times, the  failure probability is
                 estimated as PFnhV.
                 There are two main classes of the simulation methods applied for reliability analysis, 1)  the
                 zero-one indicator methods and 2) the semi-analytical conditional expectation methods.
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