# Use of Reliability Methods for the Analysis of Jack-Up Units

Prof. Guy Houlsby and Mark Cassidy

Mark Cassidy DPhil Thesis

This work was sponsored by the Rhodes Trust (support for Mark Cassidy)

In structural reliability theory the failure probability of one component is defined as:

(1)

where is the failure function. defines a "failure" state whilst a "safe" state according to the failure criterion. a set of n random basic physical variables each probabilistically defined. is the multi-variant density function of X. As the purpose in these numerical experiments is to understand a typical Jack-Up unit's non-linear dynamic behaviour and the influence of various analysis models, design "failure" criteria have to be set. Limiting conditions on the behaviour of the Jack-Up, for example maximum hull displacement or bending moment in the leg, can define failure. Therefore a general formula for the failure function g(x) can be written as g(x) = R - S, where R = resistance (or upper limit of "failure") and S = service (or value calculated).

One method of calculating the integral in Equation (1) is by Monte-Carlo simulation with full numerical experiments needed for each simulated vector of inputs Xi. For the majority of complex structural analysis problems this requires a prohibitively large number of complete runs to give an exact result. This is especially true for small probabilities of failure. One of many alternate methods requiring less computational effort is the Response Surface Methods. This technique simplifies the reliability integral by creating a response surface that is of simple mathematical form and can be solved more efficiently. g(x) is replaced by , an "equivalent" function by which the computational procedures can be simplified. Bucher (1990) suggest a generic form of a response surface as

where xi are the set of random variables and the free parameters a, bi, ci, di are constants that need to be evaluated. The response surface is evaluated by systematic variation of the basic variables, producing a simply solved set of linear equations. Monte-Carlo simulations can then be performed on the response surface with probabilities of failure and sensitivities to the basic random variables easily evaluated.

To date, sensitivity to three types of random variables have been evaluated for a single sea state. These are variables influencing environmental load, structural dynamics and Model C parameters. While Cd, the drag loading coefficient, has the highest influence upon the response (measured as hull displacement), parameters effecting the shape of the yield surface in Model C are also highly influential. Finally variability for long term sea conditions will also be investigated.

#### References

Bucher, C.G., Bourgund, U. (1990). A fast and efficient response surface approach for structural reliability problems. Structural safety, 7, 57-66