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

Prof. Guy Houlsby and Mark Cassidy

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 *X _{i}.* 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* x _{i}* are the set of random variables and
the free parameters

*a, b*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.

_{i}, c_{i}, d_{i}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* C _{d}*,
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