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)
A plasticity model capable of modelling the behaviour of a circular footing on dense sand, subjected to allpossible combinations of vertical, horizontal and moment loading has been developed at the University of Oxford. The model is known as Model C. Though Model C could be applied to any footing subjectedtocombined loading on dense sand, the primary motivation for its development was for the assessment ofJack-Up units, a type of mobile offshore drilling platform, under extreme loading. Model C is a strain-hardening plasticity model. The yield surface's shape remains constant but varies in size according to the plastic vertical displacement. Within the yield surface elastic behaviour is assumed, whilst a non-associated flow rule describes plastic deformations. The advantage of Model C is that it's strain-hardening plasticity theory can be easily implemented within a conventional structural analysis program, giving realistic soil/structure interaction. Within the offshore industry Jack-Up platform's circular spud-can footings are commonly modelled with pinned footings, or at best linear springs. Model C represents an important advance as it calculates a non-linear stiffness matrix based on measured soil behaviour under combined loads. The above figure shows an example Jack-Up unit with it's three legs idealised as a plane frame with Model C footings making up one component of a realistic representation of the structure, foundations and environmental loading. |
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| The figures to the right show a random sea surface elevation and the corresponding hull displacement this loading causes for three types of footings; pinned, Model C and fixed. Though only deck displacements have been shown, any aspect of structural response could bedetermined. As can be expected the pinned footing gives the largest horizontal deck displacement over the time period and the fixed case the lowest. The pinned case can be seen, for this example, to be rather conservative compared to the Model C footings. It should be highlighted that the Model C footings exhibited plastic displacements in the footings, causing a permanent offset in the displacement of the deck. This yielding of the sand footings occurred during the peak wave. |  |
| The dynamic response of jack-up units subjected to random ocean waves is a complex multi-variant problem. To develop an understanding of the sensitivity of the response to various components, basic reliability theory can be used to standardise the calculations. With sources of uncertaintyprobabilistically described and incorporated within a single calculation method, prediction of exceedance of various failure criteria can be evaluated and compared. Sensitivity of the basic random variables to the global response of a Jack-Up unit can be quantified, as well as the sensitivity of each of the major components of the analysis, i.e. wave loading, foundation models, structural model. In addition, different footing models (pinned, Model C, fixed, linear springs), can be compared giving a realistic quantitative description of the advantages of Model C. |  |
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