Introduction

We have seen that when the equations of motion of an offshore facility can be modeled as linear, there are sometimes analytical methods available that allow us to calculate the statistical properties of the response. In general, this possibility is lost as soon as nonlinear elements enter the dynamic model. Under such circumstances, the most general and practical option available to obtain statistical information about the response process is to perform a Monte Carlo simulation, which consists of three steps:

1. A sample of load time histories is generated.

2. For each load time history, the equations of motion are solved numerically by a time-stepping method to produce a corresponding sample of response time histories.

3. For the sample of response time histories, statistical techniques are used to derive estimates of the requested statistics. The desired accuracy of these estimates will determine the necessary sample size.

The first step is discussed in the previous section. Here, we discuss the second step. To simplify this discussion, it is expedient to start with the 1DOF case. The equation of motion is then written in the form

subject to the initial conditions

The nonlinear function f(u, u) may contain both nonlinear damping terms and nonlinear restoring force terms. p(t) is an external (deterministic) applied force.

The numerical solution of Eq. (15.1) is achieved by a recursive integration procedure in time.