Introduction

This chapter describes models for loads that have a fundamental variability and their load effects on structures. These load effects are applied in the design and reassessment of structures during operation to ensure adequate structural performance; i.e., serviceability and safety in, say, a period of 20–50 years (ISO 2394, 1998). Serviceability requirements refer to motions, deformations, vibrations, etc., that can hamper the operation, but do not represent a threat to the safety. Safety means the absence of failures and damages, and is ensured by fulfilling requirements to overall stability and ultimate strength, and fatigue life under repetitive loading to avoid ultimate consequences such as fatalities, environmental damage, or property damage. The corresponding so-called limit state criteria are defined by limit states for ultimate failure (ultimate limit state (ULS)) and fatigue failure (fatigue limit state (FLS)), respectively.

ULS criteria for overall stability of bottom-supported structures are based on overturning forces due to wave, current, wind, and stabilizing forces due to permanent and variable payloads. Stability of floating structures is considered in terms of overturning moment by wind only, and uprighting moment due to hydrostatics of the inclined body.

ULS and FLS criteria for relevant components were developed for the relevant failure modes dependent on geometry and load conditions. Permanent and variable payloads, fluid pressure loads, and environmental loads are considered. Environmental loads due to waves, current, wind, and possibly ice and earthquakes are considered.

Introduction

In Chapter 1 it is mentioned that a dynamic analysis can be carried out in two different ways, depending on how the loads are described. One alternative is a deterministic analysis, which requires that the load time history is fully known. The other alternative is a stochastic analysis, where statistical concepts are used to specify the loads. In this chapter we show why (and how) it is expedient and necessary to use statistical and probabilistic methods to describe a number of load types to which structures are subjected.

Typical examples are wind loads on a high-rise building or a suspension bridge, wave loads on an offshore structure or a floating bridge. When the loads on a structure are described in terms of statistical quantities, then the response must also be described and analyzed in terms of the same kind of quantities.

Examples of Stochastic Modeling

Atypical feature of a series of physical phenomena of engineering interest is that each one of them under seemingly identical conditions exhibit quite different behaviour from one recorded experiment to the next. In addition, each recorded time history of the quantity studied is often characterized by being highly irregular.

Figure 5.1 shows three time histories of the water surface elevation measured in a laboratory wave tank. This tank is equipped with a wave maker that can be used to generate irregular waves similar to those observed on the open ocean. The three recorded wave elevations represent the same wave condition, and the three time histories do look quite similar when the overall picture is considered. However, to mimic reality the waves were generated in such a way that when the three time series are subjected to closer scrutiny, it is seen that there are significant differences between them on a local level. Another salient feature of these time histories is that they look irregular or random, as do real waves.

Introduction

The last decade has seen a dramatic increase in the use of Monte Carlo methodsfor solving stochastic engineering problems. There are primarily two reasons forthis increase. First, the computational power available today, even for a laptopcomputer, is formidable and steadily increasing. Second, the versatility ofMonte Carlo methods make them very attractive as a way of obtaining solutions tostochastic problems. The drawback of Monte Carlo methods for a range of problemshas been that the required numerical calculations may take days, weeks or evenmonths to do. But this situation is changing, some numerical problems thatrequired several days of computer time for their solution just a few years agocan now be solved in minutes or hours. This has really opened the door for theuse of Monte Carlo-based methods for solving a wide array of stochasticengineering problems. In this chapter the focus is on adapting Monte Carlomethods for estimation of extreme values of stochastic processes encountered invarious engineering disciplines.

Simulation of Stationary Stochastic Processes

The approach to the simulation of stationary stochastic processes favored in thisbook is the spectral representation method. The main reason for this is itssimplicity and transparency for practical applications. This choice has, infact, already been implemented at several places previously in this book. Takentogether, they constitute a fairly complete description on the level needed hereof how to simulate a stationary process.

Introduction

In Chapter 9 we derive results that make it possible for us to calculate the mean value and the standard deviation of the response of a linear, time-invariant system if we know the load spectrum and the transfer function of the system. Even if such information can be important enough, it is in many cases of limited value. In connection with the design of a structure to withstand wave loads, it would be necessary to estimate the largest response and the associated stresses or strains in the structure over its specified lifetime. Such estimates cannot generally be calculated on the basis of the mean value and the standard deviation, or for that matter, on the basis of the response spectrum.

In this chapter, we provide an introduction to the calculation of extreme responses and response statistics. “Extreme” here means “the largest,” interpreted in a way that follows from the context. The background for this is that collapse of a structure, or part of a structure, is often assumed to take place either by first time exceedance of a capacity limit, or by fatigue, which is caused by accumulated damage due to repeated stress cycles at relatively moderate stress levels. We see that an important key to being able to handle these problems in practice lies in the calculation of the average number of times that the response process crosses a particular level during a given time interval.