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The nite element method is especially well suited to analyzing problems in which properties and geometries vary from place to place, but its combination with stochastic formulations presents some computational problems. Three approaches have been employed. First, the nite element method has been used as a deterministic computational tool with the uncertain properties varying uniformly in each region. This essentially assumes perfect correlation from point to point. Nevertheless, results useful for design studies have been obtained. Secondly, the random eld of properties can be represented in a series of Monte Carlo simulations. In addition to the need for large computational times to obtain adequate results, this approach requires a method for developing the properties of the random eld. The two most popular approaches are the Turning Bands Method (TBM) and Local Average Simulation (LAS). Finally, direct computation of the covariance matrices and solution for the variances by rst-order second-moment methods has been described but not widely used. The geotechnical literature contains relatively few descriptions of stochastic nite element analyses. Most of those deal with ow through porous media. Many of the results published to date tend to con rm the conservatism of conventional deterministic design approaches, in itself a welcome result. A conventional way to deal with uncertainty deterministically is to bracket the results. When this is applied to problems in which spatial correlation is important, this approach would call for analysis with uncorrelated properties and with perfectly correlated properties. However, studies of exit gradients in ow under sheet piles and of differential settlements under strip loads demonstrate that the critical condition occurs when the correlation distance is approximately the same as the signi cant dimension of the problem being studied.

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A variety of methods are available for analyzing engineering risks, but event trees have become the common approach for complex geotechnical systems. There are a number of reasons for this. Event trees provide an intuitive structure within which to organize issues about a particular site or structure. Because event trees typically progress from start to nish in chronological order, they follow a chain of events as it might unfold. They are also versatile in adapting to unique conditions at a particular site. All of this is comforting to practical engineers facing practical problems: Problems are decomposed into tractably small pieces and then brought back together to shed light on the prospect of system failures. The place where event trees have had the most extensive use in geotechnical practice is the area of dam safety. This has been driven by work at the U.S. Bureau of Reclamation, British Columbia Hydro and Power Authority (BC Hydro), and other owners of large portfolios of hydropower and irrigation dams. This chapter discusses event trees primarily around this dam safety application. The application of event trees to another type of system safety, namely the seismic safety of a petrochemical tank farm, is discussed in 22.

Listing 11.6 shows Defender s entry point function. A quick scan of the function reveals one important property the entry point is not a common runtime library initialization routine. Even if you ve never seen a runtime library initialization routine before, you can be pretty sure that it doesn t end with a call to IsDebuggerPresent. While we re on that call, look at how EAX is being XORed against itself as soon as it returns its return value is being ignored! A quick look in http://msdn.microsoft.com shows us that IsDebugger Present should return a Boolean specifying whether a debugger is present or not. XORing EAX right after this API returns means that the call is meaningless. Anyway, let s go back to the top of Listing 11.6 and learn something about Defender, starting with a call to 402EA8. Let s take a look at what it does.

mf85n:00402EA8 sub_402EA8 .h3mf85n:00402EA8 .h3mf85n:00402EA8 var_4 .h3mf85n:00402EA8 .h3mf85n:00402EA8 .h3mf85n:00402EA9 .h3mf85n:00402EAF .h3mf85n:00402EB2 .h3mf85n:00402EB5 .h3mf85n:00402EB8 .h3mf85n:00402EBB .h3mf85n:00402EBD .h3mf85n:00402EC0 .h3mf85n:00402EC1 .h3mf85n:00402EC1 sub_402EA8 proc near = dword ptr -4 push mov mov mov mov mov mov mov pop retn endp ecx eax, large fs:30h [esp+4+var_4], eax eax, [esp+4+var_4] eax, [eax+0Ch] eax, [eax+0Ch] eax, [eax] eax, [eax+18h] ecx

In the context of risk analysis, system failure means the cessation of proper functioning of what is expected of a site or structure as a whole. For example, consider a modern earth embankment dam, simpli ed in Figure 20.1. Since the function of a dam as a whole is to retain water (with some allowable seepage) functional failure occurs when the dam (i.e. the system) ceases to retain water. System means the group of interacting elements that form the complex whole of the site or structure, here, the dam, its foundation, and appurtenant mechanical equipment. The ultimate goal of Event Tree Analysis (ETA) is

The basic mechanics of manoeuvres are always the same when you break them right down. However, they change depending upon the conditions. While your hands and feet may be in the same place, the speed of the manoeuvre may change, or perhaps

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