 Probabilistic Analysis
As mentioned in the last Knowledge Base article, conventional
analysis techniques involve the use of safety factors as a way of
accounting for variation in analysis input parameters. This can
often result in overly conservative designs. By contrast,
probabilistic analysis describes a process where the variation in
input parameters can be directly represented in a model.
Two Approaches – Deterministic and Probabilistic
Traditionally, engineering analysis models have specific numerical
input values; material properties have discrete values and nominal
or minimum material dimensions are used. This is deterministic
analysis. However, the validity or conservatism in the results from
such analyses depend on the real-life variability or uncertainty of
the input values. In some situations, accounting for this
variability within analysis can be critical, or at least more
cost-effective, than over-designing products with expensive
materials or manufacturing processes.
In reality, every aspect of an analysis model is subjected to
scatter (in other words, is uncertain in some way). Material
property values, for example, have inherent scatter which itself
differs between different material types and properties - the
scatter of the Young’s Modulus for many engineering materials
could be described as a Gaussian distribution with a standard
deviation of around ±3%. Similarly, component dimensions can
only be reproduced within certain manufacturing tolerances. The
same variation can also apply to finite element model loads.
Features of a Probabilistic Analysis
Although it is possible to account for one distributed variable in
a deterministic analysis, by using some basic statistics, it is
when a number of input variables have a well understood
distribution that probabilistic analysis can be most useful.
Multiple distributed inputs can interact in unpredictable ways, in
some cases to give higher then expected probabilities of bad things
happening, like structural failure. Only probabilistic analysis can
represent this.
Variability is represented in a probabilistic analysis by using
statistical distribution functions, rather than single values, to
describe each input. The probabilities of a critical result
exceeding a certain value are obtainable from a probabilistic
analysis, but so too are the effect variations in the inputs can
have on this result. This can be useful to determine for example,
the effect of loosening tolerances and/or changing material quality
on product returns or warranty claims. Given that many analyses are
already complicated enough, there may be resistance to the idea of
introducing the greater complexity of a probabilistic analysis into
the design cycle.
Commercial justification can be made if the extra cost of
probabilistic analysis, which might allow us to accept a wider
range of scatter of an input variable, is less than the process
costs of trying to reduce this scatter.
Inherent in this justification and in the use of probabilistic
analyses in general, is the concept of acceptable levels of product
failure, however small. This may not always be appropriate for some
products or applications, and may be effectively prohibited if the
applicable standards or regulatory bodies specify failure criteria
based on a deterministic approach. Coming to terms with the concept
of acceptable levels of failure may also tend to deter managers
from accepting this approach, even though it is ultimately more
relevant to safe product design than the unreal absolutism of a
deterministic analysis. The public also (perhaps rightly) may be
concerned that, for example, aircraft components are designed for a
fixed life and should never fail prematurely, (component failure is
not the same as crashing) but the ‘laws of probability’
mean that some inevitably will, even while air travel remains a
very safe mode of transport.
Probability in Deterministic Analyses
Many design standards now appear simplistic or crude, which was a
necessary feature for safe structural design, before the ability to
solve multiple simultaneous equations with computers was widely
available. They nonetheless often have a long history of success,
instilling confidence in their ability to produce safe designs.
However, continually extending their use to more complex structures
may eventually overstretch their remit.
It is often straightforward to account for one (and only one)
distributed variable in a deterministic analysis, by using some
basic statistics - various standards for weld fatigue design in
steel and aluminium (such as BS7608, 5400, 8118) for example, allow
welds to be designed with a specified failure probability, and
these can be used in conjunction with a normal deterministic FEA.
The probability distribution of a single input parameter (in this
case weld strength) can be considered by simply comparing predicted
stresses from a deterministic analysis with a single value of
maximum allowable stress taken from the standard, as a
post-processing step. The allowable stress equates to a known
failure probability for the weld configuration in question. This is
different to a probabilistic analysis, because it considers
probability distributions for only one variable (weld strength),
while load variations, material properties etc are not considered.
An issue often neglected in the approach just described, is
accounting for the number of welds (or more specifically, the
number of critical welds) in a structure. There are obvious
problems with an approach to designing a structure with a hundred
critical welds, allowing each one a 1% probability of failure.
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