The mechanical effects of short-circuit currents in - Montefiore

5.1.1. Deterministic Methods
In deterministic methods, structural design is carried
out according to the logical approach given below:
I. Replace the always complex reality of structures
with modelling, which can be approached by
calculation,
II. Define the various actions affecting the structure, as
well as their mode of representation, which is
merely a diagrammatic simplification of physical
reality,
III. Use a computer model representative of the
structure's behavior, insofar as the computer model
is the mathematical tool making the transition
possible from knowledge of the actions to the
determination of overall stress S t , symbolized by an
elementary stress function
(5.1) S t =f(S1, ... S n )
IV. Check that the margin, characterizing the "degree
of safety", between the state of loads and the
strength criteria under consideration is sufficient.
The overall resistance is calculated on the basis of a
composition g of the various criteria.
The degree of safety is represented by a safety factor:
resistance
(5.2) k = ---------------stress
The distinction between the representation of reality
and the model fitted to computing requirements
involves a whole web of simplifications,
approximations, hypotheses and uncertainties, which
are deterministic in essence and may, for lack of a
carefully reasoned choice, distort certain conclusions.
5.1.2. First Level Probabilistic Method
In the first level of probabilistic design, illustrated in
Figure 5.1, the same computer models (for f and g) are
used, but the value of one or several parameters is
fixed on the basis of statistical data.
The classic example of this type of design is the choice
of maximum annual wind velocity on the basis of a
recurrence time obtained from meteorological
statistics. As in the deterministic methods, the stresses
must be ascertained as lower than the strengths by a set
safety factor.
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5.1.3. Second Level Probabilistic Method
These methods consist in evaluating the risk of failure
as a probability, so that the designer can choose from
among several degrees of safety, depending on
different failure probability values. The computer
models on which the design process is based are the
same as those used in the deterministic method, i.e.
functions f and g. However, the statistical distribution
of the parameters serves as the reference in
establishing a distribution of stresses and resistances
corresponding to the applied loads and the strength of
structures. Then, instead of comparing the two
distributions in terms of a safety factor, a convolution
is operated to calculate a probability of failure of a
structure or a failure recurrence time T, expressed in
annual frequency.
The hatched area in Figure 5.2 defines the theoretical
risk of failure (R), corresponding to the function:
∞
∫ 0
0
R G L f L dL
(5.3) = ( ) ( )
where: G(L) is the Cumulative Distribution Function
(C.D.F.) of the strength of the supports or components
in the same batch
and: fo(L) is the Probability Distribution Function
(P.D.F.) of the applied loads.
The theoretical risk of failure may be calculated as a
function of the relative position of curves G(L) and
fo(L).
The position of curve G(L) may also be defined as the
guaranteed statistical strength Ls and that of curve
fo(L) as the overload probability LT or as the
recurrence time T.
The risk of failure may also be denoted as a function of
the ratio:
(5.4) Γ= L
L
or as a function of the reliability index [Ref 80]
(CORNELL, ROSENBLUETH-ESTEVA, HASOFER-
LIND, DITLEVSEN).
For a given load recurrence time, the risk of failure
decreases as Γ increases.
s
T