c. Limit States. Limit states define the point at which the performance function predicts
that unsatisfactory performance will occur, or that the engineering consequences will have some
adverse economic impact. Several limit states may apply to a particular performance function.
For example, the limit state for deflection of a gravity lock wall monolith could be taken as the
event representing excessive lateral movement which causes cracking, spalling or binding of
operating equipment. Several loading conditions and the corresponding deflections may be
required to be calculated to determine the range of deflections, their impact upon the operating
equipment, barge traffic and stability, in order to establish limit states.
d. Reliability. The reliability, R, is the probability that the unsatisfactory performance,
Pr(u), will not occur. Mathematically reliability is determined as follows:
R = 1 - Pr(u).
e. Safety Ratio. A concept used in the calculation of reliability is the safety ratio.
Important parameters should be defined as random variables, then the total capacity, C,
(resistance) and the total demand, D, (load) are also random variables. The safety ratio, SR, is the
quotient of the capacity and demand; or of the resistance and load. The probability of
unsatisfactory performance can then be expressed as the probability that the safety ratio will be
less than one, or:
Pr(u) = Pr (SR<1) = Pr[(C/D) < 1]
D-3. Calculation of Probability of Unsatisfactory Performance.
a. Methods. Four methods are available to calculate the probability of unsatisfactory
performance; (1) Reliability Index, (2) Hazard Function, (3) Historical Frequency of Occurrence,
and (4) Expert Elicitation (or subjective probabilities). The selection of the method to be used to
establish the probability of unsatisfactory performance will depend on the type of component or
structure, availability of project specific data and the level of study. Each of these methods are
briefly discussed below.
b. Reliability Index. In many applications of reliability analysis, the probability of
unsatisfactory performance is discarded in favor of the reliability index, , which is a measure of
how much the expected average value of the safety ratio exceeds the limit state at a particular
point in time. A value of 3.0 implies that the expected value of the performance function lies
three standard deviations above the limit state and structures, components and performance
modes with higher indices are considered to be more reliable than those with lower indices.
Expressing reliability in terms of the reliability index has several advantages:
(1) A reliability index can be calculated knowing only the means, standard deviations and
correlation coefficients of the variables.
(2) Analysis of recurring events and replicate components (such as failure of mechanical
parts or electric power), have a measurable and easily understood frequency of failure.
Many features of navigation structures are uniquely adapted to site conditions and are not
expected to fail due to foreseeable rare events, (i.e., observations and analyses can be
made and some remedial action will be undertaken before the reliability drops below a
tolerable value).
(3) Reliability index approach is consistent with recent structural design codes and
techniques used for highway bridge evaluation.
D-2
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