Civil Engineering Reference
In-Depth Information
compute the risk of failure by accounting for all three overlap factors and selecting
the design variables so that acceptable risk of failure is achieved.
The first step in evaluation the reliability of failure of a structure is to decide on
specific performance criteria and the relevant load and resistance parameters,
called the basic variables X
i
, and the functional relationship among them corre-
sponding to each performance criterion. Then define limit state equation, which
can be explicit or implicit function of the basic random variables. The probability
of failure is calculated using first-order reliability methods (FORM) or second-
order reliability methods (SORM). Most of these methods are applicable if the
limit state equation is explicit. Estimating the probability of failure using these
methods requires a background in probability and statistics. Simple simulation
technique can be used to estimate probability of failure with minimum knowledge
of probability and statistics. So, in the simulation process, each random variable is
sampled several times to represent its real distribution. Considering each reali-
zation of all the random variables in the problem produces a set of numbers that
indicates one realization of the problem itself. Solving the problem deterministi-
cally for each realization is known as a simulation cycle. Using many simulation
cycles gives the overall probabilistic of the problem. The method commonly used
for this purpose is called the Monte Carlo simulation technique.
Several computational approaches could be pursued for the reliability analysis
of structures with implicit performance functions. These can be divided into three
categories, (1) Monte Carlo simulation, (2) response surface approach, and
(3) sensitivity-based analysis. The Monte Carlo simulation and response surface
approach could be time-consuming. Thus, combination of sensitivity analysis and
FORM/SORM methods for reliability calculations for functions with implicit
performance functions is more efficient technique. This technique is integrated
with the powerful tool of FE that realistically represents the structure problem to
estimate the reliability of failure. This methodology leads to the stochastic finite
element method (SFEM).
The developed approach consists of three main steps. First step is to develop a
finite element program using MATLAB to analyze structure under static loading
with 3D beam elements. Second step is to develop the sensitivity-based analysis
program based on Halder and Mahadeven [
3
]. The last stream is to integrate both
programs together and validate the approach on different cases already available in
the literature.
5.2 First-Order Reliability Method
The failure surface (limit state), which is the boundary between safe and unsafe
regions in the design parameter space can be defined as Z = 0. The limit state
equation plays an important role in the development of structural reliability
assessment. A limit state can be explicit or implicit function of the basic random
variables.
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