Theory of reliability analysis

In the past, the design of structural systems considered all loads and strengths as deterministic values. The strength of an element was determined in such a way that it withstood the load within a certain margin. The ratio between the strength and the load was denoted a safety factor.

This safety factor was considered as a measure of the reliability of the structure. However, uncertainties in the loads, strengths and in the modelling of the systems require that methods based on probabilistic techniques in a number of situations have to be used. A structure is usually required to have a satisfactory performance in the expected service life, i.e. it is required that it does not collapse or becomes unsafe and that it fulfils certain functional requirements.

In order to estimate the reliability by using probabilistic concepts it is necessary to introduce stochastic variables and/or stochastic processes/fields and to introduce failure and non-failure behaviour of the structure under consideration.

Generally the main steps in a reliability analysis for service life prediction are: 1. Identify the significant failure modes of the structure.

2. Decompose the failure modes in series systems or parallel systems of single components (only needed if the failure modes consist of more than one component). 3. Formulate failure functions (limit state functions) corresponding to each component in

the failure modes.

4. Identify the stochastic variables and the deterministic parameters in the failure functions. Further specify the distribution types and statistical parameters for the stochastic variables and the dependencies between them.

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5. Estimate the reliability of each failure mode (illustration of how reliability or inversely the probability of failure changes with time).

6. Evaluate the reliability result by performing sensitivity analyses.

Typical failure modes to be considered in a structural reliability analysis are yielding, corrosion, buckling (local and global), fatigue and excessive deformations.

The failure modes (limit states) are generally divided in:

Ultimate limit states correspond to the maximum load carrying capacity which can be related to e.g. formation of a mechanism in the structure, excessive plasticity, rupture due to corrosion and instability (buckling).

Conditional limit states correspond to the load-carrying capacity if a local part of the structure has failed. A local failure can be caused by an accidental action or by fire. The conditional limit states can be related to e.g. formation of a mechanism in the structure, exceedance of the material strength or instability (buckling).

Serviceability limit states are related to normal use of the structure, e.g. excessive deflections, local damage and excessive vibrations.

The fundamental quantities that characterise the behaviour of a structure are called the basic variables and can be denoted X = (X1, ..., Xn) where n is the number of basic stochastic variables. Typical examples of basic variables are loads, strengths, dimensions and material properties. The basic variables can be dependent or independent. A stochastic process can be defined as a random function of time in which for any given point in time the value of the stochastic process is a random variable.

The uncertainty modelled by stochastic variables can be divided in the following groups: Physical uncertainty: or inherent uncertainty is related to the natural randomness of a quantity, for example the uncertainty in the yield stress due to production variability.

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Measurement uncertainty: is the uncertainty caused by imperfect measurements of for example a geometrical quantity.

Statistical uncertainty: is due to limited sample sizes of observed quantities.

Model uncertainty: is the uncertainty related to imperfect knowledge or idealizations of the mathematical models used or uncertainty related to the choice of probability distribution types for the stochastic variables.

All the above types of uncertainty can usually be treated by the reliability methods. Another type of uncertainty which is not covered by these methods is gross errors or human errors. These types of errors can be defined as deviation of an event or process from acceptable engineering practice.

Generally, methods to measure the reliability of a structure can be divided in four groups, see Madsen and Krenk (1986):

• Level I methods: The uncertain parameters are modelled by one characteristic value, as for example in codes of practice based on the partial safety factor concept.

• Level II methods: The uncertain parameters are modelled by the mean values and the standard deviations, and by the correlation coefficients between the stochastic variables. The stochastic variables are implicitly assumed to be normally distributed. The reliability index method is an example of a level II method.

• Level III methods: The uncertain quantities are modelled by their joint distribution functions. The probability of failure is estimated as a measure of the reliability.

• Level IV methods: In these methods the consequences (cost) of failure are also taken into account and the risk (consequence multiplied by the probability of failure) is used as a measure of the reliability. In this way different designs can be compared on an economic basis taking into account uncertainty, costs and benefits.

Level I methods can e.g. be calibrated using level II methods, level II methods can be calibrated using level III methods, etc.

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Several techniques can be used to estimate the reliability for level II and III methods, including the following methods:

• Simulation techniques: Samples of the stochastic variables are generated and the relative number of samples corresponding to failure is used to estimate the probability of failure. The simulation techniques are different in the way the samples are generated. Monte Carlo method is the major simulation method for structural reliability analysis.

• FORM techniques: In First Order Reliability Methods the limit state function (failure function) is linearized and the reliability is estimated using level II or III methods.

• SORM techniques: In Second Order Reliability Methods a quadratic approximation to the failure function is determined and the probability of failure for the quadratic failure surface is estimated.

• Time dependent reliability techniques: when a structure is subjected to a time dependent degradation process, probabilistic time dependent methods can be used. First passage probability theory has been introduced for time dependent reliability analysis (Melchers (1999)). Gamma process concept also has the potential of usage as a model for reliability analysis of structures subject to monotonic degradation processes (van Noortwijk and Pandey (2003)). These methods are discussed and developed in Chapter 4 for reliability analysis of concrete sewers and cast iron pipes.

In level IV methods the consequences of failure can be taken into account. In cost-benefit analyses the expected total cost-benefit for a structure in its expected lifetime is maximized. For a detailed introduction to structural reliability theory references are made to the following textbooks: Melchers (1999), Thoft-Christensen & Baker (1982) and Ditlevsen & Madsen (1996).