2. Uncertainty, Reliability and Maintenance in Structural Engineering
2.3 Structural Reliability
One of the major requirements for structures usually is to have a satisfactory performance in the expected lifetime, in other words, it is required that the structure does not collapse or becomes unsafe and that it fulfils certain functional requirements over its specified period of usage, and in an economic way. Reliability
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is an efficient measure of the structural performance, the probability that the structure under consideration will perform its function during the predetermined lifetime. Reliability analysis methods provide a framework to account for numerous sources of uncertainties that should be considered in engineering design and problems in a rational and vigorous manner.
Structural analysis and design have been traditionally based on deterministic methods for decades, with the assumption and estimation that the strength of structural element is always exceeding the load with a certain margin. Contrary to this, uncertainties in the loads, strengths and in the modelling of the engineering systems require that probabilistic methods in a number of situations have to be used. In the traditional method, safety factor was defined as the ratio between the strength and the load, and in turn considered and taken as the measure of the reliability of the structure (Sorensen, 2004; Melchers, 1987; Madsen et al., 1986; Thoft-Christensen and Baker, 1982; Ditlevsen and Madsen, 1996). Furthermore, the traditional approach is based on specified minimum material properties, load intensities, and certain procedure for estimating stresses and deflections which are most of the time a deterministic prescription based on international standards and codes. It is widely accepted that the probabilistic approach seems to be well suitable for the measure of risk and reliability in structural reliability theory. It is an extension of traditional structural analysis as an art of formulating mathematical models as to how structure behaves when it’s material and geometric properties, etc. and other actions are known.
2.3.1 Basic theory and methods of structural reliability
Structural reliability estimation methods are classified according to level, moment, order and exactness of calculation result in (Huyse, 2001). The theoretical methodologies are highlighted and briefly explained on in Table 2.1
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Table 2.1: Structural reliability theoretical methods
Classification Group/Type Description
Level I II III IV Deterministic reliability methods. It uses only one characteristic value for each uncertain variable description.
These reliability methods use two values to describe each uncertain variable (such as mean, variance, coefficient of variation). e.g. FOSM
The joint probability density distribution of all the uncertain variables is used for description of each uncertain variable. Examples are - Numerical integration, approximate analytical method, FOSM, and simulation method.
This method makes a comparative analysis based on the principles of engineering economic under uncertainty between structural prospect and a reference prospect.
Moment and Order Approximate methods: FOSM, SOSM, etc.
Reliability methods here are classified into approximate methods Exactness of calculation
result
Approximate methods
Simulation methods
Direct integral method
Examples are mean-value FOSM method, etc.
Monte Carlo method is an example.
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Further classification in (Zang et al. 2002), shows that the reliability methods are roughly described and grouped into two major types, namely: mathematical-based reliability, and physics-based reliability as shown in Table 2.2.
Table 2.2: Summary and differences between two types of reliability (Zang et al. 2002)
Mathematical-based Reliability Physics-based Reliability Reliability is related to life- the time to
failure.
The state change is observed.
Reliability evaluation relies on testing or field data.
The reliability is defined by expression:
t PT t
R : T is any future time
Reliability is related to the limit state.
The state change can be mathematically modelled.
Reliability can be evaluated from physical equations (models).
The reliability is defined by expression:
0
Pg X R
The reliability is time dependent.
Typical methods include: o Fault tree analysis
o Event tree analysis
o Failure models, effects, and criticality analysis
o Markov process
o Monte Carlo simulation
The reliability may or may not be time dependent.
Typical methods include:
o First order second moment method
o First order reliability method (FORM)
o Second order reliability method (SORM)
o Design of Experiment o Monte Carlo simulation
Followed by this brief description of the theoretical methods is the overview of the structural reliability estimations, which is equivalent to the calculation of the failure probability of the structure. Let an n-dimensional vector of basic variables with continuous joint density function fX
x be associated with the existing structure, G(X) as the limit state function such that the limit state surface is G(X)=0, the safety region of the structure is denoted as G(X)>0, and the failure region is defined as G(X)<0. Then, the structural reliability is simply the calculation of the integral for estimating failure probability, Pf indicated (Rackwitz, 2001) as:23
xdx f P X G X f
0 (2.1)Assuming that a probability preserving transformation xT
u exists where uis an independent standard normal vector which transforms the probability integral into
du u U dx x f P u T G x G X f
0 0 (2.2)where U
u is the n-dimensional standard normal density with independent components. The simple result is Eq. (2.3).
f
P (2.3)
The reliability of the structure R, which is the compliment 1Pf is defined as: f
P
R1 (2.4) The reliability index, 𝛽 is defined as:
Pf 1 (2.5)is the standard normal cumulative distribution function.