Part I Background and summary
3. Structural reliability methods
3.4 Estimate of the probability of failure
The methods for estimating the probability of failure are usually grouped according to their level of sophistication from Level 1 to 3 (Thoft-Christensen & Baker 1982, Schneider 2006).
Examples of the methods and the corresponding levels of sophistication are shown in Tab. 4, and the methods are demonstrated in Fig. 5 using a known limit state function .F ~ . The probability of failure is generally calculated by the integral
4\~ -FF
jFeH , (9)
where a known joint probability distribution of the basic variables, -F, is integrated over the unsafe region, .F . An exact solution of Eq. (9) can only be found analytically in a very few cases, e.g. where -F is the normal or the rectangular distribution and the limit state function .F ~ is linear. In other cases, the integral is solved either by numerical integration or by simulation. These methods pertain to the Level 3 methods described in Tab. 4.
a) Level 3 method: 4\ is found by dividing the number of outcomes in the unsafe region by the total number of outcomes.
b) Level 2 method: a nominal reliability index, 9, is found by locating the point on .E ~ closest to the origin in the standard normal space.
c) Level 1 method: the nominal values for the basic variables, 7k^, are scaled with partial factors, :k, in order to impose an intended safety level.
Fig. 5: Demonstration of the three levels of sophistication. The solid curved line is the limit state function, .F ~ , and the shaded area is the unsafe region, .F .
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Tab. 4: The levels of sophistication in reliability assessment methods (adapted from Thoft-Christensen & Baker 1982 and Schneider 2006).
Description Representation of
basic variables Examples of methods
Level 3 the true shape of the failure domain. of failure are obtained, that should only be used for comparison purposes. in design codes where an intended level of safety on component level is attained by use of partial factors for load and resistance
The Level 3 methods will give exact estimates of the probability of failure if the analyst has full knowledge of the problem at hand (Der Kiureghian 1989). One Level 3 method is the Monte Carlo method. Here, random realizations are generated for each of the basic variables as shown in Fig. 5a, taking into account the respective distributions and possible correlation between the basic variables. The limit state function is furthermore evaluated for each of the sets of random realizations and the probability of failure is the number of limit state evaluations giving .F divided by the total number of evaluations. Further improvements to this method aiming at reducing the necessary number of limit state function evaluations include for example importance sampling (Melchers 1999) and Latin hypercube sampling (McKay et al. 1979, Olsson et al. 2003). Two important drawbacks make the Level 3 methods unsuitable for structural engineering applications:
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Full knowledge about the statistical distributions of the basic variables is in general not possible to obtain.
Since a large number of samples is necessary to obtain the necessary accuracy in the estimate of 4\, and the limit state function is often complex or time consuming to evaluate, the method will be costly or impossible to use.
Using available exact or approximated transformation rules, the vector of basic variables, F, can be transformed to a vector of uncorrelated standard normally distributed variables, E, with means equal to zero and unit-variances (Hasofer & Lind 1974). After the transformation, the vector E and the limit state function, .E ~ , will be located in the so-called standard normal space or u-space, with the origin corresponding to the mean value of the variables F, shown in Fig. 5b. In Level 2 methods, .E ~ is approximated by a first- or second-order polynomial, and an approximated value for 9 is the minimum distance from .E ~ to the origin. The point on .E ~ closest to the origin, E", is the point at the boundary between the safe and the unsafe region with the highest probability of occurring, and is often denoted as the design point. If .E ~ is far from the origin, the safety margin is large, giving a large 9 and a corresponding small 4\.
The design point can be found in an iterative manner, and several procedures are available in the literature (Hasofer & Lind 1974, Rackwitz & Fiessler 1978, Shinozuka 1983, Liu & Der Kiureghian 1991). If the limit state function is approximated by a linear polynomial, the method is denoted First Order Reliability Method (FORM), and an example of a FORM solution is shown in Fig. 5b. Also shown in the figure are the sensitivity factors, 8k, that indicate the contribution from the variation of each of the basic variables to the probability of failure. The FORM solution is exact if the limit state is linear and the basic variables can be transformed to standard normally distributed variables by a one-to-one transformation. For large 9, i.e. low 4\, FORM is a good approximation in other cases as well. The FORM solution can be improved by including second-order terms in the approximated limit state function, and this class of improved methods is denoted Second Order Reliability Methods (SORM) (Hohenbichler et al.
1987).
Assuming that the compressive strength of concrete, -Y, and the yield strength of the reinforcement, -d, are the only basic variables for the resistance of reinforced concrete, the corresponding sensitivity factors 8iw and 8ix can be estimated by a FORM analysis. In an over-reinforced cross-section, the bending moment capacity will be governed by failure of the compressive zone before the yield strength of the reinforcement is reached. In this case, 8iw
would be significantly larger than 8ix, since the concrete governs the failure mode. For an under-reinforced cross-section, where the failure mode is governed by yielding of the reinforcement steel, the opposite would be the case, i.e. 8ix# 8iw. This illustrates the added value of the results from analyses with the Level 2 methods.
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In the Level 1 methods, the design point is estimated directly by scaling characteristic or nominal values for the basic variables by partial factors. Design code developers calibrate the values of the partial factors in order to reach a target reliability level on the structural component level. By applying partial factors it is assumed that the values of the basic variables lie in the safe region, .F , see Fig. 5c. Implicit in the partial factors in the Eurocodes (CEN 2002) are assumptions of a linear limit state function following Eq. (6) and constant sensitivity factors for the resistance, 8V~ , and load, 8Q~ }, respectively. Further assuming that the load and resistance can be treated independently, the target reliability index for the resistance becomes 8V9bWa][b.
From this short summary of structural reliability methods it should be noted that the Level 2 methods are approximations of the Level 3 methods and that the Level 1 methods are calibrated to solutions obtained with the Level 2 methods. Since the Level 1 methods are based on probabilistic measures, allowing the engineer to directly incorporate a target reliability and the variability of the basic variables, but only a design point is obtained, the method is often denoted as semi-probabilistic (Cornell 1969, Ellingwood & Galambos 1982, Ellingwood 2008).