The information on this page gives a summary of the Loads and Reliability reader, which is part of the safety content of the module CIEM5000 - Structural Engineering Track Base, written by Roel Schipper. In the end an implementation in Python on reliability plots based on his GitHub repository will be given.
A structure should be sufficiently safe. This means that the probability of failure should be sufficiently low, but unfortunately enough it can never be 0 %. It is possible to determine the risk of failure of a structure (or more positively formulated: the structural reliability ) by carrying out a probabilistic calculation. There are 3 methods of calculations to deal with uncertainty, with increasing level of complexity (International Organization for Standardization 11 2015). Those are:
Semi-probabilistic approach: Safety format prescribing the design equations and the analysis procedures to be used default method in the Eurocodes, i.e. to be used for usual design situations.
Reliability-based design and assessment: Reliability requirements to fulfil unusual design situations in regard to uncertainties, code calibration.
Risk-informed decision making: Decisions are taken with due consideration of the total risks (e.g. loss of lives, injuries). Exceptional design situations in regard to uncertainties and consequences. Derivation of reliability requirements.
To prevent the design of every single structure resulting in a fully probabilistic undertaking, however, for most structures a simplified approach is allowed. Eurocode 0 (prNEN-EN-1990 2021) describes such a simplified, semi-probabilistic method, in which the stochastic distributions of material strength and load values are expressed by characteristic values, taking into account their spread, and by partial safety factors, that ensure sufficient safety margin between loads and capacity.
The Eurocode defines partial factors for the Effects side (actions) in the prNEN-EN-1990 (2021), while the partial factors on the Resistance side (materials) are defined in the various material codes for steel, concrete, timber, and so forth. This Load/Resistance Factor (LRF) or split factor format approach is used to allocate the different uncertainty of loads (effects) and uncertainty of resistance (strength) where they belong, that is in separate calculations of design values of loads and resistance, respectively.
Fig. 2.1 Partial factors on Effects-side and Resistance side combine various types of uncertainties#
In depth review of partial factors from Eurocode
How are the values of \(\gamma_{F}\) and \(\gamma_{M}\) related to the stochastic spread? From Schneider and Vrouwenvelder 1 the following equations can be derived:
\(\gamma_i\): partial safety factor for i (i can be E or R)
\(\mu_i\): mean value of i
\(\alpha_i\): weighing factor depending on how dominant the factor is for the reliability of the structure, often \(\alpha_R = 0.8\) for the most dominant resistance variable, \(\alpha_R = 0.32\) for other (non-dominant) resistance variables, \(\alpha_E = -0.7\) for the most dominant load variable or \(\alpha_E = -0.28\) for non-dominant load variables.
\(\beta_0\): required (targeted) reliability index
The general objective of calculations overseeing safety is that the resistance of the structure should be greater than the effect of the load. A useful way to express this is the unity check (2.1). This check is obtained in 4 steps. The image below gives good visual insight in the steps.
First, both effect \(E\) and resistance \(R\) are expressed in their characteristic values. For \(E\), this means the value \(E_{char}\) is calculated with a probability of exceedance (overshoot) of 5%. For \(R\), this is the value with a probability of 5% of under shoot, \(R_{char}\). This is a standard statistic operation that is done by adding or subtracting \(1.64·σ\) in case of a normal distribution. For loads or materials other distributions may apply, such as Weibull or lognormal distributions.
Second, on the Effect side, \(γ_E\) , the partial load factor, is multiplied with the characteristic value of the load or Effect, resulting in a dimensioning value of \(E\)(2.2), represented by the blue draw line, and for this example the value \(E_d\) = 163 (again, unit does not matter here, as long as both \(R\) and \(E\) are expressed in the same unit).
Third, on the Resistance side, the characteristic value of the strength or Resistance is divided by \(γ_R\), the partial material factor, resulting in a dimensioning value of \(R\)(2.3), represented by the red draw line, and for this example the value \(R_d = 201\) (same unit as \(E\)).
Fourth and finally, \(E_d\) and \(R_d\) are compared, in either of the two following ways: direct comparison \(E_d ≤ R_d\) or unity check (2.1), the latter with the advantage that it gives some numerical information on the “safety distance” between \(E_d\) and \(R_d\).
Example of a normal distribution of Effects (\(E\)) and Resistance (\(R\)) and the place of characteristic values \(E_{char}\) and \(R_{char}\) and design values \(E_d\) and \(R_d\). The characteristic values are obtained by finding from the normal distribution the values with a 5% probability of exceedance (\(μ_E\) + 1.64 · \(σ_E\) for \(E\) and \(μ_R\) − 1.64 · \(σ_R\) for \(R\)). The design values are found by multiplying /dividing the characteristic values with the partial safety factors \(γ_E\) and \(γ_R\). The distance on the horizontal axis between \(E_d\) and \(R_d\) represents a measure of the unity-check (u.c.). In this example, \(E_d\) = 0.81 · \(R_d\). The structure is more than safe.
The “safety distance” should somehow be numerically linked to the accepted level of failure probability. Understandably, the higher the standard deviation of either or both variables \(E\) and \(R\), the higher the safety distance should be to achieve the same level of reliabilty. For example, a safety margin of 1 MPa between \(E_d\) and \(R_d\) does mean less safety for a material with a standard deviation \(σ_M\) = 100 MPa than it does for a material with a standard deviation \(σ_M\) = 10 MPa. To this end, the reliability index β is defined as a function of the spread of the variables \(E\) and \(R\). First, an auxilary reliability function (2.4) is defined. Furthermore, we also need equations for the mean (2.5) and standard deviation (2.6) of the reliability \(Z\).
The reliability index of this particular example can now be calculated as the proportion between the average and the standard deviation as in equation (2.7). For our particular example β = 6.8. Using the table below, we can conclude that with this value of β the probability of failure is very small: \(P_f\) ≪ 10−8, and that we could have done with a higher \(E\) or smaller \(R\) or both and still be okay.
If you are interested in the background of the functions, implemented in python to obtain fast numerical results, you can find those in the safety calculation examples. A calculation example is provided at the bottom of this page.
