Chapter 3: Structural Reliability
3.3 Basic Reliability Theory
A model is a representation of an existing object or phenomenon in which some aspects of this representation vary somewhat from the original object or phenomenon. This is because some simplifications and assumptions have to be made in the development of the representative model.
Subsequently, uncertainties may arise from the simplifications and assumptions made in forming the model (Croce, Diamantidis and Vrouwenvelder, 2012). Other sources of uncertainty are the characteristic randomness of a physical phenomenon, as well as the predictions of states of nature made with inadequate information (Ang and Tang, 1984). With increased data and information, models representing physical phenomenon may be improved and made more accurate with inadvertent biases reduced. However, the inherent randomness of physical phenomena cannot be avoided. It cannot be guaranteed with absolute certainty that a variable will take on a particular value; instead, a range of possible outcomes may be attributed to this same variable. The likelihood of occurrence for a specific value may be determined by its probability distribution function (Holický, 2009).
As described by Ang et al. (1984), most engineering problems may be described as supply and demand problems where the safe state of the structure is where the supply exceeds the maximum amount of demand experienced over a lifetime. The supply and demand may be expressed as either random variables (Xi) or functions of random variables with their own distribution functions. In other words, the resistance of a structure (R) needs to be greater than the action effect (E) of the structure in order for the structure to remain reliable (E<R). The performance function separating the safe state of the structure or engineering process may be expressed as:
g(Xi) = R-E = 0 (3.1)
A negative value of the performance function is indicative of a failure in performance, whilst a positive answer shows that the resistance of the structure exceeds the load effect and thus the structure is safe. The limit state may be defined as the distinct separation between the safe state of the structure where it performs reliably and the unsafe state where it no longer functions. So primarily, the performance function, g(Xi), is itself a limit state.
The equivalent normal distributions of the demand and supply variables are used to approximate the failure probability. Where the reduced variate (equivalent normal variate) of the resistance (supply) may be described by the equation 3.2:
R’ =
R
-
RR
,
(3.2)
where μR and σR respectively denote the mean and standard deviation of the resistance variable.
And the reduced variate of the action effect E (demand) may be determined using the formula:
E’ =
E
-
ER
,
(3.3)
where μE and σE are the respective symbols for the mean and standard deviation of the action effect, E. Then the performance function may be rewritten as:
g(Xi) = R’-E’ = 0. (3.4)
This then equates to,
g(Xi)= σRR’ - σEE’ + μR –μE = 0
Then the linear failure distance from the origin to the failure line g(Xi) = 0 can be expressed as:
β =
2 2 R
Y R
E
=
G G
,
(3.5)
This distance β is the safety index, and describes the shortest distance from the reduced variate origin to the limit state (Wu, Lo and Wang, 2011). In other words, this distance describes the distance to the most likely point of failure along the limit state (this is illustrated in Figure 3.1).
In general, a structure is said to be in a desirable state where the limit state function is greater than zero and at values less than zero the structure will be in an undesirable state. At zero, the structure just meets the limit state as shown in Figure 3.1.
Figure 3.1: Space of Reduced Variates E’ and R’ (as adapted from Ang and Tang. (1984))
For the serviceability limit state, the performance function described above is structured such that an exceedance of a limiting design criterion (like a set deflection value, or in this case an allowable crack width) would take the form:
g(Xi) = C – S = 0, (3.6)
where C represents the serviceability criterion in question and S denotes the action effects (as described in SANS 2394:2004). Clearly, regarding cracking in liquid retaining structures, where the action effect exceeds the serviceability criterion the limit state would be exceeded and the undesirable (unsafe) state entered into. The EN 1992 cracking serviceability limit state may be similarly formulated:
g(Xi) = wlim – θw, (3.7)
where wlim describes the permissible crack limit and w represents the mean crack width based on the EN 1992 maximum characteristic crack width. In developing the crack width formula for EN 1992-1-1 it was determined through experimental data that a factor of 1.7 be applied to the average crack spacing in order to calculate the maximum crack width (Beeby and Narayanan, 2005).
However, the reliability analysis undertaken herein (described in chapter 5) requires the use of the mean crack width and so a reduction in value of 1.7 to the EN 1992 maximum characteristic crack width formula should return a function for the mean crack width.
The value, θ accounts for the model uncertainty in the EN 1992 crack model and is regarded as a random variable in the reliability analysis. An elaboration of the formation of the limit state function to be used in this investigation is given in chapter 5.
The probability of safety may then be determined by:
ps = Ф(β), (3.8)
where Ф is the standardized normal distribution function and β is the reliability index as defined in equation 3.5. The failure probability may then be determined using the relationship pf = 1 - ps, so that pf = 1 – Ф (β) = Ф (-β).
A simple illustration of the relationship between the reliability index and the failure probability may be outlined in Table 3.1:
Table 3.1: Relationship between Failure Probability and Reliability Index JCSS Part 1 (Joint Committee of Structural Saftey, 2001).
Pf 10-1 10-2 10-3 10-4 10-5 10-6 10-7
β 1.3 2.3 3.1 3.7 4.2 4.7 5.2