The failure probability is obtained from the relation between the solicitation (S) and the resistance (R). The failure event occurs when R - S < 0 or R/S < 1. The failure probability ( ^_pf ) is defined as the integral from zero to infinity of ^_FR(s) and ^_fS(s) functions’ product, as shown in eq. (1) and Fig. 2 [11,12]. The failure probability, defined by eq. (1), considers that there is no conditioning: R and S are independent variables.

Figure 2 Source: Modified from [12].

Figure 3 Source: The authors.

(1)

At point A in Fig. 2, the probability density functions ^_fR(r) and ^_fS(s) are the same. The black area in Fig. 2 represents ^_pf , being the area under the curves ^_fR(r) , to the left of the point A, and ^_fS(s) , to the right of that point. The larger the area under this curve, the higher the ^_pf , that is, the lower the reliability of the pile foundation.

The failure probability can be determined from the margin of safety linear function (MS=R-S) and the nonlinear function factor of safety (FS=R/S), which depends on the density functions ^_fS(s) and ^_fR(r) . The degree of dispersion and coefficient of variation (ν=σ/m) are represented by the shape of these curves, and they influence the magnitude of the failure probability [12].

The R and S problem can be formulated analytically in terms of the MS and FS functions through their respective reliability indexes ( ^_βMS and ^_βFS ). These measure the distance from the failure condition in terms of standard deviations.

As R and S are random variables and MS is a linear function of R and S, MS is a random variable. When MS is less or equal to zero, the failure occurs (see Fig. 3). Thus, the failure probability is ^{_{pf = P[MS≤0]}} : the hatched areas in Fig. 3.

Usually, there is not sufficient information about the extremities of the MS distribution, and the criterion ^{_{pf = P[MS≤0]}} is replaced with another that involves the MS average and standard deviation.

From the average and standard deviation of continuous random variables R and S, the average and standard deviation of MS is determined. It is known that the estimator is a linear operator and that the variance follows the next expression [13], where X is any random variable.

(2)

Whereas, ^_(RS , the correlation coefficient between R and S; the average (m); and standard deviation (σ) of MS are:

(3)

(4)

In Fig. 3, the parameter ^_βMS , called the Reliability Index, can be obtained from different distributions of R and S. This is shown in eq. (5).

(5)

From the shape of the probability density curves R and S, synthesized by their respective coefficients of variation, there is a relation between the global safety factor ( ^{_{FS = mFS = mR/mS}} ) and the reliability index. The deduction of this relation is shown below [1].

According to the definition of ^_βMS shown in Fig. 3, we have:

(6)

f_P =0,7f_y Dividing eq. (6) by ^_mR we have:

(7)

Defining ^_(MS in the previous equation, the relation between FS and ^_(MS is given by:

(8)

(9)

The Reliability Index is also found in [14,15], as shown in eq. (10)-(11), respectively. However, in these studies, the variables are mutually independent ( ^_(RS = 0) and the direct relation between FS and ^_(MS is not explained.

(10)

(11)

A knowledge of the two first probabilistic moments of the margin of safety and factor of safety functions is not enough to define the probability density function. It is also necessary to know the shape of the performance function distribution. In practice, it is usual to adopt the hypothesis of normal or lognormal distributions.

If the random variables R and S present a normal distribution, ^_βMS and ^_βFS are related to the probability of failure by the following expression, in which Φ is the standard normal cumulative distribution function.

(12)

The use of a simulation method is another way to estimate the failure probability. One of the methods most commonly used for this purpose is the Monte Carlo simulation, which consists of repeating deterministic solutions. The result is a set of values generated in accordance with their respective probability distributions, which are presumably known. This group of values is similar to a set of information derived from experimental observations. It is possible to statistically treat the generated deterministic values using all statistical inference methods.

Essentially, the Monte Carlo method is a random sampling technique, and as in any other sampling process, there is always some error involved. According to [16], in order to obtain "exact" solutions the number of simulations must approach infinity, which is one of the method’s drawbacks.

In pile foundations in which the population is finite, it has been verified in practice that the statistical sample, obtained from standard penetration test procedure and dynamic and static load tests, accurately represents the finite population parameters [1,7] (see Fig. 4). Thus, the average and standard deviation of the sample can be taken as the average and standard deviation of the finite population.

In foundations engineering, the sampling plan is basically comprised of the n size of the sample. This is the main factor responsible for the accuracy of the results between sample and population (Fig. 4). This is due to the fact that the sampling design does not meet the defined criteria because of the unknown soil variability. Contemplating that the sample size can reduce the variability of soil mass is a mistake as this is an intrinsic characteristic of the geological and geotechnical environment.

Figure 4 Source: The authors.

In practice, we work with a single sample and, therefore, we have only single estimates calculated for this sample. From the sample we can make inferences about the population values through the sampling distribution that are associated with no addiction criteria (where the estimator has an expectancy equal to the population value). We can also make inferences using asymptotic normality (when for certain sizes of samples the average sampling distribution is close to the normal distribution: meaning that the central limit theorem is valid).

The order statistic is based on ordering the elements of a set and using the extreme values: ^_Xmax and ^_Xmin . The main justification for the use of this tool is based on the fact that the number of tests (using either the in situ test or load test) will generally be small [10].

Estimates of the average value and standard deviation are given, respectively, by:

(13)

(14)

f p = 0,7f y Where η is chosen according to Table 1 below as a number (n) function of the sample:

The above relations are used to express the estimated variation coefficient E[V] (estimate of the standard deviation divided by the estimate of average value) as a function ^_Xmin/Xmax and ^_Xmin/Xmed . These relations are:

(15)

(16)

Table 1

Source: The authors.

f_P =0,7f_y The use of sample outliers to estimate population parameters may turn out to be advantageous to calculate the failure probability, by the dispersion of sample values. It is observed that when the number of the sample is small, the use of the order statistic seems to be consistent.