Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 157

where N c is the bearing capacity coefficient, S u is the undrained shear strength of the clay,
and q is the pressure at the base of the retaining wall. The pressure, q , is the normal force
per meter divided by the reduced footing width.

N
X

(3.23)

q = 2

The bearing capacity factor, N c for a long footing, with reduction for load inclination, is
found using Brinch Hansen's (Brinch Hansen, 1970) method.
As shown in Figure 3.16:

N c = 2.76 for a long footing with load inclined at 19° from vertical,
S u = 120 kPa,
q = N /2 X = 172 kPa, and
F = 1.93 against bearing capacity failure in the clay foundation.

3.7 Monte Carlo analYSIS uSIng @rISk™

The Monte Carlo method differs from the other methods discussed in this chapter in three ways:

1. It involves repeating the analysis many times (perhaps 5000 or 10,000 times) with
randomly chosen values of the variables.

2. It uses assumptions regarding the distributions of the variables involved in the calcula-
tions rather than an assumption regarding the distribution of the factor of safety. For
each calculation, randomly selected values of the variables are assigned based on the
specified distributions and a random number selection procedure.

3. Because it involves a very large number of repeated calculations, it requires use of the
computer program @Risk™, or another computer program that can automate the pro-
cess. The Monte Caro analyses described here were performed using @Risk™.

With the Monte Carlo method, the probability of failure is determined by counting, among
all the results, the number of times the computed factor of safety is less than or equal to 1.0.
If 50 of 5000 analyses performed using randomly selected values of the variables result in
F ≤ 1.0, the estimate of the probability of failure is 50/5,000 or 1.0%.
While @Risk™ can perform Monte Carlo simulations for any spreadsheet-type analysis,
Monte Carlo simulations methods are found with increasing frequency in geotechnical com-
puter programs. For example, SLIDE by Rocscience and SLOPE/W by Geostudio both allow
an engineer to perform Monte Carlo simulations for slope stability analyses within the respec-
tive program. While the following example is shown using @Risk™, the concepts shown are
also applicable to any computer program using the Monte Carlo simulation method.
The computer program @Risk™ is designed to work with spreadsheet calculations.
@Risk™ controls selection of the values of the variables for each of the many calculations,
and keeps track of the results.
The steps involved in using @Risk™, and the retaining wall spreadsheet to compute the
probability of failure for the retaining wall shown in Figure 3.16 a re

Step 1: Estimate the COV (or, alternatively, the standard deviations) of the quantities
involved in Equation 3.20 : N , μ, and E h .