Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 158

N , the normal force on the base of the footing, is proportional to the unit weight of
backfill, and has the same COV . The COV of the backfill unit weight is small, esti-
mated to be 3%, corresponding to standard deviation of γ bf = COV∗γ bf = (0.03)
(18.85 kN/m 3 ) = 0.565 kN/m 3 .
The base friction coefficient, μ = tan δ, depends on the type of granular material on which
the concrete footing is cast. According to the NAVFAC manual (NAVFAC 1986) the
value of μ for silty sand ranges from 0.45 to 0.55. The COV tanδ is estimated to be 10%,
corresponding to standard deviation = (0.10)(0.50) = 0.05.
E h , the horizontal force on the wall, is proportional to the equivalent fluid pressure: coef-
ficient γ eq . The coefficient of variation COV eq is estimated to be 15%, corresponding to
standard deviation σ eq = (0.15)(7.07 kN/m 3 ) = 1.06 kN/m 3 .
Step 2: Use a Microsoft Excel spreadsheet together with @Risk™ to compute the factor of
safety against sliding on the silty sand layer a large number of times. For the example
described here, Retaining Wall Stability Computation Sheet 2.05.1 (Yang and Duncan
2002) was used.
Step 3: One at a time, select the cells containing parameters whose values will be var-
ied during the calculations. In this example these are the cells that contain γ eq (cell
H19), γ bf (cell H20), and μ (cell H23). As each cell is selected the define distribution
option in @Risk™ provides the opportunity to define the type of distribution to be
used for the variable and its statistical parameters. Within @Risk™ the distribu-
tions of the variables are not limited to normal or lognormal. Thirty-seven different
distribution types are available in @Risk™. Unless information is available that
shows otherwise, however, a normal or lognormal distribution is probably the logi-
cal choice. The variables used in this retaining wall example are all assumed to be
normally distributed.
Normal distribution has been selected, and the average value of γ eq (7. 07 k N /m 3 ), and the
standard deviation (1.06 kN/m 3 ) have been entered at the left of the screen. The cor-
responding PDF is displayed on the screen.
Step 4: The cell in the spreadsheet that calculates the factor of safety is defined as the
output. Once the cell is selected, the “Define Output” button is pressed in @Risk™.
Step 5: When @Risk™ is run, the factor of safety is calculated for the number of times
specified. As explained above, the probability of failure is equal to the number of times
the calculated factor of safety is less than or equal to 1.0, divided by the total number
of times the factor of safety is calculated. Examining the large number of calculated
values and counting the number of times that the factor of safety is less than 1.0
would be very cumbersome. This can be avoided by adding a cell to the spreadsheet
that keeps track of the number of safety factor values that are less than 1.0. This cell
contains the following formula:
= if(referenced cell < 1,1,0)
This formula results in a value of either 0.0 (if the computed factor of safety is greater
than 1.0) or 1.0 (if the computed factor of safety is less than or equal to 1.0). At the end of
the analysis, @Risk™ computes and displays the average value of the cell for all the values
computed. Thus, for example, if the spreadsheet is run 10,000 times and the calculated fac-
tor of safety is less than 1.0 in 245 of those calculations, the cell will contain the average
of 9755 zeroes and 245 ones, or 0.0245. This is the probability of failure as computed by
Monte Carlo simulation.