Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 154

Strength envelop for Los Vaqueros claystone test fill

500

400

300

200

100

0

0 100 200 300 400 500

Effective normal stress (kPa)

600 700 800 900 1000

Figure 3.15 Development of average plus one standard deviation and average minus one standard deviation
strength envelopes using the 2 σ rule. (After Duncan, J.M., 2000, Factors of safety and reliability in
geotechnical engineering, Journal of Geotechnical Engineering , Vol. 126, No. 4, p. 307-316. Used with
permission from ASCE.)

discussed previously. Note that some points in Figure 3.14 a re outside the estimated
highest and lowest values conceivable, indicating that these data points are believed to
be erroneous.

3. Draw straight lines or curves that represent the average plus one standard deviation
and the average minus one standard deviation halfway between the average and the
lowest and highest conceivable lines. These lines are drawn halfway between the aver-
age and the LCV and HCV lines because N σ = 2.

The use of this procedure to establish average ±σ variations of undrained strength with
depth in San Francisco Bay mud is shown in Figure 3.14.
This same concept is useful in characterizing shear strength envelopes for soils. In this
case the quantity (shear strength) varies with normal stress rather than depth, but the pro-
cedure is the same. Strength envelopes are drawn that represent the average and the highest
and lowest conceivable bounds on the data, as shown in Figure 3.15. Then average plus one
sigma and average minus one sigma envelopes are drawn halfway between the average enve-
lope and the highest and lowest conceivable bounds.
These average plus one sigma and average minus one sigma envelopes are useful in cal-
culating probability of failure. For example, the Taylor Series method involves calculating
factors of safety with the values of each variable alternately increased and decreased by one
standard deviation.
Using the graphical N-sigma rule to establish average-plus-one-sigma and average-minus-
one-sigma strength envelopes is preferable to using separate standard deviations for the
strength parameters c and φ. Strength parameters (c and φ) are useful empirical coefficients
that characterize the variation of shear strength with normal stress, but they are not of fun-
damental significance or interest by themselves. The important variable is shear strength,