Environmental Engineering Reference
In-Depth Information
4
3
2
Load and resistance factors
γ
Q
= 1.10
φ = 0.38
1
0
Resistance bias statistics
μ
XR
= 1.00
COV
XR
= 0.356
Load bias statistics
μ
XQ
= 1.00
COV
XQ
= 0.283
-1
-2
-β = -2.29
-3
n = 1000 (simulations)
-4
-2
0
2
4
6
8
10
g = (γ
Q
/φ)X
R
- X
Q
Figure 8.7
Example plot of MC simulation results showing location of
β
for g
=
0.
8.5.4 Calibration
8.5.4.1 Resistance factor using MC simulation
Figure 8.7
shows the results of MC simulation using one set of load and resistance factors,
and random values taken from lognormal distributions of X
R
and X
Q
with the mean and
COV values shown in the figure. In this example, only 1000 simulations were carried out.
Nevertheless, the β value corresponding to g = 0 is easily detectable and corresponds to
β = 2.29 which is equivalent to Pf
f
= 0.011. Additional simulations with 3000 and 5000 runs
gave essentially the same value.
8.5.4.2 Resistance factor using closed-form solution
Target reliability index values of β = 2.33 (Pf
f
= 0.01) and β = 3.09 (P
f
= 0.001) were selected
and values of resistance factor computed for a range of load factors using
Equation 8.20
.
The numerical results are presented in
Figure 8.8
.
The value of φ = 0.384 shown in the figure
is in close agreement with the result from MC simulation for the same input parameters.
In practice, load and resistance factors are typically computed to two decimal places. In
LRFD design codes for geotechnical structures, the values are reported to the nearest 0.05
(AASHTO 2012; CSA 2006).
8.6 aDDItIonal ConSIDeratIonS
This chapter has highlighted the fundamental steps to carry out LRFD calibration for a
simple linear limit state design function. The example is the pullout limit state for the inter-
nal stability design of MSE walls constructed with steel strip soil reinforcement.
Figure 8.8
shows that there are detectable differences in numerical outcomes using bias
statistics for fit-to-all load bias data and fit-to-upper tail only. The differences here can be
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