LRFD calibration of simple limit state functions in geotechnical soil-structure design - Risk and Reliability in Geotechnical Engineering - page 336

these locations may be required to better estimate the true probability of failure rather than
simply fitting a normal or lognormal distribution to the entire data set. Most often it is a
lognormal CDF plot that best fits bias values using the entire data or fitting to the tails in
most soil-structure cases.
The next step is to fit a curve (or curves) to the CDF plot that matches the entire distribu-
tion of data points and/or is a good fit to the bottom of the resistance bias plot or the top of
the load bias plot based on a visual inspection.
As a starting point, the mean (μ X ) and standard deviation (σ X ) of the bias distribution X
can be computed for all data (AVERAGE and STDEVP functions in Excel). The coefficient
of variation of bias values COV X is computed as

COV X = σ X /μ X (8.11)

Approximations to the measured data assuming a normal distribution can be determined
by computing the predicted bias value as

X i = μ X (1 + z i × COV X ) (8.12)

For a lognormal approximation:

LNX i = EXP(μ LNX + z i × σ LNX ) (8.13)

Values of Xi i and LNX i can be placed in columns 6 and 7, respectively, to facilitate plotting.
A useful alternative method is to use the following approximations to estimate σ LNX and
μ LNX using normal bias statistics:

X 2

(8.14)

σ LNX = LN1 + COV ( )

LN() 1
2

2

(8.15)

µ=µ− σ
LNX X LNX

These substitutions can be used in Equation 8.13 to estimate LNXi i and these values
placed in column 8.
Values of COV X and μ X can be changed by trial and error to explore the visual fit to the
bias data over the entire range of data points or at the tails as demonstrated in the examples
given later in this chapter.

8.4.2 Selection of load factor
For a prescribed target probability of failure (or target β value), either φ or γ Q must be
assumed to give a unique solution. Typically, γ Q is selected first based on a load exceedance
criterion. For example, during the development of the AASHTO (2012) and Canadian (CSA
2006) LRFD highway bridge design codes, the load factor for vehicle loads on bridges was
selected so that the factored nominal load would not exceed measured load data in 97.7%
of cases (i.e., two standard deviations below the mean of load bias values) (Nowak 1999;
Nowak and Collins 2000). This value corresponds to a load factor computed using the fol-
lowing equation with n Q = 2 and a distribution of vehicle loads that is normal distributed:

γ Q =µ +× ( 1n COV )

(8.16)

Q Q Q