LRFD calibration of simple limit state functions in geotechnical soil-structure design - Risk and Reliability in Geotechnical Engineering - page 335

the horizontal axis at z = 0 and the slope of the line will be the inverse of the standard devia-
tion (1/σ X ) of the X distribution.
Lognormal distributions of X will plot as a curve. However, these X values can be trans-
formed to log values (LNX) and the transformed values will then plot as a straight line on a
CDF plot of zi i versus ln(X i ). The mean of log values of X, denoted as μ LNX , will intersect the
horizontal axis at z = 0 and the slope of the line will be the inverse of the standard deviation
(1/σ LNX ) of the ln(X) distribution.
8.4 CalCulatIon oF β , ϒ q , anD φ
Monte Carlo (MC) simulation can be used to compute an array of g values using random
values of X R and X Q taken from frequency distribution functions with mean and standard
deviation matching CDF plots of z versus bias values. Closed-form solutions can be used
to compute β and φ if X R and X Q are both normal distributed or both are lognormal dis-
tributed. If this is not the case, then MC simulation must be used. The following sections
describe the steps to process data and to carry out calibration.
8.4.1 generation of bias values
The first step in LRFD calibration is to gather measured load (Q m ) data and measured resis-
tance (R m ) data for the limit state under investigation. For each measured value, compute
the corresponding nominal (predicted) value (Q n or R n ). For example, for each load mea-
surement in a reinforcement layer in a reinforced soil wall structure, compute the matching
predicted load value using a closed-form solution that corresponds to that limit state. An
example for the resistance side is the measured pullout capacity (R m ) of the same type of
reinforcement material from a pullout box test (or in situ pullout test) and the predicted
pullout capacity (R n ) of the same reinforcement specimen computed using a closed-form
solution for this ultimate limit state. The ratios of matching pairs of measured and predicted
values ( Equation 8.4 ) are the load and resistance bias values.
The following steps are used to create normal and lognormal CDF plots (z versus bias) for
each set bias of values in an Excel spreadsheet (Nowak and Collins 2000):
• Place bias values (X) in column 1 of the spreadsheet.
• Sort the values from lowest to highest (X 1 to X n ) where n is the number of values.
• Calculate lognormal bias values LNX i using the Excel LN() function; column 2.
• Create (rank) column of integer values from i = 1 to n; column 3.
• Compute cumulative probability of bias values p i = i/(n + 1); column 4.
• Calculate the standard normal variable using the inverse of the standard normal
cumulative distribution for each bias value as zi i = Φ −1 (p i ); column 5 = NORM.S.INV
(column 4).
• Plot z i versus X i .
• Plot z i versus LNX i .
The two plots can be visually examined to decide whether the data are generally normal
or lognormal distributed. While it is tempting to use quantitative statistical tools and forgo
generation of a CDF plot, it is better to plot and visually examine the data because there
are often deviations from an idealized normal or lognormal plot, particularly at the tails.
It is the lower tail of the resistance bias distribution and the upper tail of the load bias
distribution that contribute largely to the estimate of Pf f (g fit 0). Hence, an accurate fit to