LRFD calibration of simple limit state functions in geotechnical soil-structure design - Risk and Reliability in Geotechnical Engineering - page 337

Alternatively, the load factor corresponding to a specified load exceedance criterion can
be computed as the bias value for which the fraction of bias values greater than that value
equals the exceedance value. This is the same as plotting a cumulative probability plot (e.g.,
p i versus X Qi ) and finding the value of X Qi that corresponds to the target exceedance value
on the vertical axis. The selection of the load factor could also be based on past practice.
Bathurst et al. (2013) studied the case of steel-reinforced soil walls where, consciously or
not, a past practice led to measured reinforcement loads that exceeded predicted loads in
37% of cases as opposed to 3% of the cases in the vehicle traffic load example described at
the beginning of this section. Clearly, these two exceedance values will give different load
factors for the same load bias distributions.

8.4.3 Selection of target reliability index
Past geotechnical design practice has led to a target probability of failure for foundations, in
general, of approximately 1 in 1000, or equivalently β = 3.09. For highly strength-redundant
systems, such as pile groups and reinforced soil walls that have multiple layers of rein-
forcement, a target reliability index value of β = 2.0-2.5 is often used (Barker et al. 1991;
Paikowsky et al. 2004; Allen 2005). The justification for the lower β value is that if one load-
carrying element in these systems fails, the load can be shed to other elements in the system.
The final choice of β is often decided by the regulators who are responsible for LRFD design
guidance documents.

8.4.4 Calculation of φ
There are two general approaches to calculate φ for a given load factor γ Q and a target β
value. The most robust method is MC simulation since this method can be used for any
distributions of load and resistance bias values. For the simple linear limit state equation
introduced earlier, there are closed-form solutions for φ if both bias value distributions are
normal distributed or both distributions are lognormal distributed.

8.4.4.1 MC simulation
MC simulation in the context of this chapter involves randomly sampling distributions of
load and resistance bias values using bias statistics that match the range on the CDF plots
that are of interest (i.e., the entire range of measured data or the tail). There are software
programs that can perform MC simulation very easily. However, in the text to follow, a
methodology that can be implemented within an Excel spreadsheet is described because
this approach highlights details that are valuable to novices and numerical outcomes can
be easily plotted.

• Select the number of simulations n and place i = 1 to n in column 1.
• Compute a column of random values of load bias X Qi and place in column 2. If the
distribution is normal, use NORM.INV(RAND(), μ XQ , σ XQ ). If the distribution is log-
normal, use LOGINV(RAND(), μ LNXQ , σ LNXQ ).
• Compute a column of random values of resistance bias X Ri and place in column 3. If
the distribution is normal, use NORM.INV(RAND(), μ XR , σ XR ). If the distribution is
lognormal, use LOGINV(RAND(), μ LNXR , σ LNXR ).
• Select values of load factor and resistance factor (γ Q and φ) and compute
g

γϕ/ − in column 4 for each pair of random numbers.
• Copy column of numbers in column 4 to column 5 and sort from minimum to maximum.

= ( ) X X

i Q Ri Qi