Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 380

4
3
2
1
0

0.8 0.85 0.9 0.95 1

-1
-2
-3
-4
-5

Limiting degree of
consolidation at t = 1 year

U £ 0.87
5%

U £ 0.94
95%

1.0
0.9
0.8
0.7

Mean U = 0.91

MC
From β
indices
= Φ(-β)

0.6
0.5
0.4
0.3
0.2
0.1
0 0.8

0.85 0.9 0.95 1

Degree of consolidation at t = 1 year

Figure 9.23 Reliability indices for different limiting U at t = 1 year, and comparison of CDF based on β indices
with CDF from Monte Carlo simulations.

9.10.5 obtaining PDF curves from β index
The CDF curves shown in the lower plots of Figures 9.22 through 9.24 were obtained eas-
ily—by Equations 9.17 through 9.19 —from the β values shown in the corresponding upper
plots. In addition, it is simple to obtain the PDF of the respective outputs ( s cf , U , and s r ), by
applying cubic spline interpolation (e.g., Kreyszig 1988) to the CDF. This is accomplished
easily in the Excel spreadsheet platform, as explained below with respect to the 19 CDF
values of Figure 9.22 :

i. Autofill a 17-cell column vector m of m i (i = 2-18) with the following formula, next to
the column of the 19 CDF values of s cf :

3

m = ( CDF − DF ), in whichisthe h s cf interval of theCDF point s ( =0. m)

i i + 1 i − 1

h

ii. A 17 × 17 tridiagonal matrix d is set up, with entries d i,i = 4, d i + 1 , i = d i , i +1 = 1, and all
other entries equal to 0.
iii. The 17 PDF values are obtained immediately and automatically upon entering
“ = mmult(minverse( d ), m )” as a spreadsheet array formula in a 17-cell column.

The PDF curve of is cf thus obtained is shown in Figure 9.25 . By the same procedure, 15
and 21 PDF values of the degree of consolidation (U) and of the settlement remaining (both