Recommended Procedure

16.8 Recommended Procedure

We list at this point some desirable attributes of a reliability-based design procedure:

1. It should account for the pertinent capacity and demand factors, their components, and their interactions.

2. It should produce outputs that can be related to the expected performance during the design life of the system under consideration.

3. It should employ as input into formulations quantities, parameters, or material characterizations that can be ascertained within the present state of the art.

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4. It should not disregard indices currently considered to be pertinent, such as factor of safety or reliability index. It should serve to supplement this knowledge and reduce uncertainty.

5. Ideally, mathematical computations should be reduced to a minimum. All of the above can be accommodated by an extension of the point estimate method. The recom-

mended procedure is as follows, where applicable:

1. Using PEM, or an equally valid probabilistic formulation, obtain the expected values and standard deviations of the capacity and demand functions: E [C], E [D], s [C], s [D].

2. Calculate the expected value and standard deviation of the safety margin, E [S], s [S].

3. Fit a beta distribution (and normal distribution, as a check) to the safety margin, using appropriate upper and lower bounds. If unknown, take them as E [S] ± 3 s [S]; see Table 16.2 .

4. Obtain the probability of failure, p (f )=P [S £ 0], the reliability, central factor of safety, and reliability index, as appropriate.

Example 16.13

The ultimate bearing capacity Q per unit length of a long footing of width B founded at a depth D below the ground surface, Fig. 16.11(a) , is

Q = --------N gB g + gDBN q + cBN c

where g is the unit weight of the soil, c is the c-parameter of strength (sometimes called the cohesion). The dimensionless factors N g ,N q , and N c , called bearing capacity factors, are functions of the friction angle f as given by the tabulated values in Fig. 16.11(b) .

a)

b) Bearing capacity factors

FIGURE 16.11 Example 16.13.

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(a) A very long footing 8 ft wide is founded at a depth of 4 ft in a soil with the parameters

Parameter, x

Expected Value

Standard Deviation

x (+)

x (–)

g 110 lb/ft 3 0 110

c 200 lb/ft 2 80 lb/ft 2 280

The correlation coefficient r ( f, c) = – 0.50. Estimate the expected value and standard deviation of the bearing capacity.

(b) If a central factor of safety (CFS) of 4 is required and it is assumed that V(D) = 50%, estimate the probability of failure.

Solution. (a) Forming the required values, as the bearing capacity factors are functions of f only. From Fig. 16.11(b) ,

N g (+) = 109.41 N g (–) = 22.40 N q (+) = 64.20 N q (–) = 18.40 N c (+) = 75.31 N c (–) = 30.14

Forming the respective values, Q( f, c) in tons,

EQ [] = Q = SQ ij ( )p ij () = 227.3 tons/ft

sQ [] = 127.40 tons,

VQ () = 56%

(b) For a CFS = 4, — D = 56.8. As V(D) = 50%, — s [D] = 28.4. Forming the characteristics of the safety — margin, with r(Q, D) = +3/4, we have E[S] = C – D = 170.50, s [S] = 143.72, S min = –152.75, S max =

493.75, b = 170.50/107.75 = 1.58.

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If S is taken at a beta variate p (f ) = 0.059 If S is taken to be normal

p (f ) = 0.057