Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 352

the Low and Tang (2004, 2007) spreadsheet procedures are available in the Low (2008a)
chapter, which will not be repeated here. Instead, the following sections discuss insights,
advantages, and subtleties associated with FORM RBD in various geotechnical engineering
examples, and the complementary roles that FORM RBD can play to LRFD or the design
based on EC7.

9.2 eXaMPle oF relIabIlItY-baSeD ShalloW
FounDatIon DeSIgn

Tomlinson (1995)'s Example 2.2 determines the factor of safety against bearing capacity
failure of a retaining wall that carries a horizontal load ( Q h ) of 300 kN/m applied at a point
2.5 m above the base and a centrally applied vertical load ( Q v ) of 1100 kN/m. The base
(5 × 25 m) of the retaining wall is founded at a depth of 1.8 m in a silty sand with friction
angle ϕ = 25°, cohesion c = 15 kN/m 2 , and unit weight γ = 21 kN/m 3 .
With respect to bearing capacity failure, the performance function ( PerFunc ) is

PerFunc

=−

qq (9.6a)

where

B ′

2

qcNsdi = + pNsdi + γ Nsdi

γγγγ

(9.6b)

u c ccc o qq qq

′ qQB

v

= (9.6c)

in which q u is the ultimate bearing capacity, q is the applied bearing pressure, c is the cohe-
sion of soil, p o is the effective overburden pressure at the foundation level, B ′ is the effective
width of the foundation, γ is the unit weight of soil below the base of the foundation, and N c ,
N q , and N γ are bearing capacity factors, which are functions of the friction angle (ϕ) of soil:

φ NN

c

= ( − 1 )cot( ) (9.7a)

q

φ

πφ tan

tan 2

Ne
q

= 45 + (9.7b)

2

N = 2 ( + 1 )tan φ (9.7c)

γ q

Several expressions for N γ exist. The above N γ is attributed to Vesic in Bowles (1996).
The nine factors s j , d j , and i j in Equation 9.6b account for the shape and depth effects of the
foundation and the inclination effect of the applied load. The formulas for these factors are
based on Tables 4.5a and b of Bowles (1996).
To illustrate RBD, Tomlinson's deterministic example is extended probabilistically here
( Figure 9.3 ). The width B of the foundation is to be determined based on a reliability index
β = 3.0 against bearing capacity failure. The parameters c , ϕ, Q h , and Q v are assumed to be
lognormal random variables with mean values equal to the values in Tomlinson's determinis-
tic example, and with coefficient of variation equal to 0.20, 0.1, 0.15, and 0.10, respectively.
The mean and standard deviation of these four variables are shown in Figure 9.3 . The random
variables are partially correlated, with the correlation matrix as shown in the figure. The
foundation width required to achieve a reliability index β of 3.0 is B = 4.51 m. The similarities
and differences between this RBD and the design based on EC7 or LRFD are discussed next.