Civil Engineering Reference
In-Depth Information
where
M
is the number of times the resisting
force exceeded the displacing force (i.e. the
factor of safety is greater than 1.0).
the intended service life, an adequate margin of
safety against collapse under the maximum loads
that might reasonably occur. Second, the struc-
ture and its components must serve the designed
functions without excessive deformations and
deterioration. These two service levels are the ulti-
mate and serviceability limit states respectively
and are defined as follows:
An example of the use of Monte Carlo ana-
lysis to calculate the coefficient of reliability of
a slope against sliding is given in Section 6.6
in Chapter 6 on “Plane failure.” This example
shows the relationship between the determin-
istic and probabilistic analyses. The factor of
safety is calculated from the mean or most likely
values of the input variables, while the probab-
ilistic analysis calculates the distribution of the
factor of safety when selected input variables are
expressed as probability density functions. For
the unsupported slope, the deterministic factor
of safety has a value of 1.4, while the prob-
abilistic analysis shows that the factor of safety
can range from a minimum value of 0.69 to a
maximum value of 2.52. The proportion of this
distribution with a value less than 1.0 is 7.2%,
which represents the probability of failure of the
slope.
•
Ultimate limit state
—Collapse of the structure
and slope failure including instability due to
sliding, toppling and excessive weathering.
•
Serviceability limit state
—Onset of excessive
deformation and unacceptable deterioration.
The basis of LRFD design is the multiplication
of loads and resistances by factors that reflect the
degree of uncertainty and variability in the para-
meters. The requirement of the design is that the
factored resistance is equal to, or greater than,
the factored loads. This is stated in mathematical
terms as follows:
η
ij
γ
ij
Q
ij
φ
k
R
nk
≥
(1.21)
1.4.5 Load and Resistance Factor Design
where
φ
k
is the resistance factor and
R
nk
is the
nominal strength for the
k
th failure mode or ser-
viceability limit state,
η
ij
is the factor to account
for the ductility, redundancy and operational
importance of the element or system,
γ
ij
is the
load factor and
Q
ij
the member load effect for the
i
th load type in the
j
th load combination under
consideration.
In the application of equation (1.21), the
load factors are greater than unity unless the
load is beneficial to stability, and the resist-
ance factors are less than unity. On this basis,
the Mohr-Coulomb equation for the shear res-
istance
This design method is based on the use of prob-
ability theory to develop a rational design basis
for structural design that accounts for variabil-
ity in both loads and resistance. The objective
is to produce a uniform margin of safety for
steel and concrete structures such as bridges,
and geotechnical structures such as foundations
under different loading conditions. The LRFD
method has been developed in structural engin-
eering and is becoming widely used in the design
of major structures such as bridges (CSA, 1988;
Eurocode, 1995; AASHTO, 1996). In order that
foundations design is consistent with structural
design, the LRFD method has been extended to
include geotechnical engineering (Transportation
Research Board, 1999).
Some of the early LRFD work in geotechnical
engineering was carried out by Myerhoff (1984)
who used the term Limit States Design, and
defined the two limit states as follows. First, the
structure and its components must have, during
of
a
sliding
surface
is
expressed
as
follows:
τ
=
f
c
c
+
(σ
−
f
U
U)f
φ
tan
φ
(1.22)
The cohesion
c
, friction coefficient tan
φ
and
water pressure
U
are all multiplied by partial
factors with values less than unity, while the nor-
mal stress
σ
on the sliding surface is calculated
