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TOPIC
FACTOR OF SAFETY AND RELIABILITY INGEOTECHNICAL ENGINEERING
ASAD SULTAN
2011-MS-CEG-12
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FACTOR OF SAFETY
Factor of safety (FOS), also known as safety factor
(SF), is a term describing the structural capacity of a
system beyond the expected loads or actual loads.
Factor Of Safety used in conventional geotechnical
practice are based on experience.
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It is common to use the same value of FOS for a
given type of application, such as long termslope stability, without regard to the degree of
uncertainty involved in its calculation.
Through regulation and tradition, the samevalue of FOS is often applied to conditions
that involve widely varying degrees ofuncertainty.
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RELIABILITYThe probability that a system performs satisfactorily
the intended function under specified operating
conditions, during its design period.
Reliability calculations provide a means of evaluating
the combined effects of uncertainties, and a means
of distinguishing between conditions where
uncertainties are particularly high or low.
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EXAMPLERETAINING WALL STABILITY
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Factor of Safety against Sliding(FSS):
The factor of safety against sliding on the sand layer beneath
the footing is given by the following formula:
Where,
W = weight of wall and backfill over the heel of the wall (lb/ft or kN/m)
tan = tangent of friction angle between base of wall and sand
E = earth pressure force on vertical plane through heel of wall (lb/ft or kN/m).
For the conditions shown in Fig.1, the value of FSS is 1.50. This is called the most
likely value of factor of safety, FMLV.
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Uncertainty in Factor of Safety against Sliding:
The terms involved in computing Fss [W, tan , and E]all involve some degree of uncertainty. Therefore the
computed value of Fss also involves some uncertainty.
It is useful to be able to assess the reliability of Fss,
as well as a best estimate of its value. This can be
done using the Taylor series method, which involvesthese steps:
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1. Estimate the standard deviations of the quantities involved in
computing Fss.
Using methods explained next, the following values of standard
deviation of the parameters involved in this example have beenestimated: efp = standard deviation of the equivalent fluidpressure = 5 pcf (0.785 kN/m); tan = standard deviation oftan = 0.05; bf= standard deviation of the unit weight of
backfill = 7 pcf (1.099 kN/m); and c = standard deviation ofthe unit weight of concrete = 2 pcf (0.314 kN/m).
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METHODS OF ESTIMATING STANDARD DEVIATION
a)-Computation from Data:
If sufficient data are available, the formula definition of canbe used to calculate its value,
= standard deviation
xi= ith value of the parameter (x)x = average value of the parameter x
N = number of values of x (the size of the sample).
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b)-Published Values:One approach to estimating values of standard deviation when
sufficient data is not available to calculate is to use estimates
based on published values, which are most conveniently
expressed in terms of the coefficient of variation,V:
from which the standard deviation can be computed:
It is important to use judgment in applying values ofVfrom published sources, and to consider as well aspossible the degree of uncertainty in the particular caseat hand.
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2-Use the Taylor series technique to estimate the standarddeviation and the coefficient of variation of the factor of
safety using these formulas:
FMLV = most likely value of factor of safety, computed usingbest estimate values for all of the parameters. For thisexample, FMLV = 1.50.
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In which F = (F - F ). F is the factor of safety calculated
with the value of the first parameter (in this case the
equivalent fluid pressure) increased by one standard
deviation from its best estimate value. F is the factor of
safety calculated with the value of the first parameter
decreased by one standard deviation. In calculating F and
F, the values of all of the other variables are kept at theirmost likely values. The values ofF, F, and F are
calculated by varying the values of the other three variables
(footing/sand friction angle, backfill unit weight, and
concrete unit weight) by plus and minus one standard
deviation from their most likely values. The results of these
calculations are shown in Table 1.
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3-Substituting the value ofF into (2a), the value of
the standard deviation of the factor of safety (F) isfound to be 0.25, and the coefficient of variation of the
factor of safety (VF), calculated using (2b), is found to
be 17%, as shown at the bottom of Table 1.
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4-With both FMLV and VF known, the probability of failure and
the reliability of the factor of safety can be determined usingTable 2. Table 2 assumes a lognormal distribution of factor of
safety values, which is usually a reasonable approximation.
A lognormal distribution is a set of numbers whose
logarithm is normally distributed. So if you take the normal
curve, then take eto the power of each value, that's what
the new curve will look like.
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The retaining wall shown in Fig. 1 has a most likely value of
factor of safety against sliding (FMLV) equal to 1.50, and a
coefficient of variation of factor of safety (VF ) equal to 17%. At
the intersection of these values in Table 2, it can be seen thatthe probability of failure is about 1%. This equates to a reliability
of about 99%.
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Inspite the fact that reliability has potential value,
reliability theory has not been used much in routinegeotechnical practice.
There are two reasons for this.
First, reliability theory involves terms and concepts which
are not familiar to most geotechnical engineers.
Second, it is commonly perceived that using reliabilitytheory would require more data, time and effort than are
available in most circumstances.
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SUGGESTION:
It is suggested that FOS and Reliability be usedtogether, as complementary measures of
acceptable design.
The reliability analyses require only modest extra
effort as compared to that required to evaluate
FOS, but they will add considerable value to the
results of the analysis.
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THANKS
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