Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

79
Frequency
Larger γ factors combined with
smaller ϕ factors place the strength
distribution R well beyond the load
distribution Q. In this illustration,
the probability of failure is P F = 0.
Frequency
distribution
of load Q
121
190
222
180
128
Even considering the inherent
variability in load effects and
member strengths, the
weakest member in this
example is strong enough to
withstand the largest load
effect if overly conservative
ϕ and γ factors are used.
210
440
Frequency
distribution of
resistance R
272
LOAd ANd RESISTANCE
FACTOR dESIgN
3
22
58
55
20
1
52
20
1 5
Load effect Q
Resistance R
FIGURE 4.4 Overly conservative load and resistance factors produce designs with a near-zero
probability of failure.
than error or incompetence.) A probability of failure P F = 0.022 represents too much risk
to be acceptable, particularly where public safety is directly concerned.
An appropriate combination of φ and γ i factors creates a small region of overlap between
R and Q (Figure 4.6). From the figure, the probability of failure is 1 out of 1,000 truss
members, or P F = 0.001. This rate might represent an acceptable trade-off between risk and
cost. (The value P F = 0.001 is known as a notional failure rate. The true failure rate is
always much less, as engineering experience has shown over years of successful practice.
In reliability analyses, often only the means and standard deviations of many variables can
be estimated and the true shape of the random variable distributions is generally not known.
These and other considerations lead to higher predicted failure rates than actually occur in
practice.)
Frequency
Smaller γ factors combined with
larger ϕ factors allow too much
overlap between the strength
distribution R and the load
distribution Q. The probability of
failure shown here is P F = 0.022.
Frequency
distribution
of load Q
121
190
222
180
128
210
440
Frequency
distribution of
resistance R
272
If the ϕ and γ factors used in
the design process are not
conservative enough, some
properly designed members
will still fail. In this example,
the strength of 22 members is
less than or equal to the load
effect acting on them.
3
22
58
55
1
20
1
52
5
Load effect Q
Resistance R
FIGURE 4.5 Unconservative load and resistance factors produce an unacceptable probability of
failure.