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

78
dESIgN CONCEPTS
Frequency
distribution
of load Q
222
190
180
121
128
58
55
22
20
3
1
–50% –30% –10%
–40% –20% Q* +10% +20% +30% +50%
+40%
FIGURE 4.2 Histogram of load effects.
210
440
272
52
20
1 5
–30%
–20%
–10% R* +10% +20% +30%
FIGURE 4.3 Resistance
histogram.
Frequency
distribution of
resistance R
a histogram showing the frequency of occurrence of differing resistance levels (Figure 4.3).
For example, in 210 out of 1,000 cases, the maximum tension strength in the truss member
was 10 percent less than the nominal strength predicted by the design calculations.
A structural component will not fail as long as the strength provided by the component
is greater than the effect caused by the loads. In LRFD, the general format for a strength
design provision is expressed as
φR
≥∑ γ Q
(4.4)
n i ni
where φ = resistance factor corresponding to the type of component (i.e., beam, column,
connection, etc.), R n = nominal component resistance (i.e., strength), γ i = load factors corresponding
to each type of load (i.e., dead load, live load, etc.), and Q ni = nominal service
load effects (such as axial force, shear force, and bending moments) for each type of load.
In general, the resistance factors φ are less than 1 and the load factors γ i are greater than 1.
In nontechnical language, the resistance of the structural component is underrated (to account
for the possibility that the actual member strength may be less than predicted)
whereas the load effect on the member is overrated (to account for extreme load events
made possible by the variability inherent in the loads).
Regardless of the design philosophy, a properly designed component must be stronger
than the load effects acting on it. In LRFD, however, the process of establishing appropriate
design factors considers the member resistance R and load effect Q as random variables
rather than quantities that are known exactly. Suitable factors for use in LRFD design equations,
as typified by Equation (4.4), are determined through a process that takes into account
the relative positions of the member resistance distribution R (Figure 4.3) and the
load effects distribution Q (Figure 4.2). Appropriate values of the φ and γ i factors are determined
through a procedure known as code calibration that uses a reliability analysis in
which the φ and γ i factors are chosen so that a specific target probability of failure is
achieved. The design strength of members is based on the load effects; therefore, the design
factors “shift” the resistance distribution to the right of the load distribution so that the
strength is greater than the load effect (Figure 4.4).
To illustrate this concept, consider the data obtained from the 1,000-bridge example.
The use of very small φ factors and very large γ i factors would ensure that all truss members
are strong enough to withstand all load effects (Figure 4.4). This situation, however, would
be overly conservative and might produce structures that are unnecessarily expensive.
The use of relatively large φ factors and relatively small γ i factors would create a region
in which the resistance distribution R and the load distribution Q overlap (Figure 4.5);
in other words, the member strength will be less than or equal to the load effect. From
Figure 4.5, one would predict that 22 out of 1,000 truss members will fail. (Note: The truss
members are properly designed. The failure discussed here is due to random variation rather