Application of Probability to Mechanical Design
131
A contradiction and for - sign
-N = -[U- 1]
0=+1
Another contradiction hence no structural factor of safety for the
Gaussian-Gaussian formulation.
The Gaussian-Weibull may be again
not be as severely limited. The conclusions and useful tools are shown
in Table 2.14 including calculation limitations for Gaussian distributions
H. Approximate Dimension Solution Using Cardsort and
Lower Material Bounds
Now attempt to formulate a means to use an upper bound on a
cardsort for s, the stress due to loads and set it equal to a lower bound
on the material properties. A cantilever beam sized in fatigue in Example
2.22 is used with the aa- 6m curve developing S for ductile titanium
Ti-16V-2.5A1 Eqs. (2.173)-(2.175) for the Weibull distribution and
(2.186) and (2.187) for the Gaussian distribution. The original tensile
strength data is Example 1.5 for 755 samples which fits both Weibull
and Gaussian curves equally well.
The Gaussian form of the stress due to the load, s, for a maximum
possible Smax is 6.0737 standard deviations above the mean, ~, with a probability of not being exceeded of approximately 1/109. This is developed
in Eqs. (2.190)-(2.192)
912.037
Smax -- {33
(2.205)
1
P(srnax >) -- 109
The Smax will be set equal to a lower value of the Gaussian and the Weibull
distributions.
Table 2.14 Calculations limitations for Gaussian
distributions
Gaussian-Gaussian
A. t < z-- Eq. (2.199)
ZS
with Pf _>10 -12 Table 2.3
N-1
B. t _< ~ Eq. (2.203)
Gaussian-Weibull
A. t not as limited here
B. again t not as limited