Page 275 - Shigley's Mechanical Engineering Design

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250 Mechanical Engineering Design
Equation (5–40) is called the normal coupling equation. The reliability associated with
z is given by
1 u
∞
2
R = √ exp − du = 1 − F = 1 −
(z) (5–41)
x 2π 2
The body of Table A–10 gives R when z > 0 and (1 − R = F) when z ≤ 0. Noting that
¯ n = μ S /μ σ , square both sides of Eq. (5–40), and introduce C S and C σ where
C S =ˆσ S /μ S and C σ =ˆσ σ /μ σ . Solve the resulting quadratic for ¯n to obtain

2
2
1 ± 1 − 1 − z C 2 S 1 − z C σ 2
¯ n = 2 (5–42)
2
1 − z C
S
The plus sign is associated with R > 0.5, and the minus sign with R < 0.5.
Lognormal–Lognormal Case
Consider the lognormal distributions S = LN(μ S , ˆσ S ) and = LN(μ σ , ˆσ σ ).Ifwe
interfere their companion normals using Eqs. (20–18) and (20–19), we obtain

μ ln S = ln μ S − ln 1 + C 2
S
(strength)

2
ˆ σ ln S = ln 1 + C
S
and

μ ln σ = ln μ σ − ln 1 + C 2
σ
(stress)

ˆ σ ln σ = ln 1 + C 2 σ
Using Eq. (5–40) for interfering normal distributions gives

2
μ S 1 + C σ
ln 2
μ σ 1 + C
μ ln S − μ ln σ S (5–43)
z =−
ˆ σ 2 +ˆσ 2 1/2 =− ln 1 + C 2 1 + C 2

ln S ln σ S σ
The reliability R is expressed by Eq. (5–41). The design factor n is the random variable
that is the quotient of S/ . The quotient of lognormals is lognormal, so pursuing the
z variable of the lognormal n, we note

2
C + C 2
μ S S σ
μ n = C n = 2 ˆ σ n = C n μ n
μ σ 1 + C
σ
The companion normal to n = LN(μ n , ˆσ n ), from Eqs. (20–18) and (20–19), has a mean
and standard deviation of

μ y = ln μ n − ln 1 + C 2 ˆ σ y = ln 1 + C 2
n n
The z variable for the companion normal y distribution is

y − μ y
z =
ˆ σ y