14-21
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 3.84051404 0.453491803 -0.076865 -0.6837712 3.035248
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.995297
T=6 H=2 L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 2 1.90054154 0.3 0.2983328 0.15 0.682022
T 6
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 1.97774941 0.298332762 -0.228812 -0.9166775 2.719496
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.10074612 0.983607058 2.1346932 1.85354006 0.405658
14-22
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 1.85354006 0.405658139 -0.354224 -0.9549809 2.804946
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.06140002 0.990412221 2.3420944 1.82499346 0.418409
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 1.82499346 0.41840877 -0.376877 -0.9621894 2.819317
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.05634405 0.991257766 2.3763641 1.81995807 0.420541
0.006759
T=6 H=3 L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 3 2.93369436 0.45 0.4474991 0.15 1.087487
T 6 6
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
14-23
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.96662412 0.447499143 -0.152542 -0.836938 3.00023
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.88891462 0.529930943 0.0750958 2.96662164 0.447533
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.96662164 0.447533388 -0.152554 -0.8369768 3.00023
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.88891428 0.529931619 0.0750981 2.96662137 0.447533
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.96662137 0.447533482 -0.152554 -0.8369771 3.000232
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.88891364 0.529932886 0.0751015 2.96662126 0.447534
14-24
Pf
0.464251
T=6 H=4 L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 4 3.95027077 0.6 0.5966655 0.15 1.375169
T 6
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.099751 3.99439614 0.519889336 0.0498756 0.99502485 0.099751
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 3.95549883 0.596665524 -0.114406 -0.7537317 3.322931
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.40658868 0.12127428 -1.168642 3.90219021 0.495669
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 3.90219021 0.495669017 -0.097655 -0.7351262 3.344071
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.42421494 0.130056005 -1.126127 3.90580834 0.498823
14-25
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 3.90580834 0.49882258 -0.098095 -0.7387732 3.341738
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.4222758 0.129068617 -1.130804 3.90541583 0.498475
T=7 H=2 L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 2 1.90054154 0.3 0.2983328 0.15 0.682022
T 7
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.149166 1.33352513 0.529726894 0.0745831 1.97774941 0.298333
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 1.97774941 0.298332762 -0.266948 -0.9367371 3.087345
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.0100596 0.998582906 2.9851799 1.71255645 0.460538
14-26
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 1.71255645 0.460537842 -0.549596 -0.979481 3.28554
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.00249464 0.999665846 3.4022378 1.61775964 0.490201
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 1.61775964 0.490201089 -0.655563 -0.9871162 3.325036
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.00187094 0.999751494 3.4825644 1.598668 0.495717
T=7 H=3 L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 3 2.93369436 0.45 0.4474991 0.15 1.087487
T 7
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
14-27
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.96662412 0.447499143 -0.177965 -0.8723162 3.354243
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.56836036 0.794698856 0.8228335 2.9425462 0.500341
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.9425462 0.500340508 -0.202249 -0.8891589 3.367436
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.55381522 0.802100895 0.8491497 2.94090091 0.502308
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.94090091 0.502307916 -0.203272 -0.8908184 3.3684
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.55275375 0.802634541 0.8510688 2.94077872 0.502452
14-28
Pf
0.169693
T=7 H=4 L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 4 3.95027077 0.6 0.5966655 0.15 1.375169
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.149166 0.66676257 0.529726894 0.0745831 3.95549883 0.596666
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 3.95549883 0.596665524 -0.133474 -0.8010189 3.655289
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.6359453 0.298205093 -0.529569 3.94403425 0.545246
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 3.94403425 0.54524634 -0.122682 -0.7890577 3.661764
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.63877741 0.302332481 -0.517704 3.94454029 0.546212
14-29
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 3.94454029 0.546212123 -0.122867 -0.7900333 3.661445
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.6386399 0.302128837 -0.518287 3.94451544 0.546165
Summary of results:
T H
Normal Lognormal
EPf EPf
5 2 1.31897 0.0936 1.35274 0.0881
Problem 14-12.
The performance function can be expressed as.
Z = L – T/(2H)
where T is a constant, and L and H are random variables. The advanced second moment method
can be used as provided below to compute for the maximum mean value of H so that the failure
14-30
For iterations:
Mean for equivalent normal = N
x
P
New design point = Mean for equivalent normal – (Directional cosine u Standard deviation of
equivalent normal u
E
)
T=5 H=? L=1
(1) (2) (3) (4) (5) (6) (7)
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 1.75 1.63633308 0.2625 0.2610412 0.15 0.54849
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.149166 1.52402872 0.529726894 0.0745832 1.73053072 0.261041
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
14-31
T SS = 0.2396622
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.20259389 0.969409913 1.8721901 1.64916753 0.34132
T=6 H=? L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 2.1 2.00527756 0.315 0.3132494 0.15 0.730812
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.149166 1.27002394 0.529726894 0.0745832 2.07663686 0.313249
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.07663686 0.313249398 -0.217917 -0.9092656 2.745822
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.16882704 0.969410156 1.8721937 1.97900064 0.409584
T=7 H=? L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 2.4 2.31711787 0.36 0.3579993 0.15 0.864343
T 7
14-32
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.37329927 0.357999312 -0.222457 -0.9124642 3.190242
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.1174306 0.976300038 1.9827079 2.24671724 0.475877
T=8 H=? L=1
Random
variable
Mean
Value
Mean for
equivalent
Normal
Standard
Deviation
Standard
Deviation of
Equivalent
Normal
Coefficient of
Variation Py
H 2.8 2.72895817 0.42 0.4176659 0.15 1.018494
Vy
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.149166 0.95251795 0.529726894 0.0745832 2.76884915 0.417666
Iteration Random
variable
Mean for
equivalent
normal
Standard
deviation for
equivalent
normal
Partial
derivative
Directional
cosine
New
design
point
H 2.76884915 0.417665864 -0.217917 -0.9092656 3.661096
Lognormal
Distribution,
fx
Lognormal
Cumulative, Fx )-1(Fx(x*)) Px
N Vx
N
0.12662 0.969410231 1.8721947 2.63866735 0.546112