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MAXIMUM RADIAL
STRESS OF A ROTATING DISK
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Description
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Limit
State Function
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Random
Variables
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Analysis
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Results
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Reference
The maximum radial stress of a
rotating disk may be obtained using the following formula:
In this
equation, (sr)max
is the maximum radial stress, w is the
rotor speed, ro is the
disk outer radius, ri is
the disk inner radius, r is density of
the disk material and n is the
Poisson’s ratio. Using this equation, perform the analyses described in the
Analysis section.
n
= X1: Poisson’s Ratio ~ Normal (m=0.30,
s=0.005)
r
= X2: Density (lb/in3) ~
Normal (m=0.284, s=0.002)
w
= X3: Rotor Speed (rpm) ~ Uniform (XL=10000, XU=11000)
ro
= X4: Outer Radius (in) ~ Normal (m=8,
s=0.02)
rI
= X5: Inner Radius (in) ~ Normal (m=2, s=0.01)
s
= X6: Stress ~ Normal (m=19600,
s=1000)
Anal1: Calculate the failure probability of Eq.
1 using different methods.
Case1: SORM-Point-Fitting Method.
Anal2: Compute the CDF of (sr)max
in Eq. 2 using different input methods
for Pf=0.000001~0.999999.
Case1: SORM-Point-Fitting Method.
Anal3: Plot (sr)max
as a function of b.
Back to Top
Results:
Analysis 1:
At MPP
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variable
names
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linearization-point
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directional
cosine in u-space
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x-space
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u-space
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Poisson
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2.99852E-01
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-2.96100E-02
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-0.02120
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Density
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2.83724E-01
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-1.37751E-01
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-0.09863
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Speed
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1.01673E+04
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-9.64813E-01
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-0.68938
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Rad_out
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7.99791E+00
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-1.04254E-01
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-0.07464
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Rad_in
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2.00013E+00
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1.30360E-02
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0.00933
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stress
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2.05988E+04
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1.00035E+00
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0.71338
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reliability index ..............................b
= 1.4008832E+00
1st-order failure probability
.............. pf1= 8.0624502E-02
generalized reliability
index ..........betag= 1.5343980E+00
second-order failure probability
......... pf2= 6.2465952E-02
Analysis 2: CDF/PDF analysis of
Limit State Function G22. range~0.000001 to 0.999999
***** SUMMARY OF CDF
ANALYSIS *****
First/Second-Order CDF of
Limit-State Function 22
Number of points in each CDF curve: 16
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1st-order
b
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1st-order Pf
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2nd-order
b
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2nd-order Pf
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G-Value
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5.1993393E+00
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9.9999086E-08
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5.3539988E+00
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4.3016139E-08
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1.9171374E+04
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4.1177859E+00
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1.9126495E-05
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4.2954358E+00
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8.7175789E-06
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1.9391860E+04
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3.1359905E+00
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8.5637363E-04
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3.3320330E+00
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4.3107215E-04
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1.9612348E+04
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2.3216950E+00
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1.0124682E-02
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2.5184647E+00
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5.8933946E-03
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1.9832832E+04
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1.5302928E+00
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6.2972129E-02
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1.5997255E+00
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5.4829803E-02
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2.0163569E+04
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1.0871957E+00
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1.3847517E-01
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1.1067049E+00
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1.3421087E-01
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2.0494299E+04
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5.3208184E-01
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2.9733466E-01
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5.3581676E-01
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2.9604267E-01
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2.1155751E+04
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1.1119670E-01
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4.5573019E-01
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1.1122957E-01
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4.5571717E-01
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2.1817221E+04
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-2.8428799E-01
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6.1190516E-01
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-2.8715080E-01
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6.1300154E-01
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2.2478683E+04
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-7.2255338E-01
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7.6502284E-01
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-7.3196846E-01
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7.6790601E-01
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2.3140145E+04
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-1.3473297E+00
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9.1106296E-01
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-1.3974104E+00
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9.1885484E-01
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2.3801603E+04
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-1.8901459E+00
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9.7063077E-01
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-2.0448389E+00
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9.7956459E-01
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2.4132333E+04
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-2.7617483E+00
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9.9712536E-01
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-2.9670011E+00
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9.9849640E-01
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2.4463058E+04
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-3.5063229E+00
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9.9977283E-01
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-3.6987562E+00
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9.9989167E-01
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2.4683554E+04
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-4.3294443E+00
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9.9999253E-01
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-4.5054323E+00
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9.9999669E-01
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2.4904042E+04
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-5.1993389E+00
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9.9999990E-01
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-5.3574080E+00
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9.9999996E-01
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2.5124530E+04
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CDF of Limit State Function G22:
**** SUMMARY OF PDF
ANALYSIS *****
First/Second-Order PDF of Limit-State Function 22
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1st-order
b
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1st-order Pf
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2nd-order
b
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2nd-order Pf
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G-Value
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-5.0675097E-03
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2.7260592E-09
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-1.6540487E-01
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3.9342947E-08
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1.9171374E+04
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-4.7180609E-03
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3.9146251E-07
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-2.4872392E-02
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9.7744884E-07
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1.9391860E+04
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-4.1361861E-03
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1.2077842E-05
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-8.6152292E-03
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1.3344795E-05
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1.9612348E+04
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-3.1916584E-03
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8.5989357E-05
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-5.8971922E-03
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9.8687742E-05
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1.9832832E+04
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-1.7133724E-03
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2.1195675E-04
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-1.7481288E-03
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1.9398908E-04
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2.0163569E+04
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-1.0729079E-03
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2.3703127E-04
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-1.1242423E-03
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2.4311345E-04
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2.0494299E+04
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-6.9482876E-04
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2.4060855E-04
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-7.0321496E-04
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2.4302742E-04
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2.1155751E+04
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-6.0013136E-04
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2.3794217E-04
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-6.0428503E-04
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2.3958816E-04
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2.1817221E+04
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-6.1083593E-04
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2.3403719E-04
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-6.1642104E-04
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2.3598398E-04
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2.2478683E+04
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-7.4306090E-04
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2.2833133E-04
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-7.5755163E-04
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2.3119563E-04
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2.3140145E+04
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-1.2792796E-03
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2.0591536E-04
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-1.4196053E-03
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2.1332518E-04
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2.3801603E+04
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-2.0974239E-03
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1.4021836E-04
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-2.4417524E-03
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1.2040364E-04
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2.4132333E+04
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-3.1339928E-03
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2.7591222E-05
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-7.5406140E-03
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3.6876431E-05
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2.4463058E+04
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-3.5857028E-03
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3.0606306E-06
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-7.9724187E-03
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3.4021448E-06
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2.4683554E+04
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-3.8579423E-03
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1.3092989E-07
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-1.5743764E-02
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2.4556363E-07
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2.4904042E+04
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-4.0212093E-03
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2.1632077E-09
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-6.3494324E-02
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1.4829412E-08
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2.5124530E+04
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PDF of Limit State Function G22:
No. of g-functions called
= 724
No. of derivatives of g called =
39
Analysis 3: Plot of Beta1 as a function of (sr)max
.
Budynas, Richard G.,
Advanced Strength and Applied Stress Analysis, McGraw-Hill Book Company,
New York, 1977, pp. 142-148.
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