|
BURST MARGIN OF A
ROTATING DISK
-
Description
-
Limit
State Function
-
Random
Variables
-
Analysis
-
Results
-
Reference
The Burst Margin, Mb, may be computed using the
equation shown below. This margin is used to evaluate the integrity of a
rotating disk (e.g., a burst margin of less than one would result in an
unacceptable design).
where:
UTS:
Ultimate Tensile Strength (lb/in2)
d
: Density (lb/in3)
w
: Rotor Speed (rpm)
R:
Outer Radius (in)
Ro:
Inner Radius (in)
MUF: Material Utilization Factor (the
Material Utilization Factor is a safety factor that accounts for uncertainties
and unknown material properties)
B:
Burst
Margin
Limit-State
Function:
Consider
a disk defined by the random variables presented here. Evaluate the integrity of
the disk by:
Case 1:
X1 =
UTS ~ Normal (m=220000,
s=5000)
X2 =
d
~ Uniform (XL=0.28, XU=0.30)
X3 =
w
~ Normal (m=21000,
s=1000)
X4 =
R ~ Normal (m=24,
s=0.5)
X5 =
Ro ~ Normal (m=8,
s=0.3)
X6 = MUF ~ Weibull (m=0.925, s=0.0722,
XL=0.)
X7 = B ~ Normal (m=0.25,
s=0.025)
Case 2:
X1 = UTS
~ Normal (m=220000,
s=5000)
X2 = d
~ Normal (m=0.29, s=0.005)
X3 = w
~ Uniform (XL=18000, XU=24000)
X4 = R
~ Uniform (XL=22.5, XU=25.5)
X5 = Ro
~ Uniform (XL=7, XU=9)
X6 = MUF ~ Uniform (XL=0.8, XU=1.05)
X7 = B ~ Normal (m=0.25,
s=0.025)
X1 = UTS
~ Normal (m=220000,
s=5000)
X2 =
d
~ Normal (m=0.29, s=0.005
X3 = w
~ Normal (m=21000, s=1000)
X4 = R
~ Uniform (XL=22.5, XL=25.5)
X5 = Ro
~ Uniform (XL=7, XU=9)
X6 = MUF ~ Uniform (XL=0.8, XU=1.05)
X7 = B ~ Normal (m=0.25,
s=0.025)
Case 4:
X1 = UTS
~ Normal (m=220000,
s=5000)
X2 = d
~ Normal (m=0.29, s=0.005)
X3 = w
~ Weibull (m=21000, s=1000,
XL=0)
X4
= R ~ Uniform (XL=22.5,
XU=25.5)
X5
= Ro ~ Uniform (XL=7,
XU=9)
X6
= MUF ~ Uniform (XL=0.8, XU=1.05)
X7
= B ~ Normal (m=0.25,
s=0.025)
Anal1:
Calculate the failure probability of the limit-state function in Eq.
1 using different methods.
Case1:
For Random Variables - Case 1
Case2:
For Random Variables - Case 2
Case3:
For Random Variables - Case 3
Case4:
For Random Variables - Case 4
Anal2:
Compute the CDF of Mb defined in Eq. 2 using
different input methods for Pf = 0.000001~0.999999.
Case1: For
Random Variables - Case 1
Case2: For Random
Variables - Case 2
Case3: For Random
Variables - Case 3
Case4: For Random
Variables - Case 4
Anal3:
Plot Mb as a function of b.
Analysis 1: Probability analysis using FORM based Mod. U Based/X
Linearized MPPL method for MPP identification
|
CASE
|
MPP
Method
|
Pf
value
|
No.
of g
|
No.
of Ñg
|
|
Case
1
|
Mod.
U Based/X Linearized MPPL
|
1.1075294E-06
|
9
|
1
|
|
Case
2
|
Mod.
U Based/X Linearized MPPL
|
4.7834045E-07
|
45
|
5
|
|
Case
3
|
Mod.
U Based/X Linearized MPPL
|
4.5081743E-08
|
48
|
5
|
|
Case
4
|
Mod.
U Based/X Linearized MPPL
|
4.1658884E-08 |
9
|
1
|
Analysis 2: CDF/PDF
analysis of Limit State Function G24. Range ~ 0.000001 to 0.999999
CASE 1
***** SUMMARY OF CDF
ANALYSIS *****
First-Order CDF of Limit-State Function 24 Number of points in each CDF curve: 14
|
1st-order b
|
1st-order Pf
|
G-Value
|
|
4.7534259E+00
|
9.9999191E-07
|
2.8259281E-01
|
|
3.9085765E+00
|
4.6420773E-05
|
3.1944725E-01
|
|
2.9940038E+00
|
1.3767126E-03
|
3.5630226E-01
|
|
1.9430977E+00
|
2.6002177E-02
|
3.9315691E-01
|
|
1.3496309E+00
|
8.8567196E-02
|
4.1169660E-01
|
|
7.3459877E-01
|
2.3129194E-01
|
4.3001165E-01
|
|
1.0043804E-01
|
4.5999828E-01
|
4.4862332E-01
|
|
-5.1503357E-01
|
6.9673522E-01
|
4.6686636E-01
|
|
-1.1283178E+00
|
8.7040714E-01
|
4.8550327E-01
|
|
-1.7081869E+00
|
9.5619917E-01
|
5.0372107E-01
|
|
-2.2787615E+00
|
9.8865938E-01
|
5.2234997E-01
|
|
-2.8142213E+00
|
9.9755522E-01
|
5.4057578E-01
|
|
-3.8280541E+00
|
9.9993542E-01
|
5.7743048E-01
|
|
-4.7534259E+00
|
9.9999900E-01
|
6.1428519E-01
|
CDF of Limit State Function G24:
***** SUMMARY OF PDF
ANALYSIS *****
First-Order PDF of Limit-State
Function 24
|
1st-order
b
|
1st-order Pf
|
G-Value
|
|
-2.2429170E+01
|
1.1098613E-04
|
2.8259281E-01
|
|
-2.3616615E+01
|
4.5370729E-03
|
3.1944725E-01
|
|
-2.6314863E+01
|
1.1873822E-01
|
3.5630226E-01
|
|
-3.0944098E+01
|
1.8690484E+00
|
3.9315691E-01
|
|
-3.2983565E+01
|
5.2926499E+00
|
4.1169660E-01
|
|
-3.4014994E+01
|
1.0360963E+01
|
4.3001165E-01
|
|
-3.4020272E+01
|
1.3503841E+01
|
4.4862332E-01
|
|
-3.3375787E+01
|
1.1661145E+01
|
4.6686636E-01
|
|
-3.2389590E+01
|
6.8369929E+00
|
4.8550327E-01
|
|
-3.1237911E+01
|
2.8971939E+00
|
5.0372107E-01
|
|
-3.0000103E+01
|
8.9215567E-01
|
5.2234997E-01
|
|
-2.8745424E+01
|
2.1862866E-01
|
5.4057578E-01
|
|
-2.6273514E+01
|
6.8921222E-03
|
5.7743048E-01
|
|
-2.3969272E+01
|
1.1860701E-04
|
6.1428519E-01
|
PDF of Limit
State Function G24:
No. of g-functions called
=
290
No. of derivatives of g called =
32
CASE 2
***** SUMMARY OF CDF
ANALYSIS *****
First-Order CDF of Limit-State
Function 24
Number of points in each CDF curve: 14
|
1st-order
b
|
1st-order Pf
|
G-Value
|
|
4.7534259E+00
|
9.9999191E-07
|
3.3071327E-01
|
|
3.5420543E+00
|
1.9851184E-04
|
3.4140753E-01
|
|
2.7114297E+00
|
3.3496869E-03
|
3.5211726E-01
|
|
2.1367601E+00
|
1.6308756E-02
|
3.6284201E-01
|
|
1.5285524E+00
|
6.3187721E-02
|
3.7883641E-01
|
|
1.0737881E+00
|
1.4145883E-01
|
3.9487698E-01
|
|
3.6060989E-01
|
3.5919555E-01
|
4.2848305E-01
|
|
-1.8253822E-01
|
5.7241982E-01
|
4.6048635E-01
|
|
-6.7453283E-01
|
7.5001369E-01
|
4.9110563E-01
|
|
-1.2354367E+00
|
8.9166599E-01
|
5.2318255E-01
|
|
-1.9316639E+00
|
9.7329950E-01
|
5.5523804E-01
|
|
-2.9536897E+00
|
9.9843000E-01
|
5.8728522E-01
|
|
-3.7274652E+00
|
9.9990329E-01
|
6.0337941E-01
|
|
-4.7534259E+00
|
9.9999900E-01
|
6.1942018E-01
|
CDF of Limit State
Function G24:
***** SUMMARY OF PDF
ANALYSIS *****
First-Order PDF of Limit-State
Function 24
|
1st-order
b
|
1st-order Pf
|
G-Value
|
|
-1.3219760E+02
|
6.5415264E-04
|
3.3071327E-01
|
|
-9.4154299E+01
|
7.0857952E-02
|
3.4140753E-01
|
|
-6.3358923E+01
|
6.4015279E-01
|
3.5211726E-01
|
|
-4.5592499E+01
|
1.8550894E+00
|
3.6284201E-01
|
|
-3.2146080E+01
|
3.9873057E+00
|
3.7883641E-01
|
|
-2.5231821E+01
|
5.6556606E+00
|
3.9487698E-01
|
|
-1.8356022E+01
|
6.8620032E+00
|
4.2848305E-01
|
|
-1.6101903E+01
|
6.3175964E+00
|
4.6048635E-01
|
|
-1.6389690E+01
|
5.2081081E+00
|
4.9110563E-01
|
|
-1.9024822E+01
|
3.5383273E+00
|
5.2318255E-01
|
|
-2.5288881E+01
|
1.5616823E+00
|
5.5523804E-01
|
|
-4.1057245E+01
|
2.0885553E-01
|
5.8728522E-01
|
|
-5.5808658E+01
|
2.1407631E-02
|
6.0337941E-01
|
|
-7.1798299E+01
|
3.5527911E-04
|
6.1942018E-01
|
PDF of Limit state Function G24:
No. of g-functions called
= 257
No. of derivatives of g called = 34
CASE 3
***** SUMMARY OF CDF
ANALYSIS *****
First-Order
CDF of Limit-State Function 24
Number of points in each CDF curve:
12
|
1st-order
b
|
1st-order
Pf
|
G-Value
|
|
4.7534259E+00
|
9.9999191E-07
|
3.2915046E-01
|
|
3.7710646E+00
|
8.1276294E-05
|
3.4612047E-01
|
|
2.9133047E+00
|
1.7881274E-03
|
3.6307256E-01
|
|
2.1706624E+00
|
1.4978352E-02
|
3.8002736E-01
|
|
1.5318613E+00
|
6.2778338E-02
|
3.9698299E-01
|
|
4.8954768E-01
|
3.1222700E-01
|
4.3089384E-01
|
|
-4.2968573E-01
|
6.6628787E-01
|
4.6602345E-01
|
|
-1.2726829E+00
|
8.9843470E-01
|
4.9872047E-01
|
|
-2.1724286E+00
|
9.8508832E-01
|
5.3263357E-01
|
|
-3.0703854E+00
|
9.9893109E-01
|
5.6648627E-01
|
|
-3.9379408E+00
|
9.9995891E-01
|
6.0044914E-01
|
|
-4.7534259E+00
|
9.9999900E-01
|
6.3437866E-01
|
CDF of Limit State Function G24:
***** SUMMARY OF PDF
ANALYSIS *****
First-Order PDF of Limit-State
Function 24
|
1st-order
b
|
1st-order Pf
|
G-Value
|
|
-6.1707642E+01
|
3.0534757E-04
|
3.2915046E-01
|
|
-5.4163040E+01
|
1.7643218E-02
|
3.4612047E-01
|
|
-4.7092199E+01
|
2.6968430E-01
|
3.6307256E-01
|
|
-4.0595422E+01
|
1.5354588E+00
|
3.8002736E-01
|
|
-3.4913324E+01
|
4.3086754E+00
|
3.9698299E-01
|
|
-2.7539832E+01
|
9.7460921E+00
|
4.3089384E-01
|
|
-2.5513991E+01
|
9.2810391E+00
|
4.6602345E-01
|
|
-2.6178226E+01
|
4.6465668E+00
|
4.9872047E-01
|
|
-2.6707053E+01
|
1.0062859E+00
|
5.3263357E-01
|
|
-2.6155898E+01
|
9.3621333E-02
|
5.6648627E-01
|
|
-2.4829211E+01
|
4.2509758E-03
|
6.0044914E-01
|
|
-2.3205913E+01
|
1.1482969E-04
|
6.3437866E-01
|
PDF of Limit State Function G24:
No. of g-functions called
=
211
No. of derivatives of g called =
28
CASE 4
***** SUMMARY OF CDF
ANALYSIS *****
First-Order CDF of Limit-State Function 24
Number of points in each CDF curve: 15
|
1st-order
b
|
1st-order Pf
|
G-Value
|
|
4.7534259E+00
|
9.9999191E-07
|
3.4716419E-01
|
|
3.6801130E+00
|
1.1656530E-04
|
3.5927187E-01
|
|
2.8288198E+00
|
2.3360000E-03
|
3.7134671E-01
|
|
2.1508625E+00
|
1.5743528E-02
|
3.8347870E-01
|
|
1.6064607E+00
|
5.4086358E-02
|
3.9552850E-01
|
|
7.5420307E-01
|
2.2536365E-01
|
4.1969145E-01
|
|
5.7283969E-02
|
4.7715949E-01
|
4.4408705E-01
|
|
-5.8704933E-01
|
7.2141471E-01
|
4.6874331E-01
|
|
-1.1694517E+00
|
8.7888916E-01
|
4.9222532E-01
|
|
-2.1870079E+00
|
9.8562903E-01
|
5.4057465E-01
|
|
-2.9415957E+00
|
9.9836737E-01
|
5.8892243E-01
|
|
-3.5254467E+00
|
9.9978862E-01
|
6.3727133E-01
|
|
-4.0017075E+00
|
9.9996856E-01
|
6.8562105E-01
|
|
-4.4044171E+00
|
9.9999470E-01
|
7.3397147E-01
|
|
-4.7534259E+00
|
9.9999900E-01
|
7.8232240E-01
|
CDF of Limit State Function G24:
***** SUMMARY OF PDF
ANALYSIS *****
First-Order PDF of Limit-State Function 24
|
1st-order
b
|
1st-order Pf
|
G-Value
|
|
-9.8516394E+01
|
4.8748811E-04
|
3.4716419E-01
|
|
-7.9098498E+01
|
3.6157871E-02
|
3.5927187E-01
|
|
-6.2536433E+01
|
4.5643914E-01
|
3.7134671E-01
|
|
-4.9899816E+01
|
1.9698828E+00
|
3.8347870E-01
|
|
-4.0981203E+01
|
4.4988281E+00
|
3.9552850E-01
|
|
-3.0921828E+01
|
9.2823306E+00
|
4.1969145E-01
|
|
-2.6926121E+01
|
1.0724358E+01
|
4.4408705E-01
|
|
-2.5444525E+01
|
8.5441650E+00
|
4.6874331E-01
|
|
-2.3892630E+01
|
4.8106050E+00
|
4.9222532E-01
|
|
-1.8007899E+01
|
6.5729005E-01
|
5.4057465E-01
|
|
-1.3535606E+01
|
7.1353596E-02
|
5.8892243E-01
|
|
-1.0800846E+01
|
8.6197310E-03
|
6.3727133E-01
|
|
-8.9982306E+00
|
1.1960365E-03
|
6.8562105E-01
|
|
-7.7171850E+00
|
1.8877890E-04
|
7.3397147E-01
|
|
-6.7594947E+00
|
3.3447969E-05
|
7.8232240E-01
|
PDF of Limit State Function G24:
No. of g-functions called
= 248
No. of derivatives of g called =
33
Analysis 3: Plot of
Beta Vs. Mb for CASE 4 above in analysis 2.
Raymond
J. Roark & Warren C. Young. Formulas
for Stress and Strain. McGraw-Hill Book Company, New York, 1975, pp. 572
| 
