SMRE - Reliability Project
Title: Tensile Strength of Fibers
Author: Brian Russell
Date: November 20, 2008
Objective:
Using data provided by “Reliability Modeling, Prediction, and Optimization”, Case
2.6, “Tensile Strength of Fibers” I will explore the tensile strength of silicon
carbide fibers after extraction from a ceramic matrix.
Description of System:
This experiment is an attempt to estimate fiber strength after incorporation into
the composite, rather than modeling composite strength and estimating each
component separately, which is the usual approach in estimating composite
strength. The difficulty with the latter is that it is necessary to account for the
interaction between the two materials in determining strength (see Fox and
Walls, 1997).
Fiber strength is measured as stress applied until fracture failure of the fiber.
The objective of the experiment was to determine the distribution of failures as a
function of gauge length of the fiber after incorporation into the composite.
Data from “Reliability Modeling, Prediction, and Optimization”, Case 2.6
0.36
0.50
0.57
0.95
0.99
1.09
1.09
1.33
1.33
1.37
1.38
1.38
1.39
1.41
1.42
265
1.93
1.96
1.97
1.99
2.04
2.06
2.06
2.08
2.11
2.26
2.27
2.27
2.38
2.39
2.47
1.25
1.50
1.57
1.85
1.92
1.94
2.00
2.02
2.13
2.17
2.17
2.20
2.23
2.24
2.30
Fiber length (mm)
25.4
2.81
1.96
2.82
1.98
2.90
2.06
2.92
2.07
2.93
2.07
3.02
2.11
3.11
2.22
3.11
2.25
3.14
2.39
3.20
2.42
3.20
2.63
3.22
2.67
3.26
2.75
3.29
2.75
3.30
2.75
12.7
3.29
3.30
3.36
3.39
3.39
3.41
3.41
3.43
3.52
3.72
3.96
4.07
4.09
4.18
4.13
5
2.36
2.40
2.54
2.67
2.68
2.69
2.70
2.77
2.77
2.79
2.83
2.91
3.04
3.05
3.06
3.81
3.88
3.93
3.94
3.94
3.94
3.70
4.04
4.07
4.08
4.08
4.16
4.18
4.22
4.24
1.42
1.45
1.49
1.50
1.56
1.57
1.57
1.75
1.78
1.79
1.79
1.82
1.83
1.86
1.89
1.90
1.92
2.48
2.73
2.74
2.33
2.42
2.43
2.45
2.49
2.51
2.54
2.57
2.62
2.66
2.68
2.71
2.72
2.76
2.79
2.79
2.80
3.34
3.35
3.37
3.43
3.43
3.47
3.61
3.61
3.62
3.64
3.72
3.79
3.84
3.93
4.03
4.07
4.13
2.89
2.93
2.95
2.96
2.97
3.00
3.03
3.04
3.05
3.07
3.08
3.13
3.20
3.22
3.23
3.26
3.27
4.14
4.15
4.29
3.24
3.27
3.28
3.34
3.36
3.39
3.51
3.53
3.59
3.63
3.64
3.64
3.66
3.71
3.73
3.75
3.78
4.35
4.37
4.50
Methodology used for Analysis:
Data was imported to Minitab so that mathematical manipulation could be
performed to produce transfer functions.
Based on the amount of data, Monte Carlo simulations will be performed to
simulate a larger population
Equations will be manipulated using Maple to produces the appropriate Reliability
functions and display the data graphically.
Expected Outcome:
The data will show if there is a significant difference in the strength of carbon
fibers after extraction of different lengths.
References:
Reliability Modeling, Prediction, and Optimization, Wallace R. Blischke and D.N.
Prabhakar Murthy, published 2000 by Wiley-Interscience Publication
Minitab Project Report
For Fiber Length 265 mm
Distribution ID Plot: C1
Goodness-of-Fit
Distribution
Weibull
Lognormal
Exponential
Normal
Anderson-Darling
(adj)
0.924
2.112
22.663
0.630
Correlation
Coefficient
0.969
0.913
*
0.988
Table of Percentiles
Percent
1
1
1
1
Percentile
0.439830
0.664794
0.0116532
0.456895
Standard
Error
0.100323
0.0648694
0.0013559
0.151184
95% Normal CI
Lower
Upper
0.281274
0.687764
0.549071
0.804908
0.0092769 0.0146382
0.160579
0.753211
Weibull
Lognormal
Exponential
Normal
5
5
5
5
0.741634
0.860701
0.0594737
0.824807
0.113691
0.0677524
0.0069202
0.119756
0.549164
0.737646
0.0473458
0.590089
1.00156
1.00428
0.0747082
1.05953
Weibull
Lognormal
Exponential
Normal
10
10
10
10
0.934107
0.987750
0.122164
1.02094
0.113298
0.0690738
0.0142146
0.104864
0.736469
0.861236
0.0972520
0.815410
1.18478
1.13285
0.153457
1.22647
Weibull
Lognormal
Exponential
Normal
50
50
50
50
1.70863
1.60536
0.803692
1.7128
0.0897044
0.0860398
0.0935155
0.0763479
1.54155
1.44528
0.639803
1.56316
1.89381
1.78317
1.00956
1.86244
Distribution
Weibull
Lognormal
Exponential
Normal
Table of MTTF
Distribution
Weibull
Lognormal
Exponential
Normal
Mean
1.71903
1.72489
1.15948
1.71280
Standard
Error
0.084254
0.095243
0.134914
0.076348
95% Normal CI
Lower
Upper
1.56158 1.89235
1.54796 1.92204
0.92304 1.45649
1.56316 1.86244
Probability Plot for Length of 265
LSXY Estimates-Complete Data
Weibull
99
90
90
50
P er cent
P er cent
C orrelation C oefficient
Weibull
0.969
Lognormal
0.913
E xponential
*
N ormal
0.988
Lognormal
99.9
10
50
10
1
0.5
1.0
C1
1
2.0
1
E xponential
N ormal
99.9
99
90
90
50
P er cent
P er cent
10
C1
10
50
10
1
0.01
0.10
1.00
10.00
1
0
C1
1
2
3
C1
The graphs show that Weibull, Lognormal and Normal distributions are good fits
to the data.
Distribution Overview Plot: C1
Goodness-of-Fit
Distribution
Weibull
Anderson-Darling
(adj)
0.924
Correlation
Coefficient
0.969
Distribution Overview Plot for Length of 265mm
LSXY Estimates-Complete Data
P robability D ensity F unction
Table of S tatistics
S hape
3.11972
S cale
1.92163
M ean
1.71903
S tDev
0.603230
M edian
1.70863
IQ R
0.844805
F ailure
50
C ensor
0
A D*
0.924
C orrelation
0.969
Weibull
99.9
90
50
P er cent
P DF
0.6
0.4
0.2
0.0
1
2
10
1
3
0.5
1.0
C1
C1
S urv iv al F unction
H azard F unction
100
4.5
Rate
P er cent
2.0
50
3.0
1.5
0
0.0
1
2
3
1
C1
2
3
C1
Probability Plot of C1
Weibull - 95% CI
99.9
Shape
Scale
N
AD
P-Value
Percent
99
90
80
70
60
50
40
3.119
1.922
50
0.773
>0.250
30
20
10
5
3
2
1
0.1
1.0
C1
10.0
This Minitab plot shows that the response at length 265 mm fits a Weibull well with
Shape of 3.119 and scale of 1.922.
The scale parameter is: a = 1.992
The shape parameter is: b = 3.119
So the Weibull function that fits this data is
F=1-exp (-(t/a) ^b)
F:=1-exp(-(t/1.992)^3.119)
To perform the Monte Carlo Simulation in Excel, this expression is first transformed to:
t=-1.992*ln(1-3.119(F))
Minitab Project Report
For Fiber Length 25.4 mm
Distribution ID Plot: 25.4
Goodness-of-Fit
Distribution
Weibull
Lognormal
Exponential
Normal
Anderson-Darling
(adj)
0.391
0.696
36.043
0.342
Correlation
Coefficient
0.997
0.980
*
0.996
Table of Percentiles
Percent
1
1
1
1
Percentile
1.20572
1.52980
0.0183122
1.26452
Standard
Error
0.127267
0.0947980
0.0018304
0.169025
95% Normal CI
Lower
Upper
0.980397
1.48284
1.35484
1.72735
0.0150542 0.0222752
0.933232
1.59580
Weibull
Lognormal
Exponential
Normal
5
5
5
5
1.68599
1.81960
0.0934589
1.72884
0.124415
0.0896237
0.0093417
0.133828
1.45896
1.65215
0.0768315
1.46654
1.94836
2.00401
0.113685
1.99114
Weibull
Lognormal
Exponential
Normal
10
10
10
10
1.95506
1.99590
0.191972
1.97637
0.117821
0.0863227
0.0191885
0.117142
1.73725
1.83368
0.157818
1.74677
2.20017
2.17247
0.233518
2.20596
Weibull
Lognormal
Exponential
Normal
50
50
50
50
2.88037
2.76578
1.26295
2.84953
0.0873777
0.0880059
0.126238
0.0851665
2.71411
2.59856
1.03825
2.68261
3.05682
2.94376
1.53627
3.01645
Distribution
Weibull
Lognormal
Exponential
Normal
Table of MTTF
Distribution
Weibull
Lognormal
Exponential
Normal
Standard
Error
0.083858
0.092411
0.182122
0.085167
Mean
2.84712
2.85686
1.82205
2.84953
95% Normal CI
Lower
Upper
2.68742 3.01632
2.68136 3.04385
1.49788 2.21637
2.68261 3.01645
Probability Plot for 25.4
LSXY Estimates-Complete Data
C orrelation C oefficient
Weibull
0.997
Lognormal
0.980
E xponential
*
N ormal
0.996
Lognormal
99.9
99.9
90
99
50
90
P er cent
P er cent
Weibull
10
50
1
10
0.1
0.1
1
1
2
2 5 .4
5
2
E xponential
N ormal
99.9
99.9
90
99
50
90
P er cent
P er cent
5
2 5 .4
10
1
50
10
1
0.1
0.001
0.010
0.100
2 5 .4
1.000
10.000
0.1
1.0
Goodness-of-Fit
Anderson-Darling
(adj)
0.391
4.0
2 5 .4
Distribution Overview Plot: 25.4
Distribution
Weibull
2.5
Correlation
Coefficient
0.997
5.5
Distribution Overview Plot for 25.4
LSXY Estimates-Complete Data
P robability D ensity F unction
0.6
90
50
P DF
P er cent
0.4
0.2
0.0
10
1
1
2
3
0.1
4
1
2
2 5 .4
2 5 .4
S urv iv al F unction
5
H azard F unction
6
100
4
Rate
P er cent
Table of S tatistics
S hape
4.86156
S cale
3.10592
M ean
2.84712
S tDev
0.669072
M edian
2.88037
IQ R
0.918007
F ailure
64
C ensor
0
A D*
0.391
C orrelation
0.997
Weibull
99.9
50
0
2
0
1
2
3
4
1
2
2 5 .4
3
4
2 5 .4
Probability Plot of 25.4
Weibull - 95% CI
Percent
99.9
99
Shape
Scale
N
AD
P-Value
90
80
70
60
50
40
30
20
4.862
3.106
64
0.178
>0.250
10
5
3
2
1
0.1
5
0.
6
0.
7
8 9 0
0. 0 . 0 . 1.
5
1.
25.4
0
2.
0
3.
0
4.
0
5.
This Minitab plot shows that the response at length 25.4 mm fits a Weibull well with
shape of 4.862 and scale of 3.106.
The scale parameter is: a = 3.106
The shape parameter is: b = 4.862
So the Weibull function that fits this data is
F=1-exp(-(t/a)^b)
F:=1-exp(-(t/3.106)^4.862)
To perform the Monte Carlo Simulation in Excel, this expression is first transformed to:
t=-3.106*ln(1-4.862(F))
Minitab Project Report
For Fiber Length 12.7 mm
Distribution ID Plot: 12.7
Goodness-of-Fit
Distribution
Weibull
Lognormal
Exponential
Normal
Anderson-Darling
(adj)
1.234
1.084
30.193
0.851
Correlation
Coefficient
0.974
0.977
*
0.983
Table of Percentiles
Percent
1
1
1
1
Percentile
1.51695
1.83419
0.0196892
1.59493
Standard
Error
0.138867
0.109073
0.0022165
0.179155
95% Normal CI
Lower
Upper
1.26780
1.81507
1.63240
2.06093
0.0157909 0.0245499
1.24380
1.94607
Weibull
Lognormal
Exponential
Normal
5
5
5
5
2.00419
2.12373
0.100487
2.03343
0.130730
0.100300
0.0113121
0.142080
1.76366
1.93597
0.0805909
1.75496
2.27751
2.32970
0.125294
2.31191
Weibull
Lognormal
Exponential
Normal
10
10
10
10
2.26652
2.29632
0.206408
2.26720
0.122440
0.0951671
0.0232359
0.124531
2.03881
2.11717
0.165540
2.02312
2.51966
2.49063
0.257364
2.51127
Weibull
Lognormal
Exponential
Normal
50
50
50
50
3.12730
3.02507
1.35792
3.0918
0.0899931
0.0920098
0.152864
0.0909962
2.95579
2.85001
1.08906
2.91345
3.30875
3.21089
1.69315
3.27015
Distribution
Weibull
Lognormal
Exponential
Normal
Table of MTTF
Distribution
Weibull
Mean
3.08449
Standard
Error
0.087560
95% Normal CI
Lower
Upper
2.91756 3.26096
Lognormal
Exponential
Normal
3.09585
1.95906
3.09180
0.095292
0.220537
0.090996
2.91461
1.57118
2.91345
3.28837
2.44270
3.27015
Probability Plot for 12.7
LSXY Estimates-Complete Data
Weibull
99.9
99
90
90
50
P er cent
P er cent
C orrelation C oefficient
Weibull
0.974
Lognormal
0.977
E xponential
*
N ormal
0.983
Lognormal
10
50
10
1
2
3
4
1
5
2
1 2 .7
E xponential
4
5
N ormal
99.9
99
90
90
50
P er cent
P er cent
3
1 2 .7
10
50
10
1
0.01
0.10
1.00
1 2 .7
10.00
1
2
Distribution Overview Plot: 12.7
Goodness-of-Fit
Distribution
Weibull
Anderson-Darling
(adj)
1.234
Correlation
Coefficient
0.974
3
1 2 .7
4
5
Distribution Overview Plot for 12.7
LSXY Estimates-Complete Data
P robability D ensity F unction
Table of S tatistics
S hape
5.85190
S cale
3.32943
M ean
3.08449
S tDev
0.611609
M edian
3.12730
IQ R
0.829597
F ailure
50
C ensor
0
A D*
1.234
C orrelation
0.974
Weibull
99.9
90
P er cent
P DF
0.6
0.4
0.2
0.0
2
3
1 2 .7
50
10
1
4
2
S urv iv al F unction
4
H azard F unction
100
6
Rate
P er cent
3
1 2 .7
50
4
2
0
0
2
3
1 2 .7
4
2
Distribution Overview Plot: 12.7
Goodness-of-Fit
Distribution
Lognormal
Anderson-Darling
(adj)
1.084
Correlation
Coefficient
0.977
3
1 2 .7
4
5
Distribution Overview Plot for 12.7
LSXY Estimates-Complete Data
P robability D ensity F unction
Table of S tatistics
Loc
1.10693
S cale
0.215072
M ean
3.09585
S tDev
0.673604
M edian
3.02507
IQ R
0.880738
F ailure
50
C ensor
0
A D*
1.084
C orrelation
0.977
Lognormal
99
0.6
P er cent
P DF
90
0.4
0.2
0.0
50
10
2
3
4
1
5
1 2 .7
2
3
1 2 .7
4
S urv iv al F unction
H azard F unction
5
100
Rate
P er cent
2
50
0
1
0
2
3
4
5
2
3
1 2 .7
4
5
1 2 .7
Probability Plot of 12.7
Weibull - 95% CI
99.9
Shape
Scale
N
AD
P-Value
Percent
99
90
80
70
60
50
40
30
20
10
5
3
2
1
1.5
2.0
3.0
12.7
4.0
5.0
5.852
3.329
50
0.911
>0.250
Probability Plot of 12.7
Lognormal - 95% CI
99
Loc
Scale
N
AD
P-Value
95
90
1.107
0.2151
50
0.873
>0.250
Percent
80
70
60
50
40
30
20
10
5
1
2
3
12.7
4
5
6
The Minitab plot shows that the response at length 12.7 mm fits a Lognormal plot well
with a correlation value of .977, but also fits a Weibull distribution with a correlation
value of .974 and shape of 5.82 and scale of 3.329.
The scale parameter is: a = 3.32943
The shape parameter is: b = 5.85190
So the Weibull function that fits this data is
F=1-exp(-(t/a)^b)
F:=1-exp(-(t/3.32943)^5.85190)
To perform the Monte Carlo Simulation in Excel, this expression is first transformed to:
t=-3.32943*ln(1-5.85190(F))
Distribution ID Plot: 5
For Fiber Length 5 mm
Goodness-of-Fit
Distribution
Anderson-Darling
(adj)
Correlation
Coefficient
Weibull
Lognormal
Exponential
Normal
0.775
1.280
33.879
0.924
0.980
0.974
*
0.983
Table of Percentiles
Percent
1
1
1
1
Percentile
1.96488
2.30149
0.0215009
2.13513
Standard
Error
0.163005
0.110021
0.0023790
0.162786
95% Normal CI
Lower
Upper
1.67002
2.31181
2.09565
2.52756
0.0173090 0.0267080
1.81607
2.45418
Weibull
Lognormal
Exponential
Normal
5
5
5
5
2.46466
2.59024
0.109733
2.53343
0.142729
0.0984042
0.0121418
0.129089
2.20021
2.40438
0.0883391
2.28042
2.76090
2.79047
0.136308
2.78644
Weibull
Lognormal
Exponential
Normal
10
10
10
10
2.72409
2.75870
0.225400
2.74576
0.128618
0.0920145
0.0249401
0.113138
2.48332
2.58412
0.181456
2.52401
2.98821
2.94507
0.279987
2.96751
Weibull
Lognormal
Exponential
Normal
50
50
50
50
3.53976
3.44533
1.48287
3.49476
0.0826591
0.0845028
0.164076
0.0826535
3.38140
3.28362
1.19376
3.33276
3.70553
3.61499
1.84198
3.65676
Distribution
Weibull
Lognormal
Exponential
Normal
Table of MTTF
Distribution
Weibull
Lognormal
Exponential
Normal
Mean
3.48921
3.49753
2.13932
3.49476
Standard
Error
0.081561
0.086448
0.236712
0.082653
95% Normal CI
Lower
Upper
3.33296 3.65279
3.33214 3.67114
1.72223 2.65742
3.33276 3.65676
Probability Plot for 5
LSXY Estimates-Complete Data
Weibull
99
90
90
50
P er cent
P er cent
C orrelation C oefficient
Weibull
0.980
Lognormal
0.974
E xponential
*
N ormal
0.983
Lognormal
99.9
10
50
10
1
2
3
5
4
1
5
2
3
E xponential
5
N ormal
99.9
99
90
90
50
P er cent
P er cent
4
5
10
50
10
1
0.01
0.10
1.00
10.00
1
2
5
Goodness-of-Fit
Anderson-Darling
(adj)
0.775
4
5
Distribution Overview Plot: 5
Distribution
Weibull
3
Correlation
Coefficient
0.980
5
Distribution Overview Plot for 5
LSXY Estimates-Complete Data
P robability D ensity F unction
Table of S tatistics
S hape
7.19240
S cale
3.72481
M ean
3.48921
S tDev
0.571679
M edian
3.53976
IQ R
0.765493
F ailure
50
C ensor
0
A D*
0.775
C orrelation
0.980
Weibull
99.9
90
P er cent
P DF
0.6
0.4
0.2
0.0
2
3
4
50
10
1
5
5
2
3
5
4
S urv iv al F unction
H azard F unction
5
100
Rate
P er cent
6
50
4
2
0
0
2
3
4
5
2
3
5
4
5
5
Probability Plot of 5
Weibull - 95% CI
99.9
Shape
Scale
N
AD
P-Value
Percent
99
90
80
70
60
50
40
7.192
3.725
50
0.512
>0.250
30
20
10
5
3
2
1
1.5
2.0
2.5
3.0
3.5
4.0
4.5 5.0 5.5
5
This Minitab plot shows that the response at length 5. mm fits a Weibull well with
shape of 7.192 and scale of 3.725.
The scale parameter is: a = 3.72481
The shape parameter is: b = 7.19240
So the Weibull function that fits this data is
F=1-exp(-(t/a)^b)
F:=1-exp(-(t/3.72481)^7.19240)
To perform the Monte Carlo Simulation in Excel, this expression is first transformed to:
t=-3.72481*ln(1-7.19240(F))
Next, enter the cumulative distribution function (F(t)) for each value of length into Maple
and solve for all of the reliability functions. The following results are obtained:
For Fibers of length 265mm
>
>
>
>
>
>
>
>
For Fibers of length 265mm
>
>
>
>
>
>
>
>
For Fibers of length 12.7mm
>
>
>
>
>
>
>
>
For Fibers of length 5mm
>
>
>
>
>
>
>
>
Combined Analysis in Maple:
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
The data shows that as the fiber length increases, the Mean Time To Failue (MTTF)
decreases. A fiber of length 5mm has a MTTF of 3.5 seconds compared to a fiber of
length 2.65 inches has a MTTF of 1.8 seconds.
Monte Carlo analysis was performed using the following equations in Excel:
265
25.4
F:=1-exp(-(t/1.992)^3.119)
t=-1.992*ln(1-3.119(F))
12.7mm
F:=1-exp(-(t/3.106)^4.862)
t=-3.106*ln(1-4.862(F))
1.784042003
2.817144112
5mm
F:=1-exp(-(t/3.32943)^5.85190)
t=-3.32943*ln(1-5.85190(F))
F:=1-exp(-(t/3.72481)^7.19240)
t=-3.72481*ln(1-7.19240(F))
3.081436469
The MTTF values in Excel match the values calculated in Maple.
Fiber Length (mm)
265
25.4
12.7
5
Maple
1.781963
2.847213
3.084489
3.489212
Excel Monte-Carlo
1.779479205
2.893593614
3.081029868
3.482595755
From this information, a conclusion can be drawn that as the length of composite fibers
increases from 5 mm to 265 mm, the tensile strength decreases.
3.486495448