tensile strength of composite fibers author: brian russell date: december 4, 2008 smre - reliability...
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Tensile Strength of Composite Fibers
Author: Brian RussellDate: December 4, 2008SMRE - Reliability Project
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:• Estimate fiber strength after incorporation into the
composite.• 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.
Methodology used for Analysis: • Data will be imported to Minitab so that
mathematical manipulation can be performed to produce transfer functions.
• Using Excel, Monte Carlo simulations was performed to simulate a larger population
• Equations were manipulated using Maple to produces the appropriate Reliability functions and display the data graphically.
• The Results of the Monte Carlo was compared to the Maple Results
321
0.6
0.4
0.2
0.0
C1
PD
F
2.01.00.5
99.9
90
50
10
1
C1
Perc
ent
321
100
50
0
C1
Perc
ent
321
4.5
3.0
1.5
0.0
C1
Rate
Correlation 0.969
Shape 3.11972Scale 1.92163Mean 1.71903StDev 0.603230Median 1.70863IQR 0.844805Failure 50Censor 0AD* 0.924
Table of StatisticsProbability Density Function
Survival Function Hazard Function
Distribution Overview Plot for Length of 265mmLSXY Estimates-Complete Data
Weibull
10.01.00.1
99.9
99
908070605040
30
20
10
5
3
2
1
C1
Perc
ent
Shape 3.119Scale 1.922N 50AD 0.773P-Value >0.250
Probability Plot of C1Weibull - 95% CI
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 = 1.992The shape parameter is: b = 3.119So the Weibull function that fits this data isF=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 Response for Fiber Length 265
mm
Minitab Response for each fiber length
321
0.6
0.4
0.2
0.0
C1
PD
F
2.01.00.5
99.9
90
50
10
1
C1
Perc
ent
321
100
50
0
C1
Perc
ent
321
4.5
3.0
1.5
0.0
C1
Rate
Correlation 0.969
Shape 3.11972Scale 1.92163Mean 1.71903StDev 0.603230Median 1.70863IQR 0.844805Failure 50Censor 0AD* 0.924
Table of StatisticsProbability Density Function
Survival Function Hazard Function
Distribution Overview Plot for Length of 265mmLSXY Estimates-Complete Data
Weibull
4321
0.6
0.4
0.2
0.0
25.4
PD
F
521
99.9
90
50
10
1
0.1
25.4
Perc
ent
4321
100
50
0
25.4
Perc
ent
4321
6
4
2
0
25.4
Rate
Correlation 0.997
Shape 4.86156Scale 3.10592Mean 2.84712StDev 0.669072Median 2.88037IQR 0.918007Failure 64Censor 0AD* 0.391
Table of StatisticsProbability Density Function
Survival Function Hazard Function
Distribution Overview Plot for 25.4LSXY Estimates-Complete Data
Weibull
5432
0.6
0.4
0.2
0.0
12.7
PD
F
5432
99
90
50
10
1
12.7
Perc
ent
5432
100
50
0
12.7
Perc
ent
5432
2
1
0
12.7
Rate
Correlation 0.977
Loc 1.10693Scale 0.215072Mean 3.09585StDev 0.673604Median 3.02507IQR 0.880738Failure 50Censor 0AD* 1.084
Table of StatisticsProbability Density Function
Survival Function Hazard Function
Distribution Overview Plot for 12.7LSXY Estimates-Complete Data
Lognormal
5432
0.6
0.4
0.2
0.0
5
PD
F
5432
99.9
90
50
10
1
5
Perc
ent
5432
100
50
0
5
Perc
ent
5432
6
4
2
0
5
Rate
Correlation 0.980
Shape 7.19240Scale 3.72481Mean 3.48921StDev 0.571679Median 3.53976IQR 0.765493Failure 50Censor 0AD* 0.775
Table of StatisticsProbability Density Function
Survival Function Hazard Function
Distribution Overview Plot for 5LSXY Estimates-Complete Data
Weibull
Monte Carlo Analysis
265 25.4 12.7mm 5mm
F:=1-exp(-(t/1.992)^3.119) F:=1-exp(-(t/3.106)^4.862) F:=1-exp(-(t/3.32943)^5.85190) F:=1-exp(-(t/3.72481)^7.19240)t=-1.992*ln(1-3.119(F)) t=-3.106*ln(1-4.862(F)) t=-3.32943*ln(1-5.85190(F)) t=-3.72481*ln(1-7.19240(F))
1.784042003 2.817144112 3.081436469 3.486495448
Using the Minitab functions transformed in Excel:
Cumulative Distribution Function Reliability Function
Probability Density Function
Hazard Function
Distribution Plots from Maple
The data shows that as the fiber length increases, the Mean Time To Failure (MTTF) decreases. A fiber of length 5mm has a MTTF of 3.5 seconds compared to a fiber of length 265 inches has a MTTF of 1.8 seconds.
Monte Carlo analysis was performed using the following equations in Excel:265 25.4 12.7mm 5mm
F:=1-exp(-(t/1.992)^3.119) F:=1-exp(-(t/3.106)^4.862) F:=1-exp(-(t/3.32943)^5.85190) F:=1-exp(-(t/3.72481)^7.19240)t=-1.992*ln(1-3.119(F)) t=-3.106*ln(1-4.862(F)) t=-3.32943*ln(1-5.85190(F)) t=-3.72481*ln(1-7.19240(F))
1.784042003 2.817144112 3.081436469 3.486495448
The MTTF values in Excel match the values calculated in Maple.
Fiber Length (mm) Maple Excel Monte-Carlo
265 1.781963 1.779479205
25.4 2.847213 2.893593614
12.7 3.084489 3.081029868
5 3.489212 3.482595755