Transcript Final Report - Tensile Strength of Composite Fibe+

Tensile Strength of
Composite Fibers
Author: Brian Russell
Date: December 4, 2008
SMRE - 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
Minitab Response for
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
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.
4.5
Rate
P er cent
2.0
H azard F unction
100
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
99
Percent
Fiber Length 265 mm
90
80
70
60
50
40
30
20
10
5
3
2
1
0.1
1.0
C1
10.0
3.119
1.922
50
0.773
>0.250
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 Response for each fiber length
Distribution Overview Plot for 25.4
Distribution Overview Plot for Length of 265mm
LSXY Estimates-Complete Data
LSXY Estimates-Complete Data
99.9
90
P er cent
0.4
0.2
0.0
1
2
50
10
1
3
0.5
1.0
C1
C1
S urv iv al F unction
0.2
0.0
1
1
2
3
2
C1
4
2
3
4
10
4
1
5
2
3
1 2 .7
4
S urv iv al F unction
H azard 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
90
0.6
0.4
0.2
0.0
5
100
2
3
4
50
10
1
5
5
3
5
2
4
S urv iv al F unction
H azard F unction
5
100
6
0
1
3
4
1 2 .7
5
50
2
3
4
1 2 .7
5
4
2
0
0
2
Rate
50
P er cent
Rate
2
P er cent
4
99.9
P er cent
50
P robability D ensity F unction
P DF
P er cent
P DF
90
1 2 .7
3
2 5 .4
LSXY Estimates-Complete Data
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
0.6
3
2
Distribution Overview Plot for 5
99
2
1
2 5 .4
LSXY Estimates-Complete Data
0.0
2
0
1
3
Distribution Overview Plot for 12.7
0.2
5
H azard F unction
50
C1
P robability D ensity F unction
2
2 5 .4
6
0
0.0
1
1
S urv iv al F unction
3.0
3
0.4
0.1
4
100
1.5
0
10
2 5 .4
P er cent
Rate
50
2
50
0.4
4.5
1
90
H azard F unction
100
P er cent
0.6
2.0
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
Rate
P DF
0.6
P robability D ensity F unction
P er cent
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
P DF
P robability D ensity F unction
0
2
3
4
5
5
2
3
4
5
5
Monte Carlo Analysis
Using the Minitab functions transformed 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
Reliability equations for all four lengths were calculated using Maple
(data for 265mm shown here)
Distribution
Plots from
Maple
Cumulative Distribution
Function
Reliability Function
Hazard Function
Probability Density
Function
MTTF Using 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
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
3.486495448
Results:
As the length of composite fibers increases
from 5 mm to 265 mm,
the tensile strength decreases.
References: Reliability Modeling, Prediction, and Optimization,
Wallace R. Blischke and D.N. Prabhakar Murthy, published 2000
by Wiley-Interscience Publication