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ME 571 Reliability Based Design

创建日期：2024-04-20 17:25:56

所属分类：数学mathematics

标签： ME 571

Reliability Based Design

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ME 571 Reliability Based Design

Introduction

Light aluminum alloys are used extensively for structural elements in aerospace and automotive
applications where weight is an important design metric. One such alloy that is commonly used
is the wrought Al 2024-T351 alloy, known for its fatigue strength. A commonly used approach
to model the life of this alloy under low cycle fatigue (less than ~10,000 cycles) is to use the
Coffin – Manson relation (Xue, 2008):

2
= (2 )

(1)
where, is the plastic strain range to which the material is subjected in each loading cycle
(see Figure 1)
is the number of load cycles to failure (2being the total number of reversals from
tension to compression)
and is the strain to failure under monotonic loading (corresponding to = 0.5).

Figure 1 Schematic of stress-strain loop for a specimen under cyclic loading. The area inside the
loop is the plastic work dissipated by the material per unit volume per cycle.
2

A set of twenty fatigue tests were performed on tension specimens of Al 2024-T351 and the
results are shown in Figure 2. Specimens were subjected to cyclic axial loads until failure or until
the stiffness dropped below a critical value. The loads were large enough to yield the specimens
and deform them plastically, to cause low cycle fatigue. The Coffin-Manson relation was used to
fit the data and the fit is shown as a dashed line. It is observed that the data has an inherent
curvature which the straight-line Coffin-Manson relation is not able to capture, resulting in a
poor fit.

Studies have shown that Eq. (1) is not very accurate for low failure cycles of below 200 cycles
(Xue, 2008). The relation is empirical and not based on foundational physics. Thus, the objective
of this project is to explore different models and develop an alternative approach for
characterizing low cycle fatigue. The models that will be studied here are:

1. Power law (Coffin-Manson) relation
2. Maximum Entropy distributions (Truncated Exponential and Left Truncated Normal)
3. Weibull distribution (also, a Max Entropy distribution, it turns out)

Figure 2: Coffin-Manson relation used to model experimental fatigue data for Al 2024-T351. The
curvature of the data is not captured by the relation
Damage Parameter

The inelastic dissipation in each loading cycle gives a measure of the damage to the specimen.
The material damage is described by a parameter, D, which ranges from 0 (undamaged) to 1
3

(failure). Following the famous Miner’s rule, for a test specimen that fails in cycles, we can
write the damage accumulated over cycles as:
=

(2)
The damage per each load reversal is then given as:
=

2
=
1
2
(3)
In this project, we will characterize fatigue failure using the damage parameter and the inelastic
dissipation per load reversal. Figure 1 shows a sample stress-strain loop, obtained from a single
loading cycle of a fatigue test. The area of each loop, , gives the plastic dissipation in each
cycle (or 2 load reversals).

= (4)
where, is the plastic dissipation in the material
and is the number of cycles to failure (2 is the number of load reversals)

Power Law Relation

One of the simplest models that can be applied is a power law relation between the damage
and the plastic dissipation per load cycle. The Coffin-Manson relation can be rewritten in terms
of the damage parameter to obtain one such relation. Rewriting Eq. (1), we get:
=
1
2
= (

2
)
−
1

(5)
This suggests an analogous power-law model in terms of the inelastic dissipation:
= (

2
)
−
1

(6)
where, is the inelastic dissipation in the material (area inside the loop in Figure 1),
is the number of cycles to failure (2 is the number of load reversals)
and is a constant related to the critical inelastic dissipation for monotonic failure.

Eq. (6) can now be used to model low cycle fatigue.

Weibull Distribution

The Weibull distribution has the form:
(; , ) = {

(

)
−1

−(

)

, ≥ 0
0 , < 0
(7)
where and are positive and called the characteristic life (scale) and shape parameters,
respectively. The corresponding CDF has the form:
4

(; , ) = {1 −
−(

)

, ≥ 0
0 , < 0
(8)

Maximum Entropy Distributions

Shannon (Shannon, 1948) proposed an information function, (), that quantifies the data
content of an event. For an event with probability :
() = − ln (9)
For a set of probable events, the expected value of the information function has a form
analogous to the Gibb’s entropy function (Jaynes, 1957) used in thermodynamics, and is hence
called Shannon’s information entropy function, ():
() = (()) = − ∑ ln (10)
Jaynes (Jaynes, 1957) proposed that the best choice of a distribution, in the absence of any
other information, is one that maximizes Eq. (10) subject to the constraint:
∑ = 1 (11)
Such a distribution will have the least amount of bias and will hence provide a superior fit. The
distribution can be computed by solving an optimization problem, with additional information
about the distribution (e.g., the mean) added as constraints.

The maximum entropy principle can be applied to the Weibull distribution discussed in the
previous section to obtain unbiased models for fatigue modeling. The entropy function of the
Weibull distribution can be evaluated from Eqs. (7) and (10), and is given by:
() = (1 −
1

) + ln

+ 1 (12)
where ≈ 0.577 is called the Euler-Mascheroni constant.

Truncated Exponential Distribution

Maximizing Eq. (12) with the constraint of the mean fixed at (()) = gives the exponential
distribution:
() = {
1

−

, ≥ 0
0 , < 0
(13)
Since the damage parameter which is being modeled reaches unity at a finite number of cycles
(or the reliability will be zero sooner than = ∞), a truncated form of Eq. (13) would be more
appropriate. If the truncation occurs at = > 0, the truncated distribution is given by:
5

() = {
1

1

−

, 0 ≤ ≤
0 , ℎ

() = {
0 , < 0
1

(1 −
−

), 0 ≤ ≤
1, >
(14)
where, = (1 −
−

) is a correction factor for the truncation. Note that is the mean of the
parent exponential distribution and not for the truncated distribution.

Left Truncated Normal Distribution

Study shows that with proper choice of moment functions, a truncated normal distribution can
also be derived from the maximum entropy principle. Young et al (Young & Subbarayan, 2019)
provided the following probability density function for left truncated normal distribution which
is truncated at t = 0 shown in Figure 3:
(; , ) = {
(
1
1−(0,,)
)
1
√2

−
−2
22 , > 0
0 , ℎ
(15)
and cumulative distribution function:
(; , ) = {
(,,)−(0,,)
1−(0,,)
, > 0
0 , ℎ
(16)
where the factor in the denominator of the CDF is the area correction factor to make up the
density lost for < 0.
6

Figure 3: Left Truncated normal distribution plotted with the parent (non-truncated) normal
distribution

Problem Statement

Twenty low cycle fatigue tests were performed on aluminum tension specimens of Al 2024-
T351, and the life to failure is summarized in Table 1. Stress-strain curves similar to Figure 1
were obtained, and the inelastic dissipation per load reversal was estimated from the area
enclosed in the stress loops and is listed in Table 1.

Question 1: Four different fatigue models have been proposed to describe the LCF failure data
listed in Table 1. These are:
i. Coffin-Manson based power law relation (Eq. (6))
ii. Weibull distribution (Eq. (8))
iii. Truncated Exponential distribution (Eq. (14))
iv. Left Truncated Normal distribution (Eq. (16))

Fit each of the above models (CDF for ii, iii and iv) to the given fatigue failure data. Summarize
your model fit parameters in a table. Discuss how well each model fits the data. Compute the
sum of the squared error (sum of the square of the difference between the fit and the data
point at each in Figure 2). Which model would you choose to best capture the features of
the data? Justify your response.
7

Question 2: Since there are only 20 data points, the Kolmogorov-Smirnov test is the preferred
test for goodness-of-fit. Use the K-S test to estimate the goodness of fit and level of significance
for the fits of Q1.

Question 3: A engine cylinder head made of aluminum (Al 2024-T351) is to be designed for
fatigue resistance. A tension specimen made of the material is subjected to 10 cycles of load
similar to that experienced in the real engine and the inelastic dissipation computed from the
stress-strain loops is 102.3 MPa per load reversal. Estimate the fatigue life of the engine
cylinder head using the models and estimated parameters of Q1.

Question 4: A set of 20 components were tested under operational use conditions. The test
was stopped once 10 components failed. The number of cycles to failure for the failed
components are listed in Table 2. Post-failure analysis using a scanning electron microscope
(SEM) found that some of the failed components contained relatively large impurities from
which fatigue cracks initiated. The components with impurities are marked in Table 2 with a √
symbol. Determine the mean time to failure for samples that contained impurities.

Report

Document the fitting results and errors in a neatly written text or typed report. Include a single
plot containing the final fits for each of the fatigue models. Give a brief summary of the
concepts learned in this project. The written report should be no more than two pages long
with additional tables and figures included as an appendix. The report should be single spaced
and use 12 pt font.

Grading rubric: Q1 50 pts, Q2 10 pts, Q3 10 pts, Q4 20 pts, Report 10 pts.

Hint: Fitting the Truncated Exponential Function

The fit for the truncated exponential function is nonlinear, and the equation is valid only for 0 <
x < a. The following tips may help with the fitting process:
1. In order to reduce the degree of nonlinearity, define
1

as a new parameter (say ).
E.g. change −

into . This improves the stability of the fitting algorithm leading
to better fits.
2. One way to handle the truncation is to define a function in MATLAB that takes in
, , as input and returns 1 if > or evaluates Eq. (14) for 0 < < (such as
myFunc(x,a,m)). This function can then be used to define the fit type as:
ftype = fittype('myFunc(x,a,m)');

8

3. To improve the stability of the fitting algorithm, the robust fitting option can be
enabled as:
fit(x,y,ftype,'Robust','on')
When robust fitting is enabled, MATLAB assigns smaller weights to the input data
outliers which reduces their influence on the fit. This can also improve the
convergence of the fit.

Table 1: Experimental data from 20 low cycle fatigue tests on Al 2024-T351 tension specimens
Test No
Fatigue Life
( )
Inelastic Dissipation per
Reversal (/)
Damage per Cycle
( = /)
1 1 292.032 1
2 1 424.008 1
3 1315 45.2088 0.00076046
4 649 33.4152 0.00154083
5 2156 31.4496 0.00046382
6 2506 30.0456 0.00039904
7 3312 26.3952 0.00030193
8 1981 21.06 0.0005048
9 7692 20.0772 0.00013001
10 9795 13.87152 0.00010209
11 16102 10.44576 6.2104E-05
12 9024 8.64864 0.00011082
13 18905 8.17128 5.2896E-05
14 13999 7.94664 7.1434E-05
15 10916 7.94664 9.1609E-05
16 49039 3.42576 2.0392E-05
17 28156 3.53808 3.5516E-05
18 102651 1.367496 9.7417E-06
19 286190 0.2552472 3.4942E-06
20 409951 0.0693576 2.4393E-06

Table 2: Number of cycles to failure for the failed components in Q4.Components with the impurities are
marked with √.
Test No. Failure Life Impurity
1 40 √
2 67 √
3 83 √
4 257 ×
5 359 √
6 470 √
7 712 ×
9

8 824 ×
9 905 √
10 1109 ×

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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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