Stress Strength Interference
-
Upload
coolguy1950 -
Category
Documents
-
view
131 -
download
8
Transcript of Stress Strength Interference
1
STRESS - STRENGTHINTERFERENCE
PROBABILITY OF FAILUREJohn Toksoy
Cummins Inc.November 7, 2002
Stress & Strength Interference 2
QUIZ 1!
§Which one of the materials would you choose for a higher fatigue strength?
§Would you pay a premium for material “D” over material “A”?
0
5
10
15
20
25
30
35
40
45
50
A B C D
Material Types
Fati
gue
Str
engt
h (k
si)
DESIGN MARGIN
=Strength
Design Stress
Stress & Strength Interference 3
VARIATION IS THE ENEMY…§ Would the additional information like the variation in
strength change your answers?
0
5
10
15
20
25
30
35
40
45
50
A B C D
Material Types
Fati
gue
Str
engt
h (k
si)
AVE
$1.5 / kg
$2.2 / kg
$2.6 / kg
$1.7 / kg
VA
RIA
TIO
N
§ Is material “C”better then “B”?
§ How would you quantify the differencebetween these materials?
§ Is design margin alone a good criteria to select a material?
Stress & Strength Interference 4
Early Practice - Design Margin§ Early practice in stress-strength relationship
dealt almost entirely along the lines of design margin. Factor of safety!
§Design margin approach use the mean value of stress & strength ignoring the natural scatter that each may possess.
§Utilization of design margin is justified whenn It is based on considerable experiencen Component design changes are not too different
than the existing design.ØGeometry, processing, function
Stress & Strength Interference 5
Recent Practice – Probability of Failure
§ The variation in stress and strength results in a statistical distribution and a natural scatter in these variables.
§When these two distributions interfere, that is when stress becomes higher than strength, failure results.
§Means of expressing these distributions in a practical engineering sense and means of calculating the resulting interference (probability of failure) is the heart of this seminar.
Stress & Strength Interference 6
Outline§Definition of failure - Unreliability
§ Reliability in simple terms
§ Part Strength & Stress
§Normal Distribution
§ Probability of failure
§ Reliability quantified
§ Example
Stress & Strength Interference 7
Definition of Failure - Unreliability
§ Failuren The inability to meet customer required function
§ Failure Moden The manner in which the item fails, not the
display of the failureØIt is very important to identify the root cause
and separate the failure modes
§Mission Disabling Failuren When mission is interrupted such that the item
cannot or should not be operated until repair occurs
Stress & Strength Interference 8
How Do Customers Talk About Reliability
§ “ A system that does what I want (function), when and where I want to use it (conditions), for as long as I want to use it (time) ”
§ “ No surprises - no unscheduled downtime”
§ “ Get me up and running quickly when failures occur ”n This is as important as not having
a failure in the first place.
Stress & Strength Interference 9
Definition of Reliability§ “The ability of an item to perform a
required function under stated conditionsfor a stated period of time”
§ “Quality over time”
§ “It is also defined and/or measured as the probability that an item performs…”n It is with this definition that
we can quantify Reliability.
Stress & Strength Interference 10
Reliability Measures§ Cumulative failure rate at a stated time
n Repairs Per Hundred (RPH) within warranty period
§ Instantaneous failure rate or hazard raten Failure rate per hour, month, mile in service
§ Time it takes to failn Mean Time Between Failures (MTBF)n B10 life – time at which 10% of the items have
failed
§ Probability of failure
Stress & Strength Interference 11
Causes of Different Failure Types§ Infant Mortality
n Manufacturing & assembly issuesn Quality control issuesn Supplier Issues
§ Random Failuresn Interference of inherent strength and
experienced stress during operationn Misapplication and/or abuse
§Wear Out Failuresn Fatigue, wear and part deteriorationn Preventive maintenance issuesn Service issues
Titanic
Stress & Strength Interference 12
Reliability Bathtub Curve
0
10
20
30
0 5 10 15 20Time
Failu
re R
ate
Deterioration Noise Factors
External Noise FactorsUnit-to-Unit Noise Factors
Warranty Period
Stress & Strength Interference 13
Outline§Definition of failure - Unreliability
§ Reliability in simple terms
§ Part Strength & Stress
§Normal Distribution
§ Probability of failure
§ Reliability quantified
§ Example
Stress & Strength Interference 14
Stress Acting on a Part§ The operating stress imposed on the part
is random.
§ Stress acting on a part changes withn TimeØClimbing a hill with full, part & no loadØCity vs. high way driving
n Ambient conditions (temperature, pressure)n Part to part – variation in geometryn User to userØ18 year old driving dad’s Porsche
Stress & Strength Interference 15
Part Strength § A given part has certain physical properties
which, if exceeded, will cause failure.
§ Part strength is a random variable that can be represented by a statistical distribution.
§ A parts strength varies fromn Lot to lot – Difference in chemical compositionn Manufacturer to manufacturer – Processn Ambient conditionsØChange in material properties with
temperature and humidityØAt low temperatures parts may shrink and
reduce sealing pressure (Space shuttle failure)
Stress & Strength Interference 16
Stress & Strength Distribution§ Random variation of stress and strength can be
expressed with different distributionsn Normal, Log Normal, Exponential, Weibull
§ Both stress and strength can be represented with any combination of the above distributions.n Normal – Normal, Normal – Weibull,
Log Normal – Exponential, Weibull – Weibull
§ For the purpose of this seminar we will assume a normal distribution for both stress and strengthn Math behind normal-normal distribution is simpler
Stress & Strength Interference 17
Outline§Definition of failure - Unreliability
§ Reliability in simple terms
§ Part Strength & Stress
§Normal Distributionn Probability Density Function
§ Probability of failure
§ Reliability quantified
§ Example
Stress & Strength Interference 18
Normal Distribution
55 75 95 115 135
0
10
20
30
40
IQ
Per
cent
IQ(Intervals of size 20)
55 75 95 115 135
0.00
0.01
0.02
IQ
Den
sity
IQ(Intervals of size 20)
55 65 75 85 95 105 115 125 135
0.00
0.01
0.02
IQ
Den
sity
IQ(Intervals of size 10)
50 60 70 80 90 100 110 120 130 140
0.00
0.01
0.02
0.03
IQ
Den
sity
IQ(Intervals of size 5)
Percent Histogram Probability = Area of Rectangle
Decrease interval size Decrease interval size more
0.39 0.0195
Area=0.0195*20
=0.39
Stress & Strength Interference 19
Normal Distribution Characteristics§ Symmetric, bell-shaped
curve.
§ Shape of curve depends on population mean µ and standard deviation σ.
§ Center of distribution is µ. 50 60 70 80 90 100 110 120 130 140
0.00
0.01
0.02
0.03
IQ
Den
sity
IQ(Intervals of size 5)
§ Spread is determined by standard deviation σ.
§ Most values fall around the mean, but some values are smaller and some are larger.
Stress & Strength Interference 20
Normal Distribution: Effect of Mean & Standard Deviation§ The mean and standard deviation affect the
shape of the normal distribution
Smaller standard deviation
Larger standard deviation
Stress & Strength Interference 21
Probability Density Function (PDF)§ The curve describes
probability of getting any range of valuesn P(X > 120), P(X>75),
P(65 > X > 75)
§ Probability is the area under the curve
40 50 60 70 80 90 100
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Grades
Den
sity
Bell-shaped curve
Mean = 70 SD = 5
Mean = 70 SD = 10
§ Area under the whole curve is 1
§ Probability of getting specific number is 0n P(X=120) = 0 55 60 65 70 75 80 85
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Grades
Den
sity
Probability student scores higher than 75?
P(X > 75)
Stress & Strength Interference 22
Probability = Area under curve
§ Probability of all grades falling between 65 & 70.n P (65 < X < 70)
55 60 65 70 75 80 85
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Grades
Den
sity
P(65 < X < 70)
55 65 75 85
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Grades
Den
sity
P(X < 65)
§ Probability of all grades falling below 65 n Is always a function of the
instructor!n Has nothing to do with
how much you study!
Stress & Strength Interference 23
Probability = Area under curve§ Integral? Calculus?! I am kidding, right?
§ But somebody did all the hard work for us!
§We just need a table of probabilities for every possible normal distribution.
§ But there are an infinite number of normal distributions (one for each µ and σ)!!
§ Solution is to “standardize”.
Stress & Strength Interference 24
Standard Normal Curve§ Take a normally distributed
value X
§ Subtract its mean µ from it
§ Divide by its standard deviation σ.
§ Call the resulting value Z.
-4 -3 -2 -1 0 1 2 3 4
0.0
0.1
0.2
0.3
0.4
Z
Den
sity
Standard Normal Curve
P(Z > z)Tail probability
Z =(X - µ)
σ§ Z is called the standard normal. Its mean µ is 0 and
standard deviation σ is 1.
§ Probability of failure, Unreliability is calculated from standardized normal distribution as Failure=P(Z).
§ Reliability = (1 – Unreliability)
Stress & Strength Interference 25
Outline§Definition of failure - Unreliability
§ Reliability in simple terms
§ Part Strength & Stress
§Normal Distribution
§ Probability of failure
§ Reliability quantified
§ Example
Stress & Strength Interference 26
Stress & Strength Interference§ X (Strength) and Y (Stress) are normally
distributed with mean values µx and µy and variances s 2
x and s 2y
§Define I = X – Y à (Strength – Stress)n Mean value µI = µx - µy
n Variance s 2I = s 2
x + s 2y
§Normalize function I=(Strength-Stress) so that standard statistical tables can be used
Z =(I - µI)
σI=
I – (µX - µY )
s 2x + s 2
y
Stress & Strength Interference 27
Stress & Strength Interference§ Part stress must be equal or exceed part
strength for failure to occurn Stress Y >= Strength Xn I (X – Y) =< 0
§ Area under the normalize function where I = (Strength-Stress) = 0 is consequently the probability of failure
Z =0 – (µX - µY )
s 2x + s 2
y
=Unreliability
Stress & Strength Interference 28
Interference of Two Normal DistributionsPart stress must exceed strength for failure to occur
Stress Strength
I = Strength - Stress
Stress & Strength Interference 29
Outline§Definition of failure - Unreliability
§ Reliability in simple terms
§ Part Strength & Stress
§Normal Distribution
§ Probability of failure
§ Reliability quantified
§ Example
Stress & Strength Interference 30
Example - Probability of Failure§ A component has a strength which is normally distributed
with a mean value of 5000 N and standard deviation of 400 N.The load it has to withstand is also normally distributed with a mean and standard deviation 3500 N and 600 N. What is the reliability of this component under the given load application?
§ Probability of failure is 1.88 out of 100
9812.00188.01Re
0188.008.2
600400
)35005000(022
=−=
==
+
−−=
liabilityityUnreliabilZ
Z
Stress & Strength Interference 31
Example – Repairs Per Hundred§ Unreliability = 0.0188 & Reliability = 0.9812
§ RPH (Repairs Per Hundred) = 100 * Unreliability= 100 * 0.0188
RPH = 1.9n 1.9 parts out of 100 will failn 19 parts out of 1000 will fail
§ RPH will be 0.4 if the load standard deviation is reduced to 400N from 600N.
§ Design Margin for both cases is 50003500
= 1.42
Stress & Strength Interference 32
Example – Cost of Robustness§ If the repair cost of such a failure is $750 and
annual engine build rate is 40,000, How much premium can the manufacturer pay for reduced standard deviation - robustness?
( )
engine$
yearengines
engine$
yearengines
eng/year
10.1140,000
750*592 Engineper Cost
592000,40100
4.088.1Failures ofNumber
==
=×−
=
§ Up to $11.1 per engine can be paid to reduce the load variability (standard deviation) from 600N to 400Nn Larger crank damper to reduce torsional amplitudes
§ What is the price for 592 happy customers?