EML 4550: Engineering Design Methods
Probability and Statistics
in Engineering Design:
Reliability, FMEA, FEMCA
Class Notes
Hyman: Chapter 5
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System reliability
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Reliability of Series Systems
Rs  R1  R2  ...Rn
0.99
0.85
n
Rs   Ri
i 1
0.98
Rs  0.825
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For constant per-unit failure rates
R i (t)  e
 i t
R sy stem   e
R sy stem  e
 i t
 t
   i
 Per-unit failure rate of series system is constant and equal to the sum of the
component failure rates
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Reliability of Parallel Systems
Rs  1  Fs  1  F1  F2    Fn
 1  1  R1   1  R2   ...1  Rn 
0.99
n
Rs  1   1  Ri 
i 1
0.85
Rs  0.99997
0.98
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Example
 Find the system reliability of the following combinational system with both serial
and parallel arrangements. Assume all sub-systems have a reliability of 0.9
1
2
3
4
6
5
Rs  ( R123 )(R45 )(R6 )
 1  (1  R1 )(1  R2 )(1  R3 )1  (1  R4 )(1  R5 )R6
 1  (0.1)(0.1)(0.1)1  (0.1)(0.1)[0.9]
 (0.999)(0.99)(0.9)  0.889
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For constant per-unit failure rates
(example: two systems in parallel)


R sy stem  1  1  e1t 1  e 2t

R sy stem  e1t  e 2t  e1  2 t
 System does not have constant per-unit failure rate even if components do
 System reliability for parallel systems is always greater than the most reliable
component
 Most systems are not designed in parallel (redundancy) due to cost
considerations (unless needed due to safety and life-protection considerations)
 Series
 Transmission line, Power train
 Parallel
 Multiple airplane engines, Two headlights
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Reliability of Large Systems
 Most systems are neither parallel nor series, but a hybrid
combination
 Calculation of overall system reliability, however, is done
following the simple principle shown before
 Parallel systems are used when extremely high reliability is
needed (by use of redundancy)
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Cost of Reliability
Minimized cost
Cost 
Total cost
Cost due to design
and manufacture
Cost to customer:
failed products, reputation, etc..
Reliability 
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FTA
 Fault Tree Analysis
 Work from the overall system backwards towards the component level (top
down approach)
 Identify system fault modes and possible causes
 Assign probabilities to each fault mode
 Build a ‘tree’ and use it to evaluate overall reliability, availability, etc.
 A Fault Tree Analysis Handbook (from US Nuclear Regulatory
Commission)
 The basic elements of a fault tree in pp. 34-44
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FMEA and FMECA
 Failure Modes and Effects Criticality Analysis
 Work from the component level and identify all possible fault modes at the
component level (a team effort and bottom-up approach)
 Assess criticality of each component fault and its effects on overall system
performance
 Build a ‘table’ with all fault modes, assign probabilities, severity, determine
interactions, possible actions, etc.
 Three factors for failure analysis: The severity of a failure (Sev), The
probability of occurrence of the failure (Occ), The likelihood of detecting the
failure (Det)
 RPN (risk priority number)=(Sev)(Occ)(Det): quantify overall risk for a specific
failure
 Use the table to asses overall reliability (see an example)
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Step-by-step Procedures
 The design is broken down into components with a block diagram showing their
interrelations.
 Identify functions for each individual components (1st column)
 List the potential failure modes (2nd column)
 Describe the consequences/effects due to the failure (3rd column); frequently
coming from customers, regulation, and/or experienced designers  Use the
severity table to determine the numerical value (Sev).
 Identify potential causes (root cause analysis, column 6)  Find Occurrence
value (Occ)
 Determine how one can detect the potential failure (colume 8)  Find
detectability (Det)
 Calculate the risk priority number (RPN)
 Determine the corrective actions to remove potential failures. Assign
responsibility to appropriate person(s) for the removal of each failure.
 Estimate the RPN after the corrective actions.
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Implications
 Incorporate availability, reliability, and maintainability on the
product specification
 Prepare a mathematical model to assess system reliability
(e.g., FMECA)
 Design with reliability and maintainability in mind
 Exercise FMECA each time a design change is needed, or to
explore incremental improvements to the design that may
improve reliability without critically affecting functionality
and cost
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