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# Reliability based design

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7. Reliability based design
Objectives
вЂў Learn formulation of reliability design problem.
Understand difference between reliability-based
design and deterministic design
вЂў Learn how to quickly approximate reliability of
component using mean value, first order, second
moment method
вЂў Be exposed to three example problems involving
analysis and design
Deterministic vs. Reliability Design
вЂў Find design variables
вЂў To minimize cost (or
weight)
вЂў so that safety factor п‚і
minimum acceptable
value
вЂў Find design variables
вЂў To minimize cost (or
weight)
вЂў so that reliability п‚і
minimum acceptable
value
Approximate calculation of reliability
вЂў Mean value, first order, second moment
method (MVFOSM) method
вЂў Assumptions:
вЂ“ Random variables are Gaussian
вЂ“ Linear Taylor approximation of performance
function, g(x1,вЂ¦,xn), is accurate (performance
function; positive implies survival, negative
implies failure)
Approximate calculation of reliability
(continued)
Linear Taylor approximation of g about mean
values of random variables:
g ( x1 ,..., x n ) п‚» g ( E ( X 1 ),..., E ( X n )) пЂ«
п‚¶gi
( x i пЂ­ E ( X i ))
пѓҐ
i пЂЅ 1,..., n п‚¶ x i
Approximate calculation of reliability
(continued)
Mean value of g
E ( g ( x1 ,..., x n )) п‚» g ( E ( X 1 ),..., E ( X n ))
Standard deviation of g
2
пЃіg п‚»
п‚¶g 2 2
п‚¶g п‚¶g
)
пЃі
пЂ«
2
пЃіX X
пѓҐ
пѓҐ
пѓҐ
i j
X
i
i пЂЅ 1,..., n п‚¶ x i
i пЂЅ 1,..., n j пЂЅ i пЂ« 1,..., n п‚¶ x i п‚¶ x j
(
The derivatives are calculated at the mean values of r.v.s
Approximate calculation of reliability
(continued)
Failure probability = пЃ†(-пЃў)
Reliability=1-failure probability
пЃў пЂЅ safety index пЂЅ
E(g)
пЃіg
Generic procedure for solving reliability
design problem
вЂў Develop equation for performance function
вЂў Develop method for calculating the reliability as a function
of the design variable(s). Since many iterations are
required in design we need an approximate method (e.g.
MVFOSM) for calculating reliability at this stage
вЂў Find the optimum values of the design variable(s) so that
the requirement for acceptable reliability as well as other
requirements are satisfied
вЂў Validate the final design, i.e. compute the reliability of the
final design using an accurate method (for example MonteCarlo simulation)
вЂў Redesign the system if needed.
Example 1
вЂў Tension element
вЂў Given probability distribution of axial load,
ultimate strength and coefficient of
variation of diameter, find mean diameter of
element that has reliability 0.9999.
P
P
d
Example 2: Design of I-beam
вЂў Find web height of I-beam to meet given
reliability target
Example 3: Design-oriented analysis
of torsion bar of a truck
Suggested reading
вЂў Frangopol, D, M., and Maute, K., вЂњReliabilitybased Optimization of Civil and Aerospace
Structural Systems,вЂќ Engineering Design
Reliability Handbook, CRC press, 2004, p. 24-1.
вЂў Mourelatos, Z., P., et al., вЂњProbabilistic Analysis
and Design in Automotive Industry,вЂќ Engineering
Design Reliability Handbook, CRC press, 2004, p.
38-1.
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