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Reliability based design

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7. Reliability based design
Objectives
• Learn formulation of reliability design problem.
Understand difference between reliability-based
design and deterministic design
• Learn how to quickly approximate reliability of
component using mean value, first order, second
moment method
• Be exposed to three example problems involving
analysis and design
Deterministic vs. Reliability Design
• Find design variables
• To minimize cost (or
weight)
• so that safety factor 
minimum acceptable
value
• Find design variables
• To minimize cost (or
weight)
• so that reliability 
minimum acceptable
value
Approximate calculation of reliability
• Mean value, first order, second moment
method (MVFOSM) method
• Assumptions:
– Random variables are Gaussian
– Linear Taylor approximation of performance
function, g(x1,…,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«
gi
( x i пЂ­ E ( X i ))

i  1,..., n  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‚»
g 2 2
g g
)
пЃі
пЂ«
2
пЃіX X



i j
X
i
i  1,..., n  x i
i  1,..., n j  i  1,..., n  x i  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
• Develop equation for performance function
• 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
• Find the optimum values of the design variable(s) so that
the requirement for acceptable reliability as well as other
requirements are satisfied
• Validate the final design, i.e. compute the reliability of the
final design using an accurate method (for example MonteCarlo simulation)
• Redesign the system if needed.
Example 1
• Tension element
• 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
• Find web height of I-beam to meet given
reliability target
Example 3: Design-oriented analysis
of torsion bar of a truck
Suggested reading
• Frangopol, D, M., and Maute, K., “Reliabilitybased Optimization of Civil and Aerospace
Structural Systems,” Engineering Design
Reliability Handbook, CRC press, 2004, p. 24-1.
• Mourelatos, Z., P., et al., “Probabilistic Analysis
and Design in Automotive Industry,” Engineering
Design Reliability Handbook, CRC press, 2004, p.
38-1.
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