Module 5 - E-Courses

Reliability Engineering Prof. G. L. Sivakumar Babu
3. Determine the reduced variates { } *
*
Z = i
x − µ
σ X i
X i
Indian Institute of Science Bangalore
*
i
Z corresponding to the design point { } *
x using
i
4. Determine the partial derivatives of
the limit state function with respect to the reduced
variates using Eq. 5.23b. For convenience, define a column vector {G} as the vector
whose elements are these partial derivatives multiplied by -1:
{ G}
⎧G1
⎫
⎪
G
⎪
⎪ ⎪
∂g
2
= ⎨ ⎬ where Gi
= −
-------------------- (27)
∂Z
i evaluated at design po int
⎪ ⎪
⎪
⎩G
⎪
n ⎭
5. calculate an estimate of β using the following formula:
{ } { }
{ }
{ } { }
⎪ ⎪
⎪
⎩
⎪
⎭
*
z ⎫ 1
T
⎪ * ⎪
G z *
⎪z
2 ⎪
β = where z * = ⎨ ⎬ -------------------------(28)
T
G G
zn
reduces to Eq. 5.18.
⎧ *
The superscript T denotes transpose. If the limit state equation is linear, then Eq 5.28
6. Calculate a column vector containing the sensitivity factors using
{ }
{ G}
T { G}
{ G}
α = -------------------------------- (29)
7. Determine a new design point in reduced variates for n-1 of the variables using
*
Z i = α iβ
i
26