Titel_SS06

'f()x (6.11)

where x( , ,..,x 12 xxn)T

XXij, ijXi

is a vector of realizations of the random variables X representing

measurement uncertainties with mean values X( XX 12 , ,.., Xn)T and covariances

Cov 1 Xj where Xi are the standard deviations and (^1) ij the correlation
coefficients. The idea is to approximate the function f()x by its Taylor expansion including
only the linear terms, i.e.:
0
0,0
1

()

() ( )

n

ii

i i

f

fxx

x

'

 



C 

  xx

x

x (6.12)

where is the point in which the linearization is performed, normally

chosen as the mean value point and

01,02,0 ,0( , ,.., )

T

x  xx xn

0

()

,1,2,..

i

f

i

x 





 xx

x

n are the first order partial derivatives

of f()x taken in xx 0.

From Equation (6.12) and Equation (6.6) it is seen that the expected value of the error E'

can be assessed by:

Ef' (X) (6.13)

and its variance Var' can be determined by:



(^000)
2
2
111,

() () ()

i i

nnn

X

iijjiiij

fff
Var
xxxij X Xj

'

/

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xx xx xx

xxx

1  (6.14)

Provided that the distribution functions for the random variables are known, e.g. Normal
distributed, the probability distribution function of the error is easily assessed. It is, however,
important to notice that the variance of the error as given by Equation (6.14) depends on the
linearization point, i.e. x01,02,0 ,0(xx x, ,.., n )T.

Example 6.1 – Linear Safety Margin

Consider a steel rod under pure tension loading. The rod will fail if the applied stresses on the
rod cross-sectional area exceed the steel yield stress. The yield stress R of the rod and the
loading stress on the rod S are assumed to be uncertain modelled by uncorrelated Normal
distributed variables. The mean values and the standard deviations of the yield strength and
the loading are given as R 350 MPa, R 35 MPa, 200S MPa and S 40 MPa

respectively.

The limit state function describing the event of failure may be written as:

gr()x s

whereby the safety margin M may be written as:

MRS