PSI - Issue 50

E.A. Afanaseva et al. / Procedia Structural Integrity 50 (2023) 1–5 Afanaseva E.A and Afanaseva O.S. / Structural Integrity Procedia 00 (2023) 000–000

2

2

Nomenclature

p vector of generalized displacement of the studied item at the initial stage of operation p 0 value vector of generalized displacement of the leader-item ˆ p forecast of generalized displacement of the studied item ε vector of unbiased, uncorrelated, normally distributed, and equal variance errors k unknown similarity coe ffi cient t n rightmost point of the observation base of the initial section of the leader-item

2. Numerical method algorithm

Let there be two similar structural elements that are under identical external loading conditions. The only di ff erence is that one of the elements has been in use for some time τ before the second. This circumstance makes it possible to make a forecast of the generalized movement of the ”lagging element” based on the behavior of the generalized movement of the first element, called the item-leader. It is known that for the same-type deformation kinetics curves of metal-made structures the values of the normal ized correlation matrix over the entire volume of the generalized irreversible displacement curves have an order of 0 . 7 ÷ 0 . 9 Radchenko et al. (2002). Given this, a simplifying hypothesis is introduced — under the same loading conditions, the curves of generalized time displacement for any pair of items will be similar . Due to the similarity hypothesis, the relationship between the generalized displacements of the leader-item and the studied item can be represented as a matrix model of linear regression p = kp 0 + ε, (1) where p = { p 1 , p 2 , ..., p n } T = { p ( t 1 ) , p ( t 2 ) , ..., p ( t n ) } T ; p 0 = { p 0 1 , p 0 2 , ..., p 0 n } T = { p 0 ( t 1 ) , p 0 ( t 2 ) , ..., p 0 ( t n ) } T ; ε = { ε 1 ,ε 2 , ...,ε n } T ; T - transpose symbol.

Using the relations of linear regression analysis, it is possible to obtain OLS estimates of an unknown parameter

n i = 1 n i = 1

p 0

i p i

ˆ k =

(2)

i

2

p 0

and an error variance

n i = 1

n i = 1

i

2

ˆ k 2

p 2

p 0

i −

s 2

0 = (3) The forecast of the generalized displacement of the studied structural element can be carried out using the ratio ˆ p ( t ) = ˆ kp 0 ( t ) , ( t > t n ) . (4) The composition N · 100(1 − α )% — confidence intervals can be carried out using the formulas: ˆ p j ± U α N , n − 1 · s 0 1 + υ j ∗ 1 2 , (5) υ j ∗ = p 0 2 (6) n − 1 .

j

+

, j = 1 , N

1 n

n i = 1

i

2

p 0

where N — number of wear curve implementations at t > t n . Use special tables to calculate U α N , n − 1 .