PSI - Issue 6

L.M. Kagan-Rosenzweig / Procedia Structural Integrity 6 (2017) 216–223 L.M. Kagan-Rosenzweig / Structural Integrity Procedia 00 (2017) 000–000

218

3

Term M ∆ is the correction due to the force P . Combining Eq. (3) – (5), we get:

M EI P M ∆ ′ + ∆ = −

EI P

0 M

.

(6)

The effect of the equilibrium conditions is the following exact equation

] [( 0 M P w w a bx ∆ = − + + , )

(7)

0 w ; a b , are

) ( P a bx + accounts for a change in support reactions 0 H ,

0 S evoked by deflection

in which the term

the constants. The deflection w is also divided into two terms: w w w = + ∆ 0 . Constants a b , coupled with deflection 0 w are denoted as 0 0 , a b , the first-order correction is introduced:

) 0 0 0 1 M P w a b x ∆ = ∆ + + , (

0 0 w w w ∆ = −

0

.

(8)

0

M ∆ is written as the sum

The total correction

M M M 2 1 ∆ = ∆ + ∆ .

(9)

0 M , according to Eq. (3), (8) we have

0

1 ( / ) M P EI M ∆ ″ = − , and Eq. (6)

As Eq. (3) is also valid for the moment

results in

EI P

EI P

M 2 2 ∆ ″ + ∆ = − ∆ M

.

(10)

M

1

The last equation is exact, but it is solved approximately. Let

c M be the form of the moment at the stability loss,

satisfying the homogeneous differential equation

 EI P

″ +

0 =

(11)

M

M





M 2 ∆ is considered proportional to

with the corresponding boundary conditions. The correction

c M :

 M CM ∆ = 2 .

(12)

According to Eq. (11), (12)

EI P 

M 2 2 ∆ ″ = − ∆

,

M

therefore approximately

EI P 

EI P

EI P

M ) ( ∆ ″ + ∆ = − + ∆ M

.

M

2

2

2

M 1 ∆ ,

M 2 ∆ :

Taking Eq. (10) into account, we have the proportionality of