PSI - Issue 17

Ivan Baláž et al. / Procedia Structural Integrity 17 (2019) 734 – 741 Ivan Baláž, Yvona Koleková, Lýdia Moroczová / Structural Integrity Procedia 00 (2019) 000 – 000

3

736

The following stability equation may be used for the number of the fields n = 2, 3, 4, … ∞:

p



  

1 cos

1 cos

  − −  − 

1 cos −

 

n

( )  =  

n w C =  2 2  ,

, for n = 1, 2, 3, 4…,

, for n = ∞ (2)

 =  ( )

− 1 cos





n

  

2

   

   

1 sin

  −  −

1 sin +

p







− 1 cos

n

where p is the number of the half-sine waves, p = 1, 2, 3, 4, … n . Results obtained from the large parametric study enable to calculate the following quantities:

2

( ) 2 2    EI

F cr 

   = ,

1 

F F

F

EI



 =

2

2

F

=

cr =

   = cr ,

F

,

,

,

=

(3)

cr

cr

2

2

E

EI





EI



2

2

E

The results of the parametric study are given in the graphical form in the figures 1, 2, 3 and 4. The diagrams show the areas I, II, III, IV, V, VI, VII in which the value of the critical force F cr is minimal and in which the buckling mode is defined by the number of the half-sine waves p = 1, 2, 3, 4, 5, 6, 7. The results for the investigated number of the fields n = 2, 3, 4 and 7 are compared in figures 1, 2, 3 and 4 with the results valid for the infinitive number of the fields n = ∞. The case n = ∞ may be used for calculation of the approximate value of the critical force of the compression member on the elastic foundation.

3. Approximate formulae for calculation of the critical force

Replacing of the curves by the straight lines the following approximate formulae were obtained.

Fig. 1. F cr / F E for n = 2 and ∞ fields. Number of half-sine waves p = 1, 2.

Fig. 2. F cr / F E for n = 3 and ∞ fields. Number of half-sine waves p = 1, 2, 3.

For n = 2:

E w C F 2 

F

F C +

0.25(

1.5 )  w

=

if

(4)

. cr a

E

E cr a F F = . if

E F C  2

 w

(5)