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

739

6

  

     

F

k

2 sin 1 2 sin 1   

2

+



2

F

k

2 2

 

− +

2

cr

(22)

cr



=

F

k

2

F

k

2

−



2

cr

cr

Approximate value of relative critical force and critical force are according to a) Hetényi (1971) and b) authors:

k

2 3.5 2  k

L EI

 

2

F

F

2 4 2 

cr  +

cr  +

F

kEI

,

a)

2

+

b)

(23)

(

)

cr

0.5 2

Formula (23a) is represented in figure 5 by bold dotted straight line. It gives F cr on unsafe side. Formula (23b) is represented by parallel line below it. Coordinates of the intersection points (figure 5): ( ) 2 1 1 2 . = + + p F cr p , ( ) 2 2 1   = +      k p p , n p 1, 2, 3... = (24)

5.2. Member with hinged ends (case B in figure 5)

Exact value of relative critical force and critical force:

2 4  p k

EI

2 kL



2

F p

2 cr p = + .

. F p cr p =

+

,

,

(25)

p 1, 2, 3... =

n

2

L

p

2 2 

2

Approximate value of relative critical force and critical force are according to a) Hetényi (1971) and b) authors:

k

2 0.5 2  k

2

F

F

cr 

cr  +

a)

F cr 2 

kEI

,

b)

(26)



2

Formula (26a) is represented in figure 5 by bold dotted straight line. It gives F cr on unsafe side. Formula (26b) is represented by parallel line above it. Coordinates of the intersection points (figure 5): ( ) 1 1 2 . = + + F p p cr p , ( ) 2 1   = +      k p p p , n p 1, 2, 3... = (27)

5.3. Member with free ends (case C in figure 5)

Exact value of relative critical force may be obtained from buckling condition:

  

  

   +    −

     

k F 2  +

2 sinh 1 2 sin 1   

2

k F 2 

k F 2  +

2

cr

cr

cr

(28)



=

  

k F 2 

k F 2  −

2

k F 2  −

2

cr

cr

cr