PSI - Issue 8

E. Marotta et al. / Procedia Structural Integrity 8 (2018) 43–55

50 8

Author name / Structural Integrity Procedia 00 (2017) 000 – 000

cos

sin 0



    

 

  

T

sin cos

0

  

(23)

12

 

0

0

1   

To complete the 6x6 stiffness matrix, which relates the forces at node 1 to the displacements at node 2, equilibrium conditions are invoked

1

4 5       F F F

1       2        3   F F F

1     2       3 F F F

1     2       3 u u u

1     2       3 u u u

1

0

0 0



  

  

    G

    II G K

  JI K

(24)

0

1





 



2 1 Y Y X X 2 ) (

(  

) 1

 

6   

1

Where X 1 , Y 1 , X 2 , Y 2 , are the node coordinates respectively of node 1 and 2. All values are expressed in the reference located at node 1. The stiffness matrix in the same reference system emerges by composition of the previous partial matrices.

T



   

II K K K K

1

JI (25) Bending stiffness is not sufficient to gives stability to the matrices, when the section loads result oriented as the local tangent. Therefore, the axial contribution is computed again by Castigl iano’s theorem. JJ JI     K



P

1

( ) 



0 

u

P

( )    d

( ) 







(27)

i

EI

F



i

The axial stiffness contribution acts in series with the bending stiffness, therefore its contribution is simply added at the flexibility matrix, derived from pure bending treatment.

1

u   

c a c a c  

1       2        3    4       5        6    F F F F F F

1

11

11

12

12

13

2      

  

(28)

u c a c a c   

21

21

22

22

23

u

c

c

c

3   

31

32

33

2

u u u

44 c a c a c c a c a c    44 45 45

  

4

46

(29)

5      

  

 

54

54

55

55

56

c

c

c

6   

64

65

66

Applying eq.s (16-25) allows to get the matrix of the bending-axial curved.

3. Test Cases The first interesting point is to highlight the family of curved functions that can be fitted by the approximated strategy provided here. To accomplish this task, a simple set of curves are represented in Fig. 5, all of them characterized by the same ending points.