PSI - Issue 6

V.A. Meleshko et al. / Procedia Structural Integrity 6 (2017) 140–145 Meleshko V. А ., Rutman Y.L. / Structural Integrity Procedia 00 (2017) 000–000

142

3

It is seen from (1), if the cross-section shape is known, a bending moment in the cross-section at simple bending can be written through the rod curvature in this cross-section (1) implies the incremental ratio

(2)

. 

    d M dM

 d dM 

is the tangent stiffness. From (1) we get

A multiplier in (2)

( ) 

T

2

3

bh

bh



     s

d d

s

( ).  (3) Using the results of Kovaleva, N.V., Skvortzov V.R., Rutman Y.L. (2007), we get the following for the rectangular cross-section 4 ( ) 2 ( )  d T dM         s

3

      

bh



3 1

s

,

1







4



  

s

,

.

T

E a E 



(4)

      1 , 1 3 a x   

3

bh



1

pl

s

1



  

  x

3

4



3



s

Having the tangent cross-sections stiffness, we can determine tangent stiffness of the planar rod at the nodes of its connection with other rods. There are displacement directions in nodes (Fig. 2).

Fig. 2. Displacement at nodes

There are stiffness factors appearing in the nodes from single displacements (Fig. 3).

Fig. 3. Stiffness factors at nodes

The differential equation of bent rod centre line is written in the standard form Volmir A.S. (2007):     .  T P x M y x V    (5) Using the initial parameters method and having integrated them two times as per x , the system of integral equations can be obtained relative to the known stiffness factors k ij . Below there is the equations system for determination of stiffness factors k 22 , k 32 , k 52 , k 62 at  2 =1: