PSI - Issue 6

V.D. Kharlab / Procedia Structural Integrity 6 (2017) 286–291 Kharlab V.D./ Structural Integrity Procedia 00 (2017) 000 – 000

290

5

mean that there are no other accurate solutions. For example, this is the solution for the round cross-section:

y Q I

2 r z 

2

1 2 8(1 )    

( ) z   xz

k

,

 

,

3

2 ( )[1 3 / ( z k ky r z    2 2 )]

( , ) y z



 

.

(18)

xz

xz

In the linear elasticity, the accurate solution of this problem is known:

Q

3 2

1 2 3 2

 

   

2 r z

2  

2

y

[

]

xz  

.

(19)

I

8(1 )  

y

r

k

10,   

0.3,

0.038

 

Fig. 2 shown the comparison of solutions (18) and (19) (for such initial data:

).

Fig. 2

When looking at this figure, it seems that our approximate solution is in fact accurate. It turned out that this is the case: The formula (18) can be reduced to (19) by algebraic transformations. Let ’ s subject the solution of (18) to the energy analysis. For this purpose, we set parameter k as

1 2

 

2       ( ) k k

2

 

(20)

0

8(1 )  

(a positive correction is added to the initial parameter). The energy is determined using the formula (14):

2 GA  0

2 [1 0,8 ( )] k   ,

(21)

(1) Q U Q   ( )

2

from which the minimum energy condition follows

(1) Q dU Q

2 GA d GA     2 0 1.6 2

0 k dk Q k

1.6 2 0  



,

(22)

d

2



0 k  , 2)

0   . The first variant was considered above (degradation into the Zhuravsky result at

0.5   ). The

that is: 1)

0 k k  to which accurate solution of the problem corresponds (19).

second variant means that

References

Kharlab, V. D., 2015. Development of elementary theory of tangent stresses at simple bending of beams (I), Digest of civil engineers. 1(48), 82-