PSI - Issue 6

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

288

3

energy and displacements from the cross force: *

2

Q

QQ GA









(7)

U

ds

ds

,

.



 

Q

Q

GA

2

2

s

s

If the vertical tangent stress is taken into account only, and it is determined by the Zhuravsky ’ s formula, as is known,

2 2 ( / ) [ ( ) / ( )] . y A A I S z b z dA   

(8)

For the rectangular cross-section, this formula results in 1.2.   Our theory substituted the Zhuravsky’s formula with another one and, therefore, it shall use other expression instead of (3).It derives directly from the first relation (2).Having taken the rod element of unit length, we get:

2

(9)

(1) / Q U Q

.

 

GA

2

As a rule, we will take into account only vertical tangent stress (energy contribution of the second tangent stress is insignificant). Then, based on (1),



(1)

2 ( , ) / 2 ( ) [1 12  

U

y z GdA

 



Q

A xz

(10)

2 2 Q S z b z I 2 ( )

2

y k k



2

/ 2 , dA G

]



2

2 b z

( )

A

y

and according to (9),

2 A S z

2

y k k

( ) [1 12  



2

dA

]

 

(11)

2

2 b z

2 b z

I

( )

( )

A

y

( ) / 2 b z  to ( ) / 2 b z ,

or after integration by y within the limits from

2 A S z

( )

2 (1 0.8 ) , k I        0 0 2

dz

.

(12)

( ) z b z

y

At 0 k  the formula (9) degenerates into the formula (3), and the formula (10) is as follows 0 .   

(13)

Contrary to the Zhuravsky ’ s theory, factor  in our theory is dependant (through parameter k ) on the Poisson ’ s ratio and the relation of general dimensions of the cross-section (it is shown in Fig. 1 by the example of the rectangular cross-section, where value / 0 h b    shall be read as h b  ). Thus, it is not quite correct to designate factor  as a cross-section shape factor, usemoreefficientlydesignationofwarping “ cross-section ” factor.

* General designations are not explained.