Проектирование трансверсально-изотропной пластинки ступенчато переменной толщины при изгибе с учётом поперечных сдвигов Design of transversely isotropic plate of stepwise variable thickness at bending, taking into account transverse shear

Ի Ի ՈՒԹ ՈՒ Ի Ի Ի Ի Ղ Ի

69, №2, 2016

539.3

-Щ Ё

. ., . .

. , , :

: - , ,

.

Keywords: transversely isotropic plate, transverse shifts, optimal design. . ., Պ . .

,

, ,

և

: Ո

, և

:

Belubekyan E.V., Poghosyan A.G.

Design of a transversely isotropic plate of stepwise variable thickness under bending accounting for transverse shifts

Based on the enhanced theory of plate bending that takes into account the effect of transverse shifts, the problem of optimal design of rectangular transversely isotropic plate, reinforced in its middle part with additional isotropic layers, is solved. The optimal geometric parameters of the additional layers are determined that provide the highest rigidity of the structure for its given dimensions and total weight.

ё , ,

- ,

ё .

ё ,

ё ё .

.

ё ,

,

ё - .

, ё ё ,

.

, ё . .

[1], ,

ё . ё ,

ё

ё . ё ё

, ё .

[2-4].

.

ё h

b

L 

2 , - ,

ё 1

2abh ,

q

( .1).

ё . .

[1] a h₁

ё , ё

, ё h₀,

.

.1. ё я и и

.

ё

ё .

:

a x

L 

 , axL ё axa ё

( xa).

0xL.

a

a

b

L

L

y

x

1

h

h

1

h

x

(p1) -

h ё (p2)

1 2h

h , x0, xa

0 

y , yb, xL.

ё [1]

(p1,2) :

 

p p

p

q

x

y

K







 





,

 





 

 



   



 

3 3 2

( ) ( )

11 3 12 66 2 11 2

2 2

66 2 12 66

2

0

p p

p p p p

p p p p

w

w

D

D

D

P

x

x y

x

P

P

P

K

y

x y





 











 



 

 









 



 

, (1)

 



   



 

 



   



 

3 3 2

22 3 12 66 2 11 2

2 2

66 2 12 66

2

0

p p

p p p p

p p p p

w

w

D

D

D

P

y

y x

y

P

P

P

K

x

x y





 











 



 

_

_

 

_

_{ }



 

.

:

 y0 yb

0 

p

w ,

M

 _yp



0

,

 

_p

0

,



p



1, 2



y0,

y



b

, (2)

 x0

2

0

 

, 2 0

 

x w

,

M

xy 2



0

x0, (3)



x



L

0 1 

w , M_x 1 0,

 

₁

0

x



L

,

(4)



x



а

, 2

1 w

w 

2 2

2 2

0 0

1 2

1 2

,

4

3

4

3

z

z

w

h

w

h

x

x

















_



_

  



_



_





_

_



_

_

  

1 2

 1  2

x

x M

M  ,M1xy Mxy 2 ,

 1  2

x x

N



N

x



a

.

(5)

(

p



1

,

2

) [1]:

    2   2    

11 2 12 2 11 12

,

p p p p

p p p p p

x

w

w

M

D

D

P

P

x

y

x

y









 







    2   2    

11 2 12 2 11 12

,

p p p p

p p p p p

y

w

w

M

D

D

P

P

y

x

y

x









 















  ₍₆₎

    2  

66 66

2

p p p

,

p p p

xy

w

M

D

P

x y

y

x











 



_



_

 

_





_

 p  p

_,

x p

N



K



N

 _yp



K

 p



_p

.

p

w



_p,



_p



p1,2



– , (6)

.

ё  p

ik

D



p1, 2



:

  _,

12 3

1 B h

D_ik  ik  



 





2







,

12

1 3 3

1 1

3

2 _{B h B h h h}

D_ik  _ik  _ik  

11 2

,

1

E

B



 

12 1



2 ,





 E

B

_

_

,

1 2 12

66   _

_

E G

B

  _,

1 ₁2 1 1

11

_



 E

B  1 1 1

12 2

1

,

1

E

B





 

   





1 1 ₁

66 12

1

,

2 1

E

B



G



 

E,



, G₁₂ –

, E₁,



₁, G₁₂ 1 – .

 p ik

P , K p :

  _,

120 5

1 B h

P_ik  ik

  ,

12 3

1 Gh

K  

   





_,

2 2

24

2 1 ₂ 1 1

3 1 2               

 h B J Rh h h

G A J B

P_{ik ik ik}

 2  1

3 12 4

2

2

,

K



Ah



G J





G J







 1



1

1 12

,

2

h

A



h



h

G



G





 1

1 1 12

_{1 ,}

4

hh h

h

G

R

G











_

_





_





,

h

h

h

h

J





























 



20

2

48

2 2 1 3 1  





, 2 2 30 1 2 2 6

2 5 5

1 3 3 1 2 1 2                                                      

 h h h h h h h h

,

h

h

h

h

J





























 



12

2

4

2 2 1 3 , 8 2 6 1 2 2 3 3 1 2 1 1 4                        

 h h h h h h

J

G –

.

const

q



,

w

_p,



_p,



_p



p1,2



, (1) (2) (3), [1]:

 axL ( , p1)





         

2

1 1 1 1

1 1 4 1 2 3 4

1,3.. ₁₁

4

1

ch

sh

sh

ch

sin

,

n

n n n n n n n n n

n _n

q

f

w

C

x C x

x C

x C x

x

y

nD

 



 









_



 

 

 



_

















   





 



       



   





       





1 1 2 11 2 4 1

1 1 1 1

1,3.. 5 6 1 1 1 2 11 2 4 3 1

1,3.. 1 1 1 1

5 6

2

sh

ch

sin

,

ch

sh

2

4

ch

sh

cos

.

sh

ch

n n n n n

n n

n

n n n n

n

n n n n n

n

n n

n n n n

B

C

x C

x

G

y

C

x C

x

B

q

C

x C

x

n

G h

G

_y

C

x C

x

   



_

_

_{ }

_

_





_







 

_

_

_



























_

_

_{ }

_

_





_{ }

_

_



 

_

_





_

_

_

_











(7)

 0xa ( ё , p2)





                              2

1 2 2

1 2 4 1 2

1,3.. 11

2

2 2 1

2 11

1 2 2 2 6

1,3..

2

2 2 2

2 11

1 2 2 2 6

1,3..

4

1

ch

sh

sin

,

2

sh

sh

sin

,

2

4

ch

ch

cos

n

n n n n n

n _n

n

n n n n n n

n _n

n n n n n n

n _n

q

f

w

C

x C

x

x

y

nD

D

C

x

C

x

y

K

D

q

C

x C

x

y

n

K

K

     



 









_



 



_

















_







 

_





 



_





















 

_





 



_



 















.

(8) : n

n

b



 

,  1 2









2

2

10 1

Eh

f

G





 

,

   2 2 11 1

K P

f  .

 p in

C (p1,2; i1, 2,...,6 ) (4) (5), (7) (8) , w_p,



_p,



_p



p1,2



,

(6) – .

a

h

₁,

ё , . .

: :

,

max

min

_p

p

x

w

w



x



 

a

,



p1,2

,

(9)

: a L h

h₁  ₀ (10)

1 1 1

0.01

  

h

0.25



,

0.01

  

₂

h

2

h

₁



0.25



₂

.

  (11)

(10) ,

(11) ё .

:

  

₁

L

a

,

L a

 

b

;

 

₁

b

,

Lab;

 

₂

2 ,

a

b

a

2 ,

 

2

b

, 2ab.

[5]. .

Ч ё ,

ё

q



const

 

2

L b



1, 2

,

20 . 0 , 10 . 0 , 05 . 0 /  h b

h , h₀ 0.1h, ё

:

12 _{12 11}

0.3

B



B

B

  

,

G

12



G

₁₂

B

₁₁

  



1



2 0.35



,



2







11

1

0, 2.6, 5, 10

G





G B



  

G E



E G





, B11 1 B₁₁ 1 B₁₁2,

  _{ }

6 . 0 11 1 12 1

12 B B 

B ,   13 1 _{1 11} 0.7

1 12

1     

 G G G B

G .

0   G

E ё

,

E G

 

2.6, 5, 10

– ё .

L a

a 2 ,

h

1



h

₁

b

,

ё 4

0 3

11h 12q b

B w

w

. ё

0

w

,

, ё



_h

E

G



_a

h

1 w0102 w102 k

1

0.05

0 0.337 0.00742 0.1654 0.1457 1 2.6 0.2915 0.00858 0.1708 0.1526 1.047 5.0 0.2915 0.00858 0.1710 0.1530 1.050 10.0 0.2915 0.00858 0.1711 0.1538 1.056

0.1

0 0.337 0.01484 0.1654 0.1457 1 2.6 0.334 0.01498 0.1745 0.1631 1.119 5.0 0.332 0.01506 0.1786 0.1644 1.128 10.0 0.323 0.01548 0.1796 0.1669 1.145

0.2

0 0.4 0.025 0.1654 0.1492 1 2.6 0.4 0.025 0.2017 0.1942 1.301 5.0 0.4 0.025 0.2083 0.1994 1.336 10. 0.4 0.025 0.2233 0.2107 1.412

2

0.05

0 0.05 0.05 0.4124 0.2909 1 2.6 0.05 0.05 0.4180 0.2986 1.026 5.0 0.05 0.05 0.4188 0.2987 1.027 10.0 0.05 0.05 0.4214 0.2989 1.027

0.1

0 0.1 0.05 0.4124 0.32186 1

2.6 0.2635 0.01897 0.4320 0.3374 1.048 5.0 0.2635 0.01897 0.4336 0.3389 1.053 10.0 0.2625 0.01905 0.4375 0.3422 1.064

0.2

0 0.267 0.03745 0.4124 0.3245 1 2.6 0.262 0.03817 0.4684 0.3743 1.153 5.0 0.26 0.03846 0.4829 0.3803 1.172 10.0 0.256 0.03906 0.5017 0.3935 1.212

ё

k



w

_cor

w

_{cl ,}

ё

, ,

ё ( 40%

 

1

20%

 

2

).

,

ё ё

ё , ё

, (11),

ё ,

 

1

10 . 0 

h

 

2,

ё (

a

=

0

.

05

). , ё

ё , ё .

,

 

2

h

=

0

.

05

:

x

=

0

.

272

,

x

=

0

.

, ё .

[4],

,

ё ё .

. ,

ё ё

,

,

, ё

ё ё .

1. . . . .: , 1987. 360 .

2. Э. ., . ., . .

- ,

. // . . . 2012. .65, № 1. .35-42.

3. . ., . . -

. // ,

, 2012, .9, № 1, .127-131.

4. Belubekyan E.V. Poghosyan A.G., Khanikyan V.M. Bending of the rectangular plate made of composite material, strengthened by additional layers or stiffening ribs. //Proceedings of 3rd _{international conference on contemporaru problems in architecture}

and construction Beijing, China Nov.20-24, 2011, P. 2-1 - 2-6.

5. . . .: , 1975. 266 .

х:

я Э и – и и , .

E-mail: ebelubekyan@yahoo.com

я и – . .- . ., . .«

»,

( ) . : 0009, , . , 105. Т .: (+37499) 099662335 E-mail: arevpoghosyan@mail.ru