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The optimal choice of location of supports in case of thermal and transverse loading of isotropic elastic rectangular plate

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Ա Ա Ա Ի Ի ՈՒԹ ՈՒ Ի Ա Ա Ի Ա Ա ԻԱ Ի Ղ Ա Ի

Ц

66, №4, 2013

539.3

Ч

л . .

л ч ы л : , , , .

Key words: plate, elastic, optimal location, temperature field.

Ա. .

և

Ա

, և :

Eloyan A. V.

The optimal choice of location of supports in case of thermal and transverse loading of isotropic elastic rectangular plate

The problem of determining the optimal location of the supports along the length of the elastic, isotropic rectangular plate in the case of simultaneous action of transverse load and temperature field.

,

.

a,b,h, ё

0

y, yb ,

с ( .1)

0.5ac 2cc 0 0.5ac x

b

0.5a

0.5a

y

z

. 1. ё ,

( )

qq y T ( , , )x y z .

c

, .

 

2 .

(2)

18

0.5

c x a c

    . ,

0 c x

   0 x 0.5ac

0 x .

( , ) 1 x y

w ( , ) 2 x y

w

,

 

1 :

2

( , )

R ( , )

1, 2

i e

D

w x y



x y

q

i

(1)

w x y

i

( , )

(

i

1, 2)

– ,

D

Eh

3

/ 12(1

 

2

)

ё , E– , – ,

/ 2

3

/ 2

3(1

)

( , , )

2

h

t h

R

zT x y z dz

e

h

 

t

– .

: –

2

2

0

1, 2

i

w

i

y

y0, yb (2)

2 2

1 2 1 2

2 2

,

w

w

w

w

y

y

x

x

x0 (3)

2 2 3 3

2 2 2 2

2 2

0,

3

(2

)

2

0

w

w

w

w

x

y

x

x y

 

  

 

x

0.5

a c

(4)

3

1 1

3

0,

0

w

w

x

x

x

 

c

(5)

,

, ,

ё .

,

T x y z

, ,

2

zT

0

h

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,

0

(1

)

4

t e

T

R x y

h

 

.

( , ) (

1, 2)

i

w x y

i

:

 

,

i

 

,

( , )

i t

w

x y

w x y

w x y

 

,

i

w x y

q y

( )

,

( , )

t

w x y

– ,

( , )

0.5

t t

(3)

19

, :

 

1 1

sin

,

sin

,

n n

n n n n

n n

q y

g

y

y b

y

g

y

 

  

 

0 0

2

2

sin

,

sin

,

b b

n n n n n

n

q

q y

y dy

g

y b

y

y dy

b

b

b

 

 

 

 

 

 

4

,

ch

ch

sin

sin

sin

1, 2

0.5

sin

n

i in n in n in n

n

in n n

t t n n

q

w x y

a

x

xb

x

c

x

D

xd

x

y

i

w

R

g

y

 



 

(1), (2), :

 

 

 

 

 

4

,

ch

ch

sin

sin

0.5

sin

sin

n

i in n in n in n

n

in n n n n

q

w x y

a

x

xb

x

c

x

D

xd

x

g

y

y

 



,

,

,

in in in in

a b c d

(3) – (5),

 

 

 

 

   

   

 

  

  

1 1 4 4

2

2 4 3 4 1 2 4 1

2 2

2 2

1 0.5

sin

,

1 0.5

(

),

,

cth

th

,

th

,

,

2 ch

sh

sh

ch

sh

ch

ch

sh

2

sh

ch

n t n n n t 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 n n n

a

a

R

g

y

b

R y b

y

a

b

c

c

a

a

y

a

a

c

b

Qb

Bb

M

x

N

x

Mx

x

x

x

x

Q

Mx

x

x

M

N

x

  

  

  

 

 

 

 

,

1

x

   

 

 

 

  

    

2 2 2 2 2 2

ch

ch

ch

sh

2

sh

ch

2

2

,

4

2

,

n n n

n n n n n

n n

M

x

x

B

Mx

x

x

M

N

x

x

M

N

 

 

    

    

 

 

   

   

2 2

2 ch n sh n n sh n ch n sh n ch n

D MxNxMx xx   xx

 

 

 

 

2

 

2

 

ch

2sh

sh

2ch

ch

sh

n n n

n n n n n n

x

x

x

x

x

x

x

x

x

  

  

 

 



 

  

    

2 2

ch

sh

2

sh

ch

/

n n n n n

x

x

x

M

N

x

x

  

 

  

    

2 2

(4)

20

( , ),

t

w x y

:

 

min max

i

,

c x

w x y

i

1, 2

x

 

c

; 0.5

a c

,

0

 

c

0.5

a

(6)

T

h

.

ё :

 

,

 

,

/

0

i i

w x

 

w x c D q

,

x

x a

/ ,

 

c a

/

.

1.

/T 1/3 1/2 1 2

-30

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 -25

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 -20

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 -15

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 -10

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 -5

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 0

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 5

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 10

=0,265

w

1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 15

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 20

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 25

=0,265

w

=1.004

=0,265

w

=1.102

=0,265

w

=1.599

=0,265

w

=1.787 30

=0,265

w

=0.367

=0,265

w

=0.615

=0,265

w

=1.599

=0,265

w

=2.011

,

 

0

c

0

0.5a

,

0,

(5)

21

0.5

0.5

x

a

x

 

a

.

 

0.5a

,

x

 

0;

a y

0;

b

ё .

(6)

/

a b

 

-300 300

.1

,

*

0.265

c

0.265

a

 

.

,

 

*

0.265

, ё

 

q

q y

, ё

y

0,

b

.

.2

/

1,

5

a b

h

 

T 250.

*

O

 

*

0.265

c

0.265

a

,

,

x y

w x

i

 

,

 

1.599

.

0 0.033 0.067 0.1 0.133 0.167 0.2 0.233 0.267 0.3 0.333 0.367 0.4 0.433 0.467 0.5 0

0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6

w1

,w

2

5.853

0 w1(λ α ) w2(λ α )

0.5

0 α

(6)

22

1. . ., . ., . .

. //

. 2009. .6. №2. .255-261.

2. .Ц., . .

. // . . . 2001. .54. №3. .14-17.

3. C. . . .: ,

1975. 704 .

и : л ич – «

», , . . , .

л.: (094) 583167;

E-mail: aeloyan@ yandex. ru.

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