# The bending of two semi-infinite beams on the edge of elastic half-plane with flat punch

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63, №4, 2010

539.3

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Key words: semi-infinite beam, punch, generalized functions, integral equations.

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Aghayan K.L.

The bending of two semi-infinite beams on the edge of elastic half-plane with flat punch

The contact problem of bending of two semi-infinite beams (plates in the condition of cylindrical bending) free lying on the border of elastic half-plane is considered. The bending is carriying out by symmetrically located , with regard to beams, rigid, smooth punch with plane foundation in the framework of classical theory of beam bending. The problem solution is brought to the system of two integral equations concerning contact stresses under the punch ( Fredholm equation of II type with continuous kernel) and deformation of boundary points of half-plane between beams (singular equation with Cauchy kernel).The solution of given equations are brought to quasicomplete regular infinite system of linear algebraic equations. It is shown that in the boundary points of punch the concentrated force and concentrated moment are appeared.

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– ,

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0 0 0

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b

− −a a b

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x

0, 0

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(1.3) :

0 0 0

x a x a x a

=± ±

=± ±

=± ±

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(1.2)

, (1.4) (1.1), :

( )

( )

3

0 3

b b

b b

(1.5)

b

b

:

2 2

0 2 0 2 0 0

0 0

0 0

b b

x b x b

x b x b

= +

=− −

= +

= −

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x b

. (1.7)

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[1,2].

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2

−∞

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– ,

,

– .

:

(1.10)

(1.9),

– (1.2),

### V

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(1.5), (1.9) (1.10) ,

:

3

( )

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0

b b

b b

(1.12)

2

1

( )

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a

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(1.13)

(4)

i x

σ

−∞

### − ∞ < σ < ∞

. (1.12) (1.13) (1.14),

( )

( )

( )

( )

3

* 3

a a

b b

− +

(1.15)

( )a

( )b

( )

±

±

±

.

,

2

3

2

*

2

0 0 0

. (1.16)

(1.15) (1.11),

( )

( )

( )

( )

*

a a

c a

s b c c

a

b c c

− − +

−∞

(1.17)

3

3

0 0

c s

(1.18)

И ё

,

( )a

( )+

(1.17),

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( )+

:

( )

( )

*

2 1

b a

a a

b b

+ +

(1.19)

1 2

c c s

c c

(1.20)

,

,

( )−a

( )a

, (1.17)

( )

*

1 2

b a

a a

b b

+

(1.21)

1

2

(1.20).

(1.19) (1.2) ,

(5)

( )

0

b b

a

+

, b

( )b

b

### −

(1.22)

,

(1.19) ( II ) (1.21) (

) (1.22).

(1.19) (1.21)

+

,

( )

b (1.17),

### )

.

2. (1.19)–(1.22) .

1

1

(2.1)

(

1

1 )

1 *

1 1 1 1

0 1

1 1

π

(2.2)

1 *

2 1 2

0 1

1 2

π

(2.3)

1

( )+

1

1

1

2

2

(2.4)

1 b 1 b 2

2 b 1 b 2

И [7,8]

( )

1 2

+

,

1 2

2 2

### 0

, (2.5) (2.2) (2.3)

1 2 1

1 0 0 1

1

n

n

∞ ε −

=

(2.6)

2

1 2 2

1 2 1 2

1

n n

n

−ε

− =

(2.7)

= ∞

=0 1 0

0

n n

n n

, ,

k

– II .

1

(2.2)

### π

. ,

(2.6) (2.7) (2.2)

2

## )

1

1

2 2

1

n n

n

+

, (2.8)

(6)

2 * 0 1

n n

∞ −ε

=

### ∑

. (2.9)

(2.6) (2.7) (2.2) (2.3) ,

∞ =0 n n

nn=1

:

( ) ( ) ( )

( ) ( ) ( )

11 12 1

0 1

21 22 2

1 1

k kn n kn n k

n n

k

kn n kn n k

n n

∞ ∞ = = ∞ ∞ = =

(2.10)

( )

( )

( )

### ( )

( ) 2 2 2 2 11 1 0 0 2 * 2 12 1 1 0 1 * 1 2 2

1 2 1

2

0 1

1

21 2

2 2 2

*

1 0

22

### 2

on kn k n kn k n k

kn k on n

kn

### a

π π π ε π − − ε − π ε − − ε

2 1 1 1 2 2 2

2 2 2 2 1

1 1

n

n

− − − ε − −

(2.11) ( )

( ) 2

1 0 1

1 0

0 0

1

2 2 2

0 2 2 2

*

1 0

k k k n

π π π ε − −

(2.12)

ik– ,

1

k

,

1

, 1

0 k

k

ε

. (2.13)

, (2.10)

=0

n

1

1 n n

=

1 2 2

,

(2.8).

(2.10) ,

( )12 kn

(2.11)

( )

2

## )

2 1 2 * 12 1 2 1

n b a kn

ε

.

(7)

( )

( )

( )

2

* *

12 2 1 2 1

1

0

n n

kn

χ

− − − − χ

ε ε

. (2.14)

(2.10) [5,9] ,

, , (2.10), ,

### λ

.

(2.10) (1.22), (1.23) (2.9),

n n 0

n n 1

b

b

∞ ∞

= =

0. И

, (1.8),

– (2.6) (1.17).

, (1.15) ,

:

( )

4 2 6 8

0 2

b b b

b j j

a

− − −

+

(2.15)

И (2.15) ,

-. И (2.15)

b

b

( )+

.

3. ,

, . .

, ( ),

, .

, (1.19),

1

2

0

b b

Q M

(3.1)

3

0

3

0

Q M

. (3.2)

b

b (1.22) :

### )

* * *

0 2 2 2 1 3 1 2 2

2 1 0

0

1 1

* 1

0 0

b M

b b

j j j j j

2

*

1 3

0

(8)

*

2 3

0

2 *

3 3 3 2

0

1

– (3.1)

−1 b

a

,

2

a

,

−2 b

### 1

. И (3.1), (3.1),

.

.2 ,

b

b

### λ

.

-0,2 0,2 0,6 1 1,4

0 0,5 1 1,5 2 2,5 3

.2

### λ

0.5 1 5 10 30 100 500

b

### Q

0.3232 0.2999 0.2674 0.2564 0.2584 0.2657 0.2763

b

### M

0.2283 0.1781 0.1226 0.0976 0.0755 0.0542 0.0331

,

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b . И

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### ( )

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P

x

500

λ =

100

λ =

10

λ =

0.5

(9)

И

1. . . , .

//И . . 1965. .5. №4. .782-785.

2. . ., .И., . . . // . 1969. №1. .167-171.

3. . ., . .

. // . 1975. .39.

.5.

4. . ., . .

, – . / .:

. – – – . И . . И .- . - , 1972.

5. . ., . .

. .: , 1983. 488 .

6. И. . . .– .: , 1949.

7. Э. . И ,

. //И . . . 1993. .46. №3-4. .20–30.

8. . ., Э. .

. //И . . . 2008. .61. №4. .5–19.

9. . . ,

.// . 1972. №5. .34-45.

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: 0019, -19, . , 24

E-mail: karo-aga@mechins.sci.am

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