Ա Ա Ա ՈՒ ՈՒ Ա Ա Ա Ա Ա Ղ Ա
В АЦ А Ь А А А А
60, №4, 2007
539.3
- ,
,
. .
лючевыеслов : , , , , .
Key words: plate, displacement, intersecting forces, flexure, exactness. . .
, ,
Ա
,
:
:
, ,
: N.G.Vasilyan
The solution of bending plates problem with uniformly normal load , which two opposite edges of the plate jointly supported, other two edges are free
This article is about the solution of plates bending problem with uniformly normal load, when two opposite edges of the plate jointly supported, other two edges are free is received. The problem is solved both by Kirchhoff theory and by improved theory.The results for displacements and intersecting forces are compared. As a result,we received, that maximal flexure of the plate for the concerned problem has order exactness relative thicknes by the theory of Kirchgov in compare with majority of problems for which order exactness is quadrate relative thicknes.
- ,
, .
. .
,
,
– .
.
1.
2
h
а:
−
a
≤
x
≤
a
,
−
b
≤
y
≤
b
,
−
h
≤
z
≤
h
.
.
D
∆
2w
=
q
(1.1),
0
,
0
22
=
∂
∂
=
y
w
2 2
2 2
3 3
3 2
0
0
0
(2
)
0
x
x
w
w
M
x
y
w
w
N
x
x y
⎛
∂
∂
⎞
=
⎜
+ ν
=
⎟
∂
∂
⎝
⎠
⎛
∂
∂
⎞
=
⎜
+
− ν
=
⎟
∂
∂ ∂
⎝
⎠
a
x
=
±
(1.3)ν– .
, , [1].
. . [2] : « ,
(
) . (1877) . (1900)» .
,
q x y
( )
,
=
const
,y
x
f
y
x
w
nn
n
(
)
cos
λ
)
,
(
1
∑
∞ ==
(1.4*)(1.2),
q x y
( )
,
:0 1
(2
1)
Cos
2
n n
n
y
q
q
a
b
∞
=
− π
=
∑
(1.4**)π − − = −
) 1 2 (
) 1 (
4 1
n a
n
n .
(1.4*) (1.4**) (1.1) (1.3) ,
0 1
sh
[1
ch
sh
]cos
(1
)
n n
n n n n n
n n
n
S
a
w
w
x
x
x
y
∞
=
ν
ν λ
=
α
+
λ
+
λ
λ
λ
− ν δ
δ
∑
(1.4)(2
1)
,
2
n
n
b
− π
λ =
0 4 10 5 5
64
( 1)
,
(2
1)
n
n
q b
w
D
n
−
−
=
α =
π
−
(1.5)
y
x
-
a
a
b
-b
0
S
n= + ν
(1
) sh
λ − − ν λ
na
(1
)
na
ch
λ
na
(1.6)
(3
δ =
n+ ν
) sh
λ
na
ch
λ − − ν λ
na
(1
)
na
(1.4) ν=0
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
∗ 2
2
2 2
0
5
1
1
b
y
b
y
w
w
(1.7)0
w
. ,5
1
5
1536
nn
∞
=
π
α =
∑
(1.8)
1
1 ,
ch
, sh
2
na
n n
a
a
a
e
b
λ
>>
λ
λ ≈
(1.9)(1.4)
x
=
a
:
2 2
0 2 2
(1
)
( , )
1
1
1
(3
)(1
)
5
y
y
w a y
w
b
b
⎛
⎞⎛
⎞
⎡
ν + ν
⎤
≈
⎢
+
⎥
⎜
−
⎟⎜
−
⎟
+ ν − ν
⎣
⎦ ⎝
⎠⎝
⎠
(1.10)y
ν=0.5
43
.
1
7
10
)
,
(
≈
=
∗
w
y
a
w
(1.11)
(1.4) a
. ,
:
0 2
1
sh2
(1
)
16
( , )
1
(1
)
(2
1)
n n
y
n n
a
a
N a b
q b
n
∞
=
⎡
ν
λ − − ν λ
⎤
=
⎢
+
⎥
− ν
−
δ
⎣
∑
⎦
(1.12), (1.9)
0
(1 3 )
( , )
(1
)
y
N a b
≈
+ ν
q b
− ν
(1.13)2.
. .
,
,
, [3]:
3
,
4
h
q
∆Φ = −
8
34
0
3(1
)
3
h
h
D w
χ
− ∆ +
∆Φ −
Φ =
− ν
, 21
0
h
∆Ψ −
Ψ =
χ
(2.1)χ=2/5 . . , χ=1/3 –
[4], Φ, Ψ :
1 , 2
x y y x
∂Φ ∂Ψ ∂Φ ∂Ψ
ϕ = + ϕ = −
∂ ∂ ∂ ∂ (2.2)
θ1, θ2
,
. [5],Φ, Ψ :
1 2
2
2
,
3
3
w
w
x
G
x
y
y
G
y
x
⎛
⎞
⎛
⎞
∂
∂Φ ∂Ψ
∂
∂Φ ∂Ψ
θ =
−
⎜
+
⎟
θ =
−
⎜
−
⎟
∂
⎝
∂
∂
⎠
∂
⎝
∂
∂
⎠
(2.3)w, Φ, Ψ [3]:
2
2
0,
0
2
0
w
x
y
w
G
y
y
x
y
∂Φ ∂Ψ
=
+
=
∂
∂
⎛
⎞
∂
−
χ ∂ ∂Φ ∂Ψ
−
=
⎜
⎟
∂
∂
∂
∂
⎝
⎠
b
y
=
±
(2.4),
:
2 2 2 2 2
2 2 2 2
2
(1
)
0
0,
0
w
w
G
x y
x
y
x
y
w
x
y
G
y
x
x
y
⎡
⎤
∂
+ ν
∂
−
χ ∂ Φ
+ ν
∂ Φ
+ − ν
∂ Ψ
=
⎢
∂ ∂
⎥
∂
∂
⎣
∂
∂
⎦
⎡
⎛
⎞
⎤
∂ ∂
−
χ ∂Φ ∂Ψ
−
=
∂Φ ∂Ψ
+
=
⎢
⎜
⎟
⎥
∂
⎣
∂
⎝
∂
∂
⎠
⎦
∂
∂
a
x
=
±
(2.5)(2.1), (2.4), :
1
( , )
n( ) cos
n,
n
w x y
f x
y
∞
=
=
∑
λ
1
( , )
n( ) cos
nn
x y
x
y
∞
=
Φ
=
∑
ϕ
λ
(2.6)1
( , )
n( ) sin
nn
x y
x
y
∞
=
Ψ
=
∑
ψ
λ
(2.6) (2.1)
1
1
4( 1)
1 cos ,
(2 1)
n
n n n
n
a y a n
− ∞
=
−
= λ =
− π
∑
(2.7):
2
0
2 2
0
2 2
2
3 , 4
2 4
3 (1 )
1
0
n n n n
n n n n n
n
n n
q a h
h h
f f q a
D D
h h
″
ϕ − λ ϕ = −
χ ″ − λ + ϕ = −
− ν + χ λ
″
ψ − ψ =
χ
(2.8)
(2.8)
0 2
3
ch
sh
4
nn n n n n
a
D
x
C
x
q
h
ϕ =
λ +
λ +
(2.9) (2.8)
2 2 0
4
2 2
ch sh ( ch sh ) (1 )
3 1
n
n n n n n n n n n n
n n
q a h
f A x B x C x x D x x h
D D
χ = λ + λ − λ + λ + + λ
λ λ − ν (2.10)
(2.8)
sh
ch
n
F
np x
nG
np x
nψ =
+
(2.11)pn=( χh)−1 1+ χλ2nh2 (2.12)
3.
A B C D F G
n,
n,
n, n,
n,
nx
= ±
a
(2.5). ,x
= −
a
0
x
=
3
3
0,
0,
0
w
w
x
x
∂
=
∂
=
Ψ =
∂
∂
(3.1)(3.1) , [3] .
(2.6), (3.1)
0,
0,
0
n n n
f
′
=
f
′′′
=
ψ =
(3.2)(2.10), (2.11) (3.2)
0
n n n
C
=
B
=
G
=
(3.3)(2.9), (2.10), (2.11) (2.6)
x
=
a
(2.5), :1
(sh
) sh
,
n n n n
F
= −
p a
−λ
a D
2 2 cth 11 3
n n n n
n
h
A D a a D
+ ν
⎛ ⎞
= ⎜λ λ − ⎟
− ν
λ ⎝ ⎠
(3.4)
2 2 0
2
1 2 2
3 1 2
3 4 (1 ) cth th
1 2 cth
th
4 (1 )
th
n
n n n n n
n n
n
n n
n
q a
D h a a a
h a
a h h
p a
−
ν ⎧ ⎡ − ν ⎤
= − ⎨ + ν + λ χ − − ν λ ⎢ λ − λ ⎥− − ν
λ λ ⎩ ⎣ ⎦
⎫ λ − λ χ + λ χ ⎬
⎭
(3.4) (2.9)-(2.11), (2.6),
. , (1.9)
x
=
a
:{
}
2 2 0
4 1
2 2 1
2 2
1
( , )
3
4
1
2
4
(1
)cth
1
cos
1
nn n
n n
n
n n n n
q
a
w a y
h
a
D
h
h
h
p a
y
∞
=
−
+ ν
⎡
≈
⎢ − ν
+ ν + λ χ − νλ −
λ ⎣
⎤
χ λ
− λ χ + λ
χ
+ +
⎥
λ
− ν ⎦
∑
(3.5)
const
q
=
(2.7). ,
0
cos
1q
=
q
λ
y
, a1=q0 an=0 n>1, q=q0(cosλ1y+αcosλ2y), 01 q
а = , a2=αq0 an=0 n>2. q=qNcosλNy, an =0
N
n< , aN =qN
n
>
N
.y q
q= 0cosλ1
,
2 21
4 0
4 2
2 (1 )
16 (1 )
( , ) 1 cos
(1 )(3 ) (1 )(3 ) 2
q b h y
w a y
D b b b
⎡ ν + ν π χν + ν ⎤ π
= ⎢ + + ⎥
π ⎢⎣ − ν + ν − ν + ν ⎥⎦ (3.7)
0
=
y :
4 0 4
2
16
(1
)
( , 0)
1
1
(1
)(3
)
(3
)
q b
h
w a
D
b
⎡
ν + ν
⎛
π χ
⎞
⎤
≈
π
⎢
+
− ν
+ ν
⎜
⎜
+
+ ν
⎟
⎟
⎥
⎢
⎝
⎠
⎥
⎣
⎦
(3.8)
(1.9)
:
0
1
2 2
2
( , ) 1 1
3 (3 )
y
h b
q b h
N a b
h b
b
⎡ ⎛π χ ⎞ ⎤
ν −
⎢ ⎜⎜ ⎟⎟ ⎛ ⎥ ⎞ χπ
⎢ ⎝ ⎠ ⎥
≈ ⎢ − ⎜⎜ + ⎟⎟⎥ π ⎢ π χ + ν ⎝ + ν ⎠⎥
⎢ ⎥
⎣ ⎦
(3.9)
, (1.9),
:
4 0
5
64 (0,0) q b
w
D
≈
π (3.10)
4 2 2
0
5 2
64 (0, 0) 1
2(1 )
q b h w
D b
⎛ π χ ⎞
≈ ⎜ + ⎟
− ν
π ⎝ ⎠ (3.11)
(3.6)
4 0
0 5
64 (0, 0) q b
w
D
≈
π (3.12)
2 2
h b . (3.8) (3.9)
2 2
h b h b.
1. . ., - . . .: ,
1963. 635 .
2. . . . .: ,
1957. 536 .
3. . . ,
. / .: “ ”, .
. . . : ” ”, 2002. .67-88.
4. . . . .: , 1987. 360 .
5. . . – .
// . . . 1998. №3. . 46-58.
6. . .
.// . . .1992. №3. .48-64 .
