Ա Ա Ա Ի Ի ՈՒԹ ՈՒ ԻԱ Ա Ի Ա Ա ԻԱ Ի Ղ Ա Ի
62, №2, 2009
539.3
. ., . .
Ключевыеслова: , , ,
, .
Keywords: plate, orthotropic, corrected coefficients, tangential stresses, the form of distribution.
. . , .Պ.
և
, և : Խ
:
:
R.M. Kirakosyan, S.P. Stepanyan
The Problem of an orthotropic round plate under the action of the surface tangential loadings taking into account the transversal shear.
An asymptotic problem of bend of an orthotropic round plate of constant thickness which is under the action of radially directed surface tangential loadings is, considered. The Problem is solved with the application of shear theory of the first arider. For the maximum bend the coefficient of corrections is calculated.
, .
.
.
. [3], [4],
, [2],
.
.
.
1.
.
,
a
h
r
θ
,z
. .0
r
θ
,.
0
z
,
τ
:2
h z
R
=+=
R
+= τ
;2
h z
, .
r
=
a
[1] :
2
1
1
r
r
N
N
N
Z
r
r
r
θ
∂
∂
+
+
= −
∂
∂θ
,
M
r1
M
rM
rM
N
rhR
1r
r
r
θ θ
∂
−
∂
+
+
=
−
∂
∂θ
, (1.2)
M
r1
M
2
M
rN
h
1r
r
r
θ θ θ
θ
∂
∂
+
+
=
−
∂
∂θ
θ
.,
r
N N
θM
r,M
θ,M
θr– . -[1]:
2 1
0
Z
= θ =
, (1.3)1
R
, (1.1), :1
2
R
R
R
+
−
−=
= τ
. (1.4)(1.2)
N
r≡
0
. (1.5). , (1.4) (1.5) :
dM
rM
rM
h
dr
r
θ
−
+
= − τ
. (1.6)[3], [4] :
55
5
6
6
r
dw
h
N
B h
dr
τ
⎛
⎞
=
⎜
ϕ +
⎟
+
⎝
⎠
,12 11
r
D
d
M
D
dr
r
ϕ
=
+
ϕ
,
M
D
12d
D
22dr
r
θ
ϕ
=
+
ϕ
. (1.7)– ,
w
ϕ
– ,.
B
55
r z
0
,D
ij–
3
12
ij ijB h
D
=
. (1.8)11 r
B
=
B
,B
22=
B
θ,B
12=
B
rθ[1] . (1.7) (1.5) :
55
5
dw
dr
B
τ
ϕ = −
−
. (1.9),
dw
dr
, ,
r
u
u
r= ϕ
z
, (1.11)
dw dr
/
=
0
55
0
5
dw
dr
B
τ
ϕ = ⇒
= −
. (1.12),
. .
(1.9) (1.7), (1.6)
3 2 2
2 2
3 2
11
5
55d w
d w
dw
hr
k
r
r
k
dr
dr
dr
D
B
2
τ
τ
+
−
=
+
, (1.13)22
11 r
E
D
k
D
E
θ
=
=
. (1.14)r
E
E
θ –, .
(1.13)
k
≠
2
:
(
)
3
1 1
1 2 3 2
55
11
5
3
4
k k
hr
r
w
c
c r
c r
B
D
k
+ −
τ
τ
= +
+
+
−
−
(1.15)– .
i
c
r
=
0
c
3=
0
. (1.16), :
(
)
3 1
1 2 2
55
11
5
3
4
k
hr
r
w
c
c r
B
D
k
+
τ
τ
= +
+
−
−
. (1.17).
(1.7) , (1.9) :
1
c
c
2r
=
a
w
2
12 11 2
55
5
rD
d w
dw
M
D
dr
r
dr
B
⎛
τ
⎞
= −
−
⎜
+
⎟
⎝
⎠
,2
22 12 2
55
5
D
d w
dw
M
D
dr
r
dr
B
θ
⎛
τ
⎞
= −
−
⎜
+
⎝
⎠
⎟
. (1.18),
k
=
2
,
w
M
rM
θk
→
2
.2. .
:
w
r a==
M
r r a==
0
. (2.1)
w
M
r,c
1c
2 :
(
)
(
)
(
)
(
)
(
)
3 2
11 12
1 2
55 11 11 12
6
2
5
3
1
4
ha
k
k
D
k D
a
c
B
D
k
k
kD
D
⎡
⎤
τ
− −
+
−
τ
⎣
⎦
=
+
+
−
+
, (2.2)
(
)
(
)(
)
(
)
2
11 12
2 2
11 12 11
2
1
4
k
ha
D
D
c
k
kD
D
D
k
−
τ
+
= −
+
+
−
. (2.3)(1.17) (1.18), :
(
)
(
)
(
)
(
)(
)
(
)
(
)
(
)
2 11 3 3 11 12 12 211 11 12
1 2
11 12
6 1
2
3 1 4
3 k k 2
k k D
k kD D r a
h
w k D
D k k kD D
r+ a− D D
⎧ ⎡ − − +⎤ ⎫
⎪ + + +⎢ ⎥ −⎪
τ ⎪ ⎪
= + − + ⎨ ⎢⎣+ − ⎥⎦ ⎬+ ⎪ ⎪ − + ⎪ ⎪ ⎩ ⎭
(
)
555
a r
B
τ −
+
(2.4)
(
)
(
11 212) (
2 2)
11
2
1
4
k k rh
D
D
r
M
a
D
k
− −
τ
+
=
−
−
r
, (2.5)(
)(
) (
)(
)
(
)
(
)
2 2
12 22 11 12 12 22 11 12
2
11 11 12
2
2
4
k ka
r
kD
D
D
D
D
D
kD
D
hr
M
D
k
kD
D
− −
θ
+
+
−
+
+
τ
=
⋅
−
+
. (2.6)
k
→
2
(2.4) –(2.6), :k
=
2
(
)
(
)
(
)
(
)
(
)
3 3 3
11 12 11 12
11 11 12 55
5
3
2
ln
36
2
5
a
D
D
a
r
r
D
D
a
r
r
w
h
D
D
D
B
⎧
+
−
−
+
⎫
⎪
⎪ τ −
= τ
⎨
⎬
+
+
⎪
⎪
⎩
⎭
, (2.7)
(
11 12)
11
2
ln
4
r
hr
D
D
a
M
D
r
τ
+
=
, (2.8)
(
)(
)
212 22 11 12 11 22 12
11 11 12
2
2
ln
4
2
a
D
D
D
D
D D
D
hr
r
M
D
D
D
θ
+
+
+
−
τ
=
⋅
+
. (2.9),
,
rz
τ
2
k
=
, . .,E
θ=
4
E
r.r
=
0
. (2.7) :
(
)
(
)
(
(
)
)
3 2
11 12 11 12 11
2 0
11 11 12 11 12 55
5
3
2
1
36
2
5
5
r
ha
D
D
D
D
B h
w
D
D
D
D
D
B a
τ
=⎡
⎤
+
+
=
⎢
+ ⋅
⎥
(
)
(
)
3
11 12 êë
0
11 11 12
5
36
2
rha
D
D
w
D
D
D
=
τ
+
=
+
(2.11).
(
)
(
)
2 11 12 11
2 11 12 55
2
3
5
5
D
D
B h
D
D
B a
+
Δ = ⋅
+
(2.12),
12
0.2
11D
=
D
B
11=
10
B
55,h
=
0.2
a
, ,10% .
0.102
r
M
.,
r
=
a e
/
max
(
11 12)
11
2
4
r r a
r e
ha
M
M
D
eD
D
τ
=
=
=
+
, (2.13), , .
M
θ ., ,
:
(
11 1211
2
4
r aha
)
M
D
D
D
θτ
=
=
−
. (2.14)
22
4
11D
=
D
,D
12=
0.3
D
11 (2.15) (2.8) (2.9)4
2.3
ln
rr
M
r
M
ha
a
a
τ
−
=
= −
⋅
r
,
4
4.6
ln
1.7
M
r
r
M
ha
a
a
a
−
θ
θ
=
= −
⋅
+
τ
r
.
(2.16). 1
M
r−
M
− θ
.1
3.
r
=
a
.:
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8
M
rr a
0.2 0.4 0.6 0.8 1r a
1.2 1.4 1.6 1.8 2 2.2 2.4
w
r a==
0
;55
0
5
r a
r a
dw
dr
B
=
=
⎛
τ
⎞
ϕ
= −
⎜
+
⎟
=
⎝
⎠
(3.1)(1.17), (3.1)
c
1c
2 :
(
)(
)
3 1
11 55
3
1
2
5
ha
a
c
D
k
k
B
τ
τ
=
+
+
+
, (3.2)
(
)
(
)
2
2 2
11
1
4
k
ha
c
D
k
k
−
τ
= −
+
−
. . (3.3)(1.17) (1.18), :
(
)
(
)
(
)
(
)
(
)
3 3 1 2
2
11 55
1
2
3
3
1
4
5
k k
k r
k a
r
a
a
r
h
w
D
k
k
B
+ −
+
+ −
−
τ −
τ
=
⋅
+
+
−
, (3.4)(
)
(
)
2 2
11 12 11 12
2 11
2
4
k k
r
r
a
kD
D
D
D
M
hr
D
k
− −
+
−
−
= τ ⋅
−
, (3.5)
(
)
(
)
2 2
12 22 12 22
2 11
2
4
k k
r
a
kD
D
D
D
M
hr
D
k
− −
θ
+
−
−
= τ ⋅
−
. (3.6):
2
k
=
3 3 3
(
)
11 55
3
ln
36
5
a r
h
a
w
a
r
r
D
r
B
τ −
τ
⎛
⎞
=
⎜
− −
⎟
+
⎝
⎠
, (3.7)(
11 12)
1111
2
ln
4
rhr
a
M
D
D
D
D
r
τ
⎡
⎤
=
⎢
+
−
⎥
⎣
⎦
, (3.8)
(
12 22)
1211
2
ln
4
hr
a
M
D
D
D
D
r
θ
τ
⎡
⎤
=
⎢
+
−
⎥
⎣
⎦
. (3.9):
2 3
11 2 0
11 55
3
1
36
5
r
B h
ha
w
D
B
=
a
⎡
⎤
τ
=
⎢
+ ⋅
⎥
⎣
⎦
. (3.10)2 11
2 55
3
5
B h
B a
Δ = ⋅
. (3.11)11
10
55B
=
B
,h
=
0.2
a
0.24
24% ..
, , ,
. ,
. :
(2.15) :
r
2.3
ln
r
r
r
M
a
a
a
−
= −
⋅
−
,
M
4.6
r
ln
r
0.3
a
a
−
θ
= −
⋅
−
r
a
.
(3.13)
. 2
M
r−
M
θ−
.
.2 :
)
,
k
=
1
; )
k
<
1
,k
=
1
, [5],
;
1
k
>
)
;
) и
N
r=
0
[2].
1. . . . .: , 1987. 360 .
2. . . – .
// . . . 1998. №3. .46-58.
3. . . ,
. // . . 2006. .106. №3. .245-251.
4. . .
. // . . 2006. .106. №4. .304–311.
5. . . .: ,1957. 463 .
19.08.2008
0.2 0.4 0.6 0.8 1
-1 -0.75
-0.5 -0.25 0.25 0.5
r
M
r a
0.2 0.4 0.6 0.8 1
-0.25 0.25 0.5 0.75 1 1.25
1.5
