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hgbeqŠh“ m`0hnm`k|mni `j`delhh m`rj `plemhh
60, 11, 2007
539.3
. .,
. .
: , , , ,
, .
Key words:stability, optimization, rib, plate, critical force, composite-material.
. . , . . Պ
,
, ,
, :
E.V. Belubekyan, A.G. Poghosyan
Optimal design of the rectangular plate of composite material, reinforced with rigid ribs by the stability criterion
The optimal geometrical and physical parameters of the plate of composite material, which secure the maximal magnitude of the critical force in case of the constant weight and given gabarit sizes of the construction are determinate. The rectangular plate is reinforced by rigid ribs at free sides and in the middle of the plate and is compressed by planar forces.
, ,
,
, ,
.
2
, , 2a b h ,
b y y =0, =
σ
x
= ±
a
x
=
0
(
α ×
h
1h
1).,
( ),
±ϕ
x
,y
.
1
,
2,
h h
α
,ϕ
,
(
2
a
3
h
1)
/
b
,
, c [1].
(
0 2)
1(
1 0)
2
a h
−
h
= α
3
h h
−
h
(1)0
h
– .,
, . ,
(
x
≥
0
).
,
(
)
4 4 4 2
11 4
2
122
66 2 2 22 4 2 20
w
w
w
w
D
D
D
D
h
x
x y
y
y
∂
+
+
∂
+
∂
+ σ
∂
=
∂
∂ ∂
∂
∂
(2)ik
D
– ,(
)
3 2
,
1, 2, 6
12
ik ikB h
D
=
i k
=
ik
B –
,
[2].
:
–
0 =
w , 2 0
2
= ∂ ∂
y w
y
=
0,
y
=
b
(3): –
0,
w
x
∂
=
∂
(
)
4 2 3 3
11 12 66
4 2
2
34
2w
w
w
w
B
A
D
D
D
y
y
x
x y
⎛
⎞
∂
∂
∂
∂
+ σ
= −
⎜
+
+
⎟
∂
∂
⎝
∂
∂ ∂
⎠
x
=
0
(4)–
3 2 2
11 12
2 2 2
0,
w
2
w
w
w
C
D
D
x y
x
y
⎛
⎞
∂
∂
∂
=
= −
⎜
+
⎟
∂ ∂
⎝
∂
∂
⎠
x
=
0
(5)–
3 2 2
11 12
2 2 2
w
w
w
C
D
D
x y
x
y
⎛
⎞
∂
= −
∂
+
∂
⎜
⎟
∂ ∂
⎝
∂
∂
⎠
(
12 66)
3 23 3 11 2 2 4
4
4
y x
w D
D x
w D y
w A y
w B
∂ ∂
∂ +
+ ∂ ∂ = ∂ ∂ + ∂
∂
σ
x
=
a
(6)4 1
/12
B
= α
E h
– ,A
= α
h
12–4 2 23
23 1 5 5
1,3,... 13
1
64
1
,
th
,
3
2
G
n
C
G
h
d
d
d
d
G
n
⎡
π
⎤
=
α β
β =
⎢
−
⎥
= α
π
⎣
∑
⎦
(7)23 13,G
G –
xoz
yoz
.(2), (3),
1 1 2 1
3 2 4 2
(
ch
sh
cos
sin
) sin
m m m m m m
m m m m m m m
w
C
x
C
x
C
x
C
x
y
=
µ λ
+
µ λ
+
+
µ λ
+
µ λ
λ
(8):
(
)
(
)
2 2 2 2
3 11 22 3 3 11 22 3
1 2 11 11
1
1
,
m m m mD
D D
k
D
D
D D
k
D
D
D
+
− +
+
− −
µ =
µ =
2 2
3 12 66 2
22
,
2
,
m m
m
h
m
D
D
D
k
b
D
σ
π
λ =
=
+
=
λ
(9)(4), (6)
(
1, 2, 3, 4 .
)
im
C
i
=
k
m( )
[
]
[
]
1 1 1 1 1 5
0 2 4 2 6
sh
ch
cos
sin
m m m m m m
m m m m
H
k
a
f
a
f
f
a
f
a
f
= µ
µ λ
−
µ λ
−
×
×
µ λ
−
µ λ
−
−
[
]
[
]
2 2 2 2 5
0 1 3 1 6
cos
sin
ch
sh
0
m m m m m
m m m m
f
a
a
f
f
a
f
a
f
−
µ λ
− µ
µ λ
−
×
×
µ λ
−
µ λ
−
=
(10):
2 2
2
1 1 1
0 2
2 22 2
m
m
h
E h
f
k
h
B
h
⎛
⎞
πα
=
⎜
−
⎟
⎝
⎠
,(
)
3 22 11 1 12
1 4
23 1
12
m
h
B
B
f
m G
h
µ −
=
π
α β
,(
)
3 2
2 11 2 12
2 4
23 1
12
m
h
B
B
f
m G
h
µ +
=
π
α β
2 12 66
11
3 1 1
22 22
4
m mB
B
B
f
B
B
⎛
+
⎞
= µ
⎜
µ −
⎟
⎝
⎠
,2 12 66
11
4 2 2
22 22
4
m mB
B
B
f
B
B
⎛
+
⎞
= µ
⎜
µ +
⎟
⎝
⎠
(
)
(
)
22 0
5 2 2
11 1 2
2
1 2
1 1 2 2 2
1 2
2
ch
sh
cos
sin
m m
m m m m m m m m m
m m
B f
f
B
f
f
a
a
a
a
=
×
µ + µ
⎛
⎞
×
⎜
µ λ
−
µ λ
−
µ λ
−
µ + υ
µ λ
⎟
µ
µ
⎝
⎠
(
)
(
)
(
)
22 0
6 2 2
11 1 2
0 1 3 1 0 2 4 2
1 2
2
1
1
sh
ch
sin
cos
m m
m m m m m m m m
m m
B f
f
B
f
a
f
a
f
a
f
a
=
×
µ + µ
⎛
⎞
×
⎜
µ λ
−
µ λ
−
µ λ
+
µ λ
⎟
µ
µ
(5), (6)
m
k
:( )
[
] [
] [
]
2 1 1 1 1 1 7
0 2 4 2 2 8 2 2
2 2 2 7 0 1 3 1 1 8
ch
sh
sin
cos
cos
sin
sh
ch
0
m m m m m m m
m m m m m m m m
m m m m m m m m
H
k
a
f
a
f
f
a
f
a
f
a
f
a
f
f
a
f
a
f
= µ
⎢
⎣
µ λ
−
µ λ
+ µ
⎥
⎦
×
×
µ λ
+
µ λ
+ µ
− µ
µ λ
+
+
µ λ
+ µ
×
µ λ
−
µ λ
+ µ
=
(11)
:
(
)
(
)
4
1 23
7 3 2 2
2 11 1 2
1 1 1 1 2 2 2 2
6
sh
ch
sin
cos
m m
m m m m m m m m m m
m
h
G
f
h B
a
f
a
a
f
a
πα β
=
×
µ + µ
× µ
µ λ
−
µ λ
+ µ
µ λ
−
µ λ
(
)
(
)
4
1 23
8 3 2 2 0 1 3 1 0 2 4 2
2 11 1 2
6
ch m m sh m m cos m m sin m m
m m
m h G
f f a f a f a f a
h B
πα β
= µ λ − µ λ − µ λ + µ λ
µ + µ
km (10)
(11) , (9),
2 2
кр
2
m m
D
k
h
λ
σ =
(12), (8) (10) (11)
m
k
>1,, ,
m
k
=1. ,m
k
. ,
,
k
m<1.(2)
(
1mch
1m m 2msh
1m m 3mch
2m m 4msh
2m m)
sin
mw
=
C
µ λ
x
+
C
µ λ
x
+
C
µ λ
x
+
C
µ λ
x
λ
y
(
)
(
)
2 2 2 2
3 3 11 22 3 3 11 22
1 2
11 11
1
1
,
m m
m m
D
D
D D
k
D
D
D D
k
D
D
+
−
−
−
−
−
µ =
µ =
(10) (11)
cos
µ λ
2m mx
ch
µ λ
2m mx
2
sin
µ λ
m mx
− µ λ
sh
2m mx
.
,
ξ
(1).:
max min
кр,
m x
σ
x
= α
{
, ,
h h
1 2,
ϕ
}
(13)( )
0
i m
H k
=
(i
=
1, 2
), 2 03
1(
1 0)
2
h
h
h
h
h
a
α
=
−
−
(14)0 1
0.2
h
≤
h
≤
b
,0.2
≤ α ≤
5,
δ ≤
h
2≤
h
0 (15)(13) (10)
(11) km, (1).
(15)
.
δ
:δ =
0.01
b
,
a
≥
b
δ =
0.01
a
a
≤
b
.
[4].
2 , 1 =
ξ
h0 =h0/b=0.015, 0.02, 0.03.:
; 04297 . 0 / ;
0196 . 0 / ;
0818 . 0 / ;
1 120 110 066 660 110
0 12 0
11 0 22 0 22 0
11 = B =B B = B =B B = B =B B =
B
. 0497 . 0 / ;
995 . 0 / ;
1
/ 13 1 1 110 23 23 110
23 G = E =E B = G =G B =
G
α
,h
1=
h b
1/ ,
h
2=
h
2/ ,
b
ϕ
0
кр кр
/
B
11σ = σ
. . 1.,
кр
∗
σ σ0кр.
. 1.
ξ
h0α
h1 h2ϕ
σ ⋅
кр10
3σ ⋅
кр∗10
3σ ⋅
0кр10
30.015 0.2 0.0822 0.01151 45° 0.9533 0.4492 0.1792
0.020 0.2 0.1003 0.01486 45° 1.6182 0.7760 0.3186
1
0.030 0.2 0.1325 0.02115 45° 3.3940 1.6775 0.7170
0.015 0.2 0.0810 0.01336 45° 0.3072 0.2122 0.1792
0.02 0.2 0.0998 0.01754 45° 0.5358 0.3713 0.3186
2
0.03 0.2 0.1336 0.02567 45° 1.1719 0.8173 0.7170
a a
ξ
=1 x 0.731.0 1.0
0.15 1.0
1.0 0.15 x
w
.1 y=b 2
2 , 1 =
ξ
, h0=0.015.1 w=w wmax
y=b 2.
, . 1, ,
кр
σ
ξ =
1
2кр
∗
σ
5
σ
0кр .ξ =
2
1.4 1.7 . ,
ξ
..1,
ξ =
1
, ( )
( ). ,
ξ
, .
2
ξ =
, .1,, ,
.
1. . ., . .
,
// . . . . 1988. .41.№ 6.
.14-18.
2. . . . : , 1987, 360 .
3. . . . : ,
1971. 40 .
4. . .
, ,
// . . . 1998. .51.№
3. .28-33.
