-
1 31.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 31.2 j . . . . . . . . . . . . . 6
1.2.1 ( ) . . . . . . . . . . . . 61.2.2 ( -
) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71.2.3 ( ) . . . . . . . . . . . . . . . 71.2.4 . . . . . . . . . .
. . . . . . . . . . . . 91.2.5 . . . . . . . . . . . . . . . . . .
. . 101.2.6 j j j . . . . . . . 111.2.7 () ,
() . . . . . . . . . . . . . . . 111.2.8 () , -
() . . . . . . . . . . 111.2.9 - . . . . . . . . . . . . . . . .
. . . . . . 13
2 152.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 152.2 j . . . . . . . . . . . . . . . . . . . . 182.3
j . . . . . . . . . . . . . . . . . 252.4 j . . . . . . . . . . . .
. . . . . . . . . . . . . . 282.5 j
, j . . . . . . . . . . . . . . . . . . 312.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 372.6.1 . . . . . . . . . . . . . . . . . . . 37
3 553.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 553.2 . . . . . . . . . . . . . . . . . . . . . . .
583.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4 .
. . . . . . . . 653.5 j . . . . . . . . . . . . . . . . . . 71
1
-
J
3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
753.6.1 . . . . . . . . . . . . . . . . . . . . 753.6.2 . . . . . .
. . . . . . . . . . . . 793.6.3 . . . . . . . . . . . . . . . . . .
. 803.6.4 . . . . . . . . . . . . . . . . . . . 81
3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
833.7.1 , . 833.7.2 .
. . . . . . . . . . . . . . . . . . . . . 903.8 . . . 95
3.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . 98
4 1114.1 j . . . . . . . 1114.2 j . . . . . . . . . . . 1134.3 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.4 j .
. . . . . . . . . . . . . . . . . . . 122
4.4.1 j . . . . . . . . . . . . . . . 1234.4.2 j . . . . . . . .
. . 1254.4.3 - . 1304.4.4 . . . . . . . . . . . . . . . . . . . . .
. . . . . 1344.4.5 . . . . . . . . . . . . . . . . . . . . . . . .
. . 1424.4.6 . . . . . . . . . . . . . . . . . . . . 143
4.5 j . . . . . . . . . . . . . . . . . . . . . . 1474.5.1 j
- . . . . . . . . . . . . . . . . . . . . . . . . 1484.5.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1524.5.3 1574.5.4 . . . . . . . . . . . 1704.5.5 j . . . . . .
1834.5.6 -
. . . . . . . . . . . . . . . . . . 1854.5.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1924.5.8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 2004.5.9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 2034.5.10 j j -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
2 j -
-
4
4.5 j
j j j . j , , , j . j j . j ( , , , , j .). j j j . , . , j . -
j , . j . j . j j. j - j. . j j -j, j j , j j j j .
j , , j j , j. j j . j , j , j . j j j j . j j , j j . j j , j .
j j.
j - 147
-
J
4.5.1 j -
. (. 4.12) -. j . , j. j , . , j - , , j - j , , . j j, , ,
.
j , (S = 0), (H = 0) (T2 = 0).
(12 = 0) j (12 = 0). j v = 0.
(. 4.26) j . j j
. 4.26:
148 j -
-
4
, , j 1 2 , ((...) = 0), j ( ) (S = H = T2 = p2 = v = 0). ,
:
- :
dd(rN)N
drd + Tr + pR1r = 0
( 1R1N +1R2N) + 1R1r
dd(Tr) + pn = 0
dd(Mr)M
drd TR1r = 0
drd = R1cos r = R2sin, :
dd (rN)NR1cos+ Tr + prR1 = 0dd (rT)Nr +NR1sin+ pnrR1 = 0dd
(rM)MR1cos TrR1 = 0
(4.61)
j (N, N,M,M T). j . , .
- ( ):
= 1R1
(dud + w
) = 1R2 (u ctg+ w)
= 1R1dd
(uR1 1R1
dwd
) = 1R1R2
(u dwd
)ctg
(4.62)
: , - , , - , u w - n.
j - 149
-
J
- :
N = Eh12 ( + )
N = Eh12 ( + )
M = D ( + )
M = D ( + )
(4.63)
D = Eh3
12(12) (4.61), (4.62) (4.63) 11 11 (N, N,M,M, T, , , , u w).
(4.63) (4.62), (4.61), j u,w T. , , , (4.62) (4.63) .
j j . j . j . j j (.4.27). j , t A jL, . j, j L
A + .
:
sin(
2
)= cos =
BC
AB=
dr
R1d(4.64)
j, r r(1 + ), AB A
B
= R1(1 + )d, :
sin[
2 (+ )
]= cos (+ ) =
BC
AB=
d [r (1 + )]R1 (1 + ) d
(4.65)
j :
(1 + ) cos (+ ) =1
R1dd [r (1 + )]
150 j -
-
4
. 4.27: j j
:(1 + ) (coscos sinsin) =
1R1d
[dr (1 + ) + rd]
j, , , :
cos = 1 ; sin =
, :
(1 + ) (cos sin) =1 + R1
dr
d+
r
R1
dd
j drd = R1cos r = R2sin, , :
(1 + ) (cos sin) = (1 + ) cos+R2R1
dd
:
cos sin+ cos sin = cos+ cos+R2R1
dd
sin.
, sin,
j - 151
-
J
, :
= ( ) ctgR2R1
dd
(4.66)
4.5.2
, j j, ,.. j j , j . j j. j j, j , j j .
(. 4.28). ,
. 4.28:
= j N,M T, j . j , j , j, j. j 0 j N, 1 j
152 j -
-
4
M T. j 0 , j 1 M T.
M T = . j, j.
R = R0 +R1
R,R0 R1, , j, j 0 j 1.
j 0 - j j, j j M T.
j, j , (4.61), (4.62) (4.63), p = pn = 0 ( j j j , j 1, (.
4.28). :
- :
dd(Nr)NR1cos+ Tr = 0dd(Tr) rN NR1sin = 0dd(Mr)MR1cos TR1r =
0
(4.67)
- ( ):
= 1R1
(dud + w
) = 1R2 (u ctg+ w)
= 1R1dd
(uR1 1R1
dwd
)= 1R1
dd
= 1R1R2
(u dwd
)ctg = 1R1ctg
(4.68)
(4.66)
= ( ) ctgR2R1
dd
(4.69)
j - 153
-
J
- ( ):
N = Eh12 ( + )
N = Eh12 ( + )
M = D ( + )
M = D ( + )
(4.70)
D = Eh3
12(12) 12 12 . j j T.
(4.67) :
N =1
R1cos
[Tr +
d
d(Nr)
](4.71)
(4.67), j , :
N = Tctg (4.72)
(4.72) j (. 4.29).
. 4.29:
OO :
Nsin 2r Tsin(
2 )2r = 0
:N = Tctg
154 j -
-
4
N (4.71), :
N =1
R1cos
[Tr +
d
d(Trctg)
] r = R2sin :
N =1
R1cos
[TR2sin+
d
d(TR2cos)
] , T R2 , :
N =1R1
d
d(TR2) (4.73)
M M (4.70) (4.67), (4.68), j T:
R2R1
d2
d2+[d
d
(R2R1
)+R2R1ctg
]d
d( +
R1R2ctg2
) TR1R2
D= 0 (4.74)
j T :
(4.70) :
= 1Eh (N N)
= 1Eh (N N)(4.75)
N N (4.72) (4.73), :
= 1Eh[Tctg R1
dd (TR2)
] = 1Eh
[1R1
dd (TR2) Tctg
] (4.76)
, (4.69) , :
j - 155
-
J
Eh = (1 + )Tctg2
(1 + ) 1R1
d
d(TR2) ctg2
R2R1
d
d
[1R1
d
d(TR2) Tctg
](4.77)
(4.74) (4.77) j T. (4.77) (4.74) j T. , , j j j . (4.74) (4.77).
. j j , j, j . j j (. 4.30). j
. 4.30: j j
j j , . , n- (n 1)-, . j T j , j j , . j , j .
(4.74) j dd ,
d2d2
, :
R2R1
d2
d2 TR1R2
D= 0
(4.77)
156 j -
-
4
T dTd j
d2Td2
:
Eh = R2R1
d
d
[1R1
d
d(TR2)
], T :
d2d2
R21D T = 0
Eh + R2R1dd
[1R1
dd (TR2)
]= 0
(4.78)
(4.78) - . (4.74) (4.77) ctg. j . > 30o, (4.78) . , > 30o
.
(4.78), j , T. T (4.72) (4.73) N N, (4.68). M M (4.70) - j dd ,
:
M = DR1dd
M = DR1dd = M
(4.79)
4.5.3
j j - , j :
R1 = R2 = a = const.
j - 157
-
J
(. 4.31) ( j), .
. 4.31:
- . .
( = ) ( = ) j. . j . j j .
j jj . j j, R1 = R2 = a = const.
(4.78) :
d2d2 a2D T = 0
Eh + d2Td2
= 0(4.80)
:
= 1Eh
d2Td2
(4.81)
Eh = const. :
d4Td4
+Eha2
DT = 0 (4.82)
158 j -
-
4
:
44 =Eha2
D=Eha2(1 2)12
Eh= 12(1 2)a
2
h2
:
4 = 3(1 2)a2
h2(4.83)
(4.82) :
d4Td4
+ 44T = 0 (4.84)
j , j 4- . :
T = e (A1cos+A2sin) + e (A3cos+A4sin) (4.85)
A1, A2, A3 A4 .
j , 1 2 (. 4.32). :
. 4.32: 1 2
= 1
= + 2(4.86)
(4.85), , :
T = e(1) [A1cos( 1) +A2sin( 1)] +
j - 159
-
J
+e(+2) [A3cos( + 2) +A4sin( + 2)] .
j , :
T = e1 (B1cos1 +B2sin1) + e2 (B3cos2 +B4sin2) (4.87)
, j C1, 1, C2 2, B1 B4 :
B1 = C1sin1 B3 = C2sin2
B2 = C1cos1 B4 = C2cos2(4.88)
(4.87) :
T = C1e1sin (1 + 1) + C2e2sin (2 + 2) (4.89)
C1, 1, C2 2 .
(4.89) T . 1, 2, , . j T . ,j T . , j j, , , j j , . j :
- j j
T = C1e1sin (1 + 1) (4.90)
- j j
T = C2e2sin (2 + 2) (4.91)
j j. T . :
dTd
= C1
2e1sin(1 + 1
4
)(4.92)
160 j -
-
4
d2Td2
= 2C12e1cos (1 + 1) (4.93)
(4.81) (4.93), :
= 1Eh
d2Td2
=2C12
Ehe1cos (1 + 1) (4.94)
j :
d
d=
2
2C13
Ehe1sin
(1 + 1 +
4
)(4.95)
(4.79) R1 = a :
M =D
a
d
d
M = M
(4.95), D (4.83), :
M =a
2C1e
1sin(1 + 1 +
4
)(4.96)
M = M
(4.82) :
N = Tctg = C1e1sin (1 + 1) ctg ( 1)
R1 = R2 = a, (4.73) :
N =dTd
= C1
2e1sin(1 + 1
4
)(4.97)
j j . :
j - 161
-
J
1. N = C1e1sin (1 + 1) ctg ( 1)
2. N =
2C1e1sin(1 + 1 4
)3. M = a2C1e
1sin(1 + 1 + 4
)4. M = M
5. T = C1e1sin (1 + 1)
6. = 22
EhC1e1cos (1 + 1)
(4.98)
j j . :
1. N = C2e2sin (2 + 2) ctg ( + 2)
2. N =
2C2e2sin(2 + 2 4
)3. M = a2C2e
2sin(2 + 2 + 4
)4. M = M
5. T = C2e2sin (2 + 2)
6. = 22
EhC2e2cos (2 + 2)
(4.99)
C1, 1, C2 2 .
j . r =asin, r.
j , j j . :
2 (r + r) 2r = 2r
:r = r (4.100)
j (4.76) R1 = R2 =
162 j -
-
4
a = const. :
=1Eh
(dTd Tctg
) , :
=1Eh
dTd
=
2Eh
C1e1sin
(1 + 1
4
) (4.100) r = asin :
r =
2aEh
C1e1sin
(1 + 1
4
)sin (4.101)
j , :
r =
2aEh
C2e2sin
(2 + 2
4
)sin (4.102)
, , j . j j M0 H0. j .
a) M0
= ,1 = 0 j M0[kNm/m]. j j , j , j . (. 4.33) j j . :
. 4.33: M0
j - 163
-
J
1 = 0 = M = M0 N = 0 (4.103)
(4.98), 3 1, , :
a
2C1sin
(1 + 4
)= M0
C1sin1ctg = 0(4.104)
:
1 = 0 C1 =2aM0 (4.105)
(4.98) (4.101) :
1. N = 2a M0e1sin1ctg ( 1)
2. N = 2
22
a M0e1sin
(1 4
)3. M =
2M0e1sin
(1 + 4
)4. M = M
5. T = 2a M0e1sin1
6. = 43
aEhM0e1cos1
7. r = 2
22
Eh M0e1sin
(1 4
)sin ( 1)
(4.106)
, 1 : 0 1 ( ). 1 = 0 :
164 j -
-
4
1. N(0) = 0
2. N(0) = 22
a M0
3. M(0) = M0
4. M(0) = M0
5. T(0) = 0
6. (0) = 43
aEhM0
7. r(0) = 22EhM0
(4.107)
) H0
j H0[kN/m] (. 4.34). j j :
. 4.34: H0
1 = 0, = :M = 0 N = H0cos
(4.98), 3 1, :
C1sin(1 +
4
)= 0 C1sin1ctg = H0cos
:
1 =
4C1 =
2H0sin (4.108)
, (4.98) (4.101) :
j - 165
-
J
1. N =
2H0sine1sin(1 4
)ctg ( 1)
2. N = 2H0sine1cos1
3. M = aH0sine1sin1
4. M = M
5. T =
2H0sine1sin(1 4
)6. = 2
22
Eh H0sine1cos
(1 4
)7. r = 2aEhH0 sine
1cos1sin ( 1)
(4.109)
(1 = 0) :
1. N(0) = H0cos
2. N(0) = 2H0sin
3. M(0) = 0
4. M(0) = 0
5. T(0) = H0sin
6. (0) = 22EhH0sin
7. r(0) = 2aEhH0sin2
(4.110)
) M0
. 4.35: e M0
166 j -
-
4
:
2 = 0, = :M = M0 N = 0 (4.111)
(4.99), 3 1, :
a2C2sin
(2 +
4
)= M0 C2sin2ctg = 0 (4.112)
:
2 = 0 C2 = 2aM0 (4.113)
, (4.99) (4.102) :
1. N = 2a M0e2sin2ctg ( + 2)
2. N = 2
22
a M0e2sin
(2 4
)3. M =
2M0e2sin
(2 + 4
)4. M = M
5. T = 2a M0e2sin2
6. = 43aEhM0e2cos2
7. r = 2
22
Eh M0e2sin
(2 4
)sin ( + 2)
(4.114)
2 = 0 :
1. N(0) = 0
2. N(0) = 22
a M0
3. M(0) = M0
4. M(0) = M0
5. T(0) = 0
6. (0) = 43aEhM0
7. r(0) = 22EhM0sin
(4.115)
j - 167
-
J
) H0
. 4.36: H0
:
2 = 0, = :M = 0 N = H0cos (4.116)
(4.99), 3 1, :
C2sin(2 +
4
)= 0 C2sin2ctg = H0cos (4.117)
:
2 =
4C2 =
2H0sin (4.118)
, (4.99) (4.102) :
1. N =
2H0sine2sin(2 4
)ctg ( + 2)
2. N = 2H0sine2cos2
3. M = aH0sine2sin2
4. M = M
5. T =
2H0sine2sin(2 4
)6. = 2
22
Eh H0sine2cos
(2 4
)7. r = 2aEhH0sine
2cos2sin ( + 2)
(4.119)
2 = 0 :
168 j -
-
4
1. N(0) = H0cos
2. N(0) = 2H0sin
3. M(0) = 0
4. M(0) = 0
5. T(0) = H0sin
6. (0) = 22
EhH0sin
7. r(0) = 2aEhH0sin2
(4.120)
j - 169
-
J
4.5.4
a, 2 h. 2b x 2d(. 4.37). .
. 4.37:
j j .
- . j , , , j . , j , . j j .
, . -, . , j , j . , j , . j , j j j. j j.
, j .
1. j
170 j -
-
4
)
4.4.4 ). j (4.36) :
N = ga
1 + cos(4.121)
(4.37)
N = ga(cos 1
1 + cos
)(4.122)
(4.75), (4.121) (4.122).
= gaEh(cos 11+cos
) =
gaEh
(1+
1+cos cos) (4.123)
, (4.100) :
r = r
r = asin , (4.123), :
r =ga2
Eh
(1 +
1 + cos cos
)sin (4.124)
j j j - (4.66):
= ( ) ctgR2R1
dd
j R1 = R2 = a :
= gaEh
(2 + ) sin (4.125)
= :
j - 171
-
J
1. N(0) = ga1+cos2. N(0) = ga
(cos 11+cos
)3. r(0) = ga
2
Eh
(1+
1+cos cos)sin
4. (0) = gaEh (2 + ) sin
(4.126)
) p
j j , 4.4.4 ), :
N = pa
2
N = pa
2cos2 (4.127)
(4.75), (4.127).
= pa2Eh (1 cos)
= pa2Eh (cos2 )(4.128)
(4.98), r = asin , (4.128):
r = pa2
2Eh(cos2 ) sin (4.129)
j j j - (4.66) R1 = R2 = a , , (4.128):
= ap2Eh
(3 + ) sin2 (4.130)
= :
172 j -
-
4
1. N(0) = pa22. N(0) = pa2 cos2
3. r(0) = pa2
2Eh (cos2 ) sin
4. (0) = ap2Eh (3 + ) sin2
(4.131)
2.
, j = , (. 4.38). (
. 4.38:
). ( ). , .
j qx j .
(. 4.39) j S .
j - 173
-
J
. 4.39:
S OY :
2S
0dPsin = 0
S =12
0qxr0sind
:S = r0qx (4.132)
:
A = 2b 2d = 4bd (4.133)
:
=S
A=r0qxA
(4.134)
j :
=E
=r0qxEA
(4.135)
:
r0 = r0
, (4.135), r0 = asin :
r0 =r20EA
qx
174 j -
-
4
:
r0 =(asin)2
EAqx (4.136)
, :
) , j
qx :qx = Ncos
N, (4.126), :
qx =qa
1 + coscos
(4.136) :
r0 =qacos(asin)2
(1 + cos)EA(4.137)
) p, j
N, (4.131), :
qx = Ncos = pa
2cos
(4.136), :
r0 = pacos(asin)2
2EA(4.138)
mT , j M. (. 4.40) r0 = asin d. ds = r0d = asind j mTds =
mTasind. j
M , .
:
mTds = Md
:mT r0d = Md
:M = mT r0 = mTasin
j - 175
-
J
. 4.40:
: =
MWprst
= 6M2b(2d)2
= 3mTasin4bd2
= 3mTasinAd
, . :
=E
=3mTasinEAd
(4.139)
:
ro0 = r0 = asin =3mT (asin)2
EAd(4.140)
:
ru0 = r0 = asin = 3mT (asin)2
EAd(4.141)
(. 4.41) j j.
j :
=|r0|d
r0, :
=3mT (asin)2
EAd2(4.142)
j ( = ) 4.5.2.
3.
176 j -
-
4
. 4.41: j
j (. 4.42).
. 4.42:
j j 0, (. 4.43).
j j 1(. 4.44). = :
f11X1 + f12X2 + 1 = 0f21X1 + f22X2 + 2 = 0
:
j - 177
-
J
. 4.43: ,j 0
. 4.44: ,j 1
f X = (4.143)
:
X = f1 (4.144)
f j , .
fij - j X1 X2.
f11 ( 1 . 4.45), X1 = 1.
(4.110), 7,
178 j -
-
4
H0 = 1 :
f(`)11 =
2aEh
sin2 (4.145)
. 4.45: X1 = 1
, j , (. 4.45).
X1 = 1, , j qx = 1 mT = 1 d = d.
qx = 1, 1 (4.136) :
(asin)2
EA(4.146)
mT = d, 1 (4.140) :
3d(asin)2
EAd= 3(asin)
2
EA(4.147)
1 X1 = 1 (4.146) (4.147), (+),j X1:
f(p)11 = +
(asin)2
EA+
3(asin)2
EA= +
4(asin)2
EA(4.148)
f11 :
f11 = f(`)11 + f
(p)11 =
2aEh
sin2+4(asin)2
EA(4.149)
f12 X1, X2 = 1.
X2 = 1
j - 179
-
J
(4.102), 7, M0 = 1 :
f(`)12 =
22
Ehsin (4.150)
X2 = 1, (4.140), mT = 1, (+), j X1:
f(p)12 = +
3(asin)2
EAd(4.151)
, f12 :
f12 = f(`)12 + f
(p)12 =
22
Ehsin+
3(asin)2
EAd(4.152)
f21 X1 = 1, X2.
X1 = 1 (4.110), 6, H0 = 1 :
f(`)21 =
22
Ehsin (4.153)
X1 = 1 (4.142) mT = 1d ( (.4.45), (+) X2:
f(p)21 = +
3(asin)2
EAd(4.154)
f21:
f21 = f(`)21 + f
(p)21 =
22
Ehsin+
3(asin)2
EAd(4.155)
j , j :f12 = f21.
f22 X2 = 1 X2.
(4.102), 6, M0 = 1:
f(`)22 =
43
aEh(4.156)
(4.142) mT = 1, (+),j j X2.
f(p)22 = +
3(asin)2
EAd2(4.157)
180 j -
-
4
f22 :
f22 = f(`)22 + f
(p)22 =
43
Eah+
3(asin)2
EAd2(4.158)
1 2 . g p.
)
1 X1, .
(4.126), 3, :
(`)1g =ga2
Eh
(1 +
1 + cos cos
)sin, (4.159)
(4.137), (-), j X1.
(p)1g = qacos(asin)2
(1 + cos)EA(4.160)
:
1g = (`)1g +
(p)1g =
ga2
Eh
(1 +
1 + cos cos
)sin qacos(asin)
2
(1 + cos)EA(4.161)
2 .
(4.126), 4, :
(`)2g = ga
Eh(2 + ) sin (4.162)
((p)2g = 0) j N C (. 4.38), :
2g = (`)2g =
ga
Eh(2 + ) sin (4.163)
) p
1p :1p =
(`)1p +
(p)1p (4.164)
(`)1p , (4.131),
j - 181
-
J
3, :
(`)1p = pa2
2Eh(cos2 ) sin, (4.165)
(p)1p , :
(p)1p = pacos(asin)2
2EA(4.166)
, 1p :
1p = pa2
2Eh(cos2 ) sin pacos(asin)
2
2EA(4.167)
2p :2p =
(`)2p +
(p)2p (4.168)
(`)2p , (4.131), 4, :
(`)2p = ap
2Eh(3 + ) sin2 (4.169)
((p)2p = 0) j N C j.
, 2p :
2p = ap
2Eh(3 + ) sin2 (4.170)
f . jj X1 X2, j :
N = N0 +N1X1 +N2X2
N = N0 +N1X1 +N2X2
M = M1X1 +M2X2
M = M
T = T1X1 + T2X2
(4.171)
182 j -
-
4
: N0 N0 - j j (4.121), (4.122) (4.127), ; N1, N1,M1 T1 - X1 = 0
(4.109), H0 = 1; N2, N2,M2 T2 - X2 = 0 (4.106), M0 = 1.
4.5.5 j
j , j -j j j .
j x, . j j - j, 1 2 x (1 x, 2 ), j ( 1 x, 2 ) :R1 =, R2 = a =
const., k1 = 0, k2 = 1a = const. :
)
Nxx +
1aS + px = 0
Sx +
1aN +
Ta + p = 0
Txx +
1aT
Na + pn = 0
Hx +
1aM T = 0
Mxx +
1aH Tx = 0
(4.172)
:
T = Hx +1aM
Tx = Mxx +1aH
(4.173)
(4.172), j j Nx, N, S,Mx,M H.
j - 183
-
J
Nxx +
1aS + px = 0
Sx +
1aN +
1a2M +
1aHx + p = 0
2Mxx2
+ 1a22M2 Na +
1a2Hx + pn = 0
(4.174)
j . , j ( ). .
. 4.46: j
. 4.46 j .
) ( )
184 j -
-
4
x = ux
= 1av +
wa
x = vx +1au
x = 2wx2
= 1a2v
1a22w2
x = 12avx
1a2wx
(4.175)
)
:
Nx = Eh12 (x + )
N = Eh12 ( + x)
S = Eh2(1+)x
Mx = D (x + )
M = D ( + x)
H = D (1 )x
(4.176)
D .
, j 15 : 3 , 6 6 , j 15 : 6 Nx, N, S,Mx,Mtheta,H; 6 x, , x, x, ,
x 3 u, v, w. j 15 , j j , j .
4.5.6 -.
- - . -
j - 185
-
J
j . j j, j j j- j , . , .
, (. 4.47).
. 4.47:
a, ` h. j .
:
px = p = 0 pn = (` x) (4.177)
( x = 0) . j(x = `) ( j) j .
j, . j j . j. , x = 0 j Mx Tx (. 4.48). j j Mx Tx (. 4.48 ).
186 j -
-
4
. 4.48: j
j :
N = apn = a (` x) (4.178)
j ( = 0), j :
=1Eh
N =a
Eh(` x) (4.179)
- , (v = 0). v = 0 (4.175) :
=w
a,
:
w = a =a2
Eh(` x) (4.180)
w j x. j , a a + w. j w j x, (4.175) x (x = 0), Mx Tx ., j j ,
j. j j j Ntheta, (4.178).
j j j j. j Mx, Tx M.
j - 187
-
J
j (4.174), (4.175) (4.176). j j, j j . j x , , , j . j , S H .
(4.177), :
)
(4.174) j, , :
dNxdx = 0
d2Mxdx2 Na + pn = 0
(4.181)
Nx x, . j , . Nx j j .
(4.173), H = 0, j M T = 0.
j Mx, Tx M (. 4.49).
)
j Nx , x u . j , v , , (4.175) :
= wa
x = d2wdx2
(4.182)
188 j -
-
4
. 4.49: j
)
( = 0), (4.176) :
N = Eh
Mx = Dx(4.183)
(4.173), H = 0, :
Tx =dMxdx
(4.184)
j 6 : , , (4.184). : N,Mx, Tx; x, x w. j j.
(4.183) (4.182) :
N = Ehw
a(4.185)
Mx = Dd2w
dx2(4.186)
j - 189
-
J
(4.181), :
d2
dx2
(Dd2w
dx2
) Eh
a2w = pn (4.187)
pn, (4.177), :
d2
dx2
(Dd2w
dx2
)+Eh
a2w = (` x) (4.188)
j D. , h, (4.188) :
Dd4w
dx4+Eh
a2w = (` x) (4.189)
j . : w1 w0:
w = w1 + w0 (4.190)
j ( (4.180)):
w0 =a2
Eh(` x) (4.191)
j (4.189) :
d4w1dx4
+Eh
a2w1 = 0
d4w1dx4
+ 44w1 = 0, (4.192)
:
44 =Eh
a2D=
12(1 2)a2h2
,
:
=4
3(1 2)ah
(4.193)
190 j -
-
4
= 0 ( ) :
=1.316ah
(4.194)
(4.192) :
w1 = ex (C1cosx+ C2sinx) + ex (C3cosx+ C4sinx) (4.195)
C1 C4 :
, (4.190), w0 w1, :
w = ex (C1cosx+ C2sinx) + ex (C3cosx+ C4sinx) +a2
Eh(` x) (4.196)
j x j w. :
dw
dx= ex [C1 (cosx sinx) + C2 (cosx+ sinx)] +
+ ex [C3 (cosx+ sinx) + C4 (cosx sinx)]a2
Eh
d2w
dx2= 22
[ex (C1sinx+ C2cosx) + ex (C3sinx C4cosx)
]
d3w
dx3= 23 ex [C1 (cosx+ sinx) + C2 (cosx sinx)] +
+23 ex [C3 (cosx sinx) + C4 (cosx+ sinx)] (4.197)
, (4.185), (4.186) (4.184) - :
N = Eha w
Mx = D d2wdx2
Tx = D d3wdx3
(4.198)
C1 C4 , j j .
j - 191
-
J
w, (4.196) j w1 w0. w1 (4.195), : ex, j, ex j. , ` , j j j w1 j
j. , j j w1 (C1 = C2 = 0), j (C3 = C4 = 0). ` j j , ` 2.5. j , ,
w1, : , j, j. , , (4.196) :
- j
w = ex (C3cosx+ C4sinx) +a2
Eh(` x) (4.199)
- j
w = ex (C1cosx+ C2sinx) +a2
Eh(` x) (4.200)
C3 C4 j, C1 C2 j.
` < 2.5, , w1 , w1 (4.196). j j j , j . (4.196), j j .
4.5.7
(` 2.5), . , M0 H0. j j ( , , -j .).
192 j -
-
4
M0
. 4.50: j
:
x = 0Mx = M0, Tx = 0
, (4.198), :
D(d2w
dx2
)x=0
= M0 D(d3w
dx3
)x=0
= 0 (4.201)
, :
w = ex (C3cosx+ C4sinx)
dwdx = e
x [C3 (cosx+ sinx) + C4 (cosx sinx)]d2wdx2
= 22ex (C3sinx C4cosx)d3wdx3
= 23ex [C3 (cosx sinx) + C4 (cosx+ sinx)]
(4.202)
(4.201) w x = 0, :
22DC4 = M0C3 + C4 = 0,
:
j - 193
-
J
C4 = M022DC3 = M022D
(4.203)
, (4.198), j (4.202) :
N = Eha w =Eh
2a2DM0e
x (cosx sinx)
Mx = M0ex (sinx+ cosx)
Tx = 2M0exsinx
(4.204)
:
w = M022D
ex (cosx sinx)dwdx =
M0De
xcosx(4.205)
x = 0, :
1) N(0) =EhM02a2D
2) Mx(0) = M0
3) Tx(0) = 0
4) w(0) =M0
22D
5) dwdx (0) = M0D
(4.206)
H0
:
x = 0Mx = 0, Tx = H0
194 j -
-
4
. 4.51: j
(4.198), :
D(d2w
dx2
)x=0
= 0 D(d3w
dx3
)x=0
= H0 (4.207)
w x = 0, :
C4 = 0
D23 (C3 + C4) = H0(4.208)
:
C4 = 0
C3 = H0D23(4.209)
, (4.198), w od (4.202):
N = Eh2a3DH0excosx
Mx = 1H0exsinx
Tx = H0ex (cosx sinx)
(4.210)
j - 195
-
J
:
w = H023D
excosx
dwdx =
H022D
ex (cosx+ sinx)(4.211)
x = 0, :
1) N(0) = Eh2a33DH0
2) Mx(0) = 0
3) Tx(0) = H0
4) w(0) = H023D5) dwdx (0) =
H022D
(4.212)
M0
. 4.52: j
:
x = `Mx = M0, Tx = 0
196 j -
-
4
:
D(d2w
dx2
)x=`
= M0(d3w
dx3
)x=`
= 0 (4.213)
, j, (4.195), :
w = ex (C1cosx+ C2sinx)
dwdx = e
x [C1 (cosx sinx) + C2 (cosx+ sinx)]d2wdx2
= 22ex (C1sinx+ C2cosx)d3wdx3
= 23ex [C1 (cosx+ sinx) + C2 (cosx sinx)]
(4.214)
(4.213), w x, x = `, :
C1sin`+ C2cos` = M022De`
C1 (cos`+ sin`) + C2 (cos` sin`) = 0(4.215)
:
C1 = M0e`
22D(cos` sin`)
C2 = M0e`
22D(cos`+ sin`)
(4.216)
, (4.198), j (4.214) :
N = Eh2a2DM0e(`x) (cos(` x) sin(` x))
Mx = M0e(`x) (sin(` x) + cos(` x))
Tx = 2M0e(`x)sin(` x)
(4.217)
j - 197
-
J
:
w = M022D
e(`x) (sin(` x) cos(` x))dwdx =
M0De
(`x)cos(` x)(4.218)
x = ` :
1) N(`) = Eh2a2DM0
2) Mx(`) = M0
3) Tx(`) = 0
4) w(`) = M022D5) dwdx (`) =
M0D
(4.219)
H0
. 4.53: j
:
x = `Mx = 0, Tx = H0
198 j -
-
4
: (d2w
dx2
)x=`
= 0 D(d3w
dx3
)x=`
= H0 (4.220)
w x, x = ` :
C1sin`+ C2cos` = 0
C1 (cos`+ sin`) + C2 (cos` sin`) = H0e`
23D
(4.221)
:
C1 = H0e`
23Dcos`
C2 = H0e`
23Dsin`
(4.222)
, (4.198), j (4.214) :
N = Eh2a3DH0e(`x)cos(` x)
Mx = H0 e(`x)sin(` x)
Tx = H0e(`x) [cos(` x) sin(` x)]
(4.223)
:
w = H023D
e(`x)cos(` x)dwdx =
H022D
e(`x) [cos(` x) + sin(` x)](4.224)
x = ` :
j - 199
-
J
1) N(`) = Eh2a3DH0
2) Mx(`) = 0
3) Tx(`) = H0
4) w(`) = H023D5) dwdx (`) =
H022D
(4.225)
4.5.8
(` 2.5), j(x = `) j (x = 0). j. j j . , x = 0 j . . (. 4.54) j j
.
j 0 , j X1 X2 . X1 X2, j :
R = R0 +R1X1 +R2X2 (4.226)
: R0 - j , - j, R1 R2 - X1 = 1, X2 = 1.
(x = 0), , j j .
:
f11X1 + f12X2 + 1 = 0f21X1 + f22X2 + 2 = 0
(4.227)
:
f X + = 0 (4.228)
200 j -
-
4
. 4.54:
:
X = f1 (4.229)
(4.227) , , .
f :
f11 - X1 = 1, X1.
j - 201
-
J
(4.212), 4, H0 = 1:
f11 =1
23D(4.230)
f12 - X2 = 1, X1. (4.206), 4, M0 = 1. X1:
f12 = 1
22D(4.231)
f21 - X1 = 1, X2. (4.212), 5, H0 = 1. j X2:
f21 = 1
22D(4.232)
f22 - X2 = 1 X2. (4.206), 5, M0 = 1:
f22 =1D
(4.233)
: 1 - X1 . (4.180), x = 0. j , (4.180) X1:
1 = a2`
Eh (4.234)
2 - X2 . (4.180) x. j X2:
2 =a2
Eh (4.235)
X1 X2, (4.226) :
N = N0 +N1X1 +N2X2
Mx = Mx1X1 +Mx2X2
Tx = Tx1X1 + Tx2X2
(4.236)
202 j -
-
4
:
N0 - , (4.178);
N1,Mx1, Tx1 - X1 = 1. (4.210) H0 = 1;
N2,Mx2, Tx2 - X2 = 1. (4.204) M0 = 1.
4.5.9
, . j j j j , . .
. 4.55:
` 2.5, . j . j , j j .
j - 203
-
J
, j: X1 X2 (. 4.56) (. 4.57).
. 4.56: - j j
X1 X2 :
f11X1 + f12X2 + 1 = 0f21X1 + f22X2 + 2 = 0
(4.237)
f , j:
fij = f(`)ij + f
(p)ij (4.238)
j j .
f (`)ij (4.230) (4.233), :
f(`)11 =
123D
; f (`)12 = f(`)21 =
122D
; f (`)22 =1D
(4.239)
jX1 = 1 X2 = 1.
j , j X1 = 1 X1 X2,
204 j -
-
4
. 4.57: - j j
X2 = 1, X1 , :
f(p)11 = f
(p)21 = f
(p)12 = 0 (4.240)
f (p)22 j X2 = 1, X2.
, M0(. 4.58), :
. 4.58: - M0
j - 205
-
J
w = M0a2Dp(1+)(1 2)
w
= = M0aDp(1+)
Mr = Mt = M0
Tr = 0
(4.241)
= 1 M0 = 1, :
f(p)22 =
a
Dp(1 + )(4.242)
: Dp = Et3
12(12) ; =ra .
= 0 ( ), :
f(p)22 =
12aEt3
(4.243)
(4.238), (4.239),(4.240) (4.243) :
f11 = 123D
f12 = f21 = 122Df22 = 1D +
12aEt3
Tr = 0
(4.244)
:
1 = (`)1 +
(p)1
2 = (`)2 +
(p)2
(4.245)
(`)1 (`)2 - j
. (4.234) (4.235) :
206 j -
-
4
(`)1 = a2`Eh
(`)2 =a2
Eh(4.246)
(p)1 - j . .
(p)1 = 0 (4.247)
(p)2 - j . , , (. 4.59):
. 4.59: -
:
w = p0a4
64Dp
(5+1+
2) (
1 2)
= p0a3
16Dp
(2 3+1+
)
Mr = p0a2
16 (3 + )(1 2
)Mt = p0a
2
16
[3 + (1 + 3) 2
]Tr = p0a2
, = 1 = 0, :
(p)2 = p0a
3
8Dp(4.248)
, j j X2.
(4.245) (4.246),(4.247) (4.248) p0 = `, :
j - 207
-
J
2 = a2`Eh
2 = a2
Eh a3`8Dp
(4.249)
= 0, :
1 = a2`Eh
2 = a2
Eh 32a3`Et3
(4.250)
t .
f , (4.237) j .
, :
N = N0 +N1X1 +N2X2
Mx = Mx1X1 +Mx2X2
Tx = Tx1X1 + Tx2X2
w = w0 + w1X1 + w2X2
(4.251)
: N0 - (4.178), w0 - (4.180), N1,Mx1, Tx1, w1- (4.210), H0 = 1,
N2,Mx2, Tx2, w2 - (4.204), M0 = 1.
j , :
Mr = Mr0 +Mr2X2
Mt = Mt0 +Mt2X2
Tr = Tr0 + Tr2X2
(4.252)
208 j -
-
4
: Mr0,Mt0, Tr0 - (4.246) p0 = `, (4.241), M0 = 1, : Mr2 = Mt2 =
1, Tr2 = 0.
. . . j j j . (. 4.60) j .
. 4.60:
f - (4.238) (4.245). f (`)ij
(`)i ,
(4.239) (4.246). (4.238) (4.245) .
, p0 = `, X1 X2. (. 4.61) . X1 j j . j j j j . j -
j - 209
-
J
. 4.61:
j :
r = t =X1t
(4.253)
j r :
r =X1E
(r t) =X1Et
(1 ) (4.254)
j :
u = ar =aX1Et
(1 ) (4.255)
, , .
j , p0 X2 . j . , . b. a r (a b), , r (a b) . , , j . ,
210 j -
-
4
. , . j j - . , , . b j , , , , . b, p0 X2. j B , b, . (. 4.62)
j j .
. 4.62:
j X2 p0 :
= X2 + p0 =X2b3EJp
p0b3
24EJp
= X2 + p0 = X2b6EJp +p0b3
24EJp
(4.256)
j - 211
-
J
B , :
= X2b6EJp
+p0b
3
24EJp= 0
:
b = 2
X2p0
(4.257)
j , j A :
- X2
X2 =2X2
3EJpp0
(4.258)
- p0
p0 = X32
3EJpp0
(4.259)
, j j . X1 = 1, (4.255) :
f(p)11 =
a
Et(1 ) (4.260)
X2 = 1, (4.258) :
f(p)22 =
23EJp
p0
- :
EJp =Et3
12(1 2), (4.261)
:
f(p)22 =
8(1 2)Et3p0
(4.262)
f (p)12 f(p)21 , j X2
X1 :f
(p)12 = f
(p)21 = 0 (4.263)
j X2 (4.259), X2 = 1 :
(p)2 = 1
EJpsqrtp0
212 j -
-
4
EJp (4.261), :
(p)2 = 4(1 2)Et3p0
(4.264)
(p)1 , j p0 X1.
, p0 = `, :
f11 = f(`)11 + f
(p)11 =
123D
+ aEt(1 )
f12 = f(`)12 + f
(p)12 = 122D
f21 = f(`)21 + f
(p)21 = 122D
f22 = f(`)22 + f
(p)22 =
1D +
8(12)Et3p0
1 = (`)1 +
(p)1 = a
3`Eh
2 = (`)2 +
(p)2 =
a2Eh
4(12)Et3p0
(4.265)
f , :
f11X1 + f12X2 + 1 = 0f21X1 + f22X2 + 2 = 0
(4.266)
j (4.251). b, X2 p0. X1 (4.253).
j (. 4.63).
j - 213
-
J
. 4.63:
p0 = `. , j .
. -, (. 4.64).
. 4.64: -
j X1 X2 .
214 j -
-
4
f , f (`)ij
(`) (4.239) (4.246). .
, j qx mT (. 4.65). C, r0 . j r0 a.
. 4.65: -
j qx, j S, (4.132)
S = r0qx (4.267)
Fp , j :
=S
Fp(4.268)
j :
=E
=S
EFp(4.269)
, r0, ,:
u = r0 =r0S
EFp=
r20EFp
qx (4.270)
j - . . (. 4.66) j .
j - 215
-
J
. 4.66:
mT mTP mTO, :
mT = mTP +mTO (4.271)
mTP , mTO - .
j mTP , j M( (4.139)):
M = mTP r0 (4.272)
M j . :
1 = MJp e1 =mTP r0Jp
e1
2 = MJp e2 =mTP r0Jp
e2
(4.273)
:
1 = 1E =mTP r0EJp
e1
2 = 2E =mTP r0EJp
e2
(4.274)
j j :
216 j -
-
4
u1 = 1r0 =mTP r
20
EJpe1
u2 = 2r0 =mTP r
20
EJpe2
(4.275)
j :
p =u1e1
=u2e2
=mTP r
20
EJp(4.276)
Jp j .
, j .
mTO, :
p1 = mTOJo c
p2 = mTOJo (b c)(4.277)
Jo j , (. 4.67).
ef 1.0. T abcd, C
j C . Jo .
1 :
p1 = kw1 (4.278)
: k - , w1 - 1.
(4.278), (4.277), :
w1 =p1k
=mTOkJo
c (4.279)
j - 217
-
J
. 4.67: j
j :
o =w1c
=mTOkJo
(4.271) :mTO = mT mTP
, :
o =mT mTP
kJo(4.280)
. (4.276) (4.280), :
mTP r2o
EJp=mT mTP
kJp,
:
mTP =EJpkJo mTr20 +
EJpkJo
(4.281)
(4.276), j
218 j -
-
4
j :
p =r20kJo mTr20 +
EJpkJo
:
=r20kJo 1r20 +
EJpkJo
: = mT (4.282)
mT = 1. p , , .
(4.270) (4.282), u , j X1 X2.
) f (p)11
f(p)11 X1 X1 = 1, (. 4.68). X1 = 1,
. 4.68: X1
u j . j d, :
f(p)11 = u+ d (4.283)
u (4.270) qx = 1 :
u =r2oEFp
(4.284)
X1 = 1, mT = 1 d. j
j - 219
-
J
j, (4.282), :
= d (4.285)
(4.284) (4.285) (4.283) :
f(p)11 =
r2oEFp
+ d2 (4.286)
) f (p)12
f(p)12 X1 X2 = 1(. 4.69).
. 4.69: X2
X2 = 1, :
= 1 =
X1 :
f(p)12 = d (4.287)
) f (p)21
f(p)21 X2 X1 = 1.
X1 = 1 j mT = 1 d (. 4.68). j , (4.282), :
f(p)21 = d (4.288)
) f (p)22
f(p)22 X2 = 1, X1, (4.282),
220 j -
-
4
mT = 1.f
(p)22 = (4.289)
, j - (. 4.70).
. 4.70:
a, :
pv = `b1
ph = `d1(4.290)
X1 X2.
:qx = ph = `d1 (4.291)
T :
mT = ph
(d2 +
d12
) pv
(c b1
2
),
:
mT = `[b1
(d2 +
d12
) d1
(c b1
2
)](4.292)
j - 221
-
J
X1 : u qx d j, mT .
(p)1 = u+ d
j u (4.270) (4.282) :
(p)1 =r2oEFp
qx + dmT (4.293)
qx mT (4.291) (4.292).
X2 :
(p)2 = mT (4.294)
:
fij = f(`)ij + f
(p)ij (4.295)
f (`)ij (4.239), f(p)ij (4.286), (4.287) (4.289).
(4.295), :
f11 = 123D +r2oEFp
+ d
f12 = f21 = 122D + d
f22 = 1D +
(4.296)
:
i = (`)i +
(p)i (i = 1, 2) (4.297)
(`)i (4.234) (4.235), (p)i (4.293) (4.294).
:
1 = a2`Eh +
r2oEFp
qx + dmT
2 = a2
Eh + mT(4.298)
222 j -
-
4
, :
f11X1 + f12X2 + 1 = 0f21X1 + f22X2 + 2 = 0
(4.299)
(4.251).
mTP (4.281)., mT , j (4.281) :
mT = m0T +mT1X1 +mT2X2 (4.300)
: m0T - , (4.292), mT1 - X1 = 1, mT1 = 1 d, mT2 - X2 = 1, mT2 =
1. :
M = r0mTP (4.301)
j :
S = r0qx (4.302)
:qx = q0x + qx1X1 (4.303)
: q0x - j (4.291), qx1 -j X1 = 1, j qx1 = 1.
4.5.10 j j -
j . j j - j. j j , j , j .
j j - , j , .
`1/`2, `1 , `2 - .
j - 223
-
J
`1/`2 > 4, , 4 `1/`2 1, `1/`2 < 1.
. 4.71: j
j j . x (. 4.72).
. 4.72:
j j, . j Mx Tx, H, . Mx, Tx H h/R, (j j j h/R 1/100). h/R, x j x
, (. 4.72). , : Mx = Tx = H = 0.
224 j -
-
4
, . j ( , sin4), j j (M), .
j j j: , j , , j j , ( j).
j j , j.
, , .
j , j j j . , , j . , , .
j , j , Mx, Tx H. .. j .
j j j. - j, j, . j, N, N = apn, . j j . j j . j j , . , j j, j j
j.
, j j .
j , ,
j - 225
-
J
j j. .
j Mx Tx.
j , sinm`1 x, j `1/m, (. 4.73).
. 4.73:
j , j . .
226 j -
MomentnaMomentna1