glava 4 momentna.pdf

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 = a2Èh

2 = a2

Eh a3`8Dp

(4.249)

= 0, :

1 = a2Èh

2 = a2

Eh 32a3Èt3

(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 -

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