,, : ;,,, (CIP) /. :, 005. ISBN 7 561 1901 6.... TB301 CIP (005) 007098 : : 17, : 71007 : ( 09 )8493844 : www.nwpup.com : : 787 mm1 09 mm 1/ 16 : 5.75 : 630 : 005 1 005 1 : 8. 00
( ) 1,,,,,,, 80100,,,,,, ;,,,,,,,,,,,,,,,,,,,,, 004 10
1 1 1 1 1 1 3 1 4 4 1 5 6 8 1 8 ( ) 9 3 ( ) 13 4 ( )14 5 19 6 ( ) 7 ( ) 5 8 ( ) 8 9 35 10 36 40 41 50 3 1 50 3 5 3 3 55 3 4 57 61 6
66 4 1 66 4 67 4 3 69 4 4 71 4 5 76 4 6 79 4 7 80 86 87 93 5 1 93 5 96 5 3 104 5 4 114 5 5 16 17 19 136 6 1 136 6 144 6 3 151 6 4 159 6 5 164 166 168 175 7 1 177 7 177 7 3 178 7 4 185 7 5 191 7 6 194 7 7 01 0
03 09 8 1 09 8 1 8 3 18 8 4 19 8 5 1 8 6 3 8 9 35 9 1 35 9 36 9 3 ( )38 9 4 39 9 5 43 46 47 53 10 1 53 10 53 10 3 55 10 4 59 10 5 63 10 6 64 * 10 7 71 10 8 74 85 86 91 11 1 91 11 9 11 3 98 11 4 301 11 5 31
14 315 31 1 1 31 1 31 1 3 35 1 4 331 333 333 338 13 1 338 13 340 13 3 341 13 4 347 13 5 351 13 6 351 353 354 357 14 1 357 14 359 14 3 363 14 4 365 374 376 379 393 404
1 1,,,,,,,,, :, ;,,,,,,,, ;,,,,,,,,,,, : (1 ),,, ( ),,, (3 ),,,,,,,, ;,,,,, 1
,,, 1,,,,,,,,,, (1 ),,,,,,, ( ),,,,,,,,,,,,,, (3 ),,,,,,,, (4 ),,,,,,,,, 1 3,,
. (1 ),, ( ),,,,,,,,,. (1 ),,,,, ; ( ),,, ;,, 1.,,,,,,,,,,,,.,,, 1 1,,,,,, FR M ( 1 ), x, y z, FR M x y z, Fx, Fy F z Mx, My Mz Fx, FN ; Fy F z ; Mx, T ; My Mz,,, 3
Fx 0, Fy = 0, Fz = 0 Mx = 0, My = 0, Mz = 0 1 1 1,,, 3 (1 ),,, ( ) (3 ), 1 4,,, 1 3, K A,A F,F A, p, 4
p F A 1 3 p, A,,, p, pk ( 1 p = lim A0 p = lim A0 F A = d F d A A (1 1) 1),, A 0 p,, p = + (1 ) N/ m, 1 N/ m = 1 Pa ( ), 1 GPa = 1 G N/ m = 10 9 Pa, 1 MN/ m = 1 MPa = 10 6 N/ m = 10 6 Pa,,,,,, ( ),,, ; 1 4( a), AB x,, A x, B x +, [ (x + + ) - ] - x =, AB, AB, m = u x = lim u x0 x = d u d x A x,, 1 (1 3) 4( b), +,,, 5
1 4 1 5,, ( ) (), (1 5 ) ; 1 5,,, 4 : (1 ), ( 1 6( a), ( b) ) ( ), (1 6 (c) ) (3 ), (1 6 ( d) ) (4 ),, ( 1 6( e) ),,,,,, ;, 6
1 6 7
( ),,,,,,,, 1, 1,,,, 3,, ;,, 1 3 8 :
1 ) :,, ( ) : ( ) 1. 4( a),, mm,, FN,, FN Fx = 0, FN - F = 0 FN = F 4,, FN, FN,,, mm,,, FN :, FN, ;, FN,, : (1 ),, ( ),,,.,,,,,, 9
,,,,, 1 5( a),, (1 ) AC : 11, ( 5( b ) ) Fx = 0 FN1 = F1 = 5 kn ( ) 5 CD :, ( 5( b) ), FN Fx = 0 FN + F - F1 = 0 FN = F1 - F = - 10 kn ( ) FN ( ),,, ( 5 (c) ) 5 (c) CB, FN max = 10 kn, CB : (1 ), (),,, ( ),, ( ),,,, ;,,, (),,,,,,, 10
, 6,, abcd,, abcd,, a b c d, :,,,,,,, 6 FN = d A = A A = FN A ( 1),, FN ( 1 ) FN, A :,,, ( ( x) = FN ( x) A( x), ( x), FN ( x) A( x),, ( max = FN max A 1) ( ) 1 ), ( 3), 7, AB 10.86 cm, F = 130 kn, = 30AB (1 ) AB A, Fy FN A B sin30= F = 0, 11
( ) A B FN AB F = 60 kn ( ) A B = FN A B A = 60 10 3 10-6 10.86 10-4 = 119.7 M Pa 7 8 3 8 (a), A1 = 3 cm, A = 4 cm ; l1 = l = 50 m A F = 1 kn = 0.08 N/ cm 3,, max (1 ) AB : 11, FN1 FN1 BC :, FN = F + A1 x1 (0 x1 l1 ) ( a), FN = F + A1 l1 + A ( x - l1 ) ( l1 x l1 + l ) ( b) ( ) (a) ( b ) x1 = 0, FN A = F = 1 kn x1 = l1, FNB = F + A1 l1 = 1 + 0.08 3 50 10 = 1.4 kn x = l1, FNB = F + A1 l1 + A ( l1 - l1 ) = 1.4 kn x = l1 + l, FNC = F + A 1 l1 + A l = 1.98 kn 8( b ) (3 ) (a) ( b), AB B, BC C ( 1 ) B = FNB A1 = 1.4 103 3 10-4 10-6 = 41.4 M Pa C = FN C A B, C, max = 41.4 MPa 1 = 1.98 103 4 10-4 10-6 = 36.8 MPa
3 ( ),,, ( ),,, mm ( 9 ) 9 F, A, ( 1 ), = FN A = F A mm A, AA A= A cos, p mm, p A cos - F = 0 p= F A cos= cos p,,,: = = pcos= cos ( 4) psin= sin ( 5) : x n, ;:, ;:,, ( 4 ) ( 5 ),,, 13
4) ( 5 ),,, () ( 4 ) ( 5) : (1 ) = 0, max ( ) = + 45, =,= 0 ; = max, = ; (3 ) = 90, = 0,= 0 : ( ) ; 45 4 ( ), E,,,,, 10,, l, l d l = 10 d, l = 5 d,, l A l = 11.3 A, l = 5.65 A 10 0.3%,,,, F,, L, F L,, F A,,, L l,, 14
( 11 ) 11, 1. ( O a ), a a, P,, = E ( 6) E,, E,, G Pa( 1 G Pa = 10 6 Pa)( 6) P, E oa E = = tan, a b,,, b e,, a, b,,,,,,. ( b c ),,,,,,,,,, s s 15
,, 1 3. ( ce ) 45,,, e b, b 1 13 4. e,,,,, 13, A, f, 5. l1, l, = l1 - l 100 % ( 7) l A, A1, = A - A1 A, : = 0% 30%,= 60% 100% ( 8), : 5% ;< 5% 6., Oa O ( 14), d,, ddd ddoa,,, dd, d, de f, d, d 14 Oabcde f d de f,, ( ), (,, 16 14 d), 14 dh
14 15,, Q345,, 4,,, 3, 15, 0.%,, P 0. ( 16 ) 17, 17
,,, 16 17, E,,,,, b,, h = (1.5 3 ) d 1., 18 : E s,,,,,, 18 18
., 19 (a), 45 55, ( 19 ( b) ) b bc bc = ( 3 5) b,, 19 5,,, s,,, b, ;, b,,,,,,,,,, u, ;,,,,, [ ], [ ] = u n ( 9), n 1, [ ] = s ns 19 ( 10)
, ns, nb ] = b nb ( 11), :, ns 1.5.5;, nb.5 5.0,, ( ),, max (), nmax max [ ] ( 1) = FNmax A [ ] ( 13) (), 1. (),,. (),, (), ( 3. A FNmax [ ] 13 ) ( 14) (),, ( 13) FN max [ ] A ( 15), () max [ ], 5%, 4 0( a), AB, CD, d = 30 mm, Q35, [ ] = 160 MPa, F = 50 kn, (1 ) 0( b ), MA = 0 FN sin30 3 000 - F3 000 = 0 FN = 150 kn 0 ( ) ( 1 ) CD
0 = FN A = FN d 4 = 4 150 103 30 10-6 = 1.3 MPa (3 ), = 1.3 MPa > [ ] = 160 M Pa, CD, 5 4, CD, CD ( 13 ), CD A = d 1 4 FN [ ] d1 4 FN1 [ ] = 4 150 10 3 160 10 6, CD 34.6 mm = 0.034 6 m = 34.6 mm 6 4, CD d = 30 mm,, FN = 3 F = FN A = 3 F d 4 [ ] F d [ ] = 30 10-6 160 10 6 1 1 F 37.7 kn = 37.7 kn 7 4, CD A B ( 1 (a) ), F AB,, M A = 0 1( b ), CD FN, A x, FN = Fx 000sin 1
, x = 3 000 ( F B ), FN 1 FNmax = 3 F sin, CD CD V AlC D = A FNmax = [ ] 3 F [ ] sin 3 F 000 [ ] sin cos =, CD V, V, sin= 1 = 45 6 000 F [ ] sin 4 7,, : ; ; ; 6 ( ) ;, l, A FN, l l1, l = l1 ( ), l l, = l l - l
, = E, FN A = El l l = FN l E A ( 16) ( 16),, l FN l, A EA (), EA,, EA () ( 16), l FN,, ( ), FN, b b1,, b = b1 = b b - b, ( ),,, =, (, 17) ( 17) = - ( 18) ( 16) A FN,,, d x, ( 3 ;,,, d(l) = 16), FN ( x) d x E A ( x), FN ( x) A( x), x 3
FN ( x) l EA ( x) d x ( 19) 3 4 8 4, BD A1 = cm, AD A = 4 cm E = 10 10 3 M Pa, F1 = 5 kn, F = 10 kn AB la B la B (1 ) BD, DC, CA FN1, FN, FN3 FN1 = - 5 kn, FN = - 5 kn, FN3 = 5 kn ( ) ( 16) lb D = l1 = ldc = l = lca = l3 = (3 ) FN1 l1 EA1 FN l EA FN3 l3 EA3 = = = - 5 10 3 0.5 10 10 9 10-4 = - 1.05 10-4 - 5 10 3 0.5 10 10 9 4 10-4 = - 0.5 10-4 5 10 3 0.5 10 10 9 4 10-4 = 0.5 10-4 la B = l1 + l + l3 = - 1.05 10-4 m 9 5 (a), BD, d = 0 mm, l = 1 m, E = 00 GPa ; BC, a = 100 mm, E = 1 GPa F = 50 kn B 4 (1 ) B( 5 (a) ) Fx = 0 Fy = 0, FN1 = 50 kn ( ) FN = 70.7 kn ( ) ( ) 5 (a), l1 = l = 1 m, l = BC BD l1 = FN1 l1 E1 l = A1 = FN l E A 50 10 3 1 00 10 9 4 0 10-6 l cos45 = 1.41 m m m m = 7.96 10-4 m = 70.7 103 1.41 1 10 9 100 10-6 = 8.31 10-4 m
5 (3 ) B, l1, l B, BD B1 D, BC B DB1 C D C, CB, B3 B3 B, B1 B3 B B3, B1 D B C, B3, B, BB1 = l1, BB = l 5 ( b) B B B4 = l1 + l B1 B3 = B1 B4 + B4 B3 = BB + B B4 = B, B B B3 l + l1 + l = 1.97 10-3 BB1 = l1 = 7.96 10-4 m m BB3 = ( B1 B3 ) + ( BB1 ) =.1 10-3 m,,,,,,, : ; ; ; ;, 7 ( ),, 5
,,,, V() 6 (a),, F, l, 6 ( b), F1, l1 6 d F1, dl1, F1 F1 dl1, dw = F1 dl1 6 ( b), dw F d F1, F F W 6 ( b), F l, W = l F1 dl1 = 1 0 Fl ( 0),,,,, W, V= W = 1 Fl ( 1) 6 FN = F,,, = FN l E A V= 1 FNl = F N l EA J( ), 1 J = 1 Nm ( ) ( )
,,,, d x, d y, d z, d x, d y, d z, ( 7( a) ), ( x ) ( ),,,,, 7( b ), 1 ( 1 d yd z), 1, d 1, d 1, d x d 1 d x, 1 d yd z d 1 d x, dw = dv= 0 7 1 d yd zd 1 d x = ( 0 1 d 1 ) dv, dv = d xd yd z, ( v= d V dv = 0 1 d 1 ( 3) 3) 1 d 1 7 ( b),, = E, v= J/ m 3 ( / 3 ) 1 v= 1 ( 4) = E = E, ( 5) ( 5) 10 8, AB A C A, A F = 35 knab A C d1 = 1 mm d = 15 mm, E = 10 GPa A (1 ) A, 1 7
FN1 = FN1 = 3 + 1 F ( ) 3 + 1 F ( ) ( ) V= F N1 l1 EA 1 + F N l EA = 4 F l1 E(1 + 3 ) d 1 + 8 F l E(1 + 3 ) d (3 ) A, F, W = 1 F 8 1 F= 4 F l1 E(1 + 3 ) d 1 + 8 F l E(1 + 3 ) d 8 Fl1 16 Fl = + = 1.368 10-3 m = 1.368 mm E(1 + 3 ) d 1 E(1 + 3 ) d,, F 8 ( ),, 9 (a), AD ( 9( b ) ), 3,,, 3,, 9 8
,,,,,,,,,, 9 ( b),,, 9( b ), 1 (), E1 A1, 3 () E3 A3, 3 l 3, FN3 ( ), 3, FN3, ( 30 (a) ) F F N3, 1 FN1 ( ), FN ( )A 30 ( b), Fx = 0, FN sin- FN1 sin= 0 Fy = 0, FN1 sin+ FN cos+ FN3 = 0 ( 6) 30,, F F N3 ( 30 (a) (c) ) 3 FN3 ( 30 ( d) ), F FN3 l; 3 FN3 l3 ( l = l3 ( 7) 7),, l1 = l = l3 = FN1 l1 E1 FN3 l3 E3 A3 A1 = E1 = FN3 l E3 A3 FN1 l A1 cos 9 ( 8) ( 9)
30 (c) ( 8) l = l1 cos = FN1 l E1 A1 cos ( 9) ( 30 ) ( 7 ), FN1 = E1 A1 ( 6) ( 31) FN1 = FN = FN3 = E3 A3 ( 30) cos FN3 ( 31) E1 A1 cos E1 A1 cos 3 + E3 A3 E3 A3 E1 A1 cos 3 + E3 A3,,, : (1 ), ( ), (3 ) () (4 ) ( ) ( 3), () 11 31( a),, B F, AB () EA 1, BC ( ) EA A, C F F 31 (1 ) F, A, C F A FC, 31 ( b) FA + FC - F = 0 ( a),, ( ) C, FC, FFC F FC l 30
,, l = 0 ( b) (3 ) AB BC, la B = FN AB l1 EA1 = FA l1 EA 1 lbc = FNBC l EA = - FC l EA l = la B + lbc = FA l1 EA1 (4 ) (c) ( b), - FC l EA (c) l = FA l1 EA1 - FC l EA = 0 ( d) (a) ( d) FA = A1 l A1 l + A l1 F A l1 FC = A1 l + A l1 1 3 (a), () EA, F, 1 3 l F 3 ( 1) 3,, F, 1 3 FN1, FN FN3, 3( b ) B Fx = 0, FN1 - FN sin30= 0 ( a) Fy = 0, FN3 + FN sin30- F = 0 ( b) ( ) 3 (c), 1 3 l3 = l sin30 + l1 = l + 3l1 (c) tan30 31
3 ), l = l1 = FN1 l1 EA = FN1 l EA FN l EA = FN l E Acos30 = FN l 3 EA l3 = FN3 l3 EA = FN3 l EA ( d) ( e) (f) (c), ( d) ( e) ( f) (4 ) (a) ( b ) ( g) 1. 3 FN3 = 4 FN + 3 FN1 ( g) FN1 = FN = 3 4 ( + 3 ) F 3 ( + 3 ) F FN3 = 8-3 3 4 ( + 3 ) F,,,,, 33,,,,,,,,,,, ;, 13, 34( a), 1 33 E1 A1, 3 E3 A3, 3 l, (1 ), 3,, 1, 34 (a) 1 3 FN1, FN FN3, 1 3 l1, l l3 A1 30 ( b), Fx = 0, FN1 sin- FN sin= 0 FN1 = FN ( a) Fy = 0, FN3 - FN1 cos- FN cos= 0 3
34 (a) FN3 = FN1 cos+ FN cos= FN1 cos FN1 = FN3 cos ( ) 30( c) ( b) (3 ) = AA1 = AA + A A1 = l3 + l1 cos l1 = l = ( d) (c), E1 = F N3 l E 3 A3 (4 ) (e) ( b ), FN1 = FN = FN3 = FN1 l FN3 l, l3 = A1 cos E3 A3 + E1 FN1 l A1 cos E1 A1 E3 A3 cos E3 A3 + E1 A1 cos l E1 A1 E3 A3 cos 3 E3 A3 + E1 A1 cos l,, ( ),.,,,,,,,, 35 (a) 35( b ), AB,, A, B,, 33 (c) ( d) ( e)
FA FB AB, FA = FB ( 3) 35 ( 3),,, FB l,, FB l = l T + lf ( 33),lT lf FB lt lf lt lf = lt ( 34) = FB l E A ( 35),, T, EA (),,, l = 0 ( 36) ( 33 )( 34) ( 35) ( 36 ), lt = FB FB l E A = EAT T = FB A = ET = 1.5 10-5 1/, E = 00 10 3 MPa, T, T = E T = 1.5 10-5 00 10 3 T =.5T M Pa T, T, :,, ;,, ; 34
, ; 5,,, ( ) ( ) (, ), 9,,,,,,,, 36,,,,,, 36,, Kt, n ; max Kt = max n, ( 37), F,, b, d, AA n = F ( b - d) ( ) ( ) 35
,,,,,,,,,,,,,, ( ), ( ), 10,, 37,,, :,, ;, 37,,,,, nn ( 37, 3 : mm 38( a) ) ; ( 38 ( b) ) ; ( 38 (c) ) 38 36
37, 39 (a),,,, mm nn ( 39 ( b) ), mm nn,, 39( c),,, FQ 39,,,, = FQ AQ ( 38), AQ = FQ AQ [ ] ( 39), [ ],, ( 38 ) b, ( 38),,,,,, ( 39),, 38,,,,, Fb s,, b s,,,,,, b s = Fbs Ab s ( 40), Abs ( ),, 37
), 40 ( b), d ( 40 (a), ( b max Fb s d 40 ), ( ), b s = Fb s Ab s [ b s ] ( 41), [ b s ],,, 40 14 41, 6, = 8 mm, b = 160 mm, d = 16 mm, F = 150 kn [ ] = 140 M Pa, [ b s ] = 330 MPa, [ ] = 170 M Pa 41 (1 ),,,, FQ = F 6 = 5 kn 38 = FQ AQ = 4 FQ d = 4 5 103 = 14.4 10 6 Pa = 14.4 MPa < [ ] 16 10-6 ( ),,
b s Fb s Ab s = F 6 d = 5 10 3 16 8 10-6 = 195.3 10 6 Pa = 195.3M Pa < [ b s ] (3 ),, 4( a),, 4 ( b) 4(c) 11, 33 3, 3-3, 33 33 33 = FN3 A3 = F b- d = 150 10 3 ( 160-16) 8 10-6 = 146.5 10 6 Pa = 146.5 M Pa < [ ] 15 43( a) (, ) d = 70 mm, bh l = 0 mm 1 mm 100 mm, [ ] = 60 M Pa, [ b s ] Me = 100 MPa 4 (1 ) nn, ( 43( b ) ), nn FQ, FQ d = Me 43 39
FQ ( ) Me = FQ AQ = Me d bl Me d = Me bdl [ ] bdl[ ] = 0 70 100 10-9 60 10 6 (4 ) 43( c) = 4 00 N = 4. kn Fb s = FQ = Me d Me dhl[ b s ] 4 b s = Fb s Ab s = Me d h l = 4 Me dhl [ b s ] = 70 1 100 10-9 100 10 6 4, Memax =.1 kn = 100 N =.1 kn, :, ( ),, ;,,,, ;, ;,, ( ),, (1 ), ( ) = FN A (3 ),, 40 (4 ) max [ ] 3 : (5 ) = E
(6 ) ( ) l FN l E A V= F N l EA v= 1 = E (7 ) () :, ; ; ; ; (8 ) : 1) ) = FQ AQ [ ] b s = Fb s Ab s [ b s ] 1 1, 1 11, 3 3 11, 33, A = 400 mm, 4 4 11, 33, A1 = 00 mm, A = 300 mm, A3 = 400 mm, 5 5 F = 10 kn, A = 100 mm, = 0, 30, 45, 60, 90 6 6, F,, 41
3 4 5 6 7 7, a = mm, b = 0 mm, l = 70 mm F = 6 kn, l = 0.15 mm, b = 0.014 mm, E 7 8 8 1,, 3, 4 : 1, A1 = A = 4 000 mm, [ w ] = 0 MPa, 3, 4 A3 = A4 = 800 mm, [ s ] = 10 MPa [ F] 4
9 9,, [ t ] = 50 M Pa, [ c ] = 60 MPa, 5 mm [ F] 8 9 10 10, 1,, B F d b F = 50 kn, [ s ] = 160 M Pa, [ w ] = 10 M Pa 10 11 11 11, F [ F] A, [ ] 1 1, F, [ ] A B, 1 13 43
13 13 [ ] = 170 MPa, AC CD 14 14, AB, A1 = cm, [ s ] = 160 MPa ; AC, A = 3 cm, [ c ] = 100 M Pa, [ F] 14 15 15 15 A E D 16 16 00 mm,, E = 10 GPa, : (1 ) ; ( ) ; (3 ) 16 17 17 17, F, f = ky, k F = 40 kn, l = 1 m, A = 64 10 3 mm, E = 10 GPak, 18 18, F, l, b1 44 b, E
18 19 19, AB, CD d = 0 mm, E = 00 G Pa, [ ] = 160 MPa, : ;, B 19 0 0 0, BC, 1 3, E = 00 G PaC F l = 1 m, A1 = A = 100 mm, A3 = 150 mm, F = 0 kn 1 1, A C, F EA, A C 1 CA, l1 = m, A1 = 00 mm, E1 = 10 5 MPa ; BD, l = 1 m, A = 800 mm, E = 1 10 5 M Pa AB,, : (1 ) AB, F DB x ; 45
) mm, F? 3 3, 3 4 4, 1 () EA, AB, F = 0 kn, [ t ] = 30 MPa, [ c ] = 90 MPa 4 5 5 5, L = 3 m, = 5 mm, d = 100 mm F [ s ] = 160 M Pa, Es = 00 GPa; [ h ] = 30 M Pa, Eh [ F] = 30 GPa 6 6, E = 10 G Pa, A1 = 3 000 mm, A = 500 mm F = 180 kn: (1 ) = 0.6 mm ; ( ) = 0.4 mm 7 7, Es = 10 GPa, cm, = 5 10 4 l, 46 (1 ) ( 7 ( a) ) ; ( ) 3 ( 7 ( b) )
6 7 8 13 43, = 110-6 /, E = 00 GPa 9 9 1, E1 = 10 GPa, 1 = 1.5 10-6 l/, A1 = 30 cm ;, E = 105 GPa, = 19 10-6 /, A = 30 cm F = 50 kn 1 9 30 30 30, F d, D, h,, [ ] = 10 MPa, [ ] = 90 MPa, [ b s ] = 40 MPa 31 31 F, b, [ ], [ b s ] l a 3 3,, b = 50 mm, F = 50 kn, [ b s ] = 10 MPa, [ ] = 1 M Pa L 47
31 3 33, a = 00 mm, l = 1 m F = 100 kn, 33, [ ] = 1.5 MPa,,? 33 34 34 F, F = 80 kn, b = 80 mm, = 10 mm, [ b s ] = 30 M Pa d = 16 mm, [ ] = 160 MPa, [ ] = 10 M Pa, 48
34 49
,,, 3 1 3 1, A, Oz y d A, ( z, y), Sy = A z d A, S z y z (3 = A yd A (3 1) 1 ),,,,,,, 3 1,,, d A d A,, Oz y yc zc yd A A = A = zd A A A yd A A = A z d A A = A, C (3, C ( 3 1) (3 ) (3 ) 1 ), 50 3 1
zc Sy A, yc = Sz A (3 3) S y = AzC, S z = AyC (3 4) ( 3 4),, yc = 0, zc = 0,, S y = 0, S z = 0 ;,, ( ),,,, Sz n = Ai yci, Sy i = 1 n = Ai zc i (3 5) i = 1, Ai yc i, zci n n (3 5 ) ( 3 3), yc = Sz A n Ai yc i = i = 1 n Ai i = 1, zc = Sy A n Ai zc i = i = 1 n Ai i = 1 (3 6) 3 1 3 S y, Sz C R (1 ) y, Sy = 0 z d A, (3 Sz d A = Rcos d y d y = Rcos d, y = Rsin =yd A A = 0 d A = R cos d 1 ), z Rsin R cos d= 3 R3 ( ) (3 ), yc = Sz A = 3 R3 1 R = 3 R 3, zc = 0 3 3 3( a) C 3 3 3 ( b) Oy z, A1 = 0.1 0.0 =.0 10-3 m 51
3 3 y1 = 0.1 (3 = 5.0 10 - m A = 0.0 0.1 =.0 10-3 m y = 0.1 + 0.0 6 ), C = 0.11 m yc =.0 10-3 5.0 10 - +.0 10-3 0.11 = 0.08 m = 80 mm.0 10-3 +.0 10-3 zc = 0,,,,,,, 3 3 4 A, d A, ( z, y), Iy =z d A, I z A =y d A (3 7) A y z (3 7 ), Iy I z, 5, I y = Ai y, Iz = Ai z (3 8)
iy = I y A, iz = Iz A (3 9) iy iz y x d A, IP = d A ( 3 10) A 3 (3 4 = y + z IP 3 4 = A ( y + z ) d A = Iz + Iy ( 3 11) 11),,,,,, 3 4 I y n = Iyi, Iz i = 1 I yz n = Izi ( 3 1) i = 1 = A y z d A ( 3 13) y, z ( 3 13 ), I yz, y z, Iy z, d A z yd A,, I yz =y zd A A 3 3 ( 1) 3, (3 10) IP 5( a), d, d d A = d = d/ 0, I y = Iz d= d4 3 (3 11), IP = I y = I z Iy = Iz = IP = d4 64 53
3 5 ( ) d, D (3 5( b) ),, IP = D4 3 - d4 3 = 3 ( D4 - d 4 ) = D4 3, y z (1-4 ) Iy = Iz = D4 64 - d4 64 = 64 ( D4 - d 4 ) = D4 64 ( 1-4 ),= d D 3 4 3 6 3, d A = hd z, ( 3 I y 6, d z h y 7) y =z d A A = b/ - b/, z z hd A = hb3 1 Iz = bh3 1,, I y n = Iyi, Iz i = 1 3 6 n = Izi ( 3 14) i = 1 3 5 3 7( a) z 3 7( b ) BH, BH z I z 1, z Iz, z I z Iz1 = Iz + Iz 54
3 7 3 4 Iz = I z1 - I z z Iz Iz1 = BH3 1 B - d = 1 Iz = BH 3 1 - h 3 = ( B - d) h3 1 ( B - d) h 3 1 3 3,,, 3 8, C, yc zc,, Iy C Iz C Iy C z C = A z C d A = A y C d A yc zc = A d A ( 3 15) y yc, a; z z C, b, y z 3 8 55
3 8 (3 17) (3 16) I yz I y Iz =z d A A =( zc A =y d A A =( yc A I y I z I yz A z d A =y d A A =y z d A A ( 3 16) y = yc + a, z = zc + b ( 3 17) + b) d A =z C d A + zc A bd A + b A d A A + a) d A =y C d A + yc A ad A + a A d A A =y z d A A =( yc + a) ( zc + b) d A yc zc A = d A + yc A bd A + zc A ad A + A abd A A zc, A d A yc A d A yc zc, yc zc, A d A = A, ( 3 15 ), Iy = Iy C + b A Iz = Iz C + a A ( 3 18) Iy z = Iy C z C + aba ( 3 18) ( 3 18),, 3 6 3 (3 3 ) T 3 3 ( b), z1, (3 I z = I z 1 + A1 a 1 = 0.0 0.1 3 1 15), z 0.0 0.13 1 + ( yc - 0.05) 0.0 0.1 = + (0.08-0.05) 0.0 0.1 = 3.47 10-6 m 4 z, ( 3 I z = I z + A a = 0.1 0.0 3 1 0.1 0.03 1 z Iz 18 ), z + (0.1 + 0.0 - yc - 0.01) 0.0 0.1 = + (0.11-0.08) 0.0 0.1 = 1.87 10-6 m 4 = I z + I z = 3.47 10-6 + 1.87 10-6 = 5.34 10-6 m 4 3 7 3 9, x I x = bh3 1, x1 x, x1 I x 1 56
,, Ix I x 1, Ix C, x xc ( 3 18 ) I x = I x C + Ix 1 - Ix = h 3 I x 1 = I x C + h 3 h 3 - h 3 A A = bh3 6 3 9 I x 1 = bh3 6 bh3 + Ix = 6 + bh3 1 = bh3 4, :,, ;,,,,,,,, 3 4 A (3 I y I z I y z =z d A A =y d A A =y z d A A 10 ), y z, ( 3 19) O,, y1, z1, y1, z1 Iy 1 =z 1 d A, Iz A 1 3 =y 1 d A, Iy z A 1 1 =y 1 z1 d A A ( 3 0) 10, d A z1 (3 0), y1 z1 = OB = OF + DE = ycos+ zsin = CB = CE - BE = zcos- ysin 3 10 ( 3 1) 57
I y 1 =, ( 3 A z 1 d A =( z1 cos- ysin ) d A = A cos A z d A + sin A y d A - sin cos A y z d A = I y cos + I z sin - I yz sin= Iy + Iz + I y - Iz 0 ) cos- I yz sin ( 3 ) Iz 1 = Iy + Iz - Iy - Iz cos+ I y z sin ( 3 3) Iy 1 z 1 = I y - Iz sin+ Iy z cos ( 3 4) I y 1, Iz 1, I y 1 z 1, (3 )(3 3) (3 4) (3 18) (3 3) Iy 1 + Iz1 = I y + Iz ( 3 5), O,,, ( 3 4),,, 0 y0, z0, I y 0 z 0,,,, (3 4) = 0 I y z 1 1 = 0 I y - I z sin 0 + I yz cos 0 = 0 ( 3 6) (3 tan 0 = - Iy z I y - Iz ( 3 7) 7 ) 0, y0 z0 Imax, Imi n (3 ( 3 7) 0, ( 3 ) ( 3 3) 7 ) cos 0 = 1 = 1 + tan 0 Iy - Iz ( Iy - Iz ) + 4 I yz sin 0 = tan 0 cos 0 = - I yz ( I y - Iz ) + 4 I y z 58
3 ) ( 3 3 ), Iy 0 = I y + Iz Iz 0 = Iy + Iz + 1-1 ( I y - Iz ) + 4 I yz ( Iy - Iz ) + 4 I y z ( 3 8) ( 3 ) (3 3), I y 1 I z 1, 0 360, Iy 1 Iz 1 ( 3 5 ), Iy 1 Iz 1, Iy 1 I z 1 = 1, d Iy 1 = 0, d = 1 d Iz 1 = 0 d = 1 I y - I z sin 1 + I yz cos 0 = 0 ( 3 9) ( 3 9 ), Iy Iz 1 1 ( = 1 ), I y z 1 1 = 0,, : (1 ),, 3 11( a), ( b), (c) ( ), 3 11( d), ( e) (3 ),, 3 11( f), ( g) (4 ), 3 11 ( h ), (i) 3 11 59
8 3 1 L, yc (1 ) 3 zc, yc = Sz A 3 1 1, Oy z C, = 10 10 60 + 70 10 115 10 10 + 70 10 zc = S y A = 70 10 40 10 10 + 70 10 = 15 mm = 80 mm ( ) y, z yc, zc Iy C, Iz C Iy C z C I y C = 10 103 1 + 10 10 15 + 10 703 1 + 10 70 5 = 10 000 + 70 000 + 85 833 + 437 500 = 1.003 10 6 mm 4 I z C = 10 103 1 + 10 10 0 + 70 103 1 1 440 000 + 480 000 + 5 833 + 857 500 = 783 333 =.783 10 6 mm 4 + 70 10 35 = I y C z C = 10 10 ( - 15) ( - 0 ) + 70 10 5 35 = 360 000 + 51 500 = 9.75 10 5 mm 4 (3 ) (3 7) tan 0 = - Iy C z C I y C - Iz C = -, yc, zc 60 (4 ) (3 8) 9.75 10 5 1.003 10 6 -.783 10 6 = 1.093 0 = 47.5 7.5 0 =.8 113.8 0 =.8
I z = Imax = Iy C + Iz C + 1.783 10 6 + 1.00310 6 3.14 10 6 mm 4 I y 0 = Imin = Iy C + Iz C - 1.78310 6 + 1.00310 6 0.574 10 6 mm 4 ( Iy C - Iz C ) + 4 I y C z C = + 1 (.78310 6-1.00310 6 ) + 4(9.7510 5 ) = ( Iy C - Iz C ) + 4 I y C z C = - 1 (.78310 6-1.00310 6 ) + 4(9.7510 5 ) = (1 ), S y = Z z d A, S z, ( ) zc = Sy A = Z yd A, yc = Sz A (3 ), Iy =z d A, I z A =y d A A (4 ), IP = A d A IP = Iy + Iz (5 ), I yz = A y z d A,, (6 ), Sz n = Ai yci, Sy i = 1 n = Ai zc i i = 1 I y n = Iyi, Iz i = 1 n = Izi i = 1 61
7 ), I y = Iy C + b A Iz = I z C + a A I yz = Iy z C C + aba (8 ), Iy 1 = I y + I z + Iy - Iz cos- Iy z sin I z1 = Iy + Iz - I y - I z cos+ I yz sin Iy 1 z 1 = I y - Iz sin+ Iy z cos (9 ), tan 0 = - Iy z I y - Iz I y 0 = Iy + Iz I z 0 = I y + I z + 1-1 ( Iy - Iz ) + 4 I y z ( Iy - Iz ) + 4 I yz 3 1 3 1 3 1 3 3 3 3 3 3 3 4 3 4 6
3 3 3 4 3 5 3 5, b = 150 mm, h = 300 mm 3 6 3 6 3 7 3 7 3 8 3 8 0a, I y = I z, a 3 9 3 9 63
5 3 6 3 7 3 8 3 9 3 10 3 10 64
10 65
,, ; ;,, 4 1 4 1( a),, F, Me = Fd,, Me= Me, 4 1( b ) Me Me, 4 1 4 66
,,,,, Me, Me, 4 3,, AB AB, AB P kw( ), 1 kw = 1 000 Nm/ s, P kw W = 1 000 P ( Nm ) Me AB, n ( r/ min), Me nme, Me ( Nm ), P1 000 = n 60 M e 4 3 Me = 9 549 P n (4 1), P ( k W, ) ; n ( r/ min) 4, 4 4( a), nn, ( 4 4 ( b) ),,,, nn T, Mx = 0, T - Me = 0 T = Me, T nn,, nn 67
4 4( c) ), T = Me,,, T : T,, T ;, 4 4, nn,,,,, 4 1 4 5( a), A 4 4, B, C, D PA = 60 kw, PB = 5 kw, P = 5 k W, PD = 10 kw, n = 300 r/ min 4 5 4 6 (1 ) (4 1 ) 68
Me B MeC = 9 549 PB n = 9 549 5 300 = 796 Nm Me A = 9 549 PA n = 9 549 60 300 = 1 910 Nm Me D = 9 549 PD n = 9 549 10 300 = 318 Nm ( ), AB : 4 5( b ), 11 T1, 4 5 ( b) T1 - MeB = 0 T1 = MeB = 796 Nm AC : 4 5( c), AD : 4 T + Me A - MeB = 0 T = MeB - Me A = - 1 114 Nm 5( d), T3 + MeD = 0 T3 = - Me D = - 318 Nm (3 ),,, 4 5( e), BC, T max = 1 114 Nm (4 ), A, 4 6 (a), 4 6 ( b), T max = 1 910 Nm,,, 4 5 4 3 4 7( a), R( R ) 10,,, (4 7 (a) abcd),,, ( 4 7( b) a b c d ),,,,,,, (4 7 (c) ) Me, Me =Rd A A = 0 RR d= R 69
Me R (4 ) 4 7 4, d x, d y, 7 ( d),,,,, (4 ) 4 7( d ),,,,, (d y) d x,,,, Fx = 0,, (d y) d x, Mz = 0 ( 4 (d y) d x = ( d x) d y = (4 3) 3 ),,,,,, 4 7 ( d),,,,, (4 7 ( d) ) 70, l, 4 7 ( b),
, r l (4 4),, = G (4 5) (4 5 ) ; G,, Pa, E,, G,, E,, G G = E (1 + ) (4 6) 4 4, 1. 4 8 (a),,,,,,, :,,, d x (4 4 8 8( b ) ), d, O b d, O b, ab, ab,, d d x, a cd, cd,, c 4 8 ( b) 71
, dd d x = d (4 7) = d d x d, (4 d x,. (4 8) 8 ),,,,, (4 = G = G d d x (4 9) 9 ),,,,, 4 9 4 9 4 10 3. 4 10, d A, d F = d A, O d A,, (4 9 ) T = A d A IP, A (4 10) (4 8 ) 7 d A T = G d A d A d d x = = d x T GI P T IP ( 4 10) ( 4 11)
1. (4 11), = R,,, max = TR I P = T IP R ( 4 1), IP R,, WP, WP = IP R ( 4 13) (4 (4 1) max = T WP 14),,. ( 4 14) Tmax,, ( 4 14 ), max max = Tmax WP [ ] ( 4 15), [ ] [ ], u, n, [ ] = u n ( 4 16),,,, (4 45(4 11 (a) ) ;,, 11 ( b) ), s ;,, b s b,, u 4 11 4 4 1 (a), AB d1 = 10 mm, BC d = 100 mm 73
Me A knm, Me B = 36 knm, MeC = 14 knm [ ] = 80 M Pa, AB, BC T1 T 4 = knm = - 14 knm 1 ( b), AB BC,,, ( 4 14 ) ( 4 15 ) AB BC 1 max = T1 W P1 = 103 = 16 ( 0.1 ) 3 64.8 10 6 Pa = 64.8 MPa [ ] 4 1 max = T WP, = 14 103 6 = 71.3 10 16 (0.1 ) 3 Pa = 71.3 MPa [ ] 4 3, Me =.0 knm, [ ] = 60 M Pa, ; = 0.8, (1 ) d, (4 15) W P = d3 16 T [ ] = Me [ ] d = 5 mm 3 16 Me d = [ ] 3 16. 10 3 80 10 6 = 51.9 10-3 m = 51.9 mm ( ) D, (4 15 ) 3 16 Me D = ( 1-4 ) [ ] D = 6 mm, d1 3 W P = 50 mm = D3 16 ( 1-4 ) Me [ ] 16. 10 3 ( 1-0.8 4 ) 80 10 6 = 61.9 10-3 m = 61.9 mm (3 ),, 74 = A1 A = 4 ( D - d 1 ) = 4 d 6-50 5 = 0.50
, 4 4 3,,, 4 4 4 13( a) ( ( 5 ) ), F D, d, 4 13 (1 ),, (4 13( b ) ),,,, FQ T, FQ = F T = FD ( ), FQ, (4 13( c) ), = T 4 max= FQ AQ T W P = FQ d 4 = 4 F d 13( d), = FD d 3 16 = 8 FD d 3,, A,, d D, D d max = max+ = 8 FD 1 + d d 3 D ( 4 17) 10, d 1, D 75
( 4 17 ) max = 8 FD d 3 ( 4 18) D d < 10,,,, max = 8 FD 4 m + d 3 4 m - 3 ( 4 19), m = D d (3 ), [ ] max [ ] 4 5, ( 4 d x d= Td x GIP, l = d= T GIP, lt, (4 GIP = Tl GIP 10 ), d x ( 4 0) 0 ) ( 4 1) ( 4 1), T l, GIP,,,, d d x, [ ] ( 4 10), 76, = d d x = T GIP ( rad/ m)
max T GIP max [ ] ( rad/ m) ( 4 ), max = Tmax GIP [ ] ( rad/ m ) ( 4 3), d rad/ m,, d x ( )/ m,, ( 4 ) ( 4 3 ), P3 4 5 4 14, n = 500 r/ min, P1 = 380 kw, P = 160 kw, = 0 k W, [ ] = 70 M Pa, [ ] = 1 / m, G = 80 GPa : (1 ) AB, BC ( ) = 0.8, (1 ) : 1) : Me A MeB = 9 549 P1 n = 9 549 P n = 7 57 Nm = 3 055 Nm Me C 4 ) : = 9 549 P3 n = 4 0 Nm 14 ( b) AB : d1 d1 = 85.3 mm 3 max = T W t 16 T = [ ] max = 3 = 16 T d 3 1 16 7 57 70 10 6 T G d4 1 3 d1 [ ] 180 [ ] 85.3 mm = 80.8 mm 4 14 BC :, d 67.4 mm; d 74.4 mm d = 74.4 mm ( ), D D 96.3 mm D 97.3 mm D = 97.3 mm, 77
d D = 0.8 97.3 = 77.8 mm, A A 47% = 4 ( D - d ) 4 d 1 = 0.47 4 6 4 15( a) AB,, C D Me, A B MA MB ( 4 16 ( b) ), Mx = 0, MA - Me + Me - MB = 0 MA = MB ( a) (a),,,, A B, A B A B,, 4 A B = AC + C D + DB = 0 ( b) 14( b), AC, CD DB T1 = - MA T = Me - MA (c) T3 = - MB, AC, CD DB ( d) ( b), - MA a GIP A C = T1 a GIP C D = T a GI P DB = T3 a GIP + = - = = - ( Me - MA ) a GI P MA a GIP ( Me - MA ) a GIP MB a GIP 4 15 - MA + Me - MB = 0 ( e) (a) (e), - MB a GIP = 0 ( d) 78 MB = Me 3 ( f)
MA MB = Me 3 4 15 (c) ( ), 4 6,,,,,, ( 4 T (4 18), 16( a) ), T1, 1 d T1, d 1, T1, T1 d 1 4 dw = T1 d 1 17( b ), dw 0 T d T1, 0 T W 4 16( b ), T, 4 16 (4 1) = Tl W = T1 d 0 1 = 1 V= W = T1 d 1 = 1 0 GIP T ( 4 4) T ( 4 5), V V= T l GIP = GIP l ( 4 6) 4 7 4 13( a), F D, d, G 4 13 ( b), FQ T, 79
FQ F T = FD ( a) F,,, s, (4 V= 6), T s GIP n, (a) ( b),, s = nd V= 4 F D 3 n Gd 4 F = 4 F D 3 n Gd 4 ( b) = 8 FD3 n Gd 4 ( 4 7) 4 7,, 4 17 (a),, (4 17( b ) ),,,,,,,,,,, ;,, 4 17 80
, K1, t n (4 18),, n, n = n n, n= 0,, n = 0,, t K, 1,, 1, 1,, 4 18 4 19,,,, 4 19, ; max, 1 max = T W t = = T hb ( 4 8) 1 = max ( 4 9) Tl = Tl Ghb 3 GIt ( 4 30), It = hb 3, h b ;, h b, 4 1 4 1, h/ b 1.0 1. 1.5.0.5 3.0 4.0 6.0 8.0 10.0 0.08 0.19 0.31 0.46 0.58 0.67 0.8 0.99 0.307 0.313 0.333 0.141 0.166 0.196 0.9 0.49 0.63 0.81 0.99 0.307 0.313 0.333 1.000 0.930 0.858 0.796 0.767 0.753 0.745 0.743 0.743 0.743 0.743 81
1, h b > 10, (4 0), = 1 3,,, ( 4 8 ) (4 30) = max = T ( 4 31) 1 3 h Tl G 1 = Tl 3 h 3 GIt 4 ( 4 3) 0 4 8 4 1, l, Me, d; h/ b = 3/ d b = Me 4 = bh = 3 b 6 d, cmax = 16 Me d 3, c = 3 Me l Gd 4 4 0 (4 8) ( 4 30 ) 4 1, r max = Me = hb r = Me l Ghb 3 = Me 0.31 hb = Me l 0.196Ghb 3 = Me 0.347 b 3 Me l 0.94Gb 4 4 1, 8
cmax r max c = r 16 0.347 3 0.94 3 6 4 6 = 0.669 = 0.81,,,,,,,,,,,,, ; ; ; (4 ) 4 4 3 T,, T1,,, T i,, 1,,, i,, (4 1 = T1 l GIt1 3) = 1 = = = i = = ( 4 33) T1 l G 1, = T l 3 h1 3 GIt 1 T = T1 + T + + Ti + ( 4 34) = T l G 1,, i = T i l 3 h 3 GI ti = G 1, 3 hi 3 i Ti l (4 33) ( 4 34 ) (4 35) T = G l 1 3 h1 1 + 1 3 h 3 + + 1 3 h1 3 i + = G l It i ( 4 35) 83
Tl G = Tl GI t t i ( 4 36) T i = It i It T ( 4 37), It = It i = 1 3 hi 3 i, ( 4 i = (4 37) (4 38), (4 Ti = 1 3 hi i Ti i 1 3 hi 3 i i = T i It 31 ), = T i i 39),, max = T max It It i ( 4 38) ( 4 39) ( 4 40),,,, It,, It = 1 3 hi 3 i ( 4 41),, = 1.00, = 1.1, T = 1.15, = 1.0 (4 4 4),, 3,,,,,, d x, abcd, 4 5 ( b), b 1, 1 ; c,,, ab cd 1, Fx = 0, 1 d x 1 - d x = 0 1 1 = i i = ( 4 4) i i (4 4),,, d s, d s ds, O d s, O, 84 T = d s = d s
4 4,O ; d s d0, d s A0 (4, T = A0 = T A0 ( 4 43) 43),,, A0,,, max = T A0 max ( 4 44) 4 9 4 5 (a), ( b), 4 5 ( b), Me, b = 50 mm = mm 4 5 4 5( a), 4 5 (c), (4 44), max= T = A0 mi n T = Me Me b = Me b 4 5 ( b), 4 5 ( d), (4 40), 85
max T max It = max T = 1 3 hi 3 i max = 3 b = 3 50 1 3 Me = 3 Me 4 b 3 4 b = 37.5,, 37.5 (1 ), Me = 9 549 P n ( ) = Me R (3 ),, ;,, = (4 ) :,, (5 ) : = G = T IP (6 ) : max = Tmax WP = Tl GIP [ ] 86 (7 ) : (8 ) : max = Tmax GIP V= T l GIP [ ] = GIP l
max T W t = T hb = Tl = Tl Ghb 3 GIt max = T max max = It T A0 min 4 1 4 1, 4 1 4 4, n = 500 r/ min, A, PA = 70 kw, BC D, PB = 10 kw, PC = PD = 30 kw (1 ) ; ( ) A C, 4 3 4 3 300 r/ min,, A 70 k W, B, C, D, E 0 k W, 10 kw, 5 kw 15 kw, 4 4 4 4, T 4 5 4 5 (a),, (4 5 ( b ) ), 87
4 3 4 4 4 5 4 6 4 6 n = 00 r/ min, P = 0 kw, [ ] = 80 MPa, d1/ D1 = 0.5 d, d1 D1 4 6 4 7 4 7, AE, D = 88
40 mm, d = 80 mm; BC, d1 = 100 mm, Me A = 0 knm, Me B = 36 knm, MeC = 16 kn m [ ] = 80 M Pa, G = 80 GPa, [ ] = 1. / m 4 7 4 8, F = 1.5 kn D = 50 mm, d = 8 mm, [ ] = 450 M Pa, 4 9, d = 0 mm, Me = 150 Nm l0 = 100 mm = 0.01 rad, G 4 10, n = 300 r/ min, P = 80 kw, [ ] = 80 MPa, [ ] = 1.0 / m, G = 80 GPa 4 11, R0, R1 R, R1/ R = n; Me s Meh, Mes Meh = 1 - n 1 + n 4 1, Me d,, l, G, = 4 Me l Gd 3 4 13 4 13, GIP 4 13 89
14 4 14 d = 60 mm Me, [ ] = 50 M Pa, [ ] = 0.35 / m, G = 80 GPa[ Me ], 4 14 4 15 4 15 4 15, d, m 4 16 10 mm100 mm 100 mm80 mm, Me, = 4 knm, Me 4 17 4 17 d = 5 mm, D = 75 mm = 1.5 mm, Me = 73.6 Nm,,,, G = 40 GPa : (1 ),? ( ) 4 17 4 18, h/ b = 4 19 4 19 90 mm60 mm, [ ] = 80 M Pa, G = 80 GPa, Me A 4 0 d d d d, Me ; 4 1 = = 300 Nm [ ] = 60 M Pa T, IP 90
(1 ) ; ( ), ; (3 ) ; 4 19 (4 ), 4, 1 max max, G1 G (1 ) 1 max > max ; ( ) 1 max < max ; ( ) G1 > G, 1 max > max ; (4 ) G1 < G, 1max < max 4 3 d1 d, D (= d/ D ), ( W1/ W ), (1 ) ( 1-4 ) 3/ ; ( ) ( 1-4 ) 3/ (1 - ) ; (3 ) ( 1-4 ) (1 - ) ; (4 ) ( 1-4 ) / 3 (1 - ) 4 4 4 4 T, l = m, Me = 00 Nm, G = 80 GPa 4 4 4 5 4 5 4 5, 4 6 4 6 ( a), 91
e 4 15 9
,,,,,,, 5 1 1. ( ),,,,, ( 5 1 (a) )(5 (a) )(5 (a) ) ( 5 3( a) ). 5 1 5,, T 93
5 5 ), ( ),, (5 6), 5 3 5 4 5 5 5 6,,,,,,,,, 1. ( ),,,,,., 94
), q( ) N/ m( / ) kn/ m( / ) q, 5 1, 5 3,,, ( ), F N( ) kn( ) 5 5 4,,,, Me Nm( ) knm () 3.,,,,, : (1 ),,,,,,, 5 7( a),,, ( ),,,,,, 5 7 ( b),, (3 ), 5 8 (c),,,,, 5 7 (c) 5 7 5 8,,, 95
,,,, (5 1( a) ) ;,,,,, ;,,,, ( 5 1 ( b) ), (5 4 (a) ),,,,,,,,, (5 4 ( b) ),, (, ), l 5 1( b )5 ( b) 5 3 ( b)5 4( b),,, 5 7, Fx,, : (1 ) :, ( 5 8( a) ) ( ) :, ( 5 8( b ) ) (3 ) :, ( 5 8( c) ),,, 5 ( ), 5 9 (a),, Ab mm, mm, ( 5 9( b) ) F1 FA y, Fy = 0 MC = 0 ( C mm ), mm : FQ 96 M
FQ,, N () kn( ) ; M,,, Nm ( ) knm(), FQ 5 9 Fy = 0, FA y - F1 - FQ = 0 FQ = FA y - F1 (5 1), MC = 0, M + F1 ( b - a) - FA y b = 0 M = FA y b - F1 ( b - a) (5 ) M C, ;,,, mm FQ, M,,, :, FQ FQ (5 ( 5 10 (a) ), ; M, 10 ( b) ),,,,,,, 5 9 ( b) mm 97
5 1) (5 ),,, : (1 ) FQ (), FQ = Fyi( ) (5 3), ; ( ) M ( ) C, M = Me C i( ) (5 4),,,, ;,, ;,, ( 5 3 ) ( 5 4 ),, ( ), (5 10) ;, 5 10,,,,, (5 3 ) ( 5 4),,, 5 1 5 11 (a), C F, C CCCF C, C, C (1 ) A B F A y, FB y,, MB = 0, FA y l - Fb = 0, FA y = bf l ; MA = 0, FB y l - Fa = 0, FB y = af l ; 98
5 11 : Fy = 0, FA y + FB y - F = bf l + af l - F,, FA y, FB y, 5 11( a) ( ) 5 11( a), C,, (5 3) (5 4 ), FQC = bf l MC = abf l FQC MC, FQ C MC ( 5 11( b) ) C,, (5 1 ) (5 ), FQ C = - bf l FC = - abf l FQC, FQC ; MC, MC 5 5 1( a), C Me, C, CA B (1 ) A B F A y, FB y A, B MB = 0, FA y l - Me = 0 M A = 0, FB y l - Me = 0, FA y = Me l, FB y = Me l 99
5 1 Me Fy : = 0, FA y + FB y = l - Me l = 0, FA y, FA y ; FB y, FB y FA y, FB y, 5 (5 1( a) ( ) 5 1( a), C,, (5 3) 4 ), FQ C = FA y = Me l MC = FA y a = FQ CMC, FQ C MC ( 5 1 ( b) ) Me l a C,, (5 1 ) (5 ), FQ C = FB y MC = - FB y b = - = Me l Me l b = - Me b l, FQ C, FQC ( 5 MC(5 1 (c) ) 1( c) ) ; MC,,,,,, A, A (5 (5 1( e) ) A, ( 5 1) (5 ), 1 ( d) ) ; B, B 100 FQ A = FA y = Me l
MA FA y 0 = 0 FQ A, MC,, FQ A (5 1 ( d) ), B, (5 3 ) (5 4 ), FQ B = FB y = Me l MB = - FB y 0 = 0 FQB MB, B, 5 1 (e) (1 ),, ( ),,, ( ) ;, ( ),, ( 5 ), ( ),, ( ) ( ), ( ) ( 5 ) ( ) 5 C,, C, C,(5 3) (5 4) C C, F QC = F QC = M C = M C - M e = - M e l F QC - F QC = 0 M C - M C = M e,:,,, ( ),,, ( ),, ( 5 3) (5 4 ),,,,,, ( ), ( 5 3) (5 4),,,, 5 3 5 13( a), q CA B,,,, ( 5 3) (5 4 ) C,,, ql/ 4,, ql/ 4 Me b l 101
A l 8 (5 13 ( b) ) (5, ( 5 3) (5 4 ), MC = - 5 13 FQ C = - ql 4 ql 4 l 8 = - ql 3 FQC MC, FQ C MC 13( b ) ) A( ), (5 3 ) ( 5 4 ) (5 13 (c) ), FQ A, MC FQ A = q0 = 0 MA = q0 0 = 0, A, B( ), ( 5 3) (5 4),, ql,, ql A l/ (5 MB FQ B = - ql = - ql l = - ql 13( d) ), FQ B, MB, B 5 13( d) 5 4, 5 14 (a) EC D (1 ),, qa,, A a(5 10 14 ( b ) )
5 14 A B FA y, FB y, A, B MB = 0, q a 3 a - FA y a + q a a - q a = 0 MA = 0, q a a + FB y a - q a a - q a = 0 FA y = q a, FB y = q a Fy : = 0, - qa + FA y - qa + FB y = - qa + qa - qa + qa = 0, FA y, FB y, FA y, FB y FA y, FB y, 5 14 (a) ( ) E, E, E, 5 14( b ) ( 5 3) (5 4 ), FQ E = - q a + FA y - q a = - q a + q a - q a = 0 ME = - q a a + FA y a - q a a = - q a + q a a - q a = - q a FQ C, 5 14 (c) ; MC, 5 14 (c) C( C ), ( 5 3 ) ( 5 4) (5 14 ( d) ), FQC = - q a, MC = - q a 0 = 0 FQ C, MC,, FQ C 5 14 ( d) 103
D D ), ( 5 3 ) (5 4) (5 14 (e) ), FQ D = 0, MD = - q a FQ D, MD,, MD 5 14 (e) (1 ) ( 5 3 ( 5 4) FQC MC, ql, FQC MC, 5 3 ( ),5 4, A, B,,,, ( 5 1 ), ( 5 ),, : ( ),,,,,,, ( ),,,,, (5 3 ) (5 4 ),, ( 5 3) ( 5 4 ) 5 3,,, x, x, x, FQ = FQ ( x) M = M( x) (),,,,, 104
,, FQ x M x,,,,,,,,,, FQ FQ x,, x, x,,,,, (5,, M x, x 10),, x, M,,, 5 5 5 11 (a) AB, q, (1 ), A, B, 5 15 (a) FA y = FB y = ql ( ) x, A x, x, 5 15( a) x, x, ( 5 3) ( 5 4), x FQ ( x) = FA y - qx = ql M( x) = FA y x - qx x = ql x - qx (a), ( b ) - qx (0 < x < l) ( a) ( 0 x l) ( b) ( 3), FQ x M x, A ; x ; FQ (5 15 ( b) ) ; M (5 (a), FQ 15( c) ) x,, A B ( AB ) (a), FQ A = - FQ B = ql/,, 105
FQ 5 x 0, FQ A x l, FQB = - = ql x, 15 ( b) ( b ), M x,,, ( b ) AB C, ql, M 5 x 0, MA = 0 x = l, MC = q l 8 x l, MB = 0 x, 15 (c) 5 15 5 15( b ) 5 15(c),, FQ max = ql ;, M max = ql 8 (1 ), (a), ( b), 5 1 5,,, x ( ), 5 15( b) 5 15 (c),, ; (5,, 10 (a) ),, ( 3),,,, ( 4), 5 15( c),,,, ( ), ( 5 ) 5 15 (b ) 5 : x,, ( 5 1) (5 ) x x,,, ( ), 106
, 5 6 5 16( a) AB, F,,, B,,,, ( 5 3) (5 4) (1 ) x, B x, x, 5 16 (a) B x, FQ ( x) = F (0 < x < l) ( a) M( x) = - Fx ( 0 x < l) ( b) ( ) ( 5 16 (a) ) FQ x M x, B, x, FQ ( 5 16 ( b ) ), M (5 16 (c) ) 5 16 (a), FQ, x, ( a) AB,, x 0, FQ A = F x l, FQB = F FQ x,, 5 16 ( b) ( b), M x,, ( b) AB,, x 0, MB = 0 x l, MA = - Fl M x,, 5 16 (c) 5 16( b) 5 16( c),, FQ max = F;, M max = Fl ( 5 3) ( 5 4), ( ),,, () ( 5 3, 5 5, 5 6),, (), ( ), ( 5 1, 5, 5 4 ),,, 107
( ) ( ),,,, ( ),, 5 7 5 17 (a) 5 17 ( 1) MB = 0 MA = 0, A, B 5 17 (a) FA y = q a, FB y = q a ( ), A, D, CA, AD DB,, (3 ) AC A D, C( 5 3 ) ( 5 4 ) CA x1 AD x, CA : FQ ( x1 ) = - qx1 (0 x1 < a) ( a) M( x1 ) = - q x 1 ( 0 x1 a) ( b) AD : FQ ( x ) = - qx + qx ( a < x a) (c) M( x ) = - qx + q a( x - a) ( a x a) ( d) 108
x3 DB B 5 3) ( 5 4 ) DB, BD : FQ ( x3 ) = - q a ( 0 < x3 < a) ( e) M( x3 ) = qx 3 - q a ( 0 < x3 a) ( f) (4 ),, : CA, ( a) ( b),,, C A FQ C = 0, MC = 0 FQC = - q a, MC = - q a AD, (c)( d),,, AD FQ A = 0, MA = - q a FQ D = - q a, MD = - q a DB, ( e) ( f),,, D B FQ D = - q a, MD = - q a FQ B = - q a, MB = - q a,,,, x, x,,,,, 5 17( b ),,, 5 17 (c) 5 17( b ) 5 17( c), A DB, FQ max = qa; B, M ma x = qa (1 ), ( ) ( ) ( ),,,, 5, 7 ( 3) ( 5 3 ) (5 4),,,,,,, ( ),, x,,, 109
, (),,,, (, ),,,,,,,,,,, : (1 ), () ( (5 ) (5 4) ),,, ( ), (),, ( ),, ;,,, ;,, 5 8 5 18( a), (1 ),, C,, ( ), BC AB (3 ) BC,,,, C, x1 ( ) (5 18 ( b ) ), ( 5 3 ) ( 5 4), x1, BC FN ( x1 ) = 0 (0 x1 a) FQ ( x1 ) = F ( 0 < x1 a) M( x1 ) = - Fx 1 (0 x1 a) AB,, B, x ( 5 18 (c) ), ( 5 3) (5 4), x, AB 110 FN ( x ) = - F (0 x a) FQ ( x ) = 0 (0 x < a)
5 18 M( x ) = - Fa ( 0 x < a) (4 ),,, 5 18( d), 5 18( e) 5 18( f) 5 9 5 19 (a) (1 ), MB = 0, MA = 0 A, B 5 19 (a) FA x = q a, FA y = 3 q a, FB y = 3q a ( ), AB BC (3 ) BC, C, AB 5 9,,, B, x1, x () ( 5 3) (5 4 ), x1 x, CB : FN ( x1 ) = 0 (0 x1 a) FQ ( x1 ) = - FC y = - q a AB : FN ( x ) = FC y = q a (0 < x1 a) M( x1 ) = FC y x1 (0 x1 < a) (0 x < a) 19 ( b), ( c) ), (5 111
5 19 FQ ( x ) = qx (0 x < a) M( x ) = FC y a - qx x = q a - qx (0 < x a) (4 ),,, 5 19( b ), (c), ( d) (1 ),,,,() ( ),, ( 3) ( 5 18 (d ) ),,,,, ( 4) D, 5 0( g),,, (,, ),, 11
,,,, 5 10 5 0( a), A, BF, R, 5 0,,,, (1 ),, 5 0 ( b) ( ), ( ), (5 0), n = 0, FN () + Fsin= 0 t = 0, FQ ( ) - Fcos= 0 M = 0, M( ) - FR sin= 0 FN ( ) = - Fsin 0 < FQ ( ) = Fcos 0 < < 113
M ) = FRsin 0 < (3 ),, 5 0 (c), ( d), ( e),, A, FN max = F, M max = FR, B, FQ max = F 5 4 5 1 (a), x, x AB, q = q( x), x x + d x, d x, 5 FQ 1 ( b )x M,,, ( FQ + d FQ ) ( M + d M), q, d x q, Fy = 0 MC = 0 ( C x + d x ), FQ + qd x - ( FQ + d FQ ) = 0 (5 5) M + d M - q d x - FQ d x - M = 0 (5 6) 5 1 (5 5 ), d FQ d x = q (5 7) FQ x q (5 6 ), q d x, dm d x = FQ (5 8) 114
M x FQ 5 8 ) ( 5 7), d M d x = q (5 9) M x q (5 7 )(5 8)( 5 9),,, ( 5 7 )(5 8 ) (5 9) x,,, (5 7) (5 8 ) q, ( 5 7) ( 5 8) (5 9 ),, ; ;, ( 5 7) ( 5 8 ) ( 5 9) x,,, (5 3) ( 5 4) (, ), : (1 ) ( q = ),,, q, FQ, M ; q, FQ, M, M q ( ),, ;, (3 ) ( q = 0 ), x,, FQ, M, FQ, M (4 ),,,,, () (5 ),,,,, ;,, (6 ), ( ),,, ;,,, ;,, 115
7 ) ( ),, 5 1, 5 1 ( ) ( ) FQ = 0 CC,,, : (1 ) ( ) (3 ), (5 3 )(5 4 ), (4 ) 5 1 5 11 5 ( a) (1 ) MB = 0 MA = 0, A B 5 (a) FA y = q a, FB y = q a ( ), AC, CD, DE EB (3 ) 116
5 1), (5 3 ) AC :,, FQ A = q a, FQ C = q a - qa = q a CD :,, FQ C = FQ C = q a FQ D = q a - q a = 0 DE :,, FQ D = FQE = q a - q a - q a = - q a EB :,, FQ E = FQB = - q a,, 5 ( b) ), ( 5 ) AC :,, MA = 0 MC = q a a - qa a = 3 q a CD :,, MC = q a a - qa a - q a = a 117
MD q a a - q aa - q a = q a DE :,, MD = MD = q a ME = q a a - 3 q a = - q a EB :,, ME = q a a = q a MB = 0,, 5 (4 ) : (c) 1) FQ () AD, C,,,, A D qa, D qa,,, qa FQ, FQ, FQ ) M (), C q a,, qa ; D,, ; E 3 qa,, 3qa, DE, EB,,,, DE EB M, FQ, M 5 1 5 3( a) (1 ),,, ( ), ( 5 3) (5 4), Fy = 0 MB = 0, 5 3 (a) ( ), AC, CD DB (3 ) FB y = 5 kn, MeB = 5 knm 1) (), ( 5 3) AC :,, FQ A = FQ C = 5 kn CD :,, FQC = 5 kn FQ D = 5-10 1 = - 5 kn DB :,, FQ D = FQB = 5-10 1 = - 5 kn 118
5 3,,, 5 3( b ), CD E, ),, (5 ) AC :,, MA = 0 MC = 5 1 = 5 knm CD :,, CD MC = 5 1 = 5 knm MD = 5-10 1 1 = 5 knm,, E, d M d x = FQ, E, EC x0, x0, FQE = FQC - 10 x0 = 0,, ( 5 x0 = FQC 10 3) E, = 5 10 = 0.5 m 119
ME 5 (1 + 0.5 ) - 10 0.5 = 6.5 knm DB :,, MD = 5-10 1 1 + 5 = 10 knm MB = 5 3-10 1 1 + 1 + 5 = 5 knm,, 5 (4 ) 3 (c) 1) FQ () CD,,, C D 10 kn FQ, FQ, FQ ) M ( ), CD, C, C D 5 knm,,, 5 knm M, FQ, M (1 ),,,, ( ) ( 5 4 ),,,,,FQ,,,, F Q (5 3 ) 5 13, 5 3( b), EC x 0, C F Q,F Q CE d F Q d x,x 0 x 0 = F QC df Q d x = 5 10 = 0.5 m ( 3),,,, 5 13 5 4( a), (1 ) MB = 0 MA = 0, A, B 5 4 (a) FA y = kn, FB y = 4 kn ( ), AC, CB BD (3 ) : 1),, (5 1 ) 10 AC :,, FQ A = kn, FQC = - = - kn
5 4 CB :,, FQ C = FQ B = 1-4 = - 3 kn DB :,, FQB = FQ D = 1 kn,, 5 4 ( b), AC E, ),, (5 4 ) AC :,, E, MA = 0 MC = - 1 = 0, ( 5 3), E A E (5 4 ), x0 = FA y q = = 1 m ME = 1-1 1 = 1 knm 11
CB,, MC = 4 1-1 - = 0 MB = - 1 1 - = - 3 knm DB :,, MB = 1 1 = 1 knm MD = 0,, 5 (4 ) : 4 (c) 1) FQ () AC,,, A C 4 kn, C 1 kn,, 1 kn ; B 4 kn,, 4 kn FQ, FQ, FQ ) M (), C 1 kn,,, ; B 4 kn knm,, 4 kn,, knm M, M, M 5 1 AB x = a x = b, (5 3 )(5 4 ) A, B b FQ ad = b qd x a b d M a = b a FQ d x FQB - FQA = A( q) A B ( 5 10) MB - MA = A( FQ ) A B ( 5 11), FQB, FQ A MB, MA A, B ; A( q) AB = b qd x a A, B ; A( FQ ) A B (5 10) ( 5 11 ), Fi, ( 5 3) ( 5 4) : = b FQ d x A, B a (1 ),, Fi 1 FQ - FQ = Fi ( 5 1) ( ),,
),, M - M = Me i A( FQ ) i ( 5 13), Me i, Me i, ; A( FQ ) i,,,, (5 ( 5 1 ), 13),, ( 5 1 )(5 13) (( 5 1 )(5 13) ) ( ) 5 14, ( 5 13) E, C ME = MA + A( FQ ) A E = 0 + 1 4 1.333 =.667 knm MC = ME + A( FQ ) EC =.667-1 ( - 1.333) = knm,,,,,, 5 14 5 5( a),, (AC), ( CE ) 5 5 ( b),,, ; ( ), 5 5, FC y, AC C (5 5 (c) ),, AC CE,, (1 ), 5 5( c), CE MC = 0 MD = 0, FC y = FD y = q a 5 5 (c) 13
5 5 AC (5 5 (c) ) Fy = 0 MA = 0, 5 5 (c) FA y = q a, Me A = 3 q a ( ) AC A B, BC, CE CD, DE (3 ) AC CE, (5 10) AB :,, 14
FQ A FQB = q a BC :,, FQB = FQ C = q a - q a = q a CD :,, FQ C = q a, FQ D = q a - q a = 0 DE :,, FQ D = q a, FQ E = 0, AC CE,, 5 5( d ), AC F, (4 ), (5 11) AB :,, MA = - 3 q a MB = - 3 q a + q a a = - q a BC :,, MB = MB = - q a MC = 0 CD :,, MD = MC = 0 1 q a a = q a DE :,, MD = MD = q a ME = q a, AC CE,,, 5 5( e) ( 1),,,5 ; 5 5(e) C 5 (d ) C ( ) (5 5 ( b) ),,,,,, (5 5( c) ),,, ( ),,, ;,,,,,,,,, 15
,,,,,,,, 5 5,,, ( 5 3 )(5 4 ),,,,,,,,,, 5 15 5 6( a) 5 6 (1 ), 5 6( b), ( c) 16
) 5 6 ( b), ( c), 5 6 (e), ( d),, 5 6( f), 1., :,., 3. (1 ), FQ M, ( ) FQ :,, M :, (3 ), : FQ, ( ), FQ = Fyi ( ) M, ( ) ( ) C, M = Me Ci ( ) (4 ) : 1) ) ( ), x ( x ) 3) (5 3 ) ( 5 4) x, 4. (1 ) (5 3 ) ( 5 4) :,,, ;,,, ; 17
, ( ), d FQ d x = q dm d x = FQ d M d x = q (3 ) FQB - FQA = A( q)a B MB - MA = A( FQ ) A B 5., (1 ),,,,,,, ( ) 3 : ; q, FQ M ;,, q, FQ M,, : 1) ) ( ), 3) (,,, ), FQ, M 4) FQ max, M max, (3 ), () 18
5 1 5 1 A, B C, D 5 1 5 5 C, C, D A, B 5 5 3 5 3, 5 3 5 4 5 4, 19
4 5 5 5 5 AB, CA, BD?? 5 5 5 6 5 6 5 6, F 130
5 7 5 7, 5 7 5 8 5 8, 5 8 5 9 5 9, 5 9 5 10 5 1 131
11 5 5 1 5 1, Fa, 5 13 5 13 q a, 5 14 5 14 5 1 5 13 5 14 5 15 5 15 5 16 5 16 5 17 5 17 5 18 5 18, 5 19 5 19, 5 0 5 0 13
15 5 16 5 17 5 1 5 1 133
18 5 19 134
0 5 1 135
,,,,,,,,, 6 1,,,,,,,,,,,,,, ;,,,,,, 1.,, cd ab, mm nn, 6 1( a),,, (6 1 ( b) ) : 136
6 1 (1 ), ( ),,,,,, : (1 ) :,,, ( ) :,,,,,,, (6 6 ),,,,,, ( ),,,,,, x (6 3( a) ), d x mm nn,,, y (), z,, z y (6 3 ( b) ) 137
6 3,, mm nn, d(6 3( c) ) O1 O,, O1 O, ab, = ab - ab ab,,, ( 6 = (+ y) d- d = y d y,, ( 6 y (6 1) 1), 1),,,, ( 6. 1),, ( ),, = E= E y (6 ), E/, ( 6 ), y,, 6 4 (a), ( b ) 6 4 138
(6 3. 6 ),, z, 5, d A,,, xy M,,, Sz (6 ) ( 6 3),, A E y d A = A d A = 0 (6 3) z d A = 0 (6 4) A A yd A = M (6 5) E A yd A = E S z = 0 = A yd A, A z E/, z, Iy z (6 ) ( 6 4), A zd A = 6 5 Sz = 0 (6 6) E A y z d A = E I yz = 0 = A y z d A, A y, z E/, Iy z = 0 (6 7), y, z y,, z, y, z ( 6 6 ) (6 7),, : (1 ) ( ),,,, Iz, x, x (6 ) ( 6 5), yd A = A E y d A = E A I z = M =y d A, A z A 1 = M E I z (6 8) 1/, (6 8 ), EI z, 1/, EIz, 139
6 8 ) ( 6 ), = My I z, (6 (6 9) ),,,, ( 6 4( a), ( b) ), ( 6 8) (6 9),, y ( T ),, ( 6 8) (6 9),,,,,,,, ( l h l/ h 5 ), (,, l/ h 5, ( 6 9 ) 1 % ),,, (6 8) ( 6 9),,, ( 6 9 ), M y, ( 6 9 ), M, y,,, 6 1 6 6 (a), ( b ) ) : ( 1) a, b, c, d; ( ) (1 ) C a, b, c, d : 1), 6 6 (c),, C, MC = 0 knm ), z, 6 6 ( b), z 140
6 6 I z = 80 1403 1-40 1003 1 = 14.96 10 6 mm 4 = 14.96 10-6 m 4 3) C a, b, c, d 6 6 ( b ), a, b, c, d ya = 70 mm, yb = 50 mm, yc = 0, yd = 70 mm MC = 0 knm, ya = 70 mm, I z = 14.96 10-6 m (6 9 ), a a = MC ya I z = 0 103 70 10-3 14.96 10-6 = 93.6 10 6 Pa = 93.6 M Pa MC = 0 knm, yb = 50 mm, I z = 14.96 10-6 m (6 9 ), b b = MC yb Iz = 0 103 50 10-3 14.96 10-6 = 66.8 10 6 Pa = 66.8 MPa MC = 0 knm, yc = 0, I z = 14.9610-6 m ( 6 9 ), c c = MC yc I z = 0 103 0 14.96 10-6 = 0 MC = 0 knm, yd = 70 mm, I z = 14.96 10-6 m (6 9 ), d d = MC yd Iz MC = 0 103 70 10-3 14.96 10-6 = 93.6 10 6 Pa = 93.6 MPa,, C,, a, a 141
b d, b d ; c, c ( ) C ( 6 9 ), C 6 6( d) 6 6 7( a), ( b ) : (1 ) B Ca, b; ( ) B C; ( 3) 6 7 (1 ) B C a, b : 1) B C, 6 7( c),, C, ( ) MC = 10 knm B, MB = 8 knm ), ( ), y (6 7( b ) ), y,, 14
yc Ai y i Ai = 10 0 10 + 10 0 80 + 60 0 150 10 0 + 10 0 + 60 0, y, 6 = 66 mm 7( b ),, 6 7( b),,, I z = 10 03 1 60 0 3 1 + 10 0 56 + 0 103 1 + 0 10 14 + + 60 0 84 = 19.46 10 6 mm 4 = 19.46 10-6 m 4 3) C B a, b(6 7 (c) ) MC,, C, a, a ; b, b 6 7( b ), a, b ya = 66 mm, yb = 94 mm MC = 10 knm, ya = 66 mm, I z = 19.46 10-6 m (6 9 ), C a C a = MC ya I z = 10 103 66 10-3 19.46 10-6 = 33.9 10 6 Pa = 33.9 M Pa ( ) MC = 10 knm, yb = 94 mm, I z = 19.46 10-6 m (6 9 ), C b C b = MC yb Iz (6 = 10 103 94 10-3 19.46 10-6 = 48.3 10 6 Pa = 48.3 MPa () 7 (c) ) MB,, B, a, a ; b, b MB = 8 knm, ya = 66 mm, Iz = 19.46 10-6 m ( 6 9 ), B a Ba = MB ya Iz = 8 103 66 10-3 19.46 10-6 = 3.7 10 6 Pa = 3.7 M Pa ( ) MB = 8 knm, yb = 94 mm, Iz = 19.46 10-6 m ( 6 9 ), B b B b = MB yb I z = 8 103 94 10-3 19.46 10-6 = 38.6 10 6 Pa = 38.6 MPa () ( ) B C ( 6 9), C B 6 7( d), ( e) (3 ),, 143
B C), B, C, + max = Bb = 38.6 M Pa max - = C b = 48.3 MPa (1 ),,,,, I z y ( ),, 6, B C,B,,,C (B C),, ( 6 8) (6 9 ),,,, ( 6 8) (6 9),, ( h/ 0., h, ) 6 (6 9 ),, ( ) (), max = Mmax ymax I z ( 6 10) W z = I z/ ymax, ( 6 10 ), W z max = Mmax W z,, [ ] 3 h b ( 6 8( a) ), ( 6 11) d ( 6 I z Iz = bh3 1, ymax = h, W z = bh 6 8( b ) ), = d4 64, ymax = d, W z = d3 3 Dd (6 144 Iz 8 (c) ), = 64 ( D4 - d 4 ), ymax = D, W z = D3 3 1 - ( d D ) 4
6 8,, max [ ], max = Mmax W z [ ] ( 6 1),,,, + max - max [ + ] [ - ], + max = M max y max + I z [ + ] ( 6 13) max - = Mmax y- max I z [ - ] ( 6 14), y + max y - max + max - max,,,,,, : (1 ),, (6 1) (6 13)( 6 14 ) ( ),, (6 1) (6 13)( 6 14 ) (3 ),, (6 1) (6 13)( 6 14 ), 145
3 6 9 (a) 0 a, [ ] = 170 MPa 6 9 (1 ), 6 B, Mmax = 4 knm 9( b), B, ( ) 6 9( a),,,, B (6 9 (c) ) ;,0 a W z = Iz ymax (6 max = Mmax ymax I z, = Mmax W z = 4. cm 3 = 4. 10-6 m 3 = 5 ), 4 10 3 = 165.3 10 6 Pa = 165.34 M Pa < [ ] 4. 10-6 (1 ),, ( ) ( ), ( ),,, 6 4 6 10 (a), [ ] = 160 M Pa (6 10( b ) ) ( 1), 6 (), B, () 146 Mmax = 8 knm 10( d),, B
6 10 ( ) 6 10 ( b),, B ( 6 10( e) ( 6 1 ), W z Mmax = [ ] 80 103 170 10 6 = 0.047 10-3 m 3 = 47 10 3 mm 3, 10, W z = 49 cm 3 = 49 10 3 mm 3, W z, 10 6 5 6 11( a), [ + ] = 40 MPa, [ - ] = 80 MPa (1 ) 6, Mmax = 6 knm 11 (c), D ( ) 6 11( b ), y, y, y, y, yc = Ai y i Zi = 160 0 100-10 160 10 160 00-10 160, y, 6 = 70 mm 10 (a) z I z = 160 003 1 + 160 00 30-10 160 3 1 + 10 160 50 = 46.5 10 6 mm 4 = 46.5 10-6 m 4 147
6 11 (3 ), [ + ] [ - ],, ( 6 13 )(6 14) 6 11 (c),,,, (6 11 ( d) ) 6 11( b) (6 13) ( 6 14 ) + max = Mmax y+ max Iz max - = Mmax y- max Iz y + max y - max = 70 mm = 130 mm = 6 103 70 10-3 = 39.1 10 6 Pa = 39.14 M Pa < [ + ] 46.5 10-6, = 6 103 130 10-3 = 7.69 MPa < [ - ] 46.5 10-6 6 6 6 1( a) a = m; y1 = 10 mm, y = 80 mm; Iz = 5 10 6 mm 4 ; [ + ] = 30 M Pa, [ - ] = 70 M Pa : ( 1) [ F] ; ( ) (6 ( 1) 6, D, 148 1 ( g) ) [ F] 1 ( b),, A
6 1 ( ) : MA = Fa, MD = Fa 4 1) [ F],,, A D, (6 13) (6 14),, : A, (6 + max = MA y+ max Iz = MA y1 I z (6 - max = M C A ymax - Iz = M C A y = 1( c) ), F 10 10-3 [ + ] = 30 10 6 5 10 6 10-1 F 13 10 3 Iz = N = 13 kn 1( c) ), F 80 10-3 [ - ] = 70 10 6 5 10 6 10-1 149
F 5.5 10 3 N = 45.5 kn D, (6 + max = MD y+ max Iz = MD y (6 - max = MD y- max Iz = Iz = F 39 10 3 MD y1 Iz 1 ( d) ), F 4 80 10-3 [ + ] = 30 10 6 5 10 6 10-1 = F 30.7 10 3 N = 39 kn 1( d) ), F 4 10 10-3 5 10 6 10-1 70 10 6 N = 30.7 kn, [ F] = 13 kn ) (6 1( g ) ) [ F] T, A, (6 + max = M C A y + max Iz = M C A y Iz = F 19.5 10 3 (6 - max = MCA y- max Iz = MC A y1 Iz = F 30.3 10 3 1( e) ), F 80 10-3 5 10 6 10-1 30 10 6 N = 19.5 kn 1( e) ), F 10 10-3 5 10 6 10-1 70 10 6 N = 30.3 kn D, (6 + max = MD y+ max Iz = MD y1 (6 - max = MD y- max Iz = Iz = F 6 10 3 MD y Iz 1 (f) ), F 4 10 10-3 5 10 6 10-1 30 10 6 N = 6 kn 1( f) ), = F 91 10 3 F 4 80 10-3 5 10 6 10-1 70 10 6 N = 91 kn, [ F] = 19.5 kn (1 ),, 150
,,,, ( ) ( y 1 / y ) > [ + ]/ [ - ],, D A,, ( D A ), (6 13 ),,,,,,, 6 3,,,,,,, (,, ), ( ),,,,,,,, ( 6 9 ),, 1. 6 13, h, b, x y,,, FQ y ( 6 13 ( b) ) h b,, : (1 ) FQ (6 13( b) ) 6 13 151
), (6 13 ( b) ), h b d x 11 (6 13 (a) ), 6 13 ( b), 11,, FQ,, M M + d M(6 13( b ) ), 11,, ( y),, 1 (6 14 (a) ) y ( y), y mn, (6 m1 n A * 6 14 (c) ) F * 1 ( x ), F * 1 = * 1 d A = A * A 14 ( b) ), m1 1 d A( My * Iz d A = M y * I z * d A = M A I z, y * d A ( 6 14 (c) ), A * S * z =y * d A * A S * z ( 6 15) 6 14, n d A F * ( x ), F * = M + d M S z * ( 6 16) I z,,,, mn ( y) (6 14( c) ), d FQ = ( y) bd x ( 6 17) F * 1, F * d FQ x (6 14( d ) ), F x = 0, F * - F * 1 - d FQ = 0 ( 6 18) (6 15) ( 6 16 )(6 17) (6 18), ( 5 8 ), ( y) = FQ S* z I z b ( 6 19) (6 19), FQ ; b 15
, 6 Iz ; S * z y 15( a), S * z = A * y * C = b h - y 1 y h + y = b h 4 - y ( 6 0) (6 0) Iz = bh3 1 ( 6 19 ), ( y) = 3 FQ bh 1-4 y ( 6 1) h 6 15,,, 6 15 ( b), (c) ( y = h/ ),, ( y = 0),, max, A = b h ( 6 1.5 = 3 FQ bh = 3 FQ A ( 6 ) ),, hb, (6 ), h = b, max, 3% ; h = b, 13% ; hb/, max, 40%, ( ),,,, ;,= 0( 6 16),,,,,,, 6 16 153
.,,,, FQ,,,,, FQ, ( 6 8), max = FQ S* z max Iz b ( 6 3), b ; Iz ; S * z max 6 17( a) d, ( 6 3), b = d, I z = d 4 / 64, S * z max 6 18 ( b), S * z max = ( d/ ) d 3 = d3 1 ( 6 4) max = 4 3 FQ A ( 6 5), A = d / 4, 1.33, max, 4% 6 17 6 7 6 18( a), F, l, b, h,, (1 ),,, ( ) Mmax FQ max = ql = ql ( ) (6 11), max = M max W z = 3 ql bh 154
(6 ), max = 3 (3 ) 6 18 FQ max A max = l max h = 3 ql bh,l 5 h, max 10 max,,, 3. T, ( ),, ( 6 6 19 ), 19( a), ( 6 19) y, Iz, b(),, S z * 6 19 (a), S * z = b h 4 - h 1 4 h 1 + 4 - y ( y) = FQ 8 Iz b( h - h 1 ) + ( h 1-4 y ) ( 6 6), ( 6 ( y = 0 ), 19 ( b) ) 155
max ( y = h/ ),, FQ 8 Iz [ bh - ( b - ) h 1 ] ( 6 7) min = FQ 8 I z ( b h - b h 1 ) ( 6 8) (6 7) (6 8),,,, 6 19( b ),, FQ, 6 19,, = FQ h1 ( 6 9),, y,,,,,,,,,,,,,,,, ( 6 19) (6 3 ),,,, (6 (6 9 ) 9), 6 8 6 0 (a) Iz = 46.5 10-6 m 4 : (1 ) a, b, c, d ; ( ) ( 1) 6 A DB, FQmax = 0 kn 0( b ),, () ( ) a, b, c, d a, d: 6 0( a), a, d, (6, ( 6 b : 6 3 ) (6 (6 156 a = d = 0 0 (a) ) 0( a),, = 40 mm; 0( b ) ) ( ) S * z = 160 40 50 = 30 10 3 mm 3 = 30 10-6 m 3 3) b
6 0 = FQ S* z I z = 0 103 30 10-6 = 3.44 10 6 Pa = 3.44 MPa 46.5 10-6 40 10-3 c : 6 (6 (6 0( a),, = 40 mm; 0( c) ) () S * z max = 0 130 75 = 338 10 3 mm 3 = 338 10-6 m 3 3) c = FQ S* zmax Iz = 0 103 338 10-6 = 3.63 10 6 Pa = 3.63 M Pa 46.5 10-6 40 10-3 (3 ) ( 6 3),,,, D 6 ( 6 0( c) 3),, ;,,, S * z ;, S z * 6 9 6 1 FQ, (1 ), R0,,, 6 1,, y y, y, 157
,,,, ( 6 19 ), max = FQ S * zmax Iz b, S * z z ; Iz ; b, b = ( ) S * z, Iz,, S * z = 3 Iz R0 = R0 4 + 3 + 4-3 R0 - R0-4 - 3 4 (3 ) S * z, Iz b, R 0 R 3 0 6 1 max = FQ S* zmax Iz b, A = 0 FQ R0 = FQ A,,6 FQ max 9,,, max ( ) ( ), max = FQmax S* zmax Iz b ( 6 30), FQmax ; Iz z ; b ; S * zmax (, Iz/ S * z max ), max, max = FQ max S* z max Iz b [ ] ( 6 31) max [ ],,,, (, ), (, ), 158
6 10 6, F18 a, [ ] = 170 MPa, [ ] = 60 MPa (1 ) F, (5 1 ) ( 5 ),,, Mmax = 30 knm,,, FQmax = F = 30 kn 6 ( ) 18a W z = 185 cm 3, (6 5 ), max = Mmax W z = 30 103 = 158 10 6 Pa = 16. M Pa < [ ] 185 10-6 (3 ) 18a ( Iz/ S * z max ) = 15.4 cm, d = 6.5 mm, (6 max = FQmax S* zmax Iz d = 31), 30 10 3 = 30 10 6 Pa = 30 MPa < [ ] 15.4 10-6.5 10-3, :,, (1 ),, ( ), max max, (3 ) max max, (6 1) ( 6 31 ),, max max 6 4 1.,,,,,,, 159
( ),, ( ),,.,,,,, (),, (6 8 ) ( 6 9), (6, ( 6 9), 9 ),, ( 6 ( a) ),, ( 6 ( b ) ), 1.,, (, ),,,,,,,, (6 3 ), ( 6 ), (6, ; S * z 3 19), = FQ S z * I z ( 6 3), FQ S * z, 6 3 160
, ( 6 3),,,, 6 11 6 4 (a) xy ( x, y, xy ) FQ,, 6 4 (1 ) (6 3),, S * z, S * z, s(6 4 ( b ) ), S z1 * = h s abcd, S * z1 bde f z, y( 6 4 ( b) ), S * z = bh + h 4 - y z I z, ( 6 1 = = FQ FQ h s I z =, 6 FQ hs Iz b h + h 4 - y Iz = 4( c) ( ) 3) FQ ( b h + h 4 - y ) Iz,,,, 6. 4( c),, 161
,, ( 6 5( a) ),,, (6 5( b ) )6 5 ( 6 10 6 4 ), 6 6 5 5( a), C, z, y C z, z y y F, x y ( 6 5 (a) ), (6 3) ( 6 10 ), 6 10 6 4( c),, d A FT FT,,, FT = b 0 d s = FQ b h 4 Iz ( 6 33), d A, FQ, 6 5 (c), A y FQ, A O ( 6 5 ( d) ) e = FT h F Q = b h 4 Iz ( 6 34) : F x y ( ), 6 5 ( b), FQ C, A, FFQ F FQ, xy F Fe( e A y ),, A, 6 4 ( b) F A y, 6 5 (a), F FQ, (6 5( a) ), A, (6 16 34), A,
.,,, : (1 ) ; ( ),, ; (3 ),, ; (4 ),, ;, ;,,,,, 6 1 6 6 6 6 6 6 (a), ( b) z, z, z,, 6 6 (a), ( b), 6 6 (a) ( b ), (6 6( a), ( b ) ) A,,, A, A 6 1,,,, ;,,,,,, 163
,,,,,,,,, 6 1 e = 1 b 3 1 h 1 b 3 1 + b 3 e = b h 4 I z 6 5,,,,,, 1., 6 7 (a),,, 6 7 164
(, 6 8 ( b) ),,.,, 6 8( a) 6 8 ( b),, 6 8,,, 1.,,, 6 5,., W z/ A W z/ A, W z/ A 6 6 Wz/ A Wz A 0.167 h 0.15 d 0.05 D (0.7 0.31) h,,,,,,, 165
,,,,,, T,,,,,,,,,,, ( 6 9 (a) ) (6 9( b ) ) 6 9 1.,,,.,,,,,,, 3.,,,,,, 4. 5. (1 ) : 166
, M I z = My I z z ; y, M, y,,,, ( ) :,,,,, max = Mmax ymax I z, W z max h b ( 6 = Mmax W z 8( a) ), d ( 6 I z = bh3 1, ymax = h, W z = bh 6 8( b ) ), Iz = d4 64, ymax = d, W z = d3 3 Dd (6 8 (c) ), Iz = 64 ( D4 - d 4 ), ymax = D, W z = D3 3 1 - d D 4 6. (1 ) :, ( y) = FQ S* z I z b, FQ ; Iz z ; b; S * z y, FQ, S * z,, T ( ) :,, 167
,,, S * z max max = FQ S zmax * Iz b, max, = 3 F Q A max = 4 3, FQ A max = FQ A 7. (1 ) :,,,, max = Mmax W z [ ],, ( ) : max = F Qmax S z * max Iz b [ ], : 1), Mmax, FQmax ; ), ; 3) max max, ( 6 1 )(6 31) 8., ; ; 6 1 d = 1 mm m,, E = 00 GPa 6 6, : ( 1) a, b, c, d ; ( ) a, b, c, d 168
6 3 T 6 ( a), ( b), : (1 ) ; ( ) (( c) ), 6 3 6 4 6 3, F, C = 4.010-4 E = 00 G Pa, : ( 1) ; ( ) F 6 4 6 5 6 5, [ ] = 160 MPa: ( 1) ( ) ; ( ) W z/ A, 6 6 6 6,, D1 = 40 mm, d/ D = 3/ 5, 169
5 6 6 6 7 6 7, b = 100 mm, h = 00 mm, [ ] = 10 M Pa d = 40 mm ( ), 6 7 6 8 6 8 [ + ] = 30 M Pa, [ - ] = 90 M Pa 6 9 6 9 z I z = 35 10 6 mm 4, y1 = 119 mm, y = 181 mm, [ + ] = 40 M Pa, [ - ] = 10 M Pa[ q] yc 6 10 T, 6 10 Iz = 9.1 10 6 mm 4, = 35 mm,, [ + ] = 30 M Pa, [ - ] = 90 M Pa 170
8 6 9 6 10 6 11 6 11, [ + ] [ - ] = 1 4 6 11 6 1 6 1, CD 10, B d = 10 mm BE [ ] = 160 MPa, [ q] 171
1 6 13 6 13, a, b, c, d 6 13 6 14 T 6 14, a, b, c, d, 6 14 6 15,, 6 15 Iz = 46.510 6 mm 4, yc = 130 mm : (1 ) ; ( ) 6 16 6 16 [ ] = 10 M Pa, [ ] = 1 MPa, [ ] = 0.34 M Pa Me 6 17 6 17 [ ] = 10 M Pa, [ ] = 1 MPah/ b = 3/, 17
15 6 16 6 17 6 18 6 18 a [ ] = 160 M Pa, [ ] = 80 M Pa 6 18 6 19 T, 6 19 I z = 8.84 10 6 mm 4, yc = 45 mm, a = 1 m, [ ] = 160 M Pa, [ ] = 80 M Pa [ q] 6 0 6 0 h1/ h = 1/,, 173
19 6 0 174
,,, 7 1, 7 1, x,, w, Ox w, w 7 1, xw,, xw,,, (6 8 ),,,,,,, 175
.,,,,,,, w 7 1, w x, x, w = w( x), 7 1,,. ( ),,, 7 1, x (7 1 ), 1 0.017 5 rad, tan= w 7 1, x ( ),, ( ) ( ) ( ) 7 (a), ( b), (6 8),, 7,, ( ),,,,,, 176
, ( 6 1 = M E I z 8), (7 1),,, x,, Iz 1 ( x) = M( x) EI z, (7 ) 1/ ( x) M( x), EI,, w = w( x) 1 ( x) d w = d x 1 + ( d w d x ), 7 3/ (7 3) 1,, M( x),, d w ; d x, M( x),, d w (7 3) M( x) d w d x d x, (7 1 ) ( 7 3), d w d x 1 + ( d w d x ) 3/ = - M( x) EI (7 4) 7 3,, d w d x ( 7 4) d w = - d x M( x) EI d w 0.017 5 rad, d x 1, (7 5) ( 7 5 ), M( x) 177