材料力學(xué)第八章_彎曲變形

1、 2 F F 2 F w撓度撓度 C C A B w x B 轉(zhuǎn)角轉(zhuǎn)角 A C C w B x w撓度（撓度（ B 撓曲線撓曲線 w A B x 轉(zhuǎn)角轉(zhuǎn)角 w撓度（撓度（ C C B ( )wf x w AB x 轉(zhuǎn)角轉(zhuǎn)角 w撓度撓度 C C B 撓曲線撓曲線 tan( )ww x w A B x 轉(zhuǎn)角轉(zhuǎn)角 w撓度撓度 C C B 撓曲線撓曲線 1M EI 1( ) ( ) M x xEI 3 2 2 1| ( ) (1) w x w 3 2 2 |( ) (1) wM x EI w O x w x O w 00 wM w M 0 0 M w 00 wM 0 0 M w M M MM 3 2

2、2 ( ) (1) wM x EI w ( ) M x w EI 2 w tan( )ww x 2 w ( )M x w EI ( )EIwM x 1 ( )dEIwM xxC 12 ( )d dEIwM xx xC xC A B A w B w A w A 0 A w0 B w 0 A w 0 A AB l ABx F max w max x l w ABx F ( )() (1)M xF lx ( ) (2)EIwM xFlFx 2 1 (3) 2 Fx EIwFlxC 23 12 (4) 26 FlxFx EIwx CC 2 1 (3) 2 Fx EIwFlxC 23 12 (4) 26

3、 FlxFx EIwx CC 0,0 0,0 xw xw 12 0 0CC 2 2 Fx EIwFlx 23 26 FlxFx EIw B wmax max x l y A F （ ） 222 max | 22 x l FlFlFl EIEIEI max max w （ ） EI Fl ww lx 3 3 max max max w A B q RR 2 AB ql FF AB q FRAFRB x 2 ( ) 22 qlq M xxx 23 46 qlq EIwxxC 2 22 qlq EIwxx 34 1224 qlq EIwxxCxD 233 (64) 24 q lxxl EI 233

4、(2) 24 qx wlxxl EI 0w x AB q FRAFRB A B 3 max 24 AB ql EI wmax 4 max 2 5 384 l x ql ww EI A B F D a b RA b FF l RB a FF l FRA FRB A B F D ab 12 x x 1R (0) A b MFxFxxa l 2 () () b MFxF xaaxl l 11 b Fx EIwM l 2 11 2 bx EIwFC l 3 111 6 b x EIwFC xD l 22 () b FxF xa EIwM l 2 2 22 () 22 bF xa x F EIwC l

5、3 3 222 () 66 bF xa x Fx EIwCD l 12 ww 12 ww 1 0w 2 0w 12 0 DD 22 12 () 6 Fb CClb l AB F D ab l 12 FRA FRB 222 11 () 3 6 Fb wlbx lEI 2 22 1 6 Fbx lwbx lEI 2 222 22 1 ()() 23 Fbl xa wxlb lEI b 3 322 2 () () 6 Fbl xxa wxlb lEI b 10 () | 6 Ax Fab lb lEI 2 () | 6 Bx l Fab la lEI max () 6 B Fab la lEI 0w

6、 1 0w 222 11 ()0 3 6 Fb wlbx lEI 22 1 (2 ) 33 lba ab x EI Fbl .bl lEI Fb ww xx 2 322 max 006420)( 39 1 2 22 (34)0.0625 48 C FbFbl wlb EIEI 1 2 22 3 max ()0 0642 9 3 x x FbFbl y |lb. w EIlEI 121122 (,)()()() nnn F FFFFF 121122 (,)()()() nnn w F FFw FwFwF B qF A C aa F = A B + AB q 2 () 4 A F Fa EI 3

7、() 6 CF Fa w EI 4 5 () 24 Cq qa w EI 3 () 3 A q qa EI 2 () 4 A F Fa EI 3 () 6 CF Fa w EI 3 () 3 A F qa EI 4 5 () 24 CF qa w EI ()() AA FA q 2 (34) 12 a Fqa EI 43 5 () 246 C qaFa w EIEI qF F =+ A A A B B B C aa q AB C A B C q (a) BA Me (c) A q (b) B )( qA )( qB () Cq w e () BM e () M A e () CM w C e

8、()( CC qMC w ww e )( A qMAA 2 4 e 5 38416 M lql EIEI ( ) 3 e () 243 M lql EIEI e )( B qMBB 3 e 246 M lql EIEI A B C q l/2 A B C q/2 C A B q/2 q/2 A B C q/2 44 1 5(2)5 384768 C qlql w EIEI 33 11 (2) 2448 BA qlql EIEI C A B q/2 q/2 C AB q/2 q/2 B q/2 A C q/2 3 3 22 ( )( ) 22 24384 AB ql ql EIEI 2 0 C

9、w 4 12 5 768 CCC ql www EI 333 12 3 48384128 AAA qlqlql EIEIEI 333 12 7 48384384 BBB qlqlql EIEIEI A B C D a a 2a 2q q A B C D a a 2a 2q q B C D q 2qa 2 B Mqa 2q A B 2qa 2 B Mqa 2 B Mqa 2qa 2qa B C D q 2 B Mqa q B C D B C D 2 B Mqa () Dq w() B q D B C 3 )()( 3 B B qBMB qa EI 4 ()() 24 B DD qDM qa ww

10、w EI 2qa B C D q 2 B Mqa () B BM () B D M w D B C 2 B Mqa 33 )( 243 qB qlqa EIEI 3 2 () 33 B B BM M lqa EIEI 44 55 () 38424 D q qlqa w EIEI 4 2 () 164 B B DM Mqa l w EIEI 2 w A 2q B 2qa 2 B Mqa A B 1 w B C 2qa B D q 2 B Mqa 122 B A wwwaw 2 2 (2 ) 8 q a w EI 444 7 3412 A qaqaqa w EIEIEI max max ww w

11、l=400mm F2=2kN A C a=0.1m 200mm D F1=1kN B F2 B C D A = + F2 B C a F2 B C D A M = + F1=1kN ADC F2=2kN C A B B l=400mm F2=2kN A C a=0.1m 200mm D F1=1kN B 2 1 1 16 B F l EI 2 1 11 16 CB F l a wa EI 2 0 B 3 2 2 3 C F a w EI 2 3 33 B MllaF EIEI 2 2 33 3 CB F la wa EI + F2 B C 圖圖2 圖圖3 + F2 B CDA M = 圖圖1

12、F1=1kN DC 2 12 163 B F lF la EIEI 232 122 1633 C F l aF aF a l w EIEIEI 44 4412 84 () 64 3.14 (8040 ) 10 64 188 10 m IDd F2=2kN =+ 圖圖1 圖圖2 l=400mm A C a=0.1m 200mm D F1=1kN B F1=1kN D B C 圖圖3 F2 B DA M AC C F2 232 6 122 5.19 10 m 1633 C F l aF aF a l w EIEIEI 2 12 -4 0.4400200 () 163210 1880163 +0.4

13、23 10 B F lF la EIEI max ww ll 65 max 5.19 10 m10 mww 4 max 0.423 100.001 F AB AB C F FRAFRB FRC FRB AB C F F AB FRAFRC A B q l 0 B w q A B FRB R ()() B BB qBF www R ()()0 B B qBF ww () B q w R () B BF w q AB 4 () 8 B q ql w EI R 3 R () 3 B B BF F l w EI B A FRB q A B FRB 43 R 0 83 B Flql EIEI R 3 8

14、 B Fql q A B FRB FRA MA R 5 8 B Fql 2 1 8 A Mql () B q w R () B BF w q AB B A FRB 0 A A B q l A B q l MA a2a A B C q 2q D l C A D B q 2q A FN FN A wl A D B C q 2q A w FN FN l B C q 2q FN A w N ()() AA qA F www N ()() A qA F wwl B C q 2q () Aq w 4 7 () 12 A q qa w EI N 3 N () AF F a w EI C FN B N ()

15、AF w N F l l EA 43 NN 7 12 F aF lqa EIEIEA 4 N 3 7 12() Aqa F Il Aa A D B C q 2q FN FN l A1 A w 4m3m2m A B D C 30kN 20kN/m 4m3m2m A B D C 30kN 20kN/m 4m3m2m A B D C 30kN 20kN/m MB B B BB 3 4 20 4 243 B B M EIEI 12804 () 243 B M EIEI 3032(52)5 653 B B M EIEI 425 3 B M EIEI 4m3m2m A B D C 30kN 20kN/m D A B 30kN 20kN/m MB B B C 12804425 2