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1、习题解答习题解答( (七七) )整理课件P94P94 50- -2t txT0 A Aa as s s sx x = = 0 0= = s s s sy ya a a a = = 3030t txWPMn=Mn = - - - - T0= - - - - 2.5 kNmWP = 16p pD3= 42.41 cm3= - - - - 58.95 MPas s30 = - -t t sin( (60) )s s- -60 = - -t t sin( (- -120) ) = - - 51.05 MPa= 51.05 MPa1Ee e3030 = ( (s s3030- -nsns- - - -6
2、060) )A A= 319106广义虎克定律广义虎克定律广义虎克定律广义虎克定律：s s s sz z = = 0 0而而而而正应变正应变 （线应变）（线应变）单元体单元体无量纲无量纲整理课件P95P95 50- -3s sxa aA AF60s sy y = 0= t tx xs sx xa a a a = = - - - - 3030AF=2s sxs s- -30 = + + cos( (- -60) )2s sx= s sx432s sxs s60 = + + cos( (120) )2s sx= s sx41s sz = 01Ee e- - - -3030 = ( (s s- -
3、- -3030- -nsns6060) ) = s sx4E3- -n ns sx = e e- -304E3- -n n= 160 MPa= 540= 540 1010 6 6F = s sxA = 50 kN则则而而广义虎克定律广义虎克定律广义虎克定律广义虎克定律：A A整理课件P95P95 50- -4 z yI I 2525 b bF FA AABF FB BCDF0.1ll0.1lqEF2 2t t1 1s s3 3t ts s1 12 23 3M 图图( (kNm) )4541.841.8210FS 图图( (kN) )21020882088危险截面危险截面危险截面危险截面： E
4、E A AC C左左左左( ( ( (D D右右右右) ) ) )危险点危险点危险点危险点： 1 12 23 3Iz = 5278 cm4 h = 250 mm t = 13 mm 查表查表查表查表Wz = 422.72 cm3 b = 118 mm tw = 10 mmSz max* * = 246.3 cm3E E截面截面截面截面：A A截面截面截面截面：C C左左左左截面截面截面截面：S Sz z3 3 * * = 183.4 cm= 183.4 cm3 3Mmax = 45 kNms s = = 106.5 MPaFS = 208 kN= 98.0 MPat t t tmaxmax =
5、 FS maxSz maxIztw* * t t t t s s s s M = 41.8 kNmFS max = 210 kNs s s smaxmax = WzMmax= 72.35 MPat t = = 88.70 MPas s1 = = 129.2 MPas s3 = = - - 40.5 MPa= 210 kN =( ( ( (B B) ) ) )= 169.7MPa= 153.5 MPa s s s s 全面校核全面校核内力图内力图不满足强度条件不满足强度条件不满足强度条件不满足强度条件：导学篇导学篇 附录附录附录附录B B- - - -3 3 整理课件P96P96 51- -1F
6、l bA l h z zy yCBFa aj jb bFy = Fcosj jFz = Fsinj j= 9.66 kN= 2.59 kN14Mz = Fy2l = 7.24 kNm14My = Fz2l = 1.94 kNm I Iz z = = 1212bhbh3 3= 1104 cm4 I Iy y = = 1212hbhb3 3= 5625 cm4s s = - - - - y - - - - z IzMzIyMy( (1) )s s = 0 令令tana a =z0y0=Iy MzIz My= tanj jbh2( )= 0.476a a = 25.5WWz z = = 6 6bhb
7、h2 2= 1103 cm3WWy y = = 6 6hbhb2 2= 750 cm3= + +WzMzWyMys st max = s sc max= 9.83 MPa( (2) )Fy( (2l) )348EIzfy =Fz( (2l) )348EIyfz = 5.43 mm= 2.59 mmf = = fy2 + + fz 2= 6.02 mmtanb b =fyfz= tana a b b = a a = 25.5f跨中截面：跨中截面：跨中截面：跨中截面：中性轴中性轴中性轴中性轴 s s s st t maxmaxs s s sc c maxmax中性轴中性轴拉压区拉压区整理课件P96
8、P96 51- -2F lA lCBF z zy ya a查表查表查表查表：b = 20 cmz0 = 5.69 cmIz = 4554.55 cm4 Iy = 1180.04 cm4 L L L L 200200 2020z z0 0s s s st t maxmax 在在在在B B点点点点：危险截面危险截面：跨中跨中Fy = Fz22= F = 17.68 kNMy =1 14 4Mz = F Fy y 2 2l l = 17.68 kNms s = - - - - y - - - - z IzMzIyMys s = 0 令令tana a =z0y0= 3.86a a = 75.5zB =
9、 - - z02yB = 0= - - 80.47 mms sB = 120.6 MPas sA = - - - - 146.2 MPas s s sc c maxmax 在在在在A A点点点点：= 60.95 mmzA = b- - z0222= 141.4 mmyA = b22中性轴中性轴中性轴中性轴 弯曲中心弯曲中心弯曲中心弯曲中心 形心主惯性平面形心主惯性平面形心主惯性平面形心主惯性平面 整理课件P97P97 51- -3zyx2FA AB BC CD DlMMz z MMy y FN = 2F = 20 kN23h2Mz = Fl - - 2F 12My = Fl + + 2F b2
10、= 7.46 kNm= 5.6 kNmA A = bh= bh = 72 cm2s s = + + + + y + + z IzMzIyMyAFNs sA = 2.78 + + 51.81 + + 77.78s sB = 2.78 + + 51.81 - - 77.78s sC = 2.78 - - 51.81 - - 77.78s sD = 2.78 - - 51.81 + + 77.78= 132.4 MPa= - - - - 23.2 MPa= - - - - 126.8 MPa= 28.8 MPaWWz z = = 6 6bhbh2 2= 144 cm3WWy y = = 6 6hbh
11、b2 2= 72 cm3 斜弯曲斜弯曲偏心受压偏心受压F FN NF3030s s角点角点角点角点 = + + + + + + + + WzMzWyMyAFN- - - - - - -整理课件P97P97 51- -4FFeez zy y5050250250FN = FA = bhM = FeWz = 6bh2s s = WzMzAFNmax+- -min= ( (1 ) ) h6ebhF+- -e e 左左 = Es s s sminmine e 右右 = Es s s smanman= ( (1- - ) ) h6eEbhF= ( (1+ + ) ) h6eEbhFF = ( ( e e左
12、左 + + e e右右 ) ) Ebh2e = h6e e右右 + + e e左左e e右右 - - e e左左= 625 kN= 25 mm讨论：讨论：( (1) )s smax 0s smin 0 0( ( e 0 0 0- - - - 整个截面：整个截面：s s s s 0 0- - - - 0 0( ( e ) )h6- - 虎克定律虎克定律偏心受拉偏心受拉整理课件P98P98 52- -2hbD风压力的合力：风压力的合力：底截面：底截面：14F = p p pD2 M = F h Mn = F b = 314 Nm= 235.6 Nms sr3 = = M2 + + Mn2W1= =
13、 h2 + + b2d3( (1- -a a4) )8pD21- -a a4a a 0.3086- - d d = = ( (1- -a a ) )d2 2 2.64 mm- - 取取圆杆内径：圆杆内径：圆杆内径：圆杆内径： 54.6 mm 54.6 mm- - - - 弯扭组合弯扭组合( (圆杆圆杆) ) W W = = 3232p p p pd d3 3( ( ( (1 1- - - -a a a a4 4) ) ) )- - s s 危险截面危险截面d d = 2.7 mm整理课件P99P99 52- -3lllABCDx xy yz zF3F4F1F24.09 kN4.09 kN15.
14、91 kN15.91 kN7.96 kN7.96 kN0.68 kN0.68 kNMMe eMMe eMn 图图0.204 kNm0.204 kNm1 kNm1 kNm1.227 kNm1.227 kNm3 kNm3 kNm1.092 kNm1.092 kNmMy 图图Mz 图图反力反力 内力图内力图危险截面危险截面B：Mz= 1.092 kNmMy= 3 kNmM M = = = = MMz z2 2 + + + + MMy y2 2s sr4 = = M2 + + 0.75Mn2W1= 3.193 kNm= 3.193 kNmMMn n= = - - - - 1 kNm 1 kNm弯扭组合
15、弯扭组合- - s s M2 + + 0.75Mn2 s s - - M2 + + 0.75Mn2 s s 323p p= 69.6 mm取取F4F1内力图内力图= = 3232p p p pd d3 3W W = = WWz z 弯扭组合弯扭组合( (圆轴圆轴) )：d = 7.0 cm整理课件P99P99 52- -4T T M M 4545A At ts sA As s = WMW = 32p pd3t t = WPTWP = 16p pd3e e 00 = Es ss s = Ee e 0 虎克定律虎克定律虎克定律虎克定律：M = Ws s = 105 MPa= 278 Nm2s ss
16、 sa a = + + cos2a a ( t t ) sin2a a2s s2s ss s 4545 = + + t t2s ss s - -4545 = - - t t1Ee e 4545 = ( (s s4545 - - nsns- -4545 ) ) 广义虎克定律：广义虎克定律：广义虎克定律：广义虎克定律：1 - -n n2= e e00 + t t 1 + +n nE1 - -n n2t t = ( (e e4545 - - e e00 ) )1 + +n nE= 40.4 MPaT = WPt t = 214 Nm整理课件P101 53- -3 5m 7m 9m2m 4m 4m(
17、( ( (m m m m l l) ) ) )2 2p p p p2 2EIEIF Fcrcr = = ( (5) )2p p2EIm m m m l lm m m m10.70.521 1150.770.592214( (4.9) )2p p2EI ( (4) )2p p2EI ( ( ( (4 4) ) ) )2 2p p p p2 2EIEI( (4.5) )2p p2EI两端铰支两端铰支两端铰支两端铰支一端固定一端固定一端固定一端固定一端铰支一端铰支一端铰支一端铰支两端固定两端固定两端固定两端固定一端固定一端固定一端固定一端固定一端自由一端自由一端自由一端自由两段两段两段两段F Fcr
18、cr最大最大最大最大F Fcrcr最大最大最大最大F Fcrcr最小最小最小最小细长压杆细长压杆细长压杆细长压杆整理课件P101 53- -4( ( ( (m m m m l l) ) ) )2 2p p p p2 2EIEIF Fcrcr = = 4 4p p p pd d2 2 = = a a2 2Fcr圆圆Fcr方方= I圆圆I方方= 64p pd4121a43p p= 0.955 细长压杆细长压杆细长压杆细长压杆整理课件P100 53- -2 Es sPl lP = p p IminAimin =查表查表查表查表：Imin = 225.9 cm4 A = 42.1 cm2 iminm
19、mll lmax = l lP= 2.32 cm= 216= 99.35属细长杆属细长杆属细长杆属细长杆( ( ( (m m m m l l) ) ) )2 2p p p p2 2EIEIminminF Fcrcr = = = 42.3 MPal l l l 2 2p p p p2 2E Es s s scrcr = = 或或或或= 178 kNFcr = s scrA= 178 kN球铰球铰球铰球铰失稳范围失稳范围失稳范围失稳范围整理课件P102 54- -1hbz zy y( (1) )( (2) )失稳方向失稳方向失稳方向失稳方向失稳范围失稳范围失稳范围失稳范围 I Iy y = = 1
20、212hbhb3 3A A = = 24 cm2 I Iz z = = 1212bhbh3 3= 32 cm4 = 72 cm4 IzAiz = IyAiy =23=3= 1.155 cm = 1.732 cm m my = 0.5 m mz = 1 l lz l lyl lz l lPiym myll ly =izm mzll lz = 132.8 = 99.6 : : 绕绕绕绕z z轴失稳轴失稳轴失稳轴失稳 Es sPl lP = p p= 99.35属细长杆属细长杆属细长杆属细长杆( ( ( (m m m m z zl l) ) ) )2 2p p p p2 2EIEIz zF Fcrc
21、r = = = 269 kN 失稳的可能性相同失稳的可能性相同失稳的可能性相同失稳的可能性相同：l l相同相同s scr相同相同iym myl=izm mzl=b12h12121= 2hbl l l l 2 2p p p p2 2E Es s s scrcr = = 整理课件P102 54- -2FFABCDaaaaF FN N1 1 = = F F2 21 1F FN N2 2 = = F FFCFN1FN1BF FN N2 2F FN N1 1F FN N1 1( ( ( (压压压压) ) ) )( ( ( (拉拉拉拉) ) ) )( ( ( (四杆四杆四杆四杆) ) ) )( ( ( (
22、中间杆中间杆中间杆中间杆) ) ) )Q Q 235235钢钢钢钢：s scr = 304 - - 1.12l l直线公式直线公式 Es sPl lP = p p= 99.35 ba- -s sSl lS = 57.1l lim m al l =稳定问题稳定问题稳定问题稳定问题强度问题强度问题强度问题强度问题 l lPl lS n nw w - - - - ABAB段段段段：BCBC段段段段: :m m = 0.7m m = 14di =a2 3i =72 3= cm = 2 cmil l1 =m m 1.5lil l2 =m m l = 157.5 = 148.5 l lP( ( ( (细长
23、杆细长杆细长杆细长杆) ) ) )l l l l 1 12 2p p p p2 2E Es s s scrcr = = = 100= 79.6 MPaFcr = As scr= 400 kN nw FcrF AB段先失稳段先失稳 n nw w = 2.5= 2.5 F = 160 kN整理课件P104 55- -1 lBCA2bbFF FC CyF FC CxF FABABS SMC = 0 F FABAB = = F F7 76 6= = 60 607 7 = = 158.7 158.7 kNkN从稳定角度考虑从稳定角度考虑从稳定角度考虑从稳定角度考虑im m ll l = 1 cm IAi
24、 = d4= 80安全系数法安全系数法安全系数法安全系数法：( ( ( (压压压压) ) ) ) Es sPl lP = p p= 99.35 ba- -s sSl lS = 57.1l l l lPl lS s scr = 304 - - 1.12l l直线公式直线公式：Fcr = As scr= 214.4 MPa= 269.4 kNn n = =F FABABF Fcrcr= 1.70 nw = 2所以所以不安全不安全整理课件P104 55- -2ABCqDl 2llF FADADF FBDBDF FBCBCS SFx = 0 F FBDBD = = 0 0S SMA = 0 S SFy
25、 = 0 F FBCBC = = 2.25 2.25 q qF FADAD = = 6.75 6.75 q q( ( ( (拉拉拉拉) ) ) )( ( ( (压压压压) ) ) )( ( ( (强度计算强度计算强度计算强度计算) ) ) )( ( ( (稳定计算稳定计算稳定计算稳定计算) ) ) )稳定计算用稳定计算用稳定计算用稳定计算用折减系数法折减系数法折减系数法折减系数法铸铁铸铁铸铁铸铁Q Q235235钢钢钢钢BCBC杆杆杆杆：ADAD杆杆杆杆：A2FBCs s =2.25q=41p pd22 s s s s - - - - q 5.59 kN/ /m- - q 8.04 kN/ /m- - im m ll l = d14i = 100j j = 0.166.75q=41p pd12 j j j j s s s s- - - - - - - - A1FADs s =| |取取 s s s s- - - - =120 MPa120 MPa s s s s =160 MPa160 MPa q = 5.59 kN/ /m整理课件
