各类梁的弯矩剪力计算汇总表-剪力计算系数

1、23456梁的简图表1简单载荷下基本梁的剪力图与弯矩图q剪力Fs图弯矩M图Z5T77nmn -I1八114，腔八 qa(ll-a)2/7X1SLJ翌QM八，a “罰7r1表2各种载荷下剪力图与弯矩图的特征某一段梁上的外力情况剪力图的特征弯矩图的特征无载荷.水平直线斜直线、或/集中力1F突变F转折集中力偶无变化 突变1/ Mq均布载荷斜直线、抛物线厂或零点极值"yTY表3各种约束类型对应的边界条件约束类型位移边界条件力边界条件（约朿端无集中载荷）固立端严w = 0 & = 0简支端漳易w = 0M=0自由端=M=OFs =0注：力边界条件即剪力图、弯矩图在该约束处的特征。序号裁

2、面閒图級面积A常用截面几何与力学特征表表25截面边缘至对主轴的惯性矩A主轴的距离yA = bhA = +(5+ bx)h(b2b)h力=3( V(b + 2b» 力二3(6&疗4 +z(*63截面抵抗矩W(Q + 4M丄十屛"2 】一 12(5 + 20(以+ 4比十曲卄2 2 一 12(26| + 0)回转半径2 0.236办序号截面简图截面枳A敘面边缘至 主轴的距离丿41P练LA二知$4y=2d5I-.2-盈） 八41 .厂卫61A = BH - bhy = *H71A = a2-a?对生轴的惯性矩I戴面抵抗矩W回转半径12討W二盘2i =丄孑42討-山）W遥（

3、宀刖i二七 J0+dfW = (BH3-6A3)-0.289 J BH-bh2吉（宀亦）W = 0.118( a3-y)F = 0.289 Ja2 a截面边缘至主轴的距离y序号就面简图般而积A10截面边缘至主轴的距离y#截面边缘至主轴的距离y1 bHJ(B-b)" 力 一 2 Bt + bh#截面边缘至主轴的距离y#截面边缘至主轴的距离yA = B|/i + 血2+ 弘r 6hu(b2 讥)於L 佝+恥弓B1 + dA * B2t211对主轴的惯性矩1截面抵抗矩W回转半径ix(Ji - r)W二丄闪2 =丄>2£=7tz = bh3-(b-)a3W二点f)/?-(B2

4、-6)(y2<2)3 (Bi - 6)(yj -仃F叭=丄截面边缘至主轴的距离y序号截面简图截面积AA = Wi + (B-/i)r14对主轴的惯性矩I截面抵抗矩W回转半径1“吉加+ (B6)円bh'mFw =*6hi o 289 x-0.289X J 弘+(Br”W = BH3-(B-b)h3：=0.289/BH3-(B-6)A5 V BH-(Bb)h2 吉BHB -b)h3W = BH3-(B-6)A32 0.289.BHlB-bW V BH-(B-b)h对主轴的惯性矩I截而抵抗矩W回转半径ix/）'卜矽;i=n"顒3专呵卜顒诸+】），产0.289H /f

5、寻册討）"哥爭】）'0.289BW 产裔(H$f 3)x=0.289 J W+Hh 七 h2/y= Y(H-A)w产普(Hf)山二 0.289&对主轴的惯性矩1截面抵抗矩W冋转半径igDi,-0.289Axy = 0.2S9fr ”5参17.785 薯I"缶17.785 誓ab - a, 儿_48ba3-btat3fy48呱-24b w W 二 WlWy24a一_ V 24("二 a2J._ / ha'-bwffy" V 24("-sbJI =曲+型6412j _曲+座+"厂 64616+血12"怦疇

6、)w =m 96(6 + ) x(3x/ + 326c/3 + 122J2 + 1663)°厂序号截面简图截而积A截面边缘至主轴的距离y对主轴的惯性矩截面抵抗矩W回转半径A-2(xR + b)t人已?十2仍尺2lynRztUbR2=0.785胡/严右加= 0.0491加 I 产着hEOWhb， 佔荷弘（沪+小速.16（hb2）少产弩00982弘2«=呼=00982肪2W0A96hb210（r吟在A点处，B点剪应力注：1I称为截而对主轴（形心轴）的截而惯性矩（mn?）o基本计算公式如下：I = ydA A2. W称为截而抵抗矩（mmJ,它表示截面抵抗弯曲变形能力的大小，基本计

7、算公式如下：W= ymax3i称截而回转半径（mm）,其基本计算公式如下:4. 上列各式中，A为截面而积（mm?）, y为截而边缘到主轴（形心轴）的距离（mm）, I为对主轴（形心轴）的惯性矩。上列各项几何及力学特征，主要用于验算构件截而的承载力和刚度。2.单跨梁的内力及变形表(表26表210)(D简支梁的反力、剪力、弯矩、挠度 表26序次图类荷载弯矩剪力荷载弯矩剪力荷载弯矩剪力荷载弯矩剪力荷敦弯矩剪力项目反力 RA=RB=-J剪力 Va=Ra； Vb=MllMlTTTnminiss弯矩挠度反力剪力F1348EJRa崽=扌FVa=Ka； Ub=_Rb挠度若a>b时，在工二(a+2b)处,

8、反力 ra=rbf剪力 Va-Ka； Vb=-b弯矩挠度反力剪力弯矩挠度反力剪力弯矩挠度Mg 二 FaWn¥,K = 24EJ 2-4/)Va=Ka; Vb= -RbMTmx=yFZ19少WraflX"384EJRa=Rb二*”Va=«a； Vb=-Rb5q广* 一 384 EJ序次图类图 示项目计算发荷载反力剪力弯矩剪力荷载弯矩挠度反力剪力弯矩弯矩剪力挠度"怡Rb晋Va=a； Vb=-RbMs晋4 +鈿式中：龙=a 晋Ra= Rb = qbJ =Ra； Vb= -RbMg二 qbax叫=谿皆¥一自悬臂梁的反力、剪力、弯矩和挠度表27序次图类图

9、示项目计算式1荷载X窗反力Rb=Fi亠剪力VB= _Rb弯矩弯矩Mx= -Fx; MjMb二- Fl剪力DT! runuT Him挠度FP2荷载,_ B反力2F剪力vb= -Rb弯矩rfTTlTnTn弯矩Mx= - F (ar-a); MzuMbu - Fb剪力llllllllllllll挠度叫=叫=黔（3-专）序次图类图示项目计 算 式3荷載,(nFLLT_l_L|b反力Rb= nF剪力VB= - Rb弯矩aiijiuiiniiinTlTn弯矩Mz=Mb二-FZ剪力挠度3n2 + 4n 十 g-叫- 24nEJ Fl4荷戟反力Rb= ql剪力Vb=-Kb弯矩11111 rrTrnTHj弯矩M

10、max=MB=-豊剪力挠度ql4- 8助5荷载Q/fllllTITI反力Rb= W:ru：:剪力Vb= - Rb弯矩rrTTTTTTTTnTl弯矩Mz=Mb=-詈(2-a)剪力挠度叫=叫=黠（3-4乡+爷）6荷裁¥4ba L b i反力Rb = qbt=±fc:剪力VB= -Kb弯矩弯矩MB=-f剪力挠度叫=鏑4-知序次图类图示项目计 算 式7荷载AB反力Rb= g剪力Pb= - Rb弯矩弯矩M b = - qcb剪力挠度Wa,= 24EI（12&S-4沪 + ”2）（3） 一端简支另一端固定梁的反力、剪力、弯矩和挠度 表28序次图类图 示项目计 算 式AC段：叫=

11、击Ra (3Z2X -X3) - qbyxBC段：Wx24E/ 4Ra (3厂兀/) -4qb3x + q (x - a)4当 x = a Bt：s严击<iRa $-疋)-qb36序次图类图 示项目计 算 式荷载理11山dii山111护反力剪力Va=/?a； F4弯矩弯矩剪力荷载弯矩剪力荷载弯矩剪力挠度当 x = 0.422/ 时，uz = 000542 晳反力剪力弯矩挠度反力剪力弯矩挠度Ra=* (8-6a + a3); RB=(6Z - a2); ° =十va=ra； vb= -rb> Mg当工AC段:U'x=4Ra (3Z2X - X3) -4qa (3bl

12、 + a2) x + x4BC段：Wx=24E/ MRa (3Z2x - x3) - qa (a3+ 126/z)+ 6ax2 - 4x3当 x = a 时.S=57 4qRa (3Z2- a2) - 3qa2 (4/6 * a2)Va=«a； Vb=-Rb当S+严时，心二心+守）AC段：叫=击Ra (3Z2X -X3) - qbyxBC段：Wx24E/ 4Ra (3厂兀/) -4qb3x + q (x - a)4当 x = a Bt：s严击<iRa $-疋)-qb36序次图类图 示项目计 算 式反力荷载剪力5 =3 vB= -kb弯矩弯矩AC段:挠度剪力+ q Q - di)

13、42 黔(】2Uf);Rb= qb、- Ra当无二5 +应旨4心(3Z2x-x3) -q» (12沪 + 蓟)xCD段:24EJ(4Ra(3/咯一工彳)-qb (1262+fef) x序次图类图 示项目计 算 式序次图类图 示项目计 算 式（4）两端固定梁的反力、剪力、弯矩和挠度 表29序次图类图 示项目计 算 式序次图类图 示项目计 算 式荷载弯矩剪力荷载弯矩剪力荷载5 弯矩剪力IIIIIIIIIIIIIIIIIIII1I<ul !)i!llllllllll , a 、l a »Tn -1BX/ZLLl山川剪力 Va=Ra； Vb= - Kb弯矩挠度Smix24=出

14、384EI反力 RRB-qa剪力 VA= Ra； Vb=挠度反力剪力弯矩挠度Ra=号(2-2a2+ a3); RB=(2Va=Ra； *=_Rb2Ma= -为(6-8a + 3a2); Q= +R2当"皿七w-RaX-3Ma/ + )BC 段：wT77 Ra* 3MaF +(a3- 4a2xL4+ 6ax2 亠 4x3)序次图类图 示项目计算式6荷载qA _ B a j b 彳 a_ 反力Ra=Rb=乎剪力Va-a； Vb=弯矩X i M /弯矩M-=岩（3-3* +書）剪力liiullrbv7inn挠度叫=眾2-2严+ J外伸梁的反力、剪力、弯矩和挠度表210序次图类fi示项目计算

15、 式1荷载f反力C 63L/.剪力Vc= - F; Vb= _ Rb-十F弯矩TnTrrrrrrrr弯矩MnMX=- fa剪力inliiiHinlJllliHJ挠度wc=3（i + f）F r 2当 z = a + 05Z 时.3仙=-0.0642 岸p2荷载f1反力R A = R B = F剪力Vb=Rb弯矩finni iiiminK弯矩Ma= Mb= - Fa剪力盯CT1I|挠度wc=wD=（3 + 2f） 当 z = a + 0.52 时，s他n=-器序次图类图 示项目计 算 式5序次图类图 示项目计 算 式荷载弯矩剪力荷载弯矩剪力荷载弯矩剪力 纟Niiiiiiii nguniCtL l

16、 a lfTRrmTTrrt1皿川111川删反力剪力弯矩挠度反力剪力弯矩挠度Va£= - qa ;= Ra-W； Vb=RbMa=wc十4珀+ 3爷）1 +专厂时,"唤=%（1 方）2Ra=Rb=（ 1 + 2 千+（l + 2a）V 曲=-曾 VA = -ygZ； V= -yg/； Vs=gaMa=Mb=-寺gab ”唤=普（1-4 食）5-24 豹反力Ra=劄2 +千）；心=-鲁剪力弯矩MmflX=MA =挠度2 2当 x = a +0.423/ 时，-0.0321 咒'5序次图类图 示项目计 算 式序次图类图 示项目计 算 式反力Ra=Rb= W荷载弯矩剪力弯

17、矩剪力荷载弯矩剪力剪力VA= -Ra； Vb=KbWTlTikx<fniTrrrrtnrcL aq弯矩剪力q JTTTT C弯矩挠度反力剪力弯矩挠度反力剪力弯矩挠度反力剪力弯矩挠度Ma= Mb=-寺 gaWc= Wp =2 +乎当 x = a +0.5Z 时，wnqa2l216EJRA=f（2 + 3f） ; Rb瞬Ma= -Fa; Mb=号3 +诗）（5-6 自Ma即 Mb=-%（ 12步）- 1+6$ + 6剧4+3f! ； Rb八欝Va&=-的；“b二Rb叱需（1屮序次图类图示项目计算式10荷载MP_B反力RIM. R 一逊 221 ' Rb- 2Z剪力VaRa；

18、*=Rb弯矩弯矩Ma = M ； Mb = _ 号剪力UMnlllilTni挠度叭二譜沂）比,IM/23.等截面连续梁的内力及变形表（1）等跨连续梁的弯矩、剪力及挠度系数表（表211表214）1）二跨等跨梁的内力和挠度系数 表211序次荷载图跨内杲大弯矩支座弯矩W 力跨度中点挠度MiMzMbVaVCW21qIIIIIIIIIIIHIHITI r戕巴c 4_L(A/i)(0.0700.070-0.1250.375-0.6250.625-0.3750.5210.5212rrE'frm41>10.096-0.0630.437-0.5630.0630.0630.912-0.3913

19、63;fziAABC j0.1560.156-0.1880.312-0.6880.688-0.3120.9110.91144-_+_L40.203-0.0940.406-0.5940.0940.0941.497一0.5865什f &产zLqaABC0.2220.222-0.3330.667-1.3331.333-0.6671.4661.4666n r飞4 l *0.278一-0.1670.833-1.1670.1670.1672.508-1.042注：1在均布荷载作用下：21 =表中系数Xg® =表中系数Xql： “,=表中系数x二匚。100 £/2.在集中荷载作用

20、下："=表中系数XF/： =表中系数XF：叩=表中系100E/2）三跨等跨梁的内力和挠度系数表2-12序次荷载图蹄内最大弯矩支座弯矩剪力跨度中点挠度Mm2MbMeVaVbsVt,W1W31qH ninpinniiiiTifTin A缈B慘C 'D0.0800.025-0.100-0.1000.400-0.6000.500-0.5000.600-0.4000.6770.0520.677次荷载图跨内最大弯矩支座弩矩剪力跨度中点挠度Mim2MbMeVAVbSVb6VufeVDWj32332jn&EipirnABCD0.101-0.050-0.0500.4500.550000

21、.550-0.4500.990-0.6250.9903irnnnf i e s0.075-0.050-0.050-0.050-0.0500.500-0.5000.0500.050-0.3130.677-0.3134qhi rinniTiF miABCD.O730.054-0.117-0.0330.383-0.6170.583-0.4170.0330.0330,5730.365-0.2085nTirnTi&、4、ABCD0.094-0.0670.0170.433-0.5670.0830.083-0.017-0.0170.885-0.3130.1046fff夕(M)倉两g(M) g -4_

22、L-_*_L3.175).100-0.150-0.1500.350-0.6500.500-0.5000.650-0.3501.1460.2081.1467rr才3BD.213-0.075-0.0750.425-0.575000,575-0.4251.615-0.9371.6158f乡舍8B0.175-0.075-0.075-0.075-0.0750.500-0.5000.0750.075*0.4691.146-0.4699£ £T $ W 30.162).1370.175-0.0500.325-0.6750.625-0.3750.0500.0500.9900.677-0.3

23、12序次祜挠阖跨内最大弯矩支座弯矩剪力跨度中点挠度Mim2MbMcV1)Wj5奶10r)-200-0.1000.0250.4000.6000.1250.125-0.025-0.0251.458-0.4690.15611竹f耳f F| F|AiS9.2440.067-0.267-0.2670.733-1.2671.000-1.0001.267-0.7331.8830.2161.88312耳产F| f&OQQABcDD.289'-0.133-0.1330.866-1.134001.134-0.8662.716-1.6672.71613F fA'"A" &

24、quot;" £$2ABCD0.200-0.1330.133-0.133-0.1331.000-1.0000.1330133-0.8331.883-0,83314fi t n ra厶ABCDD.229D.17C-0.311-0.08S0.689-1.3111.222-0.7780.0890.0891.6051.049-0.55(1.件lFg 2今艺3D27,-0.17(0.0440.822 1.17Q0.2220.222-0.044-0.04C2.438-0.83：j0.278注：1在均布荷载作用下：21 =表中系数Xg® » =表中系数Xql； vv

25、=表中系数x二匚。10() £/2.在集中荷载作用卜：21 =表中系数XF7； =表中系数XF：闪=表中系数x3)四跨等跨连续梁内力和挠度系数 表213荷载图弯矩剪 力密度中点挠度WM忡M2中MbsMe支VA%畑55nwTnTrfiiiTnrrnrn竹/知/辛C 1今Di徑叩'0.0770.036-0.1070.0710.393-0.6070,5360.4640.4640.6320.186f 产 r FQ/1 7厶B “厶C厶£)，乍E0.1690.116-0.161-0.1070.339-0.6610.554一0.4460.4461.0790.409|F f f

26、f F |F 产 A 3 y J 牛E0.2380.111_0.286-0.1910.714-1.2861.095-0.9050.9051.7640.573注：同三跨等跨连续梁。4）五跨等跨连续梁内力和挠度系数 表214荷孜简图弯矩剪力跨度中点挠度wMi中M冲M*3中Mb支Vb&畑畑B2(nimoTiRTrnrrmi0.0780.0330.046-0.105-0.0790.394-0.6060.526-0.4740.5000.6440.1510.315产 r r r F 池盲沪箸皆F0.1710.1120.132-0.158-0.1180.342-0.6580.540-0.4600.5

27、001.0970.3560.603r 亠严B , C, °D i Y 厶F ，也4也 '也卜10.2400.100D.122-0.2810.2110.719-1.2811.071-0.9301.0001.795).4790.918注：同三跨等跨连续梁。（2）不等跨连续梁的内力系数（表2-15.表2-16）1）二不等跨梁的内力系数表215荷载qqq111111111 1 IIIIHiniAx1 Jl0L - AMIAB-AL /l Le A汕L弋、4r-r-X” (Mg)MmVA畑(%)VcMm(MQVA(VmJMik：(g)Vc(VQ10-0.12500.07030.0703

28、0.3750-0.62500.62500.37500.09570.43750.0957-0.43751.1-0.13880.06530.08980.3613-0.63870.6761-0.42390.09700.44050.1142-0.47801.2-0-15500-05950.11080.3450-0.65500.7292-0.47080.09820.44320.13430.51821-3-0.17380.05320.13330.3263-0.67370.78360.51640.0993().44570.1558-0.55821.4-0.19500.04650.15720.3050-0.6

29、9500.8393-0.56070.10030.44790.1788-0.59791.5-0.21880.03960.18250.2813-0.71870.8958-0.60420.10130.45000.2032-0.63751.6-0.24500.03250.20920.2550-0.74500.9531-0.64690.10210.45190.2291- 0.6769E7-0.27330.02560.23740.2263-0.77371.011()-0.68900.10290.45370.2564-0.71621.80.30500.01900.2&690.1950-0,80501

30、-06940.73060.10370.45540.2850-0.75541.9-0.33880.01300.29780.1613-0.83871.1283-0.77170.10440.45690.3155- 0.79442.0-0.37500.00780.33010.1250-0.87501.1875-0.81250.10500.45830.3472- 0.83332.25-0.47660.00030.41700.0234-0.97661.33680.91320.10650.46150.43277.93032.5-0.5938负值0.5126-0.0938- 1.09381.4875-1.01

31、250.10780.46430.5272-1.0268注：14 =表中系数"厲；=表中系数Xqg 2（“加丿、（V；）表示它为相应跨内的最大内力。2）三不等跨梁内力系数表216弋荷载E- q ninriiiiiiiiinTfrnrnTrnq111111111XillllllllilUlillllili i |SC l u忙几/ 3数$L上"他勺古MabVAg畑Mb(MQ(%)(%x)Mar(Mmu)vA(匕QMg“0.4-0.08310.0869-0.06310.41697.58310.2000-0.0962_0.59620.46080.08900.42190.01500.

32、5-0.08040.0880-0.04910.4196-0.58040.2500-0.0947-0.59470.45020.09180.42860.02230.6-0.08000.0882-0.03500.4200-0.58000.3000-0.0952-0.59520.46030.09430.43420.03080.7-0.08190.0874-0.02060.4181-0.58190.3500-0.0979-0.59790.48250.09640.43900.04030.80.08590.0857-0.00590.4141-0.58590.40000l 021-0.60210.51160.

33、09820.44320.05090.9-0.09180.08330.00950.4082-0.59180.4500-0.1083-0.60830.54560.09980.44680.06251.0-0.10000.08000.02500.40000.60000.5000-0.1167a 0.61670.58330-10130.45000.075011-0.11000.07610-04130.3900-0.61000.5500-0.1267- 0.62670.62330.10250.45280.08851.2-0.12180.07150.05820.3782-0.62180.6000-0.138

34、5-0.63850.66510.10370.45540.10291.3-0.13550.06640.07580.36450.63550.65000.1522-0.65220.70820.10470.45760.11821.4-0.15100.06090.09400.3490-0.65100.70000.1676-0.66760.75250.10570.45970.13441.5-0.16830.05500.11300.3317-0.66830.75000.1848-0.68480.79760.10650.46150.1514L6-0.J8740.04890.13270.3127-0.68730

35、.8000-0.2037-0.70370.84340.10730.46320.16941.70.20820.04260.15310.29180.70820.8500-0.2244-0.72440.88970.10800.46480.18831.80.23080.03620.17420.2692-0.73080.9000-0.2468-0.74680.93660.10870.46620.20801.9-0.25520.03000.19610.2448-0.75520.9500-0.2710-0.77100.98460.10930.46750.22862.0-0.28130.02390.21880

36、.2188-0.78121.0000-0.2969-0.79691.03120.10990.46880.25002.25-0.35400.01060.27880.1462-0.85381.1250-0.36910.86911.15110.11110.47140.30742.5-0.43750.00190.3437-0.0625-0.93751.2500-0.45210.95211.27220.11220.47370.3701注：1皿=表中系数Xq&； =表中系数Xqg 2.（V；）为荷载在最不利布宜时的最大内力。4.双向板在均布荷载作用下的内力及变形系数表（表217表222） 符号说明

37、如下：刚度K= 少，式中E弹性模量；h板厚；v泊松比；3、G）max分别为板中心点的挠度和最大挠度；Mx为平行于厶方向板中心点的弯矩；My为平行于ly方向板中心点的弯矩；Mx°固定边中点沿/.v方向的弯矩；My°一一固定边中点沿ly方向的弯矩。正负号的规定：弯矩一一使板的受荷面受压者为正；挠度一一变位方向与荷载方向相同者为正。四边简支表2-17挠度=表中系数x甞Kmmnnniq弯矩二表中系数Xql2 式中I取厶和0中之较小者IJlyWMxMyMywMy0.500.010130.09650.01740.800.006030.05610.03340.550.009400.089

38、20.02100.850.005470.05060.03480-600.008670.08200.02420.900.004960.04560.03580.650.007960.07500.02710.950.004490.04100.03640.700.007270.06830.02961.000.004060.03680.03680.750.006630.06200.0317ZY三边简支，一边固定表218挠度二表中系数x甞K弯矩二表中系数Xq“ 式中I取人和。中之较小者WIJXSrnaxMyw0.500.005040.05830.0060-0.121210.950.003240.03240

39、.0280-0.08820.550.004920.05630.0081-0.11870.900.003680.03280.0322-0.09260.600.004720.05390.0104-0.11580.850.004170.03290.0370-0.09700.650.004480.05130.0126-0.11240.800.004730.03260.0424-0.10140.700.004220.04850.0148-0.10870.750.005360.03190.0485-0.10560.750.003990.04570.0168-0.10480-700.006050.03080

40、.0553-0.10960.800.003760.04280.01877.10070.650.006800.02910.0627-0.11330.850.003520.04000.0204*0.09650.600.007620.02680.0707-0.11660.900.003290.03720.0219-0.09220.550.008480.02390.0792-0.11930.951.001.000.003060.002850,03450.03190.02320.0243-0.0880-0.08390.500.009350.02050.0880-0.1215两边简支，两边固定表219b挠

41、度=表中系数K%K弯矩二表中系数XqN 式中/取仃和仔中之较小者tjlyIJXwIJhwMxMyM戈0.500.002610.04160.0017-0.084310.950.002230.02960.0189-0.07460.550.002590.04100.0028-0.08400.900.002600.03060.0224-0.07970.6()0.002550.04020.0042-0.08340.850.003030.03140.0266-0.08500.650.002500.03920.00577.08260.800.003540.03190.03167.09040.700.002430.03790.0072-O.O8I4I0.750.004130.03210.0374-0.09590.750.002360.03660.0088-0.07990.700.004820.03180.0441-0.10130.800.002280.03