彎曲變形doulan9

1、Longitudinal deformation l;Twist deformation ;Bending deformation?vElasticalDeflection curve/Nl EAn/pM l GI,pzEA GIEIstiffness/zzM l EIdeflection curve function v = v ( x )rotating angle function = ( x )v: deflection or displacement : rotating angle or slopedtan( )dvv xxddvx Review1zME I2 3 2(1)vv)(

2、xv ZIExMxv)()( Basic differential equation of elastical curve ( )( )dZM xvxxCEI( )( )d dZM xv xx xEICxD( )( )dZEI vxM xxC( )( )d dZEI v xM xx xCxDBoundary conditionKnown deflectionand rotating angle例例6.1 Determine deflection curve function of beam shown in fig.22)(xqxMMoment function32( )d26ZqqxEI v

3、 xxxCC 43( )d624ZqqxEI v xxxCxDCxD 0,0:vvlx63qlC 84qlD)43(24)(434xxllEIqxvz)(6)(33xlEIqxvzxl qEIz)(84maxzEIqlv)(63maxmaxzEIqlvBoundary conditonlFbRAlFaRBxlFbxRxMA)(1)0(ax)()(2axFxRxMA)(axFxlFb)(lxa1212CxlFbvEIZ11316DxCxlFbvEIZRARB332)(66axFxlFbvEIZ22DxC2222)(22CaxFxlFbvEIZ021 DD)(62221bllFbCC0:,0:02

4、1vlxvx2121,:vvvvaxabFlBoundary conditoncontinuity conditon例例6.2)(62221blxxEIlFbvZ)()(632232axblblxxEIlbFvZaFv1v2ABCbIf a = b)(483maxZEIFlv2max16ZFlEIIf a b)(39)(322maxlEIblFbvZ3220blxIf b = 0Fb = M00.5773lxl)(392maxZEIMlvZEIMllvv16)221 （中maxmax2.64vvv中（2）ACCv,)()(mvFvvCCC)(48)(3ZCEIFlFv)()(mFCCC)(24

5、)(2ZCEIFlm)()(mFAAA)(16)(2ZAEIFlF)(48)(2ZAEIFlm)(24)481161(22ZZEIFlEIFlFABCm=Fl/2l/2l/2FABCl/2l/2ABCml/2l/2例例6.3 Found vC and AExam 6.4 Found vC of beam.Exchange loading )(8)2(41EIaqvCavvBBC222)(842EIqavB)(632EIqaB)(2474EIqa21CCCvvv21CCCvvv)(24412472444EIqaEIqaEIqaABCaaqABCqABCqBqACvB2 B2Rigidizatio

6、n in SequeceAssume BC is rigid, AB deformedavvBBC111)(127234231EIqaEIaMEIaQvBBB)(12194111EIqaavvBBC)(23221EIqaEIaMEIaQBBB Assume BC deformed , AB is rigid)(842EIqavC)(2441421EIqavvvCCCABCaaqBACqvC 1aaAa B1vB1QBMBvC2ACBqvC1)(6(6dd2xaxEIxqvC)(2441d42EIqavvaaCCEquivlently replace the loading )(4881)236

7、(6)23(42EIqaaaEIaqavC2 . 1828182Element finer, precision higherin x section dF = q dx , integrateABCaaqABCaaxqdxAC3a/2F=qaExam 6.5 found vC()/2CABvvvaaABqCABEI2EICaaaaABC2ql2qlvC2qlACABq/2C0,ACCAvvvACvvCABqC4qa34ql121CCCCvvvvExam 6.6 Found support reactionEquilibrium equationqlRRAC22Constrained cond

8、ition021CCCvvv)(245384)2(5441EIqlEIlqvCPhysical equation )(648)2(332EIlREIlRvCCCComplementaryequation0624534EIlREIqlCSolved)(45 qlRC)(83qlRRBAqABEICllRARBRCCqll1Cv2CvRCllComplementaryequationABCaaFABClaaEIEADqABClaaFDqlFC Exam 6.7 Found support reaction0DvCDClvAElFlCD)()(qvFvvCCCEIaFFvC3)2()(3EIqaqv

9、C2441)(3EIqaEAFlEIFavD244138430CDClv43412483qaIFalIAConstrained conditionPhysical equation lvmaxmax（1）Change loading and support（2）Reduce span length；（3） Optimize shape of section 11()1000700l（4） Optimize shape of beam zWxMx)()(max)(xMM )()(xMxWvariable section beamF 6)(2xFhxb2 6)(hxFxb6)(bxFxhstepp

10、ed shaftsuperposed beam軸力軸力N扭矩扭矩Mn彎矩彎矩M剪力剪力QNAnpMIzMyI NA npMW zMWNllEAEnpM lGIlGzMlEI1zMEIv/l v材料的材料的力學(xué)性能力學(xué)性能拉壓實(shí)驗(yàn)拉壓實(shí)驗(yàn)扭轉(zhuǎn)實(shí)驗(yàn)扭轉(zhuǎn)實(shí)驗(yàn)截面的截面的幾何性質(zhì)幾何性質(zhì)靜矩與形心靜矩與形心慣性矩與慣性矩與慣性積慣性積變形幾何變形幾何變形物理變形物理拉壓變形拉壓變形 l扭轉(zhuǎn)變形扭轉(zhuǎn)變形 彎曲變形彎曲變形撓度撓度v, y截面形心的豎向位移截面形心的豎向位移轉(zhuǎn)角轉(zhuǎn)角 截面繞中性軸轉(zhuǎn)過的角度截面繞中性軸轉(zhuǎn)過的角度v6-1 概概 述述撓度曲撓度曲線線撓度曲線方程撓度曲線方程 v ( x )轉(zhuǎn)角

11、方程轉(zhuǎn)角方程 ( x )dtan( )dvv xxddvx1zMEI2 3 21(1)vv)(xv ZIExMxv)()( 撓曲線撓曲線基本微分方程基本微分方程( )( )dZM xvxxCEI( )( )d dZM xv xx xEICxD( )( )dZEI vxM xxC( )( )d dZEI v xM xx xCxD邊界條件邊界條件v撓度曲線撓度曲線已知的撓度和轉(zhuǎn)角已知的撓度和轉(zhuǎn)角解：解：lFbRAlFaRBxlFbxRxMA)(1)0(ax)()(2axFxRxMA)(axFxlFb)(lxa1212CxlFbvEIZ11316DxCxlFbvEIZRARB332)(66axFxl

12、FbvEIZ22DxC2222)(22CaxFxlFbvEIZ021 DD)(62221bllFbCC0:,0:021vlxvx2121,:vvvvaxabFl邊界條件邊界條件連續(xù)條件連續(xù)條件ba極值點(diǎn)在極值點(diǎn)在 AC 段段)(39)(322maxlEIblFbvZba 極值點(diǎn)在極值點(diǎn)在 C 點(diǎn)點(diǎn))(483maxZEIFlv0bFb 形成一個(gè)力偶形成一個(gè)力偶 M極值點(diǎn)極值點(diǎn)llx577.0303220blx)(392maxZEIMlv中點(diǎn)撓度中點(diǎn)撓度ZEIMllvv16)221 （中相對(duì)誤差相對(duì)誤差中64. 2maxmaxvvvaFv1v2ABCbaABbFM例例6-3 求梁指定截面求梁指定截

13、面 的撓度和轉(zhuǎn)角。的撓度和轉(zhuǎn)角。（1）Acv,解：解：)(245385)2(5441ZZCEIqaEIaqv)(648)2(432ZZCEIqaEIaFv)(83421ZCCCEIqavvv)(324)2(331ZZAEIqaEIaq)(216)2(332ZZAEIqaEIaqaaF = q aABqCqaaABCaaFABC)(65321ZAAAEIqa例例 6.4 求圖示懸臂梁的求圖示懸臂梁的 vC解一：增減載荷解一：增減載荷)(8)2(41EIaqvCavvBBC222)(842EIqavB)(632EIqaB)(2474EIqa21CCCvvv21CCCvvv)(24412472444

14、EIqaEIqaEIqaABCaaqABCqABCqBqACvB2 B2解二：逐段剛化法解二：逐段剛化法先假設(shè)先假設(shè) BC 段剛性，只有段剛性，只有 AB 段變形段變形avvBBC111)(127234231EIqaEIaMEIaQvBBB)(12194111EIqaavvBBC)(23221EIqaEIaMEIaQBBB再考慮再考慮 BC 段的變形段的變形 ( AB 段剛性段剛性 )(842EIqavC)(2441421EIqavvvCCCABCaaqBACqvC 1aaAa B1vB1QBMBvC2ACBqvC1邊界條件邊界條件連續(xù)條件連續(xù)條件撓度撓度v, y轉(zhuǎn)角轉(zhuǎn)角 彎曲彎曲變形變形的度

15、量的度量曲率曲率1/ 1zMEIZIExMxv)()( vv直接直接積分法積分法( )( )dZEI vxM xxC( )( )d dZEI v xM xx xCxD查表查表疊加法疊加法直接疊加直接疊加增減載荷增減載荷逐段剛化逐段剛化載荷代換載荷代換結(jié)構(gòu)代換結(jié)構(gòu)代換例例6-5 求圖示簡支梁的求圖示簡支梁的 vC()/2CABvvv解：aaABqCABEI2EICaaaaABC2ql2qlvC2qlACABq/2C0,ACCAvvv解：ACvvCABqC4qa34ql121CCCCvvvv(B)(C)(A)(D)頂針頂針例例6-6 求下列超靜定梁的支反力求下列超靜定梁的支反力解：解除多余約束代之以約束反力解：解除多余約束代之以約束反力平衡方程平衡方程qlRRAC22約束條件約束條件021CCCvvv)(245384)2(5441EIqlEIlqvC物理方程物理方程)(648)2(332EIlREIlRvCCC補(bǔ)充方程補(bǔ)充方程0624534EIlREIqlC解得解得)(45 qlRC)(83qlRRBAqABEICllRARBRCCqll1Cv2CvRCll解：解除多余約束代之以約束反力解：解除多余約束代之以約束反力約束條件約束條件0DvCDClvABCaaFAElFlCD)()(qvFvvCCCEIaFFvC3)2()