i-彎曲變形課件

4.梁的變形DeformationsofBeams4.1彎曲變形的概念xyoxyy(x)oxxyy(x)θ(x)ox撓度以y軸正向?yàn)檎鼵onceptofdeformationsforabeamTherearetwoformsofdeformationinabeam:(1)撓度(deflection)xyL---calledtheequationofdeflectioncurveorelasticcurve

梁有兩種變形形式撓度函數(shù)或撓曲線方程Acoordingtotheassumpationofsmalldeformationswecanget:轉(zhuǎn)角以x軸正向逆時針旋轉(zhuǎn)為正(2)轉(zhuǎn)角(slope)xyLRelationbetweenand:---calledtheequationofslope

轉(zhuǎn)角函數(shù)或轉(zhuǎn)角方程注意:材料力學(xué)所說的梁的變形實(shí)質(zhì)上指的是梁的位移!Thepositiveandnegativesymbolofcurvatureisjustcoincidentwiththatofbendingmoment,sowehavetheresultshownfollows:Itiscalledtheapproximatelydifferentialequationofthedeflectioncurve.撓曲線近似微分方程曲率的正負(fù)號與彎矩的正負(fù)號一致4.3積分法求梁的變形UsingintegrationmethodtodeterminethedeformationsofabeamBasicequationis:M(x)/EILxSinglefunction(1)M(x)/EI在梁中為單一函數(shù)

(asinglefunctioninwholebeam

)Lx0積分常數(shù)C和D由約束條件確定：AFixedpinTherearetwoboundaryconditionsforabeam.(2)M(x)/EI在梁中是分段函數(shù)

(isnotsamefunctioninonepartofthebeam

fromanother)M(x)/EIxia.梁的分段圖：xy梁的內(nèi)力和變形與載荷及抗彎剛度之間的關(guān)系:轉(zhuǎn)角集中力F均布載荷q撓度

彎矩剪力4.4對稱梁和反對稱梁Symmetricalandanti-symmetricalbeams(1)Forabeam,ifloadingsandsupportsaresymmetrical,itiscalledsymmetricalbeam.qaLLPABCqLEI對稱梁(2)Forabeam,ifloadingsandsupportsareanti-symmetrical,itiscalledanti-symmetricalbeam.qaaLLmLLPP反對稱梁1分鐘問答題:LL圖示兩梁的橫截面大小和形狀均相同,問:(1)兩梁的內(nèi)力圖是否相同?(2)兩梁的最大應(yīng)力是否相同?(3)兩梁的變形是否相同?相同相同不相同分析和討論:線彈性小變形條件下,梁的內(nèi)力,應(yīng)力及變形與下述哪些因素有關(guān)(1)梁的受力情況(2)梁材料的力學(xué)性能(3)梁的截面形狀(4)梁的約束情況分析和討論:(1)梁的受力情況(2)梁材料的力學(xué)性能(3)梁的截面形狀(4)梁的約束情況內(nèi)力分析和討論:(1)梁的受力情況(2)梁材料的力學(xué)性能(3)梁的截面形狀(4)梁的約束情況變形疊加原理

多個載荷引起的梁的變形都是可以疊加的.

Principleofsuperposition在小變形線彈性條件下,材料力學(xué)問題的內(nèi)力應(yīng)力變形都是可以疊加的,這稱為疊加原理.疊加法(1)4.5疊加法計算梁的變形疊加法

(Principleofsuperposition)mABqABmFqABFAB疊加法(2)Undertheconditionoflinerelasticityandsmalldeformations,thedeflectionsorslopesofthebeamcausedbyanyfactorcanbesuperposition.在線彈性小變形條件下,任何因素引起的梁的變形都是可以疊加的.例求圖示自由端的撓度。l/

2l/

2EIq0Al/

2l/

2θy

2y

3Bl/

2l/

2q0v

1lq0載荷的分解和重組(3)逐段剛化法是如何使用的疊加原理?

ABPFABPFABPF+實(shí)質(zhì)上是考慮梁的逐段變形然后疊加

(4)

逐段剛化法的關(guān)鍵是什么?關(guān)鍵是:1)每段梁是什么形式的梁?2)每段梁受什么載荷作用?3)每段梁的變形對所考察點(diǎn)的撓度的影響.ABPFABPFABPF5分鐘小練習(xí):求自由端的撓度和轉(zhuǎn)角?采用逐段剛化法ABPC2EIEIABPC2EIEIABPC2EIPLABPC2EIEIExample:DeterminetheslopeofsectionB.aaqEIABSolve:aaq/2EIABaq/2CBaaq/2EIABC難題解析:aaq/2EIABaq/2CB例

求

C點(diǎn)豎向位移

v。PLaABCPTTTTTφv2PPPPv1PPPPPv35.梁的剛度Stiffnessofbeams5.1

梁的剛度條件

-----AllowedmaximumslopeStiffnesscriteriaofabeam-----AllowedmaximumdeflectionInengineeringtheallowedmaximumdeflectionwasofengivedbytheformasfollows:容許轉(zhuǎn)角

容許撓度

5.2剛度條件的應(yīng)用

(Applicationsofstiffnesscriteria)(1)

校核剛度(Toexaminethestiffnessofabeam)(2)計算許可載荷(Todeterminethemaximumexternalforces)

(3)計算許可截面尺寸(To