彈性力學課件三零

2.7BOUNDARYCONDITIONInanalysingplaneproblem,thefollowingeightequationsmaybeconsideredasbasicequation:

twodifferentialequationsofequilibrium;threegeometricalequations;

threephysicalequations.Thebasicequationsinvolveeightunknownfunctionsofxandy:

threestresscomponents;

threestraincomponents;twodisplacementcomponents.Underproperboundaryconditions,theeightbasicequationscanbesolvedfortheeightunknownfunctions.Accordingtoboundarycondition,elasticityproblemsareclassifiedasdisplacementboundaryproblems,stressboundaryproblemsandmixedboundaryproblems.Foraplaneproblem,wehavexybaxabyus=uvs=va.Inadisplacementboundaryproblem,thesurfacedisplacementsofthebodyarespecified.b.Inastressboundaryproblem,thesurfaceforcesactingonthebodyareprescribedandsuchaconditioncanbetransformedintoaconditionaboutthestresscomponentsattheboundary.xyqAXN=l

x+mxyYN=m

y+lxyBP

y

yx

xy

xsYNXNThesurfaceforcesareXN=XYN=Yl

x+mxy=Xm

y+lxy=YThesetwoequations,expressingtherelationsbetweentheboundarystresscomponentsandthesurfaceforcecomponents,arethestressboundaryconditionsofaplaneproblem.Whentheboundaryisnormaltoacoordinateaxis,thestressboundaryconditionsaresimplified2xy3141and3：l=0m=+1

y=Y;xy=X++2and4：m=0l=+1

x=X;xy=Y++Usepositiveornegativesignaccordingastheoutwardnormalisalongthepositiveornegativedirectionofthecoordinateaxis.c.Inamixedboundaryproblem,someportionoftheboundaryisspecifiedwithknowndisplacementswhiletheotherportionissubjectedtoknownsurfaceforces.Forinstance：xyABCAB：v=0AC：

x=0；xy=0Mixedboundaryproblem解：AB邊：

y=-挽q角;摟

yx=0BC邊：

x=0早;衡

xy=0CD邊：

y=0幅;落

yx=0AD邊：u=憲0；盟v=豈0例1、寫出圖示懸臂梁的邊界條件，板厚為1ABCDyxq解：BC邊：

x=0略;驗

xy=0CD邊：

y=0傅;飲

yx=0AD邊：u=棕0；督v=則0AB邊：

y=

yx=0例2、寫出圖示懸臂梁的邊界條件，板厚為1xABCDyq0l解：BC邊：

y=0費;炊

yx=0DE邊:AB邊：

x=盞-gy

xy=0例3、寫出圖示結構AB、BC、DE的應力邊界條件水的重度為

ExyABCDN

co狀s(N捧,x見)=co先s=lco愁s(N厘,y引)=co夏s(9暖0似+讓)=不-s吃in北=炸ml

x+mxy=Xm

y+l

xy=Ycos

x-sinxy=0-sin

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