3 Geometric Properties ofSections 1.ppt

MechanicsofMaterials Appendix1GeometricProperties 几何性质 ofSections Appendix1 GeometricPropertiesofSections Revision where allowablestress 许用应力 max themaximumworkingstress 最大工作应力 atthecriticalpoint 危险点 Designthedimensionofthesection 截面设计 Threekindsofcalculationofstrengthmaybedoneaccordingtothecriterionofstrength Checkthestrength 强度校核 Determinetheallowableload 确定许用荷载 43 Thecriterionofstrength 强度条件 Appendix1 GeometricPropertiesofSections Revision n 1 Permissiblestress 许用应力 Limitstress 极限应力 Safetyfactor 安全因数 Appendix1 GeometricPropertiesofSections Revision Sketchofstressconcentration Sketchofthestressdistribution Stressincreasesabruptlynearthecrosssectionwithasuddenchangeindimension Appendix1 GeometricPropertiesofSections 1Firstmomentsofareaandcentroids y X y X dA O Centroid 形心 Geometriccentre 几何中心 ofaplane anarea Itisthepre requisitefordesigncalculations Firstmomentsofarea 静矩 Appendix1 GeometricPropertiesofSections 1Firstmomentsofareaandcentroids Firstmomentsofcompositeareas 组合截面 Centroidsofcompositeareas Appendix1 GeometricPropertiesofSections 1Firstmomentsofareaandcentroids Locatethecentroidofthetrapezoidsectionanddeterminethefirstmomentoftheareaaboutthebottom Thefirstmomentofthearea Locationofthecentroid x y O C1x C2x Appendix1 GeometricPropertiesofSections 2Polarmomentsofinertia momentsofinertiaandproductsofinertia Polarmomentsofinertia 极惯性矩 withrespecttopointO Momentsofinertia 惯性矩 ofaplaneareawithrespecttothexandyaxes 坐标轴 alsocalledsecondmomentsofarea 截面二次轴距 Therefore thepolarmomentofinertiacanbeconsideredasthemomentofinertiawithrespecttotheaxisperpendiculartotheplaneoftheareaandintersectingtheplaneattheoriginO Appendix1 GeometricPropertiesofSections 2Polarmomentsofinertia momentsofinertiaandproductsofinertia Productsofinertia 惯性积 withrespecttothexandyaxes Becauseofthefactthat Weobtainthefollowingexpressionfor Thus weobtaintheimportantrelationship Appendix1 GeometricPropertiesofSections 2Polarmomentsofinertia momentsofinertiaandproductsofinertia UnitforI mm4 nottousecm4 m4 Themomentsofinertiaandtheproductsofinertiaaredefinedwithrespecttoagivenaxis Thepolarmomentsofinertiaisdefinedwithrespecttoagivenpoint Themomentsofinertiaandtheproductsofinertiaarealwayspositivequantities Theproductsofinertiamaybepositive negativeorzero dependinguponthepositionofthexyaxeswithrespecttothearea Theproductofinertiaiszerowithrespecttoanypairofaxesinwhichatleastoneaxisisanaxisofsymmetry 对称轴 ofarea Appendix1 GeometricPropertiesofSections 2Polarmomentsofinertia momentsofinertiaandproductsofinertia 5 Themomentofinertiaincreasesasthecentroidsoftheareaismovedfartherfromthereferenceaxis Rectangle 矩形 Triangle 三角形 Circle 圆形 Appendix1 GeometricPropertiesofSections 2Polarmomentsofinertia momentsofinertiaandproductsofinertia Adistanceknownastheradiusofgyration 惯性半径 isoccasionallyencounteredinmechanics Radiusofgyrationofaplaneareaisdefinedasthesquarerootofthemomentofinertiadividedbytheareaitself thus Characteristicoftheradiusofgyration Theradiusofgyrationisdefinedwithrespecttoagivenaxes 2 Theradiusofgyrationhasunitsoflength m 3 Theradiusofgyrationisalwayspositivequantities Appendix1 GeometricPropertiesofSections 3Parallel axistheoremformomentsofinertiaandproductsofinertia Theparallel axistheorem 平行移轴公式 givestherelationshipbetweenthemomentofinertiawithrespecttoacentroidalaxisandthemomentofinertiawithrespecttoanyparallelaxis dA Appendix1 GeometricPropertiesofSections 3Parallel axistheoremformomentsofinertiaandproductsofinertia Themomentofinertiaaboutacentroidalaxisistheleastmomentofinertiaofanarea foragivendirectionoftheaxis buttheproductofinertiaaboutacentroidalaxismaynotbetheleast Therefore Appendix1 GeometricPropertiesofSections 3Momentsofinertiaandproductsofinertiaofcompositeareas Polarmomentsofinertia momentsofinertiaandproductsofinertiaofcompositeareas ANALYSISSTEPSFORCOMPOSITEAREAS Step1 Locatex yStep2 ApplyequationstoeachsectionStep3 SumupcontributionsfromeachsectiontoobtainIxandIyforthecomposite compoundsection Appendix1 GeometricPropertiesofSections 3Momentsofinertiaandproductsofinertiaofcompositeareas Appendix1 GeometricPropertiesofSections 3Momentsofinertiaandproductsofinertiaofcompositeareas Appendix1 GeometricPropertiesofSections 3Momentsofinertiaandproductsofinertiaofcompositeareas Appendix1 GeometricPropertiesofSections 3Momentsofinertiaandproductsofinertiaofcompositeareas Appendix1 GeometricPropertiesofS