版权说明:本文档由用户提供并上传,收益归属内容提供方,若内容存在侵权,请进行举报或认领
文档简介
StaticsStaticsofdeformablebodyChapter14
StabilityofColumns14.1Introduction14.2Criticalcompressiveforceofslendercolumnwithpinsupport14.3Criticalcompressiveforceofslendercolumnwithotherconstraints14.4Diagramofcriticalstress14.5CalculationofstabilityofcolumnsContents14.1IntroductionTheproblemofwhetherthecolumncanmaintaintheoriginallinearbalancestateiscalledthestabilityproblemofthecolumn.disturbanceforce(c)crF<FcrF=FcrF>FTheprocessofthelinearequilibriumofthecolumnfromstabletounstableiscalledbuckling.TheextremevalueFcrofpressureF,isalsoknownasthecriticalpressureorcriticalforce.(b)(a)disturbanceforcedisturbanceforceOtherformsofstabilityproblemsAthin-walledcylindersubjectedtouniformexternalcompressiveforcebecomesellipticalduetoinstability.Abeamofnarrowrectangularsectionbendslaterallyduetoinstability.F14.2CriticalcompressiveforceofslendercolumnwithpinsupportylFFxxvBytakingthecartesiancoordinatesystemshown,thedeflectionofanycrosssectionatadistancexfromtheoriginisv,andthebendingmomentis(a)whereFisanabsolutevalue.theapproximatedifferentialequationforthedeflectioncurveofcolumnis
(b)let,andthenequation(b)canbewrittenas(c)Thegeneralsolutionofthisdifferentialequationis:
(d)Boundaryconditions:
a.Bysubstitutingitintoequation(d),wegetb=0,(e)
b.v(l)=0Similarly,wecangetv(0)=0
(f)Fortheequation
(e),Ifa=0,Theremustbev=0.Thatmeansthedeflectionatallpointsontheaxisofthecolumnisequaltozero.Thiscontradictsthefactthatthecolumnremainsinequilibriuminaslightlybentstate.So,wegetsin(kl)=0.Thevalueofklthatsatisfiesthisconditionshouldbe
or(g)
whennandFiszero,itismeaningless.Whenn=1,thevalueofFistheminimum.So,FcroftheslendercolumnisThisistheformulaforcalculatingthecriticalcompressiveforceofaslendercolumnwithpinsupportatbothends,knownasEulerequation.TheFcrisproportionaltotheminimumbendingrigidityEIofthecolumnandinverselyproportionaltothesquareofthebarlengthl.Thismeansthattheslimmerthebaris,thesmallerthecriticalcompressiveforceis,andthemoreunstablethebaris.UndertheactionofFcr,wehavek=π/l.Substitutionitintoequation(e),weget
(h)Theaboveequationshowsthatthedeflectioncurveofaslendercolumnwithpinsupportatbothendsisahalf-wavesinecurve.Ifwemakex=l/2,andsubstituteitintoequation(h),wegetWhereaisthedeflectionofthemidcrosssectionofcolumn.Thereisnodefinitevalueofa.Thisisbecauseinderivingtheequationforthedeflectioncurveofthecolumn,theapproximatedifferentialequationforthebendingdeflectioncurveofthecolumnisbasis.Byexactdifferentialequationforthedeflectioncurve,WecanobtainthedeterminedvalueandthetheoreticalrelationshipbetweenthemaximumdeflectionandpressureF,asshowninthecurveOABinthefigure.Foractualmembersundercompression,duetofactorssuchasheterogeneityofmaterial,initialcurvature,orslighteccentricityduringloading,microbendingdeformationactuallyoccursbeforethepressureFreachesthecriticalpressureFcr,whichcanberepresentedbythecurveODinthefigure.FcrF14.3CriticalcompressiveforceofslendercolumnwithotherconstraintsForslendercolumnswithotherconstrainedsituations,
ThisisthegeneralformofEulerequation.
μisthelengthfactorofthecolumnunderdifferentconstraints.μlistheequivalentlength.Oneendfree,oneendfixed:
=2.0Oneendhinged,oneendfixed:
=0.7Bothendfixed:
=0.5Bothendhinged:
=1.0ExampleAcastironcolumnofcircularcrosssectionwithonefixedendandonefreeend.Ithasalengthl=3m,adiameterd=0.2mandamodulusofelasticityE=120GPa.CalculatethecriticalcompressiveforceofthecolumnfromEuler'sformula.Solution:CheckTable15-1forthelengthfactorμ=2,andthenthemomentofinertiaofthesectionisTherefore,thecriticalcompressiveforceis14.4Diagramofcriticalstress
1.Criticalstressandslendernessratiocriticalstressσcr
(a)Introducingtheradiusofgyration,weget
(b)CitatefollowingmarkTheequation(b)forthecriticalstresscanbewrittenas
whereλisadimensionlessquantity,calledtheslendernessratioorflexibilityofthecolumn.2.ApplicabilityofEuler'sformulatheconditionsfortheapplicabilityofEuler'sformulaare(c)Citatefollowingmark
Theconditions(c)fortheapplicationofEuler'sformulacanbewrittenas
Thistypeofbariscalledlargeflexibilityorslendercolumn.isonlyrelatedtothemechanicalpropertiesofthematerial,anddifferentmaterialshavedifferentvalues.TakingQ235low-carbonsteelasanexample,wehave,,andthen
ThisindicatesthattheEulerequationcanonlybeappliedtocompressionrodsmadeofQ235steelwhentheflexibility.3.CriticalcompressiveforceabovetheproportionallimitForthecommoncolumninengineering,TheircriticalstressexceedstheproportionallimitandcannotbecalculatedbyEulerequation.Weoftencalculationcriticalstressbytheempiricalformulaestablishedonexperiments,
whereaandbaretheconstantsrelatedtothemechanicalpropertiesofthematerial.Forcolumnsmadeofplasticmaterials:
or(d)theminimumvalueofslendernessratio:
Forcolumnsmadeofbrittlematerials:
Therefore,theempiricalformula(14-18)isapplicableunderλs<λ≤λp(orλb<λ≤λp).Thistypeofbariscalledmedium-flexibilitycolumnormedium-lengthcolumn.(Strengthissue)Whenλ≤λs
or≤λb,thecolumniscalledsmall-flexibilitycolumnorshortcolumn.Becauseitscriticalstresswillreachorexceedtheyieldlimitofthematerial,materialwillbedamagedduetoinsufficientstrengthwithoutbuckling,Forshortcolumnsmadeofplasticmaterials,iftheyarestillformallytreatedasastabilityproblem,theyieldstressσs
shouldbeusedasthecriticalstress.Insummary4.DiagramofcriticalstressByplottingtherelationshipbetweenthecriticalstressandtheslendernessratioofthecolumnwithinthethreeslendernessratiorangesinthecartesiancoordinatesystem(),wecangetthecriticalstressdiagramofthecolumn.sABCDOlplslssps2Eplcrs=2crabl=-scrs=ss5.Parabolicformulaanditsstressgeneraldiagram
Whenthecriticalstressexceedstheproportionallimit,weexpresstheparabolarelationshipbetweenthecriticalstressσcrandslendernessratioasfollows
China'ssteelstructurecodeprovidesaparabolicformulaestablishedbyourownexperiments:ForthecommonlyusedstructuralA2andA3steels,manganese16steelExampleAbarincompression,fixedatbothends,madeofA3steel,withacross-sectionalareaof32×102mm2.Calculatethecriticalloadwhenthecrosssectionisrectangularandcircular,respectively.F3msolution(1)rectangularsection
Minimumradiusofgyrationofthecrosssection:b2bSlendernessratioofthecolumn:
Thecolumnisaslendercolumn.F3mb2b(2)circularsectionby
weget
radiusofgyrationofthecrosssection:F3mb2bSlendernessratioofthecolumn:
Thecolumnisamedium-lengthcolumn.F3mb2bDiscussionthecriticalcompressiveforceisgreaterforcircularsectionsthanforrectangularsections,i.e.circularsectionsaremoreresistanttoinstabilitythanrectangularsections.tocalculatethecriticalcompressiveforce,itisnecessarytofirstcalculatetheslendernessratioofthecolumn,andthentoselecttheappropriateformula.Thestabilityconditionofthecolumnis
orexpressedusingsafetyfactorforstability
wherenwistheworkingsafetyfactorforstabilityofthecolumnand[nw]istheallowablesafetyfactorforstability.14.5CalculationofstabilityofcolumnsReferencevaluesof[nw]forseveralsteelcolumnsarelistedbelow.Columnsinmetalstructures---[nw]=1.8~3.0Screwsformachinetools---[nw]=2.5~4.0Tappetforlowspeedengines---[nw]=4~6Pistonbarforgrindingmachinecylinders-[nw]=4~6Liftingspirals---[nw]=3.5~5Note:Whenthereisapartialsectionweakeningofthebar,suchasoilholes,screwholes,etc.,asthecriticalcompressiveforceofthebarisdeterminedbythebendingdeformation,thelocalsectionweakeninghaslittleeffectonthevalueofthecriticalcompressiveforceandcanbeneglectedinthestabilitycalculation.Allcross-sectionalareasandminimummomentsofinertiaarecalculatedfortheunweakenedcrosssection.Strengthcheckmustbecarriedoutforcolumnwithpartialsectionwea
温馨提示
- 1. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
- 2. 本站的文档不包含任何第三方提供的附件图纸等,如果需要附件,请联系上传者。文件的所有权益归上传用户所有。
- 3. 本站RAR压缩包中若带图纸,网页内容里面会有图纸预览,若没有图纸预览就没有图纸。
- 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
- 5. 人人文库网仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对用户上传分享的文档内容本身不做任何修改或编辑,并不能对任何下载内容负责。
- 6. 下载文件中如有侵权或不适当内容,请与我们联系,我们立即纠正。
- 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
最新文档
- 2024-2025学年湖北利川文斗数学九年级第一学期开学考试模拟试题【含答案】
- 2023年南宁市第五十六中学学期编外教师招聘考试真题
- 2023年佛山市高明展达资产投资有限公司招聘考试真题
- 蛋鸡养殖技术
- 间隔是一份考卷
- 机械设计考卷2
- 特发性血小板减少性紫癫护理常规
- 汽车涂装技术 课件 项目六颜色微调
- 第七章-网络协议要点
- 2024年秋季学期新人教版三年级上册英语课件 Unit 4 Part A 第3课时 Letters and sounds
- 2024春季中国南水北调集团中线限公司招聘21人高频考题难、易错点模拟试题(共500题)附带答案详解
- 江苏省宿迁市2024年中考物理试卷【附参考答案】
- 个人简历模板 求职简历模板(12套完整版)
- 2024-2030年中国静脉识别行业发展分析及发展趋势与投资风险研究报告
- 垃圾分类知识竞赛200题(100道单选-100道多选-有答案)
- 【中考真题】重庆市2024年中考语文真题试卷A卷(含答案)
- 人教PEP六年级上册英语教学说课稿《 Unit 3 Part A 》第一课时说课稿
- 2024年太原市迎泽区小升初考试数学试卷含解析
- 食品行业数字化转型升级专题研究:技术标准与法规限制
- SL 736-2016 水文数据目录服务规范
- 陕西省咸阳市2025届高考仿真模拟生物试卷含解析
评论
0/150
提交评论