1CHAPTER10

STABILITYOFCOMPRESSEDCOLUMNS2CHAPTER10STABILIZATIONOF COMPRESSIVECOLUMNS§10–1CONCEPTSOFSTABILITYOFCOMPRESSEDCOLUMNS§10–2EULER’SFORMULAOFTHECRITICALFORCEOFSLENDERCOMPRESSEDCOLUMNS§10–3CRITICALSTRESSOFCOMPRESSEDCOLUMNSASSTRESSEXCEEDSPROPORTIONALLIMIT

§10-4STABILITYCHECKANDREASONABLESECTIONOFCOMPRESSEDCOLUMNS3§10–1

CONCEPTSOFSTABILITYOFCOMPRESSEDCOLUMNSTheload-carryingcapacityofstructuremembers：①Strength②Rigidity③Stability

Somestructuremembersinengineeringhaveenoughstrengthandrigiditybuttheyareunabletoworksafelyandreliably.STABILITYOFCOMPRESSEDCOLUMN4PSTABILITYOFCOMPRESSEDCOLUMN51、Stableandinstableequilibrium：1).instableequilibrium

STABILITYOFCOMPRESSEDCOLUMN62).StableequilibriumSTABILITYOFCOMPRESSEDCOLUMN73).Stableandinstableequilibrium

STABILITYOFCOMPRESSEDCOLUMN82、Lossofstabilityandthecriticalpressureofthecolumn：1).Idealcompressivecolumns：thematerialisabsolutelyideal；theaxisisabsolutelystraight；thecompressiveforceisabsolutelyalongtheaxisofthecolumn.2).Stableandinstableequilibriumofcompressivecolumns：STABILITYOFCOMPRESSEDCOLUMNInstableequilibriumStableequilibrium93).lossofstabilityofcompressedcolumn：4).CriticalpressureofcompressedcolumnsStableequilibriumInstableequilibriumCriticalstateCriticalpressure:Pcr

correspondingpressureSTABILITYOFCOMPRESSEDCOLUMNintermediatestate10

§10–2

EULER’SFORMULAOFTHECRITICALPRESSUREOF SLENDERCOMPRESSEDCOLUMNS

1、Criticalpressureforthecolumnwithtwohingedends:①bendingmoment：②ApproximatedifferentialequationofthedeflectioncurvePPxyPMSTABILITYOFCOMPRESSEDCOLUMNSupposethepressurehasreachedthecriticalvalueandthecolumnhasbeenintinybendingstateasshowninthefigure.Starttodeterminethecriticalforcewiththedeflectivecurve.PxL11③Solutionofthedifferentialequation：④Determinetheintegralconstants：

ThecriticalforcePcr

isthesmallestpressureundertinybending，thereforeweonlytaken=1andthecolumnwillbendabouttheaxiswiththesmallestmomentofinertia.STABILITYOFCOMPRESSEDCOLUMNThatis122、Applicationrangeoftheformula：3、Euler’sformulaofthecriticalpressureforthecolumnwithotherendconditions:1).Idealcompressivecolumns；2).Inlinearelasticrange；3).Theendsofthecolumnaresupportedbyhinges.—Lengthcoefficient（orconstraintcoefficient）Euler’sformulaofthecriticalpressureforthecompressivecolumnwithtwohingedends

GeneralformofEuler’sformulaofthecriticalpressureSTABILITYOFCOMPRESSEDCOLUMN130.5lTable10–1Euler’sformulaoftheslendercompressivecolumnundervariousconstraintconditionsSupportsTwohingedendsOnefreeendandonehingedendTwofixedendsOnefixedendandonefreeendTwofixedendsbutoneofthemismovablelaterally.

TheshapeofthedeflectivecurveinlostofstabilityPcrABlEuler’sformulaofthecriticalforcePcr

Lengthcoefficientμμ=1μ0.7μ=0.5μ=2μ=1PcrABlPcrABl0.7lCCDC:InflectionpointC、D:Inflectionpoint0.5lPcrPcrl2llC:InflectionpointSTABILITYOFCOMPRESSEDCOLUMN14Solution：Thedeformationofthecolumnisshowninthefigure.Theapproximatedifferentialequationofitsdeflectioncurveis:Theboundaryconditionsare:

Example1

Trytodeducethecriticalforceofthefollowingslendercolumnbytheapproximatedifferentialequationofthedeflectivecurve.PLxPM0PM0PM0xPM0STABILITYOFCOMPRESSEDCOLUMNLet15Inordertodeterminetheminimumcriticalpressure“k”mustbetheminimumvalueexceptzero,thatis：Thereforethecriticalpressureis：=0.5STABILITYOFCOMPRESSEDCOLUMNand16③CriticalpressureofthecolumnExample2

Determinethecriticalpressureofthefollowingslendercompressive column.=1.0，Solution：①Abouttheaxis

y，

twoendsarehinged：=0.7，②Abouttheaxis

z，theleftendisfixedandtherightendishinged：yzL1L2yzhbxSTABILITYOFCOMPRESSEDCOLUMN17Example3

Determinethecriticalpressureofthefollowingslendercompressivecolumns.Knowing:L=0.5mandE=200GPa.Solution：Fig.(a)Fig.(b)Fig.(a)Fig.(b)5010PLPL(45456)Equal-leganglesteelyzSTABILITYOFCOMPRESSEDCOLUMN18

§10–3

CRITICALSTRESSOFCOMPRESSEDCOLUMNSASSTRESS EXCEEDSTHEPROPORTIONALLIMIT

1、Basicconcepts1).Criticalstress：Meanstressinthecrosssectionofthecompressivecolumninthecriticalstate.3).Flexibility：2).Criticalstressforslendercompressedcolumns：22lpsEcr=Thatis:STABILITYOFCOMPRESSEDCOLUMNRadiusofinertiaFlexibility(orslendernessratio)ofthecolumn194).Divisionoflargeflexibilitycolumn：2、Calculationofthecriticalstressforthecolumnswithmiddleorsmallflexibility1).Empiricalformulawiththelinearvariation①AsP<<S

：STABILITYOFCOMPRESSEDCOLUMNisoneofthemiddleflexibilitycolumn.ThecolumnwithItscriticalstressmaybedeterminedbytheempiricalformula.Thecolumnswithiscalledthelargeflexibilitycolumn(orslenderclumn).ThecriticalpressureofthiskindofcolumniscalculatedbyEuler’sformula.Thecolumnswithiscalledmiddleorsmallflexibilitycolumn.ItscriticalpressureofthiskindofcolumncannotbecalculatedbyEuler’sformula.20③Totalfigureofthecriticalstress②AsS<

：bss=a-sl

PPEspl2

=STABILITYOFCOMPRESSEDCOLUMNThecriticalstressofthiscolumnisitsyieldlimit.isoneofthesmallflexibilityThecolumnwithcolumn.

212)憶Em醋pi吩ri已ca解l住fo竭rm誦ul臉a惕wi踏th槳t相he款p估ar定ab鑄ol莊ic躲v檢ar鳴ia梁ti券onTh撤e帶co糞mm御on賭f研or棄m價in達t濃hestr燦uct副ura影l(fā)e誓ngi僚nee朗rin母gi低s：①AsP<<s：②Ass<：STA騙BIL酒ITY洲OF滲CO廉MPR盤ESS西ED池COL都UMNForA3ste前el,A5ste吐el奴and午16疼Mn悉ste移el,AsThi愁sf爬orm亂ula禾is非us大ed俘to鉗det攪erm壞ine阿th孕ec訊rit撇ica使ls窩tre助ss.22Sol秧uti碎on：For鉤on納ep據(jù)iec寫eo劃fa容ngl嘴es冒tee稠l：Aft牢er克two械pi惰ece嫩so懸fa醉ngl榜es堡tee漿li扭sc純omb持ine概da昂ss籃how替ni科nt專he凍fig瞧ure敘,The惕ref套ore乒th戚ec彎rit妻ica濤lp胡res柱sur轉es懲hou甩ld拴be男det峰erm哈ine里db回y辱the證em劣pir純ica賠lf霉orm是ulayzST殖AB紡IL懸IT元Y丑OF抗C可OM任PR團ES艱SE弊D勒CO刷LU牽MNExa堅mpl糕e4A豈co貴mp腰re粉ss失iv板e糕co容lu堡mn來i惜s因ma恰de日f匙ro候m信tw壤o袖eq服ua栗l-易l問eg干a個ng咸le約s歌te育el淋o撿f貞5656哈8.截I漂ts殿t廉wo漢e喊nd也s煩ar憂e首bo箭th梳h螺in織ge職d.吸I釘ts微l吃en猴gt堤h管isL=1曠.5凱m.遺T犧he喇p劑re描ss旁ur蓄e澆isP=15驚0kN斷.T透he胡ang屬le格ste辟el鹽is碑A3ste爆el.奔Tr費yt樂od盜ete完rmi朗ne拜the組cr感iti及cal夠pr壁ess茶ure驗an武dt伏he膨saf厚ety渠co欣eff段ici償ent染of短st剃abi奶lit泊ynstby亦Eul保er’勞sf用orm壁ula共or矮th織ee摟mpi念ric祖al全for獅mul峰aw案ith衰th辜ep陽ara奇bol盲ic泄var兵iat痰ion堪.wit百ht享he兄par碎abo爭lic趣va額ria濫tio羽n.23Sa帖fe華ty新c簽oe雷ff雙ic懸ie釘ntST拾AB惱IL什IT爸Y更OF玩C饅OM晶PR尾ES片SE庫D倆CO縫LU多MN24§10掛–4STA冊BIL詳ITY層CH潑ECK論AN話DR嚇EAS淋ONA璃BLE和SE穿CTI采ONS妙OF原CO餡MPR盜ESS見ED包COL魚UMN甚S1、Per生mis圖sib孤le第str猶ess男fo欣rt柄he去com斜pre涂sse職dc撐olu尾mns即in伙st憶abi然liz頸ati渴on:1).De霧te襲rm橋in櫻e世th仗e設pe精rm關is疤si賭bl跨e演st夫re怒ss中b病y疊th擠e曲me久th顫od逃o義f冊th蛋e結sa近fe期ty賊c擔oe菌ff這ic嶄ie蠢nt派:2).De倦te獵rm粗in命e血th侵e保pe歷rm幻玉is蛋si廁bl端e趨st示re分ss詢b邪y霉th兵e院me若th錄od宋o丹f撕th嗓e午re趕du乏ct灑io粒n年co脫ef眼fi漲ci柏en慕t:2、Sta售bil箏ity聚co雖ndi津tio奧no慎ft銹he疫com斗pre扎sse視dc靜olu上mns病:ST統(tǒng)AB裝IL謠IT及Y癢OF暖C亦OM截PR房誠ES觸SE癥D藝CO何LU園MNReductioncoefficient.Itisrelatedtomaterialpropertiesandtheflexibilityofcolumns,and.25Exa圣mpl昌e5The米co昨lum型nABin腸the權si漿mpl筑ec馳ran怒es細how么ni汽nt侄he衡fig郊ureis性mad暢ef萍rom陪ci窮rcu顆lar巴de桂al.博It裝sl丑eng南th害isL=6深ma膜nd緊its躬di測ame喊ter事isd=0錘.3m啟.[]=竿11M框Pa.將Tr稅yt延od正ete脖rmi宗ne舉the逃pe貢rmi筍ssi燭ble逐pr兵ess鄙ure盯of廊th畜ec秒olu臂mn.So豎lu熟ti報on：Ap陜pl順y浙th趁e政me容th妖od潛o焦fred疲uct連ion棟co浮eff嫩ici遺ent.①Ma垮xim骨um潛fle竄xib距ili糊tyIn介pla優(yōu)nexy，=1.敬0In裂pla頌nezy，=2.干0T1ABWT2xyzOSTA支BIL黃ITY賭OF彈CO典MPR述ESS鵲ED洞COL畜UMN26②D遺et件er襲mi駝ne躲t紀he膜r允ed浪uc碰ti境on鬧c健oe我ff么ic范ie限nt③D企et遼er毀mi師ne退t層he識p應er檔mi幸ss磁ib括le僑p亭re傭ss亡ur零eST堅AB托IL衡IT裳Y棒OF挪C慢OM挑PR斑ES鍬SE饞D夾CO悲LU培MNFor法wo鋪ode眼nc淡olu城mn:烏as273、Rea馳son扮abl付es圾ect遲ion累of趙th蒸ec尺olu汽mn:It陡is民rea須son交abl占e.ST勺AB獸IL嶼IT霸Y涂OF蜓C柱OM涂PR讓ES株SE翠D聲CO泥LU形MN保國寺大殿的拼柱形式SakyamniwoodenpagodaYingzhouwoodenpagodaIt潑w廚as買b錢ui簡lt用i薪n速10距56憐w初it割h顫th病e稈do歡ub套le幅-s債le獅ev取e怨st翼ru館ct匹ur釣e.縣Th亡e利pa公go墳da拜p顛la高ne輛i輝s尋oc悼ta魯go挎n.私It含u如nd爽er嫂we丑nt狂t皮he波l眾ev汽el消8越e討ar往th凱qu栽ak鏟e戀of吼1飾30劉5.28Sol御uti燈on：For傳ea云ch育cha權nne饑ls將tee靠lo筋fn墻umb欣er唐10,殺the刃ce腔ntr梢oid估is損a健tt紅he翁poi嘉ntC1.Af漁te悠r購tw循o嘆ch晉an條ne來l免st成ee或ls瘦a英re貧c筑om司bi罪ne啦d，PLz0yy1z1C1aST少AB歇IL命IT托Y旦OF膝C洽OM念PR秤ES污SE笑D墊CO較LU匙MNExa套mpl版e6Th附e誤up敬ri覽gh慈t豪co棉lu臣mn瘦s推ho級wn陪i煮n向th設e泄fi爬gu骨re套i素s誼ma簡de墊f傳ro應m置tw育o蹄ch皺an魂ne茫l啟s罷te閱el怖s.It喊s趟le置ng因th螺i奸sL=6兇m.滴K威no忠wi書ng加th紙e腐ma耍te韻ri圈al透i乓s蹄A3st幣ee踐l,糟E乏=2輕00吃GP獻a謹a址nd.I境ts般l司ow勝er加e忙nd名i廊s哲fi粥xe有d舟an賤d潔it午s慮up印pe授r爭en俗d勢is賢s帶up敗po周rt壓ed音b淘y裹a宏sp跨he墓ri災ca治l沒hi丑ng叮e.廉H既ow獅m闖uc序h考is轟t激he牽v甩al揚ue厭o渣fawh祝en捐t快he毛c蜜ri宏ti昌ca導l趁pr醋es帝su選re舉o歌f膏th掉e濁up戴ri見gh選t仆co摧lu肝mn忍i營s薄th腎e承ma飾xi手mu李m?Th舞at該i竭s矮it滿i泊s私re蓮as提on共ab裕le兄a插sso29De銳te蠻rm級in窯e吳th洗e構cr劇it稈ic檢al魄f烏or遷ce：It懸is篩al啦arg光ef蠶lex艱ibi幸lit死yr衣od嫁.Th鄉(xiāng)豐ec艇rit播ica嚷lf鳥orc技ei尼sd已ete限r(nóng)mi抖nedby革Eul煎er’動sf牲orm服ula些.STA升BIL秩I