工程力学 第14章 压杆的稳定性与压杆设计

114§14-1§14-2§14-3*§14-42§14-5§14-6§14-7314§equilibriumconfigurationdisturbancestable(unstable)staticalcriterionforelasticstability(buckling)(loststability)(failurebybuckling)141141aFPw0FPw0FPFPw0AB141b4w00w00FPw0BD141bBC141bFPw0equilibriumpathABBDBCBDBCbifurcationofequilibriumconfigurationbifurcationofequilibriumpathcriticalpoint(criticalload)(bifurcationload),FP(bifurcationbuckling)§nn141bw0→0FP→FPcrFPFPcr142142a142b142c(x))()(PxwFxM=(a)522d)(d)(xxwEIxM-=(b)0dd222=+wkxw(141)EIFkxwwP2),(==(142)(141)kxBkxAwcossin+=(143)0)(,0)0(==lww⎭⎬⎫=⋅+⋅=+⋅0cossin00BklAklBA(c)cAB0cossin10=klkl(d)0sin=kl(144),(12,,)klnn=π=⋅⋅⋅/knl=π(142)22Pcr2nEIFlπ=(145)n=12Pcr2EIFlπ=(146)EII(c)1B0/knl=π(143)(bucklingmode)()sinnxwxAlπ=(147)A(amplitudeofbucklingmode)n(147)6()2Pcr2EIFlmπ=(148)ml143(effectivelength)mcoefficientof1ength143§7§76pPcrcrσ≤=σAF(149)crσ(criticalstress)pσFP149149complianceslendernessratioliµll(1410)IAIi=(1411)7nllpnllplsnlls144(141)144(limitedpointbuckling)§18mm24mm2773mm2896mm145d=18mml=2773mm8145146ab146147d=18mml=2773mm1489147148149d=24mml=2896mm()nPcrPFFnn149§1020λ-σcrλ-σcr14101410λ-σcrnnnλ-σcrEsσnscrσσnEsσ14111411λ-σcr111n22crλ=σE)(pλ≥λ(1412)n20crλ-σ=σk)(pλ≤λ1413,λ-σcr14121412λ-σcrl0scrσσ1413s0σσ2/scrσσ14122ps2λσ=k14141412pλλ2/scrσσs2p2σ=λE14151414Ek22s4σ=1416Q235MPa235s=σE=206Gpa16MnMPa343s=σE=206GPa(1413)(1416)pλ≤λ(Q23516Mnpλ132109nQ235scr=2350.0068l2Mpa132≤λ141712n16Mnscr=3430.00161l2Mpa109≤λ14182n22crλ=σE)(pλλ1419nscrσσbcrσσ)(sλ≤λ1420nλ-=σbacr)(psλ≤λ≤λ14211421abλ-σcr1413lplsspss(sb)lpls14lab1413λ-σcr141a/MPab/MPa332.237328.714542.150.19stabilitydesignn131412~1421ncriterionofdesignforstability[]stwnn≥(1422)wnFAFFncrPcrws==(1423)crsFA(1422)[]stn[]stn1.83.0[]stn5.05.5[]stn=2.83.2[]stn200p≤≤ll200,[]stn=1.92pll≤,[]3ppst)(818335llll-+=n(1424)[][]sjss=≤stcr(1425)[]stsj[][]stcrstnss=(1426)[][]ssjstcrn=(1427))(crcrlss[]stnj=j(λ)()(GBJ1788)(GBJ588)§1411414abdQ2351412d160mmE206GPa14141411.l=ml/iAIi/=,426444//ddidπ==πl(1414a)m1l5000mm15m20m4aldidml×==(1414b)m0.5l9000mm059m18m4.bldidml×==2p15m20m125132016m4.aldimll×=====()2626Pcrcr160102350006812510N4.FAs-⎡⎤π××==-×××⎢⎥⎣⎦3258910N2589kN=×=p059m18m1125132016m4...aldimll×=====,15N41016010)5.1120068.0235(6262crPcr⎥⎦⎤⎢⎣⎡×××××-=σ=-AFkN0003N1030003=×=ll2Q2352E142Q23514151415a1415bABl2300mmb40mmH60mmE205GPa1415142ABABABzy1415a0.1,,123=μ==bhAbhIz32hAIizz==6.132m)10(6032m)1023001(3-3=××××=μ=λ-zzil1415b5.0,,123=μ==bhAhbIy32bAIiyy==163-305230010m239948(4010m(.).)yyliml-×××===×22933Pcrcr22p2051040106010N1326.zEFAbhsl--⎛⎞π××××××==×=⎜⎟⎝⎠3276210N2762kN..=×=1431416l2.4m125mm×125mm×12mm23mmFP800kNQ235[]stn1.481416143AgAnlmlz112,2AAIIzz==1111122zzzzziAIAIAIi====11zziIA1z125mm×125mm×12mmiz1=3.83cm=38.3mmAl=28.9cm22.89×103mmiz=iz1=38.3mm66.62==zzilmlQ235141717σcr=2356.8×103×62.662×106Pa=208.3×l06Pa=208.3MpagAFcrPcrs==208.3×103×2×2.89×103N1202.2×103N1202.2kN50.1Pcrw==FFn[]stwnn2An2289×1042×23×103×12m25.288×103m25.288×103mm2[]ss=×=⎟⎟⎠⎞⎜⎜⎝⎛××==-153MPaPa10153Pa10288510800633nP.AF1441417ABNo.l4CDd20mmQ235ACDF25kNl11.25ml20.55mss235Mpans1.45[n]st1.81417144ABCDlABCm15.63kNmN1063.15mN)25.15.01025()30sin(331max⋅=⋅×=⋅×××==lFMokN65.21N1021.65N)30cos1025(30cos33PNx=×=××==ooFFNo.14Wz=102cm3=102×103mm3A21.5cm221.5×102mm218Pa105.211065.21101021063.154363maxmax⎟⎠⎞⎜⎝⎛××+××=+=--AFWMNxzsMPa2.163Pa102.1636=×=Q235[]MPa1621.45MPa235ss===nsssmax[s]smax[s]×100/[s]0.752CDkN2530sin2PPN===FFFCDo,mm54===dAIim1.0132110m105m55.00.13=××=-ilmlCD1417scr[2356.8×103×1102×106]Pa152.7×106Pa152.7MPaCD79.6MPaPa1079.6Pa1020104256623Nw=×=⎟⎠⎞⎜⎝⎛××××=-AFCDs[]8.192.179.6MPaMPa7.152stwcrw====nnss§nnn19(II)nn[]σ≤σ)(Srii[][],ww≤≤[][]stcrwstwnnnss≤≥,[][][]σσσφ≤σstcrwnwcrn141820141814191419ABABAB192001983105m6m54.2m17.25m565.4kN574.6ll2.2m0.5ml8l3.01.751.l12114641420d40mmQ235l270mm14201421I1421n()22(1422)(lateralbuckling)1422n(1423)1423n1424ab142423n1425ab1425b2126a1425nnIy=IznE24141ABCD142abcdFPmaxAFPmax(a)=FPmax(c)FPmax(b)=FPmax(d)BFPmax(a)=FPmax(c)=FPmax(b)=FPmax(d)CFPmax(a)=FPmax(d)FPmax(b)=FPmax(c)DFPmax(a)=FPmax(b)FPmax(c)=FPmax(d)142143AFPcraFPcrbFPcrcFPcrdBFPcraFPcrbFPcrcFPcrdCFPcrbFPcrcFPcrdFPcraDFPcrbFPcraFPcrcFPcrd143144FPcrπ2EI/(ml)225mAm2.0B0.7m2.0Cm0.5D0.5m0.7145AyBCCzDyz145146absasbscr(ascr(b,Asasbscr(ascr(bBsasbscr(a)scr(b)Csasbscr(a)scr(b)Dsasbscr(a)scr(b)147ABCD148[][]stcrPnAF=AA1B2Cl2DA26**149l3md20mma60mmQ235E200GPaFP1491410ABd40mml800mmQ23512FP70kNAB[n]st=2.03No.18[s]160Mpa14101411l50mmd6mm40Crt160ºCα1125×106/°Cα275×106/°C40Crss600MPaE210GPa[n]st2.0t260ºC14111412ABd80mmBCa70mmQ235E200GPaABCl3m,[n]st2.51412271413ABACd80.0mmBC4mQ235[n]st2.01FP**2FPCAqFPq14131414d40mma1mQ23S[n]st1.8l2FP114141415Q235E200GPass240MPaq=40kN/m63mm×63mm×5mm()1415*1416t120ºCl=6mmQ235E200GPaα12.5×106/°C1416