Namuhla, i-Savo Kusić igxile emafasiteleni okhuni, amafasitela okhuni-aluminium, amafasitela ngokwezifiso, umnyango kanye nezicelo zekhotheshini. Lo mbhalo usalokhu uyingobo yomlando yethiyori yesakhiwo futhi awusona isithembiso sokuklama okumile, izibalo zokuzinza noma ukuvuselelwa kwesakhiwo.
Isibalo esihlukile sokugoba ibhande
Kucatshangwa ukuthi zonke izinhloso zanelisekile, ikakhulukazi
- ukuthi induku iqonde ngokuphelele
- ukuthi umthwalo P usebenza ku-shaft yenduku
- ukuthi ama-voltage ahlala engaphansi komkhawulo wokulinganisa
- ukuthi ama-deformations amancane ngokwanele ukuvumela ukwenza lula okujwayelekile kwezibalo.
Uma induku igoba ngaphansi kwesenzo samandla P futhi mhlawumbe futhi ngenxa yomthwalo q osebenza ohlangothini oluphambene, khona-ke izimo ezaziwayo zokulinganisa phakathi kwemithwalo yangaphandle kanye namandla esigabeni kufanele zaneliswe. Ngaphandle kwalokho, kungokwemvelo ukuthi kufanele kube nobudlelwane obaziwayo phakathi kokuguga namandla esigabeni.
Sl. 1 - Ibhalansi
Emkhiwaneni.1i-elementi yenduku egobile yobude ds imelwe. Umthamo we-tangent v’ mncane ukuze kufike kuhlelo lwesibili, ubude be-elementi bungasethwa bulingane ne-projection ds=dz. Ukwengeza, ekuhlolweni okukhulunywe ngakho ngezansi, izimo lapho P eshintsha njalo eduze kwe-eksisi yenduku ngeke anakwe, okungukuthi amandla avamile N≈P=const. Ezimweni ezintathu zokulingana ezitholakalayo zesici, inani elithile lamalungu lingase linganakwa. Okwezimpawu nezimpawu ezivela emkhiwaneni.10hlala ngaleso sikhathi ekulinganisweni kokuqala kwesibalo
N=P, M’=Q, Q’=-q+Pv’’.
Amandla aguquguqukayo angaqedwa, bese M’‘=Pv’’-q ibambe. Umthetho ka Bernoulli\ M=-EI/R uhlinzeka ngesibalo esingekho. Ijika le-eksisi yenduku, njengenjwayelo, kusukela kumbono wokugoba ugongolo, lenziwa lula ngale ndlela elandelayo.
Kungenzeka ukuqeda umzuzu wokugoba; futhi nge-M=-EI*v’’ isibalo esihlukanisayo sentonga egobile
(EI*v’‘)’’ + Pv’’ = q. (1)
Esimeni esikhethekile sokuqina okugobile okungapheli EI=const. isibalo (1) yenziwe lula, ngakho-ke
EIv’‘’’ + Pv’’ = q. (1a)
Eq. (1) kanye (1a) ziwumugqa kokuthi v ngokuvumelana nokwenza lula okubalulwe.
Empeleni, isimo esilula sokuqina okungaguquki kokugoba EI=const. ibaluleke kakhulu, esebenza njengesisekelo sezibonelo ezingezansi.
Icala lika-Euler
Ukucatshangwa kwecala lakudala lokubopha, ngaphandle kwemibono eshiwo: p=0, ngakho N=const.=P, kanye nokuqina okungaguquki ngaphandle kokungqubuzana, okuku-eksisi yenduku. Isibalo esijwayelekile esihlukanisayo (1a) silandelana ngokwesine, sinomugqa futhi siyafana ngenxa yokuthi q=0. Isixazululo esijwayelekile sithi:
v(z) = A_∙_sin__αz + B∙cosαz + C∙z + D, (2)
lapho A, B, C, D okuyizinto ezine zokuhlanganisa ezingaguquki. Ipharamitha inosayizi olinganiselwe ngenxa yokuqagela okuthi EI≠0. Ngaphezu kwalokho, α≠0 kwe-P>0.
Fig. 2
Njengoba i-equation yomugqa womugqa (1a) ifana, le nkinga iphinda ixazululwe ngokuthi n_∙__v(z)_, lapho okuthi n kuwukungashintshi okuqhubekayo. Kakade kusukela kusigaba sezibalo se-equation ehlukile (1a) kungabonwa ukuthi ubukhulu bokugoba v abunakunqunywa. Ukuze unqume ukuhlanganisa okungaguquki A, B, C, D kucala elithi Euler\ (fig. 2) izimo ze-contour ezilandelayo ziyatholakala: Ngenxa yokusekelwa okushiwo, umzuzu wokugoba unguziro kuzo zombili iziphetho zenduku, i.e.
v=0 kanye ne-M=0 kwe-z=0
v=0 kanye ne-M=0 kwe-z=s.
Ngenxa yokuthi M=-EI_∙__v’‘_, kulandela ukuthi M=0 inani lika-v’’=0, ngakho imibandela emine yekhonta ifundeka kanje:
v=0 kanye nokuthi v’’=0 kokuthi z=0
v=0 kanye nokuthi v’’=0 kokuthi z=s.
Isixazululo (2) ngaleyo ndlela siholela ezilinganisweni ezine zomugqa:
Eyodwa.3
Lezi zibalo ezine ze-homogeneous zinesokuqala esibizwa ngesixazululo esincane A=B=C=D=0, esihambisana nokuthi v(z)=0. I-ungont rod v_(z_)=0 iyindawo evamile yokulinganisa yokungakhethi, ikakhulukazi amanani amancane omthwalo P. Lapha, sinentshisekelo kakhulu kuzimo lapho induku igoba khona, okungukuthi v(z)≠0. Izibalo (3) azinaso isixazululo esingasho lutho ngokuthi A, B, C, D ≠ 0 kuphela uma isicacisi se-coefficient silingana noziro.
Eyodwa. 3a
Kusukela ezimeni zokubopha Δ = α4∙s∙sinαs =0 mog ukubala amanani abalulekile PK womthwalo. Lesi sibalo saneliswa kuphela uma sin αs =0, okungukuthi αKms = mπ nge-m=1,2,3,… Amanani ahambisanayo we-Euler's buckling force athi PKm=EI_∙_α2Km = EI(mπ/s)2. Ngevelu ngayinye ye-m, kutholakala amandla athile okugoqa. Ezimweni eziningi, amancane kuphela kulawa mandla anentshisekelo, futhi lokho
PK1=EI(π/s)2.
Enguqulweni elandelayo, kuzoba khona inkomba m, noma 1 eshiywe kakhulu, ibhekisa kokuphansi kwemithwalo ebucayi. Okuthakazelisayo umugqa wokunwebeka ohambisanayo. Kusuka kokukodwa (2) okuhlala njalo B=C=D=0 kutholiwe. Umbandela osele uthi A∙sin__αs=0. Ulayini wokubopha okunwebekayo wekesi elingenhla elithi m=1, ngakho-ke, v(z) = A_∙sinπz/s_, lapho u-v(z) ethi αK resp. PK i-eigenfunction ehambisanayo. Ubukhulu be-A buhlala bungakanqunywa.
Ake siphinde sibheke yonke indawo yomthwalo. Ngamanani angenasisekelo we-P, ikakhulukazi amabutho P_<_PK1 ithi v(z)=0, induku ihlala iqondile.
Ngokulayisha “kancane kancane” u-P kushiwo kuthiyori yakudala yokuzinza uchungechunge olulandelanayo losayizi P, lapho ukuzinza kwekesi lomthwalo elibhekiwe kuzohlolwa ngokuphindaphindiwe kuleveli ngayinye yokulayisha P. Ngesikhathi sokuhlolwa kokuzinza, usayizi we-P uhlala ungashintshile. Uma isimo esisodwa P sinqunywa ukuthi sizinzile, khona-ke umthwalo ukhuphuka ngokuthile, ngakho-ke ukuhlolwa kokuzinza kuyaphindwa. Lokhu kuzimisela okunembile kuyadingeka ukuze kungabikho ukudideka ngohlobo lomthwalo, okuyisisekelo somphumela we-Shanley, lapho kukhona ukwanda komthwalo ngenkathi iziphazamiso ezincane zisebenza ngesikhathi esifanayo. Uma kuhlolwa ukuzinza ngokusho Shanley, endaweni ye-plastic buckling, amandla amancane e-bifurcation ayenzeka kunendlela yakudala yokucatshangelwa.
Ukucatshangelwa okulandelayo kuthatha indlela yokuqala yokukhulisa umthwalo: Ngokukhula kancane kancane komthwalo P ngalo mqondo, i-PK ifinyelelwa.1=EI_∙(π/s)_2okokuqala ukuya endaweni eyodwa yokuhlukanisa kabili ye-elastic equilibrium, enomugqa olula we-sine njengomugqa wokugoqa okunwebekayo
v1(z)=f_∙_sin__π__z/s.
f=A isho i-amplitude enganqunyiwe yomugqa wokunwebeka. Okokulayisha _P*__>_PK1 ihambisana nethiyori yomugqa Δ≠0, okusho ukuthi ibhalansi ayinakwenzeka lapho. Okokubopha okuphezulu kuphela PK2 njll. kuzoba Δ= futhi0futhi kanjalo v(z)≠0. Imbangela yalesi siphetho sethiyori, esingafani nokuziphatha kwangempela, ilele ekwakhiweni komugqa kwesibalo esihlukile.
Ukubopha izinti ezicindezelwe ezakhiwe izingxenye ezimbili
Sizilinganisela ezintini ezicindezelwe ezinezingxenye ezimbili, futhi ezintini ezicindezelwe ezakhiwe izingxenye ezintathu noma ngaphezulu, kuyasebenza ubuhlobo obufanayo.
Sl.3
Ngokuya ngohlobo lokuxhuma okuphambene, izinduku ze-lattice ziyehluka (isib.3a) kanye nezinduku zefreyimu (isib.3e). Ngaphandle kwezibopho eziphambanayo, ingxenye ngayinye yenduku yayizogoba yodwa (isib.3b). Uma ukugcwaliswa kubuthakathaka, khona-ke induku eyodwa yebhande ingagoba ngaphandle kokugoba induku iyonke. Owomkhiwane wenduku ye-lattice.3c ingafaniswa: kwesokunxele sohlelo olunamaphiko, kwesokudla kumabhande aqhubekayo; nekhiwane lenduku yohlaka.3f kwesokunxele ngamashidi okuxhumana athambile, kwesokudla amashidi okuxhumana aqinile. Ukusuka lapho, kungabonakala ukuthi ukuncipha λ_ kunethonya elikhulu1__=imizuzwana1/i1_. Engxenyeni ngayinye yenduku, ngabekwa1=F1_∙_i12, s1 ukuhlukaniswa kwensimu (isib.3), ngi1 ibhekisela ekseni1-1(bheka Fig.4).
Sl.4
Amandla abophayo wezinti ezihlanganisiwe ezihambisana nemigqa yokugoqa enwebekayo yekhiwane.3d, noma3g bekuyisihloko sophenyo lwetiyori oluphelele. Uma induku beyakhiwe ngezigaba ezihlukile (isb. ngombambo) noma uma ukuxhumana okuvundlayo bekuvame kakhulu kangangokuba induku egotshwe ezingxenyeni ezimbili ayihlukani nomphumela wayo entweni eyodwa, bekuyoba σ__K_=_π__2__E/λ__y__2. Lapho lichaza λ__y=sK/iy (Fig.4). Kodwa-ke, lokhu kucindezeleka kwe-buckling ngeke kufinyelelwe ngokugcwele ekusebenzeni ngoba akukho ukuxhumana okwanele okuphambene nalokho, futhi kuyanwebeka. Isilinganiso esigculisayo ezikhathini eziningi sitholakala uma sibeka ngokuvumelana nesiphakamiso F. Engesser\a ukuthi ingcindezi egobayo yenduku ecindezelwe eyakhiwe izingxenye ezimbili iyalingana
σ__K_=_π__2__E/λ__yi__2.
I-λyi yilokho okubizwa ngokuthi ubuncane obuhle.
Izibopho eziphambanayo kufanele zanele ngenani futhi ziqine ngokwanele ukuphoqa izingxenye zombili zenduku ukuba zisebenze ndawonye. Ngaphandle kokucindezeleka kwesibili, izixhumanisi eziphambeneyo zihlala zingacindezelwe inqobo nje uma induku ihlala iqondile. Kungokugoba kuphela lapho kufanele bathole amandla, futhi kufanele badlulisele amandla anqamulayo akhuphuka ngokugoba. Uma kwenzeka kuwumugqa wokunwebeka
amandla aguquguqukayo ku-I=const. liyalingana
Q = M’ = - EI∙v’‘’.
Ngomkhiwane. 5 m ingafundwa ngokushesha Q(z)=PK∙v’(z). Inani eliphakeme kakhulu ekugcineni kwenduku lithi
Sl.5
Izixhumanisi eziphambanayo kufanele zikwazi ukwamukela la mandla aguquguqukayo. Njengoba umthwalo ohambisanayo ungamandla okugoqa PK, kwanele uma umkhawulo othwala umthwalo ufinyelelwa kulokhu futhi ekuxhumekeni okuphambene - okungukuthi kanyekanye nayo yonke induku. Ngokusobala, bekungeke kube khona inzuzo uma izixhumanisi eziphambanayo bezinosayizi oqine kakhulu kunokudingekile ukuze kwamukelwe Q0. Ngakolunye uhlangothi, izixhumanisi eziphambanayo ezibuthakathaka kakhulu zingabangela ukuthi induku ihluleke ngaphambi kwesikhathi.
Izibalo ezingenhla azikwazi ukusetshenziswa ekuqaleni ngoba ubukhulu bomcibisholo ogoqayo abaziwa. Induku isekulinganisweni okungenandaba ngaphansi kwe-PK futhi ingakwazi, okungenani ngezinga lokuqala lokuqagela, ithathe inani elingafanele. Ngakho-ke iwusizo, ngokwesiphakamiso sika-F. Krohn-a, ukuze kunqunywe ubukhulu baleyo bhande maxv lapho induku egobile ilahlekelwa amandla ayo okuzala ngenxa yokwehluleka ngenxa yokugoba. Lokhu kuzokwenzeka uma iphuzu lokukhiqiza lifinyelelwa ohlangothini lwe-concave lokugoba (Fig.6). Ukugoba kwalokhu kukhula ngokuqhubekayo futhi izikhathi zokugoba ezihambisanayo azikwazi ukusetshenziswa.
Fig. 6
Ukucindezeleka okuphelele ohlangothini lwe-concave lokugoba luhambisana nomkhiwane.6for ekahle impahla plastic
Kusuka lapho kufika i-swagger enkulu kakhulu esakwazi ukubekezelelwa
ubuningiQ0 ingamandla aguquguqukayo ahambisanayo ekugcineni kwenduku.
Uma kusetshenziselwa ifomu elithi σK Tetmeier\ bese ubeke σF=3,1t/cm2kanye sK=e/2∙ λy okuhambisana nokuthi iy≈e/2, bese Krohn's inani le-maxQ litholwa0=F/28(Q kokuthi t, F ngamasentimitha2).
Ngemigqa engafanele iyatholakala
ubuningiQ0=μK∙SK.
I-coefficient μK iyashintsha ngobuncane benduku, ibuye incike ezintweni ezibonakalayo. Uma isibalo esingenhla sihlukaniswa nge-coefficient yokuphepha ngokumelene ne-buckling ingabhalwa
dozv__Q=μ∙dozvS.
Ukubopha amasistimu emigqa
Endabeni yamasistimu alayini, kuvamile ukunquma ubude bokugoba u-sK=β∙h wenduku elenga kuzo zombili iziphetho futhi amandla ayo okugoba angu-PK=π ukuze kuqhathaniswe.2EI/s2K inkulu nje. Ukuqhathanisa “ubude bokubopha” kuzonikezwa izibonelo ezingezansi.
a) Induku iboshwe kuzo zombili iziphetho (EI=const.)
Sl.7
Odabeni lomkhiwane.7ye-buckling ithi PK=4π2EI/h2=4,0∙PE, ukuqhathanisa “ubude be-buckling” sK=h/2kanye nomugqa wokubopha okunwebekayo v(z)=D(1- ngoba2πz/h).
b) Induku ekhululekile ekugcineni (isib.8)
Sl. 8
Kucatshangwa ukuthi isiqondiso sika P asishintshi njengoba induku igoba. Bese (EI=consst.)
PK=π2EI/4h2=PE/4, sK=2h
v(z)=D(1-cosπz/2h).
Izimo zekhonta yesiphetho samahhala z=h sezilapha
M=0 noma v’’=0
Q=M’-Pv’=0 noma v’’’+a2∙v’=0.
c) Induku iboshwe ekugcineni (fig. 9)
Sl.9
Ngezimo ze-contour
v=0 kanye ne-v’=0 kwe-z=0
v=0 kanye nokuthi v’’=0 kokuthi z=h
itholwa EI=const. isimo sokubopha
sin_αh - αh∙cos_αh=0_,_
noma okulinganayo
tgαh=αh. (4)
Izimpande zalesi sibalo esidlulela phezulu zitholakala kumathebula K. I-Hayashi's. Amandla okubopha aphansi ahambisana namathebula PK1=4,49342/ h2∙ EI =2,046π2EI/h2.
Ubude bokulinganisa obuqhathaniswayo ngu-sK = h / √2,046=0,699h.
d) Induku eboshwe ngakolunye uhlangothi ngokuphazamiseka (isib.10)
Fig. 10
Sizocubungula indaba yokususwa kwethiphu yosayizi ongashintshi. Izimo ze-contour zalokho ziyi
v=0kanye v =0 kwe-z=0,
v=v1futhi v’=0 kwe-z=h.
Kuyinkinga yokulingana ngomugqa wokunwebeka
Idinomineyitha yetemu ngaphambi kokuthi {}-abakaki izoba nguziro futhi ukubopha kukhuphuke ngaphezu kwayo yonke imikhawulo uma P→PK (bona umbandela we-buckling Eq. (4)). Kuyisici sokuziphatha kwenduku ukuthi ukugoba phakathi nendawo ephansi yenduku ekuqaleni kuncane ngomthwalo okhulayo P futhi ushintshe uphawu lwabo kuphela ngemithwalo ephakeme. Sl.10ibonisa ukugoba ngobude obuyingxenye.
e) Induku eqhubekayo phakathi kwezinkambu ezimbili
Sl.11
Kuzo zombili izindawo zokuxazulula (isib.11) sizosebenzisa isilungiselelo
Futhi_. v(z1) = A1∙synαz1+B1∙cosaz1+ C1∙z1+D1,_
II. v(z2) = A2∙synαz2+B2∙cosaz2+ C2∙z2+D2.
Ukwenza kube lula makube (EI)1=(EI)2=EI. Ukuze uthole ama-constants ayisishiyagalombili A1, …, A2, … isebenza njengezimo zekhonta
v=0kanye v =0 kwe z1=0,
v=0futhi v’=0 kwe z2=h,
kanye nezimo zezinguquko phakathi kwezindawo I kanye ne-II
v=0 kwe z1=h i ye-z2=0,
v’’ (z1=h)=v’(z2=0),
v’’ (z1=h)=v’’(z2=0).
Izibalo ezinemibandela zinikezwe etafuleni elingezansi
Ngokusetha i-determinant ye-coefficient elingana noziro, isimo sokubopha siyatholwa
Ubude bokugoba sK=0,878_h_, ngenkathi ikhiwa.35noma β=0,699. Amandla okubopha anciphile uma kuqhathaniswa2,046:1,297ngenxa yokwengezwa kwenkundla yesibili ehingiwe. Uma enye inkambu ibincinzwe phezulu, amandla okugoqa ayezokhuphuka futhi afike esiqongweni.2,04_PE_.
Lokhu kwethulwa komlando kwamafomula akuyona iphrojekthi noma ubufakazi bokusimama kwesakhiwo. Ukuqagela kwakudala kwelungu eliqondile elikahle, umthwalo omaphakathi nendawo enwebekayo akubandakanyi ngempela ukungapheleli kwasekuqaleni, ukucindezeleka okuyinsalela, ukugqama, ukuqina koxhumo, ukungahambisani kwezinto ezibonakalayo, ukuzinza kwezinhlangothi noma, lapho kufanele, umlilo, isenzo sokuzamazama komhlaba nezigaba zokuhlanganisa. Ukubalwa kwesakhiwo sangempela kufanele kwenziwe unjiniyela womphakathi ogunyaziwe ngokuhambisana nemithethonqubo namazinga asebenzayo.