Nhasi, Savo Kusić inotarisa pa mahwindo emapuranga, wood-aluminium Mahwindo, mahwindo etsika, gonhi uye zvikumbiro zvekotesheni Ichi chinyorwa chinoramba chiri mudura redzidziso yezvimiro uye hachisi chekupa che static dhizaini, kuverenga kwekugadzikana kana kugadzirisa kwemaitiro.
Yakasiyana equation yekukotama buckling
Zvinofungidzirwa kuti zvese zvinofungwa zvinogutsikana, kunyanya
- kuti tsvimbo yakatwasuka
- kuti mutoro P unobata musheti yetsvimbo
- kuti ma voltages anoramba ari pasi pemuganhu wekuenzana
- kuti deformations idiki zvakakwana kubvumira zvakajairwa kurerutswa kwekuverenga.
Kana tsvimbo yakakotama pasi pechiito chesimba P uye zvichida zvakare nekuda kwemutoro q unobata munzira inotenderera, ipapo iyo inozivikanwa mamiriro ekuenzanisa pakati pemitoro yekunze uye masimba ari muchikamu anofanira kugutsikana. Kunze kwezvo, ndezvemuzvarirwo kuti panofanirwa kuve nekuzivikanwa hukama pakati pekukanganisa nemasimba muchikamu.
Sl. 1 - Balance
Mumuonde.1chinhu chetsvimbo yakakotama yehurefu ds inomiririrwa. Mutsetse we tangent v’ idiki zvekuti kusvika kune yechipiri kurongeka, kureba kwechinhu kunogona kusetwa kuti kuenzane nefungidziro ds=dz. Mukuwedzera, mune zviedzo zvataurwa pasi apa, zviitiko umo P inogara ichichinja pamwe chete neaxis yetsvimbo haizofungidzirwi, i.e. normal force N≈P=const. Mune matatu equilibrium mamiriro aripo echinhu, imwe nhamba yenhengo inogona kuregeredzwa. Zvemaka nezviratidzo zvemuonde.10ramba ipapo uri mukufungidzira kwekutanga kweequation
N=P, M’=Q, Q’=-q+Pv’’.
Simba rinochinjika rinogona kubviswa, uyezve M’‘=Pv’’-q inobata. _Mutemo wa Bernoulli\ M=-EI/R unopa equation isipo. Iyo curve yeaxis yetsvimbo, semazuva ese, kubva padzidziso yekukotama kwedanda, inorerutswa sezvinotevera.
Zvinogoneka kubvisa nguva yekukotama; uye ne M=-EI*v’’ musiyano wemudonzvo wakakombama
(EI*v’‘)’’ + Pv’’ = q. (1)
Panyaya yekugara uchikotama kuoma EI=const. iyo equation (1) yakarerutswa, saka iri
EIv’‘’’ + Pv’’ = q. (1a)
Eq. (1) uye (1a) ane mutsara muna v maererano nekurerutsa kwataurwa.
Chaizvoizvo, chikesi chakareruka chekugara uchikotama kuoma EI=const. inonyanya kukosha, inova hwaro hwemienzaniso iri pazasi.
nyaya yaEuler
Mafungiro echinyakare chebuckling ndeaya, kunze kwezvakataurwa: p=0, saka N=const.=P, pamwe nekuomarara kwekusingaperi pasina kukwesha, izvo zvinogara paakiso yetsvimbo. Yakajairika musiyano equation (1a) ndeyechina kurongeka, mutsetse uye homogeneous nekuda kwe q=0. Mhinduro yacho ndeye:
v(z) = A_∙_sin__αz + B∙cosαz + C∙z + D, (2)
apo A, B, C, D ndiwo mana ekubatanidza anoramba aripo. Iyo parameter ndeye saizi inokwana nekuda kwefungidziro EI≠0. Uyezve, α≠0 ye_P_>0.
Fig. 2
Sezvo mutsara wekusiyanisa equation (1a) iri homogeneous, dambudziko iri rinogadziriswawo ne n_∙__v(z)_, apo n inongogara isingaite. Nechekare kubva muchikamu chesvomhu chekusiyanisa equation (1a) zvinogona kuonekwa kuti ukuru hwekukotama v haugone kutsanangurwa. Kuti uone zvigadziriso zvekubatanidza A, B, C, D mu Euler\ kesi (fig. 2) zvinotevera contour mamiriro aripo: Pamusana pekutsigirwa kwekutaura, nguva yekukotama ndeye zero kumativi ose emudonzvo, i.e.
v=0 uye M=0 ye z=0
v=0 uye M=0 ye z=s.
Nekuda kwe M=-EI_∙__v’‘_, zvinotevera kuti M=0 kukosha kwe v’’=0, saka mana contour mamiriro anoti:
v=0 uye v’’=0 ye z=0
v=0 uye v’’=0 ye z=s.
Mhinduro (2) nokudaro inoguma mumitsetse mana equation:
Poshi.3
Aya mana ehomogeneous equations ane yekutanga inonzi trivial solution A=B=C=D=0, inofambirana ne v(z)=0. Iyo isina kubhenda tsvimbo v_(z_)=0 ndiyo yakajairwa chinzvimbo chekupokana, kunyanya madiki emutoro P. Pano, isu tinonyanya kufarira mamiriro ayo tsvimbo inokotama, i.e. kuti v(z)≠0. Maequation (3) haana mhinduro diki ne A, B, C, D ≠ 0 chete kana chinomisikidza checoefficient chakaenzana ne zero.
Poshi. 3a
Kubva kumamiriro ekugadzirisa Δ = α4∙s∙sinαs =0 mog kuverenga zvakakosha PK yemutoro. Equation iyi inogutsikana chete kana sin αs =0, kureva αKms = mπ ine m=1,2,3,… Nyedzo dzinoenderana dze Euler's buckling force ndeidzi PKm=EI_∙_α2Km = EI(mp/s)2. Pahukoshi hwega hwega hwem, imwe simba rebuckling rinowanikwa. Muzviitiko zvakawanda, idiki chete yemasimba aya ndeyekufarira, uye izvo
PK1=EI(π/s)2.
Mushanduro inotevera, pachava nendekisi m, kana 1 kazhinji inosiiwa, ichitaura kune yakaderera yemitoro yakaoma. Inonyanya kufarira ndiyo inowirirana elastic line. Kubva kune imwe (2) zvinoramba zviripo B=C=D=0 zvinowanikwa. Chimiro chasara ndeichi A∙sin__αs=0. Elastic buckling line yekesi iri pamusoro m=1, saka, v(z) = A_∙sinπz/s_, papo v(z) ndeye αK resp. PK inoenderana eigenfunction. Hukuru hwe A hunoramba husina kutsanangurwa.
Ngatitarisei imwe nzvimbo yekutakura zvinhu. Pamaruramiro ehumwe P, kunyanya kumauto P_<_PK1 ari v(z)=0, tsvimbo inoramba yakati twasa.
Ne “zvishoma nezvishoma kuwedzera” kurodha P inorehwa mudzidziso yekare yekugadzikana mutsara unoteedzana wehukuru P, uko kugadzikana kweakacherechedzwa kesi yemutoro inoedzwa zvakare uye zvakare kune yega yega nhanho yekurodha P. Munguva yekuyedza kugadzikana, saizi ye_P_ inoramba isina kuchinjika. Kana imwe nyika P yakatsunga kuva yakagadzikana, ipapo mutoro unowedzera nechimwe chinhu, uye naizvozvo kuedza kwekugadzikana kunodzokororwa. Uku kutsunga kwakarurama kunokosha kuitira kuti pasave nekuvhiringidzika nemhando yemutoro, iyo ndiyo nheyo ye Shanley\ effect, umo mune kuwedzera kwemutoro apo kukanganisa kuduku kuri kuita panguva imwe chete. Paunenge uchiongorora kugadzikana zvinoenderana ne Shanley, munzvimbo yepurasitiki buckling, diki bhifurcation simba rinoitika pane neiyo yekare nzira yekufunga.
Izvi zvinotevera kufunga zvinotora nzira yekutanga yekuwedzera mutoro: Nekuwedzera zvishoma nezvishoma mumutoro P mupfungwa iyi, PK inosvika.1=EI_∙(π/s)_2kutanga kusvika kune imwechete bifurcation point ye elastic equilibrium, ine mutsara wesine wakapfava semutsara unomonereka
v1(z)=f_∙_sin__π__z/s.
f=A inoreva iyo isina kutsanangurwa amplitude yeelastic line. Zvekutakura _P*__>_PK1 inoenderana neiyo linearized theory Δ≠0, that is balance hazviite ipapo. Chete chepamusoro buckling masimba PK2 zvichingodaro. ichava Δ= zvakare0uye nokudaro v(z)≠0. Chikonzero cheiyi theoretical mhedziso, isingaenderane nehunhu chaihwo, chiri mumutsara wekusiyanisa equation.
Buckling yetsvimbo yakatsikirirwa inoumbwa nezvikamu zviviri
Tinozviganhurira timene kuzvitanda zvakatsikirirwa zvine mativi maviri, uye nokuda kwetsvimbo dzakatsikirirwa dzinoumbwa namativi matatu kana kuti anopfuura, ukama hwakafanana hunoshanda.
Sl.3
Zvinoenderana nerudzi rwekubatanidza kuchinjika, tsvimbo dzelatisi dzinosiyana (fig.3a) uye matanda emapuranga (fig.3e). Pasina zvisungo zvemuchinjikwa, chikamu chega chega chetsvimbo chaizokotama chega (muonde.3b). Kana kuzadza kunenge kusina simba, ipapo tsvimbo imwe yebhanhire inogona kukotama pasina kupeta tsvimbo yose. Muonde womuonde wetsvimbo.3c inogona kufananidzwa: kuruboshwe kune hinged system, kurudyi kune mabhandi anoenderera; neonde retsvimbo;3f kuruboshwe kune mashizha ekubatanidza akapfava, kurudyi kune akaomarara ekubatanidza mapepa. Kubva ipapo, zvinogona kuonekwa kuti slimness λ_ ine simba guru1__=s1/i1_. Kudivi rega rega retsvimbo, ndakaiswa1=F1_∙_i12, s1 ndiko kuparadzaniswa kwemunda (Fig.3), i1 zvinoreva axis1-1(ona Fig.4)
Sl.4
Simba remukuyu rinoenderana nerasstic buckling lines emuonde.3d, kana3g yaive nyaya yekuongorora kwakazara kwedzidziso. Kana tsvimbo yacho yaiumbwa nezvikamu zvakasiyana (semuenzaniso nembabvu) kana kuti mibatanidzwa yakachinjika yaiwanzoita zvekuti tsvimbo yakakombama kubva pamativi maviri haisiyane pakuita kwayo kubva patsvimbo imwe chete, zvino yaizova σ__K_=_π__2__E/λ__y__2. Ipapo zvinoreva λ__y=sK/iy (Fig.4) Zvisineyi, kushushikana uku hakuzosvikiki zvizere mukuita nekuti hakuna kukwana kuchinjika kwakabatana kune izvo, uye zvakare ari elastic. Kufungidzira kunogutsa kazhinji kunowanikwa kana tikaisa zvinoenderana nechirongwa F. Engesser\a kuti kubhenda kushushikana kwetsvimbo yakadzvanywa inoumbwa nezvikamu zviviri zvakaenzana
σ__K_=_π__2__E/λ__yi__2.
λyi ndiko kunonzi kwakanakira kutetepa.
Zvisungo zvemuchinjikwa zvinofanira kuva zvakakwana muhuwandu uye zvakasimba zvakakwana kuti zvimanikidze zvikamu zvose zvetsvimbo kushanda pamwechete. Kunze kwekunetseka kwechipiri, iyo inochinjika link inoramba isina kusimbiswa chero bedzi tsvimbo inoramba yakatwasuka. Kungokotama chete ndiko kwavanofanira kugashira masimba, uye vanofanira kutumira masimba anochinjika anowedzera nekukotama. Kana iri mutsara elastic
transverse force pa I=const. zvakaenzana
Q = M’ = - EI∙v’‘’.
Nemuonde. 5 m inogona kuverengwa nekukasika Q(z)=PK∙v’(z). Iko kukosha kwepamusoro pamagumo etsvimbo ndeye
Sl.5
Makonekisheni anochinjika anofanira kunge achikwanisa kugonesa iyi simba rinochinjika. Sezvo mutoro unoenderana isimba rebuckling PK, zvinokwana kana muganhu wekutakura wasvika kune izvi zvakare mune zvakachinjika zvinongedzo - i.e. panguva imwe chete netsvimbo yese. Zviripachena, hapazove nemubairo kana iwo ma-cross-link akakura akasimba kupfuura zvakafanira kugamuchira Q0. Nekune rumwe rutivi, ma-cross-link asina kusimba angaita kuti tsvimbo ikunde nguva isati yakwana.
Equation iri pamusoro haigone kushandiswa pekutanga nekuti ukuru hwemuseve unomonereka hauzivikanwi. Tsvimbo iri mukuenzanisa kusingaite pasi pePK uye inogona, zvirinani mudhigirii rekutanga rekufungidzira, kutora kukosha kwekupokana. Naizvozvo inobatsira, maererano nezano re F. Krohn-a, kuona ukuru hweiyo buckling maxv iyo tsvimbo yakakotama inorasikirwa nekubereka kwayo nekuda kwekutadza nekuda kwekukotama. Izvi zvichaitika chaizvo kana iyo goho poindi yakasvikwa padivi reconcave rekubhenda (Fig.6) Mabhendi eichi anokura zvakanyanya uye nguva dzinoenderana dzekukotama hadzigone kushandiswa.
Fig. 6
Kushungurudzika kwakazara padivi reconcave rekubhenda kuri maererano neonde.6kune yakanaka plastiki zvinhu
Kubva ipapo kunouya swagger huru iyo inogona kutsungirira
maxQ0 isimba rakachinjika rinoenderana kumucheto wetsvimbo.
Kana yakashandiswa σK Tetmeier's fomu woisa σF=3,1t/cm2uye sK=e/2∙ λy izvo zvinoenderana ne iy≈e/2, ipapo Krohn's inofungidzirwa kukosha kwe maxQ inowanikwa0=F/28(Q mu t, F mumacm2)
Kune mitsara yekupokana iyo buckling stress inowanikwa
maxQ0=μK∙SK.
Iyo coefficient μK inoshanduka nekutetepa kwetsvimbo, zvinoenderanawo nezvinhu. Kana iyo equation iri pamusoro yapatsanurwa necoefficient yechengetedzo pakurwisa buckling inogona kunyorwa
dozv__Q=μ∙dozvS.
Buckling yemutsara masisitimu
Panyaya yemasisitimu emutsara, zvakajairika kuona kureba kwekukotama sK=β∙h yetsvimbo yakasungirirwa kumagumo ese uye ine simba rekukotama riri PK=π kuenzanisa.2EI/s2K zvakangodaro. Kuenzanisa “buckling urefu” kuchapihwa kune mienzaniso iri pasi apa.
a) Tsvimbo yakasungirirwa kumativi ese ari maviri (EI=const.)
Sl.7
Nokuda kwenyaya yemuonde.7ye buckling ndeye PK=4π2EI/h2=4,0∙PE, kuenzanisa “buckling urefu” sK=h/2uye elastic buckling line v(z)=D(1- cos2πz/h).
b) Tsvimbo yakasununguka kune imwe mugumo (fig.8)
Sl. 8
Zvinofungidzirwa kuti gwara re P harichinji panokotama tsvimbo. Zvadaro (EI=const.)
PK=π2EI/4h2=PE/4, sK=2h
v(z)=D(1-cosπz/2h).
Mamiriro ezvinhu epamahara z=h aripano
M=0 kana v’’=0
Q=M’-Pv’=0 kana v’’’+a2∙v’=0.
c) Tsvimbo yakasungirirwa kune imwe mugumo (fig. 9)
Sl.9
Ne contour mamiriro
v=0 uye v’=0 ye z=0
v=0 uye v’’=0 ye z=h
inowanikwa EI=const. buckling condition
sin_αh - αh∙cos_αh=0_,_
kana zvakaenzana
tgαh=αh. (4)
Midzi yeiyi transcendental equation inowanikwa mumatafura K. Hayashi's. Simba rakaderera rinoenderana nematafura PK1=4,49342/ h2∙ EI =2,046π2EI/h2.
Kuenzanisa kureba kwebhangi ndeku sK = h / √2,046=0,699h.
d) Tsvimbo yakasungirirwa kune imwe mugumo ine kusagadzikana (fig.10)
Fig. 10
Isu tichafunga nezvenyaya yetip displacement yekugara saizi. Mamiriro ezvinhu e contour ayo ndeaya
v=0uye v =0 ye z=0,
v=v1uye v =0 ye z=h.
Iri idambudziko rekuenzana rine elastic line
Denominator yetemu pamberi pe {}-mabhuraketi ichave zero uye kusungirira kunowedzera kupfuura miganhu yese kana P→ PK (ona buckling condition Eq. (4)). Ihwo hutsika hwemafambiro etsvimbo iyo mabhendi ari pakati nepakati pekupedzisira kwetsvimbo anotanga maduku nekuwedzera mutoro P uye anoshandura chiratidzo chavo chete nokuda kwemitoro yakakwirira. Sl.10inoratidza mabhende pahupamhi hwehafu.
e) Kamuti kasingaperi muminda miviri
Sl.11
Kune ese ari maviri emhinduro nzvimbo (fig.11) tichashandisa marongero acho
Uye_. v(z1) = A1∙synαz1+B1∙cosαz1+C1∙z1+ D1,_
II. v(z2) = A2∙synαz2+B2∙cosαz2+C2∙z2+ D2.
Kuti zvive nyore ngazvive (EI)1=(EI)2=EI. Kutsvaga zvisere zvinoramba zviripo A1, …, A2, … shanda semamiriro ezvinhu
v=0uye v =0 ye z1=0,
v=0uye v =0 ye z2=h,
pamwe chete nemamiriro ekuchinja pakati penzvimbo I uye II
v=0 ye z1=h i ye z2=0,
v’’ (z1=h)=v’(z2=0),
v’’ (z1=h)=v’’(z2=0).
Maconnditional equations anopiwa mutafura iri pazasi
Nekumisa chirevo cheiyo coefficient yakaenzana ne zero, iyo buckling mamiriro inowanikwa
Hurefu hwekupeta sK=0,878_h_, ukuwo muonde.35zvimwe β=0,699. The buckling force yakadzikira zvichienzaniswa2,046:1,297nekuda kwekuwedzera kwemunda wechipiri wakasungirirwa. Kana imwe nzvimbo yakatsikirirwa kumusoro, ipapo simba rinoputika raizowedzera zvakare kusvika kumusoro.2,04_PE_.
Iyi nhoroondo yekuratidzwa kwemafomula haisi purojekiti kana humbowo hwekugadzikana kwechimiro. Iwo echinyakare fungidziro yeakanaka nhengo yakatwasuka, centric mutoro uye elastic nzvimbo haingosanganisire kusakwana kwekutanga, kusarira kunetseka, eccentricity, kuomarara kwekubatanidza, zvinhu zvisiri-mutsara, lateral kugadzikana kana, kana zvichibvira, moto, seismic chiito uye gungano zvikamu. Kuverengera kweiyo chaiyo dhizaini kunofanirwa kuitwa neane mvumo yeinjiniya yehurumende zvinoenderana nemirau uye zviyero.