Masiku ano, Savo Kusić amayang’ana kwambiri mazenera amatabwa, mazenera a matabwa a aluminiyamu, mazenera amwambo, khomo ndi zopempha za mawu. Mawuwa akadali ngati nkhokwe ya chiphunzitso cha structural ndipo sikupereka ma static design, mawerengedwe okhazikika kapena kukonzanso kamangidwe.
Kuphatikizika kosiyana kwa kupindika mabatani
Amaganiziridwa kuti malingaliro onse ndi okhutitsidwa, makamaka
- kuti ndodoyo ndi yowongoka
- kuti katundu P amagwira ntchito pamtengo wa ndodo
- kuti ma voltages amakhalabe pansi pa malire a proportionality
- kuti mapindikidwe ang’onoang’ono mokwanira kulola kuphweka mwachizolowezi kwa mawerengedwe.
Ngati ndodoyo imapinda pansi pa mphamvu ya P ndipo mwinamwake chifukwa cha katundu q akugwira ntchito modutsa, ndiye kuti zikhalidwe zodziwika za mgwirizano pakati pa katundu wakunja ndi mphamvu zomwe zili mu gawoli ziyenera kukhutitsidwa. Kupatula apo, ndizachilengedwe kuti payenera kukhala ubale wodziwika pakati pa zopindika ndi mphamvu mu gawoli.
Sl. 1 - Kusamala
Mu mkuyu.1gawo la ndodo yopindika ya kutalika ds imayimiridwa. Kutsetsereka kwa tangent v’ ndikwaling’ono kotero kuti mpaka dongosolo lachiwiri, kutalika kwa chinthucho kukhoza kukhazikitsidwa kukhala kofanana ndi ds=dz. Kuonjezera apo, mu mayesero omwe tawatchula pansipa, milandu yomwe P imasintha nthawi zonse motsatira ndodo sizingaganizidwe, mwachitsanzo, mphamvu yachibadwa N≈P=const. Pamikhalidwe itatu yofananira yomwe ilipo pagawoli, mamembala angapo amatha kunyalanyazidwa. Kwa zizindikiro ndi zizindikiro za mkuyu.10khalani ndiye mukuyandikira koyamba kwa equation
N=P, M’=Q, Q’=-q+Pv’’.
Mphamvu yodutsa imatha kuthetsedwa, ndiyeno M’‘=Pv’’-q ikugwira. Lamulo la Bernoulli\ M=-EI/R limapereka equation yosowa. Mphepete mwa nsonga ya ndodo, monga mwachizolowezi, kuchokera ku chiphunzitso cha kupindika kwa mtengo, imasinthidwa motere.
Ndizotheka kuthetsa mphindi yopindika; ndipo ndi M=-EI*v’’ equation yosiyana ya ndodo yopindika
(EI*v’‘)’’ + Pv’’ = q. (1)
Panthawi yapadera yopindika mosalekeza EI=const. equation (1) ndi yosavuta, choncho ndi
EIv’‘’’ + Pv’’ = q. (1a)
Eq. (1) ndi (1a) ndi mzere mu v molingana ndi kuphweka komwe kwatchulidwa.
M’malo mwake, njira yosavuta yokhotakhota mosalekeza EI=const. ndizofunika kwambiri, zomwe zimakhala ngati maziko a zitsanzo zomwe zili pansipa.
Nkhani ya Euler
Malingaliro a nkhani yachikale ya buckling ndi, kupatula zomwe zatchulidwa: p=0, kotero N=const.=P, komanso kuumirira kosasunthika kopanda kugunda, komwe kumakhala pakatikati pa ndodo. Kusiyanitsa wamba (1a) ndi dongosolo lachinayi, la mzere komanso lofanana chifukwa cha q=0. Yankho lake lonse ndi:
v(z) = A_∙_sin__αz + B∙cosαz + C∙z + D, (2)
komwe A, B, C, D ndi zolumikizana zinayi zokhazikika. Zoyezerazo ndi zazikulu zomaliza chifukwa cha lingaliro EI≠0. Komanso, α≠0 kwa P>0.
Sl.2
Popeza mzere wosiyanitsa mzere (1a) ndi wofanana, vutoli limathetsedwanso ndi n_∙__v(z)_, pomwe n amasinthasintha mosasintha. Kuyambira kale pamasamu a equation yosiyana (1a) zitha kuwoneka kuti kukula kwa kupinda v sikungadziwike. Kuti mudziwe zophatikizana zosakanikirana A, B, C, D mu Euler\ kesi (mkuyu 2) mizere yotsatirayi ikupezeka: Chifukwa cha chithandizo chofotokozera, mphindi yopindika ndi zero pamapeto onse a ndodo, i.e.
v=0 ndi M=0 kwa z=0
v=0 ndi M=0 kwa z=s.
Chifukwa cha M=-EI_∙__v’‘_, zikutsatira kuti M=0 ndi mtengo wa v’’=0, kotero mizere inayi imawerengedwa:
v=0 ndi v’’=0 kwa z=0
v=0 ndi v’’=0 kwa z=s.
Yankho (2) limabweretsa mizere inayi:
Mmodzi.3
Ma homogeneous equations anayiwa ali ndi choyambilira chotchedwa trivial solution A=B=C=D=0, chomwe chikufanana ndi v(z)=0. Ndodo yosapindika v_(z_)=0 ndi malo ofananirako mwanthawi zonse, makamaka mitengo yaying’ono ya katundu P. Apa, timakhudzidwa kwambiri ndi momwe ndodo imapindikira, mwachitsanzo, v(z)≠0. Ma equation (3) alibe yankho laling’ono ngati A, B, C, D ≠ 0 pokhapokha ngati chowerengera cha koyefiyiti chili chofanana ndi ziro.
Mmodzi. 3a
Kuchokera pamikhalidwe ya buckling Δ = α4∙s∙sinαs =0 mog kuwerengera zofunikira PK za katunduyo. Equation iyi imakhutitsidwa pokhapokha ngati sin αs =0, ndiko kwa αKms = mπ ndi m=1,2,3,… Zotsatira za Euler's buckling force ndi PKm=EI_∙_α2Km = EI(mπ/s)2. Pa mtengo uliwonse wa m, mphamvu inayake yolimba imapezedwa. Nthawi zambiri, chochepa chabe mwa mphamvu izi ndi chidwi, ndi kuti
PK1=EI(π/s)2.
Mu mtundu wotsatirawu, padzakhala index m, kapena 1 nthawi zambiri zosiyidwa, kutanthauza zotsika kwambiri zolemetsa. Chochititsa chidwi ndi mzere wofananira wotanuka. Kuchokera kumodzi (2) zosintha B=C=D=0 zapezedwa. Zotsalazo ndi A∙sin__αs=0. Mzere wonyezimira wazomwe zili pamwambapa m=1 ndi, v(z) = A_∙sinπz/s_, pomwe v(z) ndi αK resp. PK eigenfunction yofananira. Kukula kwa A sikunadziwikebe.
Tiyeni tionenso gawo lonse la katundu. Pazinthu zosasintha za P, makamaka pazokakamiza P_<_PK1 ndi v(z)=0, ndodo imakhala yowongoka.
Mwa “kuwonjezeka pang’onopang’ono” kukweza P kumatanthawuza mu chiphunzitso chachikale cha kukhazikika mndandanda wotsatizana wa makulidwe P, kumene kukhazikika kwa katundu wowonedwa kudzayesedwa mobwerezabwereza pa mlingo uliwonse wa kukweza P. Pakuyesa kukhazikika, kukula kwa P sikunasinthe. Ngati dziko limodzi P latsimikiziridwa kukhala lokhazikika, ndiye kuti katunduyo akuwonjezeka ndi chinachake, choncho kuyesa kukhazikika kumabwerezedwa. Kutsimikiza kolondola kumeneku ndikofunikira kuti pasakhale chisokonezo ndi mtundu wa katundu, womwe ndi maziko a zotsatira za Shanley, momwe pali kuwonjezeka kwa katundu pamene zosokoneza zazing’ono zikugwira ntchito nthawi yomweyo. Pofufuza kukhazikika kwa Shanley, m’dera la pulasitiki buckling, mphamvu yaing’ono ya bifurcation imapezeka kusiyana ndi njira yachikale yoganizira.
Malingaliro otsatirawa akuganiza njira yoyamba yowonjezerera katunduyo: Ndi kuwonjezeka pang’onopang’ono kwa katundu P m’lingaliro ili, PK ikufika.1=EI_∙(π/s)_2woyamba mpaka pagawo limodzi lokhala ndi zotanuka zofanana, wokhala ndi mzere wosavuta wa sine ngati mzere wopingasa
v1(z)=f_∙_sin__π__z/s.
f=A amatanthauza matalikidwe osadziwika a mzere wotanuka. Za katundu _P*__>_PK1 ndi molingana ndi chiphunzitso cha linearized Δ≠0, ndiko kusatheka kumeneko. Pokhapokha zamphamvu zolimba kwambiri PK2 etc. idzakhala Δ= kachiwiri0moteronso v(z)≠0. Chifukwa cha mfundo yomalizayi, yomwe sikugwirizana ndi khalidwe lenileni, ili mu mzere wa kusiyana kwa equation.
Kumanga kwa ndodo zoponderezedwa zopangidwa ndi magawo awiri
Timadziletsa tokha ku timitengo tiwiri topanikizidwa, ndipo kwa timitengo topanikizidwa topangidwa ndi magawo atatu kapena kuposa, maunansi ofanana amagwira ntchito.
Sl.3
Kutengera mtundu wa kulumikizana kodutsa, ndodo za lattice zimasiyana (mkuyu.3a) ndi ndodo za chimango (mkuyu.3e). Popanda zomangira zopingasa, gawo lililonse la ndodo limadzipinda lokha (mkuyu.3b). Ngati kudzazidwa kuli kofooka, ndiye kuti ndodo imodzi ya lamba imatha kupindika popanda kupindika ndodo yonse. Kwa ndodo ya latisi mkuyu.3c akhoza kufananizidwa: kumanzere kwa hinged system, kumanja kwa malamba osalekeza; ndi mkuyu wa ndodo ya chimango.3f kumanzere kwa mapepala ofewa olumikizirana, kumanja kwa masamba olumikizana olimba. Kuchokera pamenepo, zitha kuwoneka kuti kuwonda λ_ kumakhudza kwambiri1_=s1/ndi1_. Kwa gawo limodzi la ndodo, ndinayikidwa1=F1_∙_i12, s1 ndi malo apakati (mkuyu.3), ine1 amatanthauza axis1-1(onani mku.4).
Sl. 4
Mphamvu yomangirira ya ndodo zophatikizika zofananira ndi mizere yopumira ya mkuyu.3d, kapena3g inali phunziro la kafukufuku wokwanira wanthanthi. Ngati ndodoyo idapangidwa ndi zigawo zapadera (mwachitsanzo ndi nthiti) kapena ngati zolumikizira zopingasa zinali pafupipafupi kotero kuti ndodo yopindika kuchokera ku zigawo ziwiri sizimasiyana ndi momwe imagwirira ntchito ndi ndodo imodzi, ndiye kuti σ__K_=_π__2__E/λ__y__2. Pamenepo amatanthauza λ__y=sK/iy (Mku.4). Komabe, kupsinjika kwa buckling uku sikudzakwaniritsidwa kwathunthu chifukwa palibe kulumikizana kokwanira pa izi, komanso ndi zotanuka. Kuyerekeza komwe kumakhutitsa nthawi zambiri kumapezeka ngati tiyika molingana ndi lingaliro F. Engesser\a kuti kupindika kwa ndodo yoponderezedwa yokhala ndi magawo awiri ndikofanana
σ__K_=_π__2__E/λ__yi__2.
λyi ndi chomwe chimatchedwa slenderness yabwino.
Zomangira zopingasa ziyenera kukhala zokwanira mu chiwerengero ndi mphamvu zokwanira kukakamiza mbali zonse za ndodo kuti zigwirizane. Kupatula kupsinjika kwachiwiri, maulalo odutsa amakhalabe osakhazikika malinga ngati ndodoyo ikhala yowongoka. Ndi kupindika kokha komwe amayenera kulandira mphamvu, ndipo amayenera kutumiza mphamvu zodutsa zomwe zimawonjezeka ndi kupindika. Ngati ndi chingwe chotanuka
mphamvu yodutsa pa I=const. ndi ofanana
Q = M’ = - EI∙v’‘’.
Ndi mkuyu. 5 m itha kuwerengedwa nthawi yomweyo Q(z)=PK∙v’(z). Mtengo wapamwamba kwambiri kumapeto kwa ndodo ndi
Sl.5
Malumikizidwe odutsa ayenera kukhala ogwirizana ndi mphamvu yodutsayi. Popeza katundu wofanana ndi mphamvu ya buckling PK, ndikwanira ngati malire onyamula katundu afikira pa izi komanso muzolumikizana zodutsana - mwachitsanzo, nthawi imodzi ndi ndodo yonse. Mwachiwonekere, sipakanakhala phindu ngati maulalo opingasa akadakhala amphamvu kwambiri kuposa momwe ayenera kulandirira Q0. Kumbali ina, zolumikizira zomwe zili zofooka kwambiri zingapangitse ndodoyo kulephera msanga.
Equation yomwe ili pamwambapa singagwiritsidwe ntchito poyambira chifukwa kukula kwa muvi wobowoleza sikudziwika. Ndodoyo ili mumgwirizano wosayanjanitsika pansi pa PK ndipo imatha, osachepera pamlingo woyamba wa kuyerekezera, kutenga mtengo wokhazikika. Chifukwa chake ndizothandiza, molingana ndi lingaliro la F. Krohn-a, kuti adziwe kukula kwa buckling imeneyo maxv pomwe ndodo yopindika imataya mphamvu yake yobereka chifukwa cholephera chifukwa chopindika. Izi zidzachitikadi ngati zokolola zafika kumbali ya concave ya bend (Mkuyu.6). Kupindika kwa izi kumakulirakulira ndipo mphindi zopindika zofananira sizingagwiritsidwe ntchito.
Mth. 6
Kupsyinjika kwathunthu kumbali ya concave ya bend kumalingana ndi mkuyu.6kwa zinthu zabwino zapulasitiki
Kuchokera pamenepo kumabwera chiwombankhanga chachikulu chomwe chingathe kupirira
maxQ0 ndi mphamvu yopingasa yofananira kumapeto kwa ndodo.
Ngati atagwiritsidwa ntchito pa σK Tetmeier's mawonekedwe ndikuyika σF=3,1t/cm2ndi sK=e/2∙ λy chomwe chikufanana ndi iy≈e/2, ndiye Krohn's pafupifupi mtengo wa maxQ wapezedwa0=F/28(Q mu t, F mu masentimita2).
Kwa mizere yokhazikika, kupsinjika kwa buckling kumachitika
maxQ0=μK∙SK.
Coefficient μK imasintha ndi kuwonda kwa ndodo, zimadaliranso zakuthupi. Pamene equation yomwe ili pamwambayi yagawidwa ndi coefficient ya chitetezo motsutsana ndi buckling ikhoza kulembedwa
dozv__Q=μ∙dozvS.
Kuyika kwa ma linear system
Pankhani yamakina amizere, ndizofala kudziwa kutalika kwa sK=β∙h kwa ndodo yomwe imakongoletsedwa mbali zonse ziwiri ndipo mphamvu yake yopindika ndi PK=π poyerekeza.2EI/s2K wamkulu basi. Kufananiza “buckling length” kudzaperekedwa kwa zitsanzo pansipa.
a) Ndodo yomangidwa kumapeto onse awiri (EI=const.)
Sl.7
Kwa nkhani ya mkuyu.7ya buckling ndi PK=4π2EI/h2=4,0∙PE, kufananiza “buckling length” sK=h/2ndi mzere wa zotanuka v(z)=D(1-kos2πz/h).
b) Ndodo yaulere kumapeto kumodzi (mkuyu.8)
Sl. 8
Zimaganiziridwa kuti mayendedwe a P sasintha pamene ndodo imapindika. Kenako (EI=consst.)
PK=π2EI/4h2=PE/4, sK=2h
v(z)=D(1-cosπz/2h).
Mipangidwe ya mizere yaulere z=h ili pano
M=0 or v’’=0
Q=M’-__Pv’_=0 kapena v’’’+a2∙v’=0.
c) Ndodo yotsekeredwa kumapeto kumodzi (mkuyu 9)
Sl.9
Ndi mizere yozungulira
v=0 ndi v’=0 kwa z=0
v=0 ndi v’’=0 kwa z=h
imapezedwa pa EI=const. chikhalidwe cha buckling
sin_αh - αh∙cos_αh=0_,_
kapena zofanana
tgα=ah. (4)
Mizu ya equation yodutsa iyi imapezeka m’magome K. Hayashi’s. Mphamvu yotsika kwambiri ndiyolingana ndi ma tebulo PK1=4,49342/ h2∙ EI =2,046π2EI/h2.
Kutalika kofananirako ndi sK =h / √2,046=0,699h.
d) Ndodo yomangidwa kumapeto kumodzi ndi vuto (mkuyu.10)
Mkuyu. 10
Tikambirana nkhani ya kusamuka kwa nsonga kwa kukula kosalekeza. Miyezo ya contour ya izo ndi
_v=0ndi v =0_kwa z=0,
_v=v1ndi v =0_kwa z=h.
Ndi vuto lofanana ndi mzere wotanuka
Chiwerengero cha mawu omwe asanachitike {}-mabulaketi adzakhala ziro ndipo kukwera kumawonjezeka kuposa malire onse ngati P→ PK (onani chikhalidwe cha buckling Eq. (4)). Ndi chikhalidwe cha khalidwe la ndodo kuti mapindikira pakati ndi m’munsi kumapeto kwa ndodo poyamba ang’onoang’ono ndi kuwonjezeka katundu P ndi kusintha chizindikiro awo okha katundu apamwamba. Sl.10kusonyeza mapindikidwe pa theka kutalika.
e) Ndodo mosalekeza m’magawo awiri
Sl.11
Kwa madera onse awiri (mkuyu.11) tidzagwiritsa ntchito zoikamo
Ndipo_. v(z1) = A1∙synaz1+ B1∙cosaz1+C1∙z1+ D1,_
II. v(z2) = A2∙synaz2+ B2∙cosaz2+C2∙z2+ D2.
Kuti zikhale zosavuta zikhale (EI)1=(EI)2=EI. Kuti mupeze zokhazikika zisanu ndi zitatu A1, …, A2, … imagwira ntchito ngati mizere
_v=0ndi v =0_kwa z1=0,
_v=0ndi v =0_kwa z2=h,
komanso kusintha kwa nyengo pakati pa madera I ndi II
_v=0_kwa z1=h i kwa z2=0,
v’’ (z1=h)=v’(z2=0),
v’’ (z1=h)=v’’(z2=0).
Ma equation okhazikika aperekedwa mu tebulo ili m’munsimu
Pokhazikitsa choyimira cha coefficient chofanana ndi zero, chikhalidwe cha buckling chimapezeka
Utali wopindika sK=0,878_h_, pamene mkuyu.35kaya β=0,699. Mphamvu ya buckling idachepa poyerekeza ndi2,046:1,297chifukwa chowonjezera gawo lachiwiri lomwe limakongoletsedwa. Ngati munda winawo udatsitsidwa kwambiri, ndiye kuti mphamvu yoboola ikanawonjezekanso mpaka pachimake.2,04_PE_.
Chiwonetsero cha mbiriyakale cha mafomuwa si ntchito kapena umboni wa kukhazikika kwa kapangidwe kake. Malingaliro akale a membala wowongoka wabwino, centric load ndi zotanuka sizimaphatikizanso zolakwika zoyambira, kupsinjika kotsalira, kukhazikika, kuuma kwa kulumikizana, kusagwirizana kwazinthu, kukhazikika kwapambuyo kapena, ngati kuli kotheka, moto, zivomezi ndi magawo a msonkhano. Kuwerengera kwa kapangidwe kake kuyenera kuchitidwa ndi mainjiniya ovomerezeka molingana ndi malamulo ndi miyezo yoyenera.