Kune buckling pakukotama, danda pakuchinja kwekuenzana rinokotama mundege huru imwe.
Asi kunezve mamiriro akadaro ekugadzikana umo tsvimbo, kana danda pakuchinja kwekuenzana zvinongokombamiswa chete asiwo zvinomonereka (sl. 60a). Pamakamuri anotsigira chiitiko ichi chinonzi bočno izvijanje, uye patsvimbo dzakatsikirirwa, dzinobatwawo nekutwasanuka pakukombama pamwe nekutenderedzwa, panotaurwa nezve izvijanje pri torziji.
Kune zviitiko zviviri izvi kukosha kwekuoma pakupikisa kutenderedza C, kana kuti kupikisa kukombama C*, kwakakosha zvikuru. Kuchinja kwekuenzana kwakabatana ne torsion kunogona saka kukanganiswa pakutanga nematanho anowedzera kuoma kurwisa kumonyoroka (chimiro chakakodzera chechikamu, mafuremu ekuyambuka, kutsigira kwepadivi nezvimwewo). Kune rimwe divi, matanda akaita seribhoni ane chikamu chakazaruka anonyanya kunzwira kupfugama kana achitenderedzwa.
Differential equation yekukotama kweparutivi
Muf. 1a
Ngatitorerei danda rakatwasuka rine moments dzeinertia dzisingaenzani zvikuru, Ix >> Iy. Mutoro uri mundege y-z-saka danda richakotamiswa mundege iyi huru (u=ϑ=0). Zvakaitika pamwe chete nedzidziso zvinoratidza kuti nzvimbo iyi yekuyanana pakutanga yakagadzikana, asi kuti pamitoro mikuru inova isina kugadzikana. Iyi nzvimbo yechipiri yekuyanana yakabatana nekukotama kwemuchinjikwa mu uye netorsion ϑ.
Pazasi kuchatorwa differential equation yekukotama kwepadivi. Maequation e u ne ϑ ane chimiro chisingabvi pane chimwe chinhu. Kana mamiriro emuganhu ariwo asingabvi pane chimwe chinhu, saka kukotama kwepadivi kwedanda kunobvira chete pazvinokosha zvega zvechinetso, kureva kuti u ne ϑ zvinoramba zviri zero pakutanga padanda riri pamusoro riri pasina kukanganiswa. Chimiro chekukotama kwepadivi chinowanikwa — zvakafanana nekukotama kwetsvimbo — kana determinant ye coefficient yemaequation emamiriro asingabvi pane chimwe chinhu ema integration constants yaiswa kuti ive zero. Tichaona kuti kunyangwe pakukotama kwepadivi pane poindi chaiyo yokupatsanuka kwekuenzana kweelastiki.
Muf. 1b
Pamuf. 1b panoratidzwa chinhu chebeam chine urefu dz. Mhedzisiro yemaspaneti N1, Q1, M1, pamwe nekukombama κ__1 zvinoenderana nemutoro uye zvine kukosha kwekupedzisira. Q, M, MD ari zero panzvimbo yekutanga yekuenzana, uye achangoonekwa asiri zero kana kutanga kukotama kwepadivi kwaitika. Miganho yekupedzisira, inoreva kutanga kwekukotama kwepadivi, idiki zvisingaperi, sezvakaitawo kukanganisika kunoenderana κ uye d__ϑ/dz.
Zvese zviri zviviri zvikamu zvekupedzisira zvechinhu dz zviri mundege yakajairika paaxis yakakombama kaviri yetsvimbo uye zvichatsvetwa mundege idzi dzakajairika.
Sezvo - sezvakangoita pakukotama kwetanda - kutanga kwezvose tichifarira ukuru hwemutoro wakakosha, kureva kutanga kwekukotama kweparutivi, kwete zvakanyanya ukuru hwekukanganisika, tinogona kufuratira zvigadzirwa zvehukuru Q, M, MD ne κ kana ϑ’ sezvinhu zvidiki zverudzi rwechipiri. Naizvozvo, kuitira kuenderera mberi nekurerutsa, tichatora zvakare kukombama kukuru κ1 sekunge kuri kudiki kusingaperi uye kuchabatwa sa κ ne ϑ’.
Cherechedzo: Mhedzisiro ye κ__1 inofanirwawo kuongororwa muchidimbu mune nyaya yakapfava yakasarudzika (moment yekukotama isingachinji M semutoro). Muzvizhinji zvinodiwa zvetekiniki yekuvaka κ__1 inogona kusiiwa isina hanya. Tinogona, semuenzaniso, kufungidzira sokuti danda panyaya iyi yemutoro rakanga rakadaro ‘rakasimudzirwa’ zvekuti pamutoro wakanyanya κ__1=0.
Kubva pakuenzana kwezvikamu zvesimba munzira dze ξ, η, ζ uye kubva pamamiriro emamomenti zvichienzaniswa neaxis idzodzo, mushure mekuregerera kwakataurwa zvinowanikwa
Q’ + N1u’’ – Q1__ϑ’ + pϑ = 0_,_ (1a)
Q1’ + p = 0_,_ (1b)
N1’ = 0_,_ (1c)
M1’ – Q1 = 0_,_ (1d)
MD’ + Q + M1__ϑ’ = 0_,_ (1e)
MD’ – M1u’’ + pe ϑ = 0_._ (1f)
Kubva pamazwi.equation (1c, d i b) zvinowanikwa
N1 = const.,
Q1 = M1’, (2)
p = - Q1’ = - M1’’.
Nekubvisa Q1 kubva paeq. (1a) panowanikwa
Q’ = (M1’ ϑ_)’ – N1u’’_. (3a)
Kubva paequation. (1e) Q ichabviswa nenzira iyi, uyezve u ’’ kuburikidza ne u’’ = (MD’ + pe ϑ_)/M1_
(M+M1 ϑ_)’’ – N1/M1_ ∙ (MD’ + pe ϑ_) =_ 0_._ (3b)
Kuti tiwane equation yemusiyano ye ϑ tichaisa hukama hunotevera pakati pemamomenti nekushanduka
M1 = -B1_∙v’’,_ (4a)
M = +B_∙_u’’, (4b)
MD = +C_∙ϑ’ – EC*∙ ϑ’’’,_ (4c)
panomirira B1=EIx uye B=EIy kuomarara pakukotamisa danda, C kuomarara pakumonyoroka, uye C* kupikisa kukotama. Mukuverenga kwedu kunotevera tichasiya padivi nhengo dzine C* kuitira kurerutsa. Kana maverengero aya achifanira kushandiswa semuenzaniso kune I-profiles, ipapo zvinofanira kuwedzerwa nenzira inokodzera. Kuregeredza nhengo v’’ kunoreva zvakafanana nekufungidzira B1=∞.
Kana M ne MD zvikabviswa kuburikidza neeq. 4 rinowanikwa differential equation for lateral buckling yedanda rine muchinjiko wakafanana
Mutsara wedifferenshiyero (5) ndiwo uri homogeneous uye wechikamu chechina maererano ne ϑ. M1 ibasa rinozivikanwa re z zvinoenderana nemutoro.
Danda rine moment yekukotama isingachinji
Ngatifungei p=N1=0, uyezve Mx=ɱ=const. Chikamu hachina flange (chakamanikana) uye danda rakareba kana richienzaniswa nemiganhu yechikamu chemuchinjikwa, saka tinogona kuisa kuti C*=0. Ipapo equation (5) inoreruka, nekuda kwe M1=ɱ, saka:
ϑ’’’’ + ɱ2/BC∙ ϑ’’ = 0, (6)
kana nokuderedza muchidimbu λ2_=ɱ__2/BC_
ϑ’’’’ + λ2 ϑ = 0. (6a)
a) Danda rine zvitsigiro zvakaforokodzwa kumativi ose maviri
Fig. 2
Kutsigirwa pa z=0 uye pa z=l kuri muchimiro chemaforogo (muf. 2), izvo zvinobvumira kukotama u ne v, asi zvichidzivirira kutenderera kwemachekwa ekupedzisira; ϑ=0. Sezvo munzira ye x pasina kusungirirwa (kureva, M=0, kana maererano nechin. (4) zvakare u’’=0), zvinobuda kubva muchin. che u’’: MD’=0 uye nokudaro kubva muchin. (4c) semuganho wechipiri ϑ’’=0.
Mhinduro yakazara yeequation (6a) ndeye
ϑ(z) = A1∙sinλz + A2∙cosλz + A3∙λz + A4. (7)
Kubva kumamiriro econtour zvinowanikwa
Iyi homogeneous system yeequations ine:
- Mhinduro isina kuoma A1 = A2 = ∙∙∙ = 0, iyo inoenderana ne ϑ=0, u’’=0 kana u=0, uye danda haritenderedzi padivi.
- Mhinduro dzechisikigo maererano ne Δ=0 kana sin λ__l=0. Midzi yemamiriro aya ekukotama kwemativi ndeye λ__l=n__π (ne n=1,2,3,∙∙∙) uye mameneti akakoshesa ekukotama anoenderana nazvo
ɱ__K = _√_BC ∙ _n__π/_l. (8b)
Momenzi muduku zvikuru wekuvhunduka kudivi
min__ɱ__K = _√_BC ∙ π/l (8c)
inobva pa n=1. Mhinduro inoenderana inowanikwa nekushandisa equation (1f):
ϑ(z) = A1 ∙ sin πz/l (8d)
u(z) = C/ɱK ∙ ϑ(z).
b) Dzimwe nyaya dzekukotama nedivi
c) Kukotama kudivi kunokonzerwa nemomenti yekukotamisa uye simba rinodzvinyirira
Semuenzaniso wakaratidzwa pa sl. 3 ndi N1=-D (kudzvanya), p=0, ipapo pakaiswa C*=0. Ipapo equation yemusiyano (5) inorerutswa uye inoti
ϑ’’’’ + λ2 ϑ’’ = 0, (9)
apo λ__2 = ɱ__2_/BC + D/B_. (9a)
Mhinduro yechiyero chekusiyana (9) inopiwazve nechiy. (8), asi zvino λ inoratidza kukosha kwechipiri.
Fig. 3
Kune muenzaniso wemuf. 3 anoshanda, sezvakaitawo pa) mamiriro econtour ϑ=0 ne ϑ’’=0 pa z=0 ne z=l. Nokudaro, zvakare maequation emamiriro e A1 ari pamutemo akafanana uye zvakare mamiriro ekukotama kwepadivi Δ=sin λl=0. Mutoro wakakosha unozadzisa hukama λl=nπ (n=1,2,3) uye kukosha kudiki ndiko
ɱK2/BC + DK/B = (π/l)2. (10)
Kune D=0 kubva ipapo panowanikwa moment yekukotama kweparutivi maererano neequation (8c) uye pamutoro unongomanikidza chete (ɱ=0) simba rekuvhurika ra_Euler_ _DK=B(π/l)_2.
Pa muf. 3 nyaya yakaratidzwa inofanira kudomwa sekukotama kweexcentric mundege y-z. Nyaya yekugadzikana yatakatarisa pamberi apa saka ndeyekukotama kunobuda kunze kwendege yekukotama kwakabatana nekukotama kwepadivi u uye torsion ϑ.
Kana magumo ose maviri akasungirirwa (ϑ=0 uye ϑ’=0) zvino kubva pamamiriro emagumo, nerubatsiro rwe Δ=0, panowanikwa mamiriro ekukotama padivi
λ__l ∙ sin λ__l = 2(1-cos λl), (11)
nemidzi λ__l=m__π (m=2,4,∙∙∙) nemasimba ekukotama parutivi
ɱ__K__2/BC + DK/B = (2_π_/l)2.