Rau buckling thaum khoov, tus pas nrig thaum kev hloov ntawm qhov sib npaug raug khoov hauv ib lub dav hlau tseem ceeb.
Tab sis kuj muaj tej kis ntawm kev ruaj khov uas tus pas, los sis tus nqaj, thaum kev tshuav hloov tsis yog tsuas khoov xwb tab sis kuj xoob tig (sl. 60a). Rau cov kab ris no qhov tshwm sim no hu ua kev buckling sab nraud, rau cov pas uas raug nias thiab thaum buckling kuj raug torsion, ces hu ua kev buckling vim torsion.
Rau ob qhov tshwm sim no, qhov loj ntawm kev tawv tiv thaiv kev tig C, lossis qhov kev tiv taus rau kev khoov C*, yog qhov tseem ceeb tshaj. Yog li ntawd, kev hloov ntawm kev sib npaug uas txuas nrog kev tig tuaj yeem raug cuam tshuam thawj zaug los ntawm cov kev ntsuas uas nce kev tawv tiv thaiv kev twist (qhov zoo ntawm seem, cov thav duab hla, kev txhawb sab, thiab lwm yam). Ntawm sab tod, cov pas nrig zoo li kab txaij uas muaj seem qhib tshwj xeeb tshaj yog rhiab rau kev ntxeev thaum muaj kev tig.
Qhov sib npaug txawv rau kev nkhaus sab nraud
Daim duab 1a
Cia ib tug nqaj ncaj muaj cov moments of inertia uas sib txawv heev Ix >> Iy. Lub load nyob hauv y-z-dav hlau, yog li tus nqaj yuav raug khoov hauv lub dav hlau tseem ceeb no (u=ϑ=0). Kev paub thiab kev tshawb fawb qhia tias txoj hauj lwm sib npaug no thaum xub thawj ruaj khov, tab sis rau cov load loj dua nws dhau los ua tsis ruaj khov. Qhov no txoj hauj lwm sib npaug thib ob txuas nrog kev khoov hla kev hauv i thiab nrog kev torsion ϑ.
Hauv qab no yuav muab txoj kev sib npaug txawv rau kev nkhaus sab. Cov kab npaug rau u thiab ϑ yog homogenous. Yog tias cov ciam teb kuj yog homogenous ces kev nkhaus sab ntawm tus kab yuav ua tau tsuas yog rau cov nqi eigenvalue ntawm qhov teeb meem, txhais tau tias u thiab ϑ tseem thawj zaug yog xoom ntawm tus kab uas tau piav saum toj no tsis muaj kev cuam tshuam. Tus mob rau kev nkhaus sab tau txais – zoo ib yam li kev nkhaus ntawm tus pas – yog muab tias determinant ntawm cov coefficient ntawm cov kab npaug homogenous rau cov integration constants sib npaug rau xoom. Peb yuav pom tias txawm rau kev nkhaus sab kuj tseem muaj ib qhov chaw tseeb uas elastic equilibrium faib ua ob ceg.
Duab 1b
Hauv daim duab 1b tau qhia ib ntu ntawm tus kab laug uas muaj ntev dz. Cov txiaj ntsig ntawm cov kev ntxhov siab N1, Q1, M1, nrog rau txoj kab nkhaus κ__1 nyob ntawm lub nra hnyav thiab yog cov nqi kawg. Q, M, MD yog xoom nyob rau hauv thawj txoj hauj lwm sib npaug, thiab yuav txawv ntawm xoom tsuas yog thaum pib muaj kev txhav khoov sab. Cov nqi kawg uas hais txog kev pib ntawm kev txhav khoov sab yog me kawg nkaus, ib yam li cov kev hloov pauv uas phim κ thiab d__ϑ/dz.
Ob qho ntu kawg ntawm lub ntsiab dz nyob rau hauv ib lub dav hlau ib txwm rau lub axis nkhaus ob npaug ntawm tus pas thiab yuav raug tig hauv cov dav hlau ib txwm no.
Vim tias peb - zoo ib yam li thaum pas nrig yoog - thawj zaug xav txog qhov loj ntawm lub nra tseem ceeb, yog li ntawd pib ntawm kev yoog mus rau sab, thiab tsawg dua qhov loj ntawm kev hloov cev, peb thiaj li yuav tsis quav ntsej cov khoom ntawm qhov loj Q, M, MD nrog κ lossis ϑ’ raws li cov nqi me ntawm qib ob. Vim li ntawd, rau kev yooj yim ntxiv, peb tseem yuav kwv yees txoj kev nkhaus loj κ1 tias nws yog me kawg nkaus thiab yuav raug kho tib yam li κ thiab ϑ’.
Ceeb toom: Cov kev cuam tshuam ntawm κ__1 kuj yuav tsum tau kwv yees nyob rau hauv qhov tshwj xeeb yooj yim (lub mom bending M ua kev thauj). Rau feem ntau cov kev xav tau ntawm kev tsim kho, κ__1 tuaj yeem tsis quav ntsej. Peb tuaj yeem xav tias, piv txwv li, tus nqaj rau rooj plaub kev thauj no raug “tsa siab” kom ntau uas rau kev thauj tseem ceeb ces yog κ__1=0.
Los ntawm kev sib npaug ntawm cov ntsiab lus ntawm cov quab yuam hauv cov kev taw qhia ξ, η, ζ thiab cov xwm txheej ntawm lub moment rau tib cov axes ntawd, tom qab cov kev tsis quav ntsej uas tau hais los saum no tau txais
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)
Los ntawm equation (1c, d thiab b) tau txais
N1 = const.,
Q1 = M1’, (2)
p = - Q1’ = - M1’’.
Los ntawm kev tshem Q1 tawm hauv kab zauv (1a) tau txais
Q’ = (M1’ ϑ_)’ – N1u’’_. (3a)
Los ntawm kab zauv. (1e) yuav yog li no tshem tawm Q, tom qab ntawd u’’ los ntawm u’’ = (MD’ + pe ϑ_)/M1_
(M+M1 ϑ_)’’ – N1/M1_ ∙ (MD’ + pe ϑ_) =_ 0_._ (3b)
Txhawm rau tau txais qhov sib npaug sib txawv rau ϑ yuav qhia cov kev sib raug zoo hauv qab no nruab nrab ntawm cov moment thiab cov kev deform
M1 = -B1_∙v’’,_ (4a)
M = +B_∙_u’’, (4b)
MD = +C_∙ϑ’ – EC*∙ ϑ’’’,_ (4c)
qhov twg txhais tau B1=EIx thiab B=EIy qhov tawv tiv thaiv kev khoov ntawm tus nqaj, C qhov tawv tiv thaiv kev ntswj, thiab C* kev tiv thaiv kev nkhaus. Hauv peb cov kev suav tom ntej, peb yuav tsis suav nrog cov tswv cuab uas muaj C* kom yooj yim dua. Yog yuav tsum siv cov kab zauv piv txwv rau I-profile, ces yuav tsum ntxiv kom tsim nyog. Kev tsis suav nrog tus tswv cuab v’’ txhais tib yam li kev xav tias B1=∞.
Yog tias tshem tawm M thiab MD siv kab. 4 tau txais daim ntawv differential rau kev khoov sab ib sab ntawm ib rab nqaj uas muaj ntu tas li
Qhov sib npaug sib txawv (5) yog homogeneous thiab muaj qib plaub raws ϑ. M1 yog ib qho haujlwm uas paub ntawm z nyob ntawm qhov hnyav uas raug thauj.
Nqaj nrog lub moment khoov tas mus li
Cia p=N1=0, ntxiv rau ntawd Mx=ɱ=const. Ntu hla tsis muaj flange (nruj) thiab kab nqaj ntev piv rau qhov loj ntawm ntu hla, yog li peb tuaj yeem muab C*=0. Ces kab zauv (5) yooj yim dua, vim M1=ɱ, yog li:
ϑ’’’’ + ɱ2/BC∙ ϑ’’ = 0, (6)
los yog nrog cov lus luv λ2_=ɱ__2/BC_
ϑ’’’’ + λ2 ϑ = 0. (6a)
a) Kab teeb nrog kev txhawb nqa zoo li diav rawg ntawm ob qho kawg
Daim duab. 2
Kev txhawb nqa ntawm z=0 thiab z=l yog ua los ntawm cov diav rawg (sl. 2), uas tso cai rau kev khoov u thiab v, tab sis thaiv qhov tig ntawm cov ntu kawg; ϑ=0. Vim tias hauv kev taw qhia x tsis muaj kev txhav nres (xws li M=0, lossis raws li eq. (4) kuj u’’=0), los ntawm eq. rau u’’: MD’=0 thiab yog li ntawd los ntawm eq. (4c) ua lwm yam ciam kev mob ϑ’’=0.
Kev daws dav dav ntawm kab ke. (6a) yog
ϑ(z) = A1∙sinλz + A2∙cosλz + A3∙λz + A4. (7)
Los ntawm cov xwm txheej ciam teb tau txais
Lub system homogeneous ntawm cov kab zauv no muaj:
- Kev daws yooj yim A1 = A2 = ∙∙∙ = 0, uas sib haum rau ϑ=0, u’’=0 lossis u=0, thaum lub nqaj tsis tig sab nraud.
- Cov kev daws tshwj xeeb raws Δ=0 lossis sin λ__l=0. Cov cag ntawm qhov xwm txheej no ntawm kev khoov sab yog λ__l=n__π (nrog n=1,2,3,∙∙∙) thiab cov kev thauj lub sijhawm tseem ceeb uas phim
ɱ__K = _√_BC ∙ _n__π/_l. (8b)
Lub moment me tshaj plaws ntawm kev khoov sab
min__ɱ__K = _√_BC ∙ π/l (8c)
tshwm sim ntawm n=1. Kev daws uas haum yog siv eq. (1f):
ϑ(z) = A1 ∙ sin πz/l (8d)
u(z) = C/ɱK ∙ ϑ(z).
b) Lwm yam xwm txheej ntawm kev tig nram qab
c) Kev khoov sab vim lub moment khoov thiab lub zog nias
Rau qhov piv txwv qhia nyob rau sl. 3 yog N1=-D (nias), p=0, ces muab C*=0. Tom qab ntawd kab zauv txawv txav. (5) yooj yim dua thiab hais tias
ϑ’’’’ + λ2 ϑ’’ = 0, (9)
qhov twg λ__2 = ɱ__2_/BC + D/B_. (9a)
Kev daws ntawm qhov kev sib npaug sib txawv (9) tau muab dua los ntawm qhov sib npaug (8), tsuas yog tam sim no λ qhia txog lwm tus nqi.
Daim duab 3
Rau qhov piv txwv hauv sl. 3 siv tib yam li rau a) cov xwm txheej ciam teb ϑ=0 thiab ϑ’’=0 rau z=0 thiab z=l. Yog li ntawd cov kab lus rau A1 kuj zoo ib yam hauv txoj kev qauv thiab kuj muaj kev mob ntawm kev khoov sab nraud Δ=sin λl=0. Lub nra tseem ceeb ua raws li kev sib raug λl=nπ (n=1,2,3) thiab tus nqi tsawg tshaj yog
ɱK2/BC + DK/B = (π/l)2. (10)
Rau D=0 tau txais los ntawm qhov ntawd lub moment ntawm kev khoov sab nraud raws li kab zauv (8c) thiab rau kev thauj nias dawb huv (ɱ=0) Euler- tus quab yuam ntawm kev khoov _DK=B(π/l)_2.
Hauv daim duab sl. 3 cov rooj plaub qhia no yuav tsum raug cim tias yog kev khoov eccentric hauv txoj kab dav hlau y-z. Yog li, rooj plaub ruaj khov uas tau hais dhau los yog kev nkhaus tawm ntawm txoj kab dav hlau ntawm kev khoov txuas nrog kev khoov sab u thiab kev tig ϑ.
Yog ob qho kawg raug khov (ϑ=0 thiab ϑ’=0) ces los ntawm cov xwm txheej ciam teb, siv Δ=0, tau txais qhov xwm txheej rau kev khoov sab
λ__l ∙ sin λ__l = 2(1-cos λl), (11)
nrog cov cag λ__l=m__π (m=2,4,∙∙∙) thiab cov zog khoov sab ib sab
ɱ__K__2/BC + DK/B = (2_π_/l)2.