Ukuze ukugobeka ngesikhathi sokugoba induku, lapho ibhalansi ishintsha, igobeka endizeni eyinhloko eyodwa.
Kodwa kukhona nezehlakalo ezinjalo zokuzinza lapho induku, okungukuthi ugongolo, lapho kushintsha ibhalansi ingagobi nje kuphela kodwa futhi isonteke (sl. 60a). Kuma-bearing lesi simo sibizwa ngokuthi ukugoba eceleni, kanti ezindukwini ezicindezelwe, okuthi lapho zigobeka zibe ngaphansi nokusonteka, kukhulunywa ngo-ukugoba ngaphansi kwe-torsion.
Kuzo zombili lezi zimo, ubukhulu bokuqina bokumelana nokusonta C, noma ukumelana nokugoba C*, buyona into ebaluleke kakhulu. Ngakho-ke, ushintsho ekulinganeni oluhambisana nokusonta lungathonywa ngokuyinhloko ngezindlela ezandisa ukuqina kokumelana nokugingqika (ifomu elifanele lesiphambano, amafreyimu aqondile, ukwesekwa ohlangothini njll.). Ngakolunye uhlangothi, izinduku ezimise njengomucu ezinesiphambano esivulekile ziba sengozini ikakhulukazi yokugoba ngesikhathi sokusonta.
Isibalo esihlukile sokugoba eceleni
Umfan. 1a
Ake ibhimu eqondile ibe namamomenti e-inertia angaguquki anobukhulu obuhlukene kakhulu Ix >> Iy. Umthwalo usemkhakheni y-z-ngakho-ke ibhimu izogobeka kule ndawo eyinhloko (u=ϑ=0). Okuhlangenwe nakho kanye nethiyori kukhombisa ukuthi lesi simo sokulingana ekuqaleni sizinzile, kodwa emithwalweni emikhulu siba yisimo esingazinzi. Lesi simo sesibili sokulingana sihlotshaniswa nokugoba okuqondene nendawo kanye ne-torsion ϑ.
Ngezansi kuzokhishwa isibalo sokwahlukanisa sokugoba eceleni. Izibalo zika u no ϑ ziyahambisana. Uma nezimo zomngcele nazo ziyahambisana, khona-ke ukugoba eceleni kwebhimu kungenzeka kuphela ngamanani ayo siqu senkinga, okungukuthi u no ϑ kuqala kuhlala kuyiziro kubhimu echazwe ngenhla engenakho ukuphazamiseka. Isimo sokugoba eceleni sitholakala – ngendlela efanayo nokugoba kwenduku – uma kubekwa ukuthi ideterminanti yezibalo zokuhambisana ezifanayo zezinombolo zokuhlanganisa ilingana noziro. Sizobona ukuthi nasekugobeni eceleni kukhona iphuzu langempela lokuhlangana kabusha kokulingana okunwebekayo.
Umfanekiso 1b
Ku-sl. 1b kukhonjiswe isici sebhimu esinobude dz. Imiphumela yokucindezeleka N1, Q1, M1, kanye nokugobeka κ__1 kuncike emthwalweni futhi kuyizilinganiso zokugcina. Q, M, MD ziyi-zero endaweni yokuqala yokulingana, futhi zizophambuka ku-zero kuphela lapho kuqala ukugoba eceleni. Izilinganiso zokugcina, ezihlobene nokuqala kokugoba eceleni, zincane kakhulu ngokungenamkhawulo, njengokuguquka okuhambisanayo κ kanye no d__ϑ/dz.
Zombili izingxenye zokugcina zesici dz zilele endizeni ejwayelekile ku-eksisi yebha egobile kabili futhi zizophenduka kulezi zindiza ezijwayelekile.
Njengoba nathi – njengokugoba kwenduku – ngokuyinhloko sikhathalela ubukhulu bomthwalo obucayi, okusho ukuqala kokugoba eceleni, hhayi kakhulu ubukhulu bokuguquka, singakwazi ukunganaki imikhiqizo yobukhulu Q, M, MD no κ noma ϑ’ njengezinto ezincane zesigaba sesibili. Ngakho-ke ukuze kube lula ngakumbi sizothatha nokugoba okuyinhloko κ1 njengokuncane kakhulu futhi kuzophathwa njengo κ no ϑ’.
Qaphela: Umthelela we-κ__1 kufanele futhi uhlolwe ngokomqondo esimeni esilula esikhethekile (i-moment yokugoba engaguquki M njengomthwalo). Ezimweni eziningi zobunjiniyela bokwakha κ__1 ingashaywa indiva. Singacabanga, isibonelo, sengathi i-beam yaleyo meko yomthwalo ‘iphakanyiswe’ kanjalo kangangokuthi emthwalweni obucayi kukhona nje κ__1=0.
Kusukela ekulinganiseni izingxenye zamandla eziqondisweni ξ, η, ζ kanye nemibandela yamamomenti maqondana nalawo ma-asi afanayo, kutholakala ngemva kokushiywa okukhulunyiwe phansi
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)
Kusukela ezib. (1c, d no b) kutholakala
N1 = const.,
Q1 = M1’, (2)
p = - Q1’ = - M1’’.
Ngokususa Q1 kusuka ku-equation (1a) kutholakala
Q’ = (M1’ ϑ_)’ – N1u’’_. (3a)
Esuka ku-equation. (1e) ngakho-ke kuzokhishwa Q, bese u ’’ ngokuthi u’’ = (MD’ + pe ϑ_)/M1_
(M+M1 ϑ_)’’ – N1/M1_ ∙ (MD’ + pe ϑ_) =_ 0_._ (3b)
Ukuze kutholakale i-equation ehlukile ye ϑ kuzofakwa ubudlelwano obulandelayo phakathi kwemomenti nokuguquka
M1 = -B1_∙v’’,_ (4a)
M = +B_∙_u’’, (4b)
MD = +C_∙ϑ’ – EC*∙ ϑ’’’,_ (4c)
lapho B1=EIx no B=EIy kumelela ukuqina kokugoba kwebhimu, C ukuqina kokusonta, kanti C* ukumelana nokugoba. Ezibalweni zethu ezilandelayo sizoshiya ngaphandle amalungu ango C* ukuze kube lula. Uma izibalo kufanele zisetshenziswe, ngokwesibonelo, kuma-I-profile, kudingeka zengezwe ngendlela efanele. Ukunganaki ilungu v’’ kusho okufanayo nokuthatha ukuthi B1=∞.
Uma M no MD kususwa kusetshenziswa isib. 4 kutholakala isibalo esihlukile sokugoba eceleni kogodo olunengxenye enqamulelayo engaguquki
Isibalo esihlukile (5) siyisibalo esingalingani futhi sesigaba sesine maqondana no ϑ. M1 kuwumsebenzi owaziwayo ka z ngokuya ngomthwalo.
I-beam enomzuzwana oguqukayo ongashintshi
Makwenzeke p=N1=0, ngaphezu kwalokho Mx=ɱ=const. Isigaba asinawo ama-flange (sihlangene) futhi ugodo lude uma luqhathaniswa nobukhulu besigaba esiphambanweni, ukuze sikwazi ukubeka ukuthi C*=0. Khona-ke i-equation (5) iyalula, ngenxa M1=ɱ, ngakho-ke:
ϑ’’’’ + ɱ2/BC∙ ϑ’’ = 0, (6)
noma ngesifinyezo λ2_=ɱ__2/BC_
ϑ’’’’ + λ2 ϑ = 0. (6a)
a) Ugodo olunokwesekwa okufana nefolokhwe emikhawulweni yomibili
Fig. 2
Ukusekelwa ku-z=0 no-z=l kumiswa njengezibopho zemfoloko (sl. 2), ezivumela ukugobeka u no-v, kodwa zivimbele ukuzungeza kwezigaba zokuphela; ϑ=0. Njengoba ohlangothini lwe-x kungekho ukuqiniswa okuvalelekile (okungukuthi M=0, noma ngokwenq. (4) futhi u’’=0), kulandela ekuqondeni kuka-u’’: MD’_=0 futhi ngalokho kusukela enq. (4c) njengezinye izimo zomngcele ϑ’’=0.
Isixazululo esijwayelekile sesibalo. (6a) si
ϑ(z) = A1∙sinλz + A2∙cosλz + A3∙λz + A4. (7)
Kusukela ezimeni zomugqa kutholakala
Lolu hlelo lwama-equation olufanayo lunalezi:
- Isixazululo esilula A1 = A2 = ∙∙∙ = 0, okuhambisana naso ϑ=0, u’’=0 noma u=0, lapho ugongolo lungagobeki eceleni.
- Izixazululo eziyisisekelo ngokusho kwe Δ=0 noma sin λ__l=0. Izimpande zalo mbandela wokugoba eceleni zingu λ__l=n__π (ngo n=1,2,3,∙∙∙) futhi imithwalo yomzuzu ebalulekile ehambisanayo
ɱ__K = _√_BC ∙ _n__π/_l. (8b)
I-moment encane kakhulu yokugoba eceleni
min__ɱ__K = _√_BC ∙ π/l (8c)
ivela ku n=1. Isixazululo esifanele sitholwa ngokusebenzisa isib. (1f):
ϑ(z) = A1 ∙ sin πz/l (8d)
u(z) = C/ɱK ∙ ϑ(z).
b) Ezinye izimo zokugoba eceleni
c) Ukugoba ngaseceleni ngenxa yomzuzu wokugoba namandla acindezelayo
Esibonelweni esikhonjiswe ku sl. 3 kube N1=-D (ukucindezela), p=0, bese kufakwa C*=0. Khona-ke isibalo esiyisihlukanisi (5) siyenzeka sibe lula futhi sithi
ϑ’’’’ + λ2 ϑ’’ = 0, (9)
lapho λ__2 = ɱ__2_/BC + D/B_. (9a)
Isixazululo se-equation ehlukile (9) sinikezwa futhi nge-equation (8), kodwa manje λ imele inani elihlukile.
Isith. 3
Esibonelweni sl. 3 kusebenza njengaku-a) izimo zemingcele ϑ=0 no ϑ’’=0 ku z=0 no z=l. Ngakho-ke, izibalo ezinemibandela ze-A1 nazo ziyafana ngokomthetho, kanjalo nesimo sokugoba eceleni Δ=sin λl=0. Umthwalo obucayi wanelisa ubudlelwano λl=nπ (n=1,2,3) futhi inani elincane kakhulu lithi
ɱK2/BC + DK/B = (π/l)2. (10)
Ku-D=0 kutholakala kusukela lapho i-moment yokugoba eceleni ngokuvumelana nesibalo. (8c) futhi ngomthwalo ocindezelayo kuphela (ɱ=0) amandla okugoqa ka-Euler _DK=B(π/l)_2.
Ku-sl. 3 icala elibonisiwe kufanele libizwe ngokuthi ukugoba okungatheni endizeni y-z. Icala lokuzinza elicatshangelwe ngaphambilini, ngakho-ke, ukuchezuka ngaphandle kwendiza yokugoba okuhlobene nokugoba eceleni u kanye ne-torsion ϑ.
Uma womabili amaphethelo eboshiwe (ϑ=0 kanye ϑ’=0) khona-ke, kusukela ezimeni zomngcele, kusetshenziswa Δ=0, kutholakala isimo sokugobeka eceleni
λ__l ∙ sin λ__l = 2(1-cos λl), (11)
ezinezimpande λ__l=m__π (m=2,4,∙∙∙) namandla okugoba eceleni
ɱ__K__2/BC + DK/B = (2_π_/l)2.