Kwi ukugoba ngexesha lokugoba intonga, ngexesha lotshintsho lokulingana igotywe kwinqwelwana enye ephambili.

Kodwa zikho nezo meko zozinzo apho intonga, okanye umqadi, xa ibhalansi itshintsha ingagobi kuphela kodwa ijike nakuye (umz. 60a). Kwisixhasi esi sibonakaliso sibizwa ngokuba kukujika ecaleni, kwiintonga ezicinezelweyo, ezithi ngexesha lokujika zibe phantsi kwetorqhi kwakhona, kuthethwa ngokujika phantsi kwetorqhi.

Kuzo zombini ezi ziganeko, ubukhulu bokungqinelana nokujijeka C, okanye ukumelana nokugoba C*, bubaluleke kakhulu. Utshintsho lokulingana olunxulunyaniswa nokujijeka lunokuphembelelwa ikakhulu ngamanyathelo anyusa ukuqina ngokuchasene nokujijeka (imilo efanelekileyo yesiphambuka, izakhelo ezinqamlezileyo, ukuxhaswa ecaleni, njl.). Kwelinye icala, iintonga ezimile njengeteyiphu ezinesiphambuka esivulekileyo ziziva kakhulu ekugobeni xa kujijeka.

Umlinganiselo wahlulo wokugoba ecaleni

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Umz. 1a

Masicinge ukuba ibhimu ethe tye inemizuzwana ye-inertia engaguqukiyo enobungakanani obungalinganiyo kakhulu Ix >> Iy. Umthwalo ulele kwinqwelomoya y-z-, ukuze ibhimu igotywe kule nqwelomoya engundoqo (u=ϑ=0). Amava kunye nethiyori zibonisa ukuba le ndawo yokulingana ekuqaleni izinzile, kodwa kwiimithwalo ezinkulu iba engazinzanga. Le ndawo yesibini yokulingana inxulumene nokugoba okuthe nkqo kunye netorsion ϑ.

Apha ngezantsi kuya kuveliswa isivakalisi esahlulayo sokugoba ecaleni. Izivakalisi ze u kunye ne ϑ zingezohomogeneous. Ukuba neemeko zomda nazo zingezohomogeneous, ngoko ke ukugoba ecaleni kwentsika kunokwenzeka kuphela kumaxabiso esona sixhobo, oko kukuthi u no ϑ bahlala kuqala bengama-zero kwintsika echazwe ngasentla engenaphazamiseko. Imeko yokugoba ecaleni ifunyanwa – ngokufanayo nokugoba kwentonga – ukuba kuthathwe ukuba i-determinant yezinto ezikwizivakalisi zemeko ezihomogeneous zezikali zokuhlanganisa ilingana no-zero. Siza kubona ukuba nakukugoba ecaleni kukho inqaku lokwenene lokwahlukana kolingano oluguquguqukayo.

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Umz. 1b

Kumz. 1b kuboniswe into yomqadi enobude dz. Iziphumo zoxinzelelo N1, Q1, M1, kunye nokugoba κ__1 kuxhomekeke kumthwalo kwaye zizixa zokugqibela. Q, M, MD zingoo-zero kwindawo yokuqala yokulingana, zize zibe zahluke ku-zero kuphela xa kuvela ukugoba okusecaleni. Izixa zokugqibela, ezinxulumene nesiqalo sokugoba okusecaleni, zincinane ngokungenasiphelo, njengoko kunjalo nokuguquka okufanelekileyo κ ne d__ϑ/dz.

Omabini amacandelo okugqibela e-elementi dz alele kwindawo ethe nkqo kwi-asi yomqadi ejijekileyo kabini kwaye aya kujijwa kwezi ndawo zithe nkqo.

Kuba thina – njengakwimeko yokugoba kwenduku – kuqala kusikhathaza ubungakanani bomthwalo obalulekileyo, oko kukuthi ukuqala kokugoba ecaleni, kungekhona kakhulu ubungakanani beenguqu, singazityeshela iimveliso zobukhulu Q, M, MD nge κ okanye ϑ’ njengezinto ezincinane zomyalelo wesibini. Ngoko ke, ukuze kuqhubeke ukwenziwa lula, siza kuthatha negophe eliphambili κ1 njengelincinane kakhulu yaye liya kuphathwa njenge κ ne ϑ’.

Qaphela: Impembelelo ye κ__1 kufuneka iqwalaselwe nakwithiyori kwimeko elula ekhethekileyo (umzuzu wokugoba ongaguqukiyo M njengomthwalo). Kwiimfuno ezininzi zobunjineli bokwakha, κ__1 inokungahoywa. Singacinga, umzekelo, ngokungathi intonga kule meko yomthwalo “iphakanyisiwe” kangangokuba kumthwalo obalulekileyo kanye κ__1=0.

Kusuka ekulinganeni kwamacandelo emikhosi kwiindlela ξ, η, ζ kunye neemeko zemizuzu ngokubhekiselele kwalaa ma-asi afanayo kufunyanwa, emva kokungakhathalelwa okukhankanyiweyo,

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)

Kwizilinganiso. (1c, d i b) kufumaneka

N1 = const.,

Q1 = M1’,     (2)

p = - Q1’ = - M1’’.

Ngokuphelisa Q1 kwisib. (1a) kufumaneka

Q’ = (M1’ ϑ_)’ – N1u’’_.   (3a)

Kwisib. (1e) ngaloo ndlela Q iya kususwa, emva koko u ’’ ngokusebenzisa u’’ = (MD’ + pe ϑ_)/M1_

(M+M1 ϑ_)’’ – N1/M1_ ∙ (MD’ + pe ϑ_) =_ 0_._   (3b)

Ukuze kufunyanwe i-equation eyahlukileyo ye ϑ kuya kufakwa ezi budlelwane zilandelayo phakathi kweemomenti nokuguquka

M1 = -B1_∙v’’,_   (4a)

M = +B_∙_u’’,   (4b)

MD = +C_∙ϑ’ – EC*∙ ϑ’’’,_   (4c)

apho kubonisa B1=EIx kunye no B=EIy ukuqina ngokuchasene nokugoba komqadi, C ukuqina ngokuchasene nokujijeka, kwaye C* ukuxhathisa ekugotyweni. Kwiingxoxo zethu eziqhubekayo siya kushiya amalungu ane C* ukuze kube lula. Ukuba ezi zibalo kufuneka zisetyenziswe, umzekelo, kwiiprofiles ze-I, ngoko kufuneka zongezwe ngendlela efanelekileyo. Ukungahoyi ilungu v’’ kuthetha okufanayo nokucinga ukuba B1=∞.

Ukuba kususwa M kunye MD ngoncedo lwesigq. 4 kufumaneka isibalo esihlukileyo sokugoba ecaleni kwebhimu enqamlezileyo engaguqukiyo

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I-equation yohlobo olwahlukileyo (5) iyahambelana yaye ikwinqanaba lesine ngokwe ϑ. M1 ngumsebenzi owaziwayo ka z ngokuxhomekeke kumthwalo.

Umqadi onomzuzu wokugoba ongaguqukiyo

Masingathathe p=N1=0, kwakhona Mx=ɱ=const. Icandelo alinawo umqolo osongiweyo (lixinene) yaye umqadi mde ngokweemilinganiselo zecandelo elinqamlezileyo, ukuze sinokubeka ukuba C*=0. Emva koko isib. (5) siyenzeka lula, ngenxa ye M1=ɱ, ngoko ke:

ϑ’’’’ + ɱ2/BC∙ ϑ’’ = 0,   (6)

okanye ngesishunqulelo λ2_=ɱ__2/BC_

ϑ’’’’ + λ2 ϑ = 0.   (6a)

a) Umqadi oneenkxaso ezifana neefolokhwe kuzo zombini iziphelo

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Fig. 2

Ukuxhaswa ku z=0 kunye no z=l kusenziwa ngeefolokhwe (umz. 2), ezivumela ukugoba u no v, kodwa zithintela ukujikeleza kwamacandelo okuphela; ϑ=0. Kuba kwicala x akukho kuqiniswa okungqongqo (oko kukuthi M=0, okanye ngokwenx. (4) kwakhona u’’=0), kuvela kwis. se u’’: MD’=0 yaye ngaloo ndlela kwis. (4c) njengomqathango wesibini womda ϑ’’=0.

Isisombululo esiqhelekileyo se-equation (6a) ngu

ϑ(z) = A1∙sinλz + A2∙cosλz + A3∙λz + A4.   (7)

Kwiimeko ezikwindlela yomgca kufunyanwa

Inkqubo yezibalo efunyenwe kwiimeko zomda

Le nkqubo ilinganayo yeemeko inayo:

  • Isisombululo esilula A1 = A2 = ∙∙∙ = 0, apho kuhambelana ϑ=0, u’’=0 okanye u=0, kwaye umqadi awugobi ecaleni.
  • Izisombululo zazo ngokwe Δ=0 okanye sin λ__l=0. Iingcambu zale meko yokugoba ecaleni ziyi λ__l=n__π (nge n=1,2,3,∙∙∙) kunye nemithwalo ye-moment ebalulekileyo ehambelanayo

ɱ__K = _√_BC ∙ _n__π/_l.   (8b)

Umzuzu omncinci wokugoba ecaleni

min__ɱ__K = _√_BC ∙ π/l   (8c)

kuvela ku n=1. Isisombululo esifanelekileyo sifumaneka ngokusetyenziswa kweenj. (1f):

ϑ(z) = A1 ∙ sin πz/l   (8d)

u(z) = C/ɱK ∙ ϑ(z).

b) Amanye amatyala okujika ecaleni

ibhodi

c) Ukugoba ecaleni ngenxa ye-moment yokugoba kunye namandla acinezelayo

Kumzekelo oboniswe ku sl. 3 ngu N1=-D (uxinzelelo), p=0, emva koko kufakelwe C*=0. Ngelo xesha i-equation eyohlukeneyo (5) iyalula ibe ithi

ϑ’’’’ + λ2 ϑ’’ = 0,   (9)

apho λ__2 = ɱ__2_/BC + D/B_.   (9a)

Isisombululo se-equation ehlukana (9) sinikezelwe kwakhona ng.equ. (8), kodwa ngoku λ imele elinye ixabiso.

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Umz. 3

Kumzekelo womz. 3 kusebenza, njengakwi a), iimeko zomda ϑ=0 kunye no ϑ’’=0 ku z=0 no z=l. Ngoko ke, izibalo zemeko ze A1 zifana ngokusesikweni kwaye kunjalo nomqathango wokugoba ecaleni Δ=sin λl=0. Umthwalo obalulekileyo wanelisa ubudlelwane λl=nπ (n=1,2,3) yaye ixabiso elincinane ngulo

ɱK2/BC + DK/B = (π/l)2.   (10)

Ku D=0 kufumaneka ukusuka apho umzuzwana wokugoba ecaleni ngokomthetho. (8c) yaye kumthwalo oxinzelayo ococekileyo (ɱ=0) amandla ka_Euler_ okugoba _DK=B(π/l)_2.

Kumz. 3 imeko ebonisiweyo kufuneka ibizwe ngokugoba okungabangelwanga embindini kwiplani y-z. Ke ngoko, imeko yozinzo ebethethwe ngaphambili kukugoba okuphuma ngaphandle kwendiza yokugoba okunxulunyaniswa nokugoba ecaleni u kunye ne-torsion ϑ.

Ukuba zombini iincam zixinenwe (ϑ=0 kunye ϑ’=0) ngoko ukusuka kwiimeko zomda, kusetyenziswa Δ=0, kufumaneka imeko yokugoba ecaleni

λ__l ∙ sin λ__l = 2(1-cos λl),    (11)

ezineengcambu λ__l=m__π (m=2,4,∙∙∙) kunye nemikhosi yokugoba ecaleni

ɱ__K__2/BC + DK/B = (2_π_/l)2.