Zipolopolo ndi mmene amatchedwa zonyamulira za pamwamba, pamene pamwamba pake pa pakati si lathyathyathya koma pali kupindika koonekeratu. Khalidwe lawo la mphamvu ya static limasiyana kwambiri ndi la mbale. Akakhala ndi mawonekedwe oyenera, kupsinjika kwawo kumakhala kochepa kwambiri, kulimba kwawo kumakhala kwakukulu mogwirizana, ndipo kuwerengera kwake nthawi zambiri kumakhala kosavuta.
Kupsinjika kwa chipolopolo chilichonse kungagawidwe kukhala magawo awiri: kupsinjika kwa membrane ndi kupinda. Kupsinjika kwa membrane, mofanana ndi kupsinjika pa mbale yolemedwa mu ndege yake, kungasonyezedwe pogwiritsa ntchito mphamvu zodutsa m’mbali ya tangential – awa ndi mwachitsanzo pa gawo x=const. mphamvu yofanana _Nx=_σx*t ndi mphamvu yodutsa _Nxy=_τxy*t, pomwe t ndi makulidwe a chipolopolo. Kupsinjika chifukwa cha kupinda ndi kofanana ndi pa mbale ndipo kungayimilidwe ndi ma moment a kupinda Mx, ma moment a torsion Mxy ndi mphamvu zodutsa Qx.
Makhalidwe ofunika a zipolopolo amachokera pa mfundo yakuti, mwamalamulo, kupsinjika chifukwa cha kupinda sikuli ndi kukula kapena kufunika kwa kupsinjika kwa membrane, moti nthawi zambiri kunganyalanyazidwe kwathunthu. Pamenepa kupsinjika konse kumagawidwa mofanana pa makulidwe a chipolopolo ndipo kumagwira ntchito tangential ku malo apakati. Kuwerengera komwe kumachokera pa malingaliro amenewa kulowa mkati mwa chiphunzitso cha membrane cha zipolopolo.
Chiphunzitso cha membala cha zipolopolo zozungulira zomwe zimakhala ndi symmetry yozungulira pansi pa katundu wozungulira komanso wofanana
Mawerengedwe aakulu
Kwa mizere ya ma coordinates amatengedwa meridiani ndi mabwalo ofanana pa pamwamba lapakati pa chipolopolo, choncho ma coordinate ndi awa: azimut ϑ (“utali wa malo”) ndi kukhota φ kwa tangenti pa meridiani poyerekezera ndi ndege yokhotakhota ku axis ya kufanana (sl. 1a). Choncho mphamvu za kudutsana ndi izi: mphamvu yowongoka Nφ m’mbali mwa meridiani ndi mphamvu yowongoka Nϑ m’mphete (sl. 1b). Mphamvu za sheya Nφϑ sizimawoneka pa kupanikizika kozungulira kofanana. Katundu pa gawo la pamwamba amasonyezedwa motere (sl. 1b): Y mwa tangenti ku meridiani, yotsatira ngati φ ikukula, Z yowongoka ku chipolopolo, yotsatira kupita ku axis ya kufanana.
Chith. 1
Chith. 2
Tiyeni tiganize kuti radiyasi ya kupindika kwa meridiani ndi r1. Ndi ntchito ya ngodya φ kotero mawonekedwe a meridiani amatsimikiziridwa ndi chilinganizo _r1=r1(_φ) . Izi zitha kuonedwa ngati chilinganizo cha meridiani ndipo limati, mwachitsanzo.
za bwalo r1=a=const.
kwa parabolo r1=a/cos3__φ
za elipsi r1=a2b2(a2sin2__φ+b2cos2__φ)-3/2
Pakuwerengera mphamvu yowongoka N__φ m’njira ya meridiani, muyenera kukhazikitsa mfundo ya kusamala ya gawo la chipolopolo lomwe lili pamwamba pa bwalo lofanana φ (chith. 2). Chotsatira R cha katundu Y ndi Z womwe ukugwira gawo ili ndi chowongoka, chifukwa cha kufanana. Kukula kwake kumapezeka mwa kuphatikiza pa malo a chipolopolo kuyambira φ’=0 mpaka φ’=φ. Malo a chigawo cha chipolopolo ndi dF=r1d__φ’*rd__ϑ, choncho gawo lowongoka la katundu (Ysin__φ’+Zcos__φ’)dF. Pa gawo lonse la mphete la chipolopolo lomwe lili pakati pa mabwalo φ’ ndi φ’+dφ’, chotsatira cha katundu wakunja ndi:
dR=2__πr (Ysin__φ’+Zcos__φ’) r1d__φ’,
pomwe pothandizira kuphatikiza pa φ’ pa gawo la chipolopolo lomwe lili pamwamba pa bwalo lofanana φ’=φ limapezeka:
Nj. 1
Katundu umenewu amalandiridwa ndi mphamvu zachibadwa m’mbali ya meridiani, zomwe zimagwira ntchito kuzungulira kwa bwalo looneka; gawo loyima la chiŵerengero cha mphamvu zimenezi ndi 2__πr * N__φ sin__φ, kotero poyerekezera kuchokera pamenepa timapeza mphamvu yachibadwa yomwe ikufunika
Eq. 2
Ngati chipolopolocho chapendeketsedwa pamwamba kudutsa mozungulira φ=φ0 (mwachitsanzo, dzenje lowunikira mu dome), malire otsika a integral, ndithudi, ndi φ0, osati zero. Ngati pa m’mphepete mwa chipolopolo chomwe chawonekera motere palinso katundu R0, uwonso uyenera kuwonjezedwa ku resultant R (chith. 3).
Chith. 3
Kuti tidziwe mphamvu ya normal N__ϑ, muyenera kuyika mfundo ya kufanana kwa chinthu cha chipolopolo choyang’ana pa ndege ya tangential (chith. 4). Zotsatira za katundu wakunja m’bali imeneyi ndi Z r1 d__φ r d__ϑ. Mphamvu N__φ*r d__ϑ zomwe zili mu ndege ya meridial zimapanga pakati pawo ngodya yaying’ono d__φ, choncho zotsatira zake m’bali ya normal ku chipolopolo ndi N__φ r d__ϑ*dφ. Momwemonso, mphamvu Nϑ r1 dφ zomwe zili mu ndege ya bwalo la horizontal ndipo zimapanga pakati pawo ngodya dϑ zimakhala ndi zotsatira Nϑ r1 dφ*dϑ zomwe nazonso zili mu ndege ya bwalo la horizontal ndipo choncho zimapanga ndi normal ku chipolopolo ngodya 0,5_π-φ_, motero m’kuyesetsa kufanana limangolowa chigawo chawo Nϑ r1 dφ dϑ*sinφ. Choncho, mfundo ya kufanana ndi iyi
kapena, tikagawa ndi r r1 dϑ dφ:
Mmodzi.3
Apa ndi r2=r/sinφ yasonyezedwa ngati theka-lifupi lalikulu lachiwiri la kupindika kwa chipolopolo.
Chith. 4
Denga lozungulira
Pamene mawonekedwe a meridiyeni aperekedwa mwachizindikiro ndi equation r1=r1(φ) ndipo pamene mawu a katundu Y(φ) ndi Z(φ) aperekedwa mwachizindikiro, mphamvu za kudula mu chigoba zingathe nthawi zonse kudziwika mwa kuphatikiza mu mawonekedwe otsekedwa kapena mwa njira ya chiwerengero, pogwiritsa ntchito lamulo la Simpson-. Kwa dome yozungulira ya m’litali a, pansipa amaperekedwa mphamvu za kudula pa milandu ina ya katundu:
Chitsanzo chomaliza, chomwe pa φ=0 chimapereka mphamvu za kudula zopanda malire zazikulu, chimagwira ntchito kokha patali pang’ono kuchokera pa malo amene mphamvu ikugwirira. Pafupi kwambiri ndi malo amene mphamvu ikugwirira, ngakhale itagawidwa pa pamwamba pa bwalo lokhala ndi m’mimba mwake yomaliza koma yaying’ono, katunduyo makamaka umafalitsidwa ndi kupindika.
Maonekedwe a zipolopolo zotseguka pamwamba angachotsedwe kuchokera ku zakale, ngati pa katundu weniweni wayikidwa katundu woyerekezera P pa nsonga, kukula kwake kumatsimikizidwa motere ngati mphamvu yonse ya normal m’mbali ya meridiani yomwe imagwira pa m’mphepete wapamwamba wa chipolopolo φ=φ0 ili Nφ=0, kapena ili yofanana ndi mtengo wina wotsimikizidwa ndi mphamvu zakunja zomwe chipolopolochi chimalandira pano.
Chipolopolo cha koni
Chith. 5
Pa chipolopolo cha koni, kupendekeka kwa meridiana φ sikungagwiritsidwe ntchito ngati koordinati, chifukwa kumakhala ndi mtengo womwewo m’malo onse. M’malo mwake amalowetsedwa mtunda s kuchokera ku nsonga ya dome woyezedwa m’mbali mwa jeneratala (sl. 5). Mphamvu yopingasa m’njira ya meridiana motero imatchedwa Ns. Maonekedwe ofunikira angapezeke kuchokera ku omwe ali pamwambapa pa 1 pamene kusintha kwa malire kuchitidwa. Kuchokera ku equation (2) motero kumapezeka
a from equation (3)
Nϑ = -r2 Z = -Z s ctga.
Pa milandu yofunika kwambiri ya katundu, mafomula awa ndi amene amagwira ntchito:
Njira ya zojambula pa mawonekedwe a meridian aliwonse
Mth. 6
Pamene curve ya meridian sinaperekedwe ndi mawu a analitiki, koma, mwachitsanzo, mwa chithunzi, kudziwa radius ya kupindika kumakhala kovuta kwambiri komanso kosalondola kwambiri. Pamenepa ndi bwino kugwiritsa ntchito njira ya zithunzi (sl. 6).
Chipolopolocho chigawidwa ndi ma plane ambiri φ=const., kulemera kwa magawo a mphete Δ_R_ omwe ali pakati pa ma plane amenewo kuwerengedwa, ndipo bwino ndithu kugawa nthawi yomweyo ndi 2_π_. Mphamvu izi ΔR/2π zimayikidwa m’ndandanda wa mphamvu (chith. 6b). Ngati pambuyo pake, mwachitsanzo, kudzera pa mfundo yogawa 7 kachiwiri kaja kolunjika kofanana ndi tangenti pa meridiyani pa mfundo ya meridiyani 7, horizontal kuchokera pa mfundo yapamwamba ya ndandanda ya mphamvu idzadula pa iyo utali R1/2π sinφ7 womwe kuchokera pa equation (2) pogawa ndi r yoyenera amapeza mphamvu yoyenera ya normal m’tsogolo mwa meridiyani.
Kuti tidziwe mphamvu mu mphete, amagwiritsa ntchito lamulo la kulinganizika kwa mphamvu zopingasa pa chinthu chimodzi cha chigoba. Ngati pali katundu woyima yekha, lamulo ili limati (sl. 4):
Pamene pano dφ asinthidwa ndi kusiyana komaliza Δφ, komwe kukugwirizana ndi kugawa kovomerezeka kwa chigoba, ndipo m’malo mwa r1 dφ pakhazikitsidwa kutalika Δs la chigawo cha meridiani, kuchokera pano timapeza
M’bulaketi ya kumanja muli odsechal pa mzere wopingasa mu dongosolo la mphamvu. Choncho mbali yonse ya kumanja imaimira zigawo pakati pa mfundo zogawa pa mzerewu. Kuti tithe kuziwerenga molondola mokwanira, dongosolo la mphamvu liyenera kujambulidwa mosamala komanso pa sikelo yayikulu mokwanira.
Mphete zotambasulidwa ndi zopanikizidwa
Chipolopolo chilichonse chofanana mozungulira chimakhala cholembedwa ndi bwalo limodzi kapena awiri owongoka (chith. 7). Pa m’mphepete amenewa, mwamfundo, chingalandire kokha katundu ndi ma reaction a thandizo omwe ali ndi njira ya tangent ku meridian. Ngati mphamvu zakunja zili ndi njira ina, monga katundu P kuchokera ku gawo lapamwamba la kapangidwe kapena reaction S ya pansi pa tank yomwe yawonetsedwa pa chith. 97, mphamvu zimenezi ziyenera, monga chasonyezedwera pachithunzipo, kugawidwa kukhala zigawo m’njira ya bwalo lopingasa ndi tangent ku meridian. Gawo P/sinφ0 kapena S/sinφu likufanana ndi mphamvu yofanana m’njira ya meridian Nφ yomwe imagwira ntchito pamalo amenewo, pamene kuti kulandira mphamvu yopingasa Pctgφ0 kapena Sctgφu pamafunika mphete yomwe imalandira kukoka, kapena kukanikiza, ndipo mwa iyo katundu wa radially uwu umayambitsa mphamvu yofanana + P r0 ctgφ0, kapena -S ru ctgφu. Mphete zotere, kuwonjezera pa m’mphepete mwa chipolopolo, ziyenera kuikidwanso pa malo onse amene mzere wa meridian uli ndi kupindika. Nthawi zonse zimayambitsa kusokonezeka kwa mkhalidwe wa membrane wa kupsinjika.
Chith. 7
Mavuto chifukwa cha kupinda mu zipolopolo zozungulira symmetrically
Maonekedwe a chiphunzitso cha membrane alibe chiwerengero chokwanira cha ma constant a kuphatikiza kuti mikhalidwe yonse ya malire yomwe imagwirizana ndi ntchito yeniyeni ithe kukwaniritsidwa. Makamaka pa zipolopolo za symmetric za kuzungulira, n’zosatheka, posankha bwino ma constant a kuphatikiza, kukwaniritsa kuti kukulitsidwa εϑ = Nϑ/Et m’njira ya mphete m’mphepete mwa chipolopolo kukhale kofanana ndi kukulitsidwa kogwirizana kwa mbali ina yomanga yomwe imamangirizidwa mwamphamvu nacho pamalo amenewo. Zimenezi zimagwiranso ntchito pa zipolopolo zomwe m’bwalolo limodzi lozengereza kusintha kwa meridiana, makulidwe a chipolopolo kapena katundu pa unit ya malo zimasintha mwadzidzidzi. Ngakhale chipolopolo chili m’mphepete mwa lina sichinakhazikitsidwe ndi mphete, koma pamalopo chili ndi katundu wa mphamvu zomwe zili ndi gawo m’njira ya perpendicular ku chipolopolo (monga katundu P pa sl. 7 pamene tawona ngati mphete yachotsedwa), sichingalandire katundu wotere kudzera pa mphamvu za membrane zokha. M’mikhalidwe yotereyi mphamvu yodula ndi ma moment a kupinda (sl. 8) zimakhala ndi gawo lofunikira pa kugawa kwa mphamvu mu chipolopolo, ndipo izi ziyenera kuganiziridwa pa kuwerengera kwa ma static.
Chith. 8
Kuwerengera kolondola kwa chipolopolo pa kupindika, kupatula pa nkhani yosavuta kwambiri yapadera ya chipolopolo cha silinda, ndi ntchito yovuta kwambiri. Kwa zipolopolo zokhala ndi makoma oonda tingamange chiphunzitso chosalirana koma chothandiza kwambiri cha pafupi, chomwe chimadalira pa mfundo yodziwika m’lingaliro lolimba kuti zovuta zobwera chifukwa cha kupindika zimangokhalira m’dera laling’ono pafupi ndi m’mphepete ndipo zimachepa msanga pamene mtunda kuchokera m’mphepete ukuwonjezeka. Izi zimathandiza kuti mawu ambiri mu equation yolondola ya differential asagwiritsidwe ntchito, ndipo equationyo imakhala ndi mawonekedwe osavuta.
Ngati mkati mwa ivična tona amene ali yekha wofunikira κ alowedwa ndi mtengo wapakati, ndipo izi pafupifupi nthawi zonse zingatheke, yankho la equation ndi
Chiy. 4 ndi 5
kumene a1, b1, a2, b2 ndi ma integrala constant. Popeza yankholi limakhala losangalatsa kokha pafupi ndi m’mphepete φ=φ0, n’bwino kutenga mtunda wa ngodya ω=φ0-φ kapena ω=φ-φ0 kuchokera pamalire amenewa ngati koordinati, choncho potembenuza yankho lomwe latchulidwa pamwambapa timapeza mawu ena omwe atchulidwa. Apa pamawoneka ma constant awiri okha a integrala A, B kapena C, ψ, chifukwa gawo lina la yankho lomwe lili ndi chinthu eκω, ndipo chifukwa chake silichepa pamene ω ikukula, lingachotsedwe pa kulingalira.
Pochokera pa equation 5 amapezeka mfundo zotsatirazi za mphamvu za gawo ndi kutembenuka κ kwa tangenti ya meridiyani:
Nkhawa zomwe zimachitika chifukwa cha kupindika m’matanki a mawonekedwe a silinda yozungulira
Tiyerekeze kuti thanki ya cylindrical ya kutalika h yadzazidwa ndi madzi (chith. 9), ndiye kuti pa kutalika x pamwamba pa matanki awiri kuthamanga kwa madzi ndi p=γ(h-x). Ngati pa kutalika kumeneku, pogwiritsa ntchito magawo awiri a poyizontal omwe ali pa mtunda dx., mphete idulidwa mu thankiyo ndipo mphete imeneyi igawidwe m’magawo awiri m’lifupi mwa diamita imodzi, mkhalidwe wa kufanana wa gawo limodzi umapereka mphamvu ya membrane mu mphete Nφ=pa. Ngati makulidwe a thanki pa malo amenewa ndi t, kutambasuka kwa mphete ndi εφ=Nφ/Et ndipo motero kuwonjezeka kwa radius ya thanki (kupinda) ndi
Pa m’mphepete mwa pansi x=0 choncho
Chith. 9
Palibe limodzi kapena lina, monga lamulo, lomwe limagwirizana ndi malire, chifukwa kusintha kwa khoma kumalephereka chifukwa cha kulumikizana ndi pansi pa thanki. Pa kulumikizana pakati pa pansi ndi khoma la silinda pamapezeka ma mphamvu a transversal Qx0 ndi ma moment Mx0 (chith. 11), omwe kukula kwake kuyenera kudziwika motero kuti pamodzi ndi ma membrane forces omwe awerengedwa posachedwapa mu chipolopolo zikhale zoyambitsa kusinthika kumeneku pa m’mphepete mwa chipolopolo komwe pansi limalola.
Chith. 10, 11, 12, motsatana
Pa chith. 11 akuwonetsedwa chinthu cha shell dx*a dφ ndi mphamvu zomwe zimachitira pa icho. Chikhalidwe cha kusalingana kwa mphamvu mu njira ya perpendicular ku shell chimapereka
dQx a dφ + Nφ dx dφ = p a dφ dx,
kumene pogawa ndi dx dφ:
a Q’x + Nφ = p a.
Kuchokera ku chofunikira chakuti moment pa tangent yopingasa ya silinda ndi zero, timapeza kuti:
dM/dx = M’x = Qx
Mwa kuchotsa mphamvu ya transverse Qx m’mayeso awiri awa zikutsatira
a M’’x + Nφ = p a.
Maf. 6
Mu equation iyi, mphamvu zosadziwika zodutsa zimenezi zingasonyezedwe pogwiritsa ntchito kupinda w. Kwa Nφ kwapezeka kale Nφ=Etw/a, ndipo momentiyo ili molingana ndi kupindika kwa meridian w’’, ndiko kuti Mx=Kw’’, pomwe, monga pa ma plate, K=Et3/12(1-μ2) ndi rigidity ya chipolopolo, yomwe ingakhale yosiyanasiyana ndi x. Ngati mawu awa a mphamvu zapakati alowetsedwa mu equation (6), equation ya differential ya chiphunzitso cha thanki imapezeka:
Pa thanki yomwe makulidwe a khoma lake sasintha, yankho lake ndi ili:
Mmodzi.7
kumene λ4=3(1-μ2)/a2t2. Ngati λh ndi yayikulu (tiyeni tinene, yoposa 3), n’bwino kugwiritsa ntchito ma exponential m’malo mwa ma hyperbolic functions, choncho:
Mmodzi.8
Kenako ma konstanti A1, B1 angatsimikizidwe paokha kuchokera ku mikhalidwe ya malire pa m’munsi, ndipo ma konstanti A2, B2 kuchokera ku mikhalidwe ya malire pa m’mphepete pamwamba, pomwe kaŵirikaŵiri zimapezeka A2=B2=0. Potengera equation (8) ndi (7) mphamvu zochitirana zimasulidwa, pamene kuwerengera kuchitidwa m’njira yosinthidwa ndipo paliponse equation (8) ndi (7) ayikidwa; choncho mwachitsanzo pa A2=B2=0:
Pamene makoma a thanki amamangiriridwa molimba mu mbale, pa pansi pa thanki lomwe ndi lakuda mokwanira kuti lingawonedwe ngati lolimba, ma constants A1, B1 ayenera kudziwika kuchokera ku mfundo yakuti x=0, w=0 ndi w’=0 ndipo zimapezeka kuti:
Ngati pansi pa thankiyo ndi mbale kapena chipolopolo chotanuka, mphamvu zopingasa Qx ndi ma moment Mx m’mphepete mwa bwalo pakati pa pansi ndi khoma la silinda ziyenera kutsimikizidwa pogwiritsa ntchito njira za chiphunzitso cha machitidwe osatsimikizika mosavuta potengera zofunikira kuti Mx ndi w’ pa mbali zonse ziwiri zikhale ndi mtengo womwewo.