Matsi a kan ramukan ƙasa ya ƙunshi (sl. 1):
Fig. 1. Ƙarfin matsi da ke aiki a kan tazara
a) matsin tsaye P a saman, da ake kira matsin saman baka,
b) matsin kwance S, wanda ake kira matsin gefe,
c) martanin da ke kan tushe A, wanda ake kira matsi na ƙasa.
Girman matsi na ƙasa a kan tasha yana motsawa a cikin faɗin iyaka mai yawa sosai. A farko, a lokacin buɗe tashar, matsi na dutse yana da ɗan ƙanƙanta kuma yana daga 0 zuwa 0,50 kp/cm2, amma bayan buɗewa yana ƙaruwa kuma yana daga 0,8 zuwa 12,0 kp/cm2, dangane da nau’in ƙasa da halayenta na zahiri, tsayin layukan da ke sama da tashar da kuma girman motsin da ke faruwa a ƙasa sakamakon matsi na yadudduka na sama. Saboda dogaro da abubuwa daban-daban, ƙayyade matsi na dutse a kan tasha ya fi wahala sosai, don haka ba za a iya ƙayyade ainihin ƙimar matsi na ƙasa ta hanyar ka’ida ba, sai dai kusanci kawai. Juriya ta ciki ta ƙasa tana ƙin tasirin matsi na ƙasa a kan siffar tashar. Idan ƙasar da ake yin tasha a cikinta ta ƙunshi dutse mai ƙarfi da haɗe sosai, matsi a kan siffar tashar ba ya da yawa, domin nauyin layukan da ke sama da tasha ana ɗaukar sa ne ta hanyar damuwar yanka na dutsen mai ƙarfi. A irin wannan yanayin ba lallai ba ne a yi rufin tasha ko tallafi a lokacin buɗe tasha, don haka kuma ƙayyade matsi na ƙasa a kan siffar tashar yana faɗuwa.
A cikin duwatsu marasa haɗuwa da masu rugujewa, matsin da yake fitowa daga nauyin layukan sama a kan sashin ramin na iya wuce ƙarfin irin waɗannan duwatsu, wanda hakan ke haifar da karyewar ƙasa da manyan nakasu a sashin ramin. A irin waɗannan yanayi ya kamata a yi tallafi a lokacin hako ramin, sannan daga baya a yi rufin ramin wanda zai bijire wa matsin duwatsu kuma ya watsa shi ta hanyar tushe zuwa ƙananan layuka.
A cikin ƙasa mai ɗaure, juriya ta cikin gida ta ƙunshi haɗin kai da gogayya, yayin da a cikin ƙasa mara ɗaure tana ƙunshe ne kawai da gogayya ba tare da haɗin kai ba. Duk da haka, saboda canzawar ƙimar haɗin kai ƙarƙashin tasirin danshi, da kuma laushin ƙasa mai nau’o’i dabam-dabam, ba a ɗauki haɗin kai a matsayin juriya ta cikin gida ba, don haka ko a cikin ƙasa mai ɗaure ana ƙididdige shi ne kawai da gogayya ta cikin gida. Saboda haka, a cikin waɗannan sharuɗɗa biyu na ƙarshe, ana buƙatar rufin ramin, wanda ke nufin dole ne a ƙayyade matsin ƙasa a kan rami.
Don tantance girman matsin ƙasa a kan tazara, akwai waɗannan zato:
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Zaton cewa matsin ƙasa yana ƙaruwa daidai gwargwado da zurfi
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Tsammani da ya ginu a kan dokokin daidaiton jikin da ke rushewa
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Hasashen samuwar kwanon sauke nauyi a cikin ƙasa
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Hasashe da aka gina a kan dokokin elasticity.
Tsammanin cewa matsin lamba yana ƙaruwa daidai da zurfi
Bisa ga wannan zato, a kan ɗan tunnel ɗin (sl. 2) akwai matsin koli P saboda nauyin ƙasa a duk tsayin da ke sama da tunnel har zuwa saman ƙasa, a kan faɗin tunnel ɗin da aka huda b. Saboda haka, matsin koli shi ne
P = γ bt [Mp/m]
inda γ yake nauyin ƙima na yaduddukan ƙasa a saman bututun ƙasa [Mp/m3], t kuma tsayin yaduddukan ƙasa a saman bututun ƙasa [m].
Wannan zato ana ɗaukar sa ne kawai a cikin musamman yanayi, kamar misali idan ƙasar da ke saman tunnel ɗin ta ƙunshi laka, wacce kusan ba ta da wata juriyar ciki, ko kuma a cikin ƙasa mai haɗuwa da mara haɗuwa a cikin ruwan karkashin ƙasa, lokacin da juriyar ƙasa ta fi ƙarancin tasirin nauyin ƙasa.
Irin waɗannan yanayi suna da wuya a aikace kuma ana ɗaukar su a matsayin yanayi na musamman. A duk sauran yanayi, idan za a iya lissafa da juriyar cikin ƙasa, ƙayyade matsin ƙasa a cikin ramuka bisa wannan zato ba ta da hujja, domin tana ba da matsin da ya fi na gaske yawa sosai.
Hot. 2. Ƙayyade matsin ƙasa a kan dogo ƙarƙashin zato cewa matsin yana ƙaruwa daidai da zurfi
Tsammani da aka gina bisa ƙa’idodin daidaiton jikin da ke rugujewa
Hoto 3. Ƙayyade matsin lambar ƙasa a kan tunnel bisa zato da ya dogara da dokokin daidaiton jikin da ke ruɓewa
Bisa ga wannan zato ana ɗaukar ƙasa busasshiya kuma mai rugujewa, kuma ana ɗauka cewa matsin da ke kan baka na ramin ya fi ƙanƙanta da nauyin prism ɗin ABA’B’ saboda tasirin ƙarfin gogayya a saman tsaye AA’ da BB’ (sl. 3).
A kan wannan zato ne aka gina wasu dabaru na lissafin matsin lamba a kan kwanon rufin rami, kamar na Forcheimer, Bierbaumer da Engesser.
Jimillar nauyin ƙasar da ke sama da kumbon ramin dogo shine W= γ bt, amma prism ɗin ABA’B’ ba zai iya zamewa ƙasa da yardar rai ba, domin gogayen kayan da ke saman AA’ da BB’ za su ƙi wannan zamewar. Idan o shi ne kusurwar gogayen cikin ƙasar da ke sama da kumbon ramin dogo, to ƙimar gogayya tana daidai da tgo, kuma matsin lamba na al’ada a saman gogayya a hakikanin gaskiya shi ne ƙarfin gefe E, wato matsin ƙasa mai aiki a saman AA’ da BB’. Saboda haka a kan kowace daga cikin waɗannan saman tana aiki ƙarfin gogayya
Q = Ea tg_ϕ._
Matsa lamba P a kan rufin ramin, matsin saman, shi ne
Matsin ƙasa na aiki ana ƙididdige shi bisa Rankine
Ea = ½ γt2 tg2(45o – ϕ/2)
Ana yin tantance matsin ƙasa a kan tunnel bisa wannan zato da wuya, kuma kawai a wasu takamaiman lokuta, kamar misali idan kaurin layukan ƙasa da ke sama da tunnel ɗin ƙanƙanta ne, har ba za a iya samar da baka na rage nauyi ba.
Hasashen samuwar kwanon sauke nauyi
Bisa ga wannan hanyar, a cikin kowace ƙasa mai haɗuwa da kuma wadda ba ta haɗuwa ba, sai dai cikin laka, a saman rami ana samar da baka mai rage matsa lamba, don haka abin da ke aiki a kan siffar ramin shi ne kawai abin da ke ƙasa da wannan baka, alhali baka ɗin kansa yana kasancewa cikin daidaito saboda aikin baka, abin da aka tabbatar da gwaje-gwaje a cikin ƙasa mai yashi mai sako-sako. Don ƙayyade matsin ƙasa a kan rami bisa wannan zato akwai hanyoyi da dama, daga cikinsu shahararru su ne na Kommerell da Protođakonov. A hanyar Kommerell ana ɗauka cewa baka mai rage matsa lamba yana da siffar rabin ellipse, kuma a hanyar Protođakonov yana da siffar parabola.
Hanyar Kommerell ta rabin-ellips na matsa lamba
Bisa wannan hanyar, ana ɗauka cewa tsayin rufin rage-nauyi h yana dogara da faɗin ramin tona bi da kuma kusurwar gogayen cikin ƙasar da ake tona ramin a cikinta (hoto 4). Ana iya ƙayyade tsayin rufin rage-nauyi h bisa tsarin:
h = 100_a_/p
inda a shi ne zaman ƙasa a ƙarƙashin tushe tun daga fara hakar ramin har zuwa kammala ginin bangon ramin, p kuwa matsananciyar sassaucin ƙasa ne.
Hoto 4. Ƙayyade matsin ƙasa a kan rami bisa hanyar rabin-ellips na matsa lamba
A cikin yanayi masu kyau ana ɗaukar a = 10 – 15cm, a cikin marasa kyau a = 15 – 40cm, a cikin masu matuƙar rashin kyau a = 40 – 70cm.
Rashin ɗaurewar ƙasa na dindindin p ana ƙayyade shi a kan ƙasar da aka haƙa ta ramin tunnel a cikin % bayan haƙa da sake matsewa. Ƙimomin p suna cikin iyakokin da aka bayar a teburin 1.
Sai dai ko da yake ana amfani da wannan tsarin na sama sosai a aikace, ana ɗaukar cewa bai dace da gaskiya ba. Zama ƙasa ba shi da ƙimar cikakkiya, sai dai a cikin abu ɗaya yana bambanta gwargwadon nau’in gindin goyon baya. Idan gindin goyon bayan ƙarfe ne, zama ƙasa ya fi ƙanƙanta; idan na itace ne, zama ƙasa ya fi yawa. Don siffofin al’ada na ramin ƙasa a layukan jirgin ƙasa da hanyoyi, idan tsayin layin ƙasa a sama da ramin t > 20 m ana ɗauka cewa tsayin ƙarfen ɗaukar nauyi h=20 m.
Matsin gefe. A cewar Kommerell, matsin gefe yana tasowa ne sakamakon tasirin nauyin prism ɗin ƙasa ABC (sl. 4), wanda aka iyakance daga gefe da saman zamewa AC da farfajiyar tsaye ta bututun AB, yayin da daga sama ake ɗora masa nauyin prism ɗin ƙasa CBG. A nan ana ɗauka cewa prism ɗin ƙasa mai aiki an iyakance shi daga sama da farfajiya a kwance CB a matakin saman baka na bututun
Tab. 1. Kimanin ƙimomin ƙasa p a cikin %
Tunda jirgin AB a tsaye yake, jirgin CB a kwance yake, ana ɗauka cewa akwai yanayin Rankine na matsin ƙasa, kuma cewa hanyar matsin ƙasa mai aiki E a kwance take. A Rankine, gangaren saman zamewa AC zuwa kwance yana a kusurwar α=45o+ϕ/2, matsin ƙasa mai aiki Ea shi ne
Ea = ½ _γh_2tg2(45o-ϕ/2),
inda γ shine nauyin girman ƙasa a Mp/m3, h tsayin bangon goyon baya a m, ϕ kusurwar gogayya ta cikin ƙasa.
Yadda prism ɗin ƙasa mai aiki ABC ke ɗora wa daga sama nauyin prism ɗin CBG, a wannan yanayin zai kasance ga tsayin ramin h’
Ea = ½ γ h’ 2tg2(45o-ϕ/2),
inda γ1 = γ + 2p/h’
inda p = Q/CB = Q/h’tg(45o- ϕ/2)
Nauyin Q ana samu ne a matsayin bambanci tsakanin nauyin W na kwata na ellipse na matsa lamba CEF da nauyin W1:Q = W – W1.
Nauyin W ana samun shi daga yankin A na rabin ellipse da nauyin ƙasa na girma γ, W = Aγ. Yankin A ana samo shi ne daga daidaiton yankin ellipse: A = h·e·π/4
Ta saka ƙimar p a cikin ƙa’idar Ea muna samun:
Ea = ½ γ h’2 tg2 (45o-ϕ/2) + Q tg(45o-ϕ/2).
Matsin saman baka. Tunda baka mai sauƙaƙa nauyi yana da daidaito dangane da axis ɗin tsaye, za a iya ƙididdige matsin saman baka don rabin tsarin tukunyar rami. Nauyin W1 na prism ɗin ƙasa BEFG don tsawon ramin 1,00 m shi ne:
W1 = (z+h)/2 · b/2 · 1,00 · γ
A cikin wannan lissafi komai sananne ne sai z, wanda muke ƙayyade shi bisa ga lissafin elipsu (sl. 5): b2x2 + a2y2 = b2a2. A cikin yanayinmu b=e; a=h; x=z;
Sl. 5. Ƙayyade matsin lambar saman tsani na rami ta hanyar hanyar rabin elips na nauyi
Siffa 6. Matsin ƙasan gado p a ƙarƙashin goyon baya
Matsi na ƙasan ƙafa. Tun da ake ɗauka cewa matsi na ƙasan ƙafa A ya yi daidai da matsi na kololuwa P, don haka girmansa ya riga ya tabbata.
Idan siffar ramin tana da baka na ƙasa, matsa lambar da ke kan ƙasan ramin p ita ce: p = P/(b·1,00m) [Mp/m2].
Amma idan hakan yana da ƙarfi kuma yana da isasshen ƙarfin ɗauka, matsin saman rufi P ana watsa shi zuwa ƙasa ta hanyar rafukan goyo masu faɗin d (sl. 6), don haka matsin ƙasa a ƙarƙashin rafukan goyon shi ne: p = P/(2d·1,00m) [Mp/m2].
An yi nuni da cewa a wannan yanayin wajibi ne a gudanar da binciken ƙasa da kuma tantance ɗaukar nauyin ƙasa da zaman ƙasa, domin ya taɓa faruwa cewa ɗaukar nauyin ƙasa ya fi ƙarfin matsin ƙasan ƙafa ƙasa, sakamakon haka aka samu nakasar rigar bututun ƙasa da zaman duka bututun.
Hasashe da aka gina bisa dokokin elastisiti
A zamanin baya-bayan nan an bayyana wata sabuwar zato, wato a ɗauki ƙasa a matsayin jiki mai daidaitaccen elastik isotropik wanda za a iya amfani da dokokin elastik a kansa. Bisa ga wannan zato, axin elastik na ƙasa ana bayyana su ta ma’aunin elasticity ɗinta E da kuma ma’aunin Poisson µ wanda ke dogara da siffofin jiki na ƙasar. Sai dai a zahiri ƙasa ba isotropic ba ce, illa kuwa taro ne na anizotropic, wanda ƙimomin ma’aunin E da ma’aunin µ ke sauyawa tare da zurfi. Duk da wannan karkacewa, amfani da ka’idar elastik domin tantance matsin ƙasa yana da babbar muhimmiyar kimiyya da kuma damar ƙarin bincike a gaba.
Binciken kwanciyar hankali na rufin haƙaƙƙen bututun ƙasa ƙarƙashin tasirin matsin lambar ƙasa
Don gwada daidaiton rufin tunnel, ana zaɓar sashen giciye na rufin, kuma yana da amfani a yi amfani da misalan tunnels da aka riga aka gina a irin waɗannan yanayi da na wannan yanayin, waɗanda suka tsaya cikin kyakkyawan hali, ko kuma a ƙayyade sashen rufin tunnel ta wata hanyar ƙididdiga ta kusanci (sl. 7a). Sai a raba sashen da aka zaɓa na rufin tunnel zuwa lamellas masu yiwuwa su kasance daidai da faɗin Δs. Ga kowace lamella, a ƙayyade matsin tsaye da na gefe, da kuma nauyin lamellar ɗin kanta.
Matsin tsaye PG a kan lamela don tsawon bututun 1,00 m (sl. 7a) ana samun sa bisa ga wannan dabara:
PG = (y1+y2)/2 · Δ_x_·1,00·γ
Fig. 7. Ƙayyade matsi na ƙasa a kan tasha bisa ga hanyar Kommerll
Matsi na gefe ya ƙunshi matsi na ƙasa Eh, a kan siffar ramin dogo mai tsawo h’ da matsi Ea sakamakon aikin nauyi Q
Eh’ = ½ γ h’2 tg2 (45o-ϕ/2); Ea = Q tg(45o-ϕ/2).
Domin ƙayyade matsin gefe Eh a kan kowane lamel na rufi, ana zana jadawalin matsin gefe (7a) bisa ga ma’aunai:
Ƙarfi PG da Eh suna aiki a kan axis ɗin lamella ɗin da suka dace.
Bugu da ƙari, ana ƙayyade nauyin lamella ɗin rufi PZ, wanda ke aiki a tsakiyar nauyinsa, sannan a samo jimillar R na dukkan ƙarfi a kan lamellar ɗin da aka ba (sl. 7b). Ta wannan hanya ana ƙayyade jimillolin R1, R2, R3 ga dukkan lamellae (sl. 8a), waɗanda ake tsara su a cikin tsarin ƙarfi (sl. 8b), kuma a samu jimillar Rn. Sannan daga kowane pole O da aka zaɓa ana zana haskoki kuma a zana layin tallafi na farko ac. Daga sharadin cewa baka yana cikin daidaito idan matsin baka a kwance H, jimillar matsin ƙasa a kan rami R_n da matsin ƙasan tushe A sun haɗu a wuri ɗaya, muna samun wurin haɗuwa S. Idan muka haɗa wannan wurin da tsakiyar maƙalar haɗin baka a wurin saukarsa, muna samun layin sabon matsin ƙasan tushe A1. Idan muka haɗa ƙarfi Rn, A1 da H a cikin sabon tsarin ƙarfi (sl. 8c), muna samun girman H da sabon pole O1 daga wanda muke ja sabbin haskoki da layin tallafi na biyu ac’.
Sl. 8 Tsarin ƙarfi da layin goyon bayan matsi na ƙasa a kan rami
A kan sabon layin goyon bayan da aka samu, ana daidaita bayanin fadin sashen rufin tunnel, sannan kuma a sake ƙirƙirar layin goyon bayan, idan ya cancanta, har ma a karo na uku. An fi son a ɗauki irin wannan siffa ta rufin tunnel wadda axis ɗinta ya dace da layin goyon bayan. A wannan yanayin, sai matsalolin matsa lamba kaɗai ke rinjaye a cikin rufin. Matsin lamba da ke bayyana a kowane sashin fadin n – n na rufin tunnel (sl. 8), ana samu daga ƙa’idar
inda N shi ne ɓangaren al’ada na jimillar Rn a kan sashin rufin n – n , wanda ake samu daga shirin ƙarfi na wannan sashin; d kaurin rufin tunneli a sashin n – n; e nisan karkatar da ƙarfin hari N a wannan sashin.
Dole ne matsa lambar latsawa σ ta kasance ƙasa da matsa lambar da aka yarda da ita na rufin. Ga baka na ramin da aka yi da kankare mai ƙarafa, ana ganin cewa wannan hanyar ba ta da daidaito sosai, don haka ana ba da shawarar hanyar Spangerberg.