Giáo trình: Phân tích ứng suất Lý Trường Thành

hAu nhtr khongth~tim duoc nghiemtheaman h8t moi di~u kien bien,d~c bi~t la khi tren bien co hrct~p trung tacdung, Nham giam bot kho khan ta co th€ him "mern hoa"di~u ki~nbienv~ lireb~nge

I (

Phdn tich Ung sudt13 mQt mon khoa hoc thuQc nganh co hocv~trin bi~ndang N6 nghienciru trang thai irngsu~t, bi~ndang cuav~t th~can bfulg du&i tac dung cua cac tac dQng ben ngoai,khi rnav~t li~ucuav~t th~chuavuotquagioi handan hAL Tuy cosggi3ng nhau vemuc dlch voimonSuc bJn vdt li?uvamen ClY h9C kit cdu nhung d3i tUQ11g nghien crru thl khac nhau.N~unhirmonSuc bJn vdt li?u va mon Cahoc kit cdu lAy d3i nrongnghlen ciru chi la cac v~t th~ dangthanh,thl dAi ttrongnghien ciru cua mon Phdn tfch Ung sud!la cacv~fth~ dangthanh, dang t~m,

vovadangkh3i,tirela lien quan t&ivi~c gillibai toan 1, 2hoJc 3 chleu Men Phdn tlch fmg sudt

clingkhac voi men Suc bJn w;it li?u va ClY h9C kit cdu vetfnh chJt chScua nghiem Loi giai cua

StfC bJn Vt;lt li?uVaCCZh9Ckit cdu thlchuy~"th\gdn dilng vlkhong thoa manh~tcac dieu ki~nbiencua baitoano Khinghien ctru cac bai toan sieutinh, mon Suc bJn wJ/li?u va ClY h9C kit cdu

phai dua bd sung them cac dieuki~n tuy thuQc vao hlnh dangva lienk~t cua v~t th~ dSI~pthemcac phuong trlnhbd sung,thi monPhdn tichteng sudt chi cdn sirdung cac gillthi~tco lienquantai tinhch~tcua v~tthS giAng nhir trong cac monSuc bJn w)t li?u va mon ClY hoc kit cdu, do lacac gillthi~t:

-V~t li~u lientuc,d6ngch~tva ding huang

-V~t Ii~udim h6ituy~ntfnh vatuan theo dinhlu~tHooke

-Bi~ndang cua v~tla be

Hai gia thi~t sau lam cho cac phirong trinh mo 13 trang thai chiu hrc cua v~t th~ la cacphuong trinh tuy~ntfnh va do do co thS ap d\lng nguyen Iy ch6ng ch~t nghi~m (nguyen Iy ce)ng

Trong tili lieu nay trinh bay mon Phtin tfch Un!! sudttren casak~t hero haimonhoc laLv

TAc GIA

cua no, chang hanta goi m~tx tuc Iam~tco phaptuy~n ngoai co phirong song song voi truex.

Trongky hi~u irng suatphap, chi s6bi~u th] tencuam~tcit co ung suat phap, cling la chi phuongcua irngsu~tdo; con trongkyhi~u irng sutt ti€p, chi s6 thu' nhtt bi€u th]m~tco irngsu~t ti~p, chis6 thir 2 biSu thi phuong cua irng sudt VI du (J' x Ia trng suAt phap trenm~tx,con 1::cy hi trngSlitt

ti~ptrenm~.tx co phuong song song voitruey.

Ddu cua cac img sUdt duoc quy tree nhu sau:Niu phap tuyin ngoai ella m4U edt huang

thea chieu duong ella true fog aQ, thi cae tmg suat co ddu duong khi chung cung huang theo

PHA.NrtcaUNG SUAT

chieu duong cua h? true toa t/lj, con tren cdc m(zt co phdp tuyin ngoai nguoc chilu vai chiJuh?true toa t/lj, thi cac Ung sudt co ddu duang khi chung eo hutmg nguoc vdi ehiJu cuah~true toa

a{i,cac irng su~tve tren hlnh 1-1 d~ueod~uduong

Cac irngsu~t khong nhtrng thay ddi thea tim;sdi~m trongv~t(phu thuQe to~ dQ elladj~m)rna con phu thuQe vao phtrong cuam~teit diquadi~m'do (phu thuQe goc nghieng cuam~teit diquadi~rnd6).N~u kyhi~uehung cac (rngsu~tla S thl ta eoth~viSt: S= S(x, y, z, n) , voi n la caceosin chi phirong cuarn~teit di quadi~md6 .~-

1.1.2 PhLPCYng trint: vi phalJ canpj!}g Navier

Khao sat rnQt phan to dxdydz tach ra t11 rnQtv~t th~ cdn bAng Ngoai cac frng su~ttac dungtren cacm~tcua phan to, trong phan to eon e6 cac hrcth~tichvoi cac thanh phdn hlnh ehiSu ellan6 len cac true toa dQ Iii X, Y, Z tac.dunglen phan to ntra

N~unhirtrenm~te6 toa dQ laxta eo cac thanh phdn (mgsudtJii:·

O"x(x,y,z) 'Z"xy(x,y,z) 'Z"x:(x,y,z) ','

thl trenm~teQtQ::t dQ la(x-dx)e6 cac thanh phdn lrngsu~tla:

Dung khaitri~nTaylor vabequa cac vo cungbeb~eeaotadUQ'e:

O"x(x+dx,y,z)=O"x (x,y, z)+dcr, =O"x(x,y,z)+ a~xdx

Lam hoan toantuongtil dOivoicac thanh phdn trng sudt khacduoc cac thanh phdn(mgsu~ttren cacm~tcua phan to cho tren hlnh 1-2

crx+_Xdx Ox

+ Ot xz dx

xz ""

ox

Nhfingphwng CO'ban cua bai toen dan hOltuyentinhd~nghuting

:hant6 canb~ngduoi tac dung cua cac thanh,ph~ntrng suAt tren cacm~tva cac thanhp~~n

hrc the tich nen no thea man cac phtrong trinh can bang tinh hoc:

LMx=O

~MLJ .y =0 LMz=O

(1-1)

B~t d~u vainh6m thir nhfrtcua (1-1), chang hanvoi phuong trinh LX =0 :

( ax+oax )Ox dx dydz - a~dydz+(ot'yx ryx +ay dy dzdx -) t yx dzdx+

+ (,'" + a;;dz)dx4Y - ,,,,dxdy +Xdxdydz=0Cac phuong trlnh hlnhchi~uthea phuongj/vaphuongzlamnrongnr,Sau khirutgon ta diroc:

C larnatr~ncac toan tfr vi phan: C= [ tx ~ 3z ]

S larnatr?n irng sutt:

(1-2)

(1-3)

r; «; a z

Pia vee to' hrcth~ tich: P= {X Y zy

(1-2, ), (1-3) lit cac phuong trinh can bimg hay lit h¢ phuong trinh vi phdn can brmg Navier.

Nh6m tlur 2 cua(1-1) lit cacphirongtrinh canb~ng momen 06i voi cac true.Ch~ng hanphuong trinhd~utien lit: =0 duoc:

(,,, + a;; dz)dxdydZ-(,,,, + Ok dyYXdydz=O

Hay

(1-4) bi~u di~n lud:adi tmgcua img sudt dip.

1.1.3 (rng su§t trenm~tnghieng~ Diduki~n.bienv~Ivc

otren ta daxet phant8hlnh hQp tach ra til mQtv~t th~ can bAngva cac phuong trlnh canbAng (1-2) hoan toan thoa man d8i voicac phant8 nay Tuynhien, n~u ta chiav~t th~ bAngcacm~tphangsong songvoi cac m~tphing toa dQeach nhaunhtmgkhoang vocung nho thise chiav~tth€thanhvo s8cac phan t8 hlnh hQpva inQt s8cac phdn t5tit ai~n Cac phant3 ttrdi~nnayngoai cac m~t song songvoi cac m~t toa dQchjumcdung cua cac thanh phAn img sudt, con com~t nghieng voi cac m~t toadQ,trendo cocac 11Icb~ m~t tac dung nhu tren hlnh 1-3

Taxet rnQt tfrdi~n tach ra tit v~t th~ tren hlnh 1-3.Tfrdi~n nay cocac m~t x, y,z varn~tnghiengvchotren hlnh 1-4 Giasirluc b~ m~t toan phAntren·m~t vcocac thanhphAnhlnh chi~ulencac true toa dQ Ia Xv,Y», Z». Phaptuy~n ngoai vcuam~t nghieng hop voi cac true x,y,z cacgee a, (3,'Y.f)~t:

/T • • lnnltrfII1JI'S"'''''''~trtnn eelbaneuabai loan dan

Phant6 nayatrangthaidinb~ng nen taco:

LX=0 ~ Xv·dF-O'x·dF.l-1:Yx.dF.tn-1:zx.dF.n=0

LY =0 ~ ~.dF -:-1:x~.dF.R - O'y.dF.m-1:zydF.n=0

;

z=0 ~ Z".dF -1:x;:.dF .f.-1:yz.dF.m - O';:dF.n=0Rut gon ta duoc:

ho~c vi~t duoi dangrnatran: P,= S.L

Trong do: P, lit vecto irngsu~ttoan phdn trenm~tnghieng:

1.

d nhieu bai toan dan h6i (nhucac bai toan v~ thanh, t§m,vo, vv ) hAu nhtr khongth~tim duoc nghiemtheaman h8t moi di~u kien bien,d~c bi~t la khi tren bien co hrct~p trung tacdung, Nham giam bot kho khan ta co th€ him "mern hoa"di~u ki~nbienv~ lireb~ngeach ap dungnguyenlySaint-Venant, nguyenlynay duoc phat bieu nhir sau:

Khi mien t/(lt tai trong nho han moi kich thucc cua vdt thd, thi trang thai tmg sudt vabien dang tai nhii:ng tlidm xa nat a(lt 11!e thay tl6i rtit it khi ta thay h¢ hre nay btmg h¢ luc khac

PHA.N TieHUNG SUAT _.

H~hrcnrong duongaday duoc hiSu lah~19c c6 cung vecto 19c chlnhva vecto momenchfnh

Vi d\l thanh com~tcit ngang F lah~ngs6, chiu haih~ 19c tuong duong nhir tren hlnh 1-5,

trang thai irng suat.trong hai tnrong hQP nay chi khac nhauavung hai dAu thanh gAn noid~t I\fC,cang xa hai dAu thaeh, trang thai irng sudt cang gi6ng nhau

Ta se ap dung cu thS nguyenlynay khi giiti mQt s6 bai toanacac chuang sau

Thanh co mat cAt ngang F'

1.1.5 Trflng thai ung su§t t{li mi}t(fj~m

Nhir tren da n6i, img suat khong nhfmg phu thuQc vao tung diSm rna con phu thuecvao phirong cua m~tcit di qua diSm d6.T~p hop tat cit cac ungsudt tac dQng tai mQt diSmtheo moi phirong goi la trang thai ung sUdttai diSm d6 Nhir v~y, dS nghien ctrutrang thai trng sudt tai rnQt diSm M nao d6 trong v~t thS can bAng, ta phai bi~t h~t moi ung sudt tacdQng tai diSm M tren tat cit cac m~t di quadi~m d6~ Truceden, ta bi~u th] cac irng sudt taidiem nao d6 bAng cac irng suat tren cac m~t cua mQt phan t6 bao quanh diSm d6 Nhir v~y,

trang thai tmg su,ftt~i mQt di~m lat~p hop tat cit cac ung suat tren cac m~t cua cac phan t6bao quanh diSm d6

Nhir da chimg minha1.1.4 (cong thirc 1-6):N~u bi~ttrng suat tren 3m~tcit vuong gee voinhau di qua mQt diem thi coth~xac dinh dugc trng suat tren bat cum~tcit nao di qua diSm d6.MQt m~t rna tren d6 khong co (mg suatti~p dugcgQi la m{it chlnh tmg sudt Phap tuyenngoai cuam~t chlnhdircc goi laphuong chinh va img -suat phap tren m~t chlnh dugc gQi laung

s-M.dL,,-hinb~~gY'Q'il~_.cU ng~YngJniDltd!J' q~Ja: Q:tJ~ ITI_Qtdi~m,llaOgQ!?~~LgibCUf!g!l!!Lg!!Q'C rnQt_phan t6 hlnh hQp rna tat ca cacm~tcua n6d~u lam~tchinh Phan t6 d6 dugc gQi laphiint6chinh.

Ky hi~u c~c frng suat chfnh cua phan t6 nay Ia0'1, 0'2, 0'3 va v6'i quy uac la: O't ~0'2~0'3 (kS ca.dau) Bay giG hay xac dinh cac (mg suat chfnh

Xet phan t6 tu di~n nhu tren hlnh 1-4 Giit sirm~tABC lam~tchinh va gQi i'rng suat chfnhtrenm~tnay la 0'K ,con cac cosin chi phuang cua phuong chinh la eK' m K ,n K

(1-11)

NhCfngphurmg trinh ca ban cuabal toen dan hOI tuyen finh dAng huti'ng

(1-7) lah~ phtrong trinh thuAn nh!t M?tkhac.eK, mK, nK khongth~ d6ngthai b~ng 0nend~phuongtrinh conghiem khongtAmthirongthidjnhthirccuah~phuong trinhnayphaib~ng0:

(O'x-O'K) t ;

D =1:xy (O'y -O'K)

12= ax 1:yx +O'y 1:y;: +ax 1:;:x (1-9)

1:xy O'y 1:zy CJ;: v; 0';:

-'ax 1:yx 1:zy

ba nghiemcua(1-10)luonIii thirc,d6Iii cacirng sudtchinhO"J, 0"2, 0"3.Thay IAn hrot O"J, 0"2, 0"3

-vao(1-7) vak6thopvoi(*) gi3.ih~ do se duQ'c phuong chinh

Nhtrv~y ta da xac dinh duoc cacirngsu!t chlnhvaphirong cua cluing.Baygio tati~n hanhxac dinh cac irngsu~ttiepeiretrio Cac imgsu~t tiepcue tri n~m trong mij,tphangchua mQttruechInh va nghiengmQtgoc 45° so voi haim~tchlnh tuong (mg Cac irngsu~ttiepcue tr]ducc tinh

Xet m9tv~t th~ dim h6i au lienk6t(v~t khong

co chuyen vicung) nhu tren hlnh 1-6 Khi bien dang,

di~m Mse b]djch chuyen toi vi trf MI, tanoidi~m M

da thay d6i vi trl SI! thay dbi vi tri cuadi~m khi v~t z

1

PHA.N TIeH UNG SUAT

b] bien dang duQ'c goi lachw€n vicuadi~rn.Chungtakihi~uhlnh chi8u cua doan MMt·1en cactrue toad~lau,v, w Trong do,u,v,w dugc gel la caeth~nhphAncu~ chuyenvi "

Vi chuyen vi 130 sg dich chuyen til'di~111 nay dSndi~mkhac nen 'cac thanh phdn chuyen vicling la ham s6cua toadQ

dx+Adx, dy+~dy,dz+L\dz, va nhuv~ybien deng dai thea cac phirongx, y, zIAri Iugt se la:

x . dx , ','" y "dy , ··· ·z···"dz

con cac bien dang goc gifra cac canhdx -dy, dy - dz,dz -dx IAnlugt130:

x b)

Nhirv~yco 6 thanh phAnbi~n'dang:

{e} = [~x Eye;: r xy ry:: r zx r

Giiia bien dang va chuyen vi co m6i lienh~ voi nhau Ta hay tim m6i lien h~ nay Tnnrc h~t

ta xet hlnhchi~u ctia phan t6 tren m~t phangxoy Hlnh ABC sau bi~n dangse thanh hlnh AI B.C1

nhirtren hlnh1-7b

Diem A(x,y) sau biendangse toidi~mAIva dathuc hien cac'chuyenv] la·u,v

~ ' ~

Di~m B (x+dx.y) sau bien dang se t6'idi~m B1va da thirchi~ncac chuyen vi:

Nhu tren da: noi, chuyen vi tai mQt dl€m"ougc xac dinh b~ng 3 thanh phAn: U,V,w; con

bi~n dang thi duoc X3.C djnh b~ng 6 thanh phdn:'cx' By, e.,Yxy' Yyz' Yzx. Tir (1-13) thAy r~ng: N~u bi~t 3 thanh phdn chuyen v] u, v ;w thi hoan toan xac dinhducc 6 thanh phftn bi~n dang

PHAN TtCH (riNG SUAT·

B x ' By'S:, rJeY'rJIZ'rzx vichungchi la cacd~ohamb~c rih!t.Tuyn.hien n~u tabi€t6thanhphdnbi€n dang rna cdn phai xac dinh 3 thanhphdn chuySn vi thi nay sinh mot vdnd~ Ia 55phtrongtrinhnhieu hon,s5 in 55, nhuv~y cac thanh phdn b·i~n dang khong'thadQc I~pvci nhaudugcrna chungphai co moi rang buoc nhdt dinh,

Tli (1-13) suy ra:

Lamnrongtg ta dugc:

Vacilng til'(1-13) co:

Nhirv~ytadugc 6 phirong trinh biau diSn sg rang bUQc gifracacbi€n dang:

a2ex 02ey 02yxy

caban cua bai

1.2.4 Tr€lng thai bien dflng t€li di~m

Xet mot phant6 doan thang di qua m¢tdi~mtrong m¢tv~t th~canb~ng.Khiv~t th~b]bi~ndang thl phant6 deanth~ngnay cling b]bi~ndang,Bi~n dangnay thay d6i tuy thu¢cVaG phuongcua dean thang.T~p hoptAtca cac bi~n dangcua phant6 dean th~ng di quamot di~m thee moiphirong duocgoila trang thai biin dangtaidi~mdo

Cling gi6ng nlur trang thai irng sudt,n~utabi~ttAt ca cacbi~ndangth~ngvabi~n dang geecua 3 phan t6 doan thang vuong goc voi nhau di qua mQtdi~m nao d6 trong mQtv~t th~ canb~ngthl hoan toan xac dinhduoctrang thai biin dang~i di~mdo, Cling gicng nhu trang thai irng suAt,trangthaibi~ndang tal m¢tdi~mcling co cac tinh chdt la:luon t6n tai ba phirong vuong gee nhau

rnatheo cac plurongdochi cobi~n dangth~ngchlnh tac dung goi la truebi~ndang chlnh, v.v

1

Trong Sfrc bSnv~t li~uta da co quanh~ gifra (mg suAt yabi~n d~ng- do latljnh lu,:1t Hooke.

Trong trU'ang hgp t6ng quat,quanh~nay cod~ng:

Sx = k [ax - p(ay+az )]

Sy = k [ay- p(az+ax)]

'Yxy -_IFF'G "xy ' 'Yyz -_IFF'G "'yz ' 'Yzx -_11:G :t.x

Trang d6 Elit moaun aim h6i trang keo (nen) , p Ia h¢s6biin dflng ngangcuav~t li~u haycon gQi lah~s6Poat-xong,conG lit moauna~r:h6i tru()1.

G = E

2(1+p)

C6th~ vi~t l~i(1-15 )duaid~ngquanh~imgsu~t- bi€nd~ng:

(1-16)

d~lI cua(1-23) taduec:

Nhirv~yJchfnh Iahamd.iJuhoa.

Cac nghiern tren ngoaivi~cphai thoa man cac phuong trinh can bing conphai thoa mandi~u ki~n bien (1-5):

1.4.2 each giaithea chuy~n vi" h~phUong trinh Leme'

Khi giai theo chuyen vi, ta chon 3 thanh ph!n chuyen vi: u, v, w lam ins6. Cac in nay cAnthea man 3 phuongtrlnh can bing va diSuki~n bien.I)~ 'giai thee chuyen vi, ta vi~tcac phuong_!r1ntL~anbingJl:2)theechlly~n vi~ f)ajLl.:191.YA~(J,=-l31.YaQ,.{J.=2}_tadu:gc.:_ ,- " '-" -

(1-24)duocgciifAphucrngtrlnllLame'.

Vi~t lai (ti~u ki~n bien (1-5) dtr6i dang chuyen vi: Cling d~t (1-19) va (1-13) vao (1-5) duge:

x, =A.e~+o(~l+~m+~n)+O~l+~ m+%,n)

Yv =A.el+O(~l+~m+%n )+O(~l+~m+~ n)

"

z; =A.el+O( %' l+ ~ m+ ~ n )+O(~l+% m+~ n)18

Nhiing phut:Jng cua bal toan dan ho, tuyen tfrihd~ng'hudng

1

T~ haY,xem xet each giai theo ~g su~t C6 r~tnhi~u eondUOngd~ xac djl1rnghi~mcua'biH

toano Neu bi~t tnroc tai trong,ta co theghii tnrctieph~ pht;ong trinh vipha~can b~ng (1-2) (tftt

nhien phai keth9p, voi phuo;tg trlnh nrongthich(1-,23) ) dexac dinhe thanh phAn'(mgs~dt,r6i

buoc cac thanh phan irngsuat nayphai thea man dieu kien bien(1-5) Cach giai naygoi la'cdch

giai thudn.

Cling coth~ gia dinh tnroc cacbi~u thirc irngsu~t (ho~c chuyenvin~ugiai theo chuyen vi)

r6iki~m tra xemchungcothea mandi~u kiencanb~ng (l-2),di~u kienbien(1-5) vadi~u kien

nrongthich cuabi~ndang(1-23),' Cachgiainay d,ugc goi laeach giat nguac .

Phuong phap gild thudng~p nheng kh6 khanv~ toan khi giaih~ phirong trlnhvi phan dao

ham rieng nen giai diroc rftt it bai toano Phuong phap giaingiroc lairdt e6ng kSnh vi phai giathi~t

ratnhi~u IAn moi ra nghiem cua bai toanoD~ kh~c phuc cac nhuocdi~m cua hai phuong phap tren

ngiro! ta gia dinh truce ham (eng su~t (ho~c ham chuyen vi n~u giai theo chuyen vi)dtroi dang

mQt ham s6 nao.do nhirng chua phai bd~f.lgnrong.mlnhrna con chua mQt s6 dai hrong chuabi~t.

Sau khi buQ.c cac nghiern nay phai thoa man'cacphirong trlnh co ban: Phuong trlnh can bang,

phirong trlnh nrong thich vadi~u ki~n bien cua bai toan, ta sexacdinhn6t cac d~i hrong chua

bi~t, do do xac dinh duoc dAy du nghiem cua bai toano Cach giai nayducc goi laeach giai nira

nguochay con goi laphuong phdp Saint Venant,

Vi~ctim nghiern cuacacbai toandu6i dang cac ham s6 lien tucb~ngcacphirongphap noi

tren khong phailue nao cling thu?nti~n, th~m chinhi~u tnrong hop khongth~ lam dugc Trong

khi do l?i co th8 xac dinhduge gia trib~ng sO eua'cae hamnghi~m t~i mQt sOdi~m trongv~t th~,

tu do nQi suy cho eac di~m con l?i Lbi giai do duge gQi Ia Uti giai sO va la I(ri giai gAn dCmg

Phuang phap tim ra Ibi giai s6 dugc gQi laphuong philp s6. Cae phuong phap s6 thuo'ng dung

h·i~n nay Ia: Phuang phapsat phdn hfru h(ln,phuang phapphJn Hehfru h(ln va phuang phapphJn

(ll' bien.Trong chuang 4 cua cuOn sach nay se trinh bay ll1Qi dung co ban cua phuong phapphJn tt;

hfht hQntrng dVng vao giai cae bai toan phan deh (engsu~t

Bai toan dim h6i tuy co nhi€u each giai, nhungnghi~m cua no la duy nhftt Ta hay chung

minh di€u nay D6 ehlmg minhtadung phuong phap phan chUng

Gia sir bai to3.n khong phai conghi~m duynh~trna eo 2nghi~m Ia S va S' Ta phai chu·ng

minhr~ng n~u bai toan co nghi~m duy nhftt thi S phai b~ng S' N~u S va S' d~u la nghi~m eua ba.i

toan thi chung d~u phai thoa man phuong trinh can b~ng vadi~u ki~n bien cua bai toan, co

nghTa la:

CS+p= 0CS'+ P= 0

Pv= S'LLAn Iugttru v~vai v~cua (a) va(b) ta duge:

(a)

(b)

NhIn vao (c) tath~y ngay (S=S') chinh.langhi~m (frng suit) cua bid to8on: v~t the dan h6i

khongchilltid trQng, do d6 frng su~tnay ph:iib~ng 0:

S - S'= 0

Hay S= S' Doladi~ueAn ehung minh

1.5 Cae nguyenly vicong,va nanghJ'9'ng

Phin tren ta da xet cachgi~ibaitoal1dan hOitir,c~c phuong trlnhvi phan can bing (1-2).Tuy.nhien ta cling co th S giai.cac hai'toan nay~1Jatrep nguyen lyve cong va nang 11 1gng Cac

' ' " JC • .'" ' " ,

nguyen 19 nay lacd sacho nhieu phirongphap sonhu phuong phap phin tu htru han,phuong

1.5.1 Cae nguyenIycong kha di

XetmQtv~tthS dan h6i co thS tich V,di~ntichbem~t Schiu taitrQng'la:

ChuyringSp va Su kheng cit nhau, co nghia13.:8 :::,Sp±Suo

Gi:i su tuong ling voi cac ngoai 11Jc coh~chuySnvikha di:

(1-27)

(1-31)

Va tuong ling voi cac nQi hrc coh~cacbi~ndang khi<Ii

0&={O&x OSy O&z Orxy OryzOr;:x r (1-29)Khi thoa man cac diSuki~nrang bUQc tren bien Su, tirc Ii tren bien nay phai co:

-_ - - _ _ - W=. O-· -,· -, .- -' .

-·" -0':';10) -Thibi~utlnrcbi~udiSn cua nguyenly cong kha eli la:

JJ~O'xO&.r+O'yO&y+0'::0&::+,xyor zy +1:'yzOryz+ ,;:xoY;:x)dv

.nguyen If cong khiz di.

20

trinhcoban cuabai toan Ngoai nguyen 1:9 cong kha dl, con co nguyen if; cong bit khii di.Xet mQtv~t th~ din b~ng

chiu tac• •dungcua cac tai trong• va cotnrongbien dang• tuong thich e e ,e:if Y z'r r rxy' yz» zxthea

man cac rang bUQC nhir da: mota6 tren.Giasir co mQt tnrong luc khadl:

s; = {&Yv OY v bZvV

Vatirong ung coh~cac trng sudt khadi:

8a= {8a x 8a y 8a z orxy 8r yz 8r;:x y

Thibi~uthircbi~u di~nnguyen1:9 cong bu kha dl:

ff'exoa x +eyoay+ez8a;: +r xy8rxy+r y::8ryz+r;:xor::x)dv + V

Cong sinh ra do 19c coth~tthirchi~ntren nhtmg chuyen dci tuong tirong goila cong kha ell,

con cong sinh ra do hrc co ttrong nrong thuchi~ntren nhfrng chuyen doi coth~t goi Iacong bii kha di Chuyen vi rnatanrong tirong ragoi la chuyen vi kha di;con hrc tirong nronggoi la hie kha

di Nhirv~y,trongcong kha di, gitta lire va duong di khong lien quan gl vci nhau, nen co th8 apdung nguyenly c¢ng tac dung khi doh cong kha dl, B6i voiv~trinbi~ndang, ngoai cong kha dT

ngoai hrc con co cong kha di ntji luc, la cong sinh ra doh~ nQi luc ( img sUdt) thuc hien tren cacbi~ndang kha di tirong (mg Tirongtir,cong bu khddrntji hielit cong sinh ra doh~ m)i hie (img 'sudt) khadlthllc hi~n tren cac bi~n dangcoth~t nrong(mg.

C 1 ' : C : y ' 0 ' : 1 'xy .' 1'yz 1'i:c

B = IE.1'dC1'.1'+ JEydC1'y+ JE zdC1'z+ JYxyd1'iy+ Jryzd,1'yz+ Jyzxd1'zx (l~37)

LAyd~oham cac biSu thuc tren ta dugc:

N~uquanh~(mgsuAt-bi~ndang la tuySntfnh(dinhlu~tHooke)thi til' (1;;33) va(1-34)taeo:'"

W=B (J'.1'& z +0'yEy+(J':E:+l'xyrxy+'1'yzryz+l'zxY:x

Trong do W(f,) va W(u) IAn IUQ11a ham cua bien dang va ham ella chuyen vi

Hamth~ nang bi! sela:

B(a) = (ax +ay +aJ + 2.(l+vXT~H~""T;" -apy -apr -ap.)

B(O') Ia ham ella {mg suit

(1-43)

a} Nguyen Ifevetilu tit! nang Lagrange:

Xetv~ttha can bing nhu da rnotatrong muc 1 Gia sirv~t co chuyen vi khadivo cung be

ou, ov, owsir dung cong thirc Green thl biSu tlurc cong khadi (1-32) setrcrthanh:

Trang do J laphi~mham tangt~nangellah~ (th~nangbi~ndang vath~nang cua tai trong):

J= fffW(u}dv - .jlf(xu+ Yv+Zw}dv - ff(xvu+ Yyv+Zvw)ds

(1-44) chinh labi~uthtrc cuanguyen if; c¥c tiJu thi nang Lagrange: Khiv~t th~atrangthaidin bang thl phiem hamt6ngth~nangcuanonh~n giatridirng

b/Nguyen lj eve tiiu thi nang bit Castigliano:

Cling xet v~t th~ can b~ng nhir tren, gia su v~t eo tnrong irng su~t kha di vo cung be

oO'.r,oO'y,bO';:")o1::cy,61:y=,61::x. Thee nguyen ly C1!C ti~u th~ nang bu Castiglianofrog voi tnrong

c/lVg:l.yen!jclIc iiiu Reissner - AYlJellinger:

N~u v~t th~ rnaham chuyen vi thea man di~u ki~n rang bUQc tren ph!n bien Su, eon hamtrngsuAtl~ithea mancac,qi~u ~i~nbien tren,phdn bienSpthl phiern ham R sed~t gi~tri dung:Trongdo:

·.CHU'ONG 2

BAITOANPHANG

Bai toan phing Ii bal toan v~ trang thai t'mg su§t - bi~n dang rna trong do cac d~i hrongnghien ciruho~c la (mg su§t,ho~c Iibi~ndang chi phu thuQc haibi~nx,y trong h~ toa dQ OXYZ.TrongLy thuy~t dan h6i irng dungnguoi ta phanbiet.rahai10~i bai toan phing la: Baitoanimgsu~t phing va bai toan bi~n dang phAng

2.1.1 Ba; tosn (Png su§t phAng

Bai toan irng su§t phing lit bai toan rna imgsu~ttren clt

ca cacrn~tsong songVO;rr$ xoybing khong (O'z ='t'zy='t1x= 0)

Xet m¢tv~t th~hlnh lang trucob~dAy tr~tnho so v6'i kich

thuoccua2 m~tday.V~tchichiu hrc tren bien, cophuong

vuong gocvoi true cua langtru va phan bdd~utren be dAy

tcon 2 day khong chiu tai trong nhu tren hlnh 2-1 Chonh~

true toa dQ Oxyz sao chorn~t phAng xy song songvoim~t

phing day (xem hinh2-1) Luc nay tren hairn~t phing day

co cac irngsu~t: O'z =0, 'tzy = 'tzx =0,vib~ day t r§t be nen

sir thay d6i cua irngsu~ttheob~ dAy la khong dangk~ nen

, h.,t·, " ~", d' , " 'A-.l

-ta cot··e·-col cac thal'lfl pflan·tren-cung ung. VOlmQ.Lutenl-_.

tren v~t,cac imgsu~tcon I~i chi phu thuoc vao toadQx, y Hinh 2-1

va khong phuthuoctoa d¢z.

O'x = O'x(x,y) O'y = O'y{x,y) 'txy= 'txy(X,y) (2-1)Trongtnronghopnay O'z =°nhungbi~n dang £z*0,dfin d~n chuyenvi theophuongzclingkhac khong(w*0) vabi~ndangnay chiphuthuQcvaohaibi~nx,y Do chi coirngsu~ttheaman.djnh nghiabai toan phangn~ngoi la bai toan img sudt phiing

TIi~t v~y,fneo-djrih·h.i~t.Hooke·la co:

z

Do(j'z =° nen:

e, =1 [0", - #(0", +0",)1

2.1.2 Sa;toan b;~n d1jJng phAng

Bai toan bi~n d~ngphing la bai toan rnaehuy~nvi cua tftt ca cae diem euav~ttheo phuongsong song v6i mQtm~tphAng (xoy) bang khong ( u= u(x,y), v= v(x,y), w=0)

24

Do

N€u v~t th~ diroc;etcling nhu truang hop 1 (hlnh 2-2), nhtmgvoi di~u ki~n:la2'day eua lang tru khong dircc bien dangtgdo rna dircc giu sao cho khong c6 chuy~n vi theophuong z(w=O),va£z =0 ;con cacthanhphAn bi€ndangcon laichiphu thuoc vaotQadQx,y:

£x=cx(x,y) cy=cy(x,y) YXY=YXy(x,y)

ez =!rE~a -z r:u(ax +0' y)~=O~

Trong tnrong hop nay cac bien dang va chuyen vi chi xay ra trong rn~t phang x, y, nen ta

c6baitoan biJn dang phiing.

Hinh 2-2

Trong thuc t€ r~t it tnronghopco bai toan thoa man di€u ki~n labai toan bi€n dang ph~ng,

song n€u xet mOt each gAn dung thl l~i g~p rAt nhi€u Vi du nhu cac bai toan v~ v~t lang tru co

chieu did kha 16nsovoi kich thuccm~t day, chiu tai trong vuong g6c voi true va co cirong dO

khong d6i doc theo true: £)~p dang nuoc, nrong ch~n, cac Quang hfun V6'iMUng baitoannay,khi tinh toan nguoi ta thuong tach ra mot lat mong bci hai m~t c~t song song r~t gAn nhau va vuong goc

voi true z (xem hlnh 2-2) N€u lat c~tmong nayduqc xet mOt each bi~t l~p, nghla la hai m?t ~~t

thuoc latc~tcoi nhu hal day tir do thl chung co th€ bi€n dang theo phuong z (ez *-0) Do tinh d6i xirng cua bai toan (moim~t c~tngang d€u co th€ coilam~tphang d6i xirng cuah~), nen cacm~t c~t

ngang khong c6 dich chuyen theo phuong doc true, khi d6 1at mong dang xet duqc xern nhir khong c6 bien dang theo phuong z (cz = 0 ), luc nay bai toanducccoi nhu bai toan bi€nd~ng ph~ng

cO'

Trong chuang 1 chung ta da thiet l~p duqc cac phuong trlnh C(1 ban cho bai toan khong gian, chung duqc viet trang h¢ tQa d¢ vuong g6e Oxyz Phan nay ta h¢ th6ng cac phuong trlnhaa:thi8t

PB~NrtcaUNG SUAT '

s,: , '

l~pachuang 1 apdung cho baito~ ph~g trongmat phing to~ d9 xy Chungtanh6m.cac phuongtrlnh thanhbanhomIan

.a)Cac phuongtrinh vi phon can bdng (phuong trinh Navier)

ocrx +Utyx +X=0

Utxy ocry + ax 0'1 -.+Y=O

b)Cdc phuong trinh tliJu ki~n bien

(2-5)

r, =u;c'£ +t' y.i.m

P y =t':cy.f! +0' y.m.

(2-6)

a)Cdc phuong trinh lien hf giila cdc bien4{ln~va chuyln vi (Phuong trtnk Cauchy)

2.2.3Ca~ ph~gtrinh v;t Iy(tl/nh /u;tHook~)

Trongnhom nay,cAnd~ yd€n811khac nhaugi~abai toan (mg 8udt phingva b:ii toanbi~nd~~g phing:

(2-9')

ax =~(8x+P8y) 1-.u . ay=~(8y+P8x)1-.u

1: Xy = Gr xy

b)Ddi vai bai todn biin dang phang:

Nhirda trlnh bay, trong bai toanbien dang phingt11di~u ki~n E z = 0 ta rut ra:

f)i~unay clurngtotrang thai ungsu~tkhong phailaphing,dov~ytirdinh'lu~tHooke vi8t

thay(a) vao(b) ta duoc:

2.3 Giii bai toanphAng theo ling suit

Chon Ancuabai toan Iicachamiingsuit:

az =II(x,Y)

a y = f2{X,y)

'fzy = 'fyz = fJ (x, y)

Laigiai cua bai toan se Ii nghiemchungcuah~phuong trlnh viphdn din bing(2-5) va

phirong trlnh tuong thich (2-8)vithoaman di8uki~nbien(2-6)

Xet tnrong hQ'P 19cth~ tfch X, Y la hing s6, vai bii toan phi-ng thl phuong trlnh (1-24)

diroc vMt lai Ii:

trong do toantuLaplace:

V2=~+~

Phuongtrlnh (2-12) Ii phuong trinh lien tuc (tuong thich) cua biJn dang dugc vi~tduai

dang irngs~!t- con dugc goi h\phuong trinh kiLevy.

KSthQ'Pphuong trlnh(2~5) va(2-12)tadtigch~gArh-baphuongtrlnh v6i ba An Ia crx, cry,

txy,diid~giai bai toano

Theo Iythuy~t v8 giai cac phuong trinh vi phan,ta _nh~n duoc nghiem tAng quat cua h~

KhiX, Y Ii ~ing 56 (twang hop thuong g~p) ta coth~ chon nghiem rieng cua h~ (2-5)

theacac dangsau:

crx=-Xx

f)~ tim nghiem tAng quat cua h~ phirong trlnh thudn nh!t (2-14) ta co th~ ti~n hanh theo

eachsau:

Ddu tien, ta chon cac hamkha vi w(x,y) tuyyvadatheamanphuong trinhddutrongh~

Ta l~i coi (c) nhir m¢t plurong trlnhvi phanmoi, vade lamthoa man(c) ta chi viec chon

hamkhaviqxx.y)nao d6 vacho:

(d)

(2-17) (2-16)

0 2

1: = - + ':fxy

"'I oxoy

Trong d6 ham <p(x,y) hi ham khitvi tuY.y,nhirng no chi dugc nh~nh\ ham irngsudt khino

thea man phirong trlnh lien tuc cua Levy(1-12'), voi 19cthetfch 1fth~ng s6 va nghiem rieng cuabai toan dircc chon theo (2-15):

MQt ham<p(x,y) thea man phirong trlnh(2=18)ducc goi 1ft

giai naydoAirytim ra nam 1862).

(2-18)

img sudt hay ham Atry (Uri

'Nhirv~Y'~th\fc chAtcua vi~e giai baitoantheo (mgslldtla-di tim Ibi giai cuaphuongtrlrihvi

phan (2-18) Phuong trinh (2-18) con goi la phuong trinh song aiJuhoa. Nghiem cua phirongtrinh (2-18) la nghiem chung cuah~ phirong trinh (2-5) va (2-9) Thay cp(x,y) tim dugc vao (2-17)

vabUQC cacfrng suAt phai thoaman.di8u ki~n bien(diauki~n v~ luc tren b~ m~t cua v~t) ta se nh~nduoc Uri giai rieng cho tirng bai toan cu thS

Trong tlurc hanh, khi giai ph~ong't~inh vi' phan (2-18)'tathll<1ng ap dung phep giai rura

ngirec'ciia Saint -Venant Tnroc h~ttagiadjnh ham <p(x,y) trong do chuamQt s5d~i hrongvah~ng s5 du6'i dang An, sau do tim irng suAt theo (2-17) va bUQC cac irng suAt timducc phai thoaman dieu kien bien (2-6),tase tim dugc d~i hrong vah~ngs5 chuabi~t

D~ gia dinh ham Airy, co th~ duavao cac cO' So' sau: Dua vaovi~c phan tfch thfr nguyen,dua vao cac dieuki~n bienho~cdua vao1mgiaicuamen Sue benv~t li~uv v Chuoj ring, khi

ta themho~ebot mQt hamb~cnhAt d,5i vci x Ya.,y: (ax+by+e) vao ham Airysekhong lam thay d6igia trj cac trng suAt thu duge tu bi~uthuc (2-17) Diroi day se trinh bay mQt s5 Ibi giaie~ebaitoan

2.4.1 Sai toan dam cong son chju IIJ'C t,p trung lY d'au tv do

Xet rnQt dAm cong son coehi~udai £,m~tcit ngang hlnh chi!nh~tcoehi~u rQng bing rnQtdan vi chieu dai, chieueaobing2e va chiu rrlQt luct~p trung P 6 ddiJnr do nhir treri hlnh 2-3.Hay xac dinh trang thai irng suAt cua dAm trong tnrong hop bo qua l\fcth~tfchcuadAm.,

Day labai toan irng suAt phing Chonh~ true tea dQ xoy nhu hinh vee Ta gial bai toan naythea phuong phap rura nguoc, voivi~egiathi~tham irng sudt dua vaoIbi gicii eua Stic benv~t Ii~u

da bi€t Theo Ibi giai Suc ben ta co:

*

(a)

Phan tfch cac quanh~trong (a)ta thAy, n~u giathi~tqui lu~t bi€riddi (ni'gsudt trongd§m

gi6ng nhu Ibi giai cua Suc benv~t li~u thi ham t'rng sudt phai chua cae 55 h\lng xy3 d~ dam bao quilu~t b~c hai thea xy trong (J':c; b~c hai theay.trong'C:cy; va chua s6 h\lng xy da dam bao s6h~ng

~~ng55 trong r:cyva khong lam thay d6i quilu~tcua (J':cva (J'y' Nhu v~yta coth~ gia thi€t ham

qJ(x,Y) du6i d~ngtang quat nhu sau:

iAx,Y) = Axj + Ex) (trong do A, B lit haih~s5 chua bi€t) (b)

TathAy hamqJ(x,y) la ham Airy vi thea mandi~u ki~n tuong thieh V2V 2cp=0 Th~o, (2-16) tatinh duC)'c caebi~uthfrccua (rngsuAt:

'xy =-8xBy ~-3A y 2- B

f)~ xac dinh cac h~ngs6A, B ta dua VaG 6 Qi~u ki~nbienv~19ctren 3biensau:

Ta th§ybaQi~u ki~n bien(d.), (d3) , (ds)tgtheaman;hai di~u ki~nbien (d2) va(da)trung

nhau (vi ham 'xy la hamch~ntheay).f)i~u ki~n bien(d 6) khong thoaman Ta dung nguyenly

Xanhvonangd~ bi~nd6idi~u ki~nbien nayb~ngeachthayhrct~ptrungPb~ng h~ lucphan b6

tuongQuang ti€pxuc voi bientrai, Luc nayQi~ukien(d 6) trathanh:

c

fr xydy =-p -c

Thay'xy nr (c) vao(d7) r6i tfch phantaeo:

So sanh (t) va (a) ta th§y 1(1i giai thea irng su~t cua Sue b~n v~t li~u va loigiai cua Ly

thuyetdan h6i trungkhopnhautaivung each xad~utvdo cuad~m. Con taiQ~U tvdo loi giaila

khongchinh xac vitaaaphai bi~n06iQl~U ki~nbien

PHAN.TICHUNG SUAT

2.4.2fJ~p hay tuimgehjn co m#t cjttam giae (Ioi giai cua Levy)

X6t mQt d~p (hay nrongchin)co m~t

cittam giac chju tacdung cua ap luc mroc

tinh amQt phia va trong hrong ban than

M~t cit dugc coi nhu rna rQngva ban v~

phla duaL Trong hrong rieng cuad~pdugc

ki hi~u'Yd,trong hrong riengcua rurocIi

'Yn-Ta chon h~ true to, 49nh~n dlnh d~p lam

g6c 0, true y trung vai suon trai cua m~t

cit (hlnh 2-4), v~t li~u lamd~p dU'gc col

nhu ddng ch§.t, do docirong dQ cua l\lc thS

tlch IahAng·sA dAi voi moi diAm trong

(e)

trong d6: .e=cos(x,n) =cos~

m= cos(y,n)= cos~o0+/3 ) =-sinp (g)

.nlaphap tuyen ngoai cua m~tben OB

Thay (a) vao (d), saud6thay ti~p.(d) VaG (g),(e)va(f) d6ngthai chu yd~nlienh~ x = ytg(3,

giai h~ phirong trinh trong(e)va(f) tatim duQ'ccacgia tr] A,B,C,D:'

A = - - Y d - - + l [ ( cosa sm a - 2- - - YsinaJ 2 cosaJ C=0

Xet d~p (haynrong ch~n)co m~t cit chiinh~t, chju ap

tadua vao 10'1 giai cua mon strc bSnv~tlieu,Ta coi latc~tnhtr

mQt conson ngam ch~t vao n~n D~m co m~t cit chu nh~t

Xetm~t c~tcach dinhQ?P m9t khoang Ii y, mo men u6n

B 1 ( sina.- cosa.Je

2 tgf3 n tg2f3

Thay(h)vao bi8u tlurc(d)taduoc:

0' x = -(yn Y cos a.+1'dx sin a.)

o = [Yd (cos a _ 2sin a J_l' n 2cos a.]x+[1d (sin a -cos a.)+1n cos a.]y

Loi giai cuaLevy trongbaitoannay hiauchfnhxac d6i

voi nhtrng latc&t duxa hai bo d~p vavai nhtrng di8mxa dlnh

va dayd~p. Trong thuc t~,tai nhtrng di8mag~n dayd~p, trng

su~tcon phuthuocvaobi~ndang cuan~n d~p.

f* hiham b~c4d6i v6i x,y

f** hi hamb~c 4 d6ivoi x,y vit lit ham chin d6ivaixvay

T~i dinh y=0 thl Oy= o '

.V6inh~nxet tren ta gia dinh ham:

Oy=y(Ax3+Bxr+ Cxy + Ox)Trong d6A, B,C, Dchua xac dinh, Theo (2-16) ta co:

O.2<p

Til' day tich phan hai ldn theo x tanh~n dircc ham cp(x,y):

<p=fdxfO'ydx+x~(y)+f2(Y)Trang d6: fila hamb~c 5d6ivoiY

,-f2lith~rn b~c 6 d6i v6i y Tac6the.c.hQn~_il.(y)=fus+b:4*01 + Hy +ly (c)

' - 4fly =120Exy+2yFx+360Jy2.+~20Ky+2yL.

Thay kSt qua trenvito (2-6) ta dugc: ,

xy(6A + 128 + 120E) + x(4C + 24F) +360Jl+ 120Ky + 24L==0

D€ phuongtrinh (e) nghiem dung voi mol x, y thl:

PHANTlCHUNG'SUAT

A +2B + 20E=0

C +6F,=0J=K=L =0B~xacdinh cach~s6trenta dtra VaGcacdi~ukien bien:

- Trensirenthuonghru:

Thay cac irng suat tim dugc tu (2-16) khi chua xetd~n trong hrong banthand~p (Yd) vao

cacdi~u kien (h)tathay 7 trongs58 phirong trinh (h)d~unghiem dung,chi duy nhdt codi~u ki~n

f)~ kh~cphuc, ta bi~nddidi~u ki~nbien nay theanguyenlySaint -Venant:

· PHA.NTtCHuNd-SUAT

'

So sanh laigiai(2-21)voi lai giai cua Sucben'v~t Ii~utanh~nthAy: anhUng'm~tcit cang

xa dinh(y cangkm)cacgiahicua crykS ca 'txy tfnh duQ'ctil'hai lbi giai cangitsai khac

Ching hantatinh cry tai diSm co toa dQ x =i~'Y= a:

- Loi giai19thuyetdan hOichota:cry= O,05Yna

- Lei giai srrcbenv~t li~u chota cry = O,25Yna, k€t qua sai khac IOn!

Nhimgtaidiem cox= a; y= 10a:

- Lei giai19thuy6t dan hAi chotacry = 248Yna, luc nay sai khac chi con cO' 0,8%

Dieu nay chophep tanghiemI~inguyen19Saint -Venant

2.4.4 Saitoen dflmtuung Lui gildcuaFilon va Ribiere

X6t mQt ban chiinh~t co chieu dili £, chieu cao

h; chieu daiduocxemnhuIOn hon nhiSu so v6'i chieu

cao, ban co chieu day deu va xem nhir rit nho so voi

kfchthiroc £ va h Ban chiu tac dung cua tai tren bien

va phan bO deu tren c~ieu day, do v~y ta co the coi

ban nhu la mQt dAm chiu uOn va goi la bait()~n ddm

luang (hinh 2-6) Tuy theo kh8i lirong gOi nra, nguai

(dAm chi co hai g6inra)va dAmnronglientuc

Degiai bai toan ddm tuong Filon va Ribiere da

de nghi chon ham Airyduoi dang chu6i hrong giac,

boi vi chon ham Airyduoi dang da thuc nhuda chon

dS giai bai toand~pseg~pkho khan dOivoinhtmg bai toan colu~tphanb6tai trong plnrc tap,Trong tnronghopnay, ham Airy duQ'c chon dum dang chuOi hrong giac:

b:enCt!2ba!toan.¥",-bi:kbQ~g :?'-~~qSn l\{cthet.i.ch •.0(=Y = 0),t~bieu thfrc (2-16)taco:

" ' , _ ~ • " A.o.•.• "'.""~'_,_",_"",:"' ~., • ~• , ,,' •• ~.,.,., •.• , •.• "," ••.•.• : , ,.:;

ax = ~ =LLIm(y)sinumx+gm(y)cosumxfJy m=l

(2-24)

't xy =

36

x,y nhu hlnh 2-6 (true y thu¢cm~t c~tnuttrai cuadAm, true x trung vai bien duoi) cac di~u ki~n

otn~tcit mut trai cua dlm (x =0, 0~y~h):

obien duai (voi y =0 va0s x sf) ):

O'yly=o =-qo(x); t'xyly=o =to(x) (d)Voi cac phirong trinh nr (a)+(d)dud~tim diroc gia tr] cua caching s5 Am,Bm, ,Hm Trong

do chung ta hruyr~ng t~i hai ddu ddm, do haim~tdAmtg donentaco haidi~u ki~nbien la:

oxI~=0 = 0 va crx Ix= e = 0

B€ dam bao haidi~ukien bien nay ta phai co: Em=Fm= Gm=Hm=O va am= ~n (e)

Thay (e) vao (2-24) ta xac dinh duoc cac h~ng 55 Am, ,Dm se tir cac di~u kien atren.Nhungtucacdi~u ki~n can bang (c)va(d) tanh~nthAy v~phai cua cluingbi~u thitidtrQng phanb6 trenb~ m~t, nguai ta coth~ bi~u di~n g~n dung cac tai trQng do du&i d~ng khai tri€n Fourier

thee sin(m1tJ.e) va eos(m1t1 i). Bay gibd~xac dinh cac h~s6 cua ehu8i ta lam thea qui tic sau: tanhan hai ve cua phuongtrlnh voi sin(m7t/l)ho~e eos(1117t11)va tie~phan hai ve tir Od.Sn l~ bingeach nay ta nh~n dugc cac phuong trlnhsau(nhungd~'eho '~i:~ntrinhhay ta chi tim e~~ hing saA,B,C,Dcho sa hang thir k nao d6 trong m):

2-1.Cho ham <p=ax? +bx'y+ cxy",yeu cAu:

-~' -"""~'-' -' .'''''' -.- ,., " ··~-_· · ·_· ~ ._ ,~ ~~._ ·_v~_~.~._

a/ Ham da cho cophai IAham Airykhong?

bl Neu dung thl hay xac dinh tAi trong tren bien cua clm chitnh~t phingABCD co chi~ucao AC=h chieu dai AB =l ,di~m A(Xo,Yo), ABllox, AClloynhu tren hlnh B-2-1

Cho ham<p=ax" + bx + cxy? + dy", yeu c§.u:

I) Ham da cho co phai la ham Airy khong?

2) Hay xac djnh cacbi~uthirc cua img suftt

3) Hayxacdinhirngsuftt cua bai toanph~ngcho tren hlnh B-2-2

I,

2-3 Cho ham <p= ax' +bx2y+cxy?+dy", yeuc§.u:

Ham dachocophai la ham Airykhong? N~udung hayxac dinh taitrong trenbien cua tftmchunh~t ph~ngABCDnhu tren hinhB-2=1 X6tchotirngtnronghopsau:

1)Chi coa:¢: 0 3)Chi coc :¢: 0

2) Hayxac dinhcacbi~uthirc cuaimgsuftt Htnh B-2",3

3)Hayxacdinhirngsufttcuabaitoanph~ngcho tren hinhB-2-3

1) Chung coth~la nghiem cua bai toan danh6i phang dtroc khong?

2)N~uduochayxac dinh cacbi~uthircirngsuftt cua bai toan phang cho tren hinh B-2-4.2-50 Cho cacbi~uthirc cua imgsuftt:

2-60 Cho cacbi~uthirc cua irng suftt:

y

3q ( 2 2~

xy 4c3

Chung coth~ la loi giai cua bai toan pbAng cho tren, hlnhB -2-5 dugc ]{hong? • t ' , , ' " "",' ' '

2-7.Cho cacbi~u.thuccua imgsu~t:

qy=q

mnhB-2-6

y

YeucAu:

1)Chung eoth~la nghiem cua bai toan dan hOi phAng<lugckheng?

2)N€udirochayxac djnh cacbi~u thue img su§.t cua bai toan pbAng chotren hinh B-2-6

1) Chung coth~la nghiem cua bai toan dan hOi phAng dugc khong?

2) N€u dircc hay xac dinh cacbi~utlurc(mgsuit ella bal toan phAng cho tren hlnh B'.~2-4.2-~ C~o cac ung sui~: " i.

x x2 +y2

40

ho~e: r= ~X2 +y2 e= arctgy (3-1)

x

Chuang 2 da xet bai toan phang trongh~toa

dQ vuong gee, da giai dirocnhi~u baitoan thuong

g~p trong thirc t8.Tuy nhien, co nhi~u bai toan

phangse dtroc giaid~ dang hon n8u nhu ta dung

h~toa dQC\fC.

MQtdi~m Mco toa dQ x, y, trongh~ toadQ

vuong geeno se co toa dQ rva etrong h~ toadQ

cue (hlnh 3-1) Gitra cac toadQdocolienh~sau:

M~tphangxuyentam eo phap tuyen theo phtrongeduQ'c goi lam~tYang

Tir lienh~(3-1)tasuy ra cac lienh~ d~oham sau:

(3-3)

fi f 8A fi f 2 (1 of 10 2f) 2 (1 of 10 2f)

=-= cos e+ - - + - - - SIn B+ - - - - sIn2B

Dod6t3ng codiincliaAC doca u va vgayram: (UdB+ :;dB) (b)

Theo dinh nghia,bi~n dang dai d ddi theo phuongklnhrvathea phuong vong () duQ'c xacdinhtil"(a) va(b) la:

Bi~ndangcuagoc vuong CAB trongm~t phing·(r,B) duQ'c coi nhu la tdng cac gocxoay

arova a(Jrvibien dangbenentacoth~coi: ,.<

Y,e =Ya- =a,e +aa- = r of) + Or - r

Tilcac lienh~(c),(d), va (e)taduQ'c cac phuong trinh:

Phuong trlnh (3-8) cling coth~ vi~t du6'idang irng su~t.

b)Phudng trinh lien tuc vi biin dang:

Bang each loai cac thanhphAn chuyen dich u,vtrong cac lienh~·(3-7) ta thuduoc phirongtrinhlientuc cuabiendang: