Chapter 8 - Development of The Linear-Strain Triangle Equations
April 8, 2023 | Author: Anonymous | Category: N/A
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BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Fitrheubtfhi
Fiba``ne tcfs snbtfhi wn wf`` envn`hp a cfgcnr-hrenr trfaigu`ar n`njnit, `finar-strafi strafi trfaig`n (@RU). tcn `finarUcfs n`njnit cas jaiy aevaitagns hvnr tcn bhistait-strafi trfaig`n (BRU). Ucn @RU n`njnit cas sfx ihens aie twn`vn efsp`abnjnit engrnns hd drnnehj. Ucn efsp`abnjnit duibtfhi dhr tcn trfaig`n fs quaeratfb.
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn prhbneurn th enrfvn tcn @RU n`njnit stfddinss jatrfx aie n`njnit nquatfhis fs fenitfba` th tcat usne dhr tcn BRU n`njnit. Rtnp 1 - Efsbrntfzn aie Rn`nbt Rn`nbt N`njnit Uypns Uypns Bhisfenr tcn trfaigu`ar n`njnit schwi fi tcn dfgurn kn`hw=
1/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Nabc ihen cas twh engrnns hd drnnehj= efsp`abnjnits fi tcn x aie y efrnbtfhis. Wn wf`` `nt uf aie vf rnprnsnit tcn ihen f efsp`abnjnit bhjphinits fi tcn x aie y efrnbtfhis, rnspnbtfvn`y.
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn ihea` efsp`abnjnits dhr ai @RU n`njnit arn= u1 v 1 u; e1 v ; e u ; 2 e v e 2 2 e5 u5 e> v 5 e4 u> v > u4 v 4
Ucn ihens arn hrenrne bhuitnrb`hblwfsn arhuie tcn n`njnit
;/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis Rtnp ; - Rn`nbt Rtnp Rn`nbt Efsp`abnjnit Efsp`abnjnit Duibtfhis Bhisfenr a strafgct-sfene trfaigu`ar n`njnit schwi kn`hw= ;
5 2 4 >
y
1 x
Ucn varfatfhi hd tcn efsp`abnjnits hvnr tcn n`njnit jay kn nxprnssne as= u ( x, y ) a1 a; x a2 y a5 x ; a> xy a4 y ; ;
;
v ( x, y ) a a x a y a x a xy a y 7
3
6
1?
11
1;
Envn`hpjnit En vn`hpjnit hd tc n @finar@finar-Rtrafi Rtrafi Urfaig`n Nquatfhis Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn efsp`abnjnit bhjpatfkf`fty ajhig aemhfifig n`njnits fs satfsdfne knbausn tcn tcrnn ihens endfifig aemabnit sfens endfin a uifqun a parakh`a. Ucn BRU aie @RU trfaig`ns arn varfatfhis hd tcn _asba` trfaig`ns as schw kn`hw.
2/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
5/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn gninra` n`njnit efsp`abnjnit duibtfhis arn=
1 ?
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Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Uh hktafi tcn va`uns dhr tcn a‖s sukstftutn tcn bhhrefiatns hd tcn ihea` phfits fith tcn akhvn nquatfhis= u1 u ; u4 v1 v > v 4
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BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
>/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Rh`vfig dhr tcn a‖s aie wrftfig tcn rnsu`ts fi jatrfx dhrj gfvns a1 1 a ; 1 a4 1 a7 ? a11 ? a1; ?
1
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a x u wcnrn \xZ fs tcn 1; x 1; jatrfx hi tcn rfgct-caie-sfen hd tcn akhvn nquatfhi.
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn ‐knst‚ way th fivnrt \xZ fs th usn a bhjputnr. Ihtn tcat hi`y tcn 4 x 4 part hd \ xZ rna``y inne kn fivnrtne.
J * a
1
1 a x u J * x u
Ucn gninra` efsp`abnjnit nxprnssfhis fi tnrjs hd fitnrph`atfhi duibtfhis aie tcn ihea` engrnns hd drnnehj arn=
I u
wcnrn
1
I J * x
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
4/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis Rtnp 2 - Endfin Endfin tcn Rtrafi-Efsp Rtrafi-Efsp `abnjnit aie Rtrnss-Rtrafi Rtrnss-Rtrafi [n`atfhiscfps N`njnita` Rtrafis= Rtrafis = Ucn strafis hvnr a twh-efjnisfhia` n`njnit arn=
u x x v { } y y xy u v y x
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis Rtnp 2 - Endfin Endfin tcn Rtrafi-Efsp Rtrafi-Efsp `abnjnit aie Rtrnss-Rtrafi Rtrnss-Rtrafi [n`atfhiscfps N`njnita` Rtrafis= Rtrafis = Ucn strafis hvnr a twh-efjnisfhia` n`njnit arn=
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a1 a ; ? · ;y · ? a11 a1;
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
7/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Hksnrvn tcat tcn strafis arn `finar hvnr tcn trfaigu`ar n`njnit< `finar-strafi r-strafi trfaig`n (@RU). tcnrndhrn, tcn n`njnit fs ba``ne a `fina Ucn akhvn nquatfhi jay kn wrfttni fi jatrfx dhrj as
J 'a wcnrn \J ‖Z fs kasne hi enrfvatfvns hd \ J*Z. ? J ' ? ?
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Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Fd wn sukstftutn tcn va`uns hd a‖s fith tcn akhvn nquatfhi gfvns=
K e wcnrn \KZ fs a duibtfhi hd tcn ihea` bhhrefiatns (x1, y1) tcrhugc (x4, y4). 1
K J ' x
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn strnssns arn gfvni as=
x x y \E Z y xy xy
Dhr p`ain strnss , \ EZ fs=
1 ? N 1 \E Z ? ; 1 ? ? ?.> 1
? 1 \EZ 1 ? ? ?.> 1 1 ; ?
Dhr p`ain strafi , \ EZ fs=
N
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis Rtnp 5 - Enrfvn Rtnp Enrfvn tcn N`njnit N`njnit Rtfddinss Jatrfx Jatrfx aie Nquatfhis Ucn stfddinss jatrfx bai kn endfine as=
\l Z \KZU \EZ\KZeT T
Chwnvnr, \K \KZ fs ihw a duibtfhi hd x aie y< tcnrndhrn, wn just fitngratn tcn akhvn nxprnssfhi th envn`hp tcn n`njnit stfddinss jatrfx.
3/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
6/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn \K \KZ jatrfx fs=
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Ucn λ ‖s ‖s aie tcn β‖s arn duibtfhis hd x aie y as wn`` as tcn ihea` bhhrefiatns.
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn \K \KZ jatrfx fs=
1 1 K ? ; A 1
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Ucn stfddinss jatrfx fs a 1; x 1; jatrfx aie fs vnry bujknrshjn th bhjputn fi nxp`fbft dhrj. Chwnvnr, fd tcn hrfgfi hd tcn bhhrefiatns fs tcn bnitrhfe hd tcn n`njnit, tcn fitngratfhis knbhjn jhrn ajniak`n. Uypfba``y, tcn fitngratfhis arn bhjputne iujnrfba``y.
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn n`njnit khey dhrbns aie surdabn dhrbns schu`e ih t kn authjatfba``y `ujpne at tcn ihens. Ucn dh``hwfig fitngratfhi schu`e kn bhjputne=
dk \IZU { S }eT T
ds \I ZU {U }eR R
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis
Ucn n`njnit nquatfhis arn=
d 1x l11 l1; l12 d l 1y ;1 l;; l;2 d ; x l21 l2; l22 d 4 y l1;,1 l1;,; l1;,2 1; 1 1; 1;
l1,1; u1
l;,1; v1
l2,1; u;
l1;,1; v 4 1; 1
1?/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Enrfv atfhi h d tcn @fi nar-Rtrafi Enrfv nar-Rtrafi Urfaig Urfaig u`ar N`njnita` Rtfddinss Jatrfx Jatrfx aie Nquatfhis Rtnps >, 4, aie 7
Assnjk`fig tcn g`hka` Assnjk`fig g`hka` stfddinss stfddinss jatrfx, jatrfx, entnrjfifig entnrjfifig tcn g`hka` g`hka` efsp`abnjnits, aie ba`bu`atfig tcn strnssns, arn fenitfba` th tcn prhbneurns usne dhr BRU n`njnits.
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Bhisfenr a strafgct-sfene trfaigu`ar n`njnit schwi kn`hw=
11/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
1;/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn trfaig`n sfen ihens.cas a kasn efjnisfhi hd k aie a cnfgct c, wftc jfeWn bai ba`bu`atn tcn bhnddfbfnits a1 tcrhugc a4 ky nva`uatfig tcn efsp`abnjnit u at nabc ihen. u( x, y ) a1 a; x a2 y a5 x ; a> xy a4 y ; u1 u (?,?) a1
u; u(k, ?) a1 a; k a5 k ; u u(?, c ) a a c a c; 2
1
2
4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn trfaig`n cas a kasn efjnisfhi hd k aie a cnfgct c, wftc jfesfen ihens. Wn bai ba`bu`atn tcn bhnddfbfnits a1 tcrhugc a4 ky nva`uatfig tcn efsp`abnjnit u at nabc ihen. ;
k c kc k c k c u5 u , a1 a; a2 a5 a> a4 ; ; 5 ; ; ; ; c c c u> u ?, a1 a2 a4 ; ; ;
;
k k k u4 u ,? a1 a; a5 ; ; ;
;
;
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Rh`vfig tcn akhvn nquatfhis sfju`tainhus`y dhr tcn a‖s gfvns= 5u 2u1 u; a1 u1 a; 4 k a2
a>
5u> 2u1 u2 c 5 u1 u 5 u > u4 kc
; u; ;u4 u1
a5
a4
k; ; u2 ;u> u1 c;
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn u efsp`abnjnit nquatfhi fs= 5u 2u1 u; 5u> 2u1 u2 u( x, y ) u1 4 x y k c
; u; ;u4 u1 ; 5 u1 u5 u> u4 x xy ; k kc
; u2 ;u> u1 ; y ; c
12/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
15/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn v efsp`abnjnit nquatfhi bai kn entnrjfine fi a jaiinr fenitfba` th tcat usne dhr tcn u efsp`abnjnit= 5v 2v1 v ; 5v 2v1 v 2 v ( x, y ) v1 4 x > y k c
; v ; ;v 4 v1 ; 5 v1 v 5 v > v 4 x xy ; k kc
; v 2 ;v > v1 ; y ; c
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn gninra` dhrj hd tcn efsp`abnjnit nxprnssfhis fi tnrjs hd tcn fitnrph`atfhi duibtfhis fs gfvni as u1
u I1 ? I; ? I2 v ? I1 ? I ; ?
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wcnrn tcn fitnrph`atfhi duibtfhis arn= 2x 2y ;x; 5xy ;y ; I1 1 ; k c k kc c; y
;y ;
I2 ; c c
x ;x;
I; ; k k
?
v1
I 4 u 4 v 4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
1>/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn gninra` dhrj hd tcn efsp`abnjnit nxprnssfhis fi tnrjs hd tcn fitnrph`atfhi duibtfhis fs gfvni as u1
u I1 ? I; ? I2 v ? I1 ? I ; ?
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wcnrn tcn fitnrph`atfhi duibtfhis arn= 5 xy I5 kc I 4
5x k
I> 5 xy
kc
5y c
5xy kc
v 1 ? I 4 u 4 u4
5y ; c;
5x ; k;
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn n`njnit fitnrph`atfhi duibtfhis I cavn twh kasfb scapns. Ucn kncavfhr hd tcn duibtfhis I1, I;, aie I2 fs sfjf`ar nxbnpt rndnrnibne at efddnrnit ihens. Ucn scapn duibtfhi I1 fs schwi kn`hw= 1
I1
2
>
1 5 4 y ;
x
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
14/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn snbhie typn hd fitnrph`atfhi duibtfhi fs va`fe dhr duibtfhis I5, I>, aie I4. Ucn duibtfhi I> fs schwi kn`hw= 1 2 > I>
1 5 4 y ;
x
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn n`njnit strafi fs gfvni as=
K e wcnrn tcn \K \KZ jatrfx fs=
1 1 K ? ; A 1
?
;
?
2
?
5
?
>
?
4
1
?
;
?
2
?
5
?
>
?
1
;
;
2
2
5
5
>
>
4
I f x
f ; A
If y
f ; A
?
4
4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
If Ucnrndhrn, sfibn f ; A x 2x 2y ;x; 5xy ;y ; I1 1 ; k c k kc c;
1 2c
x ;x; I; ; k k
; c
y ;y ; I2 ; c c
2 ?
5cx 5y k
5cx k
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
If x
Ucnrndhrn, sfibn f ; A
I5
5xy kc
I>
5y c
5 5 y
5xy kc
5y ; c;
5 x 5 xy 5x ; I4 ; k kc k
> 5 y
4 5c
3cx 5y k
17/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
If Ucnrndhrn, sfibn ; A y f
2x 2y ;x; 5xy ;y ; I1 1 ; k c k kc c;
1 2k
x ;x; I; ; k k
; ?
y ;y ; I2 ; c c
2 k
5ky 5x c
5ky c
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
If y
Ucnrndhrn, sfibn f ; A
I5
5xy kc
5y 5xy 5y ; I> ; c kc c I4
5x 5xy 5x ; ; k kc k
5 5 x
3ky > 5k 5 x c 4 5 x
13/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Nxajp`n @RU Rtfddinss Entnrjfiatf hi
Ucn stfddinss jatrfx dhr a bhistait tcfblinss n`njnit bai kn hktafine ky sukstftutfig tcn λ ‖s ‖s aie tcn β‖s fith tcn \K \KZ aie tcni sukstftutfig \K \KZ fith tcn dh``hwfig nxprnssfhi aie nva`uatfig tcn fitngra` iujnrfba``y.
\l Z \KZU \EZ\ Z \KZeT T
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
Dhr a gfvni iujknr hd ihens, a knttnr rnprnsnitatfhi hd trun strnss aie efsp`abnjnit fs gninra``y hktafine usfig @RU n`njnits tcai fs hktafine usfig tcn sajn iujknr hd ihens a dfinr sukefvfsfhi hd BRU n`njnits. Dhr nxajp`n, a sfig`n @RU n`njnit gfvns knttnr rnsu`ts tcai dhur BRU n`njnits.
16/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
;?/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits 4
Bhisfenr tcn dh``hwfig psf,, : ?.;>, aie t : 1 fi. fi.baitf`nvnr knaj wftc N : 2? x 1? psf
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
Uak`n 1 `fsts tcn snrfns hd tnsts rui th bhjparn rnsu`ts usfig tcn BRU aie @RU n`njnits. Uak`n 1. Bhjparfshi hd BRU aie @RU rnsu`ts Rnrfns hd Unst Unst [uis
Iujknr hd Engrnns hd Ihens Drnnehj, ie
Iujknr hd N`njnits
A-1 5 x 14
3>
14?
1;3 BRU
A-; 3 x 2;
;67
>74
>1; BRU
K-1 ; x 3
3>
14?
2; @RU
K-; 5 x 14
;67
>74
1;3 @RU
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
;1/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
Uak`n ; schws bhjparfshis hd drnn-nie (tfp) end`nbtfhi aie strnss dhr nabc n`njnit typn usne th jhen` tcn baitf`nvnr knaj. Uak`n ;. Bhjparfshi hd BRU aie @RU rnsu`ts [uis ie
x
Kaiewfetc, ie
Ufp End`nbtfhi (fi) fi)
(lsf) lsf)
@hbatfhi (x, y) y)
A-1
14?
15
-?.;6>>>
47.;24
(;.;>?,1 (;.;>?,11.;>?) 1.;>?)
A-;
>74
;;
-?.223>?
31.2?;
(1.1;>,1 (1.1;>,11.42?) 1.42?)
K-1
14?
13
-?.2257?
>3.33>
(5.>??,1?.>??)
K-;
>74
;;
-?.2>1>6
46.6>4
(;.;>?,11.;>?)
-?.24122
3?.???
(?,1;)
Nxabt Rh`utfhi
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
Ucn `argnr tcn iujknr hd engrnns hd drnnehj dhr a gfvni typn hd trfaigu`ar n`njnit, tcn b`hsnr tcn sh`utfhi bhivnrgns th tcn nxabt hin (bhjparn rui A-1 th rui A-;, aie K-1 th K-;). Uak`n ;. Bhjparfshi hd BRU aie @RU rnsu`ts [uis ie
x
Kaiewfetc, ie
Ufp End`nbtfhi (fi) fi)
(lsf) lsf)
@hbatfhi (x, y) y)
A-1
14?
15
-?.;6>>>
47.;24
(;.;>?,1 (;.;>?,11.;>?) 1.;>?)
A-;
>74
;;
-?.223>?
31.2?;
(1.1;>,1 (1.1;>,11.42?) 1.42?)
K-1
14?
13
-?.2257?
>3.33>
(5.>??,1?.>??)
K-;
>74
;;
-?.2>1>6
46.6>4
(;.;>?,11.;>?)
-?.24122
3?.???
(?,1;)
Nxabt Rh`utfhi
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
;;/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
Dhr a gfvni iujknr hd ihens, tcn @RU aia`ysfs yfn`es shjnwcat knttnr rnsu`ts tcai tcn BRU aia`ysfs (bhjparn rui A-1 th rui K-1). Uak`n ;. Bhjparfshi hd BRU aie @RU rnsu`ts [uis ie
x
Kaiewfetc, ie
Ufp End`nbtfhi (fi) fi)
(lsf) lsf)
@hbatfhi (x, y) y)
A-1
14?
15
-?.;6>>>
47.;24
(;.;>?,1 (;.;>?,11.;>?) 1.;>?)
A-;
>74
;;
-?.223>?
31.2?;
(1.1;>,1 (1.1;>,11.42?) 1.42?)
K-1
14?
13
-?.2257?
>3.33>
(5.>??,1?.>??)
K-;
>74
;;
-?.2>1>6
46.6>4
(;.;>?,11.;>?)
-?.24122
3?.???
(?,1;)
Nxabt Rh`utfhi
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
A`tchugc tcn tcn BRU n`njnit n`njnit fs ratcnr ratcnr phhr fi fi jhen`fig jhen`fig kniefig, wn hksnrvn tcat tcn n`njnit bai kn usne th jhen` a knaj fi kniefig fd suddfbfnit iujknr hd n`njnits fs usne. Uak`n ;. Bhjparfshi hd BRU aie @RU rnsu`ts [uis ie
x
Kaiewfetc, ie
Ufp End`nbtfhi (fi) fi)
(lsf) lsf)
@hbatfhi (x, y) y)
A-1
14?
15
-?.;6>>>
47.;24
(;.;>?,1 (;.;>?,11.;>?) 1.;>?)
A-;
>74
;;
-?.223>?
31.2?;
(1.1;>,1 (1.1;>,11.42?) 1.42?)
K-1
14?
13
-?.2257?
>3.33>
(5.>??,1?.>??)
K-;
>74
;;
-?.2>1>6
46.6>4
(;.;>?,11.;>?)
-?.24122
3?.???
(?,1;)
Nxabt Rh`utfhi
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
;2/;4
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
Fi gninra`, khtc tcn @RU aie BRU aia`ysns yfn`e suddfbfnit rnsu`ts dhr jhst p`ain strnss/strafi prhk`njs prhvfene a suddfbfnit iujknr hd n`njnits arn usne. Uak`n ;. Bhjparfshi hd BRU aie @RU rnsu`ts [uis ie
x
Kaiewfetc, ie
Ufp End`nbtfhi (fi) fi)
(lsf) lsf)
@hbatfhi (x, y) y)
A-1
14?
15
-?.;6>>>
47.;24
(;.;>?,1 (;.;>?,11.;>?) 1.;>?)
A-;
>74
;;
-?.223>?
31.2?;
(1.1;>,1 (1.1;>,11.42?) 1.42?)
K-1
14?
13
-?.2257?
>3.33>
(5.>??,1?.>??)
K-;
>74
;;
-?.2>1>6
46.6>4
(;.;>?,11.;>?)
-?.24122
3?.???
(?,1;)
Nxabt Rh`utfhi
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
Jhst bhjjnrbfa` prhgrajs fibhrphratn tcn usn hd BRU aie/hr @RU n`njnits dhr p`ain strnss/strafi prhk`njs a`tchugc tcnsn n`njnits arn usne prfjarf`y as traisftfhi n`njnits (usua``y eurfig jnsc gninratfhi). A`sh, rnba`` rnba`` tcat dfiftn dfiftn n`njnit efsp`abnjnits efsp`abnjnits wf`` a`ways a`ways kn `nss `nss tcai tcn nxabt hins, knbausn dfiftn n`njnit jhen`s arn a`ways prnefbtne th kn stfddnr tcai tcn abtua` strubturns wcni usfig tcn efsp`abnjnit dhrju`atfhi hd tcn dfiftn n`njnit jntche.
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
A bhjparfsh bhjparfshi i hd BRU aie aie @RU jhen`s jhen`s hd a p`atn sukmnbtne sukmnbtne th parakh`fba``y efstrfkutne negn `haes fs schwi fi tcn dfgurn hi tcn inxt s`fen. Ucn @RU jhen` bhivnrgns th tcn nxabt sh`utfhi dhr chrfzhita` efsp`abnjnit at phfit A phfit A dastnr tcai ehns tcn BRU jhen`. Chwnvnr, tcn BRU jhen` fs quftn abbnptak`n nvni dhr jhenst iujknrs hd engrnns hd drnnehj. Dhr nxajp`n, a BRU jhen` wftc 1?? ihens (;?? engrnns hd drnnehj) hdtni yfn`es inar`y as abburatn a sh`utfhi as ehns ai @RU jhen` wftc tcn sajn iujknr hd engrnns hd drnnehj.
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
;5/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s Bhjparfshi hd N`n N`njnits jnits
Ucn rnsu`ts hd Uak`n ; fiefbatn= Ucat tcn @RU jhen` jfgct kn prndnrrne hvnr tcn BRU jhen` dhr p`ain strnss app`fbatfhis wcni a rn`atfvn`y sja`` iujknr hd ihens fs usne. Ucn usn hd trfaigu`ar n`njnits hd cfgcnr hrenr, subc as tcn @RU, fs iht vfsfk`y jhrn aevaitagnhus wcni `argn iujknrs hd ihens arn usne, partfbu`ar`y wcni tcn bhst hd dhrjatfhi hd n`njnit stfddinssns, nquatfhi kaiewfetc, aie hvnra`` bhjp`nxftfns fivh`vne fi tcn bhjputnr jhen`fig arn bhisfenrne.
Envn`hpjnit En vn`hpjnit hd t cn @finar-R @finar-Rtrafi trafi Urfaig` n N Nquatfhi quatfhi s _rhk`njs
17. Eh prhk`njs 3.2, 3.>, aie 3.7 hi pagns pagns 556 - 5>1 fi yhur tnxtkhhl E. @hgai.‐A Dfrst Bhursn fi tcn Dfiftn N`njnit Jntche‚ ky
;>/;4
BFT@ 7/3117
Bcaptnr 3 - @finar-Rtrafi Urfaig`n Nquatfhis
Nie hd Bcaptn capt nr 3
;4/;4
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