Module 6: Two Dimensional Problems in Polar Coordinate System

Transcription

1 Modl6/Lon Modl 6: Two Dimnional Poblm in Pola Coodinat Stm 6 INTRODUCTION I n an laticit poblm th pop choic o th coodinat tm i tml impotant c thi choic tablih th complit o th mathmatical pion mplod to ati th ild ation and th bonda condition In od to ol two dimnional laticit poblm b mploin a pola coodinat nc am, th ation o ilibim, th dinition o Ai St nction, and on o th t ation o compatibilit mt b tablihd in tm o Pola Coodinat 6 STRAINDISPLACMNT RLATIONS Ca : Fo Two Dimnional Stat o St Fi 6 Domd lmnt in two dimnion Conid th domation o th ininitimal lmnt ABCD, dnotin and diplacmnt b and pctil Th nal domation pincd b an lmnt ma b Applid laticit o nin TGSithaam & LGoindaRaj

2 Modl6/Lon add a compod o ) a chan in th lnth o th id, and ) otation o th id a hown in th i 6 Rin to th i, it i obd that a diplacmnt "" o id AB lt in both adial and tanntial tain Tho, Radial tain 6) and tanntial tain d to diplacmnt p nit lnth o AB i ) d d ) d Tanntial tain d to diplacmnt i in b ) d d Hnc, th ltant tain i ) ) 6) 63) 64) Similal, th hain tain can b calclatd d to diplacmnt and a blow Componnt o hain tain d to i ) d d Componnt o hain tain d to i ) Tho, th total ha tain i in b 65) 66) ) ) Applid laticit o nin TGSithaam & LGoindaRaj

3 3 Applid laticit o nin TGSithaam & LGoindaRaj Modl6/Lon 67) Ca : Fo Th Dimnional Stat o St Fi 6 Domd lmnt in th dimnion Th taindiplacmnt lation o th mot nal tat o t a in b w,, 68)

4 4 Applid laticit o nin TGSithaam & LGoindaRaj Modl6/Lon w w 63 COMPATIBILITY QUATION W ha om th tain diplacmnt lation: Radial tain, 69a) Tanntial tain, 69b) and total hain tain, 69c) Dintiatin ation 69a) with pct to and ation 69b) with pct to, w t 69d) ú û ù ê ë é \ 69) Now, Dintiatin ation 69c) with pct to and ation 69d), w t \ 69) Dintiatin ation 69) with pct to and ation 69) with pct to, w t, 3 69)

5 5 Applid laticit o nin TGSithaam & LGoindaRaj Modl6/Lon and 3 o 3 69h) Sbtactin ation 69h) om ation 69) and ation 69), w t, \ 64 STRSSSTRAIN RLATIONS In tm o clindical coodinat, th ttain lation o 3dimnional tat o t and tain a in b )] [ n )] [ n 60) )] [ n Fo twodimnional tat o t and tain, th abo ation dc to, Fo Plan St Ca ) n ) n 6) t G

6 Modl6/Lon Fo Plan Stain Ca n ) [n ) ] n ) [ n ) ] t G 6) 65 AIRY S STRSS FUNCTION With nc to th twodimnional ation o t tanomation [ation a) to c)], th lationhip btwn th pola t componnt, and t and th Catian t componnt, and t can b obtaind a blow Now w ha, t t 63) t ) t Sbtittin 64) in 63), w t t 64) 65) t Th pola componnt o t in tm o Ai t nction a a ollow 66) and t Th abo lation can b mplod to dtmin th t ild a a nction o and 67) Applid laticit o nin 6 TGSithaam & LGoindaRaj

7 Modl6/Lon 66 BIHARMONIC QUATION A dicd ali, th Ai St nction ha to ati th Bihamonic ation 4 Ñ 0, poidd th bod oc a o o contant In Pola coodinat th t 4 nction mt ati thi am ation; how, th dinition o Ñ opato mt b modiid to it th pola coodinat tm Thi modiication ma b accomplihd b 4 tanomin th Ñ opato om th Catian tm to th pola tm Now, w ha,, and tan 68) wh and a dind in Fi 63 Dintiatin ation 68) i Applid laticit o nin 7 TGSithaam & LGoindaRaj

8 8 Applid laticit o nin TGSithaam & LGoindaRaj Modl6/Lon Fi63 d d d d d d \ Alo, d d c c \ Similal, \

9 9 Applid laticit o nin TGSithaam & LGoindaRaj Modl6/Lon Now, i) Similal, ii) And, iii) Addin i) and ii), w t, Ñ i o 4 ) 0 j j j j j Ñ Ñ Ñ Th abo Bihamonic ation i th t ation o compatibilit in tm o Ai t nction d in pola coodinat tm