Module 5: Two Dimensional Problems in Cartesian Coordinate System

Transcription

1 Moule : Two Dimenionl Problem in Crein Coorine Sem Moule/Leon.. SOLUTIONS OF TWODIMENSIONAL PROBLEMS BY THE USE OF POLYNOMIALS Te equion given b will be iie b ereing Air uncion (, ) olnomil. () Polnomil o e Fir Degree Le b Now, e correoning ree re Tereore, i re uncion give re ree bo. (b) Polnomil o e Secon Degree c Le b Te correoning ree re c b 0 (.) in e orm o omogeneou Ti ow e bove re comonen o no een uon e coorine n, i.e., e re conn rougou e bo rereening conn re iel. Tu, e Alie Elici or Engineer T.G.Sirm & L.GovinRju

2 Moule/Leon re uncion rereen e o uniorm enion (or comreion) in wo ereniculr irecion ccomnie wi uniorm er, own in e Figure. below. Figure. Se o ree (c) Polnomil o e Tir Degree b c Le 6 6 Te correoning ree re c b b c Figure. Conn Sre iel Ti re uncion give linerl vring re iel. I oul be noe e mgniue o e coeicien, b, c n re coen reel ince e ereion or i iie irreecive o vlue o ee coeicien. Now, i b c 0 ece, we ge rom e re comonen 0 n 0 Ti correon o ure bening on e ce ereniculr o e i. \ A, Alie Elici or Engineer T.G.Sirm & L.GovinRju

3 Moule/Leon n A, Te vriion o wi i liner own in e Figure.. Figure. Vriion o Sree Similrl, i ll e coeicien ece b re ero, en we ge 0 b b Te ree rereene b e bove re iel will vr own in e Figure.. Alie Elici or Engineer T.G.Sirm & L.GovinRju

4 Moule/Leon Figure. Vriion o Sree In e Figure., e re i conn wi (i.e. conn long e n L o e bem), bu vrie wi riculr ecion. A, b (i.e., enile), wile, b (i.e. comreive). i ero rougou. Ser re i ero 0 n i equl o b L L. A n oer ecion, e er re i roorionl o. () Polnomil o e Four Degree Le b 6 c 6 Te correoning ree re given b c e e b c b c Now, king ll coeicien ece equl o ero, we in, 0, Alie Elici or Engineer T.G.Sirm & L.GovinRju

5 Auming oiive, e orce cing on e bem re own in e Figure.6. Moule/Leon Figure.6 Sree cing on e bem On e longiuinl ie, ± re uniorml iribue ering orce. A e en, e ering orce re iribue ccoring o rbolic iribuion. Te ering orce cing on e bounr o e bem reuce o e coule. Tereore, M Or M L L L Ti coule blnce e coule rouce b e norml orce long e ie L o e bem. (e) Polnomil o e Fi Degree b c e Le j Te correoning re comonen re given b c (c ) ( b ) b c b c (c ) Alie Elici or Engineer T.G.Sirm & L.GovinRju

6 Moule/Leon Here e coeicien, b, c, re rbirr, n in juing em we obin oluion or vriou loing coniion o e bem. Now, i ll coeicien, ece, equl o ero, we in Ce (i) Te norml orce re uniorml iribue long e longiuinl ie o e bem. Ce (ii) Along e ie L, e norml orce coni o wo r, one ollowing liner lw n e oer ollowing e lw o cubic rbol. Te ering orce re roorionl o on e longiuinl ie o e bem n ollow rbolic lw long e ie L. Te iribuion o e ree or e Ce (i) n Ce (ii) re own in e Figure.7. Ce (i) Alie Elici or Engineer 6 T.G.Sirm & L.GovinRju

7 Moule/Leon Ce (ii) Figure.7 Diribuion o orce on e bem.. BENDING OF A NARROW CANTILEVER BEAM SUBJECTED TO END LOAD Conier cnilever bem o nrrow recngulr croecion crring lo P e en own in Figure.8. Figure.8 Cnilever ubjece o n en lo Te bove roblem m be coniere ce o lne re rovie e ickne o e bem i mll relive o e e. Alie Elici or Engineer 7 T.G.Sirm & L.GovinRju

8 Bounr Coniion ( ) 0 A ± ( ) 0 A ± Moule/Leon (.) Tee coniion ere e c e o n boom ege o e bem re no loe. Furer, e lie lo P mu be equl o e reuln o e ering orce iribue cro e ree en. Tereore, P ò b (.) B Invere Meo A e bening momen vrie linerl wi, n n ecion een uon, i i reonble o ume generl ereion o e orm c (.b) were c conn. Inegring e bove wice wi reec o, we ge c ( ) ( ) (.c) 6 were () n () re uncion o o be eermine. Inroucing e u obine ino Equion (.), we ve 0 (.) Since e econ erm i ineenen o, ere ei oluion or ll n rovie 0 n 0 Inegring e bove, we ge () c c c c () c 6 c 7 c 8 c 9 were c, c., c 9 re conn o inegrion. Tereore, (.c) become c ( c c c c ) c6 c7 c8 c9 6 Now, b einiion, 6( c c6 ) ( c c7 ) Alie Elici or Engineer 8 T.G.Sirm & L.GovinRju

9 c c c c Now, ling bounr coniion o (.e), we ge c c c 6 c 7 0 n c c Alo, ò ò b c b ( ) P Solving, c P P b I were I b i e momen o ineri o e croecion bou e neurl i. Moule/Leon (.e) From Equion (.b) n (.e), ogeer wi e vlue o conn, e ree re oun o be P P, 0, ( ) I I Te iribuion o ee ree ecion w rom e en i own in Figure.8 b B SemiInvere Meo Beginning wi bening momen M P, we m ume re iel imilr o e ce o ure bening: P I ( ) (.), 0 Te equion o comibili re iie b ee equion. On e bi o equion (.), e equion o equilibrium le o 0, 0 (.g) From e econ ereion bove, een onl uon. Te ir equion o (.g) ogeer wi equion (.) give P I Alie Elici or Engineer 9 T.G.Sirm & L.GovinRju

10 P or c I Here c i eermine on e bi o ( ) ± 0 Tereore, c P I P P Hence, I I P Or ( ) I Moule/Leon Te bove ereion iie equion (.) n i ienicl wi e reul revioul obine... PURE BENDING OF A BEAM Conier recngulr bem, leng L, wi b, e, ubjece o ure coule M long i leng own in e Figure.9 Figure.9 Bem uner ure bening Conier econ orer olnomil uc i n erm give onl conn e o re. Tereore c b B einiion, Alie Elici or Engineer 0 T.G.Sirm & L.GovinRju

11 Moule/Leon,, \ Diereniing e uncion, we ge c, n b Coniering e lne re ce, 0 Bounr Coniion () A ±, 0 (b) A ±, 0 (c) A n vlue, b ò bening momen conn \b ò c bc ú 0 û Tereore, i clerl oe no i e roblem o ure bening. Now, conier irorer equion b c 6 6 Now, c b b c () (b) (c) From (b) n bounr coniion () bove, 0 ± b or n vlue o \ b 0 From (c) n e bove bounr coniion (b), 0 b ± c or n vlue o ereore c 0 ence, 0 0 Obvioul, Birmonic equion i lo iie. Alie Elici or Engineer T.G.Sirm & L.GovinRju

12 Moule/Leon i.e., 0 Now, bening momen M b ò i.e. M b ò b ò b ú û M b Or M b M were I M Tereore, I b I Alie Elici or Engineer T.G.Sirm & L.GovinRju

13 Moule/Leon.. BENDING OF A SIMPLY SUPPORTED BEAM BY A DISTRIBUTED LOADING (UDL) Figure.0 Bem ubjece o Uniorm lo Conier bem o recngulr croecion ving uni wi, uore e en n ubjece o uniorml iribue lo o ineni q own in e Figure.0. I i o be noe e bening momen i mimum oiion 0 n ecree wi cnge in in eier oiive or negive irecion. Ti i oible onl i e re uncion conin even uncion o. Alo, i oul be noe vriou rom ero c o mimum vlue o q c. Hence e re uncion mu conin o uncion o. Now, conier olnomil o econ egree wi b c 0 \ olnomil o ir egree wi c 0 b \ 6 n olnomil o i egree wi b c e 0 \ Q 6 0 ú û \ b or () Now, b einiion, Alie Elici or Engineer T.G.Sirm & L.GovinRju

14 Moule/Leon () b () b () Te ollowing bounr coniion mu be iie. (i) ( ) 0 ± c (ii) ( ) 0 c (iii) ( ) q c ò (iv) ( ) c c (v) ò ( ) c c ò (vi) ( ) c c 0 ± L ± L ± ql 0 ± L Te ir ree coniion wen ubiue in equion () n () give b c 0 bc c 0 bc c q wic give on olving q q q, b, c c Now, rom coniion (vi), we ve c ò c ú 0 û Simliing, L Alie Elici or Engineer T.G.Sirm & L.GovinRju

15 Moule/Leon Alie Elici or Engineer T.G.Sirm & L.GovinRju L q \ q L q q q q q q Now, ( ) 8 I were I Momen o ineri o e uni wi bem. ( ) \ I q L I q I q ( ) I q.. NUMERICAL EXAMPLES Emle. Sow or iml uore bem, leng L, e n uni wi, loe b concenre lo W e cenre, e re uncion iing e loing coniion i c b 6 e oiive irecion o being uwr, n 0 min.

16 Moule/Leon Figure. Siml uore bem Tre e concenre lo er re uibl iribue o ui i uncion, n o W ò on ec lleng o e bem. Sow e ree re W 0 W 8 ú û Soluion: Te re comonen obine rom e re uncion re b 0 b c Alie Elici or Engineer 6 T.G.Sirm & L.GovinRju

17 Moule/Leon Bounr coniion re (i) (ii) (iii) (iv) (v) Now, 0 or ± 0 or ò ò ± ò W or ± L 0 or ± L 0 or ± L Coniion (i) Ti coniion i iie ince 0 Coniion (ii) Coniion (iii) Coniion (iv) b 0 \ c b ò c ( ) W b W \ or b b b W W n c 8 Alie Elici or Engineer 7 T.G.Sirm & L.GovinRju

18 Moule/Leon 8 Alie Elici or Engineer T.G.Sirm & L.GovinRju ò W 0 Coniion (v) ò M ò W W M \ Hence re comonen re W 0 W W 8 ú û \ 8 W Emle. Given e re uncion H n. Deermine weer re uncion i miible. I o eermine e ree. Soluion: For e re uncion o be miible, i o i biormonic equion. Biormonic equion i given b 0 (i) Now, ú û H n ( ) [ ] H ( ) ú ú û \ H

19 Moule/Leon 9 Alie Elici or Engineer T.G.Sirm & L.GovinRju Alo, ( ) ú ú û 8 H ( ) ú ú û 0 8 H ( ) ú ú û H ( ) ú ú û 8 6 H Similrl, ( ) ú û H ( ) ú ú û H ( ) ( ) ú ú û H ( ) ú ú û H Subiuing e bove vlue in (i), we ge ( ) [ 8 6 ] Hence, e given re uncion i miible. Tereore, e ree re ( ) ú ú û ( ) ú ú û n ( ) ú ú û

20 Moule/Leon Emle. F Given e re uncion: ( ). Deermine e re comonen n kec eir vriion in region inclue in 0,, 0, on e ie oiive. Soluion: Te given re uncion m be wrien F F \ 6F F n 0 6F 6F lo 6F F Hence (i) 0 (ii) j 6F 6F (iii) VARIATION OF STRESSES AT CERTAIN BOUNDARY POINTS () Vriion o σ From (i), i i cler vrie linerl wi, n given ecion i vrie linerl wi. \ A 0 n ±, 0 6FL A L n 0, 6FL F 6FL A L n, L A L n, 6FL F 8FL L Te vriion o i own in e igure below Alie Elici or Engineer 0 T.G.Sirm & L.GovinRju

21 Moule/Leon Figure. Vriion o σ (b) Vriion o σ i ero or ll vlue o. (c) Vriion o We ve τ 6F 6F. From e bove ereion, i i cler e vriion o i rbolic wi. However, i ineenen o n i u conn long e leng, correoning o given vlue o. \A 0, 0 6F 6F A, 0 6F 6F F A, ( ) Te vriion o i own in igure below. Alie Elici or Engineer T.G.Sirm & L.GovinRju

22 Moule/Leon Figure. Vriion o Emle. Inveige w roblem o lne re i iie b e re uncion F j ú û lie o e region inclue in 0,, 0 on e ie oiive. Soluion: Te given re uncion m be wrien F F j \ 0 j. F. F F F n Hence e re comonen re j F. 0 τ Alie Elici or Engineer T.G.Sirm & L.GovinRju

23 Moule/Leon F F () Vriion o σ F. Wen 0 n 0 or ±, (i.e., conn cro e ecion) Wen L n 0, FL Wen L n,. FL Wen L n, P. Tu, L, e vriion o i liner wi. Te vriion o i own in e igure below. Figure. Vriion o re σ (b) Vriion o σ 0 i ero or ll vlue o n \ Alie Elici or Engineer T.G.Sirm & L.GovinRju

24 (c) Vriion o τ Moule/Leon F F Tu, vrie rbolicll wi. However, i i ineenen o, i.e., i' vlue i e me or ll vlue o. F \A 0, F F A ±, ( ) 0 ú ú û û F o X F L Y Figure. Vriion o er re Te re uncion ereore olve e roblem o cnilever bem ubjece o oin lo F i ree en long wi n il re o. Emle. Sow e ollowing re uncion iie e bounr coniion in bem o recngulr croecion o wi n e uner ol er orce W. W ( ) ú û Soluion: τ Alie Elici or Engineer T.G.Sirm & L.GovinRju

25 Moule/Leon W Now, [ 6 ] W W \ 6 [ 6 ] [ ] 6 0 n W 6 6 W [ ] [ ] Alo, Ñ 0 ú û Bounr coniion re () (b) (c) () (e) 0 or 0 0 or 0 n n ò.. W or 0 0 n L M ò.. 0 or 0 n L, M WL 0 ò... 0 or 0 n L 0 Now, Coniion () Ti coniion i iie ince 0 Coniion (b) W [ ] 0 Alie Elici or Engineer T.G.Sirm & L.GovinRju

26 Moule/Leon Hence iie. Coniion (c) [ ] W ò 0 W ò [ ] 0 W ú û W W W. Hence iie. Coniion () W ò 0 W 0 ú û [ 6] [ ] Hence iie. Coniion (e) W ò 0 W [ 6]. 0 0 W ú û ú û 0 Alie Elici or Engineer 6 T.G.Sirm & L.GovinRju

27 Moule/Leon W W Hence iie ú û Alie Elici or Engineer 7 T.G.Sirm & L.GovinRju