Review for the Mid-Term Exam

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Jennifer Collins
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1 Revew f the Md-Tem am A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-6

2 Date: Mach (Thusda), 8 Tme: :pm-:pm Place: Rm, Neddema Hall Smple devat Md-Tem am Pat : 4 pblems Smple cmputat Ttal Sces: 8 pts Pat I: Cmpehes: 8 pts Pat : Cmputat: pts Select stess fuct Pat : pblems(hw) Take Hme (e pblem fm HW#6) I-Class Smla t the hmewk pblems eamples gve the tet/lectue tes ach pblem ma cta seveal sub-pblems. A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-6

3 am Ctets ad Gadg At least e pblem fm each chapte quats used f the cmputat wll be pvded. Geealed Hk s law wll t be gve. Pblems wth tedus mathematcal mapulat wll be avded. Lg devat wll t appea the eam. A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-6

4 am Ctets ad Gadg (ctued) Patal sces f bth devat ad cmputats wll be gve. Patal sces wll be gve f pvdg the pcedues f devat/cmputat f the tme s ug ut. Small bus pts ma be gve f the egeeg judgmet s pvded f u fafm-make-sese esults. CHATING WILL B RPORTD TO SCHOOL. A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-6

5 Chapte Aalss f Stess A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-64

6 Chapte - Revew Stess Tasfmat Pcpal Stesses ad Mamum Shea stess Stess a gve plae Stess Ivaats Octahedal Nmal ad Shea Stesses A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-65

7 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-66 Stess Tasfmat [ ] [ ] [ ] [ ] T T T ( ) [ ] T θ. ' ' ' ' ( ) [ ] m m m m m m m T θ [ ] m l m l m l T X

8 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-67.8 Pcpal Stesses -D State f stess p p p p p p p p p m l m l m l P P P quat (-6) gves, T satsf the abve equat, we have

9 .8 P. Stesses -D State f stess(ct d) The stess vaats I I, I I ad I I I I I sm A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-68

10 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-69. Octahedal Plaes ad Stesses(Ct d) ( ) ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( ) [ ],,, 6 ct ct The mal stesses actg thse plaes ae detcal; s d the shea stesses. Thee ae gve as,

11 Dectal Cses f a Gve Plae Fm vect aalss, we have B (,b,) A A A(a,,) A v v v v v A A A j A k v v v v A AB AC Aea v v v AB ± a m bj v v v AC ± a m ck f ΔABC C(,,c) A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-7

12 Dectal Cses f a Gve Plae(Ct d) v v A Cs Cs Cs α β γ v A ( ± a m b j ) ( ± a m ck ) A A A A A A v v v v v v [ a ( ab ) j ( ac ) k bc j k ] v v ( bc ac j ab k ) v v ( bc ) ( ac ) ( ab ) bc [( bc ) ( ac ) ( ab ) ] ac [( bc ) ( ac ) ( ab ) ] ab [( bc ) ( ac ) ( ab ) ] v v v a a l m j b v k c v A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-7

13 Chapte Sta ad Stess-Sta Relatshp A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-7

14 Chapte - Revew Sta Tasfmat Pcpal Stas ad Mamum Shea sta Rgd Bd Mt Sta Ivaats Stess/Sta elatshp (cludg themal effect) Sta eg Stess/Sta wth ad wthut ctats A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-7

15 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-74. Sta Tasfmat Sta Tasfmat fllws the tasfmat law (equat -9 f stess). (-5) (-9) [ ] ' m m m l l l m l m l m l γ γ γ γ γ γ [ ] [ ][ ][ ] T T T

16 .. -D Sta Tasfmat γ ' ' ' Cs S θ θ Sθ. Csθ S Cs θ θ Sθ. Csθ Sθ. Csθ Sθ. Csθ Cs θ S θ γ (-7) [ '] [ T ( θ )][ ] Whee [ T ( θ )] m m m m m m m A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-75

17 .4 Rgd Bd Mt A elastc bd udeges a mt wthut ducg sta s gd bd mt. Rgd Bd Taslat A elastc bd udeges a dsplacemet wthut ducg sta. u v w u v w u v w A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-76

18 Rgd Bd Mt (ct d) Rgd Bd Rtat A elastc bd udeges a tatal dsplacemet wthut ducg a shea sta - plae. u v γ Cstat. u u k C C' N Taslat; ( ) v k C v ( ) C ' Rtat l C ' C ' Cstat, Cstat. C Assumg, C C C u k, v k A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-77

19 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-78 Lea lastc Stess-Sta Relat Ttal sta s a summat f all sta cmpets the same dect. N shea sta, (-)

20 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew Stess-Sta Relatshp T G G G Δ / / / α α α γ γ γ [ ] [ ] [ ] [ ] T S Δ α { Mechacal Sta { Themal Sta Cmbg equat (-) ad (-), we ca wte stess-sta elatshp f a geeal sld ude mechacal ad themal lads. (-) [ ] [ ] [ ] [ ] { } C ΔT α

21 Chapte Tw Dmesal Pblems lastct A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-8

22 Chapte - Revew Plae Stess/Sta A Stess Fuct Stess/Sta elatshp pla cdates Stess ude a ccetated Lad A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-8

23 Plae Stess Vs. Plae Sta Plae stess (-9) Plae Sta (-) [ ] [ ] [ ( )] ( ) γ γ G γ γ G γ ( ) γ A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-8

24 Plae Stess Vs. Plae Sta Cmpag equat (-) ad (-9), we bseve: The equat f a plae stess pblem ca be cveted t a plae sta pblem b usg eplaced b ad eplaced b. Plae stess Plae sta (-) A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-8

25 Stesses Due t Ccetated Lad Hece P θ θ csθ P. α s α P π If α (-) π P csθ π. (-) α A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-84

26 Chapte 4 Ctea f Mateal Falue A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-85

27 Chapte 4- Revew Falue Ctea ad the Falue velp Ma. Pcpal Stess Ma. Shea Stess Culmb Mh Ma. Dstt eg (V-Mses) Ma. Allwable Stess ad Safet Fact A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-86

28 Mamum Pcpal Stess Cte * N effect f ad teact Falue evelp t u c u SAF t u c u A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-87

29 Mamum Shea Stess Cte Falue evelp SAF (4-) A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-88

30 Culmb Mh Falue Cte I geeal, t u c u The Culmb Mh falue cte s gve as t u Falue evelp c u SAF c u t u ( all all ), t u c u A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-89

31 Mamum Dstt eg Cte Falue evelp A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-9

32 Chapte 5 Beam Bedg A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-9

33 Chapte 5- Revew Pue Bedg f Beams wth sm./asm. Csssect ( shea) Bedg f catleve ad smpl suppted Beams wth aw css-sect lemeta Beam The Tasvese Shea ad shea flw Cmpste Beam ( Beam made f multple stpc mateals bded tgethe) A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-9

34 Pue Bedg f Beam Wth Asmmetcal Css-Sect M I I I M I I A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg M I. I I M I I. (5-) The equat f eutal as s lcated at,.e. M I M I ta Φ (5-) M I M I quat (5-) ca be smplfed b tatg a cdate sstem such that M I M M ad.. (5-) M I I I Chapte Md-Tem Revew-9

35 lemeta The f Beam d dv d d M P P d d dm d I V (5-) d v d P The dsplacemet fuct, v ca be btaed fm the dect tegat f the abve fuct. A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-94

36 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-95 Tasvese Shea F Beam Wth a Geeal Smmetcal Css Sect t (5-5) h F a ectagula css sect, h t ) / ( h h t h A ). ( * t h t h h A Q ). ( * ( ). h I V t h t I V. h I V at Ma A V ht V t h Vh 4 t h h t I ) ).( ( A t h.

37 Cmpste Beams -th b k M h h da h bda cuvatue k A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg md plae sta Chapte Md-Tem Revew-96

38 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-97 Cmpste Beams (Ct d) ( )d k M Dk B M Whee, ( ) ( ) b k. ( ) b B ( ) b D (5-8) (5-8a)

39 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-98 Cmpste Beams (Ct d) ( ) h h bd k da P bd k bd P kb A ( ) A b bd A ( ) b d b B Whee, k D B B A M P (5-9) (5-9a) (5-)

40 Chapte 6 A smmetcall Laded Membes A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-99

41 Deft A stuctue ude lad ehbtg a smmetcal stess dstbut wth espect t a as s called as asmmetcall laded membe. Assumg -as t be the smmetcal as, the we ca cclude the stesses t be depedet f θ. Ths mples dsplacemet the θ dect, shea stess the θ - plae. That s, V θ θ I ths chapte, we assume f thee s aal lad. A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-

42 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew- 6.5 Stess Cmpets f Laded Membes C C d du C C u θ (6-6) Substtutg (6-6) t (6-),. C C C C C C. C C C C C C θ, θ (6-7)

43 6-6 Stesses Pessued Clde P P B.C. P P at at ute e A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-

44 Stesses Pessued Clde (Ct d) P P P P ( P P ) (6-8) F tatg clde wth agula velct, ω F ρ. ω ρ mass dest A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-

45 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-4 Case : N Rtat ad N teal Pessue (6-9) F P & P P θ P P ( ) P P (6-8) θ (6-7)

46 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-5 Case : (Ct d) Mamum stess ccus at ma. P θ P m at ma θ θ P θ P P ( ) P P (6-8)

47 Case : N-Rtatg Clde wth d Cap L Fce actg the ed cap ( ) Pπ F. A π P (6-) A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-6

48 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew-7 Case : Rtatg Clde (Ct d) The stesses have the fm 4 θ P P ( ) P P 8 ρω 4 8 ρω (6-)

49 6.7 Th-Walled Vessels (N teal Pessue) t t (wall thckess) t << P P P ( P ) P ( )( ).. t P t A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-8

50 Th-Walled Vessels (Ct d) θ P t (6-) θ θ θ π P θ s θ. dθ P t θ. t. A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-9

51 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew- Pess ad Shk Fts (Ct d) ( ) ( ) R R R R R δ δ δ δ π δ π π θ (6-6) Ttal msmatch δ R R θ θ δ δ δ R R PR R R PR (6-7) δ R δ R

52 Pess ad Shk Fts (Ct d) If ad (Tw cldes havg the same mateal) δ PR R R R R PR ( R ) ( )( R R ) (6-8) If δ s gve, the the ctact pessue, P ca be gve as P δ. R ( )( ) R R ( R ) (6-9) A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-

53 Depatmet f Mechacal & Aespace geeg A54/MA4/M 59/ Spg 8 Chapte Md-Tem Revew- 6.9 Cuved Beam a h M Stesses ae depedet f θ A Stess Fuct θ φ θ θ Φ Φ Φ θ θ θ Φ Φ Φ Φ

54 d d P Cuved Beam (Ct d) P C. C C l C l C C C C C C P H C C C l Fm equat (6-) Rewtg, θ C l C C C A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg a C C l C C C l l C ( ) C l θ C C Cl a C C C (6-5) (6-6) Chapte Md-Tem Revew-

55 Cuved Beam (Ct d) B.C. ) a a b C C a h a M C C a b C a l l b C C C C a b b b a C b a l. C a b b C l. C a b b a b a A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-4

56 6. Wkle s The a g b. h A smla t stegth f mateal appach Cetdal Le M R dθ dθ Δ( dθ ) Neutal As Ogal legth Defmed legth ( R ) dθ. Rdθ Δ( dθ ) c A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-5

57 Wkle s The (Ct d) λ A M R M ZR θ M AR Z ( R ) Wkle s Fmula At Neutal As, c Z ( R ) ZR Z A54/MA4/M 59/ Spg 8 Depatmet f Mechacal & Aespace geeg Chapte Md-Tem Revew-6

φ (x,y,z) in the direction of a is given by

φ (x,y,z) in the direction of a is given by UNIT-II VECTOR CALCULUS Dectoal devatve The devatve o a pot ucto (scala o vecto) a patcula decto s called ts dectoal devatve alo the decto. The dectoal devatve o a scala pot ucto a ve decto s the ate o