Chapter (10) lbf Ans. 3-2 Body AB: R R. Body OAC: R R. Chapter 3 - Rev. B, Page 1/100. R R 300 lbf Ans 0 R (10) 100(30) 0

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1 Chapter - M 0 8RB 6(00) 0. lbf Ans. RB F y 0 R RB 00 0 R 66.7 lbf Ans. R R. lbf Ans. C B - Bdy AB: F x 0 RAx RBx F y 0 RAy RBy M B 0 RAy (0) RAx (0) 0 R R Ax Ay Bdy OAC: 0 R (0) 00(0) 0 M O RAy Ay 00 lbf Ans. 0 R R 00 lbf Ans. F x Ox 0 R R 00 0 F y ROy Oy Ay Ax 00 lbf Ans. Chapter - Rev. B, Page /00

2 - RO RA 0.8 tan 0.9 kn Ans. 0.8 sn 0.6 kn Ans. -4 Step : Fnd R A & R E 4.5 h tan 0 M 0 A m 9R 7.794(400cs0 ) R A E R E 4.5(400sn 0 ) N Ans. F 0 R 400cs 0 0 F x y R Ax Ax N N 0 R sn 0 0 R Ay Ay 00 N Ans Step : Fnd cmpnents f R C n lnk 4 and R D M C 0 400(4.5) R 0 F F R x y D 05.4 N Ans. RCx N 0 ( R ) 400 N Cy 4 D Chapter - Rev. B, Page /00

3 Step : Fnd cmpnents f R C n lnk F 0 x F y R R Cx Cx 0 R Cy N 00 N Chapter - Rev. B, Page /00

4 -5 M C 0 500R 00(5) 00(9) 0 R 8. kn Ans. F y R 0 R 5.8 kn Ans. M 8.(00) 460 N m Ans. M (900) 740 N m Ans. M (00) 0 checks! -6 F y 0 R (6) 740 lbf Ans. M 0 0 M 0 500(8) 40(6)(7) 8080 lbf n Ans. M (8) 60 lbf n Ans. M 60 40(6) 70 lbf n Ans. M 70 (40)(6) 0 checks! Chapter - Rev. B, Page 4/00

5 -7 M B 0.R () (4) 0 R 0.9 kn Ans. 0 F y 0.9 R 4 0 R 6.9 kn Ans. M 0.9(.).09 kn m Ans. M.09.9() 4 knm Ans. M 44() 0 checks! -8 Break at the hnge at B Beam OB: Frm symmetry, R V 00 lbf Ans. B Beam BD: M D 0 00() R (0) 40(0)(5) 0 R 440 lbf Ans. F y (0) R 0 R 60 lbf Ans. Chapter - Rev. B, Page 5/00

6 M 00(4) 800 lbf n Ans. M (4) 0 checks at hnge M (6) 400 lbf n Ans. M (40)(6) 0 lbf n Ans. M 5 0 (60)(4) 0 checks! -9 q R x 9 x00 5 x00 R x V R 9 x00 5 x00 R x500 () M R x9 x00 5 x00 R x500 () At x = V = M = 0. Applyng Eqs. () and (), R95R 0 RR 4 500R 9(500 00) 5(500 00) 0 R 8. kn Ans. R kn Ans. 0 x00 : V 8. kn, M 8. x N m 00 x00 : V kn M 8.x9( x00) 0.8x700 N m 00 x500 : V kn M 8.x9( x 00) 5( x00) 5.8x8700 N m Plts f V and M are the same as n Prb. -5. Chapter - Rev. B, Page 6/00

7 q R x M x 500 x8 40 x4 40 x V R M x 500 x8 40 x4 40 x0 () M R xm 500 x8 0 x4 0 x0 () at x0 n, V M 0, Eqs. () and () gve R R 740 lbf Ans. 0 0 R (0) M 500(0 8) 0(0 4) 0 M 8080 lbf n Ans x 8: V 740 lbf, M 740x 8080 lbf n 8 x4 : V lbf M 740x ( x8) 40x4080 lbf n 4 x 0 : V ( x4) 40x800 lbf M 740x ( x8) 0( x4) 0x 800x8000 lbf n Plts f V and M are the same as n Prb q R x x. R x. 4 x V R x. R x. 4 x. () M R x x. R x. 4 x. () at x =. +, V = M = 0. Applyng Eqs. () and (), R R 4 0 RR 6 ().R() R() 0.RR 4 (4) Slvng Eqs. () and (4) smultaneusly, R = -0.9 kn, R = 6.9 kn Ans. 0 x. : V 0.9 kn, M 0.9 x kn m. x. : V kn M 0.9x( x.).9x.4 kn m. x. : V kn M 0.9x( x.) 6.9( x.) 4x.8 kn m Plts f V and M are the same as n Prb. -7. Chapter - Rev. B, Page 7/00

8 - 0 0 q R x 400 x4 R x0 40 x0 40 x0 R x V R 400 x4 R x0 40 x0 40 x0 R x0 () M Rx400 x4 R x0 0 x0 0 x0 R x0 () M 0 at x8 n 8R 400( 8 4) 0 R 00 lbf Ans. at x = 0 +, V =M = 0. Applyng Eqs. () and (), R 40(0) R 0 R R (0) 400(6) R (0) 0(0) 0 R 440 lbf Ans. R lbf Ans. 0 x 4 : V 00 lbf, M 00 x lbf n 4 x0 : V lbf, M 00x400( x4) 00x600 lbf n 0 x 0 : V ( x0) x lbf M 00x400( x4) 440( x0) 0 x0 0x 640x4800 lbf n Plts f V and M are the same as n Prb Slutn depends upn the beam selected. -4 (a) Mment at center, l a xc w l l wl l M c la a 4 At reactn, w a Mr a =.5, l = 0 n, w = 00 lbf/n M c 00(0) lbf n M r 5 lbf n Ans. (b) Optmal ccurs when Mc M r Chapter - Rev. B, Page 8/00

9 wl l wa 4 a a al l Takng the pstve rt a l l 4 0.5l 0.07 l Ans. fr l = 0 n, w = 00 lbf, a = 0.07(0) =.07 n M mn lbf n l -5 (a) 0 0 C 5 kps 0 0 CD 5 kps R kps 5 7 kps 5 7 kps 8 p tan 4.04 cw 5 R 7 kps ccw s (b) 96 C.5 kps 6 9 CD.5 kps R kps kps kps 5 p tan 7.5 ccw.5 R 6.0 kps cw s Chapter - Rev. B, Page 9/00

10 (c) 4 0 C 7 kps 4 0 CD 7 kps kps R kps kps 7 p 6 R 9. kps 90 tan 69.7 ccw ccw s (d) C 5 kps CD 7 kps R kps kps kps p 7 90 tan 7.9 cw Chapter - Rev. B, Page 0/00

11 R 0.8 kps cw s Chapter - Rev. B, Page /00

12 -6 (a) 87 C 0.5 MPa 8 7 CD 7.5 MPa MPa R MPa Mpa p R 9.60 MPa 90 tan cw cw s (b) 9 6 C.5 MPa 9 6 CD 7.5 MPa MPa R MPa MPa p tan 0.9 cw 7.5 R MPa ccw s Chapter - Rev. B, Page /00

13 (c) 4 C 4 MPa 4 CD 8 MPa MPa R MPa MPa p tan 69.4 ccw R 0.6 MPa ccw s (d) 6 5 C 0.5 MPa 6 5 CD 5.5 MPa MPa R MPa MPa 8 p tan 7.75 ccw 5.5 R 9.7 MPa cw s Chapter - Rev. B, Page /00

14 -7 (a) 6 C 9 kps 6 CD kps 4 5 kps R kps kps 4 p tan 6.6 ccw R 5 kps ccw s (b) 0 0 C 0 kps 0 0 CD 0 kps kps R kps kps 0 p tan.8 ccw 0 R.6 kps cw s Chapter - Rev. B, Page 4/00

15 (c) 0 8 C 4 kps 0 8 CD 4 kps kps R kps kps 4 p 9 R 6.64 kps 90 tan 7.6 cw cw s (d) 99 C 4 kps 9 9 CD 5 kps kps R kps kps 5 p 8 R 9.4 kps 90 tan 6.0 cw cw s Chapter - Rev. B, Page 5/00

16 -8 (a) 80 0 C 55 MPa 80 0 CD 5 MPa MPa R 0 MPa MPa MPa 87.5 MPa,.0 MPa, 4.5 MPa (b) 0 60 C 5 MPa 60 0 CD 45 MPa R MPa MPa 0 MPa MPa MPa MPa MPa Chapter - Rev. B, Page 6/00

17 (c) 40 0 C 0 MPa 40 0 CD 0 MPa R MPa MPa MPa z 0 MPa MPa, 8. MPa, 0.9 MPa (d) 50 C 5 MPa 50 CD 5 MPa MPa R MPa MPa z 0 MPa MPa, 9. MPa,.95 MPa -9 (a) Snce there are n shear stresses n the stress element, the stress element already represents prncpal stresses. x 0 kps 0 kps 4 kps y 0 ( 4) 7 kps 0 5 kps 0 ( 4) kps Chapter - Rev. B, Page 7/00

18 (b) 00 C 5 kps 0 0 CD 5 kps R kps kps 0 kps, kps R 6.40 kps, 5.70 kps, 0.70 kps (c) 8 C 5 kps 8 CD kps R 4 5 kps 550 kps, 0 kps 550 kps 0 5 kps, 0 kps, 5 kps (d) 0 0 C 0 kps 0 0 CD 0 kps R kps kps 0 kps kps kps, 6.8 kps, 6.8 kps Chapter - Rev. B, Page 8/00

19 -0 Frm Eq. (-5), ( 6 8 ) 6(8) ( 6)( ) 8( ) 9 6 ( 5) 6(8)( ) (9)(6)( 5) ( 6)(6) 8( 5) ( )(9) Rts are:.04, 5.67, 6.7 kps Ans kps kps kps Ans. - Frm Eq. (-5) (0 0 0) 0(0) 0(0) 0(0) (0)(0) (40) 0 (0) 0 0 0(0) 0(40) Rts are: 60, 0, 40 kps Ans kps kps kps Ans. Chapter - Rev. B, Page 9/00

20 - Frm Eq. (-5) ( ) 0(40) 0(40) 40(40) (40)(40) (0)( 40)( 0) 0( 40) 40( 0) 40(0) Rts are: 90, 0, 0 MPa Ans MPa Ans. - F ps 4.0 kps Ans. A FL L n Ans. 6 AE E L 60 Frm Table A-5, v = 0.9 v 0.9(0) 0 Ans. Ans 6 6 d d 0 0 (0.75) 48 0 n Ans. -4 F ps 6.79 kps Ans. A FL L n Ans. 6 AE E L 60 Frm Table A-5, v = Ans v 0.(65) 7 Ans. 6 6 d d 7 0 (0.75) 6 0 n Ans. Chapter - Rev. B, Page 0/00

21 -5 d 0.000d d d Frm Table A-5, v = 0.6, E = 9 GPa v 0.6 FL F and, s AE A E 6 9 = E (9) MPa L F A N 5.8 kn Ans. 4 S y = 70 MPa >, s elastc defrmatn assumptn s vald. -6 FL L 8() n Ans. 6 AE E FL L m 5.86 mm Ans. 9 AE E FL L 0() n Ans. 6 AE E Wth z 0, slve the frst tw equatns f Eq. (-9) smulatenusly. Place E n the left-hand sde f bth equatns, and usng Cramer s rule, E x v Ey Ex ve E y x vy x v v v v Lkewse, Chapter - Rev. B, Page /00

22 y E y v x Frm Table A-5, E = 07 GPa and ν = 0.9. Thus, 9 x vy E x 0 8 MPa Ans. v y MPa Ans Wth z 0, slve the frst tw equatns f Eq. (-9) smulatenusly. Place E n the left-hand sde f bth equatns, and usng Cramer s rule, E x v Ey Ex ve E y x vy x v v v Lkewse, E y v y v x Frm Table A-5, E = 7.7 GPa and ν = 0.. Thus, 9 Ex vy x 0 4 MPa Ans. v y MPa Ans c ac R F M Ra F l l 6 M 6 ac F F bh l Ans. bh bh l 6ac (a) (b) m bm bhm h lm l Fm ( s)( s) ( s) s F a a c c ()() s s m m Ans. Fr equal stress, the mdel lad vares by the square f the scale factr. - Chapter - Rev. B, Page /00

23 wl w l l (a) R, M l xl/ wl 8 6M 6 wl Wl 4bh W bh bh 8 4bh l Ans. (b) Wm ( m / )( bm / b)( hm / h) ( s)( s) s W l / l s m Ans. wmlm wm s s s Ans. wl w s Fr equal stress, the mdel lad w vares lnearly wth the scale factr. - (a) Can slve by teratn r derve equatns fr the general case. Fnd mum mment under wheel W. WT W at centrd f W s lxd RA WT l Under wheel, lx d M R x Wa Wa W x Wa Wa l dm W Fr mum, T l d 0ld x x dx l A T l d Substtute nt M M WT Wa W a 4l Ths means the mdpnt f d ntersects the mdpnt f the beam. Fr wheel, Nte fr wheel : l d l d x, M W W a 4l Wa 0 j j T j j j 04.4 WT 04.4, W W W W4 6. kps (00 8) Wheel : d 8 n, M (04.4) 08 kp n 4(00) Wheel : d n Chapter - Rev. B, Page /00

24 (00 54) (04.4) 6.(84) 605 kp n ns. M M A 4(00) Check f all f the wheels are n the ral. (b) x n Ans. (c) See abve sketch. (d) Inner axles -4 (a) Let a = ttal area f entre envelpe Let b = area f sde ntch Aab40()(5) mm I Ia Ib I mm Ans. (b) A 0.75(.875) n a A 0.75(.75) n b A (0.705) n Dmensns n mm. (0.70 5)(0.975) (0.6875) y n Ans (.875) 4 Ia 0.06 n.75(0.75) 4 Ib n I (0.0795) (0.705) n. (c) Use tw negatve areas. A 65 mm, A 565 mm, A 0000 mm a b c A mm ; Ans Chapter - Rev. B, Page 4/00

25 y 6.5 mm, y 50 mm, y 50 mm 0 000(50) 565(50) 65(6.5) y 57.9 mm Ans. 750 c mm Ans. I I I I I a b c a b c 50(.5) 4 88 mm 75(75) mm 00(00) n (7.9) n Ans. (d) A n a b A A A n a b A n (.875) y.88 n Ans I (4) I 5.0 n Ans. -5 (0)(40) mm 4 I A 0(40) 800 mm M s at A. At the bttm f the sectn, Mc (0) 84. MPa Ans. I Due t V, s between A and B at y = 0. V MPa Ans. A 800 Chapter - Rev. B, Page 5/00

26 -6 I ()() n A () n 4 M 0 8R 00(8)() 0 M A R 00 lbf A R 00 00(8) 400 lbf s at A. At the tp f the beam, Mc 00(0.5) 400 ps Ans. I Due t V, s at A, at y = 0. V ps Ans. A -7 I (0.75)() 0.5 n A (0.75)().5 n 4 M A 0 5RB 000(0) 0 RB. lbf R lbf M A s at B. At the tp f the beam, Mc 5000() 0000 ps Ans. I 0.5 Due t V, s between B and C at y = 0. V ps Ans. A.5 Chapter - Rev. B, Page 6/00

27 d (50) I mm d (50) A 96 mm 4 4 M B 0 6(00)(50) 00R A 0 RA 50 kn R 6(00) kn B 4 M s at A. At the tp, Due t V, s at A, at y = 0. Mc I 4 V kn/mm 509 MPa Ans. A 96-9 wl wl c 8 I M w 8 8I cl 4 (a) l 48 n; Table A-8, I 0.57 n w.8 lbf/n Ans. 48 (b) l I 60 n, n w 6.5 lbf/n Ans..560 (c) 4 l 60 n; Table A-6, I n y = 0.77 n, c =.78 n w.0 lbf/n Ans (d) l 60 n, Table A-7, I.07 n w 6.8 lbf/n Ans Chapter - Rev. B, Page 7/00

28 -40 I n, A n Mdel 500(0.5) 500(0.75 / ) M 8.75 lbf n Mc 8.75(0.5) (c) I ps 7.8 kps Ans. 4 V ps.4 kps Ans. A 0.96 Mdel (d) M 500(0.65).5 lbf n Mc.5(0.5) I ps 5.5 kps Ans. 4 V ps.4 kps Ans. A 0.96 Mdel M 500(0.475) 8.75 lbf n Mc 8.75(0.5) I.0680 (e) 7 85 ps 7.8 kps Ans. 4 V ps.4 kps Ans. A Chapter - Rev. B, Page 8/00

29 I 4 08 mm 4, A. mm 64 4 Mdel (c) 000(6) 000(9) M N mm Mc 5 000(6) I N/mm 88.4 MPa Ans. 4 V A..6 N/mm.6 MPa. Ans Mdel (d) M 000() N mm Mc I 4 000(6) N/mm 4.5 MPa. Ans 4 V N/mm.6 MPa Ans. A. Mdel (e) M 000(7.5) 5000 N mm Mc I 5000(6) N/mm 88.4 MPa. Ans 4 V N/mm.6 MPa Ans. A. Mc M d / M -4 (a) 4 I d /64 d Chapter - Rev. B, Page 9/00

30 d M (8.75) (0 000) 0.40 n Ans. V V (b) A d /4 d 4V 4(500) 0.06 n Ans. (5000) 4V 4 V (c) A d /4 4 4V 4 4(500) d 0.8 n A. (5000) ns -4 0 p p qf x p xl xl terms fr xla a p p V Fp xl xl terms fr xla a p p p M Fx xl xl terms fr xla 6a At x ( la), V M 0, terms fr x > l + a = 0 p p F F pa a 0 p p () a a pa p p 6 F( la) Fl ( a) a 0 p p () 6a a F F Frm () and () p (l a), p ( l a) () a a b a ap Frm smlar trang les b p p p p p (4) Chapter - Rev. B, Page 0/00

31 M ccurs where V = 0 x la b p p p M F( la b) ( a b) ( a b) 6a p p p Fl F( a b) ( a b) ( a b) 6a Nrmally M = Fl The fractnal ncrease n the magntude s Fa ( b) p ( a b) p p 6 a( a b) Fl (5) Fr example, cnsder F = 500 lbf, a =. n, l =.5 n () p (500).5 (.) 4 75 lbf/n. p (500) lbf/n. (4) b =.( 875)/( ) = n Substtutng nt (5) yelds = r.7% hgher than -Fl -44 Chapter - Rev. B, Page /00

32 00(0) 40 R lbf 0 00(0) 0 R lbf a n 00 M B = 800(0) = lbfn M x = 7 n = (/)900() = 5 50 lbfn 0.5().5() y.5 n 6 4 I ()( ) 0.5 n ()( ).5 n 4 I Applyng the parallel-axs therem, I z 0.5 (.5 0.5).5 (.5.5) 8.5 n (.5) At x0 n, y.5 n, x 76 ps (.5) At x0 n, y.5 n, x 594 ps (a) (.5) At x7 n, y.5 n, x 4474 ps 8.5 At x7 n, y.5 n, x 550(.5) 7456 ps 8.5 Max tensn 594 ps Ans. Max cmpressn 7456 ps Ans. (b) The mum shear stress due t V s at B, at the neutral axs. V 500 lbf Q ya.5(.5)().5 n VQ 500(.5) 875 ps Ans. V Ib 8.5() (c) There are three ptentally crtcal lcatns fr the mum shear stress, all at x = 7 n: () at the tp where the bendng stress s mum, () at the neutral axs where Chapter - Rev. B, Page /00

33 the transverse shear s mum, r () n the web just abve the flange where bendng stress and shear stress are n ther largest cmbnatn. Fr (): The mum bendng stress was prevusly fund t be 7456 ps, and the shear stress s zer. Frm Mhr s crcle, ps Fr (): The bendng stress s zer, and the transverse shear stress was fund prevusly t be 875 ps. Thus, = 875 ps. Fr (): The bendng stress at y = 0.5 n s 8000( 0.5) x 059 ps 8.5 The transverse shear stress s Q ya ()()().0 n VQ 500(.0) 800 ps Ib 8.5() Frm Mhr s crcle, ps The crtcal lcatn s at x = 7 n, at the tp surface, where = 78 ps. Ans. -45 (a) L = 0 n. Element A: My A I ( / 64)() A VQ, 0 0 A Ib (000)(0)(0.5) kps 0.9 A A (0) 50.9 kps Ans. Element B: B My, y 0 B 0 I 4r r 4r 40.5 Q ya / n 6 6 Chapter - Rev. B, Page /00

34 B VQ Ib ( / 64)() () (000)(/) kps 0 Ans kps. Element C: My C I ( / 64)() (000)(0)(0.5) kps r r r Q yda ( ) y y x dy y r y dy y y r y r r r y r / / / / r y Fr C, y = r / =0.5 n Q / n y bx r y n C VQ (000)(0.054) 4 Ib ( / 64)() (0.866) 0.7 kps 50.9 Ans (.7) 5.50 kps. (b) Neglectng transverse shear stress: Element A: Snce the transverse shear stress at pnt A s zer, there s n change kps Ans. % errr 0% Ans. Element B: Snce the nly stress at pnt B s transverse shear stress, neglectng the transverse shear stress gnres the entre stress. 0 0 ps Ans % errr *(00) 00% Ans..698 Chapter - Rev. B, Page 4/00

35 Element C: kps Ans % errr *(00) 0.% Ans (c) Repeatng the prcess wth dfferent beam lengths prduces the results n the table. Bendng stress, kps) Transverse shear stress, kps) Max shear stress, kps) Max shear stress, neglectng kps) % errr L = 0 n A B C L = 4 n A B C L = n A B C L = 0.n A B C Dscussn: The transverse shear stress s nly sgnfcant n determnng the crtcal stress element as the length f the cantlever beam becmes smaller. As ths length decreases, bendng stress reduces greatly and transverse shear stress stays the same. Ths causes the crtcal element lcatn t g frm beng at pnt A, n the surface, t pnt B, n the center. The mum shear stress s n the uter surface at pnt A fr all cases except L = 0. n, where t s at pnt B at the center. When the crtcal stress element s at pnt A, there s n errr frm neglectng transverse shear stress, snce t s zer at that lcatn. Neglectng the transverse shear stress has extreme sgnfcance at the stress element at the center at pnt B, but that lcatn s prbably nly f practcal sgnfcance fr very shrt beam lengths. Chapter - Rev. B, Page 5/00

36 -46 c R F l c M Fx 0 x a l 6M 6clFx bh bh 6Fcx h 0 xa Ans. lb -47 c Frm Prblem -46, R F V, 0 x a l V ( c/ l) F Fc h Ans. bh bh lb Frm Prblem -46, 6Fcx hx ( ). lb Sub n x = e and equate t h abve. Fc 6Fce lb lb Fc e 8 lb Ans. -48 (a) x-z plane M 0.5(0.5) (.5)sn(0 )(.5) R () O R z.75 kn Ans. F 0 R.5 (.5)sn(0 ).75 z z R z.65 kn Ans. x-y plane M 0 (.5) cs(0 )(.5) R () O R.949 kn Ans. y F 0 R (.5)cs(0 ).949 y y R kn Ans. y y z Chapter - Rev. B, Page 6/00

37 (b) (c) The transverse shear and bendng mments fr mst pnts f nterest can readly be taken straght frm the dagrams. Fr.5 < x <, the bendng mment equatns are parablc, and are btaned by ntegratng the lnear expressns fr shear. Fr cnvenence, use a crdnate shft f x = x.5. Then, fr 0 < x <.5, V x 0.5 z x M y Vzdx 0.5xC At x0, M y C M y 0.5x 0.5x Vy x x Mz x 0.649xC At x0, Mz C Mz 0.866x 0.5x0.975 By prgrammng these bendng mment equatns, we can fnd M y, M z, and ther vectr cmbnatn at any pnt alng the beam. The mum cmbned bendng mment s fund t be at x =.79 m, where M =.4 kn m. The table belw shws values at key lcatns n the shear and bendng mment dagrams. x (m) V z (kn) V y (kn) V (kn) M y (knm) M z (knm) M (knm) Chapter - Rev. B, Page 7/00

38 (d) The bendng stress s btaned frm Eq. (-7), Mzy M A yza x I I z y The mum tensle bendng stress wll be at pnt A n the crss sectn f Prb. -4 (a), where dstances frm the neutral axes fr bth bendng mments wll be mum. At A, fr M z, y A = 7.5 mm, and fr M y, z A = 0 mm. 40(75) 4(5) I z.6(0 ) mm.6(0 ) m 5(40) 5(6) I y.67(0 ) mm.67(0 ) m It s apparent the mum bendng mment, and thus the mum stress, wll be n the parablc sectn f the bendng mment dagrams. Prgrammng Eq. (-7) wth the bendng mment equatns prevusly derved, the mum tensle bendng stress s fund at x =.77 m, where M y = kn m, M z =.075 kn m, and x = 00. MPa. Ans. -49 (a) x-z plane 600 M O 0 (000)(4) (0) M 5 M Oy 84.6 lbf n Ans. 6 Fz 0 ROz (000) lbf Ans. ROz Oy x-y plane M O 0 (000)(4) (0) M 5 M Oz lbf n Ans. 4 6 Fy 0 ROy (000) lbf Ans. ROy Oz Chapter - Rev. B, Page 8/00

39 (b) ( (c) V( x) Vy( x) Vz( x) / M( x) M y( x) Mz( x) / x (m) V z (kn) V y (kn) V (kn) M y (knm) M z (knm) M (knm) (d) The mum tensle bendng stress wll be at the uter crner f the crss sectn n the pstve y, negatve z quadrant, where y =.5 n and z = n. () (.65)(.65) 4 I z.05 n () (.65)(.65) 4 I y.60 n At x = 0, usng Eq. (-7), Mz y M y z x I z I y ( 744.6)(.5) ( 84.6)( ) x 6594 ps Check at x = 4 n, ( 545.4)(.5) ( 545.4)( ) x 706 ps The crtcal lcatn s at x = 0, where x = 6594 ps. Ans. Chapter - Rev. B, Page 9/00

40 -50 The area wthn the wall medan lne, A m, s Square: A ( b t). Frm Eq. (-45) Rund: m sq m all ( ) all T A t b t t rd A ( bt) / 4 m T ( bt) t /4 Rat f Trques T sq ( b t) tall 4.7 T ( b t) t / rd Twst per unt length frm Eq. (-46) s all all TL A t L L L C GA t GA t G A A m m all m all m 4 m 4 m m Square: 4( bt) sq C ( b t ) Rund: ( bt) 4( bt) rd C C ( bt) /4 ( b t) m m Rat equals. Twsts are the same. -5 (a) The area enclsed by the sectn medan lne s A m = ( 0.065) = n and the length f the sectn medan lne s L m = 4( 0.065) =.75 n. Frm Eq. (-45), T A t (0.8789)(0.065)( 000) 8 lbf n Ans. m Frm Eq. (-46), TL ml (8)(.75) 6 l rad 4.59 Ans. 4GA t (0.8789) m (b) The radus at the medan lne s r m = (0.5)(0.065) = n. The area enclsed by the sectn medan lne s A m = ( 0.065) 4(0.565) + 4(π /4)(0.565) = n. The length f the sectn medan lne s L m = 4[ (0.565)] + π(0.565) =.48 n. Chapter - Rev. B, Page 40/00

41 Frm Eq. (-45), T Amt (0.8579)(0.065)( 000) 87 lbf n Ans. Frm Eq. (-46), TL ml (87)(.48) 6 l rad 4.7 Ans. 4GA t (0.8579) m -5 T GLc T GLc G T T T T Lc Ans. Frm Eq. (-47), G c G and are cnstant, therefre the largest shear stress ccurs when c s a mum. G c Ans. -5 (b) Slve part (b) frst snce the twst s needed fr part (a). allw MPa (a) Gc rad/m Ans (0.00) (79.) 0 (0.00)(0.00 ) GLc T.47 N m Ans. 9 GL 0.48(79.) 0 (0.00)(0.00 ) c T 7.45 N m Ans (79.) 0 (0)(0 ) GLc T 0 Ans. T TT T N m Ans. Chapter - Rev. B, Page 4/00

42 -54 (b) Slve part (b) frst snce the twst s needed fr part (a) rad/n Ans. Gc.5 0 (0.5) (a) GLc T 5.86 lbf n Ans. T GLc 6.5 lbf n Ans GLc T 4.88 lbf n Ans. T TT T lbf n Ans. -55 (b) Slve part (b) frst snce the twst s needed fr part (a). allw MPa (a) Gc rad/m Ans (0.00) (79.) 0 (0.00)(0.00 ) GLc T.47 N m Ans. 9 GL 0.48(79.) 0 (0.00)(0.00 ) c T 7.45 N m Ans (79.) 0 (0.05)(0.00 ) GLc T T TT T N m.84 N m Ans. Ans. -56 (a) Frm Eq. (-40), wth tw -mm strps, bc T.08 N m.8 / ( b/ c).8 / 0.00 / 0.00 T (.08) 6.6 N m Ans. Chapter - Rev. B, Page 4/00

43 Frm the table n p. 0, wth b/c = 0/ = 5, and has a value between 0. and 0.. Frm Eq. (-40), 0..8/(0/) Frm Eq. (-4), Tl.08(0.) bc G rad Ans. k t T N m Ans. 0.5 Frm Eq. (-40), wth a sngle 4-mm strp, T bc.9 N m Ans..8 / ( b/ c).8 / 0.00 / Interplatng frm the table n p. 0, wth b/c = 0/4 = 7.5, ( ) Frm Eq. (-4) Tl.9(0.) bc G rad Ans. k t T.9 55 N m Ans (b) Frm Eq. (-47), wth tw -mm strps, Lc T.0 N m T (.0) 6.40 N m Ans. Tl (.0)(0.) 0.5 rad Ans. Lc G k T N m Ans. t Frm Eq. (-47), wth a sngle 4-mm strp, T Lc.8 Nm Ans. Chapter - Rev. B, Page 4/00

44 Tl (.8)(0.) rad Ans. Lc G k T N m Ans. t The results fr the sprng cnstants when usng Eq. (-47) are slghtly larger than when usng Eq. (-40) and Eq. (-4) because the strps are nt nfntesmally thn (.e. b/c des nt equal nfnty). The sprng cnstants when cnsderng ne sld strp are sgnfcantly larger (almst fur tmes larger) than when cnsderng tw thn strps because tw thn strps wuld be able t slp alng the center plane. -57 (a) Obtan the trque frm the gven pwer and speed usng Eq. (-44). H (40 000) T N m n 500 Tr 6T J d 6T 65.8 d 0.0 m. mm Ans H (40000) (b) T Nm n 50 6(58) d m 48. mm Ans (a) Obtan the trque frm the gven pwer and speed usng Eq. (-4). 605H 605(50) T 6 lbf n n 500 Tr 6T J d 6T 6 6 d n Ans. (0 000) 605H 605(50) (b) T 60 lbf n n 50 6(60) d.48 n A. (0000) ns Chapter - Rev. B, Page 44/00

45 T d T 65 Nm d 6 6 Tn 65(000) Eq. (-44), H W 55.5 kw Ans T 6 T d N m d Tl d G 80 l JG T (7) l.89 m Ans. -6 6T T d lbf n d 6 6 Tl Tl (485)(4) 0.67 rad 9.57 Ans. 4 JG 4 6 dg J d J ( d d ) (a) T sld T 4 hllw r 6d r 6d sld Tsld Thllw d % T (00%) (00%) (00%) 65.6% Ans. T d (b) Wsld kd, Whllw kd d sld 6 40 Wsld Whllw d % W (00%) (00%) (00%) 8.0% Ans. W d J 4 d J d xd (a) Tsld Thllw r 6d r 6d 4 Tsld Thllw ( xd) 4 % T (00%) (00%) x (00%) Ans. 4 T d sld Chapter - Rev. B, Page 45/00

46 (b) W kd W k d xd sld sld hllw xd % Wsld Whllw W (00%) (00%) x (00%) Ans. W d Plt %T and %W versus x. The value f greatest dfference n percent reductn f weght and trque s 5% and ccurs at x. -64 Tc 6 (a) d 4 d 0.70d J d d m 6.7 mm 0(0 ) Frm Table A-7, the next preferred sze s d = 80 mm. Ans. d = 0.7d = 56 mm. The next preferred sze smaller s d = 50 mm Ans. (b) Tc 400d MPa Ans. J 4 d d Chapter - Rev. B, Page 46/00

47 -65 H (500) T N m n T 6T d = m 45 mm C 6 dc 800 Frm Table A-7, select 50 mm. Ans (a) start 7 0 Pa 7 MPa Ans (b) Desgn actvty H 6 05() T 7880 lbf n n 8 6T 6T d =.9 n C dc Frm Table A-7, select.40 n. Ans. -67 Fr a square crss sectn wth sde length b, and a crcular sectn wth dameter d, Asquare Acrcular b d b d 4 Frm Eq. (-40) wth b = c, T.8 T.8 T T (4.8) bc b / c b d d square Fr the crcular crss sectn, 6T T 5.09 crcular d d T square d.54 T crcular 5.09 d The shear stress n the square crss sectn s 5.4% greater. Ans. (b) Fr the square crss sectn, frm the table n p. 0, β = 0.4. Frm Eq. (-4), Chapter - Rev. B, Page 47/00

48 Tl Tl Tl Tl bc G b G d G 0.4 d G square 4 Fr the crcular crss sectn, Tl Tl Tl rd GJ G d d G Tl.50 4 sq dg.9 Tl rd dg The angle f twst n the square crss sectn s.9% greater. Ans. -68 (a) T 0.5T T 0 (500 75)(4) T T5 700 T 0.5T T 0 T 400 lbf Ans. T lbf Ans. (b) M 0 575(0) 460(8) R (40) (c) R R C O O lbf Ans. F 0 R O 9.5 lbf Ans. C Chapter - Rev. B, Page 48/00

49 (d) The mum bendng mment s at x = 0 n, and s M = 9.5 lbf n. Snce the shaft rtates, each stress element wll experence bth pstve and negatve bendng stress as t mves frm tensn t cmpressn. The trque transmtted thrugh the shaft frm A t B s T = (500 75)(4) = 700 lbf n. Fr a stress element n the uter surface where the bendng stress and the trsnal stress are bth mum, (e) Mc M ps = 5. kps Ans. I d (.5) Tr 6T 6(700) 44 ps = 4.4 kps Ans. J d (.5) x x 5. 5., kps Ans..9 kps Ans. xy x 5. xy kps Ans. -69 (a) T 0.5T T (00) T T (5) TT T 0 T 880 N Ans. T N Ans. (b) M 0 (0) R (50) 070(80) (c) R C R O O 794 N Ans. F y 0 R O 06 N Ans. C Chapter - Rev. B, Page 49/00

50 (d) The mum bendng mment s at x = 0 mm, and s M = 698. N m. Snce the shaft rtates, each stress element wll experence bth pstve and negatve bendng stress as t mves frm tensn t cmpressn. The trque transmtted thrugh the shaft frm A t B s T = (800 70)(0.00) = 06 N m. Fr a stress element n the uter surface where the bendng stress and the trsnal stress are bth mum, (e) Mc M Pa 6 MPa Ans. I d (0.00) Tr 6T 6(06) Pa 57.7MPa Ans. J d (0.00) x x 6 6, MPa Ans.. MPa Ans. xy x 6 xy MPa Ans. -70 (a) T 0.5T T (4) T T() TT() T 0 T 9.6 lbf Ans. T lbf Ans. (b) Chapter - Rev. B, Page 50/00

51 R Cz R Oz R Cy R Oy M Oy (6) R () 7.99 lbf Ans. z Oz.99 lbf Ans. Oz 0 50(8) R () Cy 7.7 lbf Ans. y 0 R Oy.7 lbf Ans. Cz F 0 R M F Chapter - Rev. B, Page 5/00

52 (c) (d) Cmbne the bendng mments frm bth planes at A and B t fnd the crtcal lcatn. M (98.9) ( 78.84) 05 lbf n A M B (967.84) ( 76.65) lbf n The crtcal lcatn s at B. The trque transmtted thrugh the shaft frm A t B s T = (00 50)(4) = 000 lbf n. Fr a stress element n the uter surface where the bendng stress and the trsnal stress are bth mum, (e) Mc M 50 ps =.5 kps Ans. I d () Tr 6T 6(000) 509 ps = 5.09 kps Ans. J d () x x.5.5, kps Ans..4 kps Ans. xy x.5 xy kps Ans. Chapter - Rev. B, Page 5/00

53 -7 (a) T 0.5T T (5) T T(50) TT(50) T 0 T 50 N mm Ans. T N mm Ans. (b) MOy 0 45sn 45 (00) 87.5(700) RCz(850) R 50.7 N Ans. (c) R R Cz R Oz Cy Oy Fz 07. N Ans. 0 ROz 45cs MOz 86.0 N Ans. F y 0 45sn 45 (00) RCy(850) 0 ROy 45cs N Ans. ( d ) F r m t h e b e n dng mment dagrams, t s clear that the crtcal lcatn s at A where bth planes have the mum bendng mment. Cmbnng the bendng mments frm the tw planes, M N m Chapter - Rev. B, Page 5/00

54 The trque transmtted thrugh the shaft frm A t B s T = (00 45)(0.5) =.88 N m. Fr a stress element n the uter surface where the bendng stress and the trsnal stress are bth mum, Mc M Pa 7.9 MPa Ans. I d (0.00) Tr 6T 6(.88) Pa 0. MPa Ans. J d (0.00) (e) x x , MPa Ans. 5.7 MPa Ans. xy x 7.9 xy MPa Ans. -7 (a) (b) F R B R T 0 00(cs 0º )(0) F (cs 0º )(4) Cy R Oy Cz R Oz 750 lbf Ans. M F Oz B 0 00(cs 0º )(6) 750(sn 0º )(9) R (0) 8. lbf Ans. y 0 R 00(cs 0º ) 8.750(sn 0º ) Oy 08.5 lbf Ans. M Oy 0 00(sn 0º )(6) R (0) 750(cs 0º )(9) 86.5 lbf Ans. Fz 0 ROz 00(sn 0º ) (cs 0º ) 59. lbf Ans. Cz Cy Chapter - Rev. B, Page 54/00

55 (c) (d) Cmbne the bendng mments frm bth planes at A and C t fnd the crtcal lcatn. M ( 6) ( 449) 54 lbf n M A C ( 08) ( 64) 6750 lbf n The crtcal lcatn s at C. The trque transmtted thrugh the shaft frm A t B s T 00 cs0º 0 89 lbf n. Fr a stress element n the uter surface where the bendng stress and the trsnal stress are bth mum, Mc M ps = 5. kps Ans. I d (.5) Tr 6T 6(89) 75 ps = 7.5 kps Ans. J d (.5) (e) x x 5. 5., kps Ans..47 kps Ans. xy x 5. xy kps Ans. Chapter - Rev. B, Page 55/00

56 -7 (a) (b) (c) F R B T 0 000(cs 0º )(00) F (cs 5º )(50) Cy R Oy R Cz R Oz 80 N Ans. M F Oz B 0 000(sn 0º )(400) 80(sn 5º )(750) R (050) 89 N Ans. y 0 R 000(sn 0º ) 80sn(5º ) 89 Oy 508 N Ans. M Oy 0 000(cs 0º )(400) 80(cs 5º )(750) R (050) 0 80 N Ans. F 0 R 000(cs 0º ) 80(cs 5º ) 0 80 z Oz 494 N Ans. Cy Cz (d) Frm the bendng mment dagrams, t s clear that the crtcal lcatn s at B where bth planes have the mum bendng mment. Cmbnng the bendng mments frm the tw planes, M N m The trque transmtted thrugh the shaft frm A t B s T 000cs0º 0.0 N m. Fr a stress element n the uter surface where the bendng stress and the trsnal stress are bth mum, Chapter - Rev. B, Page 56/00

57 (e) Mc M 4097 Ans I d (0.050) Tr 6T 6(0) Pa 6. MPa Ans. J d (0.050) Pa.9 MPa. x x.9.9, MPa Ans. 4.4 MPa Ans. xy x.9 xy MPa Ans. -74 (a) M 6.C.8(9.8).88(6.8) 0 Cx D z 87. lbf Ans lbf Ans. x M 6.D.(9.8).88(6.8) 0 Dx C z x.8 M D 0 Cz (808) lbf x 6. Ans.. M C 0 Dz (808) 07. lbf x 6. Ans. (b) Fr DQC, let x, y, z crrespnd t the rgnal y, x, z axes. Chapter - Rev. B, Page 57/00

58 (c) The crtcal stress element s just t the rght f Q, where the bendng mment n bth planes s mum, and where the trsnal and axal lads exst. T 808(.88) 5 lbf n M lbf n 6T 6(5) 070 ps Ans. d. M (45) b 9495 ps Ans. d. F 6.8 a 6 ps Ans. A ( / 4). (d) The crtcal stress element wll be where the bendng stress and axal stress are bth n cmpressn ps 9857 Ans ps. kps , ps 7.9 kps Ans ps 7.0 kps Ans. -75 (a) M 0 D z 6.C x.8(46.6).88(40) lbf Ans. Cx M 0 C z 6.Dx.(46.6).88(40) 0 Dx 70.9 lbf Ans..8 M D 0 Cz (406) 5.7 lbf x 6. Ans.. M C 0 Dz (406) 54. lbf x 6. Ans. Chapter - Rev. B, Page 58/00

59 (b) Fr DQC, let x, y, z crrespnd t the rgnal y, x, z axes. (c) The crtcal stress element s just t the rght f Q, where the bendng mment n bth planes s mum, and where the trsnal and axal lads exst. T 406(.88) 575 lbf n M lbf n 6T 6(575) 80 ps Ans. d M (647.) b 659 ps Ans. d F 40 a 78. ps Ans. A ( /4) (d) The crtcal stress element wll be where the bendng stress and axal stress are bth n cmpressn ps 6769 Ans ps 8.7 kps , 80 Chapter - Rev. B, Page 59/00

60 5 ps 5. kps Ans. 090 ps. kps Ans. -76 (b) M 5.6(6.8).(9.8) A 0 B z A 69.4 lbf Ans. y M.6(6.8).(9.8) B 0 A z B 76.6 lbf Ans. y 5.6 MB 0 Az (808) 5.7 lbf y Ans..6 M A 0 Bz (808) lbf y Ans. y y (c) The crtcal stress element s just t the left f A, where the bendng mment n bth planes s mum, and where the trsnal and axal lads exst. Chapter - Rev. B, Page 60/00

61 T 808(.) 050 lbf n 6(050) 7847 ps Ans M (89.8) (7) 74 lbf n M (74) b 990 ps Ans. d 0.88 F 9.8 a 5 ps Ans. A ( / 4) 0.88 (d) The crtcal stress wll ccur when the bendng stress and axal stress are bth n cmpressn ps 44 Ans ps 8.8 kps , ps.7 kps Ans ps 5.9 kps Ans. -77 F t T N c / 0.5 / Fn 600 tan N TC Ftb N m TC 00 P 667 N a 0.50 M 0 A z 450R 58.4(5) 667(75) 0 R Dy Dy 865. N MA 0 450RDz 600(5) y Fy RAy F 0 R z Az RAz R Dz 56 N R Ay 84 N 444 N Chapter - Rev. B, Page 6/00

62 AB The mum bendng mment wll ether be at B r C. If ths s nt bvus, sketch the shear and bendng mment dagrams. We wll drectly btan the cmbned mments frm each plane. B Ay Az M AB R R N m C Dy Dz M CD R R N m The stresses at B and C are almst dentcal, but the mum stresses ccur at B. M B (8.9) 6 B Pa 68.6 MPa d TB 6(00) 6 B Pa 7.7 MPa d B B B MPa Ans B B MPa Ans. Ans. -78 F t T N c / 0.5 / F T n C 600 tan N F b N m t TC 00 P 667 N a 0.50 M 0 450R 58.4(5) R 40.6 N A z Dy Dy M 0 450R 600(5) 667(75) R 7. N A y Dz Dz F 0 R R 6.8 N y Ay Ay F 0 R N z Az R Az Chapter - Rev. B, Page 6/00

63 The mum bendng mment wll ether be at B r C. If ths s nt bvus, sketch shear and bendng mment dagrams. We wll drectly btan the cmbned mments frm each plane. M AB R R N m B Ay Az C Dy Dz M CD R R N m The mum stresses ccur at B. Ans. M B (.9) 6 B Pa 50.5 MPa d TB 6(00) 6 B Pa 7.7 MPa d B B B MPa Ans B B MPa Ans. -79 T 900 Ft 80 lbf c / 0/ F 80 tan lbf T n C F b lbf n t TC 450 P 50 lbf a 6 M 0 0R 65.5(4) 50(4) R 75.9 lbf A z Dy Dy M 0 0R 80(4) R 6 lbf A y Dz Dz F 0 R R 40 lbf y Ay Ay F 0 R lbf z Az R Az Chapter - Rev. B, Page 6/00

64 The mum bendng mment wll ether be at B r C. If ths s nt bvus, sketch shear and bendng mment dagrams. We wll drectly btan the cmbned mments frm each plane. B Ay Az M AB R R lbf n C Dy Dz M CD R R lbf n The mum stresses ccur at C. Ans. MC (88) C 460 ps d.75 6TC 6(450) C 88 ps d C C C ps Ans. 460 C C ps Ans. -80 (a) Rd AB experences cnstant trsn thrughut ts length, and mum bendng mment at the wall. Bth trsnal shear stress and bendng stress wll be mum n the uter surface. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be at the wall, at ether the tp (cmpressn) r the bttm (tensn) n the y axs. We wll select the bttm element fr ths analyss. (b) Transverse shear s zer at the crtcal stress elements n the tp and bttm surfaces. / M /64 / 6T / Mc M d x 6 97 ps 6. kps I d d Tr T d xz 509 ps 5.09 kps J d d Chapter - Rev. B, Page 64/00

65 (c) x x 6. 6., xz kps Ans..46 kps Ans. x 6. xz kps Ans. -8 (a) Rd AB experences cnstant trsn thrughut ts length, and mum bendng mments at the wall n bth planes f bendng. Bth trsnal shear stress and bendng stress wll be mum n the uter surface. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be n the uter surface at the wall, wth ts crtcal lcatn determned by the plane f the cmbned bendng mments. M y = (00)(8) = 800 lbf n M z = (75)(8) = 400 lbf n M M M tt y z lbf n M y 800 = tan tan 9.7º M z 400 The cmbned bendng mment vectr s at an angle f 9.7º CCW frm the z axs. The crtcal bendng stress lcatn, and thus the crtcal stress element, wll be ±90º frm ths vectr, as shwn. There are tw equally crtcal stress elements, ne n tensn (9.7º CCW frm the z axs) and the ther n cmpressn (60.º CW frm the z axs). We ll cntnue the analyss wth the element n tensn. (b) Transverse shear s zer at the crtcal stress elements n the uter surfaces. Mttc Mtt d / M tt 6 x 6 40 ps 6.4 kps 4 I d /64 d / 6T / Tr T d 4456 ps 4.46 kps J d d Chapter - Rev. B, Page 65/00

66 (c) x x, kps Ans.. kps Ans. 6.4 x kps Ans. -8 (a) Rd AB experences cnstant trsn and cnstant axal tensn thrughut ts length, and mum bendng mments at the wall frm bth planes f bendng. Bth trsnal shear stress and bendng stress wll be mum n the uter surface. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be n the uter surface at the wall, wth ts crtcal lcatn determned by the plane f the cmbned bendng mments. M y = (00)(8) (75)(5) = 75 lbf n M z = ( 00)(8) = 600 lbf n M M M tt y z lbf n M y 75 = tan tan 6.º M z 600 The cmbned bendng mment vectr s at an angle f 6.º CW frm the negatve z axs. The crtcal bendng stress lcatn wll be ±90º frm ths vectr, as shwn. Snce there s an axal stress n tensn, the crtcal stress element wll be where the bendng s als n tensn. The crtcal stress element s therefre n the uter surface at the wall, at an angle f 6.º CW frm the y axs. (b) Transverse shear s zer at the crtcal stress element n the uter surface. Mttc Mtt d / M tt 985 x,bend 0 0 ps 0. kps 4 I d /64 d Fx Fx 75 x,axal 95.5 ps 0. kps A d /4 /4, whch s essentally neglgble x x,axal x,bend ps 0. kps Tr 6T ps 5.09 kps J d Chapter - Rev. B, Page 66/00

67 (c) x x, kps Ans..0 kps Ans. 0. x kps Ans. -8 T = ()(00) = 400 lbf n The mum shear stress due t trsn ccurs n the mddle f the lngest sde f the rectangular crss sectn. Frm the table n p. 0, wth b/c =.5/0.5 = 6, = Frm Eq. (-40), T ps 4. kps Ans. bc (a) The crss sectn at A wll experence bendng, trsn, and transverse shear. Bth trsnal shear stress and bendng stress wll be mum n the uter surface. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be at ether the tp (cmpressn) r the bttm (tensn) n the y axs. We ll select the bttm element fr ths analyss. (b) Transverse shear s zer at the crtcal stress elements n the tp and bttm surfaces. Mc M d / M 50 x 8 0 ps 8.0 kps I d 4 /64 d / 6T / Tr T d xz 5 79 ps 5. kps J d d Chapter - Rev. B, Page 67/00

68 (c) x x , xz kps Ans. 6.7 kps Ans. x 8.0 xz kps Ans. -85 (a) The crss sectn at A wll experence bendng, trsn, axal, and transverse shear. Bth trsnal shear stress and bendng stress wll be mum n the uter surface. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be n the uter surface, wth ts crtcal lcatn determned by the plane f the cmbned bendng mments. M y = (00)() = 600 lbf n M z = (50)() = 750 lbf n M M M tt y z lbf n M 750 z = tan tan 7.4º M y 600 The cmbned bendng mment vectr s at an angle f 7.4º CCW frm the y axs. The crtcal bendng stress lcatn wll be 90º CCW frm ths vectr, where the tensle bendng stress s addtve wth the tensle axal stress. The crtcal stress element s therefre n the uter surface, at an angle f 7.4º CCW frm the z axs. (b) Mttc Mtt d / M tt 450 x,bend 46 4 ps 46. kps 4 I d /64 d Fx Fx 00 x,axal 8 ps 0.8 kps A d /4 /4 x x,axal x,bend ps 46.5 kps Tr 6T ps 5. kps J d Chapter - Rev. B, Page 68/00

69 (c) x x, kps Ans kps Ans x kps Ans. -86 (a) The crss sectn at A wll experence bendng, trsn, axal, and transverse shear. Bth trsnal shear stress and bendng stress wll be mum n the uter surface. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be n the uter surface, wth ts crtcal lcatn determned by the plane f the cmbned bendng mments. M y = (00)() ( 00)() = 4700 lbf n M z = (50)() = 750 lbf n M M M tt y z lbf n M 750 z = tan tan 0.º M y 4700 The cmbned bendng mment vectr s at an angle f 0.º CCW frm the y axs. The crtcal bendng stress lcatn wll be 90º CCW frm ths vectr, where the tensle bendng stress s addtve wth the tensle axal stress. The crtcal stress element s therefre n the uter surface, at an angle f 0.º CCW frm the z axs. (b) Mttc Mtt d / M tt 5445 x,bend ps 55.5 kps 4 I d /64 d Chapter - Rev. B, Page 69/00

70 F F 00 8 ps 0.8 kps A d /4 /4 x x x,axal x x,axal x,bend ps 55.8 kps Tr 6T ps 5. kps J d (c) x x, kps Ans..9 kps Ans x 5..8 kps Ans. -87 (a) The crss sectn at A wll experence bendng, trsn, and transverse shear. Bth trsnal shear stress and bendng stress wll be mum n the uter surface, where the stress cncentratn wll als be applcable. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be at ether the tp (cmpressn) r the bttm (tensn) n the y axs. We ll select the bttm element fr ths analyss. (b) Transverse shear s zer at the crtcal stress elements n the tp and bttm surfaces. r/ d 0.5 / 0.5 D/ d.5/.5 Kt,trsn.9 Fg. A-5-8 Kt,bend.59 Fg. A-5-9 Mc M 50 x Kt,bend Kt,bend (.59) ps 44.5 kps I d Tr 6T 650 xz Kt,trsn Kt,trsn (.9) 8 ps. kps J d Chapter - Rev. B, Page 70/00

71 (c) x x , xz. 5.0 kps Ans kps Ans. x 44.5 xz. 0.7 kps Ans. -88 (a) The crss sectn at A wll experence bendng, trsn, axal, and transverse shear. Bth trsnal shear stress and bendng stress wll be mum n the uter surface, where the stress cncentratn wll als be applcable. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be n the uter surface, wth ts crtcal lcatn determned by the plane f the cmbned bendng mments. M y = (00)() = 600 lbf n M z = (50)() = 750 lbf n M M M tt y z lbf n M 750 z = tan tan 7.4º M y 600 The cmbned bendng mment vectr s at an angle f 7.4º CCW frm the y axs. The crtcal bendng stress lcatn wll be 90º CCW frm ths vectr, where the tensle bendng stress s addtve wth the tensle axal stress. The crtcal stress element s therefre n the uter surface, at an angle f 7.4º CCW frm the z axs. (b) r/ d 0.5 / 0.5 D/ d.5/.5 Ktaxal,.75 Fg. A-5-7 Kt,trsn.9 Fg. A-5-8 Kt,bend.59 Fg. A-5-9 Chapter - Rev. B, Page 7/00

72 Mc M 450 K K (.59) 7 66 ps 7.4 kps I d x,bend t,bend t,bend F 00 K ps kps A /4 x x,axal t,axal x x,axal x,bend ps 74.0 kps Tr 6T 650 Kt,trsn Kt,trsn (.9) 8 ps. kps J d (c) x x, kps Ans kps Ans x. 4.6 kps Ans. -89 (a) The crss sectn at A wll experence bendng, trsn, axal, and transverse shear. Bth trsnal shear stress and bendng stress wll be mum n the uter surface, where the stress cncentratn s als applcable. The transverse shear wll be very small cmpared t bendng and trsn, due t the reasnably hgh length t dameter rat, s t wll nt dmnate the determnatn f the crtcal lcatn. The crtcal stress element wll be n the uter surface, wth ts crtcal lcatn determned by the plane f the cmbned bendng mments. M y = (00)() ( 00)() = 4700 lbf n M z = (50)() = 750 lbf n M M M tt y z lbf n M z 750 = tan tan 0.º M y 4700 Chapter - Rev. B, Page 7/00

73 The cmbned bendng mment vectr s at an angle f 0.º CCW frm the y axs. The crtcal bendng stress lcatn wll be 90º CCW frm ths vectr, where the tensle bendng stress s addtve wth the tensle axal stress. The crtcal stress element s therefre n the uter surface, at an angle f 0.º CCW frm the z axs. (b) r/ d 0.5 / 0.5 D/ d.5/.5 Ktaxal,.75 Fg. A-5-7 Kt,trsn.9 Fg. A-5-8 Kt,bend.59 Fg. A-5-9 Mc M 5445 x,bend Kt,bend Kt,bend (.59) 8885 ps 88. kps I d Fx 00 x,axal Kt,axal ps kps A /4 x x,axal x,bend ps 88.9 kps Tr 6T 650 Kt,trsn Kt,trsn (.9) 8 ps. kps J d (c) x x,. 9.7 kps Ans kps Ans x. 49. kps Ans. -90 (a) M = F(p / 4), c = p / 4, I = bh /, b = d r n t, h = p / Chapter - Rev. B, Page 7/00

74 /4 /4 Mc F p p Fp b I bh dn p 6F b Ans. dnp r t F F 4F (b) a A d /4 d r / 6 r t / / r Ans. Tr T dr / 6T t Ans. 4 J dr / dr (c) The bendng stress causes cmpressn n the x drectn. The axal stress causes cmpressn n the y drectn. The trsnal stress shears acrss the y face n the negatve z drectn. (d) Analyze the stress element frm part (c) usng the equatns develped n parts (a) and (b). d d p n r r.5 r F x b 4584 ps = kps dnp r t 4F y a = = ps =. kps d yz 6 T 6 5 t = = 6.8 ps = 0.68 kps d Use Eq. (-5) fr the three-dmensnal stress element The rts are at 0.54, 4.584, and.476. Thus, the rdered prncpal stresses are = 0.54 kps, =.476 kps, and = kps. Ans. Frm Eq. (-6), the prncpal shear stresses are Chapter - Rev. B, Page 74/00

75 / kps Ans /.554 kps Ans /.49 kps Ans. -9 As shwn n Fg. -, the mum stresses ccur at the nsde fber where r = r. Therefre, frm Eq. (-50) r p r t, r r r r r p Ans. r r r p r r, p. Ans r r r -9 If p = 0, Eq. (-49) becmes p r r r p / r t r r pr r r r r The mum tangental stress ccurs at r = r. S pr t, r r Ans. Fr σ r, we have p r r r p / r r r r pr r r r r S σ r = 0 at r = r. Thus at r = r pr r r r, p. Ans r r r Chapter - Rev. B, Page 75/00

76 -9 The frce due t the pressure n half f the sphere s ressted by the stress that s dstrbuted arund the center plane f the sphere. All planes are the same, s /4 p d pd ( t ) av Ans. dt 4t The radal stress n the nner surface f the shell s, = p Ans. -94 σ t > σ l > σ r τ = (σ t σ r )/ at r = r r r r r r r r r r r.75 p (0 000) 597 ps Ans. r p r r p r r p r -95 σ t > σ l > σ r τ = (σ t σ r )/ at r = r r p r r p r r p r r p r r r r r r r r r r r ( p ) (5 4) mm 6 r r t r r mm Ans. -96 σ t > σ l > σ r τ = (σ t σ r )/ at r = r r p r r p r r p r r p r r r r r r r r r r r 4 (500) 49 ps Ans Frm Eq. (-49) wth p = 0, Chapter - Rev. B, Page 76/00

77 r p r t r r r r p r r r r r σ t > σ l > σ r, and snce σ t and σ r are negatve, τ = (σ r σ t )/ at r = r r r r r r r r r r r r r r.75 p (0 000) 900 ps Ans. r p r r p r r p r r p r Frm Eq. (-49) wth p = 0, r p r t r r r r p r r r r r σ t > σ l > σ r, and snce σ t and σ r are negatve, τ = (σ r σ t )/ at r = r r p r r p r r p r r p r r r r r r r r r r r 6 r r 6 ( p ) mm t r r mm Ans. -99 Frm Eq. (-49) wth p = 0, r p r t r r r r p r r r r r σ t > σ l > σ r, and snce σ t and σ r are negatve, τ = (σ r σ t )/ at r = r Chapter - Rev. B, Page 77/00

78 r r r r r r r r r r r.75 (500) r p r r p r r p r r p 69 ps Ans Frm Table A-0, S y =490 MPa Frm Eq. (-49) wth p = 0, r p r t r r r Maxmum wll ccur at r = r 0.8( 490) 5 9 r, ( ) p t r r t, p 8.8 MPa. Ans r r r (5 ) -0 Frm Table A-0, S y = 7 kps Frm Eq. (-49) wth p = 0, r p r t r r r Maxmum wll ccur at r = r r r 0.8( 7) 0.75 r p t, t, p.4 kps. Ans r r r ( ) -0 Frm Table A-0, S y =490 MPa Frm Eq. (-50) r p r t r r r Maxmum wll ccur at r = r r p r t, r r r r r t, r r p r r (5 9 ) p r r ( ) 0.8(490) (5 9 ) 05 MPa Ans. Chapter - Rev. B, Page 78/00

79 -0 Frm Table A-0, S y =7 MPa Frm Eq. (-50) r p r t r r r Maxmum wll ccur at r = r r p r p( r r ) t, r r r r r p 5.9 ks Ans. t, ( r r ) 0.8(7) ( 0.75 ) r r ( 0.75 ) -04 The lngtudnal stress wll be due t the weght f the vessel abve the mum stress pnt. Frm Table A-5, the unt weght f steel s s = 0.8 lbf/n. The area f the wall s A wall = ( /4)( ) = n The vlume f the wall and dme are V wall = A wall h = (70) = (0 ) n V dme = ( /)( ) = 5.0 (0 ) n The weght f the structure n the wall area at the tank bttm s W = s V ttal = 0.8( ) (0 ) = 4.7(0 ) lbf W 4.70 l 54 ps Awall The mum pressure wll ccur at the bttm f the tank, p = water h. Frm Eq. (-50) wth r r r p r r r t p r r r r r ft (55) ps Ans. 44 n r p r ft r 6.4(55).8 ps. p Ans r r r 44 n Nte: These stresses are very dealzed as the flr f the tank wll restrct the values calculated. Chapter - Rev. B, Page 79/00

80 Snce, = t = 5708 ps, = r = 4 ps and = l = 54 ps. Frm Eq. (-6), ps ps Ans ps -05 Stresses frm addtnal pressure are, Eq. (-5), l 596 ps 50ps ( r ) 50 ps = 50 ps Eq. (-50) t ps 50ps Addng these t the stresses fund n Prb. -04 gves t = = 768 ps = 7.7 kps Ans. r =.8 50 = 7.8 ps Ans. l = = 5709 ps Ans. Nte: These stresses are very dealzed as the flr f the tank wll restrct the values calculated. Frm Eq. (-6) ps ps Ans ps -06 Snce σ t and σ r are bth pstve and σ t > σ r t Frm Eq. (-55), t s mum at r = r = 0.5 n. The term Chapter - Rev. B, Page 80/00

81 lbf/n t (0.9) ps ps Ans. Radal stress: rr r kr r r r Maxma: d r rr k r 0 dr r r rr 0.5(.75) 0.97 n r ps Ans. -07 = (000)/60 = 09.4 rad/s, = 0 0 kg/m, = 0.4, r = 0.0 m, r = 0.5 m Usng Eq. (-55) 0.4 (0.4) 6 t 0(09.4) 0.0 (0.5) (0.5) 0.0 (0) MPa Ans. -08 = ( 000)/60 = 56.6 rad/s, 5/ lbf s / n The mum shear stress ccurs at bre where = t /. Frm Eq. (-55) (0.0) ( t ) 6.749(0 ) (0.75) ps 4 Chapter - Rev. B, Page 8/00

82 = 560 / = 680 ps Ans. -09 = (500)/60 = 66.5 rad/s, mass f blade = m = V = (0.8 / 86) [.5(0)(0.5)] =.45(0 ) lbfs /n F = (m/) r = [.45(0 )/]( 66.5 )(7.5) = 75 lbf A nm = (.5 0.5)(/8) = n nm = F/ A nm = 75/ = ps Ans. Nte: Stress cncentratn Fg. A-5- gves K t =.5 whch ncreases σ and fatgue. -0 = 0.9, E = 07 GPa, r = 0, R = 5 mm, r = 50 mm Eq. (-57), 9 07(0 ) ( )(0.05 0) 9 p 0.05(0 ) () (0.05) (0.05 0) where p s n MPa and s n mm. Maxmum nterference, [ ] 0.0 mm Ans. Mnmum nterference, mn [ ] mm Ans. Frm Eq. () p =.05(0 )(0.0) = 65. MPa Ans. p mn =.05(0 )(0.0005) =.55 MPa Ans. - = 0.9, E = 0 Mps, r = 0, R = n, r = n Eq. (-57), 6 0(0 ) ( )( 0) 7 p.5(0 ) () ( ) ( 0) where p s n ps and s n nches. Maxmum nterference, Chapter - Rev. B, Page 8/00

83 [ ] n Ans. Mnmum nterference, mn [ ] 0 Ans. Frm Eq. (), p =.5(0 7 )(0.0008) = ps Ans. p mn =.5(0 7 )(0) = 0 Ans. - = 0.9, E = 07 GPa, r = 0, R = 5 mm, r = 50 mm Eq. (-57), 9 07(0 ) ( )(0.05 0) 9 p 0.05(0 ) () (0.05) (0.05 0) where p s n MPa and s n mm. Maxmum nterference, [ ] mm Ans. Mnmum nterference, mn [ ] mm Ans. Frm Eq. () p =.05(0 )(0.095) = 9.6 MPa Ans. p mn =.05(0 )(0.009) = 7.9 MPa Ans. - = 0.9, E = 0 Mps, r = 0, R = n, r = n Eq. (-57), 6 0(0 ) ( )( 0) 7 p.5(0 ) () ( ) ( 0) where p s n ps and s n nches. Maxmum nterference, [ ] n Ans. Mnmum nterference, Chapter - Rev. B, Page 8/00

84 mn [ ] Ans. Frm Eq. (), p =.5(0 7 )(0.005) = 940 ps Ans. p mn =.5(0 7 )(0.0005) = 98 Ans. -4 = 0.9, E = 07 GPa, r = 0, R = 5 mm, r = 50 mm Eq. (-57), 9 07(0 ) ( )(0.05 0) 9 p 0.05(0 ) () (0.05) (0.05 0) where p s n MPa and s n mm. Maxmum nterference, [ ] 0.04 mm Ans. Mnmum nterference, mn [ ] 0.05 mm Ans. Frm Eq. () p =.05(0 )(0.04) = 4 MPa Ans. p mn =.05(0 )(0.05) = 69.9 MPa Ans. -5 = 0.9, E = 0 Mps, r = 0, R = n, r = n Eq. (-57), 6 0(0 ) ( )( 0) 7 p.5(0 ) () ( ) ( 0) where p s n ps and s n nches. Maxmum nterference, [ ] n Ans. Mnmum nterference, mn [ ] Ans. Frm Eq. (), Chapter - Rev. B, Page 84/00

85 p =.5(0 7 )(0.007) = 9 0 ps Ans. p mn =.5(0 7 )(0.0009) = 0 0 Ans. -6 Frm Table A-5, E = E = 0 Mps, r = 0, R = n, r =.5 n The radal nterference s n Ans. Eq. (-57), r R R r 6 E p R r r ps 8. kps Ans. The tangental stresses at the nterface fr the nner and uter members are gven by Eqs. (-58) and (-59), respectvely. R r 0 ( t) p (8) 8 ps 8. kps Ans. rr R r 0 r R.5 ( t) p (8) 670 ps.7 kps Ans. rr r R.5-7 Frm Table A-5, E = 0 Mps, E =4.5 Mps, r = 0, R = n, r =.5 n The radal nterference s n Ans. Eq. (-56), p r R R r R E r R E R r 0.00 p 4599 ps Ans The tangental stresses at the nterface fr the nner and uter members are gven by Eqs. (-58) and (-59), respectvely. R r 0 ( t) p (4599) 4599 ps Ans. rr R r 0 Chapter - Rev. B, Page 85/00

86 r R.5 ( t) p (4599) 960 ps Ans. rr r R.5-8 Frm Table A-5, E = E = 0 Mps, r = 0, R = 0.5 n, r = n The mnmum and mum radal nterferences are mn n Ans n Ans. Snce the mnmum nterference s zer, the mnmum pressure and tangental stresses are zer. Ans. The mum pressure s btaned frm Eq. (-57). r R R r E p R r r p ps Ans The mum tangental stresses at the nterface fr the nner and uter members are gven by Eqs. (-58) and (-59), respectvely. R r ( t) p ( 500) 500 ps Ans. rr R r r R 0.5 ( t) p ( 500) ps Ans. rr r R Frm Table A-5, E = 0.4 Mps, E =0 Mps, r = 0, R = n, r =.5 n The mnmum and mum radal nterferences are mn [.00.00] n Ans. [ ] 0.00 n Ans. Eq. (-56), Chapter - Rev. B, Page 86/00

87 p r R R r R E r R E R r p Ans 6 6 mn 6 6 p ps. p p mn ps. kps Ans ps 8.7 kps Ans. The tangental stresses at the nterface fr the nner and uter members are gven by Eqs. (-58) and (-59), respectvely. Mnmum nterference: R r 0 ( t) p mn mn (.). kps Ans. R r 0 r R.5 ( ) p (.) 8.09 kps Ans. t mn mn r R.5 Maxmum nterference: R r 0 ( t) p (8.7) 8.7 kps Ans. R r 0 r R.5 ( ) p (8.7) 48.6 kps Ans. t r R.5-0 d 0 mm, r 7.5 mm, r 57.5 mm Frm Table -4, fr R = 0 mm, rc mm 0 rn mm erc rn mm c rn r mm c r rn mm Ad / 4 (0) / mm M Fr c 4000(47.5) N mm Usng Eq. (-65) fr the bendng stress, and cmbnng wth the axal stress, Chapter - Rev. B, Page 87/00

88 F Mc (9.4677) 00 MPa Ans. A Aer (0.58)(7.5) F Mc (0.5) 95 MPa Ans. A Aer (0.58)(57.5) - d 0.75 n, r.5 n, r.0 n Frm Table -4, fr R = 0.75 n, rc n 0.75 rn n erc rn n c rn r n c r rn n Ad / 4 (0.75) / n M Fr c 750(.65) 8.8 lbf n Usng Eq. (-65) fr the bendng stress, and cmbnng wth the axal stress, F Mc (0.507) 7 0 ps 7. kps Ans. A Aer (0.09)(.5) F Mc (0.969) 69 ps. kps Ans. A Aer (0.09)(.0) - d 6 mm, r 0 mm, r 6 mm Frm Table -4, fr R = mm, rc 0 mm rn.8456 mm erc rn mm c rn r mm c r rn mm Ad / 4 (6) / mm M Fr c 00() 900 N mm Usng Eq. (-65) fr the bendng stress, and cmbnng wth the axal stress, Chapter - Rev. B, Page 88/00

89 F Mc (.8456) MPa Ans. A Aer (0.7544)(0) F Mc (.7544) 45 MPa Ans. A Aer (0.7544)(6) - d 6 mm, r 0 mm, r 6 mm Frm Table -4, fr R = mm, rc 0 mm rn.8456 mm erc rn mm c rn r mm c r rn mm Ad / 4 (6) / mm The angle f the lne f radus centers s Rd / 06/ sn sn 0 Rd R M F Rd / sn / sn N mm Usng Eq. (-65) fr the bendng stress, and cmbnng wth the axal stress, F sn Mc 00sn 0 950(.8456) 6 MPa Ans. A Aer (0.7544)(0) F sn Mc 00sn 0 950(.7544) 7.7 MPa Ans. A Aer (0.7544)(6) Nte that the shear stress due t the shear frce s zer at the surface. -4 d 0.5 n, r 0.5 n, r 0.75 n Frm Table -4, fr R = 0.5 n, rc n 0.5 rn n erc rn n c rn r n c r r n n Chapter - Rev. B, Page 89/00

90 Ad / 4 (0.5) / n M Fr c 75(0.65) lbf n Usng Eq. (-65) fr the bendng stress, and cmbnng wth the axal stress, F Mc (0.8686) 7 48 ps 7.4 kps Ans. A Aer (0.0064)(0.5) F Mc (0.4) 4 95 ps 5.0 kps Ans. A Aer (0.0064)(0.75) -5 d 0.5 n, r 0.5 n, r 0.75 n Frm Table -4, fr R = 0.5 n, rc n 0.5 rn n erc rn n c rn r n c r rn n Ad / 4 (0.5) / n The angle f the lne f radus centers s Rd / / sn sn 0 Rd R M F Rd / sn / sn 0.44 lbf n Usng Eq. (-65) fr the bendng stress, and cmbnng wth the axal stress, F sn Mc 75sn 0.44(0.8686) 8 76 ps 8.7 kps Ans. A Aer (0.0064)(0.5) F sn Mc 75sn 0.44(0.4) 478 ps.5 kps Ans. A Aer (0.0064)(0.75) Nte that the shear stress due t the shear frce s zer at the surface. -6 Mc (4) 0.5(0.094) (a) 80 ps 8.0 kps Ans. I (0.75) / (b) r = 0.5 n, r = r + h = = 0.44 n Frm Table -4, Chapter - Rev. B, Page 90/00