Tension, Compression, and Shear
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1 01Ch01.qxd 2/10/09 7:32 PM Page 1 1 ension, Compression, and Shear Normal Stress and Strain Problem A hollow circular post ABC (see figure) supports a load P kn acting at the top. A second load P 2 is uniformly distributed around the cap plate at B. he diameters and thicknesses of the upper and lower parts of the post are d AB 32 mm, t AB 12 mm, d BC 57 mm, and t BC 9 mm, respectively. (a) Calculate the normal stress s AB in the upper part of the post. (b) If it is desired that the lower part of the post have the same compressive stress as the upper part, what should be the magnitude of the load P 2? (c) If P 1 remains at 7.5 kn and P 2 is now set at 10 kn, what new thickness of BC will result in the same compressive stress in both parts? B A P 1 P 2 t AB d AB d BC t BC C Solution PAR (a) P kn d AB 32 mm t AB 12 mm d BC 57 mm t BC 9mm A AB p[ d AB 2 (d AB 2t AB ) 2 ] 4 PAR (b) A BC p[ d BC 2 1d BC 2t BC 2 2 ] 4 A BC m 2 P 2 s AB A BC P 1 P 2 6 kn A AB m 2 s AB 9.95 MPa s AB P 1 A AB CHECK: P 1 + P 2 A BC * 10 6 Pa 1
2 01Ch01.qxd 2/11/09 8:06 PM Page 2 2 CHAPER 1 ension, Compression, and Shear Part (c) P 2 10 kn (d BC 2t BC ) 2 P 1 + P 2 s AB A BC d 2 BC 4 p a P 1 + P 2 b s AB 2 4 d BC d A BC p a P 1 + P 2 b s AB t BC 2 t BC mm d BC 2t BC d 2 A BC 4 p a P 1 + P 2 b s AB Problem A force P of 70 N is applied by a rider to the front hand brake of a bicycle (P is the resultant of an evenly distributed pressure). As the hand brake pivots at A, a tension develops in the 460-mm long brake cable (A e mm 2 ) which elongates by d mm. Find normal stress s and strain in the brake cable. Brake cable, L = 460 mm 37.5 mm A Hand brake pivot A P (Resultant of distributed pressure) 50 mm 100 mm Uniform hand brake pressure Solution P 70 N A e mm 2 L 460 mm d mm Statics: sum moments about A to get 2P s A e d L s MPa 4.65 * 10 4 E s 1.4 * 105 MPa NOE: (E for cables is approx. 140 GPa)
3 01Ch01.qxd 2/10/09 7:32 PM Page 3 SECION 1.2 Normal Stress and Strain 3 Problem A bicycle rider would like to compare the effectiveness of cantilever hand brakes [see figure part (a)] versus V brakes [figure part (b)]. (a) Calculate the braking force R B at the wheel rims for each of the bicycle brake systems shown. Assume that all forces act in the plane of the figure and that cable tension 200 N. Also, what is the average compressive normal stress s c on the brake pad (A 4cm 2 )? (b) For each braking system, what is the stress in the brake cable (assume effective cross-sectional area of mm 2 )? (HIN: Because of symmetry, you only need to use the right half of each figure in your analysis.) D D 100 mm C E 45 DE DE mm 100 mm DC = DE DCh DC DCv C 106 mm 50 mm E B G F R B B 25 mm H A A 25 mm F R B Pivot points anchored to frame A 25 mm HA Pivot points anchored to frame V A V A (a) Cantilever brakes (b) V brakes Solution N A pad 4cm 2 A cable mm 2 (a) CANILEVER BRAKES-BRAKING FORCE R B & PAD PRESSURE a M A 0 R B (25) DCh (75) DCv (25) DCh DCv R B 4 DCh /2 R B 400 N < 2 here vs 4.25 for V brakes 2
4 01Ch01.qxd 2/10/09 7:32 PM Page 4 4 CHAPER 1 ension, Compression, and Shear s pad R B A pad s pad 1.0 MPa V BRAKES - BRAKING FORCE R B & PAD PRESSURE a M A 0 R B R B 848 N much more effective s pad R B A pad (b) SRESS IN BRAKE CABLES s cable A cable s pad 2.12 MPa s cable MPa same for both brake systems Problem A circular aluminum tube of length L 400 mm is loaded in compression by forces P (see figure). he outside and inside diameters are 60 mm and 50 mm, respectively. A strain gage is placed on the outside of the bar to measure normal strains in the longitudinal direction. (a) If the measured strain in e , what is the shortening d of the bar? (b) If the compressive stress in the bar is intended to be 40 MPa, what should be the load P? Solution Aluminum tube in compression L 400 mm d 2 60 mm d 1 50 mm (a) SHORENING d OF HE BAR d L ( )(400 mm) mm (b) COMPRESSIVE LOAD P s 40 MPa A p 4 [d 2 2 d 1 2 ] p 4 [160 mm mm2 2 ] P sa (40 MPa)(863.9 mm 2 ) 34.6 kn
5 01Ch01.qxd 2/10/09 7:32 PM Page 5 SECION 1.2 Normal Stress and Strain 5 Problem he cross section of a concrete corner column that is loaded uniformly in compression is shown in the figure. (a) Determine the average compressive stress s c in the concrete if the load is equal to 14.5 MN. (b) Determine the coordinates x c and y c of the point where the resultant load must act in order to produce uniform normal stress in the column. 500 mm 400 mm y 600 mm 500 mm 200 mm x 200 mm Solution P 14.5 MN A (600 mm mm)(500 mm mm mm) c mm22 d (500 mm) 2 A 0.94 m 2 (a) s c P A s c MPa c(600)( ) ( )(200)a b 2 + (500)( )( ) a bd (b) x c 9.4 * 10 5 x c mm x c 480 mm y c c(600)( )a200 + b + (500)( ) a b + ( )(200)(100) (2002 )a bd y c 480 mm 9.4 * 10 5 y c mm ^x c & y c are the same as expected due to symmetry about a diagonal
6 01Ch01.qxd 2/10/09 7:32 PM Page 6 6 CHAPER 1 ension, Compression, and Shear Problem A car weighing 130 kn when fully loaded is pulled slowly up a steep inclined track by a steel cable (see figure). he cable has an effective cross-sectional area of 490 mm 2, and the angle a of the incline is 30. Calculate the tensile stress s t in the cable. Solution Car on inclined track FREE-BODY DIAGRAM OF CAR W Weight of car ensile force in cable a Angle of incline A Effective area of cable R 1, R 2 Wheel reactions (no friction force between wheels and rails) ENSILE SRESS IN HE CABLE s t A Wsin a A SUBSIUE NUMERICAL VALUES: W 130 kn a 30 A 490 mm 2 s t (130 kn)(sin 30 ) 490 mm MPa EQUILIBRIUM IN HE INCLINED DIRECION F 0 Q + b W sin a 0 W sin a Problem wo steel wires support a moveable overhead camera weighing W 110 N (see figure) used for close-up viewing of field action at sporting events. At some instant, wire 1 is at on angle a 20 to the horizontal and wire 2 is at an angle b 48. Both wires have a diameter of 0.76 mm. Determine the tensile stresses s 1 and s 2 in the two wires. b 2 1 a W
7 01Ch01.qxd 2/10/09 7:32 PM Page 7 SECION 1.2 Normal Stress and Strain 7 Solution NUMERICAL DAA W 110 N d 0.76 mm a 20 deg b 48 deg cos(b) N cos (a) ENSILE SRESSES IN WIRES EQUILIBRIUM EQUAIONS A wire p 4 d2 a F h 0 a F v 0 1 cos(a) 2 cos(b) 1 2 cos(b) cos (a) 1 sin(a) 2 sin(b) W s 1 1 A wire s 2 2 A wire s MPa s MPa cos (b) 2 a sin(a) + sin (b)b W cos(a) ENSION IN WIRES 2 W a cos(b) sin (a) + sin (b)b cos(a) N Problem A long retaining wall is braced by wood shores set at an angle of 30 and supported by concrete thrust blocks, as shown in the first part of the figure. he shores are evenly spaced, 3 m apart. For analysis purposes, the wall and shores are idealized as shown in the second part of the figure. Note that the base of the wall and both ends of the shores are assumed to be pinned. he pressure of the soil against the wall is assumed to be triangularly distributed, and the resultant force acting on a 3-meter length of the wall is F 190 kn. If each shore has a 150 mm 150 mm square cross section, what is the compressive stress s c in the shores?
8 01Ch01.qxd 2/10/09 7:32 PM Page 8 8 CHAPER 1 ension, Compression, and Shear Solution Retaining wall braced by wood shores F 190 kn A area of one shore A (150 mm)(150 mm) 22,500 mm m 2 FREE-BODY DIAGRAM OF WALL AND SHORE C compressive force in wood shore C H horizontal component of C C V vertical component of C C H C cos 30 C V C sin 30 SUMMAION OF MOMENS ABOU POIN A M A 0 F(1.5 m) C V (4.0 m) C H (0.5 m) 0 or (190 kn)(1.5 m) C(sin 30 )(4.0 m) C(cos 30 )(0.5 m) 0 C kn COMPRESSIVE SRESS IN HE SHORES s c C kn A m MPa Problem A pickup truck tailgate supports a crate (W C 900 N), as shown in the figure. he tailgate weighs W 270 N and is supported by two cables (only one is shown in the figure). Each cable has an effective crosssectional area A e 11 mm 2. A horizontal force (F 450 N) is applied at the top of the crate (h 275 mm) (a) Find the tensile force and normal stress s in each cable. (b) If each cable elongates d 0.42 mm due to the weight of both the crate and the tailgate, what is the average strain in the cable? H = 300 mm ruck d c = 450 mm Cable d = 350 mm W C = 900 N F = 450 N Crate h = 275 mm ail gate W = 270 N L = 400 mm
9 01Ch01.qxd 2/10/09 7:32 PM Page 9 SECION 1.2 Normal Stress and Strain 9 Solution F h 450 N W c 900 N A e 11 mm 2 W 270 N d 0.42 mm d c 450 mm d 350 mm H 300 mm L 400 mm L c 2 L 2 + H 2 a M hinge 0 h 275 mm L c 0.5 m 2 v L W c d c W d +F h h (a) 2 v 2 + h 2 s cable A e kn s cable MPa (b) cable cable d L c v W c d c W d F h h 2L h L H v v N h * 10 3 N Problem Solve the preceding problem if the mass of the tail gate is M 27 kg and that of the crate is M C 68 kg. Assume horizontal force F = 0. Use dimensions H 305 mm, L 406 mm, d C 460 mm, and d 350 mm. he cable cross-sectional area is A e 11.0 mm 2. (a) Find the tensile force and normal stress s in each cable. (b) If each cable elongates d 0.25 mm due to the weight of both the crate and the tailgate, what is the average strain in the cable? H = 305 mm ruck d c = 460 mm Cable d = 350 mm M C = 68 kg Crate ail gate M = 27 kg L = 406 mm
10 01Ch01.qxd 2/10/09 7:32 PM Page CHAPER 1 ension, Compression, and Shear Solution M c 68 M 27 kg W c M c g W M g W c W N kg m s 2 g 9.81 m s 2 A e 11.0 mm 2 d 0.25 (a) 2 v 2 + h 2 s cable A e 819 N s cable 74.5 MPa (b) cable cable d L c d c 460 d 350 H 305 L 406 L c 2 L 2 + H 2 L c mm a M hinge 0 2 v L W c d c W d v W c d c + W d 2L v N h L H v h N Problem An L-shaped reinforced concrete slab 3.6 m 3.6 m (but with a 1.8 m 1.8 m cutout) and thickness t 230 mm is lifted by three cables attached at O, B and D, as shown in the figure. he cables are combined at point Q, which is 2.1 m above the top of the slab and directly above the center of mass at C. Each cable has an effective cross-sectional area of A e 77 mm 2. (a) Find the tensile force i (i 1, 2, 3) in each cable due to the weight W of the concrete slab (ignore weight of cables). (b) Find the average stress i in each cable. (See able H-1 in Appendix H for the weight density of reinforced concrete.) z O (0, 0, 0) Q (1.5, 1.5, 2.1) F C (1.5, 1.5, 0) 7 y 5 7 x W 1.8 m 1.8 m Coordinates of D in m m D (1.5, 3.6, 0) B (3.6, 0, 0) kn Concrete slab g = 24 m 3 c.g at (1.5 m, 1.5m, 0)
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