Axial_solutions

Axial_solutions - Problem #1 (60 points) A rigid bar BCD is...

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Unformatted text preview: Problem #1 (60 points) A rigid bar BCD is attached to a horizontal 1 in. diameter aluminum rod at B and is pinned at C. When a vertical 5 kip load is applied at D, determine: a) the normal stress in the aluminum rod AB; b) the vertical displacement of point D (assuming side BC is initially vertical and side CD is initially horizontal); c) the average shear stress in the 3/4 in. diameter aluminum pin at C. Rod materials properties: E =10 x106 psi: (3 = 4.26 X To6 psi, v = 0.17, on = 12.8 x lO'6 / 0F PAe 1411 rod 1 in. diam B b“ ZMO=O Problem 1 (90 points) The rigid bar ABCD of negligible weight is initially horizontal at a temperature of 40 0F, and the steel rods attached at A and C are stress-free. Determine the stress in each rod after a force of F = 30 kips is applied at D and the temperature changes to 90 0F. The rod attached at A has a length of 6 fl and a cross-sectional area of 0.8 inz. The rod attached at C has a length of 4 ft and a cross-sectional area of 0.5 inz. The rigid bar has dimensions of a = 5 it, b = 2 ft, and c = 1 fi. NOTE: Just set up the equations with known numbers inserted until you have a clearly solvable system (2 equations in 2 unknowns) but do not spend time solving for the numerical answer. Circle these equations and I - 5 explain what you would do with the numerical result of this system to I 1 A - o. obtain the answers requested. We’re grading on the logic of your solution, not the numencal answer, so be sure your work is neat and organized. Problem 2 (50 points) A 60 in. x 72 in. bundle is wrapped by a 1 in. wide by 1/16 in. thick reinforced plastic band under 400 lb tension. Given the material property data below, determine the: .. a) axial stress in the band, l..—— . ‘ —-| S) b) elastic elongation of the band, 72 m (0/ c) final width ofthe band (considering b T (In ‘ only the effect caused by the axial stress) (1) temperature change (in degree F) necessary to relax all tension stress so that / _L the net elongation of the band is zero. Elastic Modulus = 1 x 106 psi, Poisson Ratio = 0.4, Thermal coefficient = 9.0 x 10'5 / 0F Ah,“ gymu'IEllUAL [ands arm-'24.; {=fiafn)w=iin +4384“) @141.“ g L/oo /A a) 6:2: 2/9211:— = é-Vx/Oafsi 4/20 ép‘ A (ll-n Y%¢ fie”, AL ((4 b) 6 =3 /en A, Is 7L/7c derMm74’r/“CL 0'72 Pa g/L A : 2(72) +'Z/ao> = 204/ I}? =-0.0025(o/}1 wnl-Aw = 0.997 in a) 0-: 6-9,ka c) 8 = W9 in. d) 6) AT = 7/.I‘F Problem #2 (60 points) The support beam shown is rigidly fixed between the floor and the ceiling and is initially stress-free. Determine the maximum normal stress in the beam when a load pf P = 10 kN is applied at C and the temperature is raised by 12 0C. Indicate which section of the beam experiences this stress and whether the stress is tensile or compressive. Beam materials properties: E = 70 GPa, G = 30 GPa, I v = 0.17, on = 23.1 x 10-6 / 0C Problem #2 (70 points) The rigid bar ABC is supported by a pin at A and a 9 ft composite cable at C. The composite cable has a cross-sectional area of 0.250 in 2. The upper third of the composite cable is made of magnesium and the lower two thirds of the cable is made of bronze. QM,- ,4 = 0,25 Ihz a) If the force of P = 2000 1b is applied at B, determine the displacement of points B and C. Fab A5 J Ac, My»: p (Ma H2321“ songs a mu" J EMA :O s 0“" (YQ-Zw/A/5’ Bronze 15.0x106 41:17:: Polka 3,250 [A zooolb $75),” 'Pw‘lé) _ 5 4 § W A5 , Ac A6 = 5‘45“ ' "9 6r 5, 3, 57" ‘ ~ \\ AC . . A7; Izso/b. aw» 4 /250 M /72 I") - r: 1, /—":—— A . If : 0.04/97 "I’ 5 . AB : gAc .- In AB: l'fl AC: 00797 0.1 b) Briefly describe Saint-Venant‘s Principle M ('44. Ad W Amifama era 7‘ 0/}73 2‘ AMI/at /filo me, 2.4%. A rigid platform weighing 2000 lbs is hinged at point A and supported by a 6 it long ‘/2 inch diameter steel rod at point B and a 12 it long 1/2 inch diameter steel rod at point C. Before the platform is attached to the rods, the rod ends B and C are horizontally in line with the pin at point A. Once the platform is attached to the rods and the two loads shown in the figure are placed on the platform, find a) the normal stress in rod BD, b) the normal stress in rod CE, c) the displacement of point C. Material Properties: E = 30 x 10 6 psi, G = 11x10 6 psi, v = 0.32, a = 6.6 x 106/01T W /——~/7 41W: W=20m lb 3 [—80 = é’aadap =Z’ i" L—Cé =/Z#,o’c6‘7zifl F’IUD‘ 6-80)6252Ab €A E ab 1735 J “if/f: "7%, 2.5%, 2% =0 = mam/Pee —zooof7> -/000(S>‘500{’0) 7’ J P5,) = 3000 — /-75 F25 , I . Rn w pr / /75 9002 in) PCE (HS/,5) . 3m — 1 “LL — x z 2’ 8 EA [’75 Am” 4514— 378 000 = 367.5 pee —> Bg=/037/6 Eta/35‘”> P //85/‘> _ 5gp- £2 — ——— 2—4 35,954 A 1705) 1 1/ I I" 025 '-’ 523/ P5; A0 = 0,025 #3. AC, 3 , pcché _ /037 /6 (174,5) = — —————-——— , 7. 5 A 30x10 6 b4”: #7, (0.50,) d) Without plugging in numbers, write the new compatibility equation for this problem if rod CE experiences a temperate drop of AT while the rigid platform is attached to the rods. 2. A 4-ft concrete ost is reinforced b four steel bar - -_ S = _. 61m. _—> ac = 5.5 x 1m, "" determine: END 1 68+ (a) (25 pomts) the normal stresses induced in the steel and m the concrete by a —— temperature rise of 80°F; 6G 0 (b) (10 points) the changes in dimensions of a diameter of the steel bar if u; = 0.3 @137th ,9 Ad «y szou’?’ T r. T35 T Km M affirfi; "a: MMIM I", ', (Wm/Cog >0<¢> mp“. 5kg] 4ft 5P 55 = 56 I] I . . - ,( [Mm/e s m Cmfflsfimu 0’" m ¥WII>> L EZKMJ) T 4/an. 3” 55225: . -r P _ T P T Ban. 25:0 P‘m":Z/PS7L 85—55 -5¢+§G :2 PL 7’ W 5 Wm .144 , - I754 fl‘qlrlw/I 1L0 17“” {0.5X/O~%F> (80 F) ,VE/%Ih)229x/o°%, _—_ [SISx/O'VOFMQOOF‘) + W [(3 In )z" I}? )L](3 ‘ax )06 Il/nbz> gox/o’c' = 95,9470"? Ps+ i7s+ = 834.! I]: 5 = z 5 as") / 'r A; 2] .-: 53,4725! ( > 178/02“ I”) 1. The rigid bar BDE is supported by two links AB and CD. Link AB is made of aluminum (E = 70 GPa) and has a cross-sectional area of 500mm2; link CD is made of steel (E = 200 GPa) and has a cross-sectional area of 600mm2. For the 30-kN force shown, determine: (a) (15 points) the normal stresses in links AB and CD; (b) (5 points) the average shear stress in 24-mm-diameter pin D if the pin is in double shear. (c) (15 points) the vertical displacement at point B; 6/ MA] I EAL r. '70 GP”. r. 4. AAA = 012'" r“- ‘— ' g - “.565. - _ .7045 AB% AE=3(§CD-5AB>+éA8 A5 :3éco‘zgflfi 3 - g flab» =W®W “axiom Cb ALDELD éoox/og‘z/me/D 3:1,) 5 ‘ _éox,03u(a.’5m) : -0’5/1/x/03M— ' / AB 5mx/o“m¢/7o flown ‘> BaT/l M g . 'épe Mum "I CD WCrLAB+0h51£L AL: =Zéco -3 A5 =3/o.3Xlo'3m>— Z/—0.S/*/X/0’3m = /,928xlo m Form # '5“ Problem #3 A 6061 — T6 aluminum rod is inserted through a hole in a rigid, fixed plate % and pinned in place, as shown in the figure at a temperature of 80 ° F. The space between the rod’s rigid end cap and the plate is 0.01 in. a) Determine the axial stress in the bar when the temperature drops to 20 o F. A '- 1.51.} (Neglect the thickness of the rigid plate and assume the end cap at the .77 . 8,an bottom of the rod is also rigid.) 71 I . ’ Gwen, I t ' . ‘0 d a Axum-,1.) 7th; moi shank; mm W 0 m 7,? —20 F_ . . fix? (Muslim) 5 :0'0’ ’” 53% ’“‘ P= R "7ND: 6‘ R/vz, K/Z m b) Is the rod in tension or compression? 749,754“ c) What is the total strain in the rod at 20 ° F? FWD : 6+5“, 6 = Gama/63511 Form # Problem #2 (60 points) 50,000 lb The 2 in. diameter, 2 1/2 ft long aluminum rod supports a 50,000 lb axial load as shown. For support, the lower 1 1/2 ft section is attached to a steel sleeve having an inner diameter of 2 in. and an outer diameter of 3 in. . . _ ,4- . 5+ . 6va d”‘= 2,”,L”’=zsx;t , Has ,4!) d; -g,,,) a; :3”, a) For segment BC, determine the average normal stress in each material. 50 ki FWD" 01+ ) 0L. 6 ' ' ' 6’ 5 4.4.14 ‘ 57¢ a, $1115 I 3645—49 M 5 = [g u (536)“ 5/58c>$4 0': P/‘l u ELK/51H) : 795+ 0.5740 6' MP“ Ems/7b, 9117 ,n}-(2/n>fl(27x/0"7n’% 2; EL =a2759 x95; a,+,%r=50kioo =5 0,2757 1%; #3,; =50 by => Ps+ : 3W9 k7 p.41, '-'- /0- 3/ 3 I]; 3 6 AI 7(2m) Alummum 3 3 /I> . 55+ : E‘j‘. : /A2 = 471’ Steel 036:9:98'1234 A“ 2%,”) -/2m> J b) Calculate the vertical displacement of the free end at A. F’MD‘AA 2' .— 12/); (sax/0% 1A4 ”" 0.57 mag 1,5,4! — . lAA=§A8+5eC = Y 2/ if _,, (Li—1L4) :0.0253m 2. 7r ' 7' z. 650%) 7772,")2 mm" % 7/2”?) /ox/o°% ms Modulus of Modulus of Yield Strength (ksi) Coef. of Therm. Material ElasticityE RigidityG O'y Poisson’s Expansion a (103)1csi (103)ksi Tens. Comp. Shear Ratio (10‘)/°F Aluminum 10.0 3.7 37 37 19 0.35 13.] Steel 29.0 . 36 36 -- 0.32 6.6 ...
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Axial_solutions - Problem #1 (60 points) A rigid bar BCD is...

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