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1 EN0: Structural nalyss Exam I Wednesday, March 2, 2005 Dvson of Engneerng rown Unversty NME: General Instructons No collaboraton of any nd s permtted on ths examnaton. You may consult your own wrtten lecture notes and homewor solutons durng the course of ths exam, but no other materal. RINTED OR XEROXED NOTES RE NOT ERMITTED Wrte all your solutons n the space provded. No sheets should be added to the exam. Mae dagrams and setches as clear as possble, and show all your dervatons clearly. Incomplete solutons wll receve only partal credt, even f the answer s correct. If you fnd you are unable to complete part of a queston, proceed to the next part. Use symbols to represent results from proceedng part f need be. lease ntal the statement below to show that you have read t y affxng my name to ths paper, I affrm that I have executed the examnaton n accordance wth the cademc Honor Code of rown Unversty. -8 (6 ponts) ( ponts) TOTL (25 ponts) nalyze Ths!!

2 Multple Choce/Short nswer roblems. Wrte your answer to each problem n the space provded. Only the answer wll be graded. roblems and 2 refer to the roof truss shown below. >0. H F G C D E /2 /2. Under the loads shown, member F: (a) s n tenson (b) s n compresson (c) s a zero-force member (d) cannot be determned wthout further nformaton. NSWER (2 ponts) 2. ccordng to Maxwell s law, (a) The truss shown above s statcally determnate (b) The truss shown above s a mechansm (c) The truss shown above s statcally ndetermnate (d) ang, bang Maxwell s slver hammer came down upon her head. NSWER (2 ponts) 2

3 . You are runnng a -D fnte element truss analyss program to calculate the deflectons n the -D bcycle frame shown. Jonts D and E are constraned to have no dsplacement; ont s constraned to have no dsplacement n the drecton. The program bombs. Explan brefly how you would adust the problem statement and nput fle. W/2 W/2 W/2 C E D nswer: (2 ponts)

4 roblems 4 and 5 refer to the two trusses shown below. The cross sectonal areas of the members I I II II are shown n the fgures. The nternal forces n the members are denoted by F, F and F, F. C C E,,I E,,I E,,I 2E,,I θ θ C θ θ C d Truss I d Truss II 4. Whch of the followng s true? I II (a) F < F (b) F (c) F C I C I C > F = F C II C II C (d) Insuffcent nformaton NSWER (2 ponts) 5. Let I and II be the loads that nduce buclng n truss I and truss II, respectvely. Whch of the followng s true? (a) I < II (b) I > II (c) I = II (d) Insuffcent nformaton. NSWER (2 ponts) 4

5 6. Two nextensble cables are hung between fxed ponts (x,y)=(0,0) and (x e,y e )=(d,0). The cables have equal weght. Cable has length L cable has length L wth L <L. The maxmum tensons n each cable are gven by Tmax and T max. Whch of the followng s true? x/d y/d Cable Cable (a) Tmax (b) Tmax (c) Tmax < T > T max max = Tmax (d) Insuffcent nformaton NSWER (2 ponts) 7. n nextensble cable s hung between fxed ponts (x,y)=(0,0) and (x e,y e )=(,0.5) as shown. t whch pont s the cable tenson hghest? C (a) ont (b) ont (c) ont C (d) Insuffcent nformaton NSWER (2 ponts) 5

6 8. n nextensble cable s hung between fxed ponts (x,y)=(0,0) and (x e,y e )=(m,0.5m) as shown. The cable carres a load w(x)= Newton/meter dstrbuted evenly across ts span. The cable shape s shown below. The maxmum sag occurs at pont and s found to be 0.8 meters, occurrng a dstance 0.6 meters from the rght endpont : (x b,y b )=(0.6,0.) 0. The tenson n the cable at pont s (a) T =0.85 Newtons (b) T =0.2 Newtons (c) T =0.5 Newtons (d) T = Newton (e) None of the above NSWER (2 ponts) 6

7 . In ths problem, you wll analyze the out-of-plane stffness of the Shmano WH-M540 front wheel, whch has 8 pars of radal spoes. ssume that the wheel rm (radus R) and hub (heght 2h) are effectvely rgd compared wth the spoes. Each spoe has unstretched length L R h = +, area, and Young s Modulus E. The geometry s shown below. The orgn of the coordnate system s n the mdplane of the wheel and the hub length s 2h. The poston vectors of nodes and are therefore r =h and r =-h. Node has poston r =R, node r =R, etc. Shmano WH-M φ=tan - (h/r) 2h R load s appled to the hub (node ) n the -drecton. s a result, nodes and undergo () () () () () () correspondng dsplacements ( u, u, u ) = ( u, u, u ) = (0, 0, u ), u L. 2 2 z z 0 7

8 a. Fnd the potental energy of the loaded wheel (elastc energy of the spoes + the load) as a functon of the hub dsplacement u z and any of the parameters, E,, R, and h. ssume u L. (2 ponts) z h 5 φ=tan - (h/r) R 8

9 b. Fnd the hub dsplacement u Z as a functon of, E,, h, and R. (2 ponts) h 5 φ=tan - (h/r) R c. Fnd the force n the spoes as a functon of, E,, h, and R. (2 ponts)

10 d. Suppose you want to tae nto account the hub elastcty. The hub s made from the same materal (Young s modulus E) and has cross sectonal area of h =6. Does () () () () () () ( u, u2, u ) = ( u, u2, u ) = (0, 0, uz ) stll hold? Wrte the potental energy of the wheel (spoes, hub, and load) as a functon of the dsplacement components of nodes and. The answer may also nvolve the parameters, E,, h, and R. ( ponts) h 5 φ=tan - (h/r) R

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