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2 PROBLEM 1 (20 ponts) 1A. A 500 lbs sphere s held n place wthn ths notch shown. Determne the normal forces N 1 and N 2 from the two contact surfaces N 1 θ 60 o N 2 N 1 = lbs (2 pts) N 2 = lbs (2 pts) 1B. A person wth 200 lbs weght s standng on a ladder as shown. The ladder s held n statc equlbrum and supported by the wall and a pn support on the bottom. Suppose the wall s frcton less and the weght of the ladder s neglgble. Determne the normal forces N 1 and reacton forces Rx and Ry of the pn support. N 1 Problem dagram y x 4 ft R y 3 ft R x N 1 = lbs R x = lbs R y = lbs (2 pts) (2 pts) (2 pts) ME 270 Fnal Exam Sprng 2017 Page 2

3 ME 270 Fnal Exam Sprng 2017 Page 3

4 1C. Two blocks are stacked as shown. A short strng s attached between the upper block and the wall. Gven that µ = 0.2 for all contact surfaces, fnd the requred force F to pull the block out and the magntude of tenson T carred n the strng. F T = N = N (3 pts) (2 pts) 1D. An external load of 100 N s appled to the frame ABCD shown. Both A and D are pn supports. Determne the magntude of force actng on member BD. (5 pts) 100 N 8 m m ME 270 Fnal Exam Sprng 2017 Page 4

5 F BD N Tenson Compresson (crcle one) (5 pts) ME 270 Fnal Exam Sprng 2017 Page 5

6 PROBLEM 2. (20 ponts) GIVEN: A 5 ft x 8 ft sgn of unform densty weghts 240 lbs. The sgn s held n Statc equlbrum by a ball-an-socket support at A and cables EC and BD. FIND: a) On the sketch provded, complete the free body dagram of the sgn. (2 pts) b) Wrte vector expressons for the forces n cables EC and BD n terms of ther unknown magntudes and ther known unt vectors. (4 pts) T = (2pts) EC T = (2pts) ME 270 BD Fnal Exam Sprng 2017 Page 6

7 c) Determne the magntudes of the tensons n cables EC and BD. (10 pts) T = (5 pts) EC T = (5 pts) BD d) Determne the magntude of the reacton at the ball and socket support n the Z drecton. (4 pts) A z = (4 pts) ME 270 Fnal Exam Sprng 2017 Page 7

8 PROBLEM 3. (20 ponts) Consder the truss shown n the fgure. The truss s supported by a pn jont at A and a roller support at L and s n statc equlbrum. FIND: a) Identfy all zero force members: (2 pts) b) Determne the reactons at A and L, wrte your answer n vector form F A = F L = (2pts) (2pts) ME 270 Fnal Exam Sprng 2017 Page 8

9 c) Solve for the load n member EG and whether ts n tenson or compresson. F EG = (3pts) Tenson or Compresson: (1pt) d) Solve for the magntude of the force n member FG and determne whether t s n tenson or compresson F FG = (3pts) Tenson or Compresson: (1pt) ME 270 Fnal Exam Sprng 2017 Page 9

10 e) Member FG s made out of steel whch fals at σ fal = 250 MPa. Determne the mnmum crosssectonal area of member FG f we desgn t consderng a factor of safety of 2. Area = (4pts) f) Defne the followng: Statcally determnate truss (1 pt): Statcally ndetermnate truss (1 pt): ME 270 Fnal Exam Sprng 2017 Page 10

11 PROBLEM 4. (20 ponts) 4A Gven the shear force and bendng moment dagram draw the correspondng load on the beam (5 pts) ME 270 Fnal Exam Sprng 2017 Page 11

12 4B When a tenns player serves t creates a torsonal moment on the humerus as shown n the fgure. Assumng that the cross secton of the humerus s tubular wth the dmensons depcted n the fgure, determne the shear stresses at the nner and outer surfaces of the bone τ nner = τ outer = (3pts) (2pts) ME 270 Fnal Exam Sprng 2017 Page 12

13 4C Determne the second area moment Ix O for the T cross secton. The value of Iy 0 as compared to Ix 0 should be larger/equal/smaller (no calculatons should be necessary). I xo = (4pts) I yo s larger equal smaller (1pt) 4D Determne the second area moment I x for the gven parabolc shape wth respect to the x axs. Is the second area moment wth respect to the centrod I o greater or less than I x? I x = (2pts) I o > I x true false (3pts) ME 270 Fnal Exam Sprng 2017 Page 13

14 ME 270 Fnal Exam Sprng 2017 Page 14

15 PROBLEM 5. (20 ponts) Beam ABCD s loaded as shown and s held n statc equlbrum by a roller support at A and a roller support at C. The center pont of segments AB s labeled as P1. The beam cross-secton s Tshaped wth a second area moment of nerta Ix = 10 x 10-6 m 4. (NA refers to the neutral axs) FIND: a) Sketch a free-body dagram of the beam and determne the reactons at A and C n vector form. (Note: please use a sngle equvalent force to represent the dstrbuted loads). (6 pts) 100 N/ m A B C D P1 2 m 2 m 2 m 200 N-m NA Cross-secton 50 mm 200 mm A P1 B C D A = N C = N (2 pts) (2 pts) ME 270 Fnal Exam Sprng 2017 Page 15

16 b) On the axes provded, sketch the shear-force and bendng moment dagram of the beam. (6 pts) 100 N/ m 200 N-m A B C D P1 2 m 2 m 2 m V (N) x (m) M (N-m) x (m) ME 270 Fnal Exam Sprng 2017 Page 16

17 c) In whch segment(s) or pont(s) along the beam does pure bendng occur (2 pts)? Segments: AB BC CD None (crcle all that apply) (1 pts) Ponts: A P1 B C D (crcle all that apply) (1 pts) d) In the segment(s) or pont(s) of the beam where pure bendng exsts, determne the magntudes of maxmum tensle bendng stress (σ max ) T and maxmum compressve bendng stress (σ max ) C. (6 ponts) (σ max ) T = MPa Top Bottom (crcle all that apply) (3 pts) (σ max ) C = MPa Top Bottom (crcle all that apply) (3 pts) ME 270 Fnal Exam Sprng 2017 Page 17

18 Solutons 1A. N 1 = 471 lbs N 2 = 435 lbs 1B. N 1 = 90 lbs R x = 90 lbs R y = 200 lbs 1C. F = N T = 9. 8 N 1D. F BD = 150N (Compresson) 2A. Free body dagram 2B. T EC = T EC ( j k ) T BD = T BD ( j k ) 2C. = 280 lbs. = 90.0 lbs 2D. A Z = lbs 3A. BC, CD, HK, KJ 3B. F A = j kn F L = j kn 3C. = kn Compresson 3D. = kn Tenson 3E. Area = m 2 3F. Defntons 4A. Shear-Force and Bendng-Moment Dagrams 4B. τ nner = MPa τ outer = MPa 4C. I xo = ft 4 I yo s smaller 4D. I x = I o > I x false 5A. A = 100 j N C = 100 j N 5B. Shear-Force and Bendng-Moment Dagrams 5C. Segments: CD Ponts: P1 5D. σ max T = 1 MPa Top and Bottom σ max C = 4 MPa Bottom Only ME 270 Fnal Exam Sprng 2017 Page 18

19 Sprng 2017 Fnal Exam Equaton Sheet Normal Stress and Stran σ x = F n A σ x (y) = My I ε x = σ x E = L L ε y = ε z = ϑε x ε x (y) = y ρ FS = σ fal σ allow Shear Stress and Stran τ = V A τ(ρ) = Tρ J τ = Gγ G = E 2(1 + ϑ) γ = δ s L s = π 2 θ Second Area Moment I = y 2 da A I = 1 12 bh3 Rectangle I = π 4 r4 I B = I O + Ad OB 2 Crcle Polar Area Moment J = π 2 r4 Crcle J = π 2 (r o 4 r 4 ) Tube Shear Force and Bendng Moment x V(x) = V(0) + p(ϵ)dϵ M(x) = M(0) + V(ϵ)dϵ Buoyancy FB gv Flud Statcs p gh F p Lw eq avg Belt Frcton T T L S e 0 0 Dstrbuted Loads F eq xf L w x dx eq 0 L x w x dx 0 Centrods x x x da c da In 3D, x A c A x y x V c V Centers of Mass x x x cm x da da A cm A x y y y y A c A y cm y da c da da da y A cm A ME 270 Fnal Exam Sprng 2017 Page 19

### Please review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.

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### Please review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.

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