ME 323 FINAL EXAM FALL SEMESTER :00 PM 9:00 PM Dec. 16, 2010

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1 ME 33 FINA EXAM FA SEMESTER 1 7: PM 9: PM Dec. 16, 1 Instructions 1. Begin each problem in the space provided on the eamination sheets. If additional space is required, use the paper provided. Work on one side of each sheet onl, with onl one problem on a sheet.. Each problem is of value as indicated below. 3. To obtain maimum credit for a problem, ou must present our solution clearl. Accordingl: a. Identif coordinate sstems b. Sketch free bod diagrams c. State units eplicitl d. Clarif our approach to the problem including assumptions 4. If our solution cannot be followed, it will be assumed that it is in error. Prob. 1 _(5) Prob. _(5) Prob. 3 _(5) Prob. 4 _(5) Total (1) 1 ME 33 Final Eam, Fall 1

2 σ = Fn / A avg = V/ A FS.. = Ffail / Fallow FS.. = σ fail / σallow FS.. = fail / allow = ( Δs Δs)/ Δ s = δ / γ = ( π /) θ σ = E ν = / = Gγ avg lat long δ = α( Δ T) δ = α( ΔT) d th th 1 1 = σ ν ( σ σ ) z T, ( z) T E + + αδ = E σ ν σ + σ + αδ z = σz ν ( σ σ ) α T, γ, γ z z, γ z z E + + Δ = = = G G G δ F i Fi( ) = d ub ua AB EA δ = = + E( ) A( ) δ φ Tiρ T i Ti( ) φρ = γ = Gc = φ = φ = d B A AB J GJ φ = φ + φ G( ) J( ) πc π( co ci ) J = bar J = tube dv = p( ) n for < a d a = n=,1,,3 n ( a) for a dm = V ( ) EIv'' = M, n+ 1 d a for n ( EIv'')' = V, n Δ V = P a d= 1 n+ 1 ( EIv'')'' = p a for n> Δ M = M n 1 + E( ) M ( ) σ (, ) = =, ρ ( ) I z 3 4 rectangle: I = ( bh )/1, circle: I = ( π r )/4, 4 semicircle : I = ( π r )/8, centroid at (4 r) / (3 π ) from diameter = ( A)/( A) I = I + d A i i i i i i,i i i V ( ) Q( ) VQ (, ) =, q =, Q( ) = ' ' I zb I ηda= A A' pr pr pr spherical PV : σ s = ; clindrical PV : σ h = ; σ a = ; t t t u = (1 / )( σ + σ + σ + γ + γ + γ ) z z z z z z N T [ M( )] U truss U torsion bar U bending d i i i i elas( ) =, elas( ) =, elas( ) = EA i i GJ i i EI (1 / ) PetΔ= Uelas (1 / ) Metθ = Uelas nn i i i tt i i i m( ) M( ) mθ ( ) M( ) 1 Δ=,1 θ =,1 Δ= d,1 d EA GJ θ = EI EI i i i i ME 33 Final Eam, Fall 1

3 σ + σ σ σ σ + σ σ σ σ ' = + cosθ ' + sin θ ' σ ' = cosθ' sin θ' σ σ ' ' = sin θ ' + cos θ ' σ 1 = σ avg + R, σ = σ avg R, ma = R σ σ σ + σ σ σ σ = avg R = +, tan θ p = tan θ s = σ σ + γ + γ ' = + cosθ ' + sin θ ' ' = cosθ' sin θ' ma γ ' ' γ γ in = sin θ ' + cosθ ' 1 = avg + R, = avg R plane, = R avg + = γ R = +, tan θ = p γ tan θ s = γ abs σ ield Tresca: ma =, von Mises: σ ield = σ1 σ1σ + σ, Brittle Solids: σ1 = σultimate, σ = σ π EI Pcrit = ( ) eff ultimate PROBEM #1 (5 points) 3 ME 33 Final Eam, Fall 1

4 The beam (length a+b) shown below is loaded b a distributed load w and is supported b three simple supports. EI is constant. Determine the reactions at the three supports Sketch the shear force and bending moment diagrams. w =3 kip/ft a=1 ft b=8 ft 4 ME 33 Final Eam, Fall 1

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6 PROBEM # (5 points) The truss structure shown below is loaded b a force P=1.5 kip acting in the horizontal direction. The elements AB and BC possess cross section area A=1. in, and elastic modulus E=31 3 ksi. The distance AC is a=48 in. The angle between the two elements is β=6 o. Using work-energ principle and/or the principle of virtual work: Determine the horizontal displacement of point B, u B. Determine the vertical displacement of point B, v B. a a A C β β v B B P = 1.5 kip u B 6 ME 33 Final Eam, Fall 1

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8 PROBEM #3 (5 points) Consider that three wood planks of cross section a= in. b b=4 in. and length =6 ft are available to construct a column structure to carr an aial load. Given the three planks of wood and a few nails ou can construction the two columns shown below. Assume that the ends of the column act as pin supports, and that wood possesses an elastic modulus of E=1, ksi. Given the two options which one should be selected such that the largest load P can be sustained. Your answers need to be based on calculated numbers. P 4 P ME 33 Final Eam, Fall 1

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10 PROBEM #4a (1 points) A state of plane stress is given as: σ =1MPa,σ =MPa, =MPa. Check the correct answer If the material is a ductile metal with ield strength of σ Yield = 1.MPa : (3 pts) Following the Mises criterion, ielding occurs or does not occur. (3 pts) Following the Tresa criteriaon, ielding occurs or does not occur. If the material is a brittle ceramic with ultimate strength of (3 pts) Fracture of occurs or does not occur. σ Ultimate = 1.MPa : PROBEM #4b (5 points) Multiple Choice, no partial credit While hooking a fish, a fishing rod is bend to a quarter circle of radius r=. m. Consider that the fishing rod possesses a diameter of 5. mm and an elastic modulus of E=1 GPa. The maimum value of fleural stress is the rod is 5 MPa. The maimum value of fleural stress is the rod is 15 MPa. The maimum value of fleural stress is neither of the above. The fleural stresses cannot be determined based on the information provided. r Fish pulls PROBEM #4c (5 points) Multiple Choice, no partial credit 1 ME 33 Final Eam, Fall 1

11 A tube is subjected to torsion. From the options shown below, select the correct answer for the stress distribution in a tube cross section. r r r r PROBEM #4d (6 points) Multiple Choice, no partial credit A bar is clamped between two walls. The bar is initiall free of stress. Then, the sstem is heated to a temperature T. The stresses that develop in the bar depend on: (check all that appl) The coefficient of thermal epansion α of the material the bar is made of. The elastic modulus E of the material the bar is made of. The bar s length. The bar s cross section area A. The initial temperature T 1 of the bar. 11 ME 33 Final Eam, Fall 1