Solution ME 323 EXAM #2 FALL SEMESTER :00 PM 9:30 PM Nov. 2, 2010
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1 Solution ME 33 EXAM # FALL SEMESTER 1 8: PM 9:3 PM Nov., 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. If our solution cannot be followed, it will be assumed that it is in error. Prob. 1 _(33 Prob. _(3 Prob. 3 _(33 Total (1 T t l 1 ME 33 Eam, Fall 1
2 Fn / A avg V/ A FS.. Ffail / Fallow FS.. fail / allow FS.. fail / allow ε ( Δs Δs/ Δ s δ / L γ ( π / θ Eε ν ε / ε Gγ avg lat long L δ α( Δ T L δ α( ΔT d th th 1 1 ε ν ( T, ( T E + + αδ ε E ν + + αδ ε ν ( α T, γ, γ, γ E + + Δ G G G L δ FL i Fi( d ub ua AB EA δ + E( A( δ L φ Tiρ TL i Ti( φρ γl Gc φ φ d L J GJ φ φ + φ G( J( πc π( co ci J bar J tube dv dm p(, V(, Δ V P, Δ M M d d E( M ( (,, ρ ( I 3 rectangle: I ( bh /1, circle: I ( π r /, semicircle : I ( π r /8, centroid at ( r / (3 π from diameter iai i I I,i + di Ai A i i i B A AB V ( Q( VQ (,, q, Q( ' ' I b I ηda A A' pr pr pr spherical PV : s ; clindrical PV : h ; a ; t t t + + ' + cosθ ' + sin θ ' ' cosθ' sin θ' ' ' sin θ ' + cos θ ' 1 avg + R, avg R, ma R + avg R + (, tan θ p tan θ s ME 33 Eam, Fall 1
3 PROBLEM #1 (33 points 1 The beam shown below is loaded b a distributed load q 1 kn/m and a moment M5 knm. Determine and plot the shear force and bending moment diagrams. Your answer needs to be epressed as a V( and M( diagram with characteristic values (an local maima and minima as well as values of discontinuities clearl stated. M5 knm q 1 kn/m. m. m. m The beam possesses a T shaped cross section. Determine the location along the beam ais where the maimum value of fleural stress occurs and determine the maimum value of fleural stress; Determine the distribution of the fleural stresses on the beam cross-section. Document our answer in a drawing that depicts the distribution of these stresses in the beam cross-section and indicate minimum and maimum values ME 33 Eam, Fall 1
4 Reaction Forces RA 17.5kN R 37.5kN B R A R B V( [kn] M( [knm] ME 33 Eam, Fall 1
5 ma[ M ( ] 35kNm at.m 1 η 1 Centroid: 5( 1 + 1( 1 η 17.5mm ( 1 Second area moment: I ( (1 383mm 1 1 Fleural Stresses: ( 35 knm( 17.5 mm N mm mm ( 35 knm(1.5 mm N 1 383mm mm 1 N/mm N/mm 5 ME 33 Eam, Fall 1
6 PROBLEM # (3 points At a particular cross section of a shaft the internal resultants were determined to consist of a torque (T 5 lb.in, a bending moment (M lb.in and a shear force (V 3 lb. The shaft possesses a circular cross-section with radius r.5 in. Determine the states of stress at locations A,B,C,D on the cross-section. Document our answer b drawing the respective material elements at each location with the stress components clearl indicated b vectors and stress magnitudes. A V 3 lb B D T 5 lb.in C.5 in M lb.in The stresses at points A, B, C, and D will be due to torsion 6 ME 33 Eam, Fall 1
7 bending moment shear force A Torsion D T T 5 lb.in π 3 J c in Tc ( 5 lb.in(.5 in 1.73 ksi (CCW J in ( B C T ρ D, 1.73 ksi (towards +Z 1.73 ksi (towards -Y 1.73 ksi (towards -Z 1.73 ksi (towards +Y compressive A Bending moment M lb.in π 3 I c in M ( 5 lb.in(.5 in I @B, (.37 ksi (compressive M ( 5 lb.in( in ksi I in M I ( ( 5 lb.in(.5 in ( in +.37 ksi (tensile M ( 5 lb.in( in ksi I in ( B C D M I tensile Shear force V 3 lb A VQ I t D 7 B ME 33 Eam, Fall 1 C
8 π 3 I c in & C, Q in ±.5 in ksi r & D, Q in in i Ai r 3π 3 3 VQ ( 3 lb( in 3 I t in.5 in ' ' 3 3 ( (.51 ksi (downwards The combined state of stress at points A, B, C, and D will be the resultant linear superposition of the individual stress values due to torsion, bending, and shear force. Point element A Point element B 1.73 ksi.37 ksi (C ( ksi 13. ksi Point element C Point element D 1.73 ksi ( ksi 1. ksi.37 ksi (T PROBLEM #3 (33 points 8 ME 33 Eam, Fall 1
9 At a point in a structure the stress state is determined to be plane stress and characteried b MPa, MPa, 5MPa. Determine: The values of the first and second principal stresses; and draw a properl oriented material element for this stress state; The value of the maimum in-plane shear stress; and draw a properl oriented material element for this stress state; Draw Mohr s circle for the stress state, and on Mohr s circle mark the locations representing the stress state in (- coordinates, mark the locations representing the principal stresses and the maimum shear stress. 9 ME 33 Eam, Fall 1
10 1 ME 33 Eam, Fall 1
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