Name (Print) (Last) (First) Instructions: ME 323 - Mechanics of Materials Exam # 2 Date: Time: 8:00 10:00 PM - Location: WTHR 200 Circle your lecturer s name and your class meeting time. Koslowski Zhao Bi 8:30-9:20AM 11:30-12:20AM 1:30-2:20PM Begin each problem in the space provided on the examination sheets. Work on one side of each sheet only, with only one problem on a sheet. Please remember that for you to obtain maximum credit for a problem, you must present your solution clearly. Accordingly, coordinate systems must be clearly identified, free body diagrams must be shown, units must be stated, write down clarifying remarks, state your assumptions, etc. If your solution cannot be followed, it will be assumed that it is in error. When handing in the test, make sure that ALL SHEETS are in the correct sequential order. Remove the staple and restaple, if necessary. Prob. 1 Prob. 2 Prob. 3 Prob. 4 Total Page 1 of 12
PROBLEM #1 (24 points) Consider the following structure. The sleeve bearings at A and B support only vertical forces. The shaft AB has a solid circular cross section with diameter 2 in. (a) Find the location of maximum compressive flexural stress and determine its value. (b) Find the location of maximum shear stress and determine its value. (c) If the allowable flexural stress is allow =20 ksi, determine the smallest allowable diameter for the solid shaft AB. Page 2 of 12
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PROBLEM #2 (24 points) At a point in a structure the stress state is determined to be plane stress as shown below. It is known that xx = 60 MMMMMM (in tension), yy is a compressive stress and xxxx is in the direction shown. At this point, one of the two in-plane principal stresses is 70 MMMMMM (in tension), and the maximum in-plane shear stress is 50 MMMMMM. a) Determine yy, xxxx and the other in-plane principal stress. b) Draw Mohr s circle for the plane stress state. Locate the xx and yy faces, and mark the principal stresses and the maximum in-plane shear stress on the Mohr s circle. c) Draw a properly oriented principal stress element, indicate the magnitude and direction of the stress components. d) Draw a properly oriented maximum in-plane shear stress element, indicate the magnitude and direction of the stress components. e) Determine the absolute maximum shear stress mmmmmm,aaaaaa. Page 4 of 12
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PROBLEM #3 (26 points) The beam KC of uniform cross section is fixed to the wall at K and pinned to a vertical rod BH. The beam has uniform flexural rigidity EEEE. The length, elastic modulus, cross section area, and thermal expansion coefficient of rod BH are L, E, A, and α, respectively. I, L, and A have the numerical relationship I=A L 2. The rod BH is initially free of stress, and then is homogeneously heated with a temperature increase of ΔT. Meanwhile, a concentrated moment M 0 is applied at C to the beam. Using the integration method, determine the reactions at the supports K and B, and the deflection function vv(xx) for the beam KC. H Page 6 of 12
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_ PROBLEM #4 (26 Points): 4.1. (6 Points) A material element is subjected to the plane stress state shown below. t n 20 MPa 45 40 MPa In the following three-dimensional Mohr s circles, which point sketches the correct stress state for the inclined surface of normal vector n? (a) (b) (c) 0 20 40 0 20 40 0 20 40 (d) (e) (f ) 0 20 40 0 20 40 None of the above Page 9 of 12
_ 4.2. (8 Points) The simply supported beam is loaded with concentrated moments and concentrated forces. The loading is unknown; however, the bending moment diagram for the beam is known and provided below. M (kips ft) 20 A B C D E 12 10 6 0 1 2 4 6 x (ft) V (kips) 0 x (ft) (a) Construct the shear force diagram in the axes provided above. (b) Show the concentrated forces and concentrated moments acting on the beam, providing numerical values for the load. Page 10 of 12
_ 4.3. (8 Points) A cantilever beam is subjected to a concentrated force shown below. The beam has a triangular cross section. Two points, M and N, within the cross section aa are marked as follows. Cantilever beam Cross section view For point M, which of the following stress elements represents its stress state? For point N, which of the following Mohr s circles represents its stress state? Page 11 of 12
_ 4.4. (4 Points) The deflection function for the simply supported beam subjected to a concentrated load P is given: Pbx 2 2 2 v= ( L b x ) 0 x a 6LEI For the following statically indeterminate beam, determine the reaction force at the roller support B using the superposition method. Assuming EI of the beam is a contant. A B C Page 12 of 12