UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 20, 2011 Professor A. Dolovich

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1 UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 20, 2011 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS LAST NAME (printed): FIRST NAME (printed): STUDENT NUMBER: EXAMINATION ROOM: SIGNATURE: INSTRUCTIONS 1) The eamination consists of 8 questions. Answer all eight questions. The eam is out of a total of 100 marks. The number of marks for each question is given in brackets. PRINT YOUR NAME AT THE TOP OF EACH PAGE. For Marker s Use Onl ) This is a closed book eam. 4. Calculators are permitted. A list of formulas will be provided separatel. 5. No formula aids (other than the provided list) are permitted. No IPODs or similar devices allowed. 6. No PDAs or laptop computers allowed. No cell phones allowed. 7. Photo ID must be placed on our desk while ou write. 8. 3) SHOW YOUR WORK AND ANSWERS CLEARLY. Give all final numerical answers to 3 significant figures. Give our solutions in the space below the question. Neatl place a bo around our final answers. Continuation sheets have been provided within the eam paper. In addition, the back of each page ma be used as a continuation sheet if required. Total:

2 ME 313 Final Eam - Fall 2011 Name: Page 2 of 15 (15) 1. A strain gauge rosette has a delta configuration (i.e., with gauges forming a triangle), and is mounted to the surface of a machine part which is subjected to a static load. The measured strains for gauges A, B, and C are, respectivel, ε A = , ε B = , ε C = (a) Determine ε, ε, and γ ; (b) Using the eigenvalue method, determine the in-plane principal strains and directions. Show our work; i.e., give the quadratic equation for the principal strains, and determine the principal directions using the eigenvalue equations. Show our final answers with a sketch of the principal directions. C 60 A 60 B

3 ME 313 Final Eam - Fall 2011 Name: Page 3 of 15 Continuation Sheet - Problem 1

4 ME 313 Final Eam - Fall 2011 Name: Page 4 of 15 (15) 2. On the surface of a machine part, the state of strain is defined as follows: 55 ε = 100 μ ε = 370 μ γ = 250 μ ' ' (a) Using Mohr s circle, find the in-plane principal strains and directions. Properl label all points on our Mohr s circle. Show our final answer using a sketch of the principal directions. (b) Using Mohr s circle, find ε and γ. Again, properl label the points on our circle. (The and aes are perpendicular to each other, are in the same plane as the and aes, and are oriented as shown in the sketch above.)

5 ME 313 Final Eam - Fall 2011 Name: Page 5 of 15 Continuation Sheet - Problem 2

6 ME 313 Final Eam - Fall 2011 Name: Page 6 of 15 (15) 3. A rectangular block of plastic is subjected to a compressive load p = 20 MPa as shown, and is also subjected to a temperature increase ΔT = 5 C. The plastic block is constrained from epanding in the and z directions b smooth walls. The walls ma be treated as rigid, and are insulated from temperature effects. The plastic has Young s modulus E = 35 GPa, Poisson s ratio ν = 0.4, and thermal epansion coefficient α = / C. Determine the resulting normal stresses σ, σ, and σ z, and normal strains ε, ε, and ε z in the plastic.

7 ME 313 Final Eam - Fall 2011 Name: Page 7 of 15 Continuation Sheet - Problem 3

8 ME 313 Final Eam - Fall 2011 Name: Page 8 of 15 (15) 4. An element from a loaded bod is shown below. The bod is in a state of plane stress. 35º 35º 35º 35º 75º 50 MPa 100 MPa From smmetr, it is known that τ = 0. Using the labeled stress components and the cosine transformation law, determine σ and σ.

9 ME 313 Final Eam - Fall 2011 Name: Page 9 of 15 Continuation Sheet - Problem 4

10 ME 313 Final Eam - Fall 2011 Name: Page 10 of 15 (8) 5. Using the cosine transformation law and a 2-D analsis, prove that the principal stresses are given b the eigenvalues of the stress matri σ τ τ σ [ σ] = Do not use Mohr s circle in answering this question..

11 ME 313 Final Eam - Fall 2011 Name: Page 11 of 15 (8) 6. Determine the principal stresses σ 1, σ 2, and σ 3, and the corresponding principal directions for the following stress state: (z-ais is out of page) τ σ σ =10 =10 MPa MPa σ z =15 MPa = 0 = τz = τz Use an method ou wish. Place a neat bo around our final answer.

12 ME 313 Final Eam - Fall 2011 Name: Page 12 of 15 (8) 7. A rectangle ABCD is drawn on the surface of a machine part prior to loading. As a result of the loading, the rectangle deforms such that side AB shortens b mm and changes angle b rad clockwise, while side AD becomes longer b mm and changes angle b rad clockwise. The strain field and the rigid bod rotation is uniform over the rectangle. D C 10 mm A 20 mm B Determine strains ε, ε, and γ at point D, and the in-plane rigid bod rotation Θ at point C.

13 ME 313 Final Eam - Fall 2011 Name: Page 13 of 15 (16) 8. Forces F 1 and F 2, as well as couples M 1 and M 2, are applied at end C of the solid bar ABC. The bar s cross section has a diameter of 30 mm, and the bar is fied at end A. Determine the maimum shearing stress, τ ma, in part AB of the bar. State the location of τ ma (in terms of coordinates relative to the given coordinate sstem). You ma select the critical cross section b phsical reasoning. The origin of the given coordinate sstem is at the center of the cross section. Assume that the transverse shearing stress is negligible. M 2 B 0.2 m F 2 C F 1 M 1 F 1 = 200 N 0.75 m F 2 = 350 N M 1 = 175 N m A M 2 = 250 N m (z-ais is out of page)

14 ME 313 Final Eam - Fall 2011 Name: Page 14 of 15 Continuation Sheet - Problem 8

15 ME 313 Final Eam - Fall 2011 Name: Page 15 of 15 Continuation Sheet - Problem 8

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich

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