1 UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST NAME (printed): STUDENT NUMBER: EXAMINATION ROOM: SIGNATURE: Total: INSTRUCTIONS 1. The examination consists of 7 questions. Answer all seven questions. The exam 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. 2. This is a closed book exam. Calculators are permitted. A list of formulas will be provided separately. 3. SHOW YOUR WORK CLEARLY. Give final answers to 3 significant figures. 4. Your answers are to be given in the space below the question. Continuation sheets have been provided within the exam paper. In addition, the back of each page may be used as a continuation sheet if required.
2 ME 313 Final 2008 Name: Page 2 of 15 (15) 1. A block is initially stress free and then simultaneously subjected to a temperature increase of 50 C and a pressure of 50 MPa, as shown. The block has dimensions 0.24.m m m and is constrained from expanding in the y direction by two smooth rigid walls. The block is free to expand in the z-direction. Both the stress and strain fields are uniform and the block is made of a linear elastic, homogenous, isotropic material with Young s modulus E = 70 GPa, Poisson s ratio ν = 0.25, and thermal coefficient of expansion α.= / C. (a) Determine stresses σ x, σ y, and σ z and strains ε x, ε y, and ε z in the block. (b) Obtain expressions for displacements u and v in terms of x, y, and z. (The positive z axis comes out of the page.)
3 ME 313 Final 2008 Name: Page 3 of 15 Continuation Sheet - Problem 1
4 ME 313 Final 2008 Name: Page 4 of 15 (15) 2. Normal stresses distributed on the boundary of a 3 m by 3 m plate are shown in the figure. (The shearing stresses on the boundary are not shown.) Also, the shearing stress at every point in the plate is τ xy = 3x 2 + 7y 2 + 2x (where the numerical factors are assumed to have units such that τ xy is in megapascals). The plate is in a state of plane stress in the x-y plane and all body forces are zero; i.e., B x = 0, B y = 0, and B z = 0. (a) Determine expressions for σ x and σ y within the plate. (b) Determine the principal stresses σ 1, σ 2, and σ 3 at the origin, O. (You are not required to find the principal directions.)
5 ME 313 Final 2008 Name: Page 5 of 15 Continuation Sheet Problem 2
6 ME 313 Final 2008 Name: Page 6 of 15 (15) 3. Triangle ABC is scribed on the surface of a member prior to loading. The interior angle at A is originally 90. Following application of the load, the displacement field is given by u = c xy + c 1 2 x x + c y c v = c + c y 3 with w = 0, and where c 1 = m -1, c 2 = m -1, c 3 = , c 4.= , c 5.= m -1, and c 6 = m, and x, y, u, and v are in meters. Assuming the field to be geometrically linear, determine the following changes due to the loading: (a) the percent change in the length of a line element along the y-direction at C; (b) the change in interior angle at corner A in degrees (clearly stating whether it is an increase or decrease in angle); and (c) the total change in angle (in degrees) of a line element at B which is oriented along line BC (clearly stating whether the change in angle is clockwise or counterclockwise).
7 ME 313 Final 2008 Name: Page 7 of 15 Continuation Sheet for Problem 3
8 ME 313 Final 2008 Name: Page 8 of 15 (15) 4. The cross section of a beam carries a bending moment M z = 1200 N m. The moments of inertia I y and I z have already been calculated, and are I y *=* mm 4 I z.= mm 4 where the origin of the y-z coordinate system is at the centroid of the cross-section. Determine the bending stress σ x at the point B which has coordinates y = 45 mm, z = 45 mm.
9 ME 313 Final 2008 Name: Page 9 of 15. Continuation Sheet for Problem 4
10 ME 313 Final 2008 Name: Page 10 of 15 (15) 5. Using the cosine transformation law for stress (together with any appropriate sketches), derive the 2-D eigenvalue equation (σ xx λ) n x + τ xy n y = 0. Also, give a physical explanation (in terms of stress) as to why, for each eigenvalue λ, the system of equations (σ xx λ) n x + τ xy n y = 0 has an infinity of solutions (n x, n y ). τ yx n x + (σ yy λ) n y = 0
11 ME 313 Final 2008 Name: Page 11 of 15 Continuation Sheet for Problem 5
12 ME 313 Final 2008 Name: Page 12 of 15 (10) 6. A beam has the cross section shown and is subjected to pure bending with a bending moment M z.= in lb. The y and z axes shown in the sketch have their origin at point C, the centroid of the cross section. (The x-axis is coming out of the page.) I yy, I zz, and I yz have been calculated to be I yy.= in 4, I zz = in 4, and I yz = 10.0 in 4. The beam is made of a material for which E.= psi and ν.= Determine the magnitude of the compensating moment, M y, that would be needed to constrain the beam so that point C deflects only in the y direction and not in the z direction.
13 ME 313 Final 2008 Name: Page 13 of 15 Continuation Sheet for Problem 6
14 ME 313 Final 2008 Name: Page 14 of 15 (15) 7. A concrete beam is reinforced by three steel rods as shown. The beam is placed in pure bending with a moment M = in lb. The concrete is very weak in tension and therefore it is assumed that the steel rods carry the entire tensile force below the neutral axis. (In this case, the neutral axis is defined as the set of points for which the bending strain is zero.) It is also assumed that the beam takes the usual shape in pure bending, where lines along the length of the beam (including the steel rods) become concentric arcs due to the bending moment. The concrete has a Young s modulus E c.= psi (in compression) and the steel has a Young s modulus E s.= psi. Each steel rod has a diameter d = in. Using fundamental equations and concepts from The Chart, determine the location of the neutral axis of the beam and calculate the compressive stress at the top of the beam (i.e., where the magnitude of the compressive stress is largest). Assume that each steel rod has a uniform axial stress field.
15 ME 313 Final 2008 Name: Page 15 of 15 Continuation Sheet for Problem 7 End of Exam