# University of Pretoria Department of Mechanical & Aeronautical Engineering MOW 227, 2 nd Semester 2014

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1 Universit of Pretoria Department of Mechanical & Aeronautical Engineering MOW 7, nd Semester 04 Semester Test Date: August, 04 Total: 00 Internal eaminer: Duration: hours Mr. Riaan Meeser Instructions: Start each question on a new page. Clearl number ALL our answers according to the question. Consider the simpl supported beam in Figure. The beam is simpl supported at points A and B. a) Draw the approimate Shear-force and Bending-moment diagrams for the beam and indicate the maimum values b) Determine the maimum shear-stress and indicate its location c) Determine the maimum bending stress and indicate its location Show all calculations and relevant points. 0 Points Figure. A countershaft carring two V-belt pulles is shown in the Figure. Pulle A receives power from a motor through a belt with the belt tensions shown. The power is transmitted through the shaft and delivered to the belt on pulle B. Assume the belt tension on the loose side at B is 5 percent of the tension on the tight side. All dimensions are in mm. Page of 7

2 a) Determine the tensions in the belt on pulle B, assuming the shaft is running at a constant speed b) Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports c) Draw shear-force and bending-moment diagrams for the shaft in all relevant planes d) Determine the maimum bending moment s location and magnitude, at that point, determine the bending stress and the torsional shear stress e) At the point of maimum bending moment, determine the principal stresses and the maimum shear stress. State our assumptions. f) If the material ield strength is S 00 MPa, calculate the safet factor guarding against ielding using both the Maimum Shear Stress Theor AND the Von-Mises failure theor 0 Points Figure Page of 7

3 . Using a Mohr circle method for the two-dimensional stress element shown in Figure. Label the Mohr s circle with values, wherever needed. a) Calculate the principal normal stresses and draw the principal normal stress element (using the given - coordinate sstem as a reference) b) Calculate the maimum shear stress and draw the maimum shear stress element c) Assuming plane stress, calculate the maimum shear stress. 0 points 00 Y X 00 Figure 4. A hollow steel clinder is compressed b a force P, as is shown in Figure 4. The clinder has an inner diameter of 00mm an outer diameter of 5mm and modulus of elasticit for the material is E 00 GPa. When the force P increases from zero to 00 kn the outer diameter of the clinder increases b mm. a) Determine the increase in inside diameter b) Determine the increase in wall thickness c) Determine Poisson s ratio for the steel 5 Points Page of 7

4 Figure 4 5. A rotar lawnmower blade is shown in figure 5. This steel blade has a uniform cross section mm thick b 0mm wide, and has a mounting hole of diameter mm in the middle, as is shown in the figure. Estimate the maimum rotational speed that this blade can withstand if the material ield strength is S 00 MPa. Densit of steel 7850 kg/m. 5 Points Figure 5 Page 4 of 7

5 Universit of Pretoria Department of Mechanical & Aeronautical Engineering MOW 7 Semester 04 Semester Test I Reference Sheet Equilibrium Equations (in-plane): F 0 F M 0 z 0 F mrω Deformation: Deflection due to aial load: δ Fl AE Angular deflection due to torsional load: BMD & SFD: dv d dm q V d Stress Equations: P Normal stress due to aial load: A M c Normal stress due to bending: ma I T l θ J G Shear stress due to torsion (circular cross-section): ma Maimum shear stress due to shear force: V ( rectangular section) A 4 V ( circular section) A V ( hollow, thin walled circular section) A π d π d For a circular cross-section: I J 64 bh For a rectangular cross-section: I 4 4 T r J Page 5 of 7

6 General stress-strain relationship: ε υ ( z ) E + ε υ ( z ) E + ε z z υ ( + ) E Gγ Gγ Gγ z z z z ε -ν.ε ε δ/l G E ( + υ ) Stress Transformation Equations: θ θ θ θ Cos + Sin Cos + Sin ( ) Sinθ Cosθ + Cos θ Sin θ + Sinθ Cosθ + + Cosθ + Sinθ Sinθ + Cosθ ma,min + ± + ma,min ± + Tanθ n Tanθ s For principal stresses : Page 6 of 7

7 Static Failure Criteria Equations: S Ma. Normal Stress Theor: Distortion Energ Theor/ Von Mises Criteria: S v / v ( ) + ( ) + ( ) ut ( ) ( z ) ( z ) 6( z z ) Ma. Shear Stress Theor: S S or ma S s / Page 7 of 7

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