Cork Institute of Technology Bachelor of Engineering (Honours) in Mechanical Engineering- Stage 2 (NFQ Level 8) Autumn 2007 Mechanics of Materials (Time: 3 Hours) Instructions Answer Five Questions Question ONE is compulsory, answer FOUR other questions. Examiners: Mr. W. Corr Mr. P. Clarke Prof. M. Gilchrist Keep your answers to Q1 short and in accordance with the instruction: sketch, demonstrate, explain, etc. Q1. (a) Describe what is meant by the shear centre of a section and where it is located. (b) Identify and justify the fundamental assumption that simplifies thin-walled over thickwalled pressure vessel theory. (c) Sketch 3D Mohr s circles for the stress situation in a case of pure shear. (d) Consider two solid columns of equal length, material and end conditions: one square of side d, the other circular of diameter d. Which will buckle first? Why? (e) Natural cork has Poisson s ratio v = 0. Why use this material in wine bottles? (f) Distinguish between statically determinate and statically indeterminate systems. (g) Give one advantage and one disadvantage of fixed compared to free beams. (h) Describe the failure mechanism when a piece of chalk is twisted until it breaks. (i) Explain the terms intrinsic state of stress and stress invariance and how they realte. (j) Why do we need a theory of failure in component design?
Q2. A rubber plug has outer diameter of 30 mm and fits within a rigid sleeve having inner diameter of 32 mm. Both the plug and the sleeve are 50 mm long. (a) (b) (c) Determine the axial pressure p that must be applied to the top of the plug to cause it to contact the sides of the sleeve. (8 marks) Also, how far must the plug be compressed downward in order to do this? The plug is made from a material for which Ε = 5 MPa. v = 0.45. Finally, if a total downward compression of 10 mm is required, what additional pressure is required to achieve this? (8 marks) Hint: Treat the before-contact and after-contact situations separately. Q3. A compressed-air tank is supported by two cradles one of which is designed so that it does not exert any longitudinal force on the tank. The cylindrical body of the tank has an 800 mm outside diameter and is fabricated from a 10 mm steel plate by butt welding along a helix which forms an angle of 25 with a transverse plane. The end caps are spherical and have a uniform wall thickness of 8mm. For an internal gauge pressure of 10 bars determine: (a) The normal stress and the maximum shearing stress in the spherical caps. (5 marks) (b) The normal stress and maximum shear stresses in the cylindrical section. (5 marks) (c) The stresses in directions perpendicular and parallel to the helical weld. (d) Sketch Mohr s circles for (a) and (b) and in the case of (b), show the location of the weld on the circle. 2
Q4. (a) Define the slenderness ratio for a strut and explain the illustrations of Fig 6.1. (b) Choose any one of the configurations of Fig 6.2 and write mathematically the boundary conditions for the strut ends. (c) (i) A piece of mechanical equipment has a mass of 8000kg and is supported by four vertical steel tubes of 80mm outside diameter and with a wall thickness of 5mm. Each tube is fixed firmly at its base but unrestrained at the top end where it supports the equipment (case (d) in the table in Fig 6.2). Determine the maximum height of the tubes before buckling occurs. Take E = 200 GN/m 2. (ii) For a yield strength of 200MPa, will the strut buckle or yield first? 3
Q5. The steel beam of Fig. 5 has Young s Modulus 208 GN/m 2 and 2 nd moment of area 82 x 10-6 m 4. For the loadings shown: (i) Sketch the bending moment diagram. (5 marks) (ii) Calculate the midspan deflection. (iii) Calculate the left-hand end slope. (9 marks) Hint: For parts (ii) and (iii), Macaulay s method is probably most appropriate. Q6. (a) State and prove the concept of stress invariance. (b) The state of stress at a point on a body is shown in Figure 6. Show similarly, the same state of stress in terms of: (i) The principal stresses (5 marks) (ii) The maximum in-plane shear stress. (5 marks) (c) For part (b) (i) and (ii), demonstrate (a) above and also calculate the maximum out-ofplane shear stress. Figure 6 4
Q7. (a) Define the term Gauge Factor with respect to a strain gauge. (b) How is (i) stress (ii) direct strain (iii) shear strain, measured? (3 marks) (c) A strain gauge rosette measures microstrains 200, 250, 310, 180, 100, 70 along orientations 0, 30, 45, 90, 120, 135 degrees respectively. (i) Deduce and eliminate any suspect readings; (ii) Plot Mohr s strain circle and determine maximum strains and their orientations. (9 marks) 5