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CN213 Strength of Materials s School of Environment & Technology Semester 1 Examinations 2007-2008 CN213 STRENGTH OF MATERIALS Instructions to Candidates Time allowed: TWO hours Answer ALL questions from Section A Answer TWO questions from Section B All questions carry equal marks January 2008 Page 1 of 9 CN213 Strength of Materials 2007-08 SECTION A Answer BOTH questions from this section Question 1 (a) A steel shaft with a diameter of d = 60 mm is subjected to a torque of T= 3000 Nm and an axial force of F = 100 kN as shown in Figure 1. (i) Determine the normal stress and in-plane shear stress that act on a point on the surface of the shaft. Sketch the stress state in a typical plane element on the bar surface. (4 marks) (ii) Determine the principal stresses and the maximum in-plane shear stress that act at a point on the surface of the shaft. (4 marks) (iii) Determine the angles p and s corresponding with the maximum principal stresses and maximum shear stresses. (4 marks) F T y x T F Figure 1 Question 1 continues on next page Page 2 of 9 CN213 Strength of Materials 2007-08 Question 1 (continued from previous page) (b) A channel section with a cross section as shown in Figure 2 is subjected to a load P = 4 kN acting vertically at point A. (i) Determine the position of the shear centre of the channel section (5 marks) (ii) Determine the torque that is applied to the section by the load P (2 marks) (iii) Assuming the section is used to construct a cantilever beam of length L = 1.5 m and that the load P is applied at the free end of the beam, determine the angle of twist if G = 75 GPa. (6 marks) P A y b/2 tf = 6 mm h = 100 mm z tw = 10 mm b = 60 mm Figure 2 Page 3 of 9 CN213 Strength of Materials 2007-08 Question 2 (a) A piece of steel as shown in Figure 3 is subjected to a load P. (i) Determine the radius r of the fillets so that the maximum stress in vicinity of the fillets is equal to the maximum stress in the vicinity of the hole with diameter d = 20 mm. (Use the graphics provided in the aide memoire to solve this question) (12 marks) (ii) Determine the maximum allowable load P if the allowable normal stress is 80 MPa. (3 marks) 15 mm r 20 mm 50 mm P 80 mm Figure 3 (b) Figure 4 shows a strain gauge rosette attached to the surface of a structural member. The following strains have been measured by the strain rosette: A = 25010-6, B = -10010-6, C = 32010-6. Determine the largest in-plane normal strain. (10 marks) Figure 4 Page 4 of 9 CN213 Strength of Materials 2007-08 SECTION B Answer ANY TWO questions from this section Question 3 (a) A wooden member is subjected to plane stress as shown in Figure 5. The grain forms an angle of 35 with the horizontal. Use Mohr’s circle to determine the in- plane shearing stress parallel to the grain and the normal stress perpendicular to the grain. (15 marks) -4.5 MPa 3 MPa 2 MPa 35 Figure 5 Question 3 continues on next page Page 5 of 9 CN213 Strength of Materials 2007-08 Question 3 (continued from previous page) (b) A tube is made of aluminium and has a rectangular hollow cross-section as shown in Figure 6. The tube is subjected to the torque T. The tube is fixed at D. The shear modulus G= 26 GPa. (i) Determine the average shear stress in the tube at points A and B. (5 marks) (ii) Determine the angle of twist at end C in degrees. (5 marks) fixed 8mm A T = 50 Nm B D 5mm A 50 mm B 5mm 8mm L = 2.5 m C 35 mm Figure 6 Page 6 of 9 CN213 Strength of Materials 2007-08 Question 4 (a) Find the strain energy stored in bars (i), (ii) and (iii) with applied torques as shown in Figure 7. Use TB = 100 Nm, TC = 150 Nm, L = 1.6m, G = 80 GPa and diameter of the bars d= 30mm. (12 marks) A B (i) TB L A C B (ii) TC L/2 A C B (iii) TC TB L/2 L/2 Figure 7 Question 4 continues on next page Page 7 of 9 CN213 Strength of Materials 2007-08 Question 4 (continued from previous page) (b) A beam is made of wide flange section with a cross section as shown in Figure 8 (all dimensions a are equal, flanges and web have the same thickness t = 5mm). The beam is fixed at one end and subjected to a torque of T = 100 Nm at the free end. (i) Determine the length a that is required if the maximum allowable shear stress in the section is allow = 50 MPa. (7 marks) (ii) Determine the length a that is required if the maximum allowable angle of twist at the free end of the beam is allow = 5. (Length of the beam is L = 2 m, G = 78 GPa). (6 marks) a t = 5 mm a t = 5 mm a Figure 8 Page 8 of 9 CN213 Strength of Materials 2007-08 Question 5 (a) The concrete pier shown in Figure 9 is subjected to a load P = 500 kN applied at position a = 0.8 m and b = 1.0 m from the centroid of its cross section. (i) Determine the normal stress at the corners A and C indicating if these are compressive or tensile. Neglect the weight of the pier. (12 marks) (ii) Determine the angle between the neutral axis and the z-axis in the y-z plane. Draw a sketch of the y-z plane showing the neutral axis. (3 marks) P = 500 kN z 1.6 m b a y 1.6 m A 2.5 m 2.5 m C Figure 9 (b) The plane stress state shown in Figure 10 occurs in a structural element with yield stress Y = 450 MPa. Determine whether yield will occur in the element. If yield does not occur, determine the factor of safety. (i) Use the maximum principal stress criterion (5 marks) (ii) Use the maximum distortional energy criterion (5 marks) y = 250 MPa xy = 50 MPa x = 200 MPa Figure 10 Page 9 of 9