The University of Melbourne 436-291 Engineering Mechanics Tutorial Four Poisson s Ratio and Axial Loading Part A (Introductory) 1. (Problem 9-22 from Hibbeler - Statics and Mechanics of Materials) A short cylindrical block of 6061-T6 aluminum, having an original diameter of 20 mm and a length of 75 mm, is placed in a compression machine and squeezed until the axial load applied is 5 kn. Given that: E al = 68.9 GPa and ν = 0.35, determine: (a) the normal stress; (b) the decrease in its length; and (c) its new diameter. 2. (Problem 9-28 from Hibbeler - Statics and Mechanics of Materials) Figure 1: Shear stress-strain diagram for a steel alloy and a bolted connection The shear stress strain diagram for a steel alloy is shown in Figure 1. The bolt with 5 mm diameter is made of this material and used in the lap joint. Given that: ν = 0.3, determine: (a) the shearing modulus G (modulus of rigidity); (b) the modulus of elasticity E; and (c) the force P required to cause the material to yield. Hint: Find yielding shear from figure. 3. (Problem 8-32 from Hibbeler - Statics and Mechanics of Materials) The tapered rod shown in Figure 2 has a radius of r = 2 x 6 m and is subjected to the distributed loading of w = (60 + 40x) kn/m. Determine:. (a) the cross-sectional area at x = 3 m; (b) the tensile force at x = 3 m ; and 1
Figure 2: Tapered rod Figure 3: Two floor building (c) the average normal stress at x = 3 m. 4. (Problem 10-2 from Hibbeler - Statics and Mechanics of Materials) The A-36 steel column shown in Figure 3 is used to support the symmetric loads from the two floors of a building. Given that: P 1 = 200 kn, P 2 = 300 kn, cross-sectional area of the column is 0.015 m 2, and E A 36 = 210 GPa, determine: (a) the total force acting on the B C column; (b) the vertical displacement of B; and (c) the vertical displacement of its top A. 5. (Problem 10-54 from Hibbeler - Statics and Mechanics of Materials) Figure 4: A joint The joint shown in Figure 4 is made from three steel plates that are bonded together at their seams. Given that each plate has a thickness of 5 mm, and E st = 200 GPa, determine: Page 2 of 6
(a) the tensile force at the joint; (b) the cross-sectional areas at A, at the joint, and at B; and (c) the displacement of end A with respect to end B when the joint is subjected to the axial loads shown. 6. (Problem 10-46 from Hibbeler - Statics and Mechanics of Materials) Figure 5: Copper and aluminium rod segments The two circular rod segments, one of aluminum and the other of copper, are fixed to the rigid walls such that there is a gap of 0.1mm between them when T 1 = 15 C (see Figure 5). Given that: the diameter of each rod is 30 mm, α al = 23(10 6 )/ C, E al = 70(10 3 ) MPa, α cu = 17(10 6 )/ C, E cu = 126(10 3 ) MPa, determine: (a) the larger temperature T 2 that is required in order to just close the gap; (b) the compression force in each rod if T 2 = 85 C; and (c) the average normal stress in each rod if T 2 = 85 C. Part B 7. (Problem 9-25 from Hibbeler - Statics and Mechanics of Materials) Figure 6: Stress-strain curve While undergoing a tension test, a copper-alloy specimen having a gauge length of 50 mm is subjected to a strain of 0.40 mm/mm when the stress is 490 MPa (see Figure 6). If σ Y = 315 MPa when ɛ y = 0.0025 mm/mm, determine the distance between the gauge points when the load is released. 8. (Problem 9-27 from Hibbeler - Statics and Mechanics of Materials) The rigid beam rests in the horizontal position on two aluminium cylinders having the unloaded lengths shown in Figure 7. If each cylinder has a diameter of 30 mm, determine the placement x of the applied 80 kn load so that the beam remains horizontal. What is the new diameter of cylinder A after the load is applied? E at = 70 GPa, ν al = 0.33. Page 3 of 6
Figure 7: Rigid beam supported by two cylinders Figure 8: Truss 9. (Problem 10-10 from Hibbeler - Statics and Mechanics of Materials) The truss shown in Figure 8 is made from three A-36 steel members, each having a cross-sectional area of 400 mm 2. Determine the vertical displacement of the roller at C when the truss supports the load of P = 10 kn. E A 36 = 200 GPa. 10. (Problem 10-15 from Hibbeler - Statics and Mechanics of Materials) Figure 9: Linkage The linkage is made from three pin-connected 304 stainless steel members, each having a crosssectional area of 450 mm 2. Determine the magnitude of the force P needed to displace point B 2 mm to the right. E 304 = 193 GPa. Page 4 of 6
11. (Problem 10-24 from Hibbeler - Statics and Mechanics of Materials) Figure 10: Composite column The A-36 steel column, having a cross-sectional area of 10500 mm 2, is encased in high-strength concrete as shown in Figure 10. If an axial force of 300 kn is applied to the column, determine the average compressive stress in the concrete and in the steel. How far does the column shorten? It has an original length of 3 m. E A 36 = 210 GPa, E concrete = 29 GPa. 12. (Problem 10-29 from Hibbeler - Statics and Mechanics of Materials) Figure 11: Uniform bar The uniform bar shown in Figure 11 is subjected to the load P at collar B. Determine the reactions at the pins A and C. Neglect the size of the collar. 13. (Problem 10-31 from Hibbeler - Statics and Mechanics of Materials) Figure 12: Bolted bracket The bolt AB shown in Figure 12 has a diameter of 20 mm and passes through a sleeve that has an inner diameter of 40 mm and an outer diameter of 50 mm. The bolt and sleeve are made of A-36 steel and are secured to the rigid brackets as shown. If the bolt length is 220 mm and the sleeve length is 200 mm, determine the tension in the bolt when a force of 50 kn is applied to the brackets. E A 36 = 200 GPa. Page 5 of 6
Figure 13: Composite bar 14. (Problem 10-38 from Hibbeler - Statics and Mechanics of Materials) Three bars shown in Figure 13 each made of different materials are connected together and placed between two walls when the temperature is T 1 = 12 C. Determine the force exerted on the (rigid) supports when the temperature becomes T 2 = 18 C. The material properties and cross-sectional area of each bar are given in the figure. 15. (Problem 10-42 from Hibbeler - Statics and Mechanics of Materials) Figure 14: Bolted rigid assembly The 10 mm diameter A-36 steel bolt shown in Figure 14 is used to hold the (rigid) assembly together. Determine the clamping force that must be provided by the bolt when T 1 = 10 C so that the clamping force it exerts when T 2 = 95 C is 2500 N. E st = 200 GPa. Page 6 of 6