CHAPTER 3 THE EFFECTS OF FORCES ON MATERIALS

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1 CHAPTER THE EFFECTS OF FORCES ON MATERIALS EXERCISE 1, Page A rectangular bar having a cross-sectional area of 80 mm has a tensile force of 0 kn applied to it. Determine the stress in the bar. Stress s force F 0 50 Pa 50 MPa area A 80. A circular section cable has a tensile force of 1 kn applied to it and the force produces a stress of 7.8 MPa in the cable. Calculate the diameter of the cable. Stress s force F area A hence, cross-sectional area, A force F m stress s 7.8 Circular area π r 18. m from which, r 18. π and radius r 18. π.88 m.88 mm and diameter d r mm. A square-sectioned support of side 1 mm is loaded with a compressive force of kn. Determine the compressive stress in the support. Stress, s force F 9.44 Pa 9.44 MPa area A A bolt having a diameter of 5 mm is loaded so that the shear stress in it is 10 MPa. Determine the value of the shear force on the bolt. 58

2 Stress, s force F area A hence, force stress area stress π r 5 10 π 5 N or.5 kn 5. A split pin requires a force of 400 N to shear it. The maximum shear stress before shear occurs is 10 MPa. Determine the minimum diameter of the pin. Stress s force F area A hence, cross-sectional area, A force F 400. m stress s 10 Circular area π r. m from which, r. π and radius r. π 1.00 m 1.00 mm and diameter d r mm. A tube of outside diameter 0 mm and inside diameter 40 mm is subjected to a tensile load of 0 kn. Determine the stress in the tube. Area of tube end (annulus) ( ) ( ) D d 0 40 π π mm Stress s force F Pa 8. MPa area A

3 EXERCISE, Page 5 1. A wire of length 4.5 m has a percentage strain of 0.050% when loaded with a tensile force. Determine the extension in the wire. Original length of wire 4.5 m 4500 mm and strain Strain ε extension x originallength L hence, extension x εl ( )(4500).5 mm. A metal bar.5 m long extends by 0.05 mm when a tensile load is applied to it. Determine (a) the strain, (b) the percentage strain. (a) Strain ε extension 0.05 mm 0.05 original lengh.5 mm (b) Percentage strain %. An 80 cm long bar contracts axially by 0. mm when a compressive load is applied to it. Determine the strain and the percentage strain. Strain ε contraction original lengh 0. mm mm Percentage strain % 4. A pipe has an outside diameter of 0 mm, an inside diameter of mm and length 0.0 m and it supports a compressive load of 50 kn. The pipe shortens by 0. mm when the load is applied. Determine (a) the compressive stress, (b) the compressive strain in the pipe when supporting this load. Compressive force F 50 kn N, and cross-sectional area A ( D d ) π 4, 0

4 where D outside diameter 0 mm and d inside diameter mm. Hence, A π π (0 ) mm (0 ) m.5 m F N (a) Compressive stress, s 4 A.5 m 1. Pa 1. MPa (b) Contraction of pipe when loaded, x 0. mm m, and original length L 0.0 m. Hence, compressive strain, ε x (or 0.0%) L When a circular hole of diameter 40 mm is punched out of a 1.5 mm thick metal plate, the shear stress needed to cause fracture is 0 MPa. Determine (a) the minimum force to be applied to the punch, and (b) the compressive stress in the punch at this value. (a) The area of metal to be sheared, A perimeter of hole thickness of plate. Perimeter of hole πd π(40 ) 0.15 m. Hence, shear area, A m Since shear stress force area (b) Area of punch, shear force shear stress area πd π(0.040) m 4 4 Compressive stress force area N m compressive stress in the punch. ( )N kn, which is the minimum force to be applied 15.0 Pa 15.0 MPa, which is the to the punch.. A rectangular block of plastic material 400 mm long by 15 mm wide by 00 mm high has its lower face fixed to a bench and a force of 150 N is applied to the upper face and in line with it. The upper face moves 1 mm relative to the lower face. Determine (a) the shear stress, and 1

5 (b) the shear strain in the upper face, assuming the deformation is uniform. (a) Shear stress, τ force area parallel to the force Area of any face parallel to the force 400 mm 15 mm 150 N Hence, shear stress, τ 0.00m ( ) m 0.00 m 5000 Pa or 5 kpa (b) Shear strain, γ x L (or 4%)

6 EXERCISE, Page 5 1. A wire is stretched 1.5 mm by a force of 00 N. Determine the force that would stretch the wire 4 mm, assuming the elastic limit of the wire is not exceeded. Hooke's law states that extension x is proportional to force F, provided that the limit of proportionality is not exceeded, i.e. x α F or x kf where k is a constant. When x 1.5 mm, F 00 N, thus 1.5 k(00), from which, constant k When x 4 mm, then 4 kf i.e F from which, force F N Thus to stretch the wire 4 mm, a force of 800 N is required.. A rubber band extends 50 mm when a force of 00 N is applied to it. Assuming the band is within the elastic limit, determine the extension produced by a force of 0 N. Hooke's law states that extension x is proportional to force F, provided that the limit of proportionality is not exceeded, i.e. x α F or x kf where k is a constant. When x 50 mm, F 00 N, thus 50 k(00), from which, constant k When F 0 N, then x k(0) i.e. x ( 0) mm Thus, a force of 0 N stretches the wire mm.. A force of 5 kn applied to a piece of steel produces an extension of mm. Assuming the elastic limit is not exceeded, determine (a) the force required to produce an extension of.5 mm, (b) the extension when the applied force is 15 kn. From Hooke s law, extension x is proportional to force F within the limit of proportionality, i.e.

7 x α F or x kf, where k is a constant. If a force of 5 kn produces an extension of mm, then k(5), from which, constant k (a) When an extension x.5 mm, then.5 k(f), i.e F, from which, force F kn (b) When force F 15 kn, then extension x k(15) (0.08)(15) 1. mm 4. A test to determine the load/extension graph for a specimen of copper gave the following results: Load (kn) Extension (mm) Plot the load/extension graph, and from the graph determine (a) the load at an extension of 0.09 mm, and (b) the extension corresponding to a load of 1.0 kn. A graph of load/extension is shown below. (a) When the extension is 0.09 mm, the load is 19 kn 4

8 (b) When the load is 1.0 kn, the extension is mm 5. A circular section bar is.5 m long and has a diameter of 0 mm. When subjected to a compressive load of 0 kn it shortens by 0.0 mm. Determine Young's modulus of elasticity for the material of the bar. Force, F 0 kn 0000 N and cross-sectional area A Stress s F MPa A m π r π.874 Bar shortens by 0.0 mm m Strain ε x L Modulus of elasticity, E stress strain GPa. A bar of thickness 0 mm and having a rectangular cross-section carries a load of 8.5 kn. Determine (a) the minimum width of the bar to limit the maximum stress to 150 MPa, (b) the modulus of elasticity of the material of the bar if the 150 mm long bar extends by 0.8 mm when carrying a load of 00 kn. (a) Force, F 8.5 kn 8500 N and cross-sectional area A (0x) m, where x is the width of the rectangular bar in millimetres. Stress s F A, from which, A F 8500 N σ 150 Pa m 5.5 mm 4 Hence, 550 0x, from which, width of bar, x (b) Stress s F A MPa 7.5 mm 5.5 mm 550 mm Extension of bar 0.8 mm 5

9 Strain ε x L Modulus of elasticity, E stress strain GPa 7. A metal rod of cross-sectional area 0 mm carries a maximum tensile load of 0 kn. The modulus of elasticity for the material of the rod is 00 GPa. Determine the percentage strain when the rod is carrying its maximum load. Stress s F 0000 A 0 00 MPa Modulus of elasticity, E stress strain from which, strain stress 00 9 E Hence, percentage strain, ε % 0.% 8. A metal tube 1.75 m long carries a tensile load and the maximum stress in the tube must not exceed 50 MPa. Determine the extension of the tube when loaded if the modulus of elasticity for the material is 70 GPa. Modulus of elasticity, E stress strain from which, strain, ε stress E 70 Hence, strain, ε extension x original length L from which, extension, x εl m 1.5 m 1.5 mm 9. A piece of aluminium wire is 5 m long and has a cross-sectional area of 0 mm. It is subjected to increasing loads, the extension being recorded for each load applied. The results are: Load (kn) Extension (mm)

10 Draw the load/extension graph and hence determine the modulus of elasticity for the material of the wire. A graph of load/extension is shown below. E F x σ ε F A x L F L x A is the gradient of the straight line part of the load/extension graph. Gradient, F x BC 7000 N 1.4 N/m AC 5 m L Modulus of elasticity (gradient of graph) A Length of specimen, L 5 m and cross-sectional area A 0 mm 0 m Hence modulus of elasticity, E ( ) GPa 7

11 . In an experiment to determine the modulus of elasticity of a sample of copper, a wire is loaded and the corresponding extension noted. The results are: Load (N) Extension (mm) Draw the load/extension graph and determine the modulus of elasticity of the sample if the mean diameter of the wire is 1. mm and its length is 4.0 m. A graph of load/extension is shown below. F x E σ ε F A x L F L x A is the gradient of the straight line part of the load/extension graph. Gradient, F x BC 10 N 8.57 N/m AC 4. m L Modulus of elasticity (gradient of graph) A 8

12 Length of specimen, L 4.0 m and cross-sectional area A ( ) πd π m Hence modulus of elasticity, E ( ) GPa 9

13 EXERCISE 4, Page A steel rail may assumed to be stress free at 5 C. If the stress required to cause buckling of the rail is - 50 MPa, at what temperature will the rail buckle?. It may be assumed that the rail is rigidly fixed at its ends. Take E 11 N/m and α 14 / C. Buckling stress of steel rail 50 MPa Free expansion of rail αlt αlt Hence, strain α T where T temperature rise. L 11 Stress EαT ( )( 14 ) T T.8 T Buckling stress 50 MPa.8 T from which, T C Initial temperature at which the steel rail was stress-free 5 C Hence, the temperature at which the steel rail will buckle 17.8 C + 5 C.8 C 70

14 EXERCISE 5, Page 1 1. Two layers of carbon fibre are stuck to each other, so that their fibres lie at 90 to each other, as shown below. If a tensile force of 1 kn were applied to this two-layer compound bar, determine the stresses in each. For layer 1, E 1 00 GPa and A 1 mm For layer, E 50 GPa and A A 1 mm PE1 From equation (.8) and (.9), s 1 (A E + A E ) 1 1 PE and s (A E + A E ) 1 1 PE1 s 1 (A E + A E ) ( ) ( ) Pa i.e. the stress in the steel, s MPa PE s (A E + A E ) ( ) ( ) 14.9 i.e. the stress in the concrete, s 14.9 MPa. If the compound bar of Problem 1 were subjected to a temperature rise of 5 C, what would the resulting stresses be? Assume the coefficients of linear expansion are, for layer 1, α 1 5 / C, and for layer, α 0.5 / C. 71

15 As α 1 is larger than α, the effect of a temperature rise will cause the thermal stresses in the steel to be compressive and those in the concrete to be tensile. From equation (.5), the thermal stress in the steel, ( α1 α)e1eat s 1 (A E + A E ) ( ) MPa From equation (.), the thermal stress in the concrete, σ1a1 s A From Problem 1 above: ( 4.8 ) 4.8 MPa s MPa and s MPa EXERCISE, Page XX Answers found from within the text of the chapter, pages 47 to 1. EXERCISE 7, Page XX 1. (c). (c). (a) 4. (b) 5. (c). (c) 7. (b) 8. (d) 9. (b). (c) 11. (f) 1. (h) 1. (d) 14. (b) 15. (a) 7

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