45 Problem And Solution Mechanics of Material (Stress and Strain-Axial Loading)

1. A steel Rod is 2,2 m long and must not strecth more than 1,2 mm when a 8,5 kN load is applied to it. Knowing that E = 200 Gpa, determine (a) the smallest diameter rod which should be used, (b) the corresponding normal stress caused bya the load. Solution
2. A 4,8-ft-long steel wire od 1/4 – in diameter steel wire is subjected to a 750 - lb tensile load. Knowing that E = 29 x 10⁶ psi, determine (a) the elongation of the wire, (b) the corresponding normal stress. Solution
3. Two gage marks are placed exactly 10 inches apart in a ½ -in-diameter aluminium rod with E = 10,1 10⁶ psi and an ultimate strenght of 16 ksi. Knowing that the distance between the gage marks is 10,009 in. after a load is applied, determine (a) the stress in the rod, (b) the factor of safety Solution
4. A control rod made of yellow brass must not strecht more than 3 mm when the tension in the wire is 4 kN. Knowing that E = 105 Gpa and that the maximum allowable normal stress is 180 Mpa, determine (a) the smallest diameter that can be selected for the rod, (b) the corresponding maximum length of the rod. Solution
5. A 9-m length of 6-mm-diameter steel wire is to be used in a hanger. It is noted that the wire stretches 18 mm when a tensile force P is applied. Knowing that E = 200 Gpa, determine (a) the maginitude of the force P, (b) the corresponding normal stress in the wire. Solution
6. A 4,5-ft aluminium pipe should not strecht more than 0,05 in. when it is subjected to a tensile load. Knowing that E = 10,1 x 10⁶ psi and that the allowable tensile strenght is 14 ksi., determine (a) the maximum allowable length of the pipe, (b) the required area of the pipe if the tensile load is 127,5 kipsSolution
7. A nylon thread is subjected to a 8,5-N tension Force. Knowing that E = 3,3 GPa and that the length of the thread increases by 1,1 %, determine (a) the diameter of the thread, (b) the stress in the thread. Solution
8. A cast-iron tube is used to support a compressive load. Knowing that E = 10 x 10⁶ psi and the maximum allowable change in the length is 0,025 %, determine (a) the maximum normal stress in the tube, (b) the minimum wall thickness for a load of 1600 lb if the outside diemeter of the tube is 2,0 in Solution
9. A block of 10-in, length and 1,8 x 1,6 in. Cross section is to support a centric compressive load P. The material to be used is a bronze for which E = 1,4 x 10⁶ psi. Determine the largest load which can be applied, knowing that normal stress must not excedd 18 ksi and that the decrease in length of the block should be at most 0,12 % of its original length. Solution
10. A 9-kN tensile load will be applied to a 50-m length of steel wire E = 200 GPa. Detemine the smallest diameter wire which can be used, knowing that the normal stress must not excedd 150 Mpa and that the increase in the length of the wire should be at most 25 mm. Solution
11. The 4-mm-diameter cable BC is made of a steel with E 200 GPa. Knowing that maximum stress in the cable must not exceed 190 Mpa and that the elongation of the cable must not exceed 6 mm, find the maximum load P that can be applied as shown. Solution
12. Rod BD is made of steel E = 29 x 10⁶ psi and is used to brace the axially compressed member ABC. The maximum force that can be development in member BD is 0,02 P. If stress must not exceed 18 ksi and the maximum change in length of BD must not exceed 0,001 times the length of ABC, determine the smallest diameter rod that can be used for member BD. Solution
13. A single axial load of magnitude P = 58 kN is applied at end C of the brass rod ABC. Knowing that E = 105 Gpa, determine the diameter d of portion BC for which the deflection of point C will be 3 mm. Solution
14. Both portion of the rod ABC are made of an aluminium for which E = 73 Gpa. Knowing that the diameter of portion BC is d = 20 mm, determine the largest force P that can be aplied if σ_all = 160 MPa and the corresponding deflection at point C is not to exceed 4 mm. Solution
15. The specimen shown in made from a 1-in-diameter cylindrical steel rod with two 1,5-in-outer-diameter sleeves bonded to the rod as shown. Knowing that E = 29 x 10⁶ psi, determine (a) the load P so that the total deferomation is 0,002 in., (b) the corresponding deformation of the central portion BC. Solution
16. Both portion of the rod ABC are made of an aluminium for which E = 70 Gpa. Knowing that the magnitude of P is 4 kN, determine (a) the value of Q so that the deflection at A is zero, (b) the corresponding deflection of B. Solution 
17. The rod ABC is made of an aluminium for which E = 70 Gpa. Knowing that P = 6 kN and Q = 42 kN, determine the deflection of (a) point A, (b) point B. Solution
18. The 36-mm-diameter steel rod ABC and a brass Rod CD of the same diameter are joined at point C to form the 7,5-m rod ABCD. For the loading shown, and neglecting the weight of the rod, determine the deflection of (a) point C, (b) point D. Solution
19. The brass tube AB (E = 15 x10⁶) has a cross-sectional area of 0,22 in² and is fitted with a plug at A. The tube is attached at B to rigrid plate which is itself attached at C to the bottom of an aluminium cylinder E = 104 x 10⁶psi with a cross-sectional area 0,40 in². The cylinder is then hung from a support at D. In order to close the cylinder, the plug must move down through 3/64 in. Determine the force P that must be applied to cylinder. Solution
20. A 1,2-m Section of aluminium pipe of cross-sectional area 1100 mm² rest on a fixed support at A. the 15-mm-diameter steel rod BC hangs from a rigid bar that rests on the top of the pipe at B. Knowing that the modulus of elasticity is 200 GPa for steel and 72 GPa for aluminium, determine the deflection of point C when a 60 kN force is applied at C. Solution\
21. The steel frame E 200 GPa shown has a diagonal brace BD with an area of 1920 mm². Determine the largest alllowable load P if the change in length of member BD is not to exceed 1,6 mm, Solution
22. For the steel truss E = 200 GPa and loading shown, determine the deformations of members AB and AD, knowing that their cross-sectional areas are 2400 mm² and 1800 mm², respectively. Solution
23.  Member AB and BC are made of steel E = 29 x 10⁶ psi with cross-sectional areas of 0,80 in² and 0,64 in², respectively. Form the loading shown, determine the elongation of (a) member AB, (b) memebr BC. Solution
24. Member AB and CD are 1 1/8 –in-diameter steel rods, and members BC and AD are 2/8-in-diameter steel rods. When the tumbuckle is tightened,, the diagonal member AC is put in tension. Knowing that E = 29 x 10⁶ psi and h = 4 ft, determine the largest allowable tension in AC so that the deformations in member AB and CD do not exceed 0,04 in. Solution
25. For the structure in Problem 2.24, determine (a) the distance h so that the deformation is members AB, BC, CD and AD are all equal to 0,04 in., (b) the corresponding tension in member AC Solution
26. Member ABC and DEF are joined with steel links E = 200 GPa. Each of the links in made of pair of 25 x 35-mm plates. Determine the change in length of (a) member BE, (b) member CF Solutio
27. Each of the links AB and CD is made of aluminium E = 75 GPa and has a cross-sectional area of 125 mm². Knowing that they support the rigid member BC, determine the deflection of point E.Solution
28. Link BD is made of brass E = 15 x10⁶ psi and has a cross-sectional area of 0,04 in². Link CE is made of aluminium E = 10,4 x 10⁶ psi and has a cross-sectional area of 0,05 in². Determine the maximum force P that can be applied vertically at point A if the deflection of A is not to exceed 0,014 in. Solution
29. A homogeneous cable of length l and uniform cross section is suspended from one end. (a) Denoting by  r the density (mass per unit volume) of the cable and by E its modulus of elasticity, determine the elongation of the cable due to its own weight. (b) assuming now the cable to be horizontal, determine the force that should be applied to each end of the cable to obtain the same elongation as in part a.Solution
30. Determine the deflection of the apex A of a homogeneous circular cone of height h, density r, and modulus of elasticity E, due to its own weight. Solution
31. The volume of a tensile specimen is essentially constant while plastic deformation occurs, if the initial diameter of the specimen is d₁, shown that when the diameter is d, the true straisn is Î_t = 2 in (d₁/d). Solution
32. Denoting by Î the “engineering strain” in a tensile specimen, shown that the true strain is Î_t = In  (1 + Î). Solution
33. An axial force of 60 kN is applied to the assembly shown by means of rigid end plates. Determine (a) the normal stress in the brass shell, (by) the corresponding deformation of the assembly.Solutio
34. The length of the assemby decreases by 0,15 mm when an axial force applied by means of rigid end plates. Determine (a) the magnitude of the applied force, (b) the corresponding stress in the steel core.Solution
35. The 4.5-ft concrete post is reinforced with six steel bars, each with a 1 1/8-in. diameter Knowing that E_s = 29 x 10⁶ psi and E_c = 4,2 x 10⁶ psi, determine the normal stresses in the steel and in the concrete when a 350-kip axial centric force P is applied to the post.Solution
36. An Axial centric force of magnitude P = 450 kn is applied to the composite block shown by means of a rigid end plate. Knowing that h =10 mm, determine the normal stress in (a) the brass core, (b) the aluminium plates.Solution
37. Form the composite block shown in Problem 2.36, determine (a) the value  of h if the portion of the load carried by the aluminium plates is half the portion of the load carried by the brass core, (b) the total load if the stress in the brass is 80 Mpa.Solution
38. For the post of problem 2.35, determine the maximum centric force which may be applied if the allowable normal stress is 20 ksi in the steel and 2,4 ksi in the concrete.Solution
39. Three steel rods (E = 200 Gpa) support a 36-kN load P. Each of the rods AB and CD has a 200-mm² cross-sectional area and rod EF has a 625-mm² cross-sectional area. Determine the (a) the change in length of Rod EF, (b) the stress in each rod.Solution
40. A brass bolt  (E_b = 15 x 10⁶ psi) with a 3/8-in. diameter is filted inside a steel tube (Et = 29 x 10⁶ psi) with a 7/8-in. outer diameter and 1/8-in wall thickness. After the nut has been fit snugly, it is tighened on quarter of a full turn. Knowing that the bolt is single-threaded with a 0,1-in. pitch, determine the normal stress (a) in the bolt, (b) in the tube.Solution
41. Two cylindrical rods, CD made of steel (E = 29 x 10⁶ psi) and AC made of aluminium (E = 10,4 x 10⁶ psi), are joined at B and restrained by rigid supports at A and D, determine (a) the reactions at A and D, (b) the defection of point C.Solution
42. A steel tub (E = 200 GPa) with a 32-mm outer diameter and a 4-mm thickness is placed in a vise that adjusted so that its jaws just touch the ends of the tube without exerting any pressure on them. Two forces shown are then applied to the tube. After these forces are applied, the vise is adjusted to decrease the distance between its jaws by 0,2 mm. Determine(a) the forced exerted by the vise on the tube at A and D, (b) the charge in length of the portion BC of the tube.Solution
43. Solve problem 2.42, assuming that after the forces have been applied, the vise is adjusted to decrease the distance between its jaws by 0,1 mm.Solution
44. Three wires are used to suspend the plate shown. Aluminium wires are used at A and B with a diameter of 1/8 in. and steel wire is used at C with a diameter of 1/12 in. Knowing that the allowable stress for steel (E = 29 x 10⁶ psi) is 18 ksi, determine the maximum load P that may be applied Solution
45. The rigid Bar AD is supporteed by two steel wires of 1/16 in. diameter (E = 29 x 10⁶ psi) and a pin and bracket at D. knowing that the wires were initially taught, determine (a) the additional tension in each wire when a 220-lb load P is applied at D, (b) the corresponding deflection of point D Solution

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