TM
HOMEWORKS - 02 2.01 Knowing that d =1.2 in., determine the torque T that causes a maximum shearing stress of 7.5 ksi in the hollow shaft shown. 2.02 Knowing that the internal diameter of the hollow shaft shown is d = 0.9 in., determine the maximum shearing stress caused by a torque of magnitude T = 9 kip.in. 2.03 The solid spindle AB has a diameter d s = 1.5 in. and is made of a steel with an allowable shearing stress of 12 ksi, while sleeve CD is made of a brass with an allowable shearing stress of 7 ksi. Determine the largest torque T that can be applied at A.
F ig. 2.01 and 2.02
F ig. 2.03 2.03 and 2.04
F ig. 2.05 2.05
2.04 The solid spindle AB is made of a steel with an allowable shearing stress of 12 ksi, and sleeve CD is made of a brass with an allowable shearing stress of 7 ksi. Determine (a) the largest torque T that can be applied at A if the allowable shearing stress is not to be exceeded in sleeve CD, (b) the corresponding required value of the diameter d s of spindle AB. 2.05 The torques shown are exerted on pulleys A and B. Knowing that both shafts are solid, determine the maximum shearing stress in (a) in shaft AB, (b) in shaft BC.
F i g. 2.06
F i g. 2.07
F i g. 2.08 and 2.09 2.09
2.06 The aluminum rod AB (G = 27 GPa) is bonded to the brass rod BD (G 539 GPa). Knowing that portion CD of the brass rod is hollow and has an inner diameter of 40 mm, determine the angle of twist at A. 1
TM 2.07 The solid cylinders AB and BC are bonded together at B and are attached to fixed supports 6
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at A and C. Knowing that the modulus of rigidity is 3.7 10 psi for aluminum and 5.610 psi for brass, determine the maximum shearing stress (a) in cylinder AB, (b) in cylinder BC. 2.08 Each of the two aluminum bars shown is subjected to a torque of magnitude T = 1800 Nm. Knowing that G = 26 GPa, determine for each bar the maximum shearing stress and the angle of twist at B. 2.09 Determine the largest torque T that can be applied to each of the two aluminum bars shown
and the corresponding angle of twist at B, knowing that [ ] = 50 MPa and G = 26 GPa.
F ig. 2.10
F i g. 2.11
F i g. 2.12 and 2.13
2.10 The design of a machine element calls for a 40-mm-outer-diameter shaft to transmit 45 kW. (a) If the speed of rotation is 720 rpm, determine the maximum shearing stress in shaft a. (b) If the speed of rotation can be increased 50% to 1080 rpm, determine the largest inner diameter of shaft b for which the maximum shearing stress will be the same in each shaft. 2.11 A steel pipe of 3.5-in. outer diameter is to be used to transmit a torque of 3000 lb.ft without exceeding an allowable shearing stress of 8 ksi. A series of 3.5-in.-outer-diameter pipes is available for use. Knowing that the wall thickness of the available pipes varies from 0.25 in. to 0.50 in. in 0.0625-in. increments, choose the lightest pipe that can be used. 2.12 The stepped shaft shown must transmit 45 kW. Knowing that the allowable shearing stress in the shaft is 40 MPa and that the radius of the fillet is r = 6 mm, determine the smallest permissible speed of the shaft. 2.13 The stepped shaft shown must rotate at a frequency of 50 Hz. Knowing that the radius of the fillet is r = 8 mm and the allowable shearing stress is 45 MPa, determine the maximum power that can be transmitted.
F ig. 2.14
F i g. 2.15
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F i g. 2.16
TM 2.14 Two shafts are made of the same material. The cross section of shaft A is a square of side b and that of shaft B is a circle of diameter b. Knowing that the shafts are subjected to the same
torque, determine the ratio A/B of maximum shearing stresses occurring in the shafts. 2.15 A torque T = 750 kNm is applied to the hollow shaft shown that has a uniform 8-mm wall thickness. Neglecting the effect of stress concentrations, determine the shearing stress at points a and b. 2.16 Two solid steel shafts (G = 77.2 GPa) are connected to a coupling disk B and to fixed supports at A and C. For the loading shown, determine (a) the reaction at each support, (b) the maximum shearing stress in shaft AB, (c) the maximum shearing stress in shaft BC.
F ig. 2.17
F ig. 2.18
F ig. 2.19
2.17 A 36-kip.in. torque is applied to a 10-ft-long steel angle with an L8 81 cross section. From the tables we find that the thickness of the section is 1 in. and that its area is 15 in 2.
Knowing that G = 11.2106 psi, determine (a) the maximum shearing stress along line a-a, (b) the angle of twist. 2.18 A 3-m-long steel angle has an L20315212.7 cross section. From the tables we find that
the thickness of the section is 12.7 mm and that its area is 4350 mm 2. Knowing that [] = 50 MPa and that G = 77.2 GPa, and ignoring the effect of stress concentrations, determine (a) the largest torque T that can be applied, (b) the corresponding angle of twist. 2.19 A torque T = 5 kNm is applied to a hollow shaft having the cross section shown. Neglecting the effect of stress concentrations, determine the shearing stress at points a and b.
F i g. 2.20
F ig. 2.21
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F ig. 2.22
TM 2.20 Knowing that the couple shown acts in a vertical plane, determine the stress at (a) point A, (b) point B. 2.21 Knowing that a beam of the cross section shown is bent about a horizontal axis and that the bending moment is 50 kip.in., determine the total force acting (a) on the top flange (b) on the shaded portion of the web. 2.22 The beam shown is made of a nylon for which the allowable stress is 24 MPa in tension and 30 MPa in compression. Determine the largest couple M t hat can be applied to the beam. 2.23 A W20031.3 rolled-steel beam is subjected to a couple M of moment 45 kNm. Knowing
that E = 200 GPa and = 0.29, determine (a) the radius of curvature , (b) the radius of curvature of a transverse cross section. ’
F ig. 2.23
F ig. 2.24
F ig. 2.25
2.24 A steel bar and an aluminum bar are bonded together to form the composite beam shown. The modulus of elasticity for aluminum is 70 GPa and for steel is 200 GPa. Knowing that the beam is bent about a horizontal axis by a couple of moment M = 1500 Nm, determine the maximum stress in (a) the aluminum, (b) the steel. 2.25 The reinforced concrete beam shown is subjected to a positive bending moment of 175 kNm. Knowing that the modulus of elasticity is 25 GPa for the concrete and 200 GPa for the steel, determine (a) the stress in the steel, (b) the maximum stress in the concrete. 2.26 Semicircular grooves of radius r must be milled as shown in the sides of a steel member. Using an allowable stress of 60 MPa, determine the largest bending moment that can be applied to the member when (a) r = 9 mm, (b) r = 18 mm.
F ig. 2.26
F ig. 2.27
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F ig. 2.28
TM 2.27 Knowing that the allowable stress for the beam shown is 90 MPa, determine the allowable bending moment M when the radius r of the fillets is (a) 8 mm, (b) 12 mm. 2.28 Knowing that the magnitude of the horizontal force P is 8 kN, determine the stress at (a) point A, (b) point B. 2.29 A milling operation was used to remove a portion of a solid bar of square cross section.
Knowing that a = 30 mm, d = 20 mm, and [ ] = 60 MPa, determine the magnitude P of the largest forces that can be safely applied at the centers of the ends of the bar. 2.30 through 2.34 The couple M is applied to a beam of the cross section shown in a plane
forming an angle with the vertical. Determine the stress at (a) point A, (b) point B, (c) point D.
F ig. 2.29
F ig. 2.32
F ig. 2.30
F ig. 2.31
F ig. 2.33
F ig. 2.34
2.35 Two vertical forces are applied to a beam of the cross section shown. Determine the maximum tensile and compressive stresses in portion BC of the beam. 2.36 Determine the maximum stress in each of the two machine elements shown.
F ig. 2.35
F ig. 2.36
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TM 2.37 through 2.42 For the beam and loading shown, draw the shear and bending-moment diagrams.
F ig. 2.37
F ig. 2.38
F ig. 2.40
F ig. 2.39
F ig. 2.41
F ig. 2.42
2.43 through 2.48 Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value (a) of the shear, (b) of the bending moment.
F ig. 2.43
F ig. 2.44
F ig. 2.45
F ig. 2.46
F ig. 2.47
F ig. 2.48
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TM 2.49 through 2.52 For the beam and loading shown, determine the maximum normal stress due to bending on a transverse section at C.
F ig. 2.49
F ig. 2.50
F ig. 2.51
F ig. 2.52
2.53 Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value (a) of the shear, (b) of the bending moment. 2.54 Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending. 2.55 For the beam and loading shown, design the cross section of the beam, knowing that the grade of timber used has an allowable normal stress of 1750 psi. 2.56 Determine the largest permissible value of P for the beam and loading shown, knowing that the allowable normal stress is +8 ksi in tension and -18 ksi in compression.
F i g. 2.53
F ig. 2.54
F i g. 2.55
F ig. 2.56
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TM 2.57 Three boards, each of 1.53.5 in. rectangular cross section, are nailed together to form a beam that is subjected to a vertical shear of 250 lb. Knowing that the spacing between each pair of nails is 2.5 in., determine the shearing force in each nail. 2.58 Three boards, each 2 in. thick, are nailed together to form a beam that is subjected to a vertical shear. Knowing that the allowable shearing force in each nail is 150 lb, determine the allowable shear if the spacing s between the nails is 3 in. 2.59 For the beam and loading shown, consider section n-n and determine (a) the largest shearing stress in that section, (b) the shearing stress at point a. 2.60 A square box beam is made of two 20 80-mm planks and two 20120-mm planks nailed together as shown. Knowing that the spacing between the nails is s = 30 mm and that the vertical shear in the beam is V = 1200 N, determine (a) the shearing force in each nail, (b) the maximum shearing stress in the beam.
F ig. 2.57
F ig. 2.58
F ig. 2.59
F ig. 2.60
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