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Therefore, the shear modulus G of the rubber used for the blocks is
G
␥
0.4375 MPa
0.3805 rad 1.150 MPa
⫽ ⫽ ⫽ Ans.
FIGURE M3.1 M3.1 Three basic problems requiring the use of Hooke’s Law.
MecMovies Exercises
P3.1 At the proportional limit, a 2-in. gage length of a 0.375-in.-diameter alloy rod has elongated 0.0083 in. and the 0.375-in.-diameter has been reduced 0.0005 in. The total tension force on the rod was 4.75 kips.
Determine the following properties of the material:
(a) the modulus of elasticity (b) Poisson’s ratio
(c) the proportional limit
P3.2 A solid circular rod with a diameter of d ⫽ 16 mm is shown in Figure P3.2. The bar is made of an aluminum alloy that has an elastic modulus of E ⫽ 72 GPa and Poisson’s of v ⫽ 0.33. When subjected to the axial load P, the diameter of the rod decreases by 0.024 mm. Determine the magnitude of load P.
P d P
FIGURE P3.2
PROBLEMS PROBLEMS
P3.3 At an axial load of 22 kN, a 45-mm-wide by 15-mm-thick polyimide polymer bar elongates 3.0 mm while the bar width contracts 0.25 mm. The bar is 200 mm long. At the 22-kN load, the stress in the polymer bar is less than its proportional limit. Determine
(a) the modulus of elasticity.
(b) Poisson’s ratio.
(c) the change in the bar thickness.
P3.4 A 0.75-in.-thick rectangular alloy bar is subjected to a ten-sile load P by pins at A and B as shown in Figure P3.4/5. The width of the bar is w ⫽ 3.0 in. Strain gages bonded to the specimen mea-sure the following strains in the longitudinal (x) and transverse ( y) directions: x⫽ 840 and y⫽ ⫺250 .
(a) Determine Poisson’s ratio for this specimen.
(b) If the measured strains were produced by an axial load of P ⫽ 32 kips, what is the modulus of elasticity for this specimen?
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P3.5 A 6-mm-thick rectangular alloy bar is subjected to a tensile load P by pins at A and B, as shown in Figure P3.4/5. The width of the bar is w ⫽ 30 mm. Strain gages bonded to the specimen mea-sure the following strains in the longitudinal (x) and transverse (y) directions: x⫽ 900 and y⫽ ⫺275 .
(a) Determine Poisson’s ratio for this specimen.
(b) If the measured strains were produced by an axial load of P ⫽ 19 kN, what is the modulus of elasticity for this specimen?
P3.6 A nylon [E ⫽ 2,500 MPa; ⫽ 0.4] bar is subjected to an axial load that produces a normal stress of . Before the load is ap-plied, a line having a slope of 3:2 (i.e., 1.5) is marked on the bar as shown in Figure P3.6. Determine the slope of the line when ⫽ 105 MPa.
tube (2) as shown in Figure P3.7. The inside diameter of the steel tube is d2⫽ 2.52 in. An external load P is applied to the nylon rod, compressing it. At what value of P will the space between the nylon rod and the steel tube be closed?
P3.8 A metal specimen with an original diameter of 0.500 in.
and a gage length of 2.000 in. is tested in tension until fracture occurs. At the point of fracture, the diameter of the specimen is 0.260 in. and the fractured gage length is 3.08 in. Calculate the duc-tility in terms of percent elongation and percent reduction in area.
P3.9 A portion of the stress–strain curve for a stainless steel alloy is shown in Figure P3.9. A 350-mm-long bar is loaded in ten-sion until it elongates 2.0 mm, and then the load is removed.
(a) What is the permanent set in the bar?
(b) What is the length of the unloaded bar?
(c) If the bar is reloaded, what will be the proportional limit?
P3.10 The 16 by 22 by 25-mm rubber blocks shown in Figure P3.10 are used in a double U shear mount to isolate the vibration of a machine from its supports. An applied load of P ⫽ 285 N causes the upper frame to be defl ected downward by 5 mm. Determine the shear modulus G of the rubber blocks.
P3.11 Two hard rubber blocks are used in an anti-vibration mount to support a small machine as shown in Figure P3.11. An applied load of P ⫽ 150 lb causes a downward defl ection of
0.002 0.004 0.006 0.008 0.010 0
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0.25 in. Determine the shear modulus of the rubber blocks. Assume a ⫽ 0.5 in., b ⫽ 1.0 in., and c ⫽ 2.5 in.
P3.12 Two hard rubber blocks [G ⫽ 350 kPa] are used in an anti-vibration mount to support a small machine as shown in Figure P3.12. Determine the downward defl ection that will occur for an applied load of P ⫽ 900 N. Assume a ⫽ 20 mm, b ⫽ 50 mm, and c ⫽ 80 mm.
P3.13 A load test on a 6-mm-diameter by 225-mm-long alumi-num alloy rod found that a tension load of 4,800 N caused an elastic elongation of 0.52 mm in the rod. Using this result, determine the elastic elongation that would be expected for a 24-mm-diameter rod of the same material if the rod were 1.2 m long and subjected to a tension force of 37 kN.
P3.14 The stress–strain diagram for a particular stainless steel alloy is shown in Figure P3.14. A rod made from this material is initially 800 mm long at a temperature of 20°C. After a tension force is applied to the rod and the temperature is increased by 200°C, the length of the rod is 804 mm. Determine the stress in the rod and state whether the elongation in the rod is elastic or inelas-tic. Assume the coeffi cient of thermal expansion for this material is 18 ⫻ 10⫺6Ⲑ°C.
P3.15 In Figure P3.15, rigid bar ABC is supported by axial member (1), which has a cross-sectional area of 400 mm2, an elas-tic modulus of E ⫽ 70 GPa, and a coeffi cient of thermal expansion of ␣ ⫽ 22.5 ⫻ 10⫺6Ⲑ°C. After load P is applied to the rigid bar and the temperature rises 40°C, a strain gage affi xed to member (1) measures a strain increase of 2,150 . Determine
(a) the normal stress in member (1).
(b) the magnitude of applied load P.
(c) the defl ection of the rigid bar at C.
P3.16 A tensile test specimen of 1045 hot-rolled steel having a diameter of 0.505 in. and a gage length of 2.00 in. was tested to fracture. Stress and strain data obtained during the test are shown in Figure P3.16. Determine
(a) the modulus of elasticity.
(b) the proportional limit.
(c) the ultimate strength.
(d) the yield strength (0.20% offset).
(e) the fracture stress.
(f) the true fracture stress if the fi nal diameter of the specimen at the location of the fracture was 0.392 in.
FIGURE P3.11
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P3.17 A tensile test specimen of stainless steel alloy having a diameter of 0.495 in. and a gage length of 2.00 in. was tested to fracture. Stress and strain data obtained during the test are shown in Figure P3.17. Determine.
(a) the modulus of elasticity.
(b) the proportional limit.
(c) the ultimate strength.
(d) the yield strength (0.20% offset).
(e) the fracture stress.
(f) the true fracture stress if the fi nal diameter of the specimen at the location of the fracture was 0.350 in.
P3.18 A bronze alloy specimen having a diameter of 12.8 mm and a gage length of 50 mm was tested to fracture. Stress and strain data obtained during the test are shown in Figure P3.18. Determine (a) the modulus of elasticity.
(b) the proportional limit.
(c) the ultimate strength.
(d) the yield strength (0.20% offset).
(e) the fracture stress.
(f) the true fracture stress if the fi nal diameter of the specimen at the location of the fracture was 10.5 mm.
P3.19 An alloy specimen having a diameter of 12.8 mm and a gage length of 50 mm was tested to fracture. Load and deformation data obtained during the test are given. Determine
(a) the modulus of elasticity.
(b) the proportional limit.
(c) the ultimate strength.
(d) the yield strength (0.05% offset).
(e) the yield strength (0.20% offset).
(f) the fracture stress.
(g) the true fracture stress if the fi nal diameter of the specimen at the location of the fracture was 11.3 mm.
Load (kN)
Change in
Length (mm) Load (kN)
Change in
41.0 1.00 45.1 fracture
0.025
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P3.20 A 1035 hot-rolled steel specimen with a diameter of 0.500 in. and a 2.0-in. gage length was tested to fracture. Load and deformation data obtained during the test are given.
Determine
(a) the modulus of elasticity.
(b) the proportional limit.
(c) the ultimate strength.
(d) the yield strength (0.05% offset).
(e) the yield strength (0.20% offset).
(f) the fracture stress.
(g) the true fracture stress if the fi nal diameter of the specimen at the location of the fracture was 0.387 in.
Load (lb)
Change in
Length (in.) Load (lb)
Change in Length (in.)
0 0 12,540 0.0209
2,690 0.0009 12,540 0.0255
5,670 0.0018 14,930 0.0487
8,360 0.0028 17,020 0.0835
11,050 0.0037 18,220 0.1252
12,540 0.0042 18,820 0.1809
13,150 0.0046 19,110 0.2551
13,140 0.0060 19,110 0.2968
12,530 0.0079 18,520 0.3107
12,540 0.0098 17,620 0.3246
12,840 0.0121 16,730 0.3339
12,840 0.0139 16,130 0.3385
15,900 fracture
P3.21 A 2024-T4 aluminum test specimen with a diameter of 0.505 in. and a 2.0-in. gage length was tested to fracture.
Load and deformation data obtained during the test are given.
Determine
(a) the modulus of elasticity.
(b) the proportional limit.
(c) the ultimate strength.
(d) the yield strength (0.05% offset).
(e) the yield strength (0.20% offset).
(f) the fracture stress.
(g) the true fracture stress if the fi nal diameter of the specimen at the location of the fracture was 0.452 in.
Load (lb)
Change in
Length (in.) Load (lb)
Change in Length (in.)
0 0.0000 11,060 0.0139
1,300 0.0014 11,500 0.0162
2,390 0.0023 12,360 0.0278
3,470 0.0032 12,580 0.0394
4,560 0.0042 12,800 0.0603
5,640 0.0051 13,020 0.0788
6,720 0.0060 13,230 0.0974
7,380 0.0070 13,450 0.1159
8,240 0.0079 13,670 0.1391
8,890 0.0088 13,880 0.1623
9,330 0.0097 14,100 0.1994
9,980 0.0107 14,100 0.2551
10,200 0.0116 14,100 0.3200
10,630 0.0125 14,100 0.3246
14,100 fracture P3.22 A 1045 hot-rolled steel tension test specimen has a diam-eter of 6.00 mm and a gage length of 25 mm. In a test to fracture, the stress and strain data below were obtained. Determine (a) the modulus of elasticity.
(b) the proportional limit.
(c) the ultimate strength.
(d) the yield strength (0.05% offset).
(e) the yield strength (0.20% offset).
(f) the fracture stress.
(g) the true fracture stress if the fi nal diameter of the specimen at the location of the fracture was 4.65 mm.
Load (kN)
Change in
Length (mm) Load (kN)
Change in
11.16 0.04 20.56 2.26
12.63 0.05 20.67 2.78
13.02 0.06 20.72 3.36
13.16 0.08 20.61 3.83
13.22 0.08 20.27 3.94
13.22 0.10 19.97 4.00
13.25 0.14 19.68 4.06
13.22 0.17 19.09 4.12
18.72 fracture
P3.23 A concentrated load P is supported by two bars as shown in Figure P3.23. Bar (1) is made of cold-rolled red brass [E ⫽ 16,700 ksi; ␣ ⫽ 10.4 ⫻ 10−6 Ⲑ⬚F] and has a cross-sectional area of 0.225 in.2. Bar (2) is made of 6061-T6 aluminum [E ⫽ 10,000 ksi; ␣ ⫽ 13.1 ⫻ 10 ⫺6Ⲑ⬚F] and has a cross-sectional area of 0.375 in.2. After load P has been applied and the temperature of the entire assembly has increased by 50⬚F, the total strain in bar (1) is measured as 1,400 (elongation). Determine
(a) the magnitude of load P.
(b) the total strain in bar (2).
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P3.24 The rigid bar AC in Figure P3.24 is supported by two axial bars (1) and (2). Both axial bars are made of bronze [E ⫽ 100 GPa; ␣ ⫽ 18 ⫻ 10⫺6Ⲑ⬚C]. The cross-sectional area of bar (1) is A1⫽ 240 mm2 and the cross-sectional area of bar (2) is A2⫽ 360 mm2. After load P has been applied and the temperature of the entire assembly has increased by 30⬚C, the total strain in bar (2) is measured as 1,220 (elongation). Determine
(a) the magnitude of load P.
(b) the vertical displacement of pin A.
P3.25 The rigid bar in Figure P3.25/26 is supported by axial bar (1) and by a pin connection at C. Axial bar (1) has a cross-sectional area of A1⫽ 275 mm2, an elastic modulus of E ⫽ 200 GPa, and a coeffi cient of thermal expansion of ␣ ⫽ 11.9 ⫻ 10 ⫺6Ⲑ⬚C. The pin at C has a diameter of 25 mm. After load P has been applied and the temperature of the entire assembly has been increased by 20⬚C, the total strain in bar (1) is measured as 925 (elongation). Determine (a) the magnitude of load P.
(b) the shear stress in pin C.
P3.26 The rigid bar in Figure P3.25/26 is supported by axial bar (1) and by a pin connection at C. Axial bar (1) has a cross-sectional area of A1⫽ 275 mm2, an elastic modulus of E ⫽ 200 GPa, and a coeffi cient of thermal expansion of ␣ ⫽ 11.9 ⫻ 10 ⫺6 Ⲑ⬚C. The pin at C has a diameter of 25 mm. After load P has been applied and the temperature of the entire assembly has been decreased by 30⬚C, the total strain in bar (1) is measured as 925 (elongation). Determine (a) the magnitude of load P.
(b) the shear stress in pin C.
(1)
(2)
A C
B
P
500 mm 900 mm
1,300 mm
2,000 mm
FIGURE P3.24
40° 55°
(1)
(2)
C
A
B
P FIGURE P3.23
FIGURE P3.25/26 C
Connection detail
275 mm 540 mm
260 mm 135 mm
P
A B
C
Rigid bar (1)
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