Engineering M12 Solutions Chapter 6

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Homework Chapter #6 Solutions
A specimen of copper having a rectangular cross section 15.2 mm X 19.1 mm (0.60 in. X 0.75
in.) is pulled in tension with 44,500 n (10,000 lbf) force, producing only elastic deformation.
Calculate the resulting strain.
This problem calls for us to calculate the elastic strain that results for a copper specimen stressed in tension. The
cross-sectional area is just (15.2 mm) x (19.1 mm) = 290 mm2 (= 2.90 x 10-4 m2 = 0.45 in.2); also, the elastic
modulus for Cu is given in Table 6.1 as 110 GPa (or 110 x 109 N/m2). Combining Equations 6.1 and 6.5 and
solving for the strain yields
44,500 =
= 1.39 x 10 - 3
Ao E 2.90 x 10 m 110 x 10 / m
A cylindrical rod of steel (E=207 GPA, 30X106 psi) having a yield strength of 310 MPa (45,000
psi) is to be subjected to a load of 11,100 N (2500 lbf). if the length of the rod is 500 mm (20.0
in.), what must be the diameter to allow an elongation of 0.38 mm (0.015 in.)?
This problem asks us to compute the diameter of a cylindrical specimen of steel in order to allow an elongation of
0.38 mm. Employing Equations 6.1, 6.2, and 6.5, assuming that deformation is entirely elastic
σ =
 d2 
π o 
= E
 4 
do =
4 loF
π E ∆l
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Consider a cylindrical specimen of a steel alloy (figure 6.21) 8.5 mm (0.33 in.) in diameter and
80 mm (3.15 in.) long that is pulled in tension. Determine its elongation when a load of 65,250
N (14,500 lbf) is applied.
This problem asks that we calculate the elongation ∆l of a specimen of steel the stress-strain behavior of which is
shown in Figure 6.21. First it becomes necessary to compute the stress when a load of 65,250 N is applied using
Equation 6.1 as
σ =
d 
π o 
 2
65, 250 N
 8.5 x 10 −3 m
π 
= 1150 MPa (170, 000 psi)
Referring to Figure 6.21, at this stress level we are in the elastic region on the stress-strain curve, which
corresponds to a strain of 0.0054. Now, utilization of Equation 6.2 to compute the value of ∆l
∆ l = ε lo = (0.0054)(80 mm) = 0.43 mm (0.017 in.)
A cylindrical metal specimen 15.0 mm (0.59 in.) in diameter and 150 mm (5.9 in.) long is to be
subjected to a tensile stress of 50 MPa (7250 psi); at this stress level the resulting deformation
will be totally elastic.
(a) if the elongation must be less then 0.072 mm (2.83 X 10-3in.), which of the metals in Table
6.1 are suitable candidates? Why?
This part of the problem asks that we ascertain which of the metals in Table 6.1 experience an elongation of less
than 0.072 mm when subjected to a tensile stress of 50 MPa. The maximum strain that may be sustained, (using
Equation 6.2) is just
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Engineering M12
6.20 cont
(b) If, in addition, the maximum permissible diameter decrease is 2.3 X 10-3 mm (9.1 X 10-5 in.)
when the tensile stress of 50 MPa is applied, which of the metals that satisfy the criterion in
part (a) are suitable candidates? Why?
This portion of the problem further stipulates that the maximum permissible diameter decrease is 2.3 x 10 mm
when the tensile stress of 50 MPa is applied. This translates into a maximum lateral strain εx (max) as
But, since the specimen contracts in this lateral direction, and we are concerned that this strain be less than
1.53 x 10 , then the criterion for this part of the problem may be stipulated as
Now, Poisson’s ratio is defined by Equation 6.8 as
For each of the metal alloys let us consider a possible lateral strain,
. Furthermore, since the deformation
is elastic, then, from Equation 6.5, the longitudinal strain, εz is equal to
Substituting these expressions for εx and εz into the definition of Poisson’s ratio we have
which leads to the following:
Using values for ν and E found in Table 6.1 for the six metal alloys that satisfy the criterion for part (a), and for σ =
50 MPa, we are able to compute a
for each alloy as follows:
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6.20 cont
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Thus, the brass, copper, and titanium alloys will experience a negative transverse strain greater than 1.53 x 10 .
This means that the following alloys satisfy the criteria for both parts (a) and (b) of this problem: nickel, steel, and
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A cylindrical speciman of stainless steel having a diameter of 12.8 mm (0.505 in) and a gauge
length of 50.8 mm (2.000 in) is pulled in tension. Use the load-elongation characteristics
tabulated to complete parts a-f.
This problem calls for us to make a stress-strain plot for stainless steel, given its tensile load-length data, and then to
determine some of its mechanical characteristics.
(a) The data are plotted below on two plots: the first corresponds to the entire stress-strain curve, while for the second,
the curve extends to just beyond the elastic region of deformation.
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6.28 cont
(b) Compute the modulus of elasticity.
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The elastic modulus is the slope in the linear elastic region as
(c) Determine the yield strength at a strain offset of 0.002.
For the yield strength, the 0.002 strain offset line is drawn dashed. It intersects the stress-strain curve at
approximately 750 MPa (112,000 psi ).
(d) Determine the tensile strength of this alloy.
The tensile strength is approximately 1250 MPa (180,000 psi), corresponding to the maximum stress on the
complete stress-strain plot.
(e) What is the approximate ductility, in percent elongation?
The ductility, in percent elongation, is just the plastic strain at fracture, multiplied by one-hundred. The total fracture
strain at fracture is 0.115; subtracting out the elastic strain (which is about 0.003) leaves a plastic strain of 0.112.
Thus, the ductility is about 11.2%EL.
(f) Compute the modulus of resilience.
From Equation (6.14), the modulus of resilience is just
Ur =
which, using data computed in the problem, yields a value of