
(a)
Find the residual stress at
(a)

Answer to Problem 89P
The residual stress is
Explanation of Solution
Given information:
The yield stress for the beam is
The Young’s modulus of steel is
Calculation:
Show the cross-section of the beam as shown in Figure 1.
Refer Figure 1.
Refer to Figure 1.
Calculate the area of the cross section
Here, b is the width of the cross section and d is the depth of the cross section.
Calculate the area of the portion (1)
Substitute
Calculate the area of the portion (2)
Substitute
Calculate the moment of inertia
Calculate the moment of inertia of portion (1)
Substitute
Hence,
Calculate the moment of inertia of portion (2)
Substitute
Calculate the total moment of inertia
Substitute
Calculate the centroid (c) as shown below.
Substitute
Sketch the stress acting on the cross-section of the beam as shown in Figure 2.
Refer Figure 1.
Calculate the area of the portion (2)
Substitute
Calculate the reaction applied to portion (1)
Substitute
Calculate the reaction applied to portion (2)
Substitute
Calculate the moment
Substitute
Calculate the stress
Substitute
Calculate the stress
Substitute
Calculate the residual stress at
Substitute
Calculate the residual stress at
Substitute
Sketch the stress distribution as shown in Figure 3.
Hence, the residual stress is
(b)
Find the point where the residual stress is zero.
(b)

Answer to Problem 89P
The point where the residual stress is zero is
Explanation of Solution
Given information:
The yield stress for the beam is
The Young’s modulus of steel is
Calculation:
Consider that the residual stress
Calculate the yield stress
Calculate the point where the residual stress is zero as shown below.
Substitute
Substitute
Therefore, the point where the residual stress is zero is
(c)
Find the radius of curvature corresponding to the permanent deformation of the bar.
(c)

Answer to Problem 89P
The radius of curvature is
Explanation of Solution
Given information:
The yield stress for the beam is
The Young’s modulus of steel is
Calculation:
Refer to part (a).
The residual stress
Calculate the radius of curvature
Calculate the point where the residual stress is zero as shown below.
Substitute
Therefore, the radius of curvature is
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Chapter 4 Solutions
Mechanics of Materials, 7th Edition
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