Concept explainers
A circle of a diameter d = 200 mm is etched on a brass plate (see figure). The plate has dimensions of 400 x 400 x 20 mm. Forces are applied to the plate, producing uniformly distributed normal stressescr^ =59 MPaander^ = —17 MPa. Calculate the following quantities: (a) the change in length Aac of diameter at: (b) the change in length Abd of diameter bd; (c) the change At in the thickness of the plate; (d) the change AV in the volume of the plate; (e) the strain energy U stored in the plate; (f) the maximum permissible thickness of the plate when strain energy £/must be at least 784 J; and (g) the maximum permissible value of normal stress axwhen the change in volume of the plate cannot exceed 0.015% of the original volume. (Assume E = 100 GPa and v = 0.34
(a)
The change in the length
Answer to Problem 7.5.14P
The change in the length
Explanation of Solution
Given information:
The normal stress acting on the x-direction is
Explanation:
The figure below shows the element.
Figure (1)
Write the expression of the strain along x-direction.
Here, the normal stress along x-direction is
Write the expression of the change in the length.
Here, the change in the length is
Calculation:
Substitute
Substitute
Conclusion:
The change in the length
(b)
The change in the length
Answer to Problem 7.5.14P
The change in the length
Explanation of Solution
Given information:
The length of the
Write the expression of the strain along y-axis.
Write the expression of the change in the length
Calculation:
Substitute
Substitute
Conclusion:
The change in the length
(c)
The change in thickness of the plate.
Answer to Problem 7.5.14P
The change in the thickness is
Explanation of Solution
Given information:
Write the expression of the strain along z-direction.
Write the expression of the change in thickness.
Here, the change in the thickness is
Calculation:
Substitute
Substitute
Conclusion:
The change in the thickness is
(d)
The change in the volume of the plate.
Answer to Problem 7.5.14P
The change in the volume is
Explanation of Solution
Write the expression for the volumetric strain.
Here, the change in the volume is
Calculation:
Substitute
Conclusion:
The change in the volume is
(e)
The strain energy stored in the plate.
Answer to Problem 7.5.14P
The strain energy stored in the plate is
Explanation of Solution
Write the expression for the strain energy stored in the plate.
Here, the energy in the plate is
Calculation:
Substitute
(f)
The maximum permissible thickness of the plate.
Answer to Problem 7.5.14P
The maximum permissible thickness is
Explanation of Solution
Given information:
The strain energy stored must be at least
Write the expression of the strain energy directly proportional to the thickness.
Calculation:
Substitute
Conclusion:
The maximum permissible thickness is
(g)
The maximum permissible volume of the plate.
Answer to Problem 7.5.14P
The maximum permissible volume is .
Explanation of Solution
Write the expression for the volumetric strain.
Calculation:
Substitute
Conclusion:
The maximum permissible volume is .
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Chapter 7 Solutions
Mechanics of Materials (MindTap Course List)
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- Mechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage Learning