Concept explainers
-12 A square plate of a width h and thickness t is Loaded by normal forces Pxand P and by shear forces V, as shown in the figure. These forces produce uniformly distributed stresses acting on the side faces of the plate.
(a) Calculate the change AV in the volume of the plate and the strain energy U stored in the plate if the dimensions are ft = 600 mm and f = 40 mm; the plate is made of magnesium with E = 41 GPa and v = 0,35; and the forces are Pv= 420 kN, P, = 210 kN, and V = 96 kN. (b) Find the maximum permissible thickness of the plate when the strain energy U must be at least 62 J. [Assume that all other numerical values in part (a) are unchanged.]
(c) Find the minimum width b of the square plate of thickness / = 40 mm when the change in volume of the plate cannot exceed 0.018% of the original volume.
(a)
The change
The strain energy
Answer to Problem 7.5.12P
The change
The strain energy
Explanation of Solution
Given information:
The normal force acting on the x-direction is
Figure (1)
Write the expression for the volumetric strain.
Here, the change in the volume of the plate is
Write the expression for the strain energy stored in the plate.
Here, the strain energy stored in the plate is
Write the expression of the original volume of plate
Here, the width of the plate is
Write the expression for the stress along x-direction.
Here the normal force along x-direction is
Write the expression for the stress along y-direction.
Here the normal force along y-direction is
Write the expression for the area of the plate.
Write the expression for the shear modulus.
Write the expression of the volumetric strain.
In the following plate we have two surface area at which shear force is applied so, the surface area for the shear stress is
Write the expression of the shear stress.
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
The change
The strain energy
(b)
The maximum permissible thickness of the plate.
Answer to Problem 7.5.12P
The maximum permissible thickness of the plate is
Explanation of Solution
Given information:
The strain energy
Write the expression of the strain energy stored in the plate.
Calculation:
Substitute
Conclusion:
The maximum permissible thickness of the plate is
(c)
The minimum width
Answer to Problem 7.5.12P
The minimum width
Explanation of Solution
Given information:
The thickness of square plate is
Write the expression of the condition of the change in the length.
Calculation:
Substitute
Substitute
Substitute
Conclusion: The minimum width
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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