A cube of cast iron with sides of length a = 4.0 in. (see figure) is tested in a laboratory under triaxialsire.ss. Gages mounted on the testing machine show that the compressive strains in the material arc a
= -225 X l06and,ay = 37.5 X l0_.
Determine the following quantities: (a) the norm al stresses i. r,.. and acting on the x, y, and z faces of the cube; (b) the maximum shear stress r in the material; (C) the change ..W in the volume of the cube: (d) the strain energy U stored in the cube; (e) the maximum value of s when the change in volume must be limited to O.O28%; and (f) the required value of when the strain energy must be 38 in.-lb. (Assume £ = 14,000 ksi and v = 0.25.)
(a)
The normal stresses acting on the
Answer to Problem 7.6.5P
The normal stress acting on the
The normal stress acting on the
The normal stress acting on the
Explanation of Solution
Given information:
A cube of cast iron having side
Explanation:
Write the expression for the stress along
Here, the stress along
Write the expression for stress along
Here, stress along
Write the expression for stress along
Here, stress along
Calculation:
Substitute
Substitute
Substitute
Conclusion:
The normal stress acting on the
The normal stress acting on the
The normal stress acting on the
(b)
The maximum shear stress in the material.
Answer to Problem 7.6.5P
The maximum shear stress in the material is
Explanation of Solution
Write the expression for the shear stress in the
Here, the shear stress in
Write the expression for the shear stress in the
Here, the shear stress in
Write the expression for the shear stress in
Here, the shear stress in the
Calculation:
Substitute
Substitute
Substitute
From the above values of shear stresses, the maximum shear stress be
Therefore,
Conclusion:
The maximum shear stress in the material is
(c)
The change in the volume of the cube.
Answer to Problem 7.6.5P
The change in the volume is
Explanation of Solution
Write the expression for the total volumetric strain in the cube.
Here, the total volumetric strain in the cube is
Write the expression for the volume of cube.
Here, the volume of cube is
Write the expression for change in volume.
Here, the change in the volume is
Calculation:
Substitute
Substitute
Substitute
Conclusion:
The change in the volume is
(d)
The strain energy stored in the cube.
Answer to Problem 7.6.5P
The strain energy stored in the cube is
Explanation of Solution
Write the expression for the strain energy stored in the cube.
Here, the strain energy is
Calculation:
Substitute
Conclusion:
The strain energy stored in the cube is
(e)
The maximum value of normal stress along the
Answer to Problem 7.6.5P
The maximum value of normal stress along the
Explanation of Solution
Given information:
The change in volume is limited to
Explanation:
Write the expression for the change in volume.
Write the expression for the stress along
Calculation:
Substitute
Substitute
Conclusion:
The maximum value of normal stress along the
(f)
The required value of strain along the
Answer to Problem 7.6.5P
The required value of the strain along the
Explanation of Solution
Given information:
The strain energy of the system is
Explanation:
Write the expression for the strain energy.
Calculation:
Substitute
Now solve the quadratic equation for obtaining the value of
Conclusion:
The required value of the strain along the
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Chapter 7 Solutions
Mechanics of Materials (MindTap Course List)
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