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Solve the preceding problem if the cube is granite (E = 80 GPa, v = 0.25) with dimensions E = 89 mm and compressive strains E = 690 X l0-6 and = = 255 X 10-6. For part (c) of Problem 7.6-5. find the maximum value of cr when the change in volume must be limited to 0.11%. For part. find the required value of when the strain energy must be 33 J.

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Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347

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Section
BuyFindarrow_forward

Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 7, Problem 7.6.6P
Textbook Problem
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Solve the preceding problem if the cube is granite (E = 80 GPa, v = 0.25) with dimensions E = 89 mm and compressive strains E = 690 X l0-6 and = = 255 X 10-6. For part (c) of Problem 7.6-5. find the maximum value of cr when the change in volume must be limited to 0.11%. For part. find the required value of when the strain energy must be 33 J.

  Chapter 7, Problem 7.6.6P, Solve the preceding problem if the cube is granite (E = 80 GPa, v = 0.25) with dimensions E = 89 mm

(a)

To determine

The normal stresses acting on the x , y and z faces of the cube.

Explanation of Solution

Given information:

A cube of cast iron having side 89 mm is tested under triaxial stress. The strain in the x direction is 690 × 10 6 , strain in y direction and z direction is being equal which is 255 × 10 6 . The modulus of elasticity is 80 GPa and the Poisson’s ratio is 0.25 .

Explanation:

Write the expression for the stress along x axis.

   σ x = E ( 1 + ν ) ( 1 2 ν ) [ ( 1 ν ) ε x + ν ( ε y + ε z ) ] ...... (I)

Here, the stress alon x axis is σ x , modulus of elasticity is E , the Poisson’s ratio is ν , strain along x axis is ε x , strain along y axis is ε y and strain along z axis is ε z .

Write the expression for stress along y axis.

   σ y = E ( 1 + ν ) ( 1 2 ν ) [ ( 1 ν ) ε y + ν ( ε x + ε z ) ] ...... (II)

Here, stress along y axis is σ y .

Write the expression for stress along z axis.

   σ z = E ( 1 + ν ) ( 1 2 ν ) [ ( 1 ν ) ε z + ν ( ε x + ε y ) ] ...... (III)

Here, stress along z axis is σ z .

Calculation:

Substitute 690 × 10 6 for ε x , 255 × 10 6 for ε y , 255 × 10 6 for ε z , 80 GPa for E and 0.25 for ν in Equation (I).

   σ x = 80 GPa ( 1 + 0.25 ) ( 1 ( 2 × 0.25 ) ) [ ( 0.75 × 690 ) + ( 0.25 × ( 255 255 ) ) ] × 10 6 = 128 GPa ( 645 × 10 6 ) = 82560 GPa × ( 10 3 MPa 1 GPa ) × 10 6 = 82

(b)

To determine

The maximum shear stress in the material.

(c)

To determine

The change in the volume of the cube.

(d)

To determine

The strain energy stored in the cube.

(e)

To determine

The maximum value of normal stress along the x axis.

(f)

To determine

The required value of strain along the x axis.

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