   # An element of aluminum is subjected to tri-axial stresses. Calculate the strains in the element in x, y, and z directions lithe stresses a x a y and a z are -2000 psi, 3500 psi, and 2700 psi, respectively. Assume E = 10,000 ksi and v = 0.33. Also, find the unit volume change of the element. ### Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 7, Problem 7.6.1P
Textbook Problem
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## An element of aluminum is subjected to tri-axial stresses. Calculate the strains in the element in x, y, and z directions lithe stresses axayand azare -2000 psi, 3500 psi, and 2700 psi, respectively. Assume E = 10,000 ksi and v = 0.33. Also, find the unit volume change of the element. To determine

The strains in the element in x , y and z directions and the unit volume change of the element.

### Explanation of Solution

Given information:

The normal stress acting in x direction is 2000 psi , the normal stress acting in y direction is 3500 psi ,the normal stress acting in z direction is 2700 psi ,the modulus of elasticity is 10 , 000 ksi and the Poisson’s ratio is 0.33 .

Explanation:

Write the expression for the normal strain in the x direction.

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

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

Write the expression for the normal strain in y direction.

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

Here, the strain in y direction is ε y .

Write the expression for the normal strain in z direction.

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

Here, the strain in z direction is ε z .

Write the expression for the unit volume change.

e = ε x + ε y + ε z .....(IV)

Here, the unit volume change is e .

Calculation:

Substitute 2000 psi for σ x , 3500 psi for σ y , 2700 psi for σ z , 10 , 000 ksi for E and 0.33 for ν in Equation (I).

ε x = 1 10000 × 10 3 psi ( 2000 psi 0.33 ( 3500 psi+2700 psi ) ) = 4046 psi 10000 × 10 3 psi = 4.046 × 10 4

Substitute 2000 psi for σ x , 3500 psi for σ y , 2700 psi for σ z , 10 , 000 ksi for E and 0

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