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.

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

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