A solid circular rod that is 600 mm long and 20 mm in diameter is subjected to an axial force of P = 50 kN The elongation of the rod is Ii = l. 40 mm. and its diameter becomes d' = 19.9837 mm. Determine the modulus of elasticity and the modulus of rigidity of the material. Assume that the material does not yield.

a) V = 0.354

b) G = 25.34 GPA

Explanation:

Solution:

We first determine Modulus of Elasticity and Modulus of rigidity

Elongation of rod ΔL = 1.4 mm

Normal stress, δ = P/A

Where P = Force acting on the cross-section

A = Area of the cross-section

Using Area, A = π/4 · d²

= π/4 · (0.0020) ² = 3.14 * 10⁻⁴m²

δ = 50/3.14 * 10⁻⁴ = 159.155 MPA

E (long) = Δl/l = 1.4/600 = 2.33 * 10⁻³mm/mm

Modulus of Elasticity Е = δ/ε

= 159.155 * 10⁶/2.33 * 10⁻³ = 68.306 GPA

Also final diameter d (f) = 19.9837 mm

Initial diameter d (i) = 20 mm

Poisson said that V = Е (elasticity) / Е (long)

= - (19.9837 - 20 / 20)

2.33 * 10⁻³

= 0.354,

∴ v = 0.354

Also G = Е/2. (1+V)

= 68.306 * 10⁹ / 2. (1 + 0.354)

= 25.34 GPA

⇒ G = 25.34 GPA