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28 September, 08:54

A cylindrical bar of metal having a diameter of 20.2 mm and a length of 209 mm is deformed elastically in tension with a force of 50500 N. Given that the elastic modulus and Poisson's ratio of the metal are 65.5 GPa and 0.33, respectively, determine the following: (a) The amount by which this specimen will elongate in the direction of the applied stress. (b) The change in diameter of the specimen. Indicate an increase in diameter with a positive number and a decrease with a negative number.

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  1. 28 September, 09:18
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    A) ΔL = 0.503 mm

    B) Δd = - 0.016 mm

    Explanation:

    A) From Hooke's law; σ = Eε

    Where,

    σ is stress

    ε is strain

    E is elastic modulus

    Now, σ is simply Force/Area

    So, with the initial area; σ = F/A_o

    A_o = (π (d_o) ²) / 4

    σ = 4F / (π (d_o) ²)

    Strain is simply; change in length/original length

    So for initial length, ε = ΔL/L_o

    So, combining the formulas for stress and strain into Hooke's law, we now have;

    4F / (π (d_o) ²) = E (ΔL/L_o)

    Making ΔL the subject, we now have;

    ΔL = (4F•L_o) / (E•π (d_o) ²)

    We are given;

    F = 50500 N

    L_o = 209mm = 0.209m

    E = 65.5 GPa = 65.5 * 10^ (9) N/m²

    d_o = 20.2 mm = 0.0202 m

    Plugging in these values, we have;

    ΔL = (4 * 50500 * 0.209) / (65.5 * 10^ (9) * π * (0.0202) ²)

    ΔL = 0.503 * 10^ (-3) m = 0.503 mm

    B) The formula for Poisson's ratio is;

    v = - (ε_x/ε_z)

    Where; ε_x is transverse strain and ε_z is longitudinal strain.

    So,

    ε_x = Δd/d_o

    ε_z = ΔL/L_o

    Thus;

    v = - [ (Δd/d_o) / (ΔL/L_o) ]

    v = - [ (Δd•L_o) / (ΔL•d_o) ]

    Making Δd the subject, we have;

    Δd = - [ (v•ΔL•d_o) / L_o]

    We are given v = 0.33; d_o = 20.2mm

    So,

    Δd = - [ (0.33 * 0.503 * 20.2) / 209]

    Δd = - 0.016 mm
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