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8 August, 08:43

A concentrated load P is applied to the upper end of a 1.47-m-long pipe. The outside diameter of the pipe is D = 112 mm and the inside diameter is d = 101 mm.

(a) Compute the value of Q for the pipe.

(b) If the allowable shear stress for the pipe shape is 83 MPa, determine the maximum load P that can be applied to the cantilever beam.

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  1. 8 August, 12:13
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    Pmax = 38251.73 N

    Explanation:

    Given info

    L = 1.47 m

    D = 112 mm ⇒ R = D/2 = 112/2 mm = 56 mm

    d = 101 mm ⇒ r = D/2 = 101/2 mm = 50.5 mm

    a) We can apply the following equation in order to get Q (First Moment of Area):

    Q = 2 * (A₁*y₁-A₂*y₂)

    where

    A₁ = π*R² = π * (56 mm) ² = 3136 π mm²

    y₁ = 4*R / (3*π) = 4*56 / (3*π) mm = 224 / (3*π) mm

    A₂ = π*r² = π * (50.5 mm) ² = 2550.25 π mm²

    y₂ = 4*r / (3*π) = 4*50.5 / (3*π) mm = 202 / (3*π) mm

    then

    Q = 2 * (3136 π mm²*224 / (3*π) mm-2550.25 π mm²*202 / (3*π) mm)

    ⇒ Q = 62437.833 mm³

    b) If τallow = 83 MPa = 83 N/mm²

    P = ?

    We can use the equation

    τ = V*Q / (t*I) ⇒ V = τ*t*I / Q

    where

    t = D - d = 112 mm - 101 mm = 11 mm

    I = (π/64) * (D⁴-d⁴) = (π/64) * ((112 mm) ⁴ - (101 mm) ⁴) = 2615942.11 mm⁴

    Q = 62437.833 mm³

    we could also use this equation in order to get Q:

    Q = (4/3) * (R³-r³)

    ⇒ Q = (4/3) * ((56 mm) ³ - (50.5 mm) ³) = 62437.833 mm³

    then we have

    V = (83 N/mm²) * (11 mm) * (2615942.11 mm⁴) / (62437.833 mm³)

    ⇒ V = 2942.255 N

    Finally Pmax = V = 38251.73 N
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