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Homework Help: Principle stresses and maximum shearing stress in a cylindrical shaft. Pl help!

  1. Dec 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Nevermind the rough picture, the shaft is cylindrical. Knowing that the post has a diameter of 60mm, determine the principle stresses and the maximum shearing stress at point K.

    problem-1.jpg

    2. Relevant equations

    These are equations for your reference:

    [itex]\sigma = My/I[/itex]
    [itex]\tau = Tc/J[/itex]
    Where M is bending moment, T is twisting torque. J is polar moment of inertia. c is radius of cylindrical shaft. y is distance of point from neutral axis.


    3. The attempt at a solution

    I find:

    J = pi/2 . c^4 = 1.27e-6 m^4; I = J/2 = 6.36e-7 m^4

    [itex]\tau_K = Tc/J = M_yc/J = 1.125e3\cdot0.03/ 1.27e-6 = 26.56MPa[/itex]

    [itex](\sigma_K)_y= M_zc/I = -2.7e3\cdot0.03/ 6.36e-7 = -127.32 MPa[/itex]

    [itex](\sigma_K)_z = 0[/itex] right?!?!

    Because of the shearing force, [itex]P_z, \tau = 3/2 \cdot P/A = 3/2 \cdot 7.5e3/(\pi /2 \times .03^2) = 7.965 MPa[/itex]
    So, [itex]\tau_{total}= 26.57MPa + 7.965MPa = 34.535MPa[/itex]
    Now, because of [itex]M_x[/itex] there won't be any normal longitudinal stress at K because K lies on neutral axis, right? So the only normal longitudinal stress is due to [itex]M_z[/itex].

    Now, because of P_y there will be compressive stress = [itex]P_y/A = -18e3/(\pi / 2 \times .03^2) = -12.73MPa[/itex]
    This I add with the normal longitudinal stress [itex]\sigma_y=-127.36MPa[/itex] calculated earlier.
    So, [itex](\sigma_y)_{total} = -127.36 - 12.73MPa = - 140.09MPa[/itex]

    So this is the diagram for the stress distribution on an element at point K:
    probl2-1.jpg

    So the principle stresses are found to be:

    [itex]\tau_{max} = \sqrt{((\sigma_y-\sigma_z)/2)^2 + (\tau_{yz})^2} = \sqrt{((-140.09-0)/2)^2 + 34.535^2} = 78.09 MPa.[/itex]


    [itex]And, \sigma_{max,min} = \sigma_{ave} \pm \tau_{max} = (-140.09/2 \pm 78.09) MPa = -148.135MPa, 8.045MPa[/itex]


    The answers in the back of the text are : 6.45MPa, -140.0MPa for 73.3MPa. I am not sure which value is for which stress. I guess the first and last values are for the principle stresses and the middle value is for the maximum shearing stress.
     
    Last edited: Dec 28, 2009
  2. jcsd
  3. Dec 28, 2009 #2

    nvn

    User Avatar
    Science Advisor
    Homework Helper

    taureau20: Excellent work. Double-check the given question in your text book and see if you erroneously omitted or supplied a negative sign on Py, Pz, My, or Mz in your above diagram. Secondly, check your formula for cross-sectional area; it is currently incorrect. See if this resolves the problem. Also, per the international standard (ISO 31-0), always leave a space between a numeric value and its following unit symbol; e.g., 73.3 MPa, not 73.3MPa.
     
  4. Dec 29, 2009 #3
    O Christ! How can I write area = pi/2 r^2 when its pi.r^2!!!
     
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