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Stress in a concrete support column

  1. Oct 3, 2009 #1
    1. The problem statement, all variables and given/known data

    As quoted from the question sheet:
    When building a tall support, often the self weight of the support must be considered. For an optimal support, the volume of material, and hence the cost, will be a minimum. If the maximum allowable stress in concrete is 12 MPa, determine the optimal geometry of a column 100 metres tall made of concrete to support a mass of 1000 tonnes at its top. (Hint: think of the shape of the CN tower)

    (from a table of values) The weight per cubic meter of concrete is 24 kN/m^3, or 24000 N/m^3.

    Use 9.81 m/s^2 as the value of gravitational acceleration.

    2. Relevant equations

    Stress = Force per area = F/A

    A of a circle = pi(d^2)/4, where d is the diameter

    Volume of a cylinder = Ah, A = pi(d^2)/4

    3. The attempt at a solution

    1000 tonnes = 1.0 E6 kg, so the weight of the mass = 1E6 kg x g = 9.81E6 N
    Maximum stress of concrete is 12 MPa = 12E6 Pa
    12E6 Pa = (9.81E6 N + (100m)pi(r^2)(24000 N/m^3))/(pi*r^2)
    Radius of column = 0.57 m

    Is that the correct approach? Thanks in advance.
     
    Last edited: Oct 3, 2009
  2. jcsd
  3. Oct 3, 2009 #2

    tiny-tim

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    Hi comicnabster! :smile:

    Hint: the CN tower is tapered (different cross-sections all the way up). :wink:
     
  4. Oct 3, 2009 #3
    Thanks, but now I have another question - how do I find the volume of a trapezoidal cylinder prism? I know how to find the volume of a trapezoidal straight-edge prism but not for the type where the bases are two circles of different areas.
     
  5. Oct 4, 2009 #4
    All right, I think I got it this time!

    So the stress is actually uniform throughout the column, thus it must also be 12 MPa at the top.

    I used integration to get the volume of the column (revolve around x-axis).

    Conclusion: Lower radius = 0.527 m, upper = 0.510 m
     
  6. Oct 4, 2009 #5

    tiny-tim

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    That's right … if the tower is to be minimal, the stress will be the safe maximum all the way up! :smile:

    btw, the question asks for the "optimal geometry" … so what is the shape? :wink:
     
  7. Oct 4, 2009 #6
    I described it as a cone with the top end cut off
     
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