Stress in a concrete support column

comicnabster
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Homework Statement



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.

Homework 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

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.
 
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Hi comicnabster! :smile:

Hint: the CN tower is tapered (different cross-sections all the way up). :wink:
 
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.
 
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
 
comicnabster said:
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).

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:
 
I described it as a cone with the top end cut off
 
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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