Stress in a concrete support column

[comicnabster]

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.

 

[tiny-tim]

Hi comicnabster!

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

 

[comicnabster]

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.

 

[comicnabster]

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

 

[tiny-tim]

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!

btw, the question asks for the "optimal geometry" … so what is the shape?

 

[comicnabster]

I described it as a cone with the top end cut off