MHB What Is the Optimal Geometry for a Concrete Column Supporting 1250 Tonnes?

[Dethrone]

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 15 MPa, determine the optimal geometry of a column 100 metres tall made of concrete to support a mass of 1250 tonnes at its top. Hint: think of the shape of the CN tower
There's no textbook for this course, so we're expected to use common sense to answer these questions...
$$\sigma = \frac{F}{A}=\frac{\pi D^2/4\cdot H\cdot W+1250\cdot 10^3 \cdot 9.81}{\pi D^2/4} $$
W=25 Kn/m, and putting all the information in, I get diameter is 1.1m. Is that right, and I was told that the radius is different at the top, not sure what to do.

 

[I like Serena]

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 15 MPa, determine the optimal geometry of a column 100 metres tall made of concrete to support a mass of 1250 tonnes at its top. Hint: think of the shape of the CN tower
There's no textbook for this course, so we're expected to use common sense to answer these questions...
$$\sigma = \frac{F}{A}=\frac{\pi D^2/4\cdot H\cdot W+1250\cdot 10^3 \cdot 9.81}{\pi D^2/4} $$
W=25 Kn/m, and putting all the information in, I get diameter is 1.1m. Is that right, and I was told that the radius is different at the top, not sure what to do.

Hey Rido! (Smile)

It appears you have already made the assumption that the support is a massive cylinder.
Suppose we make no assumptions about the shape yet, and just assume a cross section of $A$.
And suppose we do not make the assumption that the cross section is constant, but instead is a function of height. That is $A=A(y)$.

What formula would yet get for the stress at height $y$? (Wondering)

And since we want to use a minimal amount of material, suppose this stress is equal to the maximum stress at each $y$.
What can you deduce from that?