Concrete Cone: Maximum Height Given Radius at Base

I don't know of one. There is an iterative method where a deflected shape is assumed for a proposed configuration. The cone would be divided into equal lengths and the masses placed at the joints, the moment due to the deflection (pΔ) calculated. New deflected shape calculated from these moments, and newer momentsts calculated.... repeat until there is no significant change.

I suspect though, that you will have more trouble with vortex shedding than with Euler buckling with differing diameters and wind velocity increasing with height.

Thank you. I would be mastering this from scratch, so was hoping that there exist some elementary pre-canned results.

My understanding of vortex shedding is that it can be averted by adding appropriate aerodynamic structures, at a fairly low cost in weight. If you suspect that optimism to be misguided, please say so.

1. Admittedly, of the few billions it would cost, I am currently short by a few billions. But please allow an academic-interest semi-theoretical question, even on an engineering bulletin board.

2. Office towers are more difficult. They require lots of empty space to hold people and plumbing; they require elevator shafts to get those people to the top and back again; and then those same people become litigiously unwell if the thing sways too much. Plus fire escapes, windows, cool-looking entrances, cables and more. Happily for my budget this question is about a chimney.

Sorry: should have marked my original comment off topic. I'm just wondering: a cylinder is clearly not the optimum shape, but, is a cone?

A "concave" shape like the Eiffel tower is clearly best for a material with high strength to weight ratio, and, conversely a "convex mound" is the best you can manage with sand. I have no idea how to determine the optimal shape when the constraints are just height and cost.

"… a cylinder is clearly not the optimum shape, but, is a cone?": probably not, but it's probably nearer to the optimum than a cylinder, and for my approximate purposes, that would be sufficient.

Obviously the optimum shape depends on the properties of the material: concrete withstands compressive forces well, whereas Eiffel's steel has a higher relative tensile strength. For a shape filled with concrete the optimum shape is likely to be wider at the base than at the top. Let's call that, as a first order approximation, a cone (or perhaps a truncated cone).

But how wide would it need to be at the base?

For a first run, I would use the formula, but working at increasing distances from the top to find the diameter; that is, rearrange the formula so that distance from the top is the input and diameter is the output. There are conservative and liberal offsets in this, each level would think it was the same diameter all the way to the top although it reduces, but it would also assume a rigid base, not knowing that the base is further down, albeit with increasing diameter.

They were trying methods to break up vortex shedding before I retired, but they were not sure that they worked in real life (outside the wind tunnel), they could not be sure that the tower had seen, and survived, a circumstance where it would have failed in the absence of the protection; I was thinking of the calculation, with wind layers in more than one direction.

"… rearrange the formula so that distance from the top is the input and diameter is the output": the formula quoted in the first post has r ∝ h^3/2. But this can't be used to determine the shape, as anti-buckling strength needs to be maximal in the middle, not at the base.

Or I have misunderstood you?

For a freestanding tower, the greatest moment is at the bottom.

The radius of curvature is least where the moment is greatest, therefor the cone shape.

Divide the tower into say ten pieces, H/10. Find the diameter required for the top H/10 section only, then the top H/5 section, then the top 3H/10 section... finish with the full H length. I don't know how to calculate the critical buckling though except for the iterative process I mentioned. I only suggest this as a slight improvement on the cone.

On thinking a little, I don't think this will be significant when compared with the stresses due to wind loads and the PΔ effect from wind load deflection.

GA

How's this for a first shot at a model. I've ignored wind; the potential difficulty of "joining" one section to the next, and, the fact that there needs to be a hole up the middle. I haven't done exactly what you said: Ive started with the top section then worked out how much I need to chop off the next section to compensate for the weight of the top one etc etc.

C9 = =C4*C1/(C2*C3)
C3 = =C12+0.5 (change this to change the iterations and thinnest section)
D12 = =($C$9*C13^2)^(1/3)
E13 = =D13
E14 = =D14-(C13^2/C14^2)*D13
F13 = =F12+E13

Back to Randall's model, above.This appears to be modelling the cone as a serious of towers, higher towers assuming that all lower towers are rigid and stationary, and lower towers assuming that all higher towers are a mass rigidly connected to the current tower.

Please, is that understanding correct?

Yes.

Except that it comes out at the more optimum shape.

If only compression is considered then it can be assumed that the request is that compression stress is constant all over the column.

If we consider 2 sections (A and A + dA) separated by dz (z is the vertical coordinate) then the weight of this slice should load the additional area of section (dA) by no more than σ [N/m^2^] allowable stress.

The equation is thus: dA*σ = (A+dA/2)*ρ*g*dz where ρ is the specific mass[kg/m^3^] and g the earth gravitational acceleration[m/s^2^].

Separating the variables and neglecting the terms in dA*dz being small→ dA/A=(ρ*g/σ)*dz which has as solution ln(A)=(ρ*g/σ)*z+Co. From the boundary value at z=0 A=Ao, Z considered from the top we determine Co=ln(Ao) so that the final equation is : A=Ao* exp((ρ*g/σ)*z). Ao is the smallest area at top of chimney. For a given height "H" the base section will have an area A=Ao*exp(C1*H) with C1=ρ*g/σ.

It is possible also to start with A and Ao and define H.

The form is near conical if concrete is high resistance and as low as possible as specific mass.

The problem is MUCH more complex if the wind effect is also considered, the equation becomes a complex integral equation much more difficult to solve than the simple differential equation above. However the stability of a high structure is more function of wind and possible seismic loads. If wind can be influenced (reduced but not concealed as effect, in this respect the Taipei tower with profile and dynamic stabilization based on oscillating masses is a very good example) the effect of an earthquake cannot be avoided so that the higher the structure the more this load becomes the determinant factor.

Taipei tower showing structure for wind effect reduction.

Spherical swinging mass made of thick steel plates

Inertial mass suspended on cables for free movements and stabilized by hydraulic dampers.

If wind is also considered the profile will be similar to the Eiffel Tower. The difference will be the higher specific area on which the wind will apply its pressure. The Eiffel Tower has a structure designed to minimize the wind total pressure.

Compression stress: "compression stress is constant all over the column". So take a disc of top area A and thickness ∂z, bottom thickness A+∂A, with a volume of tower above the disc equalling V. We want to be constant the above volume per unit area.

So (V)/(A) = (V+A×∂z)/(A+∂A)
⇒ V×A+V×∂A = A×V+A²×∂z
⇒ V×∂A = A²×∂z
⇒ ∂z/∂A = V/A²
⇒ z = –V/A + constant

Reversing the direction of z
⇒ r ∝ √z.

Not quite sure whence comes the Exp[…].

Thank you: compression stress fixed. But would that buckle?

Buckling occurs because nothing is perfect, material is not completely homogenious, geometry is not perfect, etc. so we assume perfection and then give it a sideways poke to introduce a PΔ and then see how large P needs to be to buckle the thing.

V is also a function of z and A your integration is not correct.

Compression stress: "compression stress is constant all over the column". So take a disc of top area A and thickness ∂z, bottom thickness A+∂A, with a volume of tower above the disc equalling V. We want to be constant the above volume per unit area.

So (V)/(A) = (V+A×∂z)/(A+∂A)
⇒ V×A+V×∂A = A×V+A²×∂z
⇒ V×∂A = A²×∂z
⇒ ∂z/∂A = V/A²

⇒ z = –V/A + constant

Your error is that you consider "V" as constant and V= ∫A.dz (∂ is for a partial derivative!)

So that you do NOT separate the variables and cannot proceed to the integration as you did. Elementary mathematics!

What I did was to write the DIFFERENTIAL equation → dV*ρ*g= dA*σ and with dV= (A+1/2*dA)*dz = A*dz+ 1/2*dA*dz as mentioned I neglected the second term since the product dA*dz<<Adz and obtained the equation→ A*dz*ρ*g= dA*σ the rest is in my comment.

So as sorry as I am your equation is wrong and also the results.

Now with respect to buckling. It could be possible to use same approach as for the concentrated load and since the conicity is not very important is the σ-Value is high make an approximation with a cylindrical column.

I shall have a look and let you know.

Apologies: my error, and an elementary one at that. If A[z] ∝ ∫A[z]∂z, then (of course!) A[z] ∝ Exp[z]. Sorry.

Back to the problem. Exp[z] is not a helpful shape, as it quickly grows rather wide.

Consider compression. Young's modulus of concrete ≈ 30GPa = 30×10^9 kg/(m·s²). Density of concrete is ≈ 2400 Kg/m³, so a column of height z and cross-section 1m² exerts a downward force of z×23500kg/m²/s². So z = (30×10^9 kg/(m·s²)) / (23500kg/m²/s²) ≈ 1.27×10^6 m. Hence a tower of concrete <1000km tall won't have compressive failure. That is more than sufficiently tall.

Hence buckling is the constraint, not compressive failure (the buckling perhaps caused by a small earthquake, as you suggest).

Edit: posts seem to have overlapped in time.

I'm a little confused here, the formula for h_critical is for elastic buckling and is not stress related, I don't see any way that it will control the design, I think you should find a design that supports the dead + wind loads and then check that for dead + seismic and then check the result for buckling if you wish.

If this were my job, I would be tempted to build a minimum diameter chimney with a structural steel lateral support system; imagine the Eiffel Tower with a chimney running up through the middle, magnified many times over of course. Some of the flame stacks at oil refineries are built this way although the chimneys are of steel pipe.

Again a simple error: the stress limit for concrete does not go over 800 MPa so that for compression the limit is h<σ/(ρ*g)=8E8/(2.5E3*9.81)≈ 35 km not 1000! From the other point of view buckling has a limit of ≈ 70*D for the conditions above so that the D corresponding to 35 km would be 500m. Stress limit and Young modulus are not really related.

A last remark an exponential function grows fast or slow depending how big is the exponent, if it is small the exp is near to a cone. ρ*g/σ = 2.5E3*9.81/8E8 ≈ 3E-5 and e^x=1+x/1!+...+x^n/n! if x<<1 then e^x≈1+x.

I took the number for concrete from the Wikipedia page on Young's modulus, which gives "High-strength concrete (under compression)" as 30 GPa.

Of course, Wikipedia may well be wrong.

No, the text is correct, YOU mix notions the loading limit is not the Young modulus of elasticity. Those are 2 (two) DIFFERENT notions!

In the text they give the value for the modulus of Young in compression but this not means that you can moad up this value. The concrete will be destroyed by shear stresses a lot earlier! If you look at properties of concrete you will find values for specific mass ( 1800...3000 kg/m^3) AND stress limits (up to 800 MPa).

I have the feeling that you are not very familiar with notions related to material behavior.

For materials as concrete which cannot be loaded in tension the Young modulus is specified for compression only.

> I have the feeling that you are not very familiar with notions related to material behavior.

Correct. Thank you for the clarity of correction.

In fact I did not try to find a solution for buckling because passingto green gave already one but if you insist. If I consider a cylindrical column with a constant area, ration m=Di/De=0.5 and with the own weight as load I come to the equation:

l [m]=0.7886*(De^2*(1+m^2)*E/(ρ*g))^(1/3) De[m] ρ [kg/m^3^] g=9.81 [m/s^2^]

With ρ=1800kg/m^3 and E= 2 E10 N/m^2 the result is l≈70*De^(2/3)

Based on indications in Roark.

I maintain that the limit will NOT be buckling but stresses due to an earth quake. the most dangerous would be an horizontal wave.

Progress: The Shape of the Tallest Column and a small correction thereof.

Also www.springerlink.com/content/r0144745m67m6plm/ linking to www.springerlink.com/content/r0144745m67m6plm/fulltext.pdf, which says that:

"This paper discusses the problem of achieving the weight distribution of a column, along its height, that optimizes the buckling load of the column, taking into consideration the column self-weight. The field equations of optimality and stability were generated through variational principles and solved by applying the method of perturbation. Results obtained have shown that for the optimal buckling load of a plane-tapered column, the distribution of the cross-sectional area along the column axis follows a perturbed parabolic law. The optimal column has a buckling load which is 21.58% larger than that of the corresponding uniform column with the same volume and height."

So the optimal shape gives same volume and height as if concrete were 21.58% stronger. That's less than the excess that should be built into the system anyway, so my back-of-the-envelope calculation isn't materially different to the cylinder.

Thank you everybody for your help and suggestions.

HI, All.

I Think, this problem can be easy solved by numerical method, for example in ANSYS. If You want, I can create ANSYS input file for this problem. How I know, buckling length for isotropic cone is not exist in engineer's books of general purpose.
In ANSYS this problem may be solved by solid FE or beam FE. I solve reinforced concrete problem by solid with material nonlinearity, but main factor of your problem is geometrical nonlinearity (or deformed shape) and I can't solve this problem with material and geometrical nonlinearity.
I can solve it with account geometrical nonlinearity for isotropic material and calculate safety factor and critical force.

You excuse me for my English. I am from Russia and my English is very bad, but I want discuss and collaborate with civil engineers from other country for any problem.