Just bricks or blocks of squared stone piled on one another, without wooden or metal supports? Now obviously you could have a simple pile of rubble as high as you wanted. So let’s say it has to be an oblisk with the same taper as the Washington Monument: about 1:25 if my figures are correct*. I believe the Washington Monument uses lead soldier instead of mortar between the stones, so I won’t quibble about an absolute absence of metal. So how high could you go?
( 55 feet wide at base, minus 34 feet wide at top of shaft, equals 21 feet in. Height from ground to pyramidion ~500 feet.)
Look up the density and maximum compression strength of your chosen material.
Since your geometry is known and has only one independent variable, calculate the volume as a function of height, V(h), and the Base Area as a function of height, A(h). Multiply V(h) by the density to get the mass with respect to height, m(h).
The function m(h)/A(h) is the Base Pressure with respect to height, P(h). When P(h) exceeds the maximum compressive strength of your material – or that of the ground you’re building on – your tower will collapse, crack, lean, or otherwise fail.
This approach assumes perfectly cut, smoothly lapped blocks of your material; it ignores imperfections such as cracks; and it assumes that you’re stacking in the orientation most advantageous to any “grain” that exists in your stone.
The compressive strength of average concrete is about 4000 psi. High strength concrete (used for prestressed beams) can run as high as 15,000 psi. Brick can run the same range. So I’d use 15,000 psi as your upper end.
What’ll be the death of your structure will be the horizontal forces. The pyramids have such a shallow slope they don’t need to worry about it. The structure you’re proposing will. Horizontal forces (I should look up average wind pressures but I’m eating lunch, maybe I will later) will result in the structure bending. That’s what the reinforcing is for. Concrete and mortor have very little tensile strength which will result in an unreinforced structure cracking on the tension side. As the crack spreads, you’ll have less area to resist the crushing. Then the structure will just topple over.
WIth a given horizontal force, this would be pretty easy to figure out. Maybe I’ll work on it a little later…
Critical buckling length in a structure of that shape will probably be exceeded well before the material fails due to yield (compressive strength). This problem is very geometry dependent.
OK, the worse case wind loading we use for bridge is 100 psf so I’d use that. From here, you can set up a series of equations:
Wind force = 100 psf * area of a side (assume wind is blowing on one side only)
Downward force = weight of material (use 490 pcf which is concrete) * volume
Downward force cannot exceed 15000 psi. (we’ll ignore factors of safety)
The moment (combined action of the downward force and the horizontal force) about the base must equal zero.
Set up these two equations and solve for your unknowns, the length of a side and height. To make this simpler, I’d assume the obelisk keeps the same slope all the way up to the tip.
Isn’t buckling on the macro scale a result of some point failing compressively at the micro scale, due to incidental moments? If I remember that right, then you can account for sideways forces pretty easily.
Use my “tallest tower” from above as the ideal maximum limit, and start calculating the moment and resulting compressive stress that a 5mph crosswind would put on the lee side of your tower. Better yet, assume a 200mph maximum crosswind (that accounts for the Florida case), hitting squarely against one face. The rotational moment will have a resulting compressive load on the leeward side of your tower. You should be able to express that pressure as a function of height (take wind as a constant, side area as a function of height, and column thickness as a function of height). Ensure that P[sub]wind/sub + P[sub]gravity/sub do not exceed the material’s limits.
tremorviolet is essentially correct: your tower is neither tipping nor moving, so you can assume that the sum of forces and sum of moments are both zero, as in any classical statics problem. Do I smell another engineering major?
I still maintain that your failure will be due to the design exceeding the material’s compressive strength at some point. The mechanism – buckling, side forces, etc. – can probably be accounted for and reduced to a height-dependent variable.
Yep, I’m a structural engineer. I do bridge design tho’, never looked at a tower before.
OK, I worked through this several times and I’m wrong. The weight of the concrete is enough to counteract the wind load. After the tower is around 12’ high, the increased weight more than compensates for the wind load. Probably why this shape was popular for obelisks even if the designers didn’t know it.
If you just consider crushing using 10,000 psi concrete, you could theoretically build a tower 9876’ high.
So I think Fuji Kitakyusho is correct, buckling will control. And the buckling equations are really annoyngly complicated and will involve calculations of the moment of inertia and stuff. (for this weird shape) Bleh. I’m looking in the Amer. Conc. Inst. handbook but most of the calcs assume reinforcing.
The thing to consider in all of this discussion is that I check the structure as if it were built exactly as planned, i.e. all the weight is centered. We really can’t build stuff exactly, there will always be a small amount of error. And as the structure’s height increases, the amount of error increases until there’s enough to cause a non-trivial eccentricity (at this point the weight will no longer be over the exact center but to the side) and, at some point, the structure will fail.
We really don’t know exactly when this will occur so we have equations that have a considerable factor fo safety built in. How did we get the equations? We ran lots and lots of tests to see when things failed and then looked for an equation we could force to fit with lots o’fudge factors. That’s the thing about real world engineering, it’s not really as scientific as you woud like to believe.
Philadelphia’s City Hall is the world’s tallest masonry building if that helps at all, 548 ft if we include the (bronze) William Penn statue at the top, which is also the world’s largest statue on top of a building. It was also the world’s largest city hall until Tokyo’s new one surpassed it. “The weight of the building is borne by stone walls 11 ft thick, rather than steel.”
Huh, Wikopedia has it wrong. The walls at the base are twenty-two feet thick at the base. cite Still impressive tho’. Think how much space is wasted by such huge walls and you can see why using steel reinforcing was such a huge innovation…
In general I see the point about the waste of space. It may be helpful to know that only about 25 years before the building was designed there were massive Protestant nativist vs Catholic immigrant riots (think Gangs of New York ) in the city (once a mob even used an artillery piece) and it is in part designed no doubt as a partial city government fortress if necessary. The ground it sits on was the former Centre Square, a park area, and there was plenty of room to build big.
If you ever get to take a tour (recommended), check out the anti-riot gates outside of the City Council chambers.