## Thursday, August 11, 2011

### Arches at Ninety Degrees

Back in 2000, I had an insight into vaulted arches.  I had been developing triangular interlocking block as a masonry system, and had devised of a method for making cylinders from these triangular blocks.

A cylindrical section can be turned horizontally to create a vaulted arch, like a roman or circular or barrel vault.

People are largely used to living in a square-cornered building or house.  There is something fundamental to the human psyche that looks to straight walls and square corners within a living space as a standard which sets people at ease.  Domes and round buildings make up a tiny fraction of habitable structures: almost all buildings have straight walls and square corners.  As you read this now, look around you at the building you are in.  I'd be willing to bet that it has straight walls and square corners.  If not, you are in a very tiny minority, living in an "alternative" structure.

It is a challenge to make a vaulted arch roof system sit atop a square cornered building.  In my own masonry approach, which uses triangular block to assemble a cylinder or arch, the triangles can be assembled to provide a helical or spiral edge.  A helix or spiral has translation and rotation, and the 'helicity' or angle of 'spiralness' can be varied.  Think of stretching a slinky: the spiral angle of translation and rotation can be varied, from shallow to steep angles of helicity.

If the translation versus rotation of a given cylindrical section is equal to the radius of the cylinder versus 90 degrees, then this helicicty can be placed atop a right-angled base.  It is thus possible to place vaulted arches atop a right-angled structure.  The right angle can be a "turn-in" as shown below, where the angle between walls is simply 90 degrees, as shown below. (taken from one of my patents)

Below is shown a structure where the vaulted arches are shown as a "turn out" where the walls meet at 270 degrees.  In the case of both a "turn-in" and a "turn-out" a gap is created between the helical edges of the abutting arches where they meet at a corner.  This gap is satisfied by a larger spherical section.  Specifically, if the arches are taken as having a raius of 1.0, then the larger spherical section that fills the gap has a radius of 1.5  (If you'd like to see these images better, just click on them to see a larger view).
This sort of arrangement provides extensive design flexibility.  One, two, three or four vaulted arches can meet at right angles to each other, merging seamlessly into a larger dome or spherical section.

This arrangement creates some very interesting possibilities far outside the realm of masonry.  Could this arrangement have something to do with the structure of DNA and the "magic" of reproduction?  We'll take a look at this next time, including a discussion of centromeres and telomeres.

1. I seek your advice on thickness of the wall for hemispherical shaped 2storey house. Would single skin of trapizum (don't know a batter name)shaped interlocking bricks suffice? Thanks for your interesting articles

2. As with any engineering question, the answer is often: "it depends." How strong is your material? How dense? Assuming manufactured concrete block, 8 inches thick should suffice. You can use a thinner wall if your structure is not a full hemisphere, but you must resolve the thrusting forces at the base by having a substantial anchor. I discussed this somewhat more here: http://masonrydesign.blogspot.com/2010/07/lessons-learned-from-galileos-mistake.html
and here: http://masonrydesign.blogspot.com/2010/07/more-on-galileo-and-master-masons.html

3. Thanks for your reply and all interesting articles you wrote. The reason behind hemispherical shape is opportunity to use the same size 'trapizium' block throwout entire construction. House would have large round opening at the top for round chimney and skylight and all covered with earth

4. Thanks for your reply and all interesting articles you wrote. The reason behind hemispherical shape is opportunity to use the same size 'trapizium' block throwout entire construction. House would have large round opening at the top for round chimney and skylight and all covered with earth

5. ..and a few windows around. The reason behind covering it with earth is to anchor the entire house, gain some insulation and heat retention. How to calculate minimum wall thickness where radius of the house is 7m and skylight radius is 2m?

6. It is smart to cover the dome with earth, this will actually make it stronger by keeping the thrust force lines within the wall thickness. "Wall thickness" can be thought of in different ways: (1) the thickness between inside and outside of the dome (2) The thickness of the trapezoidal ("trapizium") masonry unit. To help you visualize this, take a look at the ancient Roman Pantheon:

Note that there is more stress at the bottom of the dome, requiring thicker walls toward the bottom; less stress toward the top, meaning thinner walls. Here the actual trapezoidal unit has thinner walls toward the top.

Beyond this, the formula for stress in the shell of a dome of constant wall thickness increases as a reciprocal cosine function (not to be confused with inverse cosine function). Here's a decent discussion and some drawings. You may want to register (I did).

There are two different types of forces in a dome: hoop forces and meridional forces. Meridional forces push 'down' and hoop forces push 'out.' (compression and tension)

In terms of calculating your minimum thickness, you must determine an appropriate safety factor. I generally use a factor of around ten. In other words, if your calculated strength is 0ne (1) [whatever your units are: e.g., psi, etc.] then with a safety factor of ten (10) your walls should have a strength of ten.

My email is: roberts.peter01@gmail.com

Thanks for your interest and good luck!

7. There has been small misunderstanding. The blocks in the Roman Pantheon are getting smaller toward the top indeed, but I'm proposing use of trapezoidal shaped blocks of THE SAME SIZE in the whole dome (by the way I'm polish and in our language word for the term house is 'dom')By knowing radius of the dome and thickness (depth I mean rather then width) of the block I could mass produce it.

8. Yes, I understand how you want to build with mass-produced blocks. This is what I've done also. See this entry for a photo:
http://masonrydesign.blogspot.com/2010/03/introduction.html

The thickness of the blocks can vary from top to bottom (which I did NOT do here) while using the same mold, just changing the core.

In my design, the abutting edge of the blocks is trapezoidal, but the inside & outside face of the block is triangular. Is this what you plan?

It is impossible to give a precise answer to your question of wall thickness without knowing: (1) strength of material (2) size of sphere from which dome is made (3) safety factor (4) density of material.

Please note that the size of the sphere from which the dome is made might be different from the span of the building.

I have a powerpoint presentation I could send you which might be helpful. I do not send spam or advertisements, I simply offer to help.

Please let me know if I can be of further assistance, I am happy to help.

Your English is excellent my Polish friend!

Peter