## Monday, January 30, 2012

### Interlocking Triangular Block and Shell Theory

In my last entry I introduced shell theory as pertaining to masonry domes.  The essential aspects of this theory include a thin-walled, doubly curved structure.  “Thin walled” is not exceeding one tenth of the radius of the arch or dome.

Much work has been done on Finite Element Analysis of domes and arches.  A couple good examples of research which include literature surveys of this topic include “Limit state analysis of hemispherical domes with finite friction” (D. D'Ayala and C. Casapulla) and “Structural Assessment of Guastavino Domes” (H.S. Atamturkur).  These two academic papers provide a rather complete analysis of FEA methods used by the authors and include a decent look at research by others in the field.

As I stated before, FEA is –in my view- an imperfect method for analyzing masonry domes.  I find Thrust Line Analysis to be a more insightful approach to understanding the forces, stresses, strains, and failure mechanisms within masonry domes.  The visual representation of thrust lines within shell thickness gives a direct representation of stress within the wall thickness, whereas FEA typically use different shades of color for different stress levels over the entire dome.  Excellent work on thrust line analysis in masonry domes has been done by Philippe Block, Thierry Ciblac and John Ochsendorf at Massachusetts Institute of Technology, as described in their paper “Real-time limit analysis of vaulted masonrybuildings.”

A thrust line is a visual representation of the force present in a given block or voussoir within an arch.  If the thrust line is kept within the wall thickness, then the structure is stable and will remain intact (as shown below).  If the thrust line leaves the wall thickness or even touches the wall boundary, then a hinge is created, the structure can buckle and fail or collapse (as shown above).

A masonry shell is strongest if the thrust line is kept at the center of the wall thickness.   The masonry unit, or voussoir, or block is strongest and best able to bear its load if the thrust line is in the middle of the block.

If an individual block moves or slides out of the shell’s curved plane, then the thrust line will be closer to the inside of the block.  This can create a hinge if the thrust line touches the inner surface of the block, possibly resulting in failure and collapse of the dome or arch.  It is critical to keep individual masonry units in the curved surface of the shell.

The symmetry of the dual inverse mirror plane (dimp) blocks as discussed in this blog creates an interlocking feature which prevents blocks from moving outside of the curved shell surface.  Blocks are effectively locked into the tangential plane of an assembled spherical section.

This symmetry also creates a “line-of-sight” exactly in the middle of the block, halfway between the inside and outside of a dome.  This line of sight allows for a wire, or cable, or other tensile element to be woven between adjacent blocks.  This geodesic tensile web within the center of a masonry shell (feature 660 below) serves the purpose of keeping blocks located within the tangential plane of the dome, arch, etc.

Shell theory in a masonry dome analyzes deflection of the shell surface under various loading conditions.   Deflection or movement of the shell weakens the shell because the thrust lines can then reach the inner and outer surface of the masonry wall, resulting in a hinge being created; failure and collapse.

Since the “dimp” blocks are configured with key and keyway symmetry; together with the line-of-sight at the center of the abutting edge of each block which allows a woven tensile element, all blocks are effectively prevented from deflecting or moving outside of the tangential curved surface of the masonry shell.  If the blocks cannot move out of the shell, they are much less prone to failure or collapse.

1. Fantastic! I didn't understand very well. Do you mean you have created one kind element for build sphere? Can I build sphere if I have only several equal elements?

1. Yes, it is possible to build a sphere from one type of element; if it is an icosahedron or dodecahedron, or trapezoidal hexecontahedron, or something similar. Some of these are shown here: http://masonrydesign.blogspot.com/2012/02/tension-cables-in-masonry-dome-or.html

2. Any plans on taking these blocks into production?

1. Yes Syed, we're having molds made. Hope to be in production within the year. I've already run test molds on a block machine; they performed very well. Thanks for your interest.

3. Peter, the last post I see indicates that you may have produced molds. Is this now available?

1. Hello Pro Gadget Fix, We have had several types of molds made, and are currently having the latest design for our molds made. In order for us to commercialize this technology, we are undergoing an evaluation by the International Code Council- Evaluation Services. This process is expected to take around 2.5 years. Following this evaluation, we expect to have our technology available as a product in accordance with the International Building Code (IBC). ICC writes the IBC. This will allow us to sell this technology as a product globally. These things take time! (and money!) Thank you for your interest, keep following the progress here, it's coming.

4. i dont see how its possible to get the last block piece in place. as the under and over places in the previous block would be prevented from fitting the last blocks over and under places.

5. There is a way, if you want a complete dome. Leave a gap in the bottom course, build up, around and over this gap, then the final block goes up into the bottom course last. More commonly though, a gap or an 'oculus' is left at the top, for a cupola, or venting, etc. where several block are left out, at the very top of the dome.