Assembling a self-supporting masonry dome

Here are the basic steps to assemble a first frequency truncated icosahedron from 'pent' and 'hex' concrete block, as provided by Spherical Block. Trace out a circle of radius 3 ft. 1 inch.  Arrange the block in a five-fold, or pentagonal pattern, one pent, two hex, [repeat 5X]. Arrange all blocks with tips pointing up. 

Next, place blocks with tips pointing down, two pent, one hex, [repeat 5X]. FRP, Fiber Reinforced Plastic rebar is used here, #3, or 3/8 inch diameter. Rebar is 8 ft. 10 inches, and goes from center hex block and is easily bent  past two pent and into the next center hex, as shown. 

Additional rebar is provided horizontally, as shown. Length is 41 inches. Rebar is secured with zip ties, to help align the structure.

Hex block are placed, tip pointing up, as shown.  

Because this is a first frequency structure, and is made from block designed for a second frequency structure, wooden shims were used during this dry-stack assembly.  This gave the blocks the increased 'wedge' required for first order arrangement. When using mortar, the mortar will be tapered for a first frequency dome; thicker outside.

Additional rebar are placed vertically, as shown, attached with zip ties. These are also 41 inches long. Additional pent blocks are placed, as shown

Two more hex blocks are placed, tips down, as shown.

An additional course of hex block are added, as shown.

Finally, 5 pent blocks are added to the top and final course. All of these block edges would be in close alignment if this were mortared together. Assembly occurs without any additional support scaffolding or centering. This first frequency dome has an outer diameter of around 8 ft.

The first frequency test was easily disassembled.  A second frequency dome is now being assembled.

Here are the same basic steps, except that the radius is doubled and 4 times as many block are used. This second frequency dome has an outer diameter of around 16 feet. 2 pent, 4 hex, 5X, etc., tips up.

And so on. It is all self-supporting as it is assembled. Larger domes of higher frequencies can also be made, all from the same block.

A new engineering model for a new block

Contemporary engineering analysis of masonry arches provides a model which is not adequate for analysis of the masonry system I’ve been describing on this blog (dual inverse mirror plane, or ‘dimp’).  A new model is required to analyze this triangular interlocking system, which I shall attempt to describe.

The currently accepted engineering model makes three assumptions about masonry arches.   (1)  Masonry units have no tensile strength (2) Masonry units are infinitely strong in compression (3) Blocks (or voussoirs) never slide against each other.  An arch modeled on these 3 assumptions is then viewed in cross section, and a catenary thrust force line is imposed on the wall thickness of the arch.  If the thrust force line touches or exits the wall thickness, then a hinge is formed at that point (between two adjacent blocks or voussoirs) and the arch will buckle and collapse.  If a large force is applied to the arch, the thrust force line will eventually touch or exit the inside (intrados) or outside (extrados) of the masonry arch, and failure will result in a hinging mechanism which causes the arch to buckle and collapse.

The dimp design can employ a tensile element, like a wire or cable within the wall thickness of the block.  This feature gives the arch some tensile strength.  When a large force is applied to this arch, the tensile action of the cable or wire counters this force and keeps the imaginary thrust force line more toward the center of the arch thickness.  In addition to this tensile containment, another feature of the dimp comes in to play.

A large force applied to a dimp arch will first be contained by some of the tensile web, woven as great circle arcs.  Instead of hinges forming when the thrust force line touches the intrados or the extrados, conjugate shearing occurs (as described here).  Control joints allow block faces to slide against each other; they are actually designed to.  This deformation is a strain (movement) resulting from excessive stress (applied force).  The strain relieves the stress, and when the applied force is removed, the structure returns to its original state.  The forces which restore a deformed arch are from gravity and the tensile elements.  There is of course a limit to an applied force, beyond which a dimp arch will collapse, but it is greater than that of a conventional arch constructed from voussoirs of the same thickness.  

Thus the currently accepted method of engineering analysis for masonry arches does not appear to work for the dimp design.  First, an arch made of dimp blocks has tensile strength.  Second, the blocks move (slide) against each other.  Finally, instead of a hinging mechanism there is a conjugate shearing mechanism between blocks.  It is a whole different model.

I am currently working toward a computer model to reflect this different engineering analysis.   I hope to have it available to post here eventually.