Tuesday, February 21, 2012

Catenary domes

The catenary form is key to understanding the design of masonry arches.  As discussed here, here, and here, “catena” is Latin for “chain.”  This word origin serves as a useful tool in analyzing the catenary arch.  This is because a chain hung slack is an exact analogy for a masonry catenary sprung arch, only it’s the opposite.  “Up” in a slack chain segment is “down” in sprung masonry arch; “tension” in a slack chain segment is “compression” in a sprung arch segment.  This fact provides insight by simply hanging a chain (or chains) and observing.

Some architects and designers use the term “funicular” when describing catenary arches.  Funicular comes from the Latin “funis” meaning rope or cable.  In terms of masonry, catenary is more accurate than funicular because the individual links of a chain are analogous to the individual blocks (voussoirs) of an arch; rather than the smooth continuity of a rope or cable.

A small chain will behave exactly as a large chain, in a proportional sense.  This reflects the scaleability of masonry arches, as discussed in earlier entries on Galileo’s wrong application of his Square Cube Law to masonry arches.  This means that application of Galileo’s Square Cube Law to chains is also wrong.  Chains are scaleable and do not need to be redesigned to be made larger.  Small models using small chains have direct application for larger models using larger chains.  Since the chain is a direct representation of the catenary arch, large masonry arch structures can be represented by small chain models.  Antoni Gaudi used this method to model the Sagrada Familia, as shown below.

Thrust force lines are the imaginary lines that indicate where the compressive force in a voussoir or block is located in the thickness of the arch.  In an arch, the thrust lines always describe a catenary curve.  If the thrust lines touch the inside (intrados) of the arch or leave the wall thickness, then a hinge is created and the arch or dome will buckle out and collapse.  If the thrust lines touch the outside (extrados) of the arch or exit the wall thickness, then a hinge is created and the arch or dome will buckle in and collapse.   In any masonry arch that stands, the catenary thrust lines are kept within the wall thickness; if thrust lines touch or leave the wall thickness then the arch collapses.

If thrust lines can be kept within thin walls, then wall thickness can be reduced.  There are different ways to keep the thrust lines within a thin masonry shell:

·         A smaller section of a dome can be used.  This smaller dome section creates an outward thrust or splay which must be contained by either buttressing the outside of the arch, or by having a tensile element inside the arch.  

·         If a half round arch is used, the thrust lines will be either close to, or touching, (or beyond) the intrados at a location in the arch known as the haunch.  By applying an external load to the extrados (outside) of the arch at the haunch, the thrust line is brought back toward the middle of the wall thickness.  By applying an external load (e.g., fill, rubble, etc.) to the outside of the haunch in a round arch, the arch can be kept relatively thin (this is somewhat counter-intuitive: that adding weight makes it stronger).   

·         Wall thickness can be reduced dramatically by simply making the dome as a catenary shell.  That is, instead of building spherical domes, they are made more “pointy” like the small end of an eggshell.  By making a catenary arch, the catenary thrust lines are kept within a very thin wall.

From a masonry perspective, the difficulty in assembling a catenary dome lies in the large number of different unit shapes required to make it.  The advantage of a spherical masonry system lies in the small number of triangular unit shapes needed to build a dome; they are all interchangeable (unlike a catenary dome).  Catenary arches can be made with different profiles, depending on how much slack the chain has.  Selection of the catenary arch which comes closest to a circular arch is advantageous.  By accounting for the slight differences in a true spherical dome and a catenary dome, assembly of a structure which comes very close to a true catenary can be achieved from triangular masonry units.  This is done by varying the thickness of mortar (or gasket material, as described here) and dihedral angles between voussoirs slightly as they are assembled.  This is most easily achieved by using a catenary-shaped form or support or bracing for the voussoirs to be assembled against.

In earlier entries (here and here) I discussed using an inflatable bladder as a form to support masonry units as they are assembled into a dome.  It is a simple modification to change a hemispherical bladder into a catenary-shaped bladder which can be used to support block as they are assembled into a catenary dome.  A dome with the thinnest walls, using the fewest number of unit shapes, in the strongest possible configuration is provided by using this method.

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