In my last blog entry, I began a discussion of an article by Santiago Huerta, Galileo was Wrong! the Geometrical Design of Masonry Arches, Nexus Network Journal, Volume 8, No. 2, 2006. The link to this article is worth taking a look at; Dr. Huerta uses many excellent illustrations to demonstrate his salient points.
Dr. Huerta provides some valuable insight into the engineering analysis of masonry domes and arches. He argues that the proportional design of arches, as used by master masons of antiquity, has provided a tool for arch construction which has proven successful and is more accurate than the approach first described by Galileo in his famous paper “Dialogues Concerning Two New Sciences” of 1638.
If we look at the components that comprise a masonry arch, known as voussoirs, and perform an analysis of their thrusting forces under gravity, the resulting thrust line analysis provides critical insight into the strength of masonry arches. I briefly referred to this method of analysis in an earlier blog entry while discussing Gothic arches, using the illustration below.
The thrust lines may be drawn to represent the gravitational forces acting on the individual voussoirs within the arch. As long as these thrust lines are within the wall thickness of the arch, the structure is stable, and will remain in a static state of equilibrium.
Master masons of antiquity, in using their geometrical approach (as discussed in my last entry) consistently provide a structure wherein the thrust line analysis yields a stable structure in equilibrium. This analysis remains consistent even when the structure is scaled up, or made much larger. This scale ability is in direct contradiction to Galileo’s “Square Cube Law” which states that as scale increases, the design must account for the fact that while cross sectional area of material increases as a squared function, the volume (and hence mass) increases as a cubed function. This relates directly to the concept of Allometry, or the relationship between size and shape.
This insight –and refutation of Galileo- has some profound implications for analysis and design of masonry arches. Thrust line analysis is different than other methods of stress analysis within a structure. One common method of stress analysis today is Finite Element Analysis, as performed by students who did work on my masonry system and provided their own Finite Element Analysis, as shown here several entries ago. A thrust line analysis is superior to Finite Element Analysis in providing specific tools and insight into masonry arch construction.
In performing a thrust line analysis on a masonry arch, it becomes obvious that if a circular arch (or barrel vault) or hemisphere (or dome) is built, the thrusting forces (pushing out) are greatest toward the bottom of the vault, or dome, or hemisphere. This is accommodated by simply making the wall thicker, so as to keep the thrust lines within the (thicker) wall. Taking this approach a step further, if the arch, or dome, or hemisphere is truncated at the base, or taken as a smaller section which is less than a full hemisphere, then the entire arch wall may be made substantially thinner: because the thrust lines no longer go to the bottom of a full semi-circle (where they splay out) and can be kept within the wall thickness of a thinner wall. The key to performing this arch truncation in a structurally sound manner is to provide a thick abutment at the base of the dome section, so that thrusting forces are resolved here. This insight allows for arches and domes to be made substantially thinner.
The insight of thrust line analysis is ultimately a succinct and quantified summary of the reasons for building arches as catenary arches, as discussed earlier in this blog.
Another important insight provided by thrust force analysis is the effect of adding weight (or loading) to parts of a masonry arch in order to affect the thrust force lines occurring within the arch. Adding weight to the top of an arch changes the direction of thrust force lines, and keeps them more toward the center of the masonry arch wall. In other words, adding weight to the top of an arch can make the arch stronger. This is somewhat counter-intuitive and is a valuable insight. Below is an image of a cupola I built atop one of my domes, this cupola weighs around 6 tons (12,000 pounds!).
I have really just begun to look at the implications of this insight. It is fascinating that master masons of antiquity obtained this knowledge through their empirical experience, rather than through any theoretical knowledge. In the sense of “trial and error” their approach was one of experimentation, and their approach is essentially scientific: simply empirical as opposed to the theoretical approach of Galileo.
There is yet more to this, and we’ll continue this investigation next time.