Catenary reconsidered

Application of catenary structures must be carefully considered for use in different environments.  A few examples worth looking at include:

·        An environment without gravity.  If a structure is built in space, or acts as a satellite, or is built in conditions of very low gravity (like on an asteroid, or the moon, or even a buoyant ball) then the reasons for a catenary structure practically disappear.  Under these conditions a sphere or spherical dome is the optimal structure.

·         Very high external pressure.  If a structure is submerged to any depth, then an outside compressive force acts on the entire structure.  Under these conditions, again we find that a sphere is the strongest and most stable structure.  A catenary structure under great external pressure is weaker than a spherical structure.  (Do you ever crack an egg at the tip? No, you crack it on the weak side.)

·         Extreme loading from high velocity winds.  Such conditions are found in extreme storms, including hurricanes, typhoons, and tornadoes.  Under these conditions, the exposed surface area per unit volume is minimized by using a spherical form.  The profile is further minimized by using only a smaller segmental section of the spherical form, further reducing the profile of the structure.  A woven tensile geodesic web will help blocks resist suction forces in very high winds.

·         Earthquakes.  A catenary arch results from acceleration due to gravity.  In an earthquake, the ground can move in a sudden sideways fashion.  This results in acceleration in a sideways or lateral sense.   If a chain hangs from a rod, and the rod is tipped or inclined away from horizontal, then the catenary changes relative to the rod: the same way thrust force lines in a dome change relative to the horizontal ground movement during an earthquake. 

This situation is as if the arch was built on an inclined surface; the catenary still exists, but it is like a catenary on an inclined surface.  The direction of the inclined surface is relative to the motion of the ground.  The result of this sideways acceleration is that the catenary arch may eventually touch or exit the wall thickness; a hinge is created and the structure will buckle and collapse. 

Catastrophic failure of masonry arches during earthquakes can be prevented by using tensile elements woven into the arches as great circle arcs.  This geodesic tensile web will prevent the creation of hinges, catenary thrust force lines will not exit the wall thickness due to lateral acceleration.   Structural integrity is maintained if the hinges cannot open.   Tension is provided.

Tensile elements woven into a dome will help hold it together during an earthquake.

Galileo's Thirst for Knowledge: Priceless. Quenched: $470 Billion.

Over the past three entries I wrote about Galileo’s mistaken approach toward analyzing masonry domes and arches, where he invoked his Square Cube Law to wrongfully criticize the work of ancient master masons. One of the insights gained from a proper and correct stress analysis, which involves thrust force lines describing a catenary curve within the thickness of an arch wall, is that extra weight, or loading, applied to the outside of an arch actually makes the arch stronger by keeping the thrust force lines closer to the center of the arch wall.

The implications of this correct structural analysis are far-reaching and insightful, for numerous applications of masonry structures: some of which I have already been discussing in this blog.

If a complete sphere is assembled, and this complete sphere is submerged below water, the water applies a load to the outside of the sphere. Water pushes in on the round sphere fairly equally, all the way around the sphere from all directions. This external loading keeps the thrust line forces equally distributed around the entire sphere, and it keeps these thrust line forces located in the center of the wall thickness, resulting in an optimal loading of compressive forces. Any other shape, whether it is a cube, rectangular, elliptical, etc., will not distribute this external loading in an equal, symmetrical manner as a sphere does.

This attribute of a masonry sphere subject to external compressive forces bearing such loading equally and symmetrically about its surface means that a below ground storage tank, built as a sphere, is an ideal configuration for any below ground tank. If a below ground tank is used to store water, then the weight of the water will apply an interior force, or head pressure, against the inside of the sphere, so that this force weakens the sphere and must be countered by an external force. Given that the density of water is 1.0 grams/cubic cm, and that average soils have a density of around between 2 and 3 g/cc, it is obvious that the external forces of the surrounding soil are 2 or 3 times the internal pressure of the water held in the tank. In other words, there is substantially greater compressive force acting on the outside of the sphere from the surrounding soil than there is acting on the inside of the tank by the water stored there.

These examples further illustrate that a masonry sphere used as either a below-ground water storage tank or as a means of desalination, as discussed here and there in this blog, are ideal solutions to the growing global problem of potable water use, storage and procurement. This water problem can be addressed by the existing manufacturing capability of the concrete block industry, using its existing methods, materials, and infrastructure. This can be done in an economical, sustainable and easily implemented manner.

This represents a huge market, and the proposed technology could be a big part of the answer to a pressing problem which is expected to worsen with climate change and the growing needs of humanity. Currently, the size of the market for potable water is estimated at $470 billion. One in eight people around the globe lack access to fresh water; that’s almost one billion people. The solution described here could help address this problem.

To see a completed prototype for water storage, please look here.

More on Galileo and Master Masons

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.