Fiction v. Engineering: Part 1, The Wall, a Structural Analysis (From Game of Thrones and A Song of Ice and Fire)

This is going to be the first in a number of posts that look at a number of structures, vehicles, weapons, and other of technologies from fiction in terms of whether they could exist in our real, not fictional world. First up: the Wall, from the book series A Song of Ice and Fire by George R.R. Martin, and its TV adaption Game of Thrones (cue the epic music).

For those who have been living under a rock for the past few years not read the books or watched the show, here is an attempt to describe the Wall without too many plot specific details: it is a giant wall of ice and magic that protects the people living in the Seven Kingdoms, which are south of the wall, from the people (refereed to as Wildlings or Freefolk) and/or ice zombies (White Walkers or Others) living north of the Wall. The closest real world examples would be Hadrian's Wall (which protected Roman-occupied England and Wales from the Scottish) and the Great Wall of China (which protected the Chinese from the Mongolians, rather unsuccessfully). But unlike the 16-20 foot tall Hadrian's Wall (1) or the 16-25 foot tall Great Wall (2), the Wall is 700 feet tall, with a width of "enough for a dozen armored knights to ride abreast," which I am going to estimate, using the 4 ft 8.5 in width of Roman roads designed to handle 2 harnessed sources as a guide, as 28.25 feet (3). So much, much larger. But could the Wall stand (i.e. not collapse under its own weight)  much like it's real life counter parts?

Before I start, I'm going to establish a number of assumptions:

1. Ignoring the effects or air pressure and wind. Air pressure would be negligible in terms of other forces. And while wind could provide a torque on the wall that would cause structural issues, my goal is to do a simple, back of envelope calculation for feasibility on a structure of that height.

2. Uniform properties.

3. Objects on the Wall having no weight. Much like air pressure, the weight of objects (and people) would be small compared to the weight of the call itself.

4. Ignoring plastic flow. This is a big one. Normally, when a solid near its melting point (such as an ice wall that one can see melting on as described in the books) has force applied to it, it deforms and "flows." That can be combated with one of two options: cooling the solid sufficiently or adding structural support to counter act the force. For ice, the temperature required would be -58 deg F (4). The way I'm interpreting "magic" is to mean that through magical means this refrigeration could be achieved (the magnitude of that magic may be the topic of another post). There is also visual evidence in the show of wooden bracing in the Wall; hence I am going to say plastic flow can neglected due to "magic" refrigeration and bracing. While this runs contrary to the observations of melting sides of the Wall, I am going to assume the Wall has a structural "cold" core and a warmer section that can melt to make climbing more difficult.

5. Gravity in Westeros is the same as that on Earth (32.2 ft/s/s or 9.81 m/s/s). Because humans, animals, and human structure seem to be similar to those that have existed on Earth, it is a good guess.

As you may have noticed, I am using US Customary units. This is for two reasons:1. the measurements given in the books and show are in the related Imperial system, and 2. I'm an American, and often many resources and building conventions I am used to use the Customary System. 

So, let's begin with a sketch of the Wall showing how I am labeling dimensions:

The height, h, is 700 feet and the width at the top, wt, is 28.25 feet. In the books, the angle of the Wall sides (φ) is not specifically stated but said to be great than zero. In the show, the sides appear to be vertical. Because a smaller width that the bottom would mean the stress at the bottom would be higher, I'm going to use a vertical side (φ=0), as if the Wall can stand then, it can stand with angled sides.

Now to do some math. I am doing everything symbolically to start, and in a general form that can apply with a non-zero φ.

Equation 1 is for the width of the base, as seen in the sketch.

Equation 2 shows θ in terms of φ, as seen in the sketch.

(also full disclosure: I will use Greek letters a lot. I figure having to memorize them for joining ΚΚΨ means I should use them for something. Also, they show up in science and engineering for some reason. Something like the Greeks providing many of the earliest Western writings on the subjects.)

Equation 3 shows the volume of the Wall for the arbitrarily long section we are looking at. The length/depth variable ends of cancelling out.

Equation 4 shows the mass of the Wall section in terms of the volume and the density of the Wall, ρ.

Equation 5 shows the weight of the Wall, from the mass and gravitational accelerations (32.2 m/s/s).

Equation 6 shows the area of the base.

Equation 7 shows the stress at the base of the wall, as a result of the force generated by the weight of the Wall. For the Wall to not collapse under its own weight, this stress must be less than that of the yield stress of the material, which for ice is 499.94508 psi (5). 

The factor of safety, in equation 8, is simply how many times the load actually applied is the yield strength. Various industries tend to have different accepted factors of safety, although all greater than one (or else it would fail). In aerospace, the factor of safety is lower, as every ounce counts, while in infrastructure it tends to be higher, as structures need to stand despite corroding over a long lifetime, disasters can be devastating, and weight isn't super important for something standing still.

After some substituting and cancelling out of variables, the final form of the equation for stress is:

For ice ρ=1.7764 slugs/cubic feet. In case you were wondering, slug is a unit of mass that works with the Customary system. It essentially fills the same role as kilogram in the metric/SI system (5).

So, plugging in these values, the stress on the base of the Wall, if made of ice, is 278.0559 psi, which would mean the Wall would hold, with a factor of safety of 1.7980. Which is good. But ice has a few problems. One is the plastic flow issue. And again, it is not the strongest of potential materials. Many kinds of stone and metal are stronger, which might be helpful if you are being attacked by the supernatural... or people with trebuchets.

But what if you could have a stronger version of ice? Something that could stand up to more of a beating, and maybe even doesn't have as low of a temperature to avoid plastic flow. If this sounds too good to be true, you haven't heard of Project Habakkuk, the attempt by the Allies during World War II to build an aircraft carrier out of ice to defend shipping against U-boats (6).

No, seriously, this happened. Due to the shortage of aluminum and steel, and invention of one Geoffrey Pyke of a substance called pykrete, which was a frozen mix of water and wood pulp, it was actively explored by the Allies starting in 1942. Pykrete, with a yield strength of 1099.9662 psi, density of 1.9015 slugs/cubic feet and a higher melting point (and plastic flow occurring 3 deg F rather than -58 deg F for ice), was an improvement over ice and the floating aircraft carrier (with refrigeration, some metal structural supports, and cork insulation) was considered a viable idea. Scale models were in fact built in Canada (5, 6).

Something this ridiculous was not meant to be. Because of the need to keep the pykrete cold and apply structural support, as well as engines and control surfaces to propel the proposed 2.2 ton ship, a pykrete carrier would actually be more expensive than a conventional carrier. And when a full sized ship may have been built in 1944, aircraft had enough range to patrol the Atlantic shipping routes from preexisting bases in Britain and North America, and new bases in Portugal and the Azores (6).

But, since pykrete is solely wood pulp and water, it could feasibly be made by a Westerosi builder. And once we remember the Westerosi also have magic to cool the Wall, a pykrete Wall could be a reasonable proposition. Because of the higher density, the stress is higher,  297.6376 psi, but because the yield strength is higher, the factor of safety is 3.6957. Considering testing for Habakkuk included determining pykrete is, in fact, bulletproof, it would have better performance against the weapons used by the Wildlings and White Walkers against the Wall.

So, the Wall isn't a completely impossible fortification. Impractical compared to conventional structures in the real world (the energy required for cooling would be enormous), but Westeros isn't in the real world. It's in a fictional universe. And within that universe's rules, it could be built. 

In that case, night falls, and now my watch begins...

Sources:

(1) https://en.wikipedia.org/wiki/Hadrian%27s_Wall

(2) https://en.wikipedia.org/wiki/Great_Wall_of_China#Characteristics

(3) http://awoiaf.westeros.org/index.php/Wall

(4) http://courses.washington.edu/me354a/chap8.pdf

(5) https://en.wikipedia.org/wiki/Pykrete

(6) https://en.wikipedia.org/wiki/Project_Habakkuk