Ice wall and creep

According to this article in the Washington Post, a glaciologist, Martin Truffer from the University of Alaska, has written a conference poster about the ice wall in Game of Thrones and how it couldn’t exist. There’s some stress analysis and material science in the article, so I thought worth at quick review.

The stress analysis is simple:

Truffer calculated that the weight of an ice wall about 655 feet high would generate nearly 1.8 megapascals (261 pounds per square inch) of stress at its base.

The compressive stress in the vertical direction at the base of a column (or wall or heap) of material is equal to the product of the unit weight of the material, \(\gamma\) and its height, \(h\). The easiest way to convince yourself that’s the case is to imagine a cube of material sitting on a floor. The pressure the cube exerts on the floor is equal to the weight of the cube divided by the cross-sectional area of the bottom face of the cube. And by Newton’s third law, that’s equal to the pressure the floor exerts back onto the bottom of the cube. Which, by definition, is the compressive stress at the base of the cube.

By definition, the cube weighs \(\gamma a^3\), where \(a\) is the side length of the cube, and the cross-sectional area of the bottom is \(a^2\). So the pressure at the bottom of the cube is \(\gamma a\). Now imagine another cube stacked on top of the first; the pressure at the bottom is now \(2 \gamma a\). With \(n\) cubes stacked up, the pressure at the bottom is \(n \gamma a\), and because the height of the stack is \(h = n a\), the pressure at the bottom is \(\gamma h\).

What’s the unit weight of ice? I don’t have that memorized, but I do have the unit weight or water memorized, even in US Customary units: 62.4 pounds per cubic foot (pcf).1 And I remember that water expands 9% when it freezes, so the unit weight of ice must be

\[\frac{62.4}{1.09} = 57.25 \,\mathrm{pcf}\]

So an ice wall 655 feet high will have a compressive stress at its base of

\[\frac{(57.25 \,\mathrm{\frac{lb}{\,ft^3}})(655 \,\mathrm{ft})}{144 \,\mathrm{\frac{in^2}{ft^2}}} = 260.4 \,\mathrm{\frac{lb}{\,in^2}}\]

which is, as the article says, just under 261 psi or about 1.8 MPa.

While this is a pretty low stress for everyday construction materials, it’s high for ice. Ice flows readily under this kind of stress, and Truffer makes the point that this is why you don’t see 600-foot ice cliffs—the ice would just slump to a lower height with gradual slopes.

This type of material behavior is typically called creep, and it’s something engineers try to avoid when designing structures and equipment.

Typical low-carbon steel, for example, is a great material for buildings and bridges because it doesn’t creep under normal temperatures. But it’s a terrible material to use for steam piping in a boiler. Under high temperatures, regular steel will creep under the stresses imposed by a pipe’s internal pressure, and eventually it will burst open in a failure mode known as creep rupture. Metallurgists get around this problem by alloying the steel with elements like chromium, nickel, and molybdenum, raising the temperature at which the steel can operate without creep rupture. And raising its price.

Concrete creeps at room temperature. This typically isn’t a big deal for reinforced concrete, but it is for prestressed concrete, which gets its strength through a clever trick. Concrete, while pretty strong in compression is ridiculously weak in tension. The ratio of compressive to tensile strength is in the neighborhood of 10:1. This makes concrete a poor material for structural members that get bent, like beams and floor slabs, because bending induces tensile stresses on the convex side of the bent member.

This is where the prestressing trick comes in. By running steel cables (called tendons) through the concrete and stretching them to a high tension before anchoring them to the concrete, we induce compressive stresses in the concrete member before (that’s the pre part) any building loads are applied. When the loads are applied and bend the member, they reduce the pre-existing compression, but don’t cause any tension in the concrete.

Unfortunately, creep makes this clever trick less effective than we’d like. Under the stress from the prestressing tendons, the concrete creeps and gets shorter. That makes the tendons shorter, too, and they lose some of their tension in the process. And as the tension in the tendons goes down, so does the compression in the concrete, making it more likely to go into tension when bent. The design rules for prestressed concrete have provisions that account for this.

As its name implies, creep is usually thought of as a slow process, mainly because engineers avoid conditions that would make it advance quickly. But Prof. Truffer says that’s not the case with the ice wall:

Special magical powers would be necessary to maintain its shape, even for just a few days

  1. Why do I have this memorized? If you use a number regularly for 40 years, you tend to remember it.