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## Wall Collapse

NOTE: This is a more detailed explanation of wall collapse than what is found on the page on Natural Arch Formation.

Consider the forces acting on any one point in a tall, thin wall of rock. These include the force of gravity, the force supplied by the cementing grains that hold the rock together (its tensile strength), and the normal force all the grains supply to resist compression (its compressive strength). The first of these forces, gravity, acts in the vertical direction. The other two do not have a preferred direction.

Obviously, these forces exactly balance each other at every point in the wall. Otherwise, the wall would not stay together. Any unbalanced forces would tend to pull the wall apart. But a key point is that the magnitude of these forces must vary from point to point in the wall in a systematic way in order to maintain this balance.

The vertical component of the normal force must be greater at the bottom of the wall than at the top of the wall to resist the added weight of the rock above it. Indeed, this component of force must increase smoothly from top to bottom as the weight above increases. This balance of greater vertical forces near the bottom of the wall means that the rock there is more constrained from moving vertically than it is horizontally. But that is not the case for rock at the top of the wall where the forces balancing each other vertically are of equal magnitude to the forces balancing each other horizontally.

Now consider what happens when temperature variation causes the rock to expand and contract. At the top of the wall, expansion and contraction can happen equally in any direction. At the bottom of the wall, however, expansion and contraction will be mostly horizontal movement because of the weight of rock above.

But now we must consider another dimension. There are two horizontal directions, one perpendicular to the wall and one parallel to it. Since the rock wall is thin, there is less normal force to resist movement along an axis perpendicular to the wall than there is along an axis parallel to it. Thus, horizontal expansion and contraction will be less constrained (greater thermal movement will occur) perpendicular to the wall than parallel to it. This is even truer if the wall widens into massive abutments at its ends, a frequent occurrence.

Therefore, temperature fluctuation will cause a tall, thin wall of rock to expand and contract mostly in the direction perpendicular to itself, i.e., the wall will become thicker as the temperature rises and thinner as it gets colder. Moreover, this tendency will be much greater at the bottom of the wall than at the top. Heat will cause the base of the wall to swell more than the top. Similarly, cold will cause the base to constrict more than the top. If one likens a cycle of expansion and contraction to breathing, it could be said that the base of the wall takes much deeper breaths than the top does during each daily temperature cycle.

Because the magnitude of thermal flexing varies with height, the rock wall is impacted in two important ways. The first impact is the preferential weakening of the rock at its base and at its thinnest point, usually the central part of the wall. The second impact is that stress relief produces a regular pattern of fractures throughout the wall. Although these two impacts occur together and are coupled, let's first consider them separately.

What does it mean to say that the rock at the base of the wall has become weaker? The rock is weaker because the lattice of fused small grains that cements the rock together has become weaker. Micro-fractures in this lattice have formed at random throughout the rock in this area. These micro-fractures are the direct result of the stress due to the constrained thermal flexing described above. They weaken the cement and hence lower both the rock's tensile and compressive strengths. Rock with lower tensile strength will be more susceptible to many processes of erosion. Rock with lower compressive strength will provide less support for the weight of the rock above.

One might guess that the rock would slump at this point, but that is not what happens. Instead, the support for the weight is redistributed along force lines in the shape of an inverted catenary. A catenary is the shape a rope takes when suspended freely from its two ends. An inverted catenary is that shape turned upside-down. It's the shape of an arch. The weight of the wall of rock is now supported by normal forces acting along force lines that have taken on the shape of an arch.

Now let's look at the second impact. The variable thermal flexing creates a strain or stress throughout the wall. If it is of sufficient magnitude, this stress is relieved by a very specific three dimensional pattern of fractures in the rock. In the plane of the wall, these fractures would look like a series of congruent inverted catenaries. In other words, the fractures align themselves with the pattern of weight distribution. In the third dimension, however, the fractures are a series of planes parallel to the face of the wall.

At this point we can see that the stage is set for a rapid cycle of erosion at the base of the wall. This is most dramatic at the wall's thinnest point, usually at its mid point. Stress relief exfoliation (and freeze expansion if water is present) will cause sheets of rock to slough off the face of the wall at its base. Because of the fracture pattern described above, these sheets will have a specific shape - inverted catenary on the top and sides, flat on the bottom. Of course, this will leave an arch shaped cavity or alcove in the wall. If the wall is exposed on both sides, the sheets will slough off (exfoliate) from both sides. If the wall is fully occluded by a cliff on one side, this will act as a brace and exfoliation will only occur on the exposed side.

In both cases, the wall becomes thinner at that point (the base, usually near the center). This acts to accelerate the erosion cycle. Eventually, the process will create an opening through the base of the wall. A classic semicircular aperture has formed. Or, if the thinnest point of the wall is near one of its ends (where it becomes an abutment) rather than its center, a half-semicircular aperture will result.

Finally, since the weight of the wall over this opening has been redistributed in an inverted catenary shape, gravity will compress the rock along these force lines. Compression strengthening will occur, making the remaining wall more resistant to erosion. A natural arch has formed and will begin to age under other processes of erosion.

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