The Bridge of Khazad-dûm:
A structural analysis of a bridge of Dwarven engineering*
A structural analysis of a bridge of Dwarven engineering*
I don't know which is more impressive--that an Ivy League professor actually approved the project proposal or that I found someone in the class who was willing to go along with my wacky idea. As I recall, the professor was game from the start. I consulted her at office hours before writing the proposal, since I wanted at least some assurance that the proposal had a hope of being approved, and she was enthusiastic about the idea (my good grades in the class up to that point may have helped convince her that I could pull it off). My friend in the class who had originally agreed to be my partner, and who liked the LotR movies and was initially excited about my idea, backed out before the proposal was due saying, "I want to do this project on something that I can tell an interviewer about down the road--you know?--impress them, saying, 'I analyzed this important structure' or something." When she suggested a traffic light support structure, it was clear our partnering for this project was not going to work out. Luckily, I managed to find another acquaintance in the class who said, yes, she's a big fan of the LotR movies and is still looking for a partner. I owe her a lot, for keeping my dream alive. In reality, she got a good deal; we scored an A+ on the paper. And we had some fun, watched Fellowship of the Ring as "research", and came away with a good story to tell.
So how does one make an engineering term paper out of the Bridge of Khazad-dûm? We started by estimating the dimensions of the bridge. Tolkien, being the thorough author he is, specifically says that the bridge is 50 feet long. However, knowing that we had more to go on overall (curvature, width, depth, material) with the bridge depicted in the movie, we decided to defer to the film and based all our estimates on the bridge's appearance in the movie. The beauty of this project being on an imaginary bridge, of course, was that our estimates only had to be just that--our professor wasn't going to take off points because the Bridge of Khazad-dûm was actually one foot wider than we thought. Anyway, approximating Gandalf's height and shoulder width and using those as units of measurement, we estimated that the bridge was 80 feet long, 3.5 feet wide, and 6 feet thick in the center and 10 feet thick at its supports. With these dimensions, we constructed equations for the top and bottom edge of the bridge, assuming a slight parabolic curvature.
Top edge: y = -0.00125(x - 40)^2 + 12
Bottom edge: y = -0.00375(x - 40)^2 + 6
(x goes from 0 to 80 ft)
Next, we had to make an educated guess of the material used to make the bridge. This involved some awkward questioning. I consulted a geologist and a geological engineer (I still remember the subject of the email I sent to this professor I'd never met: "An odd question...") as well as did a little reading on my own. Given the bridge's location in a mountain mine and considering what types of rock would be appropriate for a stable bridge, I concluded that the bridge was made of a rock with properties similar to quartzite. And thus we used the density and maximum compressive, tensile, and shear stress strength of quartzite for our calculations. With the material and dimensions determined, we provided a delightful analysis of the self-weight of the bridge. I won't bore you with the details, but in summary, the bridge was structurally very sound (the Dwarves obviously knew what they were doing).
Next, we delved into the bridge's critical moment** in the story: Gandalf's confrontation with the Balrog. The first question we had to ask for this was How much does a Balrog weigh? By judging the Balrog to have the approximate proportions of a 30-foot gorilla and the density of basalt, I estimated the weight of the Balrog to be around 185,000 pounds (seven times the weight of the largest elephant on record--it is made of rock, remember). Though the Balrog was clearly very hot, considering that the heat did not harm the fellowship members in close proximity and that the bridge was composed of heat-resistant rock, we judged the thermal effects of the Balrog to be negligible.
Armed with the weight of the Balrog and the self-weight and structural properties of the bridge, we went about solving what exactly it was that Gandalf's spell did to the bridge to make it break under the Balrog's weight. Our calculations showed that the bridge would have been able to hold the weight of the Balrog on its own (it would have been sort of amusing if we had found that the bridge was going to break under the Balrog anyway, without Gandalf casting any spell). So what kind of failure did the bridge experience after Gandalf cast his spell? A stone bridge could easily suffer a tensile failure. Picture a heavy rock on a flimsy shelf: the shelf bends downwards, squeezing the top surface of the shelf (compressive stress) while stretching the bottom surface of the shelf (tensile stress). The same happens when a load is placed on a stone beam, though the bending may not be visible, and while stone can take a lot of compression, it is not strong against tension. However, this type of failure would have been more likely to lead to a total collapse of the bridge and threaten Gandalf's footing as well as the Balrog's. What appeared to happen in the movie was a shear break--a section of rock cleanly breaking and sliding straight down from the adjacent rock. So Gandalf must have wisely chosen to weaken the shear strength of the rock. To make a long story short, we found that Gandalf's spell could have lowered the maximum allowable shear stress of the quartzite directly in front of him to under 65 pounds per square inch, causing a shear collapse under the Balrog when it stepped forward on the bridge. You... shall not... pass!!!
We concluded our paper with an analysis of the cantilevered remains of the broken bridge (it would still stand) and a summary of our findings.
And there you have it--more than you ever wanted to know about the structure of the Bridge of Khazad-dûm. In later years, when I'd become a disillusioned engineering upperclassman bitter about my choice of major (I switched into engineering? What was I thinking?!), I could always look back fondly on this class and this project and remember, Oh yeah, that's why I chose this major. I have no regrets about that class. And to the friend that dumped me as a partner because she wanted a project she could boast about to interviewers: In an interview for the job that lead me to my current career, when the interviewer asked me, "Can you give me an example of something unique and creative you've done, or a problem you've solved in a particularly creative way?" I proudly answered, "In my core engineering class on the Mechanics of Solids, I wrote a paper on the bridge in The Lord of the Rings!"
*The subtitle kind of cracks me up (I think my partner may have come up with it). It's so plain and straightforward and dull, but at the same time totally absurd.
**Sort of an engineering pun!! (and not one I'd originally intended.) Yeah, I'm a total nerd.