Wednesday, October 7, 2009

Trucks and the Fourth Power Rule

There are many hidden subsidies for select American industries. One of the most blatant is the gasoline tax and registration fees for motor vehicles. Both of these are supposed to pay for the expenses of maintaining rights-of-way (roads, tunnels, bridges) and they are supposed to be fair in the usual "Republican" sense: everyone pays his way. I'm sure you've seen the messages on trucks reading something like: "This vehicle pays over $[so many thousands] of dollars in taxes each year." It all seems so impressive and expensive, since we car drivers know that this is far more than we pay. Poor truckers trying to make ends meet!

Well, it turns out that large vehicles are not really paying their fair share. The heavier they are, the worse the discrepancy. The reason is an empirical fact widely known among civil engineers called the "Fourth Power Rule," which I'll simply call FPR.

Take the state and federal gas taxes for example, which are a certain number of pennies per gallon. First of all, these are not automatically adjusted for inflation, so as gas (and other expenses) increase, they are a smaller and smaller percentage of the fuel actually used. But it gets worse. You figure: Wow, a big truck uses lots of gas, so it must really pay a lot in taxes. If we assume the gas used by a vehicle is in rough proportion to its weight (which it is, roughly), then a truck weighing 10 times my car uses 10 times as much fuel, so pays ten times the taxes. OK, still roughly true. The same for registration fees, which are mostly proportional to the weight of the vehicle. (Here in Massachusetts they go up $20 for each thousand pounds). However, the main reason for the gas tax and registration fees is to maintain roads, and here is the catch. If you think that a truck weighing 10 times my car would wear out the roads only 10 times as fast, you'd be very wrong, very wrong. That's what the FPR is all about.

FPR says that, roughly speaking, the rate at which a vehicle destroys a road is proportional not to its weight but to the FOURTH POWER of its weight. This law has to be applied carefully, though. The more the weight is distributed evenly, the less wear. It is customary to divide the weight by the number of axles to normalize this rule. Thus, if my car weighs T tons and has 2 axles, its adjusted weight it T/2. The truck then weighs 10T and has, say, 5 axles. The adjusted ratio is then (10T/5) divided by (T/2), or 4. Since rate of road wear is proportional to the fourth power of this adjusted ratio, the damage due to the truck is roughly 4 to the 4th power or 256 times the damage due to my car. If the trucks owners were truly paying their share, they would have to pay 256 dollars for every dollar I pay. This is not even close to being true in any state I checked.

Why does the FPR hold? The answer is that roads wear out not so much by tires rubbing against them (though this is a factor) but by flexing under load. The more flexing, the greater the wear. But flexing is not a simple function of weight, nor is damage a simple function of flexing. A very rigid road will hardly flex at all under a small weight: it takes a lot of tons to produce any flex in a well-constructed highway. But, once flexing occurs, the results are disastrous. It's a little like a paper clip: you can basically put a few sheets in and out for years without any noticable effect, since the flex is negligible. But even a few dozen real sharp bends and its metal will fatigue, crack and break. In the case of roads, heavy trucks flex them until they start to develop cracks. These cracks increase the flexing even more. The elements, in the form of rain but especially snow and ice, exacerbate these cracks until they widen and cause failure, like an overstressed paper clip. If only passenger cars drove on interstate roads, they would probably last for a century. It's the trucks that kill them.

(In rural Central Maine where I often drive and bike, the state has little money to prepare rigid roads. Their basic roadbeds are not excavated and filled deeply enough, and they are pretty much thin layers of asphalt-like covering over gravel. Several years after repaving, the flexing of logging and other heavy commercial traffic has combined with the Maine winters to make them laced with cracks and often premature pot-holes. In half a dozen years they are at death's door.)

As I said, the FPR is universally accepted among civil and highway engineers, but is ignored by politicians, many of whom listen only to the big truck and gas lobbies. You can read about the FPR in many places (mostly technical); here's one: Fourth Powers and pavement

Next time you read a sign on a truck saying how much tax revenue it generates, remember that, effectively, it is getting a free ride.

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