Mathematics of street maintenance

Summary: Street maintenance requires planning, preparedness, and risk assessment, all of which involve mathematics.

Stone paved roads date back thousands of years and mathematicians and architects have long investigated ways to lay paving stones. Another connection between mathematics and streets dates to when Hermann Minkowski proposed numerous metric spaces, one of which is referred to in the twenty-first century as “taxicab geometry.” Some streets are laid out on a grid system, leading to mathematical investigations in taxicab geometry. The surface curvature of roads is also mathematically interesting and important in drainage and safety issues. Street maintenance is a combination of services that includes resurfacing of streets and curbs, pothole patching, sweeping, snow removal, and maintenance of drains. Mathematical problems that arise within street maintenance have to do with engineering, applied physics and chemistry, logistics, budgets, and communication.

Types of Streets

Different types of roads call for different maintenance. Civil engineers can use tools such as falling weight deflectometers to measure properties of street coverings—in this case, deformation under dropped weight. A heavily loaded truck can damage the street surface approximately 10,000 times more than a small passenger car. This fact explains why streets with industrial traffic require more frequent maintenance, owners of trucks pay more taxes, and trucks are not allowed on most streets.

There are many materials used to cover streets, and the choice of material provides interesting mathematical optimization problems. For example, rubberized asphalt contains recycled tires, which is an environmental bonus and can reduce the noise of the road by about 10 decibels, which is valuable for nearby homes. However, it can only be laid in certain temperatures. Concrete is more durable than asphalt but is more expensive and harder to repair. Brick and cobblestone coverings do not form potholes and can hold heavy loads. However, they are noisy, they require manual installation and maintenance, and they can damage cars.

Potholes and Fatigue

Most potholes happen because of what is known in the materials science as “fatigue” of the surface. Fatigue occurs when materials are subject to periodic forces, such as heavy cars passing through. Small cracks start to appear, which then aggregate into networks of cracks, which then give way to a pothole. Calculus, differential equations, and statistics models are used to test road surface materials for resistance to fatigue and to predict fatigue’s time through the statistically derived fatigue curves (S-N curves). Cycles of heating and cooling can quickly extend existing cracks and make potholes larger as well as freezing water that has seeped into cracks.

Cleaning

The mathematics of cleaning schedules involves balance among many random variables, such as traffic or seasonal leaves removal. In a typical city, urban streets with heavy pedestrian traffic are swept daily, and other streets are swept every week or two. Statistical data on street use determines where to place garbage cans and how often to empty them, when to send heavy sweep machines for cleaning, and how to avoid disrupting regular street use and events with cleaning activities.

Some street maintenance measures prevent street dirt. Highly visible trash cans can drastically reduce littering. In many communities, residents are invited to participate in street cleaning and maintenance to some degree, from sites where they can report potholes to street cleaning celebrations on holidays or weekends. Birds can be attracted to appropriate places, and dog owners are guided to special parks and runs. Mathematical models behind such measures come from studies of human and animal behavior.

Accidents and disasters—from dust storms to spilled poisons—may require special cleaning activities. Because such events are rare but require special knowledge and equipment, it usually makes sense to maintain tools and specialists for these special events only in large cities and to send teams to smaller places that need help.

Snow Maintenance

Streets under snow require special maintenance, including mechanical removal of the snow by snowplows, snow-blowers, or shovels; inert surface treatment for traction with sand or sawdust; and chemical surface treatment. The mathematics of dealing with snow includes economical and environmental factors. When snow immobilizes traffic, productivity and sales are lost. However, snow-removal measures cost money and take time. In cities where it snows infrequently, it is usually cheaper to wait for the snow to melt rather than to maintain a fleet of removal machines.

Most of the chemical treatment of snow is done with sodium chloride (table salt). Salt makes snow melt at about 10 degrees Fahrenheit less than usual (freezing-point depression). Switzerland uses more than a pound of salt a year for every square yard of its roads. Chemical treatments can damage plants and animals throughout the watershed. Safe amounts of chemicals can be determined based on ecological models. Chemicals also cause vehicle damage and faster road deterioration. These costs are part of the decision of which type of snow maintenance is more economically sound.

Bibliography

Kelly, James, and William Park. The Roadbuilders. Reading, MA: Addison-Wesley, 1973.

Krause, Eugene. Taxicab Geometry: An Adventure in Non-Euclidean Geometry. New York: Dover Publications, 1987.

Perrier, Nathalie, Andrew Langevin, and James Campbell. “A Survey of Models and Algorithms for Winter Road Maintenance. Part IV: Vehicle Routing and Fleet Sizing for Plowing and Snow Disposal.” Computers & Operations Research 34, no. 1 (2007).