Dam engineering and mathematics

Summary: Mathematics is vital to the design, monitoring, maintenance, and safety of dams.

Dams are embankments across a waterway for control of water or for water storage; they have served many functions in societies throughout history. The earliest dams were primarily used for irrigation and as a water source for livestock. Today, smaller dams provide water for livestock, fish and wildlife habitat, and recreation. Larger dams can provide flood control in places below sea level, like New Orleans and the Netherlands; municipal and industrial water supply; irrigation for crops; hydroelectric power; commercial navigation; and recreation. They are typically earthen dams, concrete structures, or some combination. Older dams were sometimes made of timber, masonry, or steel. Mathematicians and engineers investigate many aspects of the construction and maintenance of dams using geometry, trigonometry, and stochastic and limit-state analyses. For instance, Boris Galerkin, who had degrees in applied mathematics and mechanics, studied stress in dams, and Pelageia Polubarinova, who had a degree in mathematics, contributed to the theory of seepage flow of groundwater through porous materials that included earth dams. Some well-known dams are the Itaipú Dam in Brazil and Paraguay, the Hoover Dam in the United States, the Aswan Dam in Egypt, and the Dneproges Dam in the Ukraine.

Considerations for building a dam must take into account both positive and negative impacts. There are a variety of benefits of a dam that are closely related to its uses—providing water supply, flood control, hydroelectric power, and navigation. Hydroelectric power provides an important source of electrical power around the world. Commercial navigation through river systems provides efficient and economical transportation of agricultural products and commercial goods. Many dams that control flood plains provide farmers with an increased crop yield because land that would once have been flooded is now controlled upstream by the dam. Negatively, some dams may hinder fish movement; for example, along some streams, salmon are not able to get back to their native spawning areas because of the dam. Additionally, dams affect the natural order of a stream—its sediment load and flooding characteristics.

Purposes and Design

Dams are constructed with a definite purpose in mind based on the function(s) they are to serve. Dams are built to control watershed areas (all the area upstream of the dam, which provides runoff to the structure). Engineers use a variety of mathematics skills as they plan, design, construct, and operate a dam. During the planning stage, engineers work with sponsors to scope out the needs and develop a basic design for the structure including design issues such as location, height, and base flow of the structure. Base flow is calculated with the formula Q =v × A where Q is the base flow rate, v is the velocity of water, and A is the area. Another important part of the planning stage is determining the economic feasibility of building the dam by calculating a benefit-to-cost ratio. Using a mathematical model, both the benefits of the dam over its life and the total cost of building and maintaining the dam are calculated. Ideally, for the construction of a dam to be feasible, the benefit-to-cost ratio needs to be greater than 1.

As a part of the design process, engineers must create detailed blueprints for the structure and an accompanying cost sheet that includes items such as quantities or volumes of a variety of materials (for example, cubic yards of concrete) and the cost of the removal and placement of earthen materials, which can be millions of cubic yards in the case of large dams. During the construction of the dam, the blueprints must be followed with precision and detail to ensure the integrity of the dam. Once the dam is constructed, regular monitoring is important to ensure the most efficient use of the available storage. Engineers monitor the amount of water leaving the dam through its spillway, as well as the amount of water entering the watershed. These inflows and outflows must be balanced in order to maintain storage needs and prevent flooding or low flows in the river downstream.

Safety

A major consideration in the planning, design, construction, and maintenance of any dam is safety. Engineers determine a hazard rating for each dam, with the highest hazard rating dealing with potential loss of human life. A breach in a dam can be catastrophic. A breach in a dam can be caused by a flaw in the design of the structure, extreme rainfall, lack of or poor maintenance of the structure, or a geological occurrence. Regular inspection and maintenance are important to ensure the safety of those downstream from the dam.

Bibliography

Hiltzik, Michael. Colossus: Hoover Dam and the Making of the American Century. New York: Free Press, 2010.

Macy, Christine. Dams. New York: W. W. Norton & Company, 2009.

Prabhu, N. U. Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication. 2nd ed. New York: Springer, 1998.