What Is a Beam Equation?

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  • Written By: C.B. Fox
  • Edited By: Susan Barwick
  • Last Modified Date: 14 October 2019
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A beam equation is any mathematical equation used to describe the behavior of beams when they are placed under stress. The equations come out of beam theory, which was first developed in the 1700s. Scientists and engineers use beam equations to predict how much a beam will be displaced when a force is applied to a section of it. There are often many variables in beam equations, and a knowledge of calculus is needed to solve them.

Though the notable Renaissance-era scientists, Leonardo da Vinci and Galileo Galilei, had both attempted to mathematically describe the properties of beams using a beam equation, it wasn't until the middle of the 18th century that scientists first developed beam theory. Once the equations had been formulated, it took another hundred years for engineers to trust the mathematics of beam theory enough to put them into practice. Beam theory is sometimes referred to as Euler-Bernoulli beam theory, after the 18th century scientists, Leonhard Euler and Daniel Bernoulli. The Ferris wheel and the Eiffel tower, both of which were created in the 19th century, were the first large structures to utilize the beam equation.


Modern scientists and engineers use beam theory to predict the behavior of beams in many different situations. A beam equation may be used to predict how far a beam will be displaced or bent when a section of the beam is subjected to a certain amount of force. These equations are particularly useful for determining how much weight a beam can bear without bending so far that the integrity of a structure is compromised. There are also beam equations to describe stress on a beam, both from the force of another object acting on it and from any displacement in the beam itself. These equations are used to determine whether a beam could be in danger of breaking.

There are many different variables when working with a beam equation. Beams that are attached at one end behave differently than beams attached at both ends. The effect of a stress or weight is different depending on where it acts on the beam. Large and small beams may also react to stress in different ways. Given all these variables, and that many of them are expressed as coordinates, a sophisticated level of mathematical knowledge is needed to solve a beam equation. The equations in beam theory build upon the principles of calculus.


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