What is the Euler-Bernoulli beam equation?

The Euler-Bernoulli beam equation is a mathematical model used in engineering to calculate the load-carrying capacity and deflection of beams. First enunciated around 1750, it simplifies the linear theory of elasticity for beams experiencing small deflections and lateral loads. By applying calculus and boundary conditions, it predicts internal shear forces, bending moments, and structural behavior.

What we know

• Euler-Bernoulli beam theory is a simplification of the linear theory of elasticity that calculates load-carrying capacity and deflection in beams. [2]
• The theory states that shear force is the cumulative sum of applied loads, bending moment is the sum of shear forces, and deflection is the fourth integral of applied loads. [2]
• The model is limited to small deflections of beams subjected strictly to lateral loads causing elastic bending, and can be evaluated using the Euler-Bernoulli beam equation alongside I-beam geometry. [2, 5] · medium confidence
• It was first enunciated around 1750 but did not see large-scale application until the construction of the Eiffel Tower and Ferris wheel in the late 19th century. [2]

Where to go next

• compared withHow does Timoshenko beam theory differ from Euler-Bernoulli?
• a case in pointWhy are I-beams shaped the way they are?
• whoWhat role did Leonhard Euler play in mathematics?
• how we knowHow did the Eiffel Tower drive structural engineering forward?
• the bigger pictureWhat is the linear theory of elasticity?
• an unexpected connectionHow did a 1750 beam theory enable the Second Industrial Revolution?Although first enunciated around 1750, the theory was not widely applied until late-19th-century marvels like the Eiffel Tower proved its utility, subsequently becoming a cornerstone of modern industrial engineering.

Sources

• [1]Euler–Bernoulli beam theory (Wikidata Q805099) · CC0 1.0
• [2]Euler–Bernoulli beam theory · CC BY-SA 4.0
• [3]List of topics named after Leonhard Euler · CC BY-SA 4.0
• [4]Daniel Bernoulli · CC BY-SA 4.0
• [5]I-beam · CC BY-SA 4.0