EulerBernoulliTheorie

The Euler-Bernoulli beam theory, also known as the classical beam theory, is a fundamental concept in solid mechanics used to calculate the load-carrying and deflection characteristics of beams. It was developed by Leonhard Euler and Jakob Bernoulli. The theory makes several simplifying assumptions, most notably that the beam is slender and that plane cross-sections remain plane and perpendicular to the neutral axis after bending. This means shear deformation is neglected.

According to the Euler-Bernoulli theory, the bending stress in a beam is directly proportional to the distance from the neutral axis and the curvature of the beam. The neutral axis is an imaginary line within the beam where the material experiences no tensile or compressive stress during bending. The theory also states that the deflection of the beam is small compared to its length.

The key equation derived from this theory relates the bending moment to the beam's material properties and geometry, specifically its Young's modulus (E) and area moment of inertia (I). The relationship is expressed as M = EI(d^2v/dx^2), where M is the bending moment, E is the Young's modulus, I is the area moment of inertia, and d^2v/dx^2 represents the curvature of the beam. This equation forms the basis for analyzing beam behavior under various loading conditions. While simplified, the Euler-Bernoulli beam theory provides accurate results for many practical engineering applications where slender beams are subjected to relatively small deflections.