Euler–Bernoulli beam theory

Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case for small deflections of a beam that are subjected to lateral loads only. It is thus a special case of Timoshenko beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.the corresponding Euler–Lagrange equation is Q C = P a L {displaystyle Q_{C}={ frac {Pa}{L}}} at x = L 2 − b 2 3 {displaystyle x={sqrt { frac {L^{2}-b^{2}}{3}}}} (b) Linearly distributed load with maximum q0 Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case for small deflections of a beam that are subjected to lateral loads only. It is thus a special case of Timoshenko beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution. Additional analysis tools have been developed such as plate theory and finite element analysis, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering. Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. Da Vinci lacked Hooke's law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made. The Bernoulli beam is named after Jacob Bernoulli, who made the significant discoveries. Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750.At the time, science and engineering were generally seen as very distinct fields, and there was considerable doubt that a mathematical product of academia could be trusted for practical safety applications. Bridges and buildings continued to be designed by precedent until the late 19th century, when the Eiffel Tower and Ferris wheel demonstrated the validity of the theory on large scales. The Euler–Bernoulli equation describes the relationship between the beam's deflection and the applied load: The curve w ( x ) {displaystyle w(x)} describes the deflection of the beam in the z {displaystyle z} direction at some position x {displaystyle x} (recall that the beam is modeled as a one-dimensional object). q {displaystyle q} is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of x {displaystyle x} , w {displaystyle w} , or other variables. E {displaystyle E} is the elastic modulus and I {displaystyle I} is the second moment of area of the beam's cross-section. I {displaystyle I} must be calculated with respect to the axis which passes through the centroid of the cross-section and which is perpendicular to the applied loading. Explicitly, for a beam whose axis is oriented along x with a loading along z, the beam's cross-section is in the yz plane, and the relevant second moment of area is where it is assumed that the centroid of the cross-section occurs at y = z = 0. Often, the product E I {displaystyle EI} (known as the flexural rigidity) is a constant, so that This equation, describing the deflection of a uniform, static beam, is used widely in engineering practice. Tabulated expressions for the deflection w {displaystyle w} for common beam configurations can be found in engineering handbooks. For more complicated situations, the deflection can be determined by solving the Euler–Bernoulli equation using techniques such as 'direct integration', 'Macaulay's method', 'moment area method, 'conjugate beam method', 'the principle of virtual work', 'Castigliano's method', 'flexibility method', 'slope deflection method', 'moment distribution method', or 'direct stiffness method'.