Macaulay's method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay's technique is very convenient for cases of discontinuous and/or discrete loading. Typically partial uniformly distributed loads (u.d.l.) and uniformly varying loads (u.v.l.) over the span and a number of concentrated loads are conveniently handled using this technique. The first English language description of the method was by William Macaulay. The actual approach appears to have been developed by Clebsch in 1862. Macaulay's method has been generalized for Euler-Bernoulli beams with axial compression, to Timoshenko beams, to elastic foundations, and to problems in which the bending and shear stiffness changes discontinuously in a beam.
Method The starting point is the relation from Euler-Bernoulli beam theory
± E I d 2 w d x 2 = M {\displaystyle \pm EI{\dfrac {d^{2}w}{dx^{2}}}=M}
Where w {\displaystyle w} is the deflection and M {\displaystyle M} is the bending moment. This equation is simpler than the fourth-order beam equation and can be integrated twice to find w {\displaystyle w} if the value of M {\displaystyle M} as a function of x {\displaystyle x} is known. For general loadings, M {\displaystyle M} can be expressed in the form
M = M 1 ( x ) + P 1 ⟨ x − a 1 ⟩ + P 2 ⟨ x − a 2 ⟩ + P 3 ⟨ x − a 3 ⟩ + … {\displaystyle M=M_{1}(x)+P_{1}\langle x-a_{1}\rangle +P_{2}\langle x-a_{2}\rangle +P_{3}\langle x-a_{3}\rangle +\dots }
where the quantities P i ⟨ x − a i ⟩ {\displaystyle P_{i}\langle x-a_{i}\rangle } represent the bending moments due to point loads and the quantity ⟨ x − a i ⟩ {\displaystyle \langle x-a_{i}\rangle } is a Macaulay bracket defined as
⟨ x − a i ⟩ = { 0 i f x < a i x − a i i f x > a i {\displaystyle \langle x-a_{i}\rangle ={\begin{cases}0&\mathrm {if} ~x<a_{i}\\x-a_{i}&\mathrm {if} ~x>a_{i}\end{cases}}}
Ordinarily, when integrating P ( x − a ) {\displaystyle P(x-a)} we get
∫ P ( x − a ) d x = P [ x 2 2 − a x ] + C {\displaystyle \int P(x-a)~dx=P\left[{\cfrac {x^{2}}{2}}-ax\right]+C}
However, when integrating expressions containing Macaulay brackets, we have
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