Fourier Series Based on the Deflection Equation of a Simple Beam Bearing the Linear Load

Beginning with the total potential energy functional of the beam bearing the distributed load, the differential equation of the beam deflection curve is obtained using the variational method. The Fourier series of the equation of deflection curve for the simple beam carrying the linear load is given and the equation of the deflection curve for the simple beam is generalized. The generalized equation is expanded into the corresponding Fourier series and a series of summations of the infinite series are obtained. It is found that the results are related to Bernoulli numbers and π. The relations among the coefficients of the beam, Bernoulli numbers and Euler numbers are found, and the relative mathematical formulas are advanced.