Fourier Series Solution for Lateral-Torsional Buckling (LTB) of Laterally Braced Beams

This study investigates I-beams with discrete lateral bracings subjected to uniform end moments. Initially, the derivation of critical moment formulas is conducted, employing both simple harmonic waves and Fourier series based on trigonometric functions as displacement functions. Subsequently, the impact of the number of terms in the displacement functions on both the LTB strength and threshold stiffness is examined, leading to recommendations for an appropriate number of terms. Ultimately, the calculation formulas are simplified. The findings show that, when the beam is “fully braced” (meaning the brace stiffness is sufficient to allow buckling between the braces of the beam), the derived LTB strength expressions coincide, irrespective of whether a simple harmonic wave or Fourier series is adopted as the displacement function. Divergences in LTB strength formulas emerge solely in scenarios of “partially braced,” where the brace is inadequate to permit buckling between braces. The simple harmonic wave form constitutes a special instance within the Fourier series, with the latter’s critical moment results offering a refinement over the former’s results. The precision of these refined results hinges upon the number of terms in the Fourier series, with the optimal number intimately tied to the number of braces. Consequently, this study advocates that in deriving the LTB strength of laterally braced beams via the energy method, the selection of terms in the displacement function ought to be 2 × (the number of brace +1).