<p>This work explores an archetypal one-dimensional fourth-order model equation governing buckling instabilities in bars, plates, and shells, with a particular emphasis on modulated and localised deformations. A modified multiple-scale asymptotic approach is employed to develop a general theoretical framework for systematically approximating critical loads in terms of a small parameter when buckling patterns are concentrated near a boundary point. Previously established results for cases where localisation occurs at an interior point are shown to align with the methodology adopted here. Necessary conditions for the validity of the derived formulae are provided, and the general theory is illustrated on a small selection of concrete examples. Comparisons with direct numerical simulations provide additional validation of the accuracy of our results.</p>