Freixial Argente dos Santos, Hugo AlexandreSilberschmidt, V. V.2026-05-112026-05-112026-05Santos, H. A. F. A., & Silberschmidt (2026). Finite element analysis of non-uniform functionally graded multi-cracked Timoshenko beams using an equilibrium-based formulation. European Journal of Mechanics – A/Solids, 117, 1-14. https://doi.org/10.1016/j.euromechsol.2025.1060080997-7538http://hdl.handle.net/10400.21/22870A novel finite element formulation is introduced for the static analysis of non-uniform functionally graded multi-cracked Timoshenko beams with small deformations. The cracks, assumed to remain open, are modelled using the so-called discrete spring approach, in which Dirac delta generalized functions are introduced into the bending flexibility of the beams. The formulation is derived on the basis of a complementary variational approach that involves only the elements' shear forces and bending moments as the fundamental unknown fields. The corresponding element flexibility matrix is obtained in closed-form, with the crack contributions explicitly separated from the standard bending and shear flexibility terms. The numerical solutions produced by the formulation are strictly equilibrated, i.e., they satisfy all equilibrium conditions of the associated boundary-value problem in strong form. The effectiveness and accuracy of the formulation are numerically assessed through its application to several benchmark problems. The obtained results are analysed and compared, where possible, to exact (or reference) solutions and solutions given by the standard displacement-based finite element formulation, clearly illustrating the capability of the formulation to deliver highly accurate results for both thin and thick beams, even on meshes with only a few degrees-of-freedom.engFGM beamsNon-uniform cross-sectionsTimoshenko theoryCracksFinite element methodEquilibrium-based formulationresearch article10.1016/j.euromechsol.2025.1060081873-7285