From displacement-based VIV prediction to internal-force assessment of ultra-slender risers: A two-degree-of-freedom mixed Timoshenko wake-oscillator framework

Semi-empirical wake-oscillator models efficiently predict vortex-induced vibration (VIV), but many beam formulations eliminate sectional rotations and reconstruct internal forces from displacement derivatives. This study develops a two-degree-of-freedom mixed Timoshenko beam–wake oscillator framework for circular cylinders in non-uniform currents. In-line and cross-flow displacements and rotations are solved as primary variables, enabling consistent recovery of bending moment, shear force, and beam-based stress indicators from the displacement–rotation fields. Pinned, semi-rigid, and clamped supports are represented by an elastic-boundary formulation, and vibration-induced axial-tension feedback is introduced through centreline stretching. Validation uses a tensioned riser with L / D ≈ 470 and an ultra-slender cylinder with L / D = 2000 . A dry forced-response benchmark against ANSYS BEAM188 verifies the structural load-to-internal-force recovery, whereas the coupled VIV internal-force predictions are evaluated as relative screening indicators; grid refinement shows that displacement RMS convergence alone can under-resolve bending and shear fluctuations. The mean additional axial tension reaches 310.7 N , about 1.27 N top , with 97.7% generated by in-line deformation. Boundary stiffness strongly changes end-region internal-force hotspots while weakly affecting global displacement RMS. A same-load Euler–Bernoulli/mixed Timoshenko diagnostic shows moderate global displacement differences but localised travelling-band differences in structural demand. The framework extends reduced-order VIV prediction toward internal-force-oriented assessment and relative cyclic stress-demand screening.