A WFP-Squares model for prediction of wave force on bridge piles: Theory, implementation and validation

Extreme wave loading on vertical piles is commonly estimated either by semi-empirical formulations or by high-fidelity CFD and SPH methods. However, a quantitative relationship between local wave run-up and structural load remains insufficiently established in the former, whereas the latter remains prohibitively expensive for repeated design-stage evaluation. In this study, a WFP-Squares (Wave Force of Piles) model is proposed, in which iSquares serves as the underlying flow solver and the total wave force is reconstructed as the superposition of a run-up governed hydrostatic component and a near-pile velocity governed hydrodynamic component. Both the run-up difference and the representative near-pile velocity are extracted from the same iSquares flow-field outputs, ensuring internal consistency in force reconstruction. The identified vertical distribution function exhibits a stable piecewise pattern with the still-water level as the interface, showing a linear decay above the still-water level up to the maximum run-up elevation and a nonlinear decrease below it toward the bed. Within the examined parameter space, the hydrodynamic contribution, quantified by R d , accounts for a relatively small proportion, with peak R d values ranging from 2.63% to 14.10% among the investigated cross-sections. Using the adopted run-up reference and calibrated coefficients, the absolute peak-force errors for cross-sections M1-M5 remain within 10%. These findings indicate that, under the tested isolated-pile configurations and wave conditions, the WFP-Squares model provides a physically interpretable and computationally efficient reduced-order framework for repeated wave-force evaluation.