Rock avalanches, triggered by slope failures and characterised by large-scale and well-graded mixtures, pose severe threats to bridge piers in mountainous regions. Existing methods for estimating impact forces primarily consider bulk flow properties, while the influence of natural particle-size distributions remains poorly understood. To address this, we developed an analytical model that integrates elastoplastic contact theory with statistical particle-size distributions, explicitly capturing how the temporal and spatial overlap of individual particle impulses governs cumulative impact forces. The model was validated using field observations, laboratory experiments, and high-fidelity numerical simulations across different scales. Results reveal that asymmetric particle-size distributions lead to substantial variability in impact forces, particularly in well-graded distributions. Overlapping elastoplastic impulses cause nonlinear force amplification, especially at high impact rates, which can rapidly exceed the dynamic shear capacity of bridge piers and lead to structural collapse. The amplification is further dependent on the waveform shape of individual impact pulses. This method enables identification of critical conditions where impact demand equals or exceeds structural resistance across a multi-parameter space, supporting performance-based bridge design and risk mitigation in avalanche-prone regions.