Worn Stone Stairs Read as Mechanical Records of Centuries of Footsteps

A worn stone staircase is usually treated as a blemish, a sign of age that conservators note and visitors step over without a second thought. A new study argues that those shallow bowls and rounded edges are something far more interesting: measurable mechanical records of how people actually used the stairs, sometimes over nearly a thousand years. Researchers in China have built a framework that turns three-dimensional scans of stone treads into quantitative statements about contact stress, walking direction, and even how long a staircase has been in service.

The work, published in the journal Mechanical Sciences, was carried out by Jun Yao, Changdong Yin and colleagues at Wuhu Vocational Technical University, Anhui Normal University, Anhui Polytechnic University and East China University of Science and Technology. Their approach combines point-cloud surface reconstruction, polynomial curve fitting, finite-element contact simulation and an extended version of a classical wear equation. When tested on four real staircases with very different ages and traffic levels, the inverse model estimated service ages with a mean relative accuracy of about 94.4 percent.

The field sites were all in Wuhu, in China’s Anhui Province. One staircase came from Guangji Temple, a historic Buddhist sanctuary, and is estimated to have been built around 1065 CE, receiving roughly 200 pedestrian transits per day today. A second, from the Ming Dynasty Wuhu Ancient City complex, dates to about 1575 CE and now sees around 50 transits daily. Two modern staircases in Zheshan Park, both built in 2005, provided a sharp contrast in use intensity: one carries roughly 800 people per day, the other only about 30. The authors are careful to note that these daily counts are short-term field observations, useful as approximate descriptors of current traffic rather than reconstructions of historical pedestrian flow.

Capturing the wear itself required a non-destructive scanning protocol built around an Intel RealSense D455 depth camera. Raw point clouds were denoised, cropped to the region of interest and aligned into a unified coordinate system, with surface normals estimated for reconstruction and sparse or occluded regions interpolated where necessary. Each tread was then divided into transverse sampling intervals, and depth values within each interval were averaged to produce stable longitudinal profiles. Wear depth was defined in a standardized way: the vertical depression of the present surface relative to an estimated unworn reference plane, with larger values indicating stronger material loss and anomalously small values hinting at preserved geometry, local repair or material heterogeneity.

To convert those irregular surfaces into continuous, comparable curves, the team fitted polynomial functions to each longitudinal profile, selecting the polynomial degree objectively with the Akaike Information Criterion rather than by eye. For the representative staircase from Guangji Temple, a 20th-degree polynomial emerged as the preferred compromise between fitting accuracy and model complexity. The fitted profiles captured central concavity, edge rounding and asymmetric deformation, the signature of repeated loading along a relatively stable central path consistent with long-term concentrated use at a religious site’s entrance. Minor oscillations appeared near some boundaries where the point cloud was sparse or edges were affected by spalling, a limitation the authors say should be weighed against point-cloud quality and field observations.

The mechanical interpretation came from finite-element analysis. The stone was modeled as a homogeneous isotropic elastic medium with a Young’s modulus of 60 gigapascals, a Poisson’s ratio of 0.22 and a density of 2,700 kilograms per cubic meter, values adopted as representative for comparative analysis rather than site-specific calibrated properties. The static foot–tread contact model showed that stress is far from uniform across a tread: it peaks in the central contact region and falls off toward the sides, matching the bowl-shaped depressions seen in the fitted profiles. The static model produced a single-step wear index of 4.29 × 10⁻⁷ and, when used to back out daily pedestrian frequencies, captured the general order of loading intensity across the four sites, though it did not reproduce observed counts consistently for every staircase. The authors treat this as a mechanical plausibility test rather than a traffic census, since wear accumulates over decades under variable conditions while the field counts reflect only a short measurement window.

Dynamic simulations added a directional dimension. During ascent, foot contact and push-off concentrate stress near the anterior region and nosing of the tread; during descent, the body’s downward and forward motion shifts loading toward the posterior and middle regions. A multi-person model, built by combining finite-element analysis with Monte Carlo sampling of randomly placed foot contacts, produced a broader, less localized stress field than single-person directional loading. These distinct stress signatures gave the team a mechanical basis for reading wear patterns as records of movement mode.

To test that idea on real data, the researchers applied K-means clustering to derivative features extracted from the fitted profiles, which encode the transitions between concave, convex and flat regions of the worn surface. The classification agreed with field-observed use patterns in all four cases: the Guangji Temple staircase and the busy Zheshan Park staircase were identified as upward single-person loading cases, the Wuhu Ancient City staircase as a downward single-person case, and the lightly used second park staircase as a universal multi-person case. The authors stress that the sample is small and the result should be seen as a field validation of feasibility, not a universal accuracy estimate, since mixed circulation, crowding, material heterogeneity and surface damage can all modify the wear gradient field.

The centerpiece of the framework is an improved Archard wear model, extended from the classical 1953 formulation to handle heritage conditions. It incorporates time-dependent material degradation through decay functions for the wear coefficient and hardness, plus an environmental correction factor driven by temperature and humidity treated as periodic variables. Contact pressure from a single pedestrian is modeled as a dynamic quantity modulated by walking frequency, with the effective contact area estimated using Hertzian contact theory because the real contact area between a shoe sole and rough stone is smaller than the apparent one. A diffusion term allows localized wear to spread to adjacent surface regions, and an adaptive time-step strategy keeps the numerical scheme stable. By back-extrapolating the simulated wear evolution from the observed morphology to the initial unworn state, the model estimates service age.

The results were strongest where conditions are best constrained. The two modern staircases showed errors of less than 0.3 years, with individual accuracies of 98.6 and 99.6 percent, while the older cases reached 85.8 and 93.6 percent. The largest deviation came at Guangji Temple, where the model underestimated the reference age by roughly 136.5 years, a gap the authors attribute to substantial variation in loading frequency, environmental exposure, material condition and use history over long periods. They therefore present the inverted age as a probabilistic mechanical estimate rather than deterministic dating. Robustness checks were encouraging: predicted service ages stayed within a variation range of less than 5 percent when the material decay coefficient was perturbed across four increasingly accelerated aging scenarios, and 200 Monte Carlo simulations with 5 percent artificial noise added to the wear depth field kept overall inversion errors below 4 percent. The team cautions that the noise test represents random measurement disturbance only, not systematic errors such as biased depth measurement or poor registration, and that highly porous, strongly anisotropic or salt-affected stones may show stronger sensitivity to degradation parameters. Beyond dating, the framework can also flag anomalies: a step whose inferred wear time deviates from its neighbors may signal repair, replacement, local damage or material heterogeneity, though the authors emphasize this is a screening indicator requiring corroboration from material comparison and historical records. Future work, they write, should expand the dataset to other stone types and environmental regimes and incorporate laboratory abrasion calibration to improve generality.

Subject of Research: A computational framework that infers pedestrian wear patterns and service age from 3D scans of stone stair treads

Article Title: Mechanically informed three-dimensional wear inversion of stone stair treads under pedestrian loading

Article References: Yao, J., Yin, C., Fei, B., Xu, Z., Wu, Y., Liu, Q., & Ye, D. (2026). Mechanically informed three-dimensional wear inversion of stone stair treads under pedestrian loading. Mechanical Sciences, 17(2), 951-965. https://doi.org/10.5194/ms-17-951-2026

Image Credits: AI Generated

DOI: 10.5194/ms-17-951-2026

Keywords: stone stairs, wear modeling, finite-element analysis, 3D scanning, point cloud, Archard wear model, heritage structures, contact mechanics, service age inversion, pedestrian loading, K-means clustering, Monte Carlo simulation