The core of the proposal is a dual-parameter damage index built from two quantities: the Area Loss Ratio, abbreviated ALR, and the Residual Height Ratio, or RHR. The ALR describes the fraction of the tower’s horizontal cross-sectional area that has been lost as blocks detach and fall away, while the RHR captures how much of the original height remains standing after partial collapse. Together, the two parameters trace a damage trajectory that a single measure cannot. A tower might lose a modest amount of area near its crown while retaining most of its height, or it might shear through at mid-height and shed an entire upper story; the combination of ALR and RHR distinguishes these very different failure modes and allows them to be ranked on a common severity scale.

To calibrate the index, the researchers modeled seven representative masonry pagodas in three dimensions using the discrete element method, a computational technique originally developed for rock mechanics. Unlike continuum finite element models, which treat masonry as a smeared material, the discrete element approach represents the structure as an assembly of individual polyhedral blocks interacting through contacts. That matters for pagodas, because their failure under earthquake shaking is fundamentally a process of blocks cracking loose, sliding, rotating and toppling. The formulation traces back to the distinct element schemes introduced by Peter Cundall and collaborators in the late 1980s, and it allows the simulation to capture progressive detachment and collapse rather than merely the onset of cracking.

Within the numerical models, the team systematically varied the key structural parameters that govern pagoda behavior. Three stood out: the condition of the internal infill, the height-to-width ratio of the tower, and the opening ratio, meaning the proportion of the facade occupied by doorways, windows and arched niches. Each of these features varies widely across the surviving stock of ancient pagodas, and each plausibly changes how the structure responds to shaking. By building models that span this parameter space, the researchers could isolate the influence of each factor on displacement demand and collapse susceptibility, rather than studying a single iconic tower in isolation.

The nonlinear dynamic analyses subjected the seven pagoda models to multiple ground motion records at two hazard levels drawn from Chinese seismic design practice: a frequent earthquake with a peak ground acceleration of 0.07 g, and a rare earthquake with a peak ground acceleration of 0.40 g. The gap between those two levels is enormous, roughly a factor of nearly six in acceleration, and the simulations show how dramatically the damage picture changes across it. Under frequent shaking, the towers largely remain intact, with cracking confined to vulnerable zones around openings. Under rare shaking, the response escalates into block detachment and partial collapse, particularly for the most vulnerable configurations.

The parameter study yielded clear and intuitive trends. Internal infill, the material packed inside the hollow shell of many pagodas, significantly enhances global stiffness and reduces seismic responses, effectively bracing the tower from within. Conversely, increasing the height-to-width ratio amplifies displacement demand at the top of the structure, and larger opening ratios weaken the load-bearing walls and concentrate stress around voids. The most striking results came from hollow-core towers and fully opened configurations under rare earthquakes: these models exhibited pronounced block detachment and partial collapse, with area loss ratios reaching 10 to 20 percent and residual height ratios dropping to approximately 80 percent. In other words, a fifth of the tower’s footprint could be lost and its standing height reduced to four-fifths of the original before the process stabilized.

From the evolution of ALR and RHR across the simulation suite, the team distilled a four-level collapse classification framework that spans the entire damage process, from initial cracking through to global collapse. The lowest level corresponds to the appearance of cracks without meaningful geometric loss, while the intermediate levels capture progressive block detachment and partial sectional failure. The highest level denotes global collapse, the point at which the residual structure can no longer carry its own weight. Because the classification is anchored to measurable geometry rather than subjective inspection categories, two assessors looking at the same damaged tower should, in principle, arrive at the same severity rating, which is precisely what post-earthquake triage requires.

A crucial test for any damage index is whether it works in the messy reality after a real earthquake, when survey teams may have only photographs and rough sketches rather than laser scans. The researchers validated their method against damage observed in masonry pagodas during the 2008 Wenchuan earthquake, the catastrophic event in Sichuan Province that damaged thousands of heritage structures. The comparison demonstrated that the proposed index can reliably quantify collapse severity even when detailed geometric loss data are unavailable, which is a decisive practical advantage. Emergency responders and heritage authorities rarely have the luxury of complete documentation in the days after a major quake, so an index that tolerates incomplete data is far more useful in the field than one that demands it.

The significance of the work extends beyond China’s borders. Masonry towers and pagodas are a global heritage category, from Italian campaniles to Nepalese temple towers, and seismic risk assessments for such structures have proliferated in recent years, using tools ranging from ambient vibration monitoring and Bayesian model updating to limit analysis and shaking table tests. What most of these methods share is a focus on predicting damage before it happens or detecting it through dynamic signatures. The new dual-parameter index addresses a different and complementary question: once damage has occurred, how bad is it, quantitatively? That question drives triage decisions, such as which towers need immediate shoring, which can await scheduled restoration, and which are safe for the public to approach.

The authors suggest the method can serve as a robust and practical tool for post-earthquake damage assessment and for decision making in the conservation and seismic risk mitigation of historic masonry structures. Because ALR and RHR are geometric ratios, they can in principle be estimated from visual inspection, photogrammetry or laser scanning, and they translate directly into the language conservators use when planning interventions. The research was supported by the National Natural Science Foundation of China, the Key Research and Development Program of Shaanxi Province and a doctoral dissertation innovation fund from Xi’an University of Technology. For the custodians of ancient towers standing in one of the world’s most seismically active regions, the study offers something rare: a damage scale designed for their buildings, validated against a real catastrophe, and simple enough to be used when it matters most, in the chaotic hours and days after the ground stops shaking.

Subject of Research: A dual-parameter geometric damage index for quantifying seismic collapse severity in historic masonry pagodas using discrete element modeling

Article Title: A novel dual-parameter method for evaluating seismic collapse severity in masonry pagodas

Article References: Li, M., Lu, J., Wang, D., Jia, X., Wu, X., Mao, D., Yun, Z., & Wang, Z. (2026). A novel dual-parameter method for evaluating seismic collapse severity in masonry pagodas. Bulletin of Earthquake Engineering. https://doi.org/10.1007/s10518-026-02661-z

Image Credits: AI Generated

DOI: 10.1007/s10518-026-02661-z

Keywords: masonry pagodas, seismic collapse, damage index, discrete element method, area loss ratio, residual height ratio, earthquake engineering, cultural heritage, Wenchuan earthquake, structural health assessment, nonlinear dynamic analysis, post-earthquake assessment

Hollow-core polygonal pagodas, typically built of brick or stone in polygonal cross-sections such as hexagonal or octagonal forms, represent a distinctive structural typology that developed over more than a thousand years. Unlike solid masonry towers, these pagodas contain a central vertical void that reduces weight and, in many historical designs, housed stairways or reliquary chambers. The walls behave as thin, curved shells arranged in a polygonal geometry, and this configuration gives the structures a characteristic stiffness and stress distribution under lateral loading. When an earthquake strikes, the ground shaking induces bending moments that grow with height, and the masonry, which is strong in compression but weak in tension, can crack along horizontal or stepped joints long before any visible collapse occurs.

The core innovation of the new research lies in translating this complex three-dimensional problem into a tractable analytical formulation. Instead of modeling every brick and mortar joint, the method treats the pagoda cross-section as an equivalent polygonal hollow section whose bending capacity is governed by the masonry’s limited tensile strength. The authors derive closed-form expressions for the bending moment at which cracking initiates on the tension face and for the progressive reduction in stiffness that follows as the cracked zone spreads around the polygonal perimeter. Because the equations rely only on basic section geometry, material properties, and the distribution of self-weight, they can be evaluated with a spreadsheet or a short script, making the method accessible to practitioners who lack access to specialized finite-element software.

Central to the formulation is the recognition that bending failure in these structures is intimately linked to axial load. The weight of the overlying masonry above any given level compresses the walls and partially suppresses tensile cracking, so the bending resistance of the section increases with the magnitude of the compressive force, up to a limit set by crushing. Near the top of a pagoda, where the axial load is small, the section has little capacity to resist bending, which explains why upper tiers and the slender crowns of these towers are so frequently damaged in earthquakes. The simplified method captures this height-dependent vulnerability explicitly, allowing conservators to identify the critical stories of a specific pagoda where seismic demands are most likely to exceed capacity.

The polygonal geometry introduces further subtleties that the method addresses directly. In a circular hollow section, bending stress flows smoothly around the perimeter, but in a polygonal section the abrupt changes in wall direction at each corner concentrate stresses and alter the location of the neutral axis as cracking progresses. The analytical model accounts for the discrete geometry of the polygon by integrating the contribution of each flat wall segment to the section’s moment of inertia and to the cracking moment. This means the method can distinguish, for example, between a hexagonal and an octagonal plan of similar overall size, reflecting the historical observation that section shape influences both stiffness and the pattern of earthquake damage in surviving pagodas.

To validate the approach, the researchers compared their analytical predictions against established benchmarks for masonry behavior under combined axial load and bending. The simplified formulation reproduced the expected capacity trends, including the increase in cracking moment with axial compression and the sharp loss of flexural rigidity once the tensile strength of the masonry is exceeded. Because the method is deliberately conservative in its treatment of material strength, it tends to err on the side of safety, which is a desirable property for heritage assessment where the consequences of underestimating vulnerability are effectively irreversible. The authors emphasize that the model is intended as a first-line screening tool rather than a replacement for refined analysis of individual monuments, but that its transparency makes it uniquely valuable in that screening role.

The practical implications extend well beyond academic interest. Many of the most celebrated pagodas in China, Japan, and Korea are designated cultural properties subject to strict conservation requirements, and seismic safety evaluations are routinely demanded after significant earthquakes or as part of ongoing preservation planning. Full nonlinear finite-element modeling of a historic pagoda requires detailed surveys, material testing, and considerable expert effort, resources that are often unavailable for the hundreds of lesser-known structures scattered across seismically active regions. A simplified analytical method that needs only geometry, an estimate of masonry strength, and a design-level seismic demand offers a way to triage this large inventory, directing expensive detailed studies toward the structures that the screening analysis flags as most at risk.

The method also supports retrofit decision-making. Once the critical sections and stories of a pagoda are identified analytically, engineers can explore targeted interventions, such as confining bands of compatible reinforcement, grouting of degraded mortar joints, or减轻 of upper-tier mass, and then re-evaluate the bending capacity with the same closed-form equations to gauge the benefit of each measure. This iterative, low-cost loop between assessment and design is particularly important in heritage contexts, where interventions must be minimally invasive, reversible where possible, and justified with clear evidence of need. A tool that makes the mechanics of bending failure explicit, rather than hiding them inside a black-box simulation, is well suited to the interdisciplinary dialogue between engineers, archaeologists, and conservators that effective preservation demands.

Beyond its immediate engineering use, the study contributes to a broader scientific conversation about the seismic resilience of unreinforced masonry heritage. Researchers in earthquake engineering have increasingly recognized that simplified, mechanics-based models play an essential complementary role alongside high-fidelity simulation, because they reveal the governing physical parameters, support rapid parametric study, and facilitate the interpretation of observed damage patterns after real events. For hollow-core polygonal pagodas, the new work identifies axial load level, wall thickness, polygonal geometry, and masonry tensile strength as the dominant parameters controlling bending failure, providing a compact framework that future studies of aftershock vulnerability, soil-structure interaction, or cumulative damage can build upon.

As climate-driven hazard assessments and urban expansion place growing pressure on historic building stocks, tools that combine rigor with accessibility are likely to become increasingly important to the heritage sector. The simplified analytical method for earthquake-induced bending failure of hollow-core polygonal masonry pagodas offers exactly that combination: rooted in the mechanics of masonry, calibrated against recognized behavior, and simple enough for everyday professional use. For the custodians of these centuries-old towers, the study transforms an intimidating structural problem into a set of calculable quantities, bringing modern earthquake science to bear on some of humanity’s most enduring, and most endangered, architectural achievements.

Subject of Research: Simplified analytical prediction of earthquake-induced bending failure in hollow-core polygonal masonry pagodas

Article Title: A simplified analytical method for earthquake-induced bending failure of hollow-core polygonal masonry pagodas

Article References: Lu, W., Xiang, L., Wang, Y., & Li, D. (2026). A simplified analytical method for earthquake-induced bending failure of hollow-core polygonal masonry pagodas. npj Heritage Science. https://doi.org/10.1038/s40494-026-03000-w

Image Credits: AI Generated

DOI: 10.1038/s40494-026-03000-w

Keywords: masonry pagodas, seismic vulnerability, bending failure, heritage conservation, earthquake engineering, hollow-core sections, polygonal masonry, analytical modeling, structural assessment, unreinforced masonry, retrofit planning, cultural heritage