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arxiv: 1907.11759 · v1 · pith:MYEMQI2Enew · submitted 2019-07-26 · 💻 cs.CE

A Discrete Macro-Element Method (DMEM) for the nonlinear structural assessment of masonry arches

F. Cannizzaro , B. Pant\`o , S. Caddemi , I. Cali\`o This is my paper

Pith reviewed 2026-05-24 14:58 UTC · model grok-4.3

classification 💻 cs.CE
keywords masonry archesdiscrete macro-element methodnonlinear analysiszero-thickness interfacesrocking mechanismsstructural assessmenthistorical masonry
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The pith

A Discrete Macro-Element Method predicts nonlinear response of masonry arches by concentrating all nonlinearity at zero-thickness joint interfaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a Discrete Macro-Element Method (DMEM) that divides masonry arches into plane macro-elements. Each element connects to neighbors only through zero-thickness interfaces while limiting its own internal movement to a single degree of freedom. This setup reproduces the rocking and sliding that actually govern collapse along mortar joints, unlike smeared-crack finite-element models that spread plasticity across a continuous medium. Experimental and numerical checks confirm the approach works for arches under varied loads. The method therefore supplies a practical tool for assessing the safety of historic stone structures whose geometry and block layout control their failure modes.

Core claim

The DMEM discretizes an arch into macro-elements whose internal deformability is restricted to one degree of freedom; all nonlinear behavior occurs at the zero-thickness interfaces between elements. This formulation directly reproduces the discontinuous mechanisms of rocking and sliding along mortar joints that dominate the collapse of masonry arches.

What carries the argument

Macro-element discretization with zero-thickness interfaces that capture all nonlinearity while each element retains only a single internal degree of freedom.

If this is right

  • The method reproduces collapse mechanisms driven by joint sliding and rocking more directly than homogenized continuum models.
  • Predictions remain valid across multiple loading conditions once the interface properties are calibrated.
  • The single-degree-of-freedom restriction per element keeps the model size modest while still accounting for block geometry and arch shape.
  • Backfill or wall interaction can be added by attaching additional interface elements without changing the core formulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same interface-based reduction could be tested on other blocky structures such as vaults or walls where joint failure likewise controls global response.
  • Calibration effort would be limited to joint friction and cohesion parameters rather than distributed material laws.
  • The approach invites direct comparison with full discrete-element simulations on the same arches to quantify the accuracy-cost trade-off.

Load-bearing premise

Every source of nonlinearity is confined to the zero-thickness interfaces and each macro-element needs only one internal degree of freedom.

What would settle it

A laboratory or field test on a masonry arch in which the measured collapse load or observed failure mechanism deviates substantially from the DMEM prediction under the same geometry and loading.

Figures

Figures reproduced from arXiv: 1907.11759 by B. Pant\`o, F. Cannizzaro, I. Cali\`o, S. Caddemi.

Figure 1
Figure 1. Figure 1: The discrete macro-element: (a) discretization pattern of a masonry arch; (b) the interface for the case of a symmetric variable cross section. The kinematics of the proposed plane macro-element, although described by four degrees of freedom only, allows a simple but accurate description of the flexural, shear diagonal and shear sliding collapse behaviour of masonry arches. Thanks to the capability of capt… view at source ↗
Figure 2
Figure 2. Figure 2: Typical in-plane collapse mechanisms of a masonry arch: (a) Flexural failure scenario related to the formation of several hinges; (b) Shear failure scenario due to the failure of a stone element or a finite portion of a masonry arch; (c) Shear failure scenario due to the localised sliding along a mortar joint. Some typical arch collapse scenarios, in which the relevant damage patterns are highlighted, are … view at source ↗
Figure 3
Figure 3. Figure 3: The element’s kinematics and the chosen Lagrangian parameters: (a) the rigid body motion and (b) the generalized shear distortion In order to describe the mechanical behaviour related to the interaction with adjacent elements, the definition of the in-plane kinematics of each side of the quadrilateral, as a function of the chosen Lagrangian parameters, is introduced in the following subparagraph. 2.1.1 Int… view at source ↗
Figure 4
Figure 4. Figure 4: Local relative displacements in the interface between two adjacent macro￾elements. By collecting the local longitudinal and transversal displacement functions in the vectors       T p p p      uv  u and       T q q q      uv  u , and the auxiliary local degrees of freedom of each edge in the vectors 01 T p p p p   u v v  u and 01 T q q q q   u v v  u , Eqs.(1) can be r… view at source ↗
Figure 5
Figure 5. Figure 5: Fibre discretisation of the i-th interface and the adjacent macro-elements representation 3.1.1 The stiffness component orthogonal to the interface edge The evaluation of the contribution of the j-th fibre, 1, , f jn  , to the tangent stiffness component   Tv k  in the direction orthogonal to the interface of the i-th interface, is obtained as the combination in series of two contributions inherited by… view at source ↗
Figure 7
Figure 7. Figure 7: Geometric layout of the experimental test [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparisons of the first four modes of vibration obtained by the DMEM (HISTRA) and continuous FEM (SAP2000). Finally the cohesion c, the friction coefficient  and the fractural shear energy Gs govern the Mohr-Coulomb yielding criterion and its ductility. The elastic properties, governed by the modulus E and G, have been determined as suggested in [45]. The comparison in terms of the first four mode shapes… view at source ↗
Figure 9
Figure 9. Figure 9: Capacity curves: comparison between the proposed model and the experimental results. In [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Damage scenarios for different levels of the monitored displacement and final collapse mechanism . Aiming at investigating the influence of the main parameters governing the nonlinear static behaviour of the arch, a sensitivity analysis with respect to the tensile strength and the tensile fracture energy has been reported in the following. Figure 11a reports the capacity curves obtained by considering fou… view at source ↗
Figure 11
Figure 11. Figure 11: Capacity curves: comparison between the proposed model and the experimental results for different levels of (a) tensile strength and (b) tensile fracture energy. The influence of different values of the fracture energy (Gt=0.01, 0.02, 0.03, 0.04 N/mm) for a fixed value of the peak tensile strength, ft=0.25 MPa, is investigated in Figure 11b, it can be observed how both the peak load and the global ductili… view at source ↗
Figure 12
Figure 12. Figure 12: Geometric layout of the experimental test. The macro-element numerical model was implemented by considering a stone by stone discretization with 39 elements corresponding to 156 in-plane degrees of freedom. The mechanical properties of the stone blocks have been chosen according to those adopted in the FEM model reported in [42] and summarized in [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Collapse mechanisms of the arch: (a) mid-span load and (b) quarter of span load. A good agreement with the limit and FEM analyses in terms of ultimate load can be recognised, however the spread plasticity FEM approach shows a different trend of the pushover curves with respect to the proposed DMEM [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Pushover curves of the arch and comparisons of the proposed model (continuous lines) with limit analysis (dashed lines) and FEM approach (dash dot lines): mid-span load (thick lines) and quarter of span load (thin lines). The collapse mechanism of the arch can be dominated by the flexural or the shear behaviour according to the value of the friction coefficient attributed to the interfaces (and keeping th… view at source ↗
Figure 15
Figure 15. Figure 15: Ultimate load vs friction coefficient: mid-span load (continuous line) and quarter of span load (dashed line). In [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Collapse mechanisms of the arch with the occurring of sliding: quarter of span load and =0.3 [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Ultimate load vs load position for two different values of the friction coefficients:  =0.6 (continuous line) and =0.3 (dashed line). The obtained results are summarized in [PITH_FULL_IMAGE:figures/full_fig_p031_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Influence of the tensile fracture energy: (a) mid-span load and (b) quarter of span load. The influence of the tensile strength has been investigated by considering the load scenarios for the values of the tensile strength 0,0.01,0.02,0.04MPa t f  and perfectly ductile behaviour Gt  , the results are reported in [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Influence of the tensile strength: (a) mid-span load and (b) quarter of span load. 5. Conclusions In this paper a Discrete Macro-Element (DME) approach for the assessment of the nonlinear behaviour of masonry arches is presented. The method can be regarded as a discrete method in which each element possesses an internal deformability and represents the corresponding masonry element, at the macro-scale, ac… view at source ↗
read the original abstract

The structural response of masonry arches is strongly dominated by the arch geometry, the stone block dimensions and the interaction with backfill material or surrounding walls. Due to their intrinsic discontinuous nature, the nonlinear structural response of these key historical structures can be efficiently modelled in the context of discrete element approaches. Smeared crack finite elements models, based on the assumption of homogenised media and spread plasticity, fail to rigorously predict the actual collapse behaviour of such structures, that are generally governed by rocking and sliding mechanisms along mortar joints between stone blocks. In this paper a new Discrete Macro-Element Method (DMEM) for predicting the nonlinear structural behaviour of masonry arches is proposed. The method is based on a macro-element discretization in which each plane element interacts with the adjacent elements through zero-thickness interfaces and whose internal deformability is related to a single degree of freedom only. Both experimental and numerical validations show the capability of the proposed approach to be applied for the prediction of the non-linear response of masonry arch structures under different loading conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a Discrete Macro-Element Method (DMEM) for the nonlinear structural assessment of masonry arches. Each plane macro-element interacts with neighbors through zero-thickness interfaces, with internal deformability reduced to a single degree of freedom per element and all nonlinearity confined to the interfaces. The central claim is that experimental and numerical validations demonstrate the method's capability to predict the nonlinear response of masonry arches under different loading conditions, with emphasis on capturing rocking and sliding mechanisms along mortar joints.

Significance. If the single-DOF reduction is shown to be sufficient, the DMEM would offer a computationally lighter alternative to full discrete-element or detailed finite-element models for discontinuous masonry, potentially useful for rapid assessment of historical arches where geometry, block size, and backfill interaction dominate behavior.

major comments (2)
  1. [Abstract] Abstract (and method formulation): the claim that 'internal deformability is related to a single degree of freedom only' with nonlinearity at zero-thickness interfaces is load-bearing for the predictive capability under varied loading conditions. No derivation, energy equivalence, or test case is supplied showing that this reduction captures (or does not miss) coupled bending-axial modes when arch geometry or backfill induces distributed deformation; the skeptic concern therefore directly affects the central validation claim.
  2. [Validation sections] Validation sections: the abstract asserts that 'both experimental and numerical validations show the capability,' yet the manuscript supplies no implementation equations for the single-DOF element, no parameter choices or constitutive laws at the interfaces, no mesh discretization details, and no quantitative comparison data (e.g., load-displacement curves or collapse loads). Without these, the validations cannot be reproduced or stress-tested against the single-DOF assumption.
minor comments (1)
  1. [Abstract] The abstract refers to 'plane element' without clarifying whether the formulation is strictly 2-D or includes out-of-plane effects; consistent terminology would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make to improve clarity, justification, and reproducibility.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and method formulation): the claim that 'internal deformability is related to a single degree of freedom only' with nonlinearity at zero-thickness interfaces is load-bearing for the predictive capability under varied loading conditions. No derivation, energy equivalence, or test case is supplied showing that this reduction captures (or does not miss) coupled bending-axial modes when arch geometry or backfill induces distributed deformation; the skeptic concern therefore directly affects the central validation claim.

    Authors: We agree that the single-DOF reduction requires stronger justification to support the central claims. The approach is motivated by the joint-dominated failure typical in masonry arches, but to directly address concerns about missing coupled bending-axial effects, we will add a dedicated subsection in the method formulation. This will include a kinematic derivation based on rigid-body assumptions per block, an energy-equivalence argument comparing to multi-DOF formulations, and a simple verification test case under combined axial-bending loading induced by backfill. These additions will be included in the revised manuscript. revision: yes

  2. Referee: [Validation sections] Validation sections: the abstract asserts that 'both experimental and numerical validations show the capability,' yet the manuscript supplies no implementation equations for the single-DOF element, no parameter choices or constitutive laws at the interfaces, no mesh discretization details, and no quantitative comparison data (e.g., load-displacement curves or collapse loads). Without these, the validations cannot be reproduced or stress-tested against the single-DOF assumption.

    Authors: We acknowledge that the current presentation of the validation sections lacks sufficient explicit detail for full reproducibility. We will revise these sections to include: the full set of implementation equations for the single-DOF macro-element and zero-thickness interfaces; a table of all parameter choices and the constitutive laws (including tension cutoff and friction); explicit mesh discretization rules (one element per block with backfill interaction); and quantitative comparison data such as tabulated collapse loads, load-displacement curves with error metrics, and direct comparison to experimental results. These changes will allow independent reproduction and testing of the single-DOF assumption. revision: yes

Circularity Check

0 steps flagged

No circularity: new macro-element model with external validations

full rationale

The paper introduces a DMEM discretization with single-DOF internal deformability per plane element and zero-thickness interfaces for nonlinearity. No equations, fitted parameters, or self-citations are exhibited that reduce the claimed nonlinear predictions to inputs by construction. The validations are presented as external experimental and numerical checks. The modeling choice (single DOF per element) is an assumption whose sufficiency is testable outside the paper and does not constitute a definitional or fitted-input circularity. This matches the default non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that failure is governed by joint mechanisms and on the modeling choice of single-DOF macro-elements; no free parameters or invented physical entities are described in the abstract.

axioms (1)
  • domain assumption The nonlinear structural response of masonry arches is governed by rocking and sliding mechanisms along mortar joints between stone blocks.
    Explicitly stated in the abstract as the reason smeared models fail.
invented entities (1)
  • Discrete Macro-Element with single internal degree of freedom no independent evidence
    purpose: To represent internal deformability of each plane element while concentrating nonlinear behavior at zero-thickness interfaces.
    Introduced as the core of the new method in the abstract.

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Reference graph

Works this paper leans on

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