Fast Seismic Vulnerability Evaluation of Historical Masonry Aggregates through Local Analyses: an Adaptive NURBS-based Limit Analysis Approach

An efficient computational tool for the local failures analysis in historical masonry aggregates is proposed. A NURBS (Non-Uniform Rational B-Spline) representation of geometry is adopted. NURBS entities, which are common in commercial CAD packages, have the great advantage to describe complex geometries (such as curved elements and walls with a high number of holes) with very few elements. An upper bound limit analysis formulation is implemented, in which the adopted NURBS elements are idealized as rigid bodies with dissipation allowed only along interfaces. The mesh of NURBS elements is progressively adjusted through a genetic algorithm in order to minimize the live load multiplier. This procedure is applied in the evaluation of the horizontal load multiplier associated with the activation of local mechanisms during a seismic event. Some case studies, referring to masonry aggregates located in the historical centers of Arsita (Abruzzo region, Italy) and Sora (Lazio region, Italy), are here presented. A quick evaluation of the seismic vulnerability is performed through the presented NURBS-based computational tool, showing the high importance of the local response in the study of the seismic behavior of masonry aggregates

UB-ALMANAC: An adaptive limit analysis NURBS-based program for the automatic assessment of partial failure mechanisms in masonry churches

As well known, masonry churches fail upon formation of partial failure mechanisms, usually activating at very low levels of horizontal accelerations, which are responsible for the collapse of specific macro-blocks, typically the façade, the apse, lateral naves long walls, etc. Such collapses are a sort of fingerprint for a church and are dependent on the peculiar geometric features of each structure. In order to cope with such unique behavior, the Italian Guidelines on Cultural Heritage for the safety assessment of historical masonry churches require the separate analysis of 28 pre-assigned failure mechanisms by means of the application of the upper bound theorem of limit analysis in presence of a no-tension material. The utilization of an arbitrary subset of mechanisms, whilst fully justified by past earthquakes experience, could in principle lead to an overestimation of the load carrying capacity and force practitioners to calculations that are still not fully automated. In this context, we present here an efficient and straightforward automatic Upper Bound Adaptive LiMit ANAlysis program for masonry Churches: UB-ALMANAC. The code proposed in this paper relies into a rough finite element discretization constituted by few NURBS rigid elements joined by elasto-plastic interfaces. The mesh is directly prepared within a CAD environment based on the 3D model of the whole church, thus being immediately conceived at architectural level. Limit analysis is then performed automatically under the desired horizontal loads distribution, using the kinematic theorem of limit analysis with dissipation allowed only along interfaces and progressive adaptation of the mesh through a Genetic Algorithm, leading to a quick estimation of the first activating failure mechanism and the most vulnerable macro-block. Three small-medium size churches damaged by the recent Emilia Romagna (2012) and Monti Sibillini (2016) seismic sequences are analyzed and results are compared with both alternative numerical approaches and the actual damages observed. Very good match is systematically found, meaning that the proposed tool could represent a breakthrough toward the full automation of the limit analysis assessment of partial failure mechanisms for churches

Seismic assessment of masonry aggregates: A NURBS-based limit analysis computational tool

Masonry structures represent one of the most common worldwide structural typologies for buildings. To date, many historical towns and villages in Italy are built from aggregates of masonry buildings, in which each architectural unit interacts with others forming a complex structural system. Local failure mechanisms are particularly important for the assessment of the safety level of masonry buildings and, especially, of aggregates: recent seismic events have provided evidence that masonry buildings are particularly vulnerable to out-of-plane actions, triggering a wide number of local collapse mechanisms. For this reason, this contribution proposes a new fast computational tool for the automated seismic assessment of local failure mechanisms in masonry aggregates through limit analysis. The proposed tool is based on the NURBS geometric description of the aggregate, which is the common output of many free form modelers. This peculiarity makes the tool easily interfaced with CAD design environments and requires no advanced computational skills to the user. The historical center of Arsita (Italy), a beautiful example of masonry aggregate, which was hit by the 2009 L'Aquila earthquake, is analyzed as a case study and the results are used to validate the proposed tool

Adaptive limit analysis of historical masonry structures modeled as NURBS solids

In this work, we propose an adaptive upper bound limit analysis based on the representation of geometry through NURBS-solids. A NURBS-solid is a closed volume identified by boundary NURBS surfaces (Non-Uniform Rational Bezier Spline). Differently from using NURBS surfaces representing masonry shell-elements, NURBS-solids allow an accurate representation of masonry three-dimensional macro-blocks, such as vaults or walls with variable thickness. The initial model is subdivided into very few macro-elements, each one is still a NURBS solid and is considered as a rigid block. Since dissipation occurs only along interfaces, NURBS boundary surfaces represent possible fracture zones. An upper bound limit analysis is applied. The minimum kinematic multiplier is found by modifying the initial subdivision of solids until the real collapse mechanism is reproduced. This automatic research is performed through a Genetic Algorithm. A simple numerical example is finally reported

A Class of Nonlocal Hypoelliptic Operators and their Extensions

In this paper we study nonlocal equations driven by the fractional powers of hypoelliptic operators in the form Ku = Au -partial_t u = tr(Q nabla^2 u) + - partial_t u, introduced by Hormander in his 1967 hypoellipticity paper. We show that the nonlocal operators (-K)^s , (-A)^s can be realized as the Dirichlet-to-Neumann map of doubly-degenerate extension problems. We solve such problems in L^infty, in L^p for 1 = 0. In forthcoming works we use such calculus to establish some new Sobolev and isoperimetric inequalities

Nonlocal isoperimetric inequalities for Kolmogorov-Fokker-Planck operators

In this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of Kolmogorov-Fokker-Planck operators which are of interest in physics. These operators are very degenerate and do not possess a variational structure. The prototypical example was introduced by Kolmogorov in his 1938 paper on Brownian motion and the theory of gases. Our work has been influenced by ideas of M. Ledoux in the local case

Hardy\u2013Littlewood\u2013Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroups

In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper H\uf6rmander discussed a general class of degenerate Ornstein\u2013Uhlenbeck operators that includes Kolmogorov\u2019s as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy\u2013Littlewood\u2013Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting

Functional inequalities for a class of nonlocal hypoelliptic equations of H\uf6rmander type

We consider a class of second-order partial differential operators A of H\uf6rmander type, which contain as a prototypical example a well-studied operator introduced by Kolmogorov in the \u201930s. We analyse some properties of the nonlocal operators driven by the fractional powers of A, and we introduce some interpolation spaces related to them. We also establish sharp pointwise estimates of Harnack type for the semigroup associated with the extension operator. Moreover, we prove both global and localised versions of Poincar\ue9 inequalities adapted to the underlying geometry