PSI - Issue 29

R. Gagliardo et al. / Procedia Structural Integrity 29 (2020) 48–54 R. Gagliardo, G. Terracciano, L. Cascini, F. Portioli, R. Landolfo/ Structural Integrity Procedia 00 (2019) 000 – 000

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2. Modellingapproach andnumerical formulation The numericalmodel a ims a t analyzing settled structures assembled as a collection of polyhedral elements using rigid block limit analysis. The settlement is simula ted by an additional block loca ted in the settled area moving downward, whose action produces a gradually loss of the base reactionuntil the collapse. The geometricalmodel is drawnusingComputerAidedDesign (CAD) tools, such as AutoCAD®or Rhinoceros®, where the overa ll model is discretized into several block types a ttached by some a ttributes. The a ttributes are essentia lly the cartesian coordinates of the vertexes andof the centroid of the blocks, the labels of the quadrangular contact surfaces and the amount of the block volume. The GUI was designed to define the masonry mechanical properties (friction angle and weight per unit volume), the boundary and loading conditions (both live and constant loads) and thesettlement direction. A point contact model is adopted to describe the blocks interaction at the interface. In such a model, the internal forces are located a t the corner of the blocks k . They are essentially the normal force n k and shear forces t 1k and t 2k , collected in the vector c . There are at least 4 contact points per contact interfaces, i.e. 12 vectors (Figure 1b). It is worth noting that the number of vectors per each element is not fixed, dependingon thecontact interfaces. The numerical model assumes two contact failure conditions: opening and sliding a t interfaces. The procedure solves an optimization mathematical programming problem based on the lower bound theorem of the limit analysis ca lculating the maximum value of the collapse load factor (or base reaction) subjected to equilibrium and failure conditions constraints. The output is representedby the collapse mechanismand the value of the load factor (or base reaction) a t collapse. Theproblem is described by the following formulation: max α s.t. The equation (1) describes the optimization problem. The calculation of the maximum admissible load factor is subjected to twoconstraints. Thefirst constraint is the equilibriumconditionbetween the contact forces c and external loads, where A is the equilibrium matrix, f D is the vector of the dead loads and f L is the vector of live loads, which is coincidentwith thevaryingcomponent of the basereactionat themoving support. Then the model assumes a Coulomb friction fa ilure criterion, where the collapse condition is represented by a convex cone as a function of the contact forces and of the friction coefficient µ . The numerical procedure a llows to assume both associative and non-associative friction model. The associative solution represents an upper bound value for the load factor. An iterative procedure is implemented to calculate the non-associative solution in order to obtain a zero-dila tancy slidingbehavior. Ac=f D +α f L c ∈ { c k ∈ R 3 : μ n k ≥ √ t 1k 2 +t 2k 2 , n k ≥ 0 } (1)

rigid block i

contact interface j

n k

t 2k

f Ds

t

1k

contact point k

Support block

α f

Ls

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Fig. 1. (a) Rigid block assemblage; (b) internal forces at contact point k .