Issue 51

M. Pepe et alii, Frattura ed Integrità Strutturale, 51 (2020) 504-516; DOI: 10.3221/IGF-ESIS.51.38 506 results shows the efficacy of Limit Analysis for a fast and reliable assessment of the in-plane failure analysis of masonry walls. L IMIT A NALYSIS ollowing and reinterpreting the model formulated in [30, 31], masonry is described as a system of n rigid blocks and m joints unable to carry tension and resistant to sliding by friction. The blocks can translate and rotate about the edges of the contact blocks (hinging) as well as slide along the joints. Let introduce   1 2 3 T e e e  e the orthonormal basis in the three-dimensional space. We consider the two blocks in Fig. 1. Loads are applied to the centroid of each rigid block th i : static dead loads are collected in vector   0 01 02 03 ,  ,   T i i i i f f m  f , static live loads are collected in vector   1 2 3 ,   ,   T i i i i L L L L f f m  f . For the whole structure it results   0 0 i  f f and   i L L  f f , with i = 1, …, n . The vector of the load over the whole system is 0 L    f f f , where live loads are proportional to the dead loads through a non-negative coefficient  called collapse multiplier. Let   1 2 3 ,  ,   T i i i i u u   u denote the vector of generalized displacement of the centroid of each th i block. The vector   i  u u , with i = 1, …, n , collect the displacements for the whole structure which correspond in a virtual work sense to loads f . The static variables are the internal forces acting at each th j contact surface between blocks, whose components are the normal force j N , the shear force j T and the moment j M . For each joint they are collected in vector   ,  ,   T j j j j N T M  σ . The vector, with j = 1, …, m , refers to the whole structure. The kinematic variables, or generalized strain, are the relative displacement rates at joints, that is normal displacement j  , tangential displacement j  and rotation j  . For each joint j = 1, …, m they are collected in the vector   ,  ,   T j j j j     ε . The vector   j  ε ε refers to the whole structure and corresponds in a virtual work sense to the vector of static variables σ . Figure 1 : Simple two-block structure: dead and live loads, kinematic and static variables Masonry is described as a system of rigid blocks directly interacting through contact surfaces unable to carry tension and resistant to sliding by friction. The set of equations for the model are represented by:         T T T T L ε Bu B σ f 0 y N σ 0 ε Mλ y λ 0 f u 1 (1.1) F