Issue 51

A. Gesualdo et alii, Frattura ed Integrità Strutturale, 51 (2020) 376-385; DOI: 10.3221/IGF-ESIS.51.27 379 A NALYTICAL FORMULATIONS OF THE VARIATIONAL PROBLEM 2-D Minimum complementary energy approach he analysis considers a NT masonry panel loaded with a constant vertical force and an increasing horizontal one. In the present approach the equilibrium solution is determined by minimizing the complementary energy (7) according the general method reported in [26]. The rectangular domain defined by the panel is traction free on the lateral sides. The body forces are null and the displacements are prescribed at top and bottom bases: the relative displacements of the two bases are defined by the triplet   { , } U V . A reference system  { , } O x y is defined with origin in the panel centroid, see Fig. 2(a). Figure 2 : Masonry panel and displacement vectors (a); angle  with the vector rays (b); vector rays and coordinates system (c) The problem is solved considering an approximate solution over a reduced definition domain, i.e. 2 3    , that is to search the statically admissible stress field 0 T in 2  when the free boundary between 2  and 3  is determined. As above remarked, when the body forces are null, the isostatic compressive curves are straight lines, named compression rays. They form an angle  with the y axis and cross the two bases of the panel, since the vertical edges are part of the free boundary. The curvilinear reference system for the compression rays is defined by the coordinate system    1 2 { , } O , see Fig. 2(c). Within the panel area an uniaxial thrust region is recognized. This compression region is determined by means of the complementary energy minimum. Reference is made to a normalized rectangle of unitary base and height / H B H  where B and H are respectively the real base and height of the masonry panel as in Fig. 2(a). The compression rays are defined by means of the slope function 1 ( ) g  , subjected to the geometrical constraints: 1 1 1 1 1 1 1 2 1 2 ( ) 1 2 1 2 ( ) g H H g H H                 (8) where 1  is the intersection of the slope function with the horizontal axis. The constitutive conditions: T (a) (b) (c)