Issue 51

A. Gesualdo et alii, Frattura ed Integrità Strutturale, 51 (2020) 376-385; DOI: 10.3221/IGF-ESIS.51.27 380 1 1 ( ) , 0 V g U U U        (9) correspond to the existence of compression rays corresponds to a kinematic constraint on the function 1 ( ) g  . The Complementary Energy (7) assumes in this case the form (see Fortunato, 2010):                  1 1 1 2 2 1 2 ( ) 2 1 / 2 (1 )ln 1 / 2 c U g V g E E d g H g g H (10) where the triplet  ( , , ) U V represents the given set of relative rigid displacements, 1  and 1  are the geometrical bounds of the compression region and  g is the first derivative of 1 ( ) g  with respect to  1 . The minimum of c E is obtained solving the Euler equation associated to (10) adopting a multiple shooting technique, in other words looking for the function 1 ( ) g  that minimizes the functional (10) with the constraints (9) and the boundary conditions: 1 1 ( ) , ( ) g g g g     . (11) The conditions (11) correspond to the upper and lower load conditions. 1-D Minimum potential energy approach Like in the previous paragraph, body forces are null and the analysis is performed considering a NT masonry panel loaded with a constant vertical force and an increasing horizontal one. As the horizontal load increases, the resultant force R is that corresponding to the triangular distribution with base A in Fig. 3(a). The straight line connecting the middle points of the triangular distribution forms and angle 1  with the vertical axis. A partition of the entire rectangular domain due to the constitutive model adopted is recognized, so that the compressive stress area  2 in (5) can be assumed that enclosed in the polygonal domain represented in Fig. 3(b), whose geometry is defined by:   2 cos min B A ,   2 cos max B B where  2 is the angle that the symmetry axis of the domain forms with the vertical one and in general it is distinct from 1  . The problem is skew-symmetric with respect to the vertical axis, and the entire problem can be reduced to one- dimensional model, i.e. a masonry strut with variable cross section and symmetric shape. The resulting problem is an Euler-Bernoulli cantilever beam with variable cross section as in Fig. 4(a), loaded by R . The internal forces on the beam are:                   1 2 1 2 1 1 2 cos( ) , sin( ) , . tan 2 3 2 B A H N R T R M N (12) The variational formulation of the problem in terms of potential energy ( ) p u E in (6) is used. The boundary conditions are defined at the beam ends of Fig. 4 (b) and in the cross section where there is a stiffness first derivative variation. The problem solution is in this case the triplet 2 ( , , ) B C  v v satisfying the displacements boundary conditions and minimizing the ( ) p u E expressed by: 1 2 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 0 0 0 0 ( ) ( , , ) 1 1 ( ) ( ) ( ) ( ) 2 2 p p B C L L L L C EI z dz EI z dz EA z dz EA z dz                                  u v v F v E E (13)