Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 340 Given the isostatic nature of the problem, for any given set of descending z i ’s the state of stress is completely defined by simple equilibrium equations, so allowing the construction of any corresponding axisymmetric membrane surface. In other words, the value of z i ’s are only bounded by the sliding constraint. Finding the state of stress of the equilibrated model, this constraint is determined, as described in the following First consider the section with an angle of embrace  j of half dome divided into a number of rows equal to the number of blocks along the meridional length. Being ω the weight for unit surface, the total weight of this section is:   π 2 j  ω π  1 cosα j W R   (2) passing through the centre of mass of the half dome section (Fig. 4a). The x coordinate of this centre is:   π π 1 π π j i i i wj j W x x W    (3) where W i  is the weight of the corresponding row on the half dome and x i π is the x coordinate of block i on the thrust-line for  =  (Figs. 4b). The z coordinates of this point and control point i on the base thrust-line are identical. (a) (b) (c) Figure 4 : Thrust-lines and details for the half dome with an angle of embrace  j . The angle  j π between the meridional force resultant on interface j ( S j π ) of the half dome and the horizontal line at point P j  ( x j  , z j ), shown in Fig. 4a, can be defined both from the global equilibrium of the section and from the construction of the corresponding thrust-line. In fact, the rotational equilibrium of the section  j of half dome about point P j   requires the horizontal hoop force resultant H j  to be applied to a point with z Hj coordinate, so that: π π π π π π tanγ j wj j j j Hj j j x x W H W z z     (4) from which the angle  j π can be derived. On the other hand,  j π also corresponds to the inclination of the thrust-line between the control points of block i and i +1 when  =  (Fig. 4c), i.e.: π 1 π π 1 tan γ i i j i i z z x x      (5)