Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 346 D ISCRETE NETWORK OF FORCES FOR HEMISPHERICAL DOMES s demonstrated above, the geometric properties of the interfaces between the blocks can be ignored in formulating the sliding constraint within the limit analysis framework. By contrast, using interlocking blocks, the geometric properties of the interfaces must be considered within the sliding constraints because the interface behaviour is not isotropic and the stress state at each interface must be found separately. To this aim, a discrete network of forces can be modelled with reference to meridional thrust-lines instead of base thrust-lines, while parallel polygons can still be assumed in compression or tension if the sliding resistances are met in different directions. To construct the meridional thrust-line for the lune with horizontal angle  first the relation between its meridional force S j  and its corresponding force S j b on the base thrust-line is determined. Given     2 ω π  1 cosα / π  b b j j j W R x       and H j b as the two components of S j b for the base thrust-line, the vertical and horizontal internal forces applied on interface j of the lune with horizontal angle  (as components of S j  ) can respectively be found as (Fig. 9): θ   θ b b j j j W W x  (20) θ θ 2 sin 2 b b j j j H H x        (21) where x j b is the x coordinate of the point on interface j at which forces are applied. Fig. 10 illustrates the relation between H j  and H j b (Eqn. (21)), graphically. Figure 9 : Relation between the meridional force S j   for the lune with horizontal angle  and the force S j b for the base thrust-line. Finding W j  and H j   the x coordinate of centroid i , through which the thrust-line of the lune with horizontal angle  passes ( x i  ), can be computed through the following steps. Knowing the x and z coordinates of centroid i , the network of force can be constructed using these formulations: θ θ θ   θ θ              tan γ   tan γ θ θ 2 sin 2sin 2 2 b b j j j b j j b b j j j W W x H H x                (22) A