Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 338 of a discrete network of forces. Then, the proposed convex contact model and the heuristic method to find the sliding resistance of the assumed interlocking blocks are developed and the conclusions of the work are finally provided. Figure 1 : A hemispherical dome, a lune depicted in blue and two interlocking blocks presented in red. An interlocking block has several locks on its hoop face whose geometric properties including width g , height h and length b affect the sliding resistance of the block. L IMIT ANALYSIS OF HEMISPHERICAL DOMES COMPOSED OF CONVENTIONAL RIGID BLOCKS he objective of the optimization is to find the minimum constant thickness of a stable hemispherical dome under its own weight, with assigned radius. To develop the optimization method, first a number of control points representing the blocks are defined and the axisymmetric membrane surface is constructed. Applying the discrete approach of O’Dwyer [5] within the limit analysis framework, a network of forces can represent the structural model of the dome. A discretized dome includes a set of similar lunes (Fig. 1), each of them spanning horizontal angle  = 2  / nl , where nl is the number of lunes (Fig. 1). Each lune also includes a set of similar rigid blocks stacked over each other. The network of forces is constructed using a set of control points located on the blocks. A set of horizontal parallel polygons and a set of similar meridional thrust-lines pass through these control points. Decreasing the horizontal angle  , and therefore the length of the lune support, the centroids of the blocks move horizontally and radially towards the mid- surface of the dome, while the parallel polygons tend to become circular. For unit length of parallels (of the hemisphere), the network of forces becomes very close to a membrane surface for the hemispherical dome [1] and the generating meridional trust-line can be called base thrust-line . Given the symmetry of the problem, only half of the dome is considered in the proposed procedure. The generic trust- lines with the coordinates of two subsequent control points for a lune with horizontal angle  =  (i.e. for half dome), with  <  , and the base thrust-line are represented in Fig. 2 by a broken line, a dashed line, a dash-dotted line, respectively. The half dome is modelled by assuming that its mass is distributed along its mid hemispherical surface of radius R (its projection on the plane XZ is represented by the central-arc C in Fig. 3). To determine the coordinates of the control points of the base thrust-line, first the central-arc C (and the unknown thickness as well) is divided into the desired number of blocks r . Considering the mid-points of all the blocks (empty red circles in Fig. 3a), their x coordinates coincide with the x coordinates of the control points (filled red circles in Fig. 3b). Two different thrust-lines for the lunes with horizontal angle  =  and  <  are also constructed through interpolation of the points (depicted by orange and yellow dots in Fig. 2a and b, respectively) whose x coordinates are different from the control points on the base thrust-line. Therefore, a different superscript of x is used for the different lengths of the considered lunes. On the other hand, the z coordinates of the control points of the base thrust-line are the same as those of the thrust-lines of lunes  and  , and no T