Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 344 S.T.   tan α γ μ b j j   meridional sliding constraint ,lim 1,lim b b b i j j H H H    parallel sliding constraint It is worth highlighting that the parallel sliding constraint is particularly suitable to the stability of a dome composed of an assemblage of blocks with running bond pattern, while it does not work in presence of stacked bond pattern, where zero hoop forces must be considered. In this case, in fact, the problem reduces to the one of an arch of increasing width from the crown to the spring, obtained by cracks along the meridional interfaces between adjacent lunes, and the value of minimum thickness to span ratio as a safe solution was first provided by Heyman[17] under the assumptions of infinite compressive strength and friction resistance and zero tensile strength. The corresponding minimum thickness represents an upper bound of the real solution. On the other hand, the classical membrane theory with unlimited hoop forces provides zero thickness of the hemispherical dome, representing the lower bound solution. The solution provided by the approach herein proposed is a membrane surface with limited hoop forces and the minimum thickness required is expected to be in the range of the lower and upper bounds. R ESULTS case study of a hemispherical dome with 10m centreline radius, containing 20 rows and 20 lunes under its own weight was optimized. The obtained results were then compared to the results obtained by other existing methods. To model the dome, construct the membrane and implement part of the structural analysis, C# component of Grasshopper was applied. The outputs of the optimization including the final model were obtained through the Grasshopper environment as well. Grasshopper is a visual programming language which runs within Rhinoceros 3D. The core of structural analysis and optimization was done by MATLAB used as backend. To solve the minimax problem (19), the MATLAB’s fminimax method was implemented [46]. Fig. 8 graphically shows the relationship between the normalised minimum thickness of a structurally feasible dome and the coefficient of friction at block interfaces obtained by the proposed approach and by other existing methods introduced by [1, 17, 31]. The curve “Heyman’s cracked model” [17] represents the classical solution of a dome already cracked along the meridians (zero hoop forces) but satisfying the meridional sliding constraint according to Eqn. (14). In fact, this model yields a constant value of the t/R ratio of 0.0425 if the friction coefficient  is greater than 0.25. For  < 0.25, the resultant of meridional stresses reaches a limit value on some block interface (according to Eqn. (4)) and the minimum thickness increases sharply. As expected, this upper bound solution is not very far from the proposed curve “Revised membrane with zero hoop forces”, which is based on the same assumptions. Instead, the curves “Membrane with limited hoop forces” obtained by D’ayala and Casapulla [1] and “Revised membrane with limited hoop forces” herein proposed represent the solutions obtained by a similar optimization procedure but with a slightly different sliding constraint on the hoop forces, as described above. In particular, the approach herein proposed provides more conservative results with respect to the previous one, still included in the described range of solutions, whose lower bound is represented by the horizontal curve “Classical membrane with unlimited hoop forces”. Lastly, the “Discrete element approach” [31] still considering hoop resistances at interfaces (unlike the Heyman’s cracked approach), provides results closer to the Heyman’s model than to the models with limited hoop forces, even if for  > 0.23 the curve decreases similarly. On the other hand, the results of all approaches present the same pattern, characterized by three domains of the friction coefficient. First, no equilibrium is possible when  is so small (  <  1 ), where  1 is the minimum value. For larger values of  so that  1 <  <  2 (where  2 is the value marking a clear change in the curve inclination), the ratio of minimum thickness to radius t / R decreases (linearly or non-linearly) and the dome behaves as a cracked dome with mixed sliding and rocking mechanisms governed by the meridional sliding constraint (three hinges at the extrados and intrados of each lune are formed and the sliding mechanism occurs at the lowest interface of the lune). Instead, when  is large enough (  >  2 ), sliding along the lower parallels can occur for the models considering the presence of limited hoop forces or, in case of zero hoop forces, pure rocking mechanism can be observed on the optimal result so that four hinges are formed at the intrados and extrados of each cracked lune. A