Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 345 Using different approaches, different  1 and  2 were calculated. The results in terms of both values of  and the corresponding t / R are reported in Tab. 1, starting from the proposed approach. It is first interesting to observe that while the thickness of dome for any  between  1 and  2 given by Heyman’s method is larger than the corresponding result obtained by the proposed approach for the dome with stacked bond pattern (zero hoop forces), the minimum thicknesses for  >  2 are almost similar using the two approaches. Moreover, using discrete element analysis of Simon and Bagi [31], the results for  1 <  <  2 are closer to the proposed approach than to the Heyman’s method, while those for  >  2 are similar to the decreasing curves for limited hoop forces. The last remark is that the results obtained by the proposed method for domes with running bond pattern (limited hoop forces) are slightly different from those given by the membrane approach [1], herein revised. The reason is to be found in the different parallel sliding constraint that governs this part of the curve, as described above. Instead for  >  2 these are in good agreement and both shows that for values of   0.7 the minimum thickness required by that formulation is infinitely small as the membrane surface tends to coincide with the mean hemispherical surface of the dome, which is considered the lower bound of possible solutions. Figure 8 : Relationship between the minimum thickness of a dome composed of conventional blocks and the coefficient of friction. Approaches  1  2 t / R (  =  1 ) t / R (  =  2 ) Revised membrane (zero hoop forces) 0.19 0.22 0.08 0.043 Revised membrane (limited hoop forces)  0.13 0.17 0.06 0.018 Membrane (limited hoop forces) [1] 0.09 0.11 0.20 0.016 Heyman’s model [17] 0.20 0.25 0.10 0.043 DEM approach [31] 0.20 0.23 0.10 0.045 Table 1 : Compared results for the optimized hemispherical dome composed of conventional rigid blocks.