Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 342 Eqns. (9) and (10) easily allow the calculation of the meridional and hoop forces of the constructed membrane surface. Finding the stress state through the equilibrium equations, the sliding constraint is defined by limiting the maximum values of the internal shear force to be not greater than the frictional resistance at each block interface. This limitation can be imposed independently on the two contributions obtained by the meridional and hoop force resultants so that two states of presence and absence of hoop forces can be analysed. With reference to the meridional sliding constraint (in the absence of hoop forces), it is observed that the internal force S j b at interface j per unit length of parallels is distributed uniformly on the intersected circular arc. According to the Coulomb’s friction law, the sliding constraint can be written as (Fig. 6): μ b b j j T N  (11) where  is the friction coefficient and T j b and N j b are the components of S j b tangential and normal to interface j , respectively, expressed by the formulations:   sin α γ b b b j j j j T S   (12)   cos α γ b b b j j j j N S   (13) As a result, in Eqn. (11) can be rewritten as:   tan α γ μ b j j   (14) Figure 6 : Meridional sliding constraint at interface j governed by tangential and normal components of S j b .  With reference to the parallel sliding constraint, first the horizontal resultant of the hoop stresses per unit length of parallels H i b for each block i can be found by taking into account Eqns. (4) and (8), i.e. (Figs. 7a and 7b): 1 b b b i j j H H H    (14) which is applied at the control point at level z i . This force is applied horizontally outwards when positive (showing compressive hoop forces) and inwards when negative (presenting tensile hoop forces). Then, due to the geometry of the hemispherical dome and its composing blocks, inward movement of the upper portion of the dome over the lower one at interface j is not possible. This means that only limiting hoop force H b j,lim against outward movements must be considered as frictional resistance at this interface and that, including the contribution of the meridional force, the normal and tangential components of the resultant at interface j can be respectively written as: