Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 341 Using Eqns. (4) and (5) the following expression can be obtained: π 1 π π π π 1 tan γ Hj j i i j j wj i i z z z z x x x x         (6) from which a value of z Hj can also be derived for each  j . Moreover, considering that x i  = (2/  x i b ), the following relation between the angle  j π and the corresponding one  j b on the base thrust line (representing the inclination of the meridional force resultant S j b on interface j ) is: π 1 1 π π tan γ  tanγ 2 2 b i i j j b b i i z z x x       (7) as represented in Figs. 5a to 5c which refer to the same dome section but per unit length of parallels. It is worth noting that z Hj is the same for both representations of the dome section. Similarly, it can be demonstrated that: π 2 b b j j j H x H  (8) (a) (b) (c) Figure 5 : Thrust-lines and details for the lune per unit length of parallels (base thrust-line). In sum, taking into account Eqns. (6) and (7), the equilibrium of the dome section with angle of embrace  j simply requires that:   tanγ b b b Hj j j wj j z z x x    (9) where: 1 1 tanγ b i i j b b i i z z x x      tanγ 1 tanα tanγ b b i i j j b j j z x z    and   tan α b j j j x z  (10)