Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25

Using Eqns. (4) and (5) the following expression can be obtained:

π z z z z x x x x        π π Hj j i i wj i 1 j

π

1



(6)

tan γ

j

π

i

from which a value of z Hj  = (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: can also be derived for each  j . Moreover, considering that x i

1 z z x x     i i b i

π 2

π 2

b

π

1





(7)

tan γ

 tanγ

j

j

b

i

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 H x H  j j

(8)

(a) (c) Figure 5 : Thrust-lines and details for the lune per unit length of parallels (base thrust-line). (b)

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:

tanγ 1 tanα tanγ b b i i j z x  

1 z z x x     i i b i

tanγ b

  tan α b j j j



z

1



x z 

and

(10)

j

j

b

b

j

j

i

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