Issue 51

C. Anselmi et alii, Frattura ed Integrità Strutturale, 51 (2020) 486-503; DOI: 10.3221/IGF-ESIS.51.37

Finally, if hoops are inserted, the possible action of them is represented by a F h plane parallel to the dome springing and oriented towards the inside. Having fixed a suitable N h

force applied on extrados face, lying in a

pretension effort, the

= N h · tan(  /8) is positioned at half the height with respect to the height of the generic parallel ring with

resultant force F h

hoop (point C in Fig. 5).

Z

C

z

O

Y

X

(a)

F h

Nh

A

F h

C

B

π/8

Fh

Nh

z A

z B

z C

=(z A

+z B

)/2

π/8

π/8 π/8

Nh

D

x

Fh

hT

Fh

(c)

(b)

Fh

Nh

Figure 5 : Force depending by the hoop. (a) The resultant force F h

; (b) N h

pretension effort and F h

resultant force; (c) Level of C

application point of F h

force.

E QUILIBRIUM CONDITIONS FOR THE SINGLE BLOCK

where T

he equilibrium conditions for the generic block, in the global reference frame X , Y and Z are so formulated:

e e

e e

e e

e e

e

e

0 F F F SD SC YB XA        e



(1)

0

1

0

h

1

e A and

e B are (6x3) matrices of the coefficients of the redundant unknowns

e X and

e Y on the meridian interfaces j

e D are (6x6) matrices of the unknowns e 0 F is the (6x1) vector of the dead loads, e h

S and e 1

and j +1 respectively, listed in (3x1) vectors, while interfaces i and i +1 respectively, listed in (6x1) vectors; e 0 the possible action of the hoop, and e 1 e C and

S on the radial

F is the (6x1) vector of

F the (6x1) vector of the live load increasing through the α collapse multiplier 

It is necessary to specify that, in the problem under examination, only the unknowns 1

S remain on the radial faces,

S ones are zero on the face of the block at the top and, as regards to every other block, they coincide with

because the 0 the unknowns 1

S of the i +1 face of the previous block.

Y IELD DOMAINS FOR THE GENERIC INTERFACE

W

ith reference to a generic quadrilateral interface (Fig.6), the six stress resultants on generic radial interface i +1 have to respect the yield conditions related to rocking and sliding domains; whereas, on generic interfaces j and j +1 lying on meridian symmetry planes, the stress resultants only are three (the normal force and the two bending moments) and just have to respect the conditions related to rocking domain. In the following, linearized yield domains circumscribing the actual nonlinear ones are shown.

490