Issue 51

C. Anselmi et alii, Frattura ed Integrità Strutturale, 51 (2020) 486-503; DOI: 10.3221/IGF-ESIS.51.37 493 0 TN tan t 0    (6) 0 TN tan r 0    (7) Thus, in Fig. 10,  coincide with the angle of friction  0 . G d 3 R d 4 d 1 d 2 k 3 k 2 k 4 k 1 (a) (b) φ 0 N T r T t φ 0 Figure 11 : Yield domain for sliding. (a) Yield domain normal force N - shear force T ; (b) Yield domain normal force N - twisting moment M n , "equivalent" circular section. Instead, for the yield domain normal force N - twisting moment M n reference was made to an "equivalent" circular section having radius R equal to the mean of the distances of G by the sides of quadrilateral section (Fig.11b): )d d d (d 4 3 2 1     4 1 R (8) Therefore two conditions are imposed: 0 MN Rtan n 0 3 2    (9) being, in the yield domain by sliding of Fig. 10, tanφ = ⅔ R tanφ 0 . In matrix form, the conditions on the generic meridian interface and on radial one can be expressed respectively by: 0 TN XD Y    mf mf mf x mf (10) 0 TN SD Y    rf rf 1 rf s rf (10′) where mf Y is a (12x1) vector, mf x D is a (12x3) matrix of the coefficients of the redundant unknowns mf X (that is ) f( j X or 1) f(  j Y on the meridian faces j and j +1 respectively), listed in (3x1) vector, while mf TN is a (12x1) vector of known terms. Moreover rf Y is a (18x1) vector, rf s D is a (18x6) matrix of the only rf 1 S unknowns on the radial faces i +1, listed in a (6x1) vector, while rf TN is a (18x1) vector of known terms. G OVERNING EQUATIONS OF WHOLE STRUCTURE AND ASSESSMENT OF THE COLLAPSE MULTIPLIER  amed m the number of balance equations for whole structure and n the number of the unknowns on the interfaces, in matrix form we have: 0 F F AX    1 α (11) where A is a ( m x n ) matrix of the coefficients of the unknowns X on the interfaces, listed in the ( n x1) vector, while F is the ( m x1) vector of the dead loads and of the possible actions of the hoops, and 1 F the ( m x1) vector of the live load increasing by the unknown collapse multiplier α  N