Issue 51

C. Anselmi et alii, Frattura ed Integrità Strutturale, 51 (2020) 486-503; DOI: 10.3221/IGF-ESIS.51.37 494 Moreover, named d the number of the yield domain conditions, in matrix form we have: 0 TN XDY    (12) where Y is a ( d x1) vector, D is a ( d x n ) matrix and finally TN is a ( d x1) vector of known terms. Now the static theorem of limit analysis can be applied to evaluate the collapse multiplier α  by implementing the following problem of mathematical optimization: maximize α subject to: (13) 0 F F AX   1 α (balance equations) 0 TN XDY    (yield domain conditions) 0  α (not negativity of α) being X and α the unknowns of the problem. R ECONSTRUCTION OF THE COLLAPSE MECHANISM nce the unknowns α and X have been defined through the static theorem of the limit analysis (paragraph 5), is possible pursue the kinematic problem to identify the corresponding collapse mechanism. The unknowns of problem are: - ( m x1) vector u which collects the degrees of freedom, six for every block of the discretized structure; - ( n x1) vector Δ which collects the possible displacements between the interfaces (six for every radial interface and three for every meridian one); - ( d x1) vector λ which collects the generalized strain rates associated to the yield conditions Y (Eq.12). These unknowns are bounded by kinematic conditions : ΔuA  T (14) and by the flow rule λDΔ T  (15) from which, by eliminating Δ , we obtain the compatibility conditions : 0 λDuA   T T (16) being T A and T D the transposed matrices of A and D respectively (defined in paragraph 5). If we denote with ΔXuF uF T T 1 T v α L    the virtual work done by the forces through the associated displacements and by the stress resultants through the mutual displacements between the interfaces, the problem of mathematical optimization is set in the following way: It is imposed 0 L v  subject to: (17) 0 λDuA   T T (compatibility conditions) 0 λ  (not negativity of λ ) Once solved the problem (17), the kinematic mechanism is represented at the instant in which the collapse is reached through suitable matrices of directional cosines. O