Issue 51

R. Landolfo et alii, Frattura ed Integrità Strutturale, 51 (2020) 517-533; DOI: 10.3221/IGF-ESIS.51.39

vector c with normal forces k t . The limit analysis problem for settlement-induced collapse is formulated in terms of a cone programming problem, in the context of the lower bound limit analysis theorem, as follows [3]: n and shear forces 1 k t and 2 k



max s.t.

Ac f

f

 



(1)

D

S





3

2  

2



c

c  



n t

t

n

: μ

 , 

0 

k

k

k

k k

1

2

Figure 1 : Blocks i , contact interfaces j and contact points k adopted in the rigid block model for limit analysis (a); internal contact forces at contact points and dead loads at blocks centroid (b). The unknowns of the optimization problem are the load factor  and the vector of internal forces c . In the problem (1), the first constraint represents the equilibrium condition between internal static variables associated to contact interactions and external loads, being A the equilibrium matrix of the rigid block model. D f and S f are, respectively, the vector of dead loads and the vector of the support reaction at the moving support which is used to model the settlement. The second constraint in problem (1) represents the Coulomb failure condition and corresponds to a convex cone which is a function of variables 1 k t , 2 k t , k n and of the friction coefficient µ . The failure mode and the relevant kinematic variables corresponding to the upper bound (dual) formulation of the limit analysis problem are derived from the Lagrange multipliers f of the dead load is also assigned to the additional rigid block which is used to model the moving support, and which is associated to the single degree of freedom governing the ground movement, i.e. the vertical displacement. As such, the vector D f can be expressed follows:   T 0 0 0 0 0 i V    Di f ;   T 1 ... ... Ds f  D D Di f f f . (2) is intended to be representative of the reaction at the base support in the initial configuration (i.e. the configuration before settlement) and is assigned as a factor of the total weight of the model. Differently from lateral loads analysis, in the case of settlement analysis, a varying load  S f is applied to the centroid of the moving support block only. This load is directed downwards and is expressed as a factor of the initial value Ds f . As such, the vector S f can be expressed as follows.   T 0 ... 0 ... , Ds f   S f (3) and the magnitude of the reaction at the moving support corresponding to the onset of the failure mechanism is   1 Ds f   associated to the solution of the static problem (1). The vector of dead loads collects the gravity loads Di f associated to the unit weight  and volume i V of the block i in the wall panel (or to the permanent loads associated to floor loads). An initial value Ds The value Ds f

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