PSI - Issue 44

A. Floridia et al. / Procedia Structural Integrity 44 (2023) 504–511 Author name / Structural Integrity Procedia 00 (2022) 000–000

508

5

2.2. Internal forces corresponding to point P V Point Pv is characterized by a null bending moment and by an assigned axial force. The shear force corresponding to a value of the angle θ is evaluated taking into account six combinations of values of the variables y 1 , y 2 , 1 σ  , 2 σ  , 3 σ  , c1 σ , c2 σ (see Table 1). In each combination, the values of one or two variables are not fixed as they are to be calculated case by case to ensure a null bending moment on the cross-section. These free variables are highlighted in Table 1 by grey hatches. The proposed combinations intend to maximize the size of zone F 3 for assumed limit stresses of concrete or steel. The corresponding axial resistances are identified by ( ) V, i R N (i=1 to 6) and increase with the number in the superscript of the parameter. In particular, the first combination of the variables identifies the maximum axial compressive resistance whereas the latter identifies the maximum axial tensile resistance of the cross-section. If the maximum stress resultant in the longitudinal reinforcement is symmetric with respect to the x-axis, the limits of zone F 3 and the distributions of the normal stresses are symmetric with respect to the x-axis. If this is not the case, the proposed combinations allow limits of zone F 3 and distributions of normal stresses that are non-symmetric. In particular, if the maximum contribution of A slf1 to the rotational equilibrium is not lower than that of A slf2 , i.e. slf1 1,lim y A f y   ≥ slf2 2,lim y A f y  , the proposed combinations of y 1 , y 2 , 1 σ  , 2 σ  , 3 σ  , c1 σ , c2 σ are reported in Table 1. The axial resistance (V, 1) R N is calculated assuming that 2 σ  and 3 σ  are equal to the compressive yield stress of steel while c2 σ is equal to the compressive strength of concrete f c . The variables ( ,1) 1 V y and ( ,1) 2 V y are null if slf1 1,lim y A f y  = slf2 2,lim y A f y  while they are equal to 1,lim y −  if slf1 1,lim y A f y  > slf2 2,lim y A f y  . The free variables c1 σ and 1 σ  must be specified to provide a null bending moment. To this end, the stress 1 σ  is first set equal to the compressive yield stress of steel and c1 σ is calculated by the following equation to ensure a null bending moment ( ) { } ( ) 2 2 2 2 c1 2 3 1 slf1 lw 1 1 1,lim 1,lim c,lim 0.5 M M A y b y y b y y     σ = + + σ + ρ − −        (11) where the bending moment contributions M 2 and M 3 are reported in Equations (8) and (9). If the value of c1 σ resulting from Equation (11) is positive and lower than f c , it is accepted as the value of c1 σ and 1 σ  is assumed equal to the compressive yield stress of steel. If this is not the case, c1 σ is set equal to the limit of the concrete strength (0 or f c ) nearer to the value obtained by the above calculation and 1 σ  is calculated by means of the rotational equilibrium equation ( ) ( ) 2 2 2 2 1 2 3 c1 slf1 lw 1 1 c,lim 1,lim 1,lim 0.5     σ = − − + σ − + ρ −        M M b y y A y b y y (12) 3 σ  are equal to the compressive yield stress of steel and the stress of concrete in zones F 1 and F 2 is null. The free stress 1 σ  is first assumed equal to the compressive yield stress of steel and the value of the variable y 2 is calculated by rotational equilibrium, i.e. The axial resistance (V,2) R N is determined assuming that the variable y 1 is equal to 1,lim y −  , the stresses 2 σ  ,

1 2

(

)

0.5 + ρ

slf2 A y

b y

   

    

0.5 − σ

2 y M − σ ρ − σ − 3 lw c3, l 0.5 b

b y

σ

2

 

 

2

lw

2

c2

1





2,lim

2,lim





1

c,lim

(13)

y

= 

(

)

2

c2 3 lw c3, l − σ + σ ρ − σ ρ − σ 0.5 0.5 0.5 b b b 2 lw





where the bending moment contribution M 1 is reported in Equation (7). If the value of y 2 resulting from Equation (13) is not higher than 2,lim y  , it is assumed as the value of the variable y 2 . If this is not the case, y 2 is fixed equal to 2,lim y  and 1 σ  is calculated by Equation (12). The axial resistance (V,3) R N corresponds to null values of variables 2 σ  , 3 σ  , c1 σ , c2 σ and to a value of y 1 equal to 1,lim y −  . The free stress 1 σ  is first assumed equal to the tensile yield stress of steel and the value of the variable y 2 is calculated by Equation (13). If this value of y 2 is higher than 2,lim y  , y 2 is fixed equal to 2,lim y  and 1 σ  is calculated by Equation (12).