PSI - Issue 44

A. Floridia et al. / Procedia Structural Integrity 44 (2023) 504–511

507

4

A. Floridia et al./ Structural Integrity Procedia 00 (2022) 000–000

Finally, the shear force V is given by the integral of the shear stresses in F 3 , i.e. ( ) 2 1 V b y y = τ −

(10)

2. N-M-V ultimate interaction domain The N-M-V ultimate domain is identified by the envelope of M-V interaction domains characterized by different values of the axial force. The construction of the single quadrant of each M-V domain is based on five points, later called P V , P M , P 1 , P 2 and P 3 . As shown in Figure 2 (where the interactions curves are derived by means of the proposed simplified method with the design values of the mechanical properties of the materials and cotg θ in the range from 1 to 2.5), points P V and P M are at the ends of the quadrant as corresponding to a null bending moment or shear force, respectively. Point P 2 identifies the point with the maximum shear strength. If more than one point is characterized by the same maximum shear strength, point P 2 identifies the point of this group with the maximum bending moment. Point P 3 identifies the point of the interaction domain with the same shear strength as point P 2 but with the lowest bending moment. Point P 1 does not correspond to the maximum value of either the internal forces. However, it identifies a change in the cross-section behaviour that has been noted when examining the results of the reference nonlinear mathematical programming problem (Rossi and Recupero 2013, Rossi 2013). The central zone F 3 , which is generally quite large in P 2 and has a null area in P M , does not shrink linearly from P 2 to P M . In particular, in the first part of this route from P 2 to P M one of the ending lines of the central zone F 3 tends to remain close to one of the limit positions of F 3 (i.e. the y-coordinate of this line is either 1,lim y −  or 2,lim y  ) while the other ending line slowly tends to the first one and reduces the area of the central zone. Then, the two ending lines move closer to each other and reach the common position corresponding to point P M . The point corresponding to the end of the first behaviour and to the beginning of the second is reported here as P 1 . In some cases, some basic points are equal. Owing to this, the generic M-V ultimate interaction curve is characterized by a number of distinct basic points ranging from two to five. 2.1. Ultimate state of stress in zone F3 With the sole exception of P M , the shear force of the single basic point is calculated as the maximum of the shear forces corresponding to angles θ variable in an assigned range of values. Once an angle θ has been selected, the normal stresses of concrete ( c3 σ ) and hoops ( s3 σ ) in zone F 3 are calculated by Equation (3). To this end, the normal stress c3 σ is first fixed equal to the concrete strength under biaxial stress state and the normal stress s3 σ is obtained by Equation (3). If this latter value is lower than the yield strength f yw of the transverse reinforcement, the above values of c3 σ and s3 σ are assumed as the normal stresses of concrete and hoops in zone F 3 . If this is not the case, s3 σ is assumed equal to f yw and c3 σ is obtained by Equation (3). The normal stress of concrete c3, l σ is calculated as 2 c3 cos σ θ . As an example, this paper shows how the internal forces corresponding to P V are calculated. For the internal forces corresponding to the other points and to the intermediate points, readers are referred to (Rossi, 2021).

V [kN]

V [kN]

30

30

A slf2 = u A slf1

A slf2 = u A slf1

P 3 ≅ P 2

600

600

A slw

A slw

60

60

hoops φ 8/15

hoops φ 8/5

P V

A slf1

A slf1

400

400

P 3

P V

P 2

A slf1 =12.57 cm²

A slf1 =12.57 cm²

200

200

u =0.50

u =0.00

P 1

A slw =3.14 cm²

A slw =3.14 cm²

P 1

0

0

P M

P M

M [kNm]

M [kNm]

0

150

300

450

0

150

300

450

Figure 2. Simplified ultimate M-V interaction curves ( f c =30 MPa; f c2 =15.8 MPa; f y = f yw =450 MPa; N =0)