PSI - Issue 44

Vanni Nicoletti et al. / Procedia Structural Integrity 44 (2023) 371–377 Vanni Nicoletti et al. / Structu al Integrity Procedia 00 (2022) 00 – 000

375

= √( ℎ ℎ + ) ( + ) = ( ℎ 1− 0 .8 ) 1 ℎ

(11)

(12)

Eq. 12 derives from both Eq.s 7 if some simplifying assumptions are considered: the shear V C acting on the upper column is (conservatively) neglected in the calculation on V jhd , as well as A s1 = A s2 = A s,max is assumed; A s,max represents the maximum amount of rebar (in terms of steel area) among the upper and lower layers of the beam framing into the joint. Both hypotheses configure as conservative assumptions because they contribute to increase the shear demand. Assuming τ Ed the dependent variable and σ Ed the independent one, Eq.s 10, 11 and 12 represent functions of the type τ Ed = f ( σ Ed ), fixing all the other parameters, namely the concrete grade, the B-CJ typology (interior or exterior), the steel design mechanical properties, and two other non-dimensional parameters defined as follows: = ℎ ℎ ⁄ (13a) ̅ = ℎ ℎ ⁄ (13b) The and ̅ coefficients are defined as the lateral reinforcement ratio and the plane reinforcement ratio, respectively, because they represent the hoop area amount spreads all along the joint lateral area ( b j h jw ) and the horizontal effective joint area ( b j h jc ). As an example, by fixing the concrete grade C35/45 and the steel characteristic yield stress f yk = 450 MPa, Eq.s 10, 11 and 12 can be drawn as reported in Fig. 1; this diagram is the so called nomogram. Lines I and E are the graphical representation of Eq. 10 in case of interior (I) or exterior (E) joints and, hence, they represent the resisting domain edges considering the concrete compressive bearing capacity. As expected, the domain relevant to exterior joints is smaller than that for the interior ones because of the 20% reduction of the resistance. The eleven curves on the right hand-side of the nomogram are the graphical representation of Eq. 11, drawn by varying the lateral reinforcement ratio , while the six ones on the left-hand-side are the graphical representation of Eq. 12, by varying the plane reinforcement ratio ̅ . They represent the domain boundaries of the diagonal tensile stress assessment following the Approach 1 (on the right side) and Approach 2 (on the left side), by varying the hoop area amount and/or the joint dimensions. It is worth noting that for the Approach 1, the curve for = 0 (i.e. null transverse reinforcement) provides a non-null tensile resistance for the B-CJ due to the tensile resistance of the concrete; however, a minimum amount of hoops within the joint is always required by the code, so this curve is not representative of a real case. The nomogram provides a chart in which all three B-CJ verifications are put together and considered at the same time; in this way, a fast and direct comparison between their outcomes can be achieved and the joint at hand can be quickly sized. The nomogram use and, hence, the joint sizing can be summarized as follows: 1. the shear ( τ Ed ) and normal ( σ Ed ) stresses acting on the joint are calculated after the seismic analysis (adopting actions obtained from a numerical model of the considered structure); 2. the point P with coordinates equal to ( τ Ed , σ Ed ) is positioned on both sides of the nomogram; 3. the concrete compressive bearing capacity is assessed verifying that point P is placed below (or at least on) the I or E curves of Eq. 10 (for interior and exterior joints, respectively). If this condition is not satisfied, then the B-CJ must be re-sized (or the concrete grade must be increased); 4. the reinforcement ratios ( for the Approach 1 and ̅ for the Approach 2) needed to satisfy the verifications can be directly read on the nomogram (a linear interpolation can be performed, or the curve above the point can be conservatively considered);