PSI - Issue 78

Pier Paolo Rossi et al. / Procedia Structural Integrity 78 (2026) 726–733

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(a) (b) Fig. 3 Directions of the compressive forces from (a) the top of the stirrups or (b) the bottom of the loading plate

To identify the limits of the fan of the stresses along lines L C1 and L C2 , other two lines must be determined on the right and left sides of each of L C1 and L C2 lines. Starting from the centroid of the cross-section of the top horizontal leg of the transverse rebar (stirrup or hanger), two lines - located on the right side of lines L C1 and L C2 and later named L R1 and L R2 - are first determined as characterized by angles   /2 and   /2 with respect to line L V (Fig. 4a). A similar procedure is applied with reference to lines L C1 and L C2 starting from the bottom of the loading plate (Fig. 4b). Owing to the above geometric construction, at an arbitrary vertical distance d e from the centroid of the cross-section of the top horizontal leg of the transverse rebar (or from the bottom of the loading plate), the distance along the longitudinal axis from line L V to line L Ri (i=1 or 2) is equal to the distance from line L Ri to line L Ci (i=1 or 2); this distance is indicated in Figure 4 as e and is equal to 0.5 d e tg  . As previously reported, to completely identify the fan of the compressive stresses that originate from point P T another line is necessary, which is on the left side of L C1 or L C2 and is named L L1 or L L2 . This line connects point P T to a point that, at a transverse distance d e from point P T , is at a longitudinal distance e from line L Ci (i=1 or 2) (see Fig. 4a-b). To consider the width of the elements from which the fans originate, the above lines L Ri and L Li (i=1 or 2) are finally translated to the end sides of the cross-section of the top horizontal leg of the transverse rebar or bottom of the loading plate, as shown in Figure 4. This translation identifies lines L F and thus the boundaries of the fans of the uniformly distributed compressive stresses. Owing to the above consideration, the amplitude, along the z -axis, of the single stress fan may be calculated by means of the following relation 0 σi e i i     z z d tg (i=1 or 2) where 0  z is the diameter of the transverse rebar or the length of the loading plate (along the longitudinal direction) depending on the element from which the stress fan originates. As also evident in Figure 4, the final translation of the limit lines of the fans has marginal effects on the size of the fan that originates from the transverse rebars. However, it is important to identify the fan that originates from the loading plate. The vertical force that develops in the single transverse rebar or at the bottom of the loading plate is equilibrated by the compressive stresses of the two fans that are directed to points P B1 or P B2 . The magnitude of the resultants of the compressive forces of the two struts is calculated as reported in Rossi and Spinella (2025). It is dependent on the stiffness of the struts to a vertical relative displacement of the ends, i.e. on the cross-section of the struts, on the angle of inclination of the resultants and on the distance between point P T and point P B1 or P B2 .

(a) (b) Fig. 4 Fan of the compressive stresses originating from (a) the top of the stirrups or (b) the bottom of the loading plate