PSI - Issue 78

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

729

y,i   N N 

sin

0

   V V

(2)

dy,i

c

where  is the angle of inclination of the diagonal rebars, V is the shear force, N c is the resultant of the normal (longitudinal) compressive stresses of the uncracked concrete, V c is the resultant of the shear stresses of the uncracked concrete, N z is the resultant of the stresses in the longitudinal rebars, N y is the resultant of the stresses in the transverse rebars, N dz and N dy are the longitudinal and transverse components of the resultant of the stresses in the diagonal rebars, respectively. In the above equations, the summation is assumed to be extended to all the rebars that are intersected by the assumed failure surface. In addition, axial forces are positive when tensile, whereas shear forces and reactions are positive when directed as reported in Figure 2a. The equation of rotational equilibrium about point P C , where the stress resultant N c is applied, may be written as

Ny,i    z N y N z N y,i Nz,i z,i Ndy,i  



sin

cos

0

Ndz,i y N

  V z

(3)





d,i

d,i

R

where z is the distance, along the z -axis, from point P C to the resultant indicated in subscript and y is the distance, along the y -axis, from point P C to the resultant indicated in subscript (Fig. 2b).

(a) (b) Fig. 2 Internal forces and distances involved in the translational equilibrium

2.5. Transverse normal stresses on the uncracked concrete The uncracked concrete is considered to be subjected to longitudinal and transverse normal stresses, combined with shear stresses. The transverse compressive stresses are caused by stress fans that are assumed here to originate from the top of the transverse rebars (stirrups of hangers) and from the bottom of the loading plates. The transverse compressive stresses originating from the single transverse rebar are not equal on the two sides of the rebar. Indeed, they mostly tend to spread in the direction of the lines connecting the top of the transverse rebar to the support or lower end of the hangers. To evaluate the transverse normal stresses in the compressed zone of the failure surface, a procedure is formulated which sums the effects of the loading plate and those of all the transverse rebars located in the length of the beam from the failure surface to the loading plate. Referring to the single transverse rebar, the centroid of the top cross-section of this element (point P T in Fig. 3a) is connected to either the intersection of the vertical line passing through the centroid of the support and the horizontal line passing through the centroid of the tensile reinforcement in the nib (point P B1 ) or the intersection of the centroidal axis of the outermost hanger and that of the tensile reinforcement in the full-depth section (point P B2 ). The line connecting P T to P B1 is later named L C1 , whereas the one connecting P T to P B2 is named L C2 . Similarly, the centroid of the bottom of the loading plate is connected to either P B1 or P B2 and the lines connecting the bottom of the loading plate to either P B1 or P B2 are named L C1 and L C2 , respectively (Fig. 3b). Lines L C1 and L C2 are inclined at angles   and   with respect to line L V , which is the vertical line passing through P T .