PSI - Issue 72

Albena Doicheva / Procedia Structural Integrity 72 (2025) 243–251

249

Where f vd [kN/cm

2 ] — design value of the yield strength of steel;

γ Rd should not be taken less than 1.2. The comparison of Equations (10) with (1) and (11) is expressed in the comparison shown in Equation (12).



M M

?

?





;

(12)

V H H H

3 s yd H H H A A f      2 1 1 2 Rd s

b     

b



3

2

1

j

j

j

b

b

where b M is the moment of the beam on the face of the column. It was calculated using the RuckZuck program, Version 4.0 (2009). 2 21kN/ cm yd f  — design value of the yield strength of steel

h=25 cm; g=0m

h=25 cm; g=1,0m

2*(Mb/jb)/P Eurocode/P (H1+H2+H3)/P, E=1700 (H1+H2+H3)/P, E=4000

10.00 11.00

3 5 7 9 11

Eurocode/P

2*(Mb/jb)/P

7.00 8.00 9.00

(H1+H2+H3)/P, E=1700 (H1+H2+H3)/P, E=4000

H1+H2+H3/P

H1+H2+H3/P

2.0

2.4

2.9

3.8 h/b

5.6

2.0

2.4

2.9

3.8 h/b

5.6

10.0

50.0

10.0

50.0

a)

b)

0 g m  ; ( b )

1.0 g m  .

Figure 5. Comparison of the parameters on Equation (12) calculated by Equations (5) – (7), symmetric cross-section: ( a )

Figure 5 shows the variation in the sum with respect to the parameters of the three support reactions, H 1 / P + H 2 / P + H 3 / P , calculated by Equations (5) – (7), while the crack between the beam and the column grows. The comparison is made by Equation (12). The results of Fig.5 show the differences between the exact method ( 3 2 1 H H H   ) and the approximate method ( M b / j b + M' b / j' b ) used in Equation (1) for the symmetrical section, and with Eurocod calculated by Equation (11). For sections with E 1 = 1700 kN/cm 2 for concrete and 0 g m  , the differences between the two first methods exceed 27% before the opening of a crack and at its final size. The biger difference is at h/b = 50 – 29.85%. For concrete with E 1 = 4000 kN/cm 2 , the biger difference is at h/b = 50 – 48.45 %. The comparison of the exact method with Eurocode shows close results at the extreme values of H 3 + H 2 + H 1 22% to 38.27% for both moduli. 7. Conclusions A solution for a beam with a special arrangement of the supports was developed. The actual dimensions of the beam were taken into account. The beam was loaded with two symmetrically located moments. The derived expressions for the reactions of the horizontal supports yielded results that clearly show the distribution of the forces along the height of the beam. The formulas were derived for the limit stage and enabled us to determine the distribution of forces before and after the appearance of a crack between the beam and the column. The resulting expressions for the support reactions take into account the influence of both the geometry of the beam and the material properties of all its components. This makes it possible to trace the variation in the forces acting on the beam and entering the beam – column connection with different data combinations of the included quantities. A comparison was made between the contribution of beam forces to the shear force value in RC internal beam – column connections and those known from the literature and from Eurocode. The results showed that the proposed exact method gives results that differ from the adopted one by 27% to 48%, depending on the stage of crack development. The difference between the new exact method and that of the Eurocode ranges from 22% to 38.27%, based only on the largest shear force value determined by the exact method for g=0 m.