PSI - Issue 66

Albena Doicheva et al. / Procedia Structural Integrity 66 (2024) 433–448 Author name / Structural Integrity Procedia 00 (2025) 000–000

444 12

The data on the material properties are reported into Doicheva et al. (2023c). The following results are for a beam with a symmetrical and asymmetrical cross-section. A size of 25/25 cm is accepted for both sections (Figure 3 а and 3b). 2 2 3 25cm A А = = and 2 2 3 18,500 kN / cm E Е = = —the areas of the bottom and top reinforcement and the moduli of elasticity, respectively; 3cm e = and 3cm c = —the cover of reinforcement; 200cm L = —the length of the beam; 2 1 3310kN/ cm E = —modulus of elasticity for normal concrete; and 2 1 4500kN/ cm E = for high-strength concrete; 2 21kN/ cm yd f = —design value of the yield strength of steel

h=25 cm

h=25 cm

9 10 11 12 13 14

13

Eurocode (H1+H2+H3)/qL 2×(Mb/jb)/qL

12

Eurocode (H1+H2+H3)/qL 2×(Mb/jb)/qL

11

10

(H1+H2+H3)/qL

(H1+H2+H3)/qL

9

h/b

2,0 2,4 2,9 3,8 5,6 10,0 50,0

2,0 2,4 2,9 3,8 5,6 10,0 50,0

h/b

a)

b)

Figure 10. Comparison of the parameters on Equation (33) calculated by Equations (12)–(14), symmetric cross-section: ( a ) 2 1 4500kN/ cm E = . Figure 10 shows the variation in the sum with respect to the parameters of the three support reactions, 2 3 H qL H qL H qL + + , calculated by Equations (12)–(14), while the crack between the beam and the column grows. The comparison is made by Equation (33). The graphs show that the proposed new model for calculating shear force gives us not only its most unfavorable value but also makes the shear force traceable throughout the crack development process. The results of Table 1 show the differences between the exact method ( 3 2 1 H H H + + ) and the approximate method ( ) b b b b M j M j + ′ ′ used in Equation (3) for the symmetrical section, and with Eurocod calculated by Equation (32). For sections with E 1 = 3310 kN/cm 2 for concrete, the differences between the two first methods exceed 10% before the opening of a crack and at its final size. The biger difference is at h/b = 2.9–16.08%. For concrete with E 1 = 4500 kN/cm 2 , the biger difference is at h/b = 2.9 –20.23%. The comparison of the exact method with Eurocode shows close results at the extreme values of 3 2 1 H H H + + -0.45 to -4.51% for both moduli. Table 1. The comparison by Equations (33) for symmetric cross-section 25/25cm. 2 1 3310kN/ cm E = ; ( b ) 1

(

) ( s A A H H H H H H + − + + + + 1 2 3 2 s

)

γ

( ′   +  − + + ′   + + 3 j H H H 2 b b b b M M H H H j 

)

100%

1

Rd

(

)

/ h b

1

Section

100%

3

2

1

(

)

3

2

1

2.0 2.9 50 2.0 2.9 50

4.25

24.78 0.45 34.58 23.91 -4.51 36.99

E 1 = 3310 kN/cm 2

-16.08 12.44 -20.23 14.45 3.52

E 1 = 4500 kN/cm 2