PSI - Issue 66

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

446 14

Table 3. The comparison by Equations (33), neglecting the normal force in the strain potential energy expression, calculated by Equations (15)–(17), symmetric cross-section.

(

) ( 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 2.6 50 2.0 2.6 50

0.00

19.7

E 1 = 3310 kN/cm 2

-25.58 16.97 -31.02 18.75 0.00

-10.92 40.01 -17.42 42.14 19.7

E 1 = 4500 kN/cm 2

1 H H H + + ) and the approximate method

The results of Table 3 show the differences between the exact method ( 3 2

b b b b M j M j + ′ ′ for symmetrical section, calculated by Equations (15)–(17), neglecting the axial force in the expression of the potential energy of the deformation. For sections with E 1 = 3310 kN/cm 2 for concrete, the differences between the two methods exceed 16% before the opening of a crack and at its final size. The biggest difference is at h/b = 2.6 – 25.58%. For concrete with E 1 = 4500 kN/cm 2 , the biggest difference is at h/b = 2.6–31.02%. The difference between the exact solution and Eurocode is between 10% and 40%. The comparison shows that the calculated exact magnitudes of the forces for the considered static loading forces are close to those predicted by Eurocode. Designing according to Eurocode includes several calculated safety factors and values of the exact method greater than those set in Eurocode cause concern. 7. Conclusions A solution for a cantilever 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 uniformly distributed load. 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 15% to 31%, depending on the stage of crack development. The difference between the new exact method and that of the Eurocode ranges from 0.5% to 18%, based only on the largest shear force value determined by the exact method. Eurocode specifies the force that the intended reinforcement could withstand, while the new formulas show how big the forces actually are. The new formulas can protect us from choosing an inappropriate combination of data, as well as oversizing the reinforcing bars, as this will overestimate the horizontal forces at the joint. The new approach will prevent us from displacing the reinforcing bars towards the axis of the beam, as we will obtain smaller forces in the reinforcement, whereas the forces in the concrete will increase, Doicheva (2021), Doicheva (2022). The new formulas allow us to calculate the contribution of concrete to the shear force and the capacity of the concrete section. The results show that the values of all parameters included in the formulas have a serious impact on the magnitudes of the support reactions. The model allows to track how and by how much the change in just one parameter will affect the final results. The derived expressions for the support reactions can also be used for steel structures in the design of bolted connections. The obtained results can be useful for both researchers and practicing engineers. The research from this paper can help to interpret the results obtained from construct analyses and experimental studies.