PSI - Issue 79

Albena Doicheva et al. / Procedia Structural Integrity 79 (2026) 370–378

372

where T , T  and

S C  , S C are the tensile and the compressive forces in the top and bottom longitudinal reinforcing

bars in the beam passing through the connection, respectively; C C and

C C  are the compressive forces in concrete on the bottom and top edge of the beam;

C V is the column shear force. The forces are shown in Fig. 1. The determination of T and T  presents a difficulty, and acceptance is suggested



M

M

 

and

(2)

T

T



b

b

j

j

b

b

where b M and b M  are the moments at the column face, and b j and b j  are the lengths of the bending moment arms at the column face. According to the proposed acceptance, the forces in the top and bottom reinforcement should be equal to each other. However, these forces are not equal, as was shown in Doicheva (2023a), Doicheva (2023b), Doicheva (2024a), Doicheva (2024b), Doicheva (2025a), Doicheva (2025b), Doicheva (2025c), Doicheva (2025d). In this article, the following tasks are set: 1. to determine the expressions for the forces from Figure 1, at the column face, when the cantilever beam is loaded with a linearly distributed load and has an asymmetric cross-section; 2. to compare the obtained results for shear force with the results from Equations (1) and Eurocode 8 (2004). 3. Mathematical model of cantilever beam The cantilever beam from Fig 2 is considered. The beam is statically indeterminate. The beam under the conditions of special bending with tension/compretion and Bernoulli-Euler hypothesis is considered.

h/5

h/5

h

d e c a

(34h-60)h 72h-120

h

(38h-60)h 72h-120

h

a)

b)

Figure 2. Mathematical model of the beam; (a) Supports of cantilever beam; (b) Cross-section of the beam - asymmetrical

The beam is loaded with a vertical linearly distributed load   kN/cm' q . In vertical support 1 a vertical support reaction   kN А occurs. At the level of the reinforcing bars, elastic supports 2 and 3 , with linear spring coefficients 2 k and 3 k , are introduced. They are set as the reduced tension/compression stiffness of the reinforcing bar by the multipliers 2  and 3  , respectively. The connection of the concrete on the beam with that of the column is taken into account by linear spring supports, The forces in all springs are reduced to a single force   1 kN H , which is symmetrically located with respect to the intact lateral edge (   2 cm b ).The force moves along the height of the beam as the crack length increases. The coefficient of the linear spring is 1 k . It is given as the reduced tension/compression stiffness of the concrete section by the multiplier 1  .

3 3 E A

1 1 EA

2 2 E A

and

(3)

k

k

k

;

1  

 

 

1

2

2

3

3

L

L

L

where: 2 3 kN/cm E     are the elastic moduli of the bottom and top longitudinal reinforcing bars of the beam passing through the beam-column connection; 2 2 cm A   and 2 3 cm A   are the area of the cross-section and 2 2 kN/cm E     and