PSI - Issue 72

Albena Doicheva / Procedia Structural Integrity 72 (2025) 235–242

237



M

M

 

and

(2)

T

T



b

b

j

j

b

b

where M b and M' b are the moments at the column face, and j b and j' b 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) and Doicheva (2025a). 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; 2. to compare the obtained results with the results from Equations (1) and Eurocode 8 (2024). 3. Mathematical model of cantilever beam A cantilever beam is considered, Fig 2. The beam is statically indeterminate. The beam under the conditions of special bending with tension/compretion and Bernoulli-Euler hypothesis is considered.

h a e

e a

h

a)

b)

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

The beam is loaded with a vertical linearly distributed load q [kN/cm']. In vertical support 1 a vertical support reaction A [kN] occurs. At the level of the reinforcing bars, elastic supports 2 and 3 , with linear spring coefficients k 2 and k 3 , are introduced. They are set as the reduced tension/compression stiffness of the reinforcing bar by the multipliers ζ 2 and ζ 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 H 1 [kN], which is symmetrically located with respect to the intact lateral edge (2 b [cm]).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 ζ 1 .

3 3 E A

1 1 E A

E A

and

(3)

k

k

k

;

1  

 

 

2 2

1

2

2

3

3

L

L

L

where: L [cm] is the length of the beam; A 2 [cm] and A 3 [cm] are the area of the cross-section and E 2 [kN/cm

2 ] and E

2 [kN/cm

2 ] are the moduli of elasticity

of bottom and top longitudinal reinforcing bars in beam passing through the connection; A 1 [cm] is the area of the cross-section of the concrete on the beam; E 1 [kN/cm 2 ] is the modulus of elasticity of the concrete on the beam. The following notations have also been introduced, Fig. 2: H [cm] - the heigth of the beam;