PSI - Issue 66

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

436 4

The beam is loaded with a vertical uniformly 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 ζ and 3 ζ , respectively.

3 3 E A

E A

and

(4)

k

k

ζ =

ζ =

2 2

3

3

2

2

L

L

[ ] cm L is the length of the beam; 2 3 cm A   are the area of the cross-section and

where:

2 2 cm A   and

2 2 kN/cm E     and

2 3 kN/cm E     are the moduli of

elasticity of bottom and top longitudinal reinforcing bars in beam passing through the connection. The supporting reactions that occur here are [ ] 2 kN H and [ ] 3 kN H . The connection on the concrete of the beam with that of the column is taking into account by linear spring supports act along the vertical wall of the beam. The forces in all the springs are reduced to one force [ ] 1 kN H . In the case of large deformations, part of the vertical edge is destroyed. The unbroken edge has length of [ ] 2 cm b . The reaction [ ] 1 kN H , which is symmetrically located with respect to the intact lateral edge, moves along the height of the beam as the crack length increases. For convenience, it has been transferred [ ] 1 kN H to the support along the bottom edge (support one), after applying Poinsot's theorem concerning the transfer of forces in parallel to their directrix. This necessitated the introduction of compensating moments [ ] 1 kN.cm Hb . The coefficient of the linear spring is 1 k . It is set as the reduced tensile/compressive stiffness of the concrete section by the multiplier 1 ζ .

1 1 E A

(5)

k

1 ζ =

1

L

[ ] cm L is the length of the beam;

where:

2 1 cm A   is the area of the cross-section of the concrete on the beam 2 1 kN/cm E     is the modulus of elasticity of the concrete on the beam As a consequence of the linear deformations in the cantilever beam, a normal axial force occurs [ ] kN N , which is introduced at the free end, Fig.2.

h/5

h/5

h

h a e e a

d e c a

(34h-60)h 72h-120

h

(38h-60)h 72h-120 z

h

h

a)

b)

Figure 3. Cross-section of the beam; a) symmetrical cross-section; b) asymmetrical cross-section

The following notations have also been introduced, Fig. 3: [ ] cm h - the heigth of the beam; [ ] cm e and [ ] cm a - offset of the reinforcing bars from the top and bottom edges of the beam and from the axis of the beam, respectively; 1 1 E A ; 2 2 E A and 3 3 E A - tensile (compressive) stiffness of the concrete cross-section and the reinforcing bars, respectively; 1 1 E I ; 2 2 E I and 3 3 E I - bending stiffness of the concrete cross-section and the reinforcing bars, respectively; ( ) 1 1 y I I , ( ) 2 2 y I I and ( ) 3 3 y I I are the moment of inertia of the concrete cross-section and of the top and bottom reinforcing bars relative to the principal axis of inertia у ;