PSI - Issue 72

Albena Doicheva / Procedia Structural Integrity 72 (2025) 243–251

247

Neglecting the normal force in the strain potential energy expression, the support reactions become:

















 D a k k k k D         2 1 2 1 3 4 5

 

2 2 3 MkL gEIbL g agk k gL    24 2 

1

2

3

(8)

,

H



1

3

 



 











 D a k k k k D      2 2 1 2 1 3 2 96 Mak L g EI kD 1 4 5 3

 D a k k k k D      3 2 1 2 1 3 2 96 Mak L g EI kD 1 4 5 3

(9)

,

H

H





2

3

where additionally is introduced:  3 3 5 1 192 8 D EI  











2 61212 6636 24 EI kL g Lb Lg Lb Lg bg Lbg Lak k          The solution was performed in the symbolic environment of the MATLAB R2017b program. 5. Results and discussion The numerical results shown in Section 5 are for symmetrical cross-section of the beam from Fig. 2 with dimensions 25/25 cm. 2 2 3 10,05cm A А   for 5 16  reinforcing bars and 2 2 3 39,000 kN / cm E Е   represent the areas of the bottom and top reinforcement for two sections, and the moduli of elasticity, respectively; 3cm e  — the cover of the reinforcement; 700cm L  — the length of the beam; E1 = 1700 kN/cm 2 — the modulus of elasticity for normal concrete and E1 = 4000 kN/cm 2 for high-strength concrete; ζ 1 = 10 — multiplier for reduced tensile/compressive stiffness of the concrete section; ζ 2 = 20 and ζ 3 = 20 — multiplier for reduced tensile/compressive stiffness of the reinforcing bars; M = 5000 kN·cm — for all numerical results. The distance b [cm] varies in the interval [12.5, 0) and is monitored by the ratio / h b . All results are in parametric form. For this purpose, a force 50kN P  from a couple with a arm 100cm d  was used. 2 2 2 2 2 2 3 96

h=25 cm, g=1,0m

h=25 cm, g=1,0m

0.00 1.00 2.00

0.00 1.00 2.00

H1/P H2/P H3/P

H1/P H2/P H3/P

H1/P; H2/P; H3/P

H1/P; H2/P; H3/P

1.0

3.0

5.0

7.0 h/b

9.0

1.0

3.0

5.0

7.0 h/b

9.0

11.0

13.0

11.0

13.0

a)

b)

Figure 3. The parameters of the three support reactions for a beam with symmetrical cross-section 25 /25 cm and a) 2 1 4000kN/cm E  , calculated by Equations (5) – (7). On Fig. 3. the parameters of the three support reactions for a beam with symmetrical cross-section 25/25cm and concrete elastic moduli E 1 = 1700 kN/cm 2 and E 1 = 4000 kN/cm 2 are calculated by Equations (5) – (7). Support reaction H 1 is constant and there are values smaller than those of the other two support reactions, H 2 and H 3 . They decrease slightly with the opening of the crack between the beam and the column. 2 1 1700kN/cm E  b)