PSI - Issue 66

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

439

7

{

}

[

]

2 6 qLka EA EI Lkh Lkd EIL k k Lkkn EI EAD LD   + + + + +   + 2 2 2 2 1 2 3 1 3 1 3 3 3 3

2

(23)

,

H

=

{

}

2

4

3

{

}

[

]

[

]

2 6 qLk EAd EI Lka EILkn ka Lkkan a h H EI EAD LD   − + + − +   = + . 2 2 3 2 1 3 2 1 2 3 2 3 3 3

(24)

{

}

3

4

C h b z = − ; 3

where 2

2 n d h = −

2 3 1 2 D k a k d k h = + + ( ) 1 2 2 2 3 2

2

(

)

(

)

2

4 2 2 3 D kkaah kkdh kkaad = + + − + + . Neglecting the normal force in the strain potential energy expression, the support reactions become ( )( ) 2 2 1 2 2 1 3 3 6 qLk EI Lk a h H EID − − = , 1 3

(25)

2 6 qLk а EI Lk d Lk h EID    + +  2 2 2 3 1 2 3

H

(26)

=

,

2

3

(

)

2 6 qLk d EI Lk a EID − 3 2 3

2

H

(27)

=

.

3

3

By substituting d a = , c e = and

in Equations (22)–(24) and Equations (25)–(27), the formulas

/ 2

C z a c h = + =

for the symmetric cross-section Equations (12)–(14) and Equations (15)–(17) are obtained, respectively. 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 two cross-sections of the beam, both symmetrical and asymmetrical indicated in Figure 3. For both cross-sections, the following is accepted, and dimensions are 25/25 cm; 75/75 cm. 2 2 3 25cm A А = = for cross section 25/25 cm, 2 2 3 75cm A А = = for cross section 75/75 cm 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 = and 3cm c = —the cover of the reinforcement; 200cm L = —the length of the beam; 2 1 1700kN/ cm E = —the modulus of elasticity for normal concrete and 2 1 3700kN/ cm E = for high-strength concrete; 1 1 ζ = —multiplier for reduced tensile/compressive stiffness of the concrete section; 2 1 ζ = and 3 1 ζ = —multiplier for reduced tensile/compressive stiffness of the reinforcing bars; 0,05 kN/cm' q = —for all numerical results. 5.1. Symmetrical cross section А beam with a cross -section of 25/ 25cm and 75/ 75cm , as shown in Figure 3a, is introduced. The distance [ ] cm b varies in the interval [ ) 12.5, 0 and is monitored by the ratio / h b .