PSI - Issue 79

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

373

2 1 cm A   is the area of the cross-section and

2 1 kN/cm E     is the modulus on elasticity to the concrete of the

beam; The following notations have also been introduced on Fig. 2:   cm L and   cm h - are the length and the height 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 2 2 3 3 EA EA E A E A    - tensile (compressive) stiffness of the composite section; 1 1 2 2 3 3 EI EI E I E I    - bending stiffness of the composite section. 4. Support reactions

The solution is based on Menabria’s theorem about statically indeterminate systems in first-order theory. The potential energy of deformation in special bending, combined with tension (compression) and with the effects of linear springs, taken into account, will be as follows:     2 2 2 2 2 L 3 1 2 1 1 L M x N x H H H It is a well-known fact that, according to Menabria’s theorem, the desired hyperstatic unknown is determined by the minimum potential energy condition with respect to it, Equation (4). The solutions give the formulas of the horizontal support reactions, provided below:         2 2 2 1 2 1 3 1 2 2 3 1 1 1 2 4 4 24 qLk EA EI Lak h EIL kn ka Lakk n a d H EI EAD LD                  , (5)       2 2 2 2 2 2 1 1 3 1 3 1 3 1 2 1 2 4 4 24 qLka EA EI Lkh Lkd EIL k k Lkkn H EI EAD LD            , (6)         2 2 2 3 2 1 1 2 1 2 1 1 3 1 2 4 4 24 qLk EAd EI Lka EILkn ka Lkkan a h H EI EAD LD            . (7) 0  0  1 2 3 2 2 2 2 2 k k k dx dx EI EA      ; 1 2 3 0; H         0; 0 H H  (4) Neglecting the normal force in the strain potential energy expression, the support reactions become   2 2 1 2 1 1 1 4 24 qLk EI Lk a h H EID    ; 2 2 2 2 3 1 1 2 1 4 24 qLk а EI Lk d Lkh H EID        ;   2 2 3 2 3 1 4 24 qLk d EI Lk a H EID   The solution was performed in the symbolic environment of the MATLAB R2017b program. 5. Results and discussion The numerical results are for the cross-section of the beam, indicated in Figure 2b) with two values of the dimension h - 30 cm and 75 cm for sections - 30/30 cm and 75/75 cm. The following are also accepted for the calculations: (9) where 1 C h b z   ; 1 1 n d h   2 3 1 1 D k a k d k h    ; 2 2 1 2       2 2 1 2 3 D kkaah kkdh kkaad       1 2 1 1 3 (8)