PSI - Issue 72

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

238

  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 EA    - 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. The solutions give the formulas of the horizontal support reactions, provided below:           2 2 2 2 1 2 1 2 3 2 2 3 2 1 1 2 4 2 4 2 12 qLk EA EI Lka h Lakkn EIL ak kn H EI EAD LD            , (5)         2 2 2 2 2 2 3 1 1 1 3 2 1 3 2 1 2 16 4 16 24 qLka EA EI Lka Lkh Lkkn EIL k k H EI EAD LD            , (6)         2 3 2 2 2 3 2 1 2 1 1 2 2 3 1 2 16 4 4 8 2 24 qLk EA EIa Lka Lakk a h EIL kn ka 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 12 qLk EI Lk a h H EID    ; 2 2 2 2 3 1 1 2 1 16 4 24 qLk а EI Lk a Lk h H EID        ;   2 2 3 2 3 1 2 4 12 qLk а EI Lk a H EID   (9) 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, indicated in Figure 2b). For both cross-sections, the following is accepted, and dimensions are 30/30 cm; 75/75 cm. A 2 = A 3 = 30 cm 2 for cross section 30/30 cm, A 2 = A 3 = 75 cm 2 for cross section 75/75 cm and E 2 = E 3 = 39,000 kN/cm 2 represent the areas of the bottom and top reinforcement for two sections, and the moduli of elasticity, respectively; e = 3 cm - the cover of the reinforcement; L = 200 cm - the length of the beam; where 1 2 h b h   ; 1 1 1 4 D k k a k h    ;   2 2 1 2 3 1 2 n a h   ; 2 1 2 n a h   ; 2 2 (2 2 ) (2 2 ) 8 D kkaa bh kk a bh kka        2 2 1 2 1 3 2 3 (8)