PSI - Issue 72

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

246

The following notations have also been introduced, Fig. 2:   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; EA = E 1 A 1 + E 2 A 2 + E 3 A 3 - tensile (compressive) stiffness of the composite section; EI = E 1 I 1 + E 2 I 2 + E 3 I 3 - 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

0; H         0; H H

0



(4)

0 

0 

dx

dx

;





  

2

2

EI

EA

k k k

1

2

3

1

2

3

According to Menabria’s theorem, the desired hyperstatic unknown is determined by the minimum potential energy condition with respect to it, Equation (4). A system of three linear equations with respect to the three unknowns ( H 1 , H 2 and H 3 ) is obtained. The solutions give the formulas of the horizontal support reactions, provided below:













    

2 EAEIbL g agk k gL        2 4 2 2 3

 

 







2

3

12

2 Mk L g



 















1

2 EILkk gLb akk      4 4

  

2 La gk k g L  2 3 3 4



 





2

3

2

3

H



(5)

,

1

2 EAD EID LakkkD   1 2 3 3 2 1 2









   

   

96 EA EI аkaD EILk g L a b ak        24 12 48 48 96

   









1 4

1

3

2 Mk L g

 









2

24 3 Lak k D ag g L  



1 3

4

H

(6)

,



2

2 EAD EID LakkkD   1 2 3 3 2 1 2









   

   

1 4 EA EI а k aD EIL k Lak k D ag g L        96 24 3

   

24 12 48 48 96 g L a b ak     







1

2

2 Mk L g

 









3

1 2

4

(7)

H

.



3

2 EAD EID LakkkD   1 2 3 3 2 1 2

where:





















2

2 gkgL bg Lb Lkk abkbL          6 4 48 2 2

 

3 192 8 48 EI Lk  

D EI 



;

1

1

1

2

3

1





4 108 120 36  

2 4 k k k k a L g  

3 Lg L g 2 2

 

1 2  



1 3





 La kkLL a b g LaLbLgabagbgLg g a b kkLL a b g LaLbLgabagbgLg g a b                              ; 4 4 3 2 2 3 108 120 36 D L g Lg L g     ; 3 3 2 2 2 4 12 6 6 72 24 DL g Lg Lg bg Lbg          2 2 192 (4 48 48 48 24 24 24 96 96 96) (4 48 48 48 24 24 24 96 96 96) 2 1 2 3 2 3 2 2 2 2 2 2 1 2 2 2 2 2 2 2 1 3 96 D EIL k k k k k 24 144 144 24 144 144