PSI - Issue 66

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

437 5

1 1 2 2 3 3 EA E A E A E A = + + - tensile (compressive) stiffness of the composite section; 1 1 2 2 3 3 EI E I 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

0 ∫

0 ∫

dx

dx

(6)

Π=

+

+ + +

2

2

2 2 2 k k k

EI

EA

1

2

3

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, or

0; H ∂Π ∂Π ∂Π = = ∂ ∂ ∂ 0; H H

0

=

(7)

1

2

3

4.1. Mathematical model of beam with symmetrical cross-section Consider the cantilever beam with a symmetrical section from Fig. 2a). The three equilibrium conditions of statics give us respectively: A qL = ,

(8)

(9)

1 2 3 N H H H =− − + ,

2 qL h

H Hb

1 a − + 2

2

1

∑

0 M H

H

(10)

= → =

−

.

4

2

3

The bending moments for the beam will be:

h

qx

2

1 M qLx H а H а H b   = − − − − −     3 2 1 2

(11)

.

2

Substitute Equations (8)–(11) in Equation (6). We apply the first and third conditions from Equation (7). A system of two linear equations with respect to the two unknowns ( 1 3 and H H ) is obtained. We substitute them in Equation (10). 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 3 2 3 2 3 qLk EA EI Lka h Lakkn EIL ak kn H EI EAD LD   − − + + −   = + , (12) ( ) [ ] { } { } 2 2 2 2 2 2 3 1 1 1 3 2 1 3 2 1 2 12 4 12 6 qLka EA EI Lka Lkh Lkkn EIL k k H EI EAD LD   + + + + +   = + , (13)