PSI - Issue 66

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

438

6

{

}

(

)

[

]

2 6 qLk EA EIa Lka Lakk a h EIL kn ka H EI EAD LD     − + − + + +     = + . 3 2 2 2 3 2 1 2 1 1 2 2 3 12 4 4 6 2

(14)

{

}

1

2

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

Neglecting the normal force in the strain potential energy expression, the support reactions become ( )( ) 2 2 1 2 1 1 1 3 3 qLk EI Lk a h H EID − − = ,

(15)

2 12 4 6 qLk а EI Lk a Lk h EID  + +  2 2 3 1 1

 

2

H

(16)

=

,

2

1

(

)

2

2 3 qLk а EI Lk a EID − 3 2 3

2

H

(17)

=

.

3

1

4.2. Mathematical model of beam with asymmetrical cross-section Consider the cantilever beam with an asymmetric section from Fig. 2b). The three equilibrium conditions of statics give us respectively: A qL = ,

(18)

(19)

1 2 3 N H H H =− − + ,

qL

2

(

)

H z b H d − −

−

2

1

3

C

∑

0 M H

(20)

= → =

.

4

2

a

The bending moments for the beam will be:

2 qx M qLx H d H а H z b = − − − − − .

[

]

(21)

1

3

2

1

C

2

Substitute Equations (18)–(21) 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 (20). The solutions give the formulas of the horizontal support reactions, provided below: [ ] ( ) { } { } 2 2 2 1 2 2 3 3 2 2 3 3 1 3 4 3 3 6 qLk EA EI Lak h EIL kn ka Lakk n a d H EI EAD LD     − − + − + + +     = + , (22)