PSI - Issue 72

Victor Rizov / Procedia Structural Integrity 72 (2025) 120–127

124

1

(15)

.





P

m 

e

In Eq. (14), n is the number of revolutions per minute of the motor (the periodic load acting on the frames is induced by a motor placed in section, D 3 , of the horizontal member of the frames). The maximum and minimum SERR, G max and G min , induced by the periodic load are given in terms of G ef and G dn by Eqs. (16) and (17), respectively.

(16) (17)

st dn G G G   max , st dn G G G   min .

f F G 2

The SERR, G st and

, are derived also by Eqs. (18) and (19) for check-up.

U



F

1 ,

G

(18)



st

A



U



F f 2

G

(19)



,

F f 2

A



where

A ab  .

(20)

The strain energy, U E , due to F 1 is given by Eq. (21).   i F U U 1 .

(21)

The strain energy, U i is defined in terms of the specific strain energy, u 0 i , by Eq. (22).

2 1

i u

0 

i i  

(22)

,

where σ i and ε i are defined by Eqs. (4) and (5), respectively. The strain energy,

F f U 2 , due to F 2f is obtained by replacing of F 1 with F 2 f in Eq. (22) (analogical replacement is

done also in Eqs. (1), (2) and (3)). The SERR, G st and

F f G 2 , determined through Eqs. (18) and (19) confirms the SERR derived by Eq. (10). The frame in Fig. 1b is explored next. The SERR is determined first for the load, F 1 . The support conditions of the frame in Fig. 1b indicate that this frame has one degree of static indeterminacy. Therefore, the static indeterminacy has to be resolved before determining of the SERR. The horizontal reaction, R D,H , in the pinned support in point, D 6 , is taken as a hyperstatic unknown. Thus, the static indeterminacy is resolved by Eq. (23) (this equation expresses the fact that the strain energy has extremum for the actual value of the hyperstatic unknown) (Mladenov et al. (2012)).

U



6 D H F

1 

(23)

0

.

R



Then the SERR is derived through Eq. (10) (before this,  , i  , ni z , i  and i  are obtained from Eqs. (1), (2), (3), (4) and (5) by using the values of the axial forces and bending moments in the frame in Fig. 1b). A check-up is done