PSI - Issue 19

Giovanni M. Teixeira et al. / Procedia Structural Integrity 19 (2019) 175–193 Author name / Structural Integrity Procedia 00 (2019) 000–000

185

11

frequency and load. Fig. 13 schematically shows the first 10 generalized displacements that result from a given system being excited to a constant amplitude harmonic load. Note how these curves behave as the frequency of this harmonic load is swept from 0Hz to 500Hz. The peaks correspond to the maximum contribution of every mode, which happens when load and natural frequencies coincide.

Fig. 12. Generalized displacements (amplitude) for the 10 first modes.

The frequency response functions for each load can then be evaluated by combining generalized displacements and modal results. This combination is known as the modal superposition technique. In the case of welds the modal bending and membrane stresses are combined to the generalized displacements in order to define the frequency response functions as Eq. 20 shows:

K K

  f L ,

  f

  f L ,

  f

  f L ,

  f

  f L ,













 





GD

GD

GD



m

m

1

1

m

2

2

mM

M

(20)

  f L ,   

  f

  f L ,

  f   

  f L ,

  f

  f L ,



 





GD

GD

GD

b

b

1

1

b

2

2

bM

M

For loads L=1… N . Similarly to Eq. 10, the bending ratios are then calculated according to the following expression:

  , f L

b 

(21)

r

  , f L



  , f L

  , f L



b 



m

And finally the equivalent structural stresses are evaluated from Eq. 22:

  , f L t I r  2 2    m m m

  , f L

b 

S

  , f L

(22)



S

1

    , f L

m

    1 ,

m

The in Eq. 22 is basically Eq. 8 and Eq. 9 defined at every frequency for each load channel. In a multiple channel scenario there will be as many equivalent structural stress functions as load channels. Every equivalent structural stress function has its own row in the dynamic response matrix Q (Eq. 12). These functions are defined over the same frequency range of the loads. f L I r