PSI - Issue 19

Giovanni M. Teixeira et al. / Procedia Structural Integrity 19 (2019) 175–193 Author name / Structural Integrity Procedia 00 (2019) 000–000   2 2 m R 1 1 1 2 2 0 2 x S 1 D D Z D D 1 1 R 2 m            

184 10

.

m m

1 m m m m

 

(17)

3 1 2 D 1 D D E P   

x





4

2

m

2

0

4





3 2 1.25 D D R   

2

x D

m

  



.

R

Q







m 1

2 

2 1 1 1 D D    

D

m m

1

0 4

The integration of Equation 16 over the domain of stress ranges ( S R ) yields the damage accumulated in the time T (exposure time):     0    DIR R R R h E P T D S p S dS C (18)

Where C, m' are obtained from Verity SN Master Curve (see also Fig. 5):

h

1 

C S N

  S

(19)

It is well understood how the equivalent structural stress is defined in the time domain (presented in Eq. 7). The next paragraphs explain how the same parameter can be evaluated in the frequency domain. As already discussed the structural stress methods require the stress state at the weld line to be expressed as a combination of bending and membrane stresses. The Volvo and Verity methods evaluate those stresses from nodal forces and moments (Eq. 1-3), improving the consistency of fatigue results across mesh sizes and finite element formulations. And that is the reason these methods are sometimes referred as mesh insensitive methods. In the frequency domain the bending and membrane stresses are evaluated for all the relevant vibration modes included in the finite element results. In other words, for every mode and for every node in the finite element model there is a pair of bending and membrane stresses as Fig 11 shows. As these stresses are derived from nodal forces and moments, the latter must be requested in the frequency extraction analysis (aka modal analysis).

Fig. 11. Bending and moment stresses evaluated for all the modes at every node.

Every linear vibratory system can be characterized by its mode shapes and associated natural frequencies. The contribution of every mode shape depends on the types and frequencies of the loads exciting the system. These contributions are known by the name of modal coordinates or generalized displacements and they are function of