Page 366 - Handbook of Materials Failure Analysis

364 CHAPTER 14 Fatigue failure analysis of welded structures

Forces and moments were determined from linear elastic FE simulations.
As mentioned earlier, in the structural stress approach, sheets and spot-welds are
modeled with shell and beam elements, respectively.
According to Equation 14.6, four values were obtained for the structural stress
range at each spot-weld. Maximum structural stress range,
△S max ¼ max △S 11 ,△S 12 ,△S 21 ,△S 22 Þ; (14.7)
ð
was used in the Neuber’s rule to estimate the maximum local stress and strain ranges,
that is, △σ max and △ε max (assuming cyclic Ramberg-Osgood behavior),
2
ð K f △S max Þ
¼ △σ max △ε max ; (14.8)
E
0 1=n 0
△ε max ¼ △σ max =E +2 △σ max =2Kð Þ ; (14.9)
0
where K f is the fatigue notch factor, E is the elastic modulus, K is the cyclic strength
0
coefficient, and n is the cyclic strain hardening exponent. The effect of mean stress,
σ m , on the crack initiation process was accounted for in this model and was obtained
from
σ m ¼ σ max 0:5△σ max ; (14.10)
in which △σ max was determined by solving Equations 14.8 and 14.9, and σ max was
found from the following equations

2
+ σ rs =E ¼ σ max ε max ; (14.11)
K f max S ij peak
1=n
ε max ¼ σ max =E + σ max =Kð Þ ; (14.12)
¼ △S ij = 1 RÞ, R is the load ratio, σ rs is the residual stress, K is the
where S ij peak ð
strength coefficient, and n is the strain hardening exponent. Crack initiation life
and early growth, N i , was found using the Smith-Watson-Topper’s [24] formulation,
1=b
0 ; (14.13)
2N i ¼ 0:5△σ max = σ σ m
f
where σ f and b are the fatigue strength coefficient and fatigue strength exponent, respec-
0
tively. σ f and b are material properties which theoretically correspond to the region
0
where crack initiation occurs. Residual stress, σ rs , in the as-welded condition was
assumedto be equaltothe yieldstrengthin the crack initiation region. The stress con-
centration factors reported by Radaj [40]andKuang etal.[41] for tensile-shear spec-
imens were adopted with an adjustment due to the different nominal stress definition.
Sheppard in 1993 [25] extended the application of the structural stress to include the
fatigue crack propagation. The mode I SIF range was obtained in terms of the structural
stresses due to the membrane load and bending moment. The structural stress relation-
ship was later modified in 1996 to include the effect of forces normal to the sheet [20].
Some advantages and shortcomings are associated with this model. Both the
crack initiation and propagation phenomena are explicitly accounted for, and the
mode I SIF is updated during the crack propagation. On the other hand, the effects
of modes II and III loading during crack propagation are ignored. Also, there is no