Problem Merc-1
Title: Crack
Growth Prediction of Center Wing Component with Emphasis on b-Factor Determination
Objective:
To illustrate the process of applying finite
element and boundary element analyses to the determination of b-factors for crack growth simulations
General Description:
This
problem focuses on the determination of b-factors for crack growth analysis of a longeron
in the center wing area. Both 2-D and
3-D numerical methods are applied in the analysis. FRANC2D/L, a 2-D finite element (FE) analysis program, is used
for the b-factor determination of through cracks. FRANC3D, a 3-D boundary element (BE) program
is used to determine b-factors of a corner crack.
Topics Covered: finite
element analysis, boundary element analysis, b-factor
calculation
Type of Structure: center
wing, longeron
Relevant Sections of Handbook: Sections 2, 5, 11
Author: Robert D. McGinty
Company
Name: Mercer
Engineering Research Center
Structures Technology Group
Warner Robins, GA 31088-7810
478-953-6800
www.merc.mercer.edu
Contact Point: Robert D. McGinty
Phone: 478-953-6800
e-Mail: bmcginty@merc.mercer.edu
Critical Area
This problem focuses on a critical area of an
angle longeron in a center wing. As
shown in the finite element mesh below, the crack starts in, and grows across
the horizontal leg of the angle longeron.
It begins as a corner crack in a fastener hole where the longeron and
upper splice bar are attached.
The dominant loading mode of the longeron is
tension during banking and turning maneuvers, but due to the complex geometries
and interactions of the components, the details of the stress state are more
complicated than simple, uniform tension near the critical fastener. The determination of b-factors through analytical means can, therefore, introduce significant
errors in the crack growth analysis.
Numerical methods such as finite and boundary element analysis provide a
powerful alternative to the analytical solutions.
Crack
Growth Phases and Modeling Approach
As shown in Figure
MERC-1.2 below, crack growth behavior can be divided into phases, usually
alternating between "corner crack growth" and "through crack growth". The odd- numbered phases are corner cracks,
and the even-numbered phases are through cracks. Also, it will be assumed here that the phases of crack growth
take place sequentially. Corner crack
growth starts in Phase 1 and continues until it grows through the part's
thickness, at which point, through crack growth begins in Phase 2. Finally, Phase 3 does not begin until Phase
2 growth reaches the free edge of the longeron
(Analysis has shown that the crack reaches critical length during
Phase 3, therefore analysis of subsequent phases is not necessary.).
The b-factors for each phase of crack growth will
be determined somewhat differently, due to the increasing complexity of each
phase. Emphasis is placed on the
numerical methods used for Phases 2 and 3.
Phase 1 - Corner Crack at Fastener Hole
This
represents a corner crack emanating from a loaded fastener hole that is offset
from the center of the longeron leg.
AFGROW provides pre-programmed solutions of b-factors for this case.
Therefore, relatively little user expertise is required to obtain b-factors here, at least when compared to
subsequent phases. Figure MERC-1.3 shows the AFGROW input form for
defining part geometry, including the crack, from which AFGROW automatically
computes b-factors.
Figure MERC-1.3. AFGROW input form for
defining part geometry, including the crack, from which AFGROW automatically
computes b-factors.
Phase 2 - Through Crack Growing from
Fastener Hole
Part
geometry and loading conditions introduce complexities into the second phase of
crack growth that are not fully incorporated into existing b-factor solutions. Therefore, numerical methods, i.e., finite elements, are used to
estimate the b-factors. Figure MERC-1.4
shows a 2-D FE model of the longeron's horizontal leg developed in
FRANC2D/L. FRANC2D/L is chosen because
of its ability to automatically remesh while "growing" the
crack. The boundary conditions, tractions
and fastener forces, are determined from the NASTRAN FE model shown in Figure MERC-1.1.
The crack path is also shown in Figure MERC-1.4. Note that it passes nearby a fastener whose
presence introduces additional stress concentrations that further complicate
the problem.
FRANC2D/L computes K values versus crack length as it
automatically propagates the crack and remeshes. b-factors are determined
by defining a reference stress, which is usually a remote tensile value, and
solving for b in Eq. (1).
|
(1)
|
where K is the
stress intensity factor, a is crack
length, and sref is the reference stress, in this case 6,800
psi. Both K and b values are shown in the graph below.
Phase 3 - Corner Crack Growing from
Opposite Side of Fastener Hole
In Phase
3, a corner crack is assumed to grow from the opposite side of the fastener
hole after the through crack reaches a free edge in Phase 2. But unlike Phase 1, AFGROW does not provide b-factor solutions for this situation. It is necessary to compute them numerically
as in Phase 2. The process is
substantially more involved because the problem is 3-D, however, it does begin
just as the Phase 2 case. The NASTRAN
and FRANC2D/L models are developed just as before, but the Phase 2 crack is
present. The FRANC2D/L model is shown
in Figure MERC-1.6 below.
The FRANC2D/L model results in Figure
MERC-1.6 are used as boundary conditions for another, still more refined,
numerical model. It is a 3-D boundary
element model, developed in FRANC3D, of a corner crack at the fastener
hole. This is shown in Figure MERC-1.7 below. FRANC3D reports K
values along the corner crack front, and these are used to calculate b-factors using Eq. (1) just as in Phase
2.
Figure MERC-1.7. FRANC3D model and principal stresses for
Phase 3 corner crack
at fastener hole. FRANC2D/L stresses serve as boundary
conditions
for the FRANC3D model
Several FRANC3D analyses are, in fact, necessary for each
critical area in order to cover the possible combinations of thickness and
surface lengths of the corner crack. As
an example, Figure MERC-1.8 illustrates nine (3x3)
corner cracks of various lengths that are analyzed in order to provide
sufficient b-factor information for
corner crack growth studies.
Figure
MERC-1.7. FRANC3D model and
principal stresses for Phase 3 corner crack at fastener hole. FRANC2D/L stresses serve as boundary
conditions for the FRANC3D model
|
|
Tables MERC-1.1 and MERC-1.2 give b-factors
for various corner crack geometries.
Two tables are required to determine crack growth through the part
thickness and along the part surface. Table MERC-1.1 gives b-factors
that apply to crack growth through the part's thickness. Table MERC-1.2
corresponds to crack growth along the part's surface. AFGROW uses this information to independently predict growth
rates along the two directions. Though
independent, it is generally true that the growth rates produce a corner crack
whose thickness length is approximately 50% greater than its surface length.
|
|
Thickness
Crack Length (in)
|
0.05"
|
0.264"
|
0.488"
|
Surface
Crack
Lengths (in)
|
0.05"
|
9.739
|
4.715
|
3.219
|
0.200"
|
11.26
|
8.172
|
7.940
|
0.400"
|
11.30
|
11.48
|
14.62
|
|
|
|
Thickness Crack Length (in)
|
0.05"
|
0.264"
|
0.488"
|
Surface Crack Lengths (in)
|
0.05"
|
7.590
|
12.88
|
14.03
|
0.200"
|
1.381
|
6.824
|
11.25
|
0.400"
|
1.211
|
3.813
|
10.57
|
Predicting Crack Growth
AFGROW is used to predict crack growth in the longeron using the b-factors presented here. The crack is assumed to start as a
0.05" radius corner crack at the fastener hole and grow according to the
discussion above. Only surface crack
length is plotted. Details of the
stress spectra and material da/dN
data are not presented. Note that
Willenborg retardation is applied.