3.0
COMPLETED WORK
3.1
Selection of Structural Models with Holes
FET began its work by searching for structural models with holes. One book that
was recommended to them was Peterson’s Stress Concentration Factors compiled by
Walter D. Pilkey [3]. The book catalogues various structural components with holes that
have been tested as well as the stress concentration factors associated with them. The
team searched through the book for models that were representative of the types of
discontinuities that are typically found in aerospace structures. Because the book gives
the stress concentration factor for each discontinuity configuration catalogued, FET could
compare their finite element analysis results with the theoretical results from the book.
Three configurations were selected and are shown below along with the stress
concentration factor associated with each (All figures and equations courtesy of
Peterson’s Stress Concentration Factors.):
Figure 3.1 Tension of a finite width thin element with a circular hole. [3]
K tg 
 max
2
d
d


 0.284 
 0.6001    1.321  

1 d H
 H
 H
2
(2.1)
15
Figure 3.2 Tension of a finite width thin element with an
infinite row of circular holes. [3]
K tn 
 max
d 
 d  d  
 d  d 
 1.949  1.476   0.916  2.845     1.926  1.069  
 

H 
 H  l  
 H  l 
2
(2.2)


1 d H 
Figure 3.3 Biaxially stressed infinite thin element with an
infinite row of circular holes, 1 = 2. [3]

d
d
d
K tg  max  1.9567  1.468   4.551   9.6867 
1
l
l
l
2
3
(2.3)
16
3.2
Comparing Finite Element Analysis Results with Theory for Structural
Models with Holes
After the three discontinuity configurations were selected, a finite element model
of each was generated and run in ABAQUS. Because all of the physical structures
exhibited symmetry about the horizontal and vertical axis, FET was able to obtain
accurate results in ABAQUS by modeling only one quarter of each structure and
imposing the symmetry boundary condition. The results of the analyses are presented
below.
Tension of a Finite Width Thin Element with a Circular Hole:
The stress distribution for the first model undergoing a tensile pressure load of unity is
shown below in Fig. 3.4:
Figure 3.4 Stress distribution for the tension of a finite width
thin element with a circular hole, d/H = 0.5.
17
As expected, the presence of the hole introduced a stress concentration at the top edge of
the hole as indicated by the red. Because the right edge of the model has a uniform stress
distribution of 1.0, the hole is sufficiently far away from the lateral edge for the plate to
be considered infinite in length.
The model was run with various diameter- to-width ratios (d/H), and the resulting stress
concentration factor, Ktg, was plotted against this ratio and compared with Peterson’s
theoretical values (see Fig. 3.5).
5
4.5
4
3.5
Ktg
3
2.5
2
ABAQUS
1.5
1
Theoretical
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
d/H
Figure 3.5 Stress concentration factor for the tension of a finite
width thin element with a circular hole.
18
From Fig. 3.5 above, it can be seen that the ABAQUS results deviate from Peterson’s
values by a maximum of 1%. Therefore, it is safe to conclude that the finite element
analysis of this model was accurate.
Tension of a Finite Width Thin Element with an Infinite Row of Circular Holes:
The stress distribution for the second model undergoing a tensile pressure load of unity is
shown below in Fig. 3.6:
Figure 3.6 Stress distribution for the tension of a finite width thin element
with an infinite row of circular holes, d/l = 0.6.
Again, the stress concentrations for this configuration appear at the top edges of the
holes. The plot of stress concentration factor as a function of the diameter-to-hole
distance ratio (d/l) is given below in Fig. 3.7:
19
2.5
2
Ktn
1.5
1
ABAQUS
Theoretical
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
d/l
Figure 3.7 Stress concentration factor for the tension of a finite width
thin element with an infinite row of circular holes.
The figure above indicates that the ABAQUS results deviate from theory by a maximum
of 2% again showing good agreement with theory.
Biaxially Stressed Infinite Thin Element with an Infinite Row of Circular Holes:
Fig. 3.8 shows the stress distribution for the biaxially stressed infinite thin element with
an infinite row of circular holes. Once again the model was run with a pressure load of
unity.
20
Figure 3.8 Stress distribution for a biaxially stressed infinite thin
element with an infinite row of circular holes, d/l = 0.6.
When undergoing biaxial loading, the stress concentrations have now moved from the top
edges of the holes to the left and right edges. Again, notice the uniformly distributed unit
stress at the edges of the plate indicating that the edges can be considered to extend to
infinity. A plot of the stress concentration factor as a function of d/l is shown below in
Fig. 3.9:
21
3.5
3
2.5
Ktg
2
1.5
ABAQUS
1
Theoretical
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
d/l
Figure 3.9 Stress concentration factor for a biaxially stressed infinite thin
element with an infinite row of circular holes.
Like the first two models, the ABAQUS results for the third model are virtually identical
to the theoretical results from Peterson indicating accurate finite element modeling.
3.3
Reducing the Stress Concentration Factor Around Holes
After running the previous three models, FET attempted to determine a way to
alleviate the stress levels seen around a hole. At the recommendation of their advisor,
FET added an additional hole of radius r at the locations seen in Fig. 3.10 and 3.11.
22
Figure 3.10 Tension of a finite width thin element with a circular hole and additional
holes in a plane perpendicular to the loading direction.
Figure 3.11 Tension of a finite width thin element with a circular hole and additional
holes in a plane parallel to the loading direction.
Since theoretical results could not be obtained for the above two cases, FET had to trust
that their finite element modeling skills were proficient enough to achieve accurate
results.
23
Tension of a finite width thin element with a circular hole and additional holes in a
plane perpendicular to the loading direction:
FET first determined the effect of adding additional holes in a plane perpendicular to the
loading direction (see Fig. 3.12).
Figure 3.12 Stress distribution for the tension of a finite width thin element with a center
hole and additional holes in a plane perpendicular to the loading direction.
By adding a 0.25 in. radius hole a distance 5 in. away from the original hole, it was
discovered that the stress concentration factor was now 3.411, a 14% increase from the
original stress concentration factor of 3. By varying the radius of the additional holes,
FET determined that the stress concentration factor for this configuration was always
greater than 3. Therefore, they could conclude that adding holes in a plane perpendicular
to the loading direction would not lower the stress concentration factor of the original
configuration.
24
Tension of a finite width thin element with a circular hole and additional holes in a
plane parallel to the loading direction:
Next, FET investigated the effect of adding holes in a plane parallel to the loading
direction (see Fig. 3.13).
Figure 3.13 Stress distribution for the tension of a finite width thin element with a center
hole and additional holes in a plane parallel to the loading direction.
Adding a 1.0 in. radius hole a distance 4 in. away from the original hole resulted in a
stress concentration factor of 2.894, a 3.5% decrease from the original stress
concentration factor of 3. Fig. 3.14 shows how the stress concentration factor varies with
the hole radius r.
25
3.1
2.9
2.7
Ktg
2.5
2.3
2.1
1.9
1.7
1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
r (in)
Figure 3.14 Variation in stress concentration factor with additional hole radius for the
tension of a finite width thin element with a center hole and additional holes in a plane
parallel to the loading direction, l = 4 in.
It should be noted that for the cases represented in the above two figures (l = 4 in.), the
maximum value that r can take is 2 in. due to the spacing of the holes. As r approaches 2
in. the material between the two holes becomes so thin that buckling would be a problem
due to compression in the lateral direction.
Fig. 3.15 and 3.16 show the same plot as Fig. 3.14; however, the distance between the
holes, l, has been increased to 5 in. and 7.5 in., respectively.
26
3.2
3
Ktg
2.8
2.6
2.4
2.2
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
r (in)
Figure 3.15 Variation in stress concentration factor with additional hole radius for the
tension of a finite width thin element with a center hole and additional holes in a plane
parallel to the loading direction, l = 5 in.
2.9
2.7
2.5
Ktg
2.3
2.1
1.9
1.7
1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
r (in)
Figure 3.16 Variation in stress concentration factor with additional hole radius for the
tension of a finite width thin element with a center hole and additional holes in a plane
parallel to the loading direction, l = 7.5 in.
27
Fig. 3.14 – 3.16 indicate that the stress concentration factor for this geometry is always
less than 3. Therefore, FET concluded that adding holes in a plane parallel to the loading
direction reduces the stress concentration factor associated with a hole. By plotting the
optimal (lowest) stress concentration factor for the cases represented in Fig. 3.14 – 3.16,
the variation in optimal stress concentration factor with the hole distance, l, can be seen
(see Fig. 3.17).
2.6
Lowest K tg
2.55
2.5
2.45
2.4
2.35
0
1
2
3
4
5
6
7
8
l (in)
Figure 3.17 Variation in optimal (lowest) stress concentration factor with hole spacing.
The above figure shows that stress concentration factor increases with increasing l.
However, remember that for small values of l, buckling of the material between the holes
would be a problem. Therefore, it would be best to choose the lowest l where buckling
could be avoided.
28
3.4
Selection of Structural Models with Cracks
FET began their finite element analysis of cracks again by selecting cases that
could be compared with theory. The following three models were selected because of
their applicability to virtually any aerospace structure (All stress intensity factor, K,
equations courtesy of The Stress Analysis of Cracks Handbook [4]).
K   a
K  1.12 a
a R h
K   a F  , , 
b b b
Figure 3.18 Geometry
Figure 3.19 Geometry
Figure 3.20 Geometry and stress
and stress intensity factor
and stress intensity factor
intensity factor for a tension element
for a tension element with
for a tension element with
with a center hole and a center crack
a center crack.
a crack at each edge.
through the hole. [4]
29
3.5
Comparing Finite Element Analysis Results with Theory for Structural
Models with Cracks
Finite element models of the three test cases shown above were run in ABAQUS,
and the results were compared with theory. All of the finite element models were loaded
with a uniform tensile stress of unity (1 psi), and the results are presented below.
Note: The crack tips have been magnified for clarity.
Tension of a Thin Element with a Center Crack:
Fig. 3.21 shows the stress distribution for a thin element in tension with a center crack of
length a = 3.0 in. Notice the concentration of high stress at the crack tip.
Figure 3.21 Stress distribution for a thin element in tension with a center crack.
30
The theoretical stress intensity factor for above crack configuration is K  3.07 psi in
while the finite element model yielded a value of K  3.10 psi in showing good
agreement with theory.
Tension of a Thin Element with a Crack at Each Edge:
Fig. 3.22 shows the stress distribution for a thin element in tension with cracks at both
edges of length a = 3.0 in.
Figure 3.22 Stress distribution for a thin element in tension with cracks at both edges.
Comparing theoretical stress intensity factor of K  3.44 psi in with the finite element
result of K  3.41 psi in for this configuration shows once again that the finite element
modeling was performed correctly.
31
Tension of a Thin Element with a Hole and a Center Crack Through the Hole:
The stress distribution for the final crack test case is shown below in Fig. 3.23.
Figure 3.23 Stress distribution for a thin element in tension with a hole and a center
crack through the hole.
For the particular case shown above, the stress intensity factor depends not only on the
crack size but also on the length and width of the structure as well as the radius of the
hole. Therefore, the geometric factor, F, was plotted against the ratio of the crack size to
the width of the element, a/b, for the finite element results (shown in red) and compared
to a theoretical plot to check accuracy (see Fig. 3.24). Once again it can be seen that
finite element results correspond well with theory.
32
Figure 3.24 Comparison of finite element result with theory for the tension of a thin
element with a hole and a center crack through the hole.
(Original image courtesy of The Stress Analysis of Cracks Handbook)
3.6
Crack Repair Methods
After running the three crack test cases shown above, FET was confident in their ability
to model cracks with finite elements. Therefore, they proceeded to investigate crack
repair methods. Four repair methods were investigated in an attempt to reduce the stress
intensity factor for a thin tension element with a center crack: stop holes, an array of
holes placed near the crack tip, a composite patch, and composite arrester strips (see Fig.
3.25 – 3.28).
33
Figure 3.25 Thin tension element with a
Figure 3.26 Thin tension element with a
center crack and stop holes at the crack
center crack and an array of holes near the
tips.
crack tip.
Figure 3.27 Thin tension element with a
Figure 3.28 Thin tension element with a
center crack and a composite patch.
center crack and composite arrester strips.
34
Tension of a Thin Element with a Center Crack and Stop Holes at the Crack Tips:
Since the stress intensity factor only applies to cracks and not holes, the effectiveness of
stop drilling was determined by observing how much the maximum stress in the structure
was reduced. Fig. 3.29 shows the stress distribution for the tension element with a center
crack of length a = 1.0 in. and stop holes of radius r = 0.1 in.
Figure 3.29 Stress distribution for a thin element with in tension with a center crack and
stop holes at the crack tips.
The maximum stress without the stop holes was 27.81 psi while the maximum stress with
the stop holes was 8.132 psi, a 71% reduction. Also, since the crack tip has now been
replaced by a hole, crack propagation will not be a concern unless another crack forms at
the edge of the hole.
35
Tension of a Thin Element with a Center Crack and an Array of Holes Near the
Crack Tip:
Fig. 3.30 shows the stress distribution of a thin tension element with a center crack of
length a = 1.0 in and an array of holes, each with a radius of r = 0.1 in., near the crack tip.
Figure 3.30 Stress distribution for a thin element in tension with a center crack and an
array of holes near the crack tip.
For this case, the stress intensity factor without the array of holes was K  1.78 psi in
while the stress intensity factor with the holes was K  1.68 psi in , a reduction of 6%.
36
Tension of a Thin Element with a Center Crack and a Composite Patch:
The stress distribution for the thin tension element with a center crack of length a = 2.0
in. and composite patch of longitudinal tensile modulus of E1 = 15,000 psi is shown in
Fig. 3.31.
Figure 3.31 Stress distribution for a thin element in tension with a center crack and
composite patch.
Comparing the stress intensity factor of the crack without the patch, K  2.765 psi in to
the stress intensity factor with the patch, K  0.7351 psi in , shows a 73% reduction.
Tension of a Thin Element with a Center Crack and Composite Arrester Strips:
Fig. 3.32 shows the stress distribution of a thin tension element with a center crack of
length a = 2.0 in. and composite arrester strips with a longitudinal tensile modulus of E1 =
15,000 psi and width b = 1.0 in.
37
Figure 3.32 Stress distribution for a thin element in tension with a center crack and
composite arrester strips.
In the above figure, the distance from the crack tip to the nearest edge of the arrester strip
is denoted by L. Since the stress intensity factor is dependent upon this distance, the
finite element model was run for various values of L, and the stress intensity factor for
each case was recorded. By plotting each of these values, FET was able to determine
how the stress intensity factor was affected by the position of the arrester (see Fig. 3.33,
Note that the variable L has been non-dimensionalized by dividing by the crack length,
a).
38
2.9
2.8
2.7
0.5
K (psi-in )
2.6
2.5
2.4
2.3
K (with arrester)
2.2
K (without arrester)
2.1
2
0
0.5
1
1.5
2
2.5
3
3.5
L/a
Figure 3.33 Effect of arrester distance on stress intensity factor for a tension element
with a center crack and composite arrester strips.
Fig. 3.3 above shows how increasing the distance between the crack tip and arrester strips
results in an increase in stress intensity factor. Also, it can be seen that the stress
intensity factor of the model with the arrester strips converges toward the stress intensity
factor of the model without the arrester strips as the distance of the arrester strips from
the crack tip is increased. If the arresters are placed sufficiently far from the crack tip,
then they will have no affect on the crack and thus the stress intensity factor will be the
same as the model without arresters.
39