Senior Project Report

By Davila, Kuhl, Kieu, & Sanchez 1 EGR 481&482 Senior Project
California State Polytechnic University Pomona
EGR 481&482 SENIOR PROJECT June 09, 2016
ULTIMATE STRENGTH OFAIRCRAFT STRUCTURES, ANALYSIS & TEST
PROJECT ADVISOR: DR. TODD COBURN
Davila, Francisco; franciscor_d@yahoo.com
Kuhl, Ethan; ethankuhl914@gmail.com
Kieu, Tai, kieuchitai1208@gmail.com
Sanchez, Victor; v.sanchez91@yahoo.com
Aerospace Engineering
California State Polytechnic University Pomona
OBJECTIVE
In this paper, results of a study of Ultimate Strength of Aircraft
Structures are presented. A group of four students worked
countless hours on analyzing and testing multiple beams with
different shape to check the local crippling as the theory from
Dr. Todd Coburn‟s paper. This project consists of analyzing
specimen‟s critical section, which resort to plastic bending
analysis, and Cozzone‟s method.
ULTIMATE STRENGTH OF AIRCRAFT STRUCTURES
OVERVIEW
Analysis of the critical sections of aircraft structures often use
the methods of plastic bending from E.F Bruhn‟s Iterative Slice
Method, or Cozzone‟s Simplified Method for Symmetric
Sections. However, these methods fall short when the critical
section includes thin flanges that buckle or cripple prior to
ultimate failure.
The comparison of an elastic stress to an ultimate material
allowable is valid for a member subjected to axial loading, but
it is only an approximation because it only approximates the
strength of member subjected to bending since it ignores the
local yielding effects of the material between the yield and
ultimate strengths of the material.
The paper form Dr. Todd Coburn presents a solution to the
above shortfall, and introduces a hybrid procedure for
calculating the ultimate strength of a cross section that accounts
for material non-linearity, flange stability, and other effects.
As addressed from Ultimate Strength of Aircraft Structure from
Dr. Todd Coburn, the key relationship to “plastic” or “ultimate”
analysis of members subjected to bending:
When the ultimate strain of a material is known, the strain at
any point y in the cross section can be determined. Based on the
above understanding, we can create a table with LIMITS
column to insert different allowable values to evaluate the
ultimate moment allowable of the inelastic section.
Figure 1: Example of a straight beam in bending
FUNDAMENTALS & APPROACH
This is a fundamental relation for plastic analysis when
members are subjected to bending.
When the ultimate strain of the material used is known, then the
strain at any point of the cross section can be determined,
where:
= strain at point i
= ultimate strain of the material

= maximum distance from the neutral axis to the
extreme fibers
= distance above the neutral axis to the element
The young modulus E is constant across the cross-section when
the stresses are below the elastic limit. Also the stresses vary in
a linear fashion along the cross-section when they are under the
elastic limit. When the stresses increase above the elastic limit,
then the stress distribution becomes nonlinear but the strain
distribution continues with its linear distribution.
Based on the understanding previously mentioned, the
allowable of the inelastic section can now be evaluated. This is
done by enforcing a linear strain distribution on the cross
sectional, starting with the maximum strain at the maximum
fiber.
1. An algebraic expression for the stress-strain curve
shape is of great usefulness for structural analysis. To
begin the analysis, a stress-strain curve was created by
Excel using Ramberg-Osgood formula. The Excel was
made in a way that can easily modify the engineering
values for different materials. The stress-strain curve
of 6061 Aluminum was created as shown in Appendix.
2. Idealize the section by dividing it into selected number
of elements from top to bottom.
3. Record the height hi the width bi and the centroid
relative to the lower surface of the member Yi of each
strip section into a table.
4. The area of each strip Ai is calculated and tabulated,
then the center of gravity YCG of the element section is
determined.
(1)
5. Inside the inelastic range, YNA = YCG . This assumption
stays valid for symmetric sections with materials that
have strain and stress curves with equal tensions and
compressions. If either condition is not present, then
the neutral axis moves from the center of gravity to a
location where the strain is zero.
(2)
6. The strain on each element ɛi can be determined by
(3)
inserting the maximum strain ɛmax and distance ymax
from the neutral axis. For a symmetrical cross section
with an identical strain-stress curve in tension and
compression it becomes simple. It involves reading the
maximum strain ɛmax from the strain-stress curve and
implementing it with the maximum distance y to the
farthest fiber into the equation above for each strip.
This will mean that ymax will be equal to YNA and YNA =
H-YNA for a symmetrical section.
When the stress-strain curve is different in tension and
compression, additional precautions are taken. The
elements in tension are limited by one max strain, and
the elements in compression are limited as well.
Correct results will be obtained by implementing the
smaller value of ɛmax into equation 3 with the
corresponding y of either YNA or H-YNA.
If the section property is not clear, then evaluate both
conditions and take the one producing minimum
ultimate moment as the preferred solution.
7. The strain for each element is determined and
tabulated, the stress is found from the stress-strain
curve. This can be done by hand or by writing a code
using the Ramberg-Osgood coefficients for a specific
material.
8. The force in each element is calculated by Pi = σi Ai
and moment Mi = Pi yi
9. When the section is symmetric and the stress-strain
curve is the same for tension and compression, then
this completes the process, and the ultimate moment
allowable of the section can be obtained by summing
the moment contribution of each element Mult = ΣMi
Table 1: Breakdown table for analysis
Properties of Section Strain Iteration Value
NO.
1
2
3
4
.
.
.
n-2
n-1
n
Σ
bi hi Yi Ai yi ɛi σi Pi Mi
10. If the section is not symmetric or if the stress-strain
curve in compression and tension are different, then
additional steps are taken in order to find the correct

ultimate moment. This becomes apparent when the
forces Pi are added on each element. if the section is in
pure bending, then this value is zero. If the sum is not
zero, then the neutral axis is different from the center
of gravity and it need to be moved to the correct
location. This is done by iterations.
If the net force is positive, then the neutral axis moves
upwards. If the net force is negative, then it must move
downwards.
The correct ultimate moment is found by adjusting the
assumed location of the neutral axis YNA until the sum
of the force is equal to zero, or until the net force is
below some error parameter. In order to find the error
parameter, one can take the absolute ratio of the sum
of the forces to the sum of the moments | ΣP / ΣM |.
This can be done by hand, excel, Mathlab to drop the
error below a negligible range.
The above step can be used to find the allowable
ultimate moment Mult for any cross section, where the
stress and strain curve is the same or different in
tension and compression.
The procedure can also be used to evaluate across
sections subjected to axial loads and bending. The
difference is that the neutral axis for this case is moved
until the sum of the forces equals the applied load,
instead of zero in the pure bending case. The ultimate
moment is then the moment that can be applied in
addition to some force Paxial, and should perhaps be
identified as such.
Figure 3: C Channel divided into sections
Ultimate Moment Including Stability Effects
Weakness: The procedure explained above fails to precisely
account for the stress when the element exceeds the stability
allowable of the webs or flanges, as is in the case for aircrafts
and other lightweight structures.
Case: In aircrafts and other lightweight structures, the thickness
of the flanges and webs are quite thin that they experience
stability failure prior to reaching the stresses otherwise
supportable by the material. Buckling, crippling, and Euler-
Johnson allowable frequently limit the strength of these thin
flanges and webs.
Solution: Use a variation of the method above. It is only needed
to add one more column with the limits Fcc/Fcr and to write
down any cut off or limiting values on the stresses for those
elements.
Procedure:
1. Determine the crippling allowable for each flange and
web. Enter the values of Fcc into the column „limits‟.
The web buckling Fcrw should also be checked, and
flange buckling to the nearest support Fcrf, and if these
produce lower values than Fcc, then these should be
entering in the limits column.
a. First, we calculate the Fcc for each elements
Crippling Failures - Where the allowable
compression stress of a structure member is based
on the structural shape of the cross sectional are of
the members.
(4)
Table 2: Material and Edge Support Coefficient
Material Edge Condition Ce
Aluminum
2 free edges
1 free edges
no free edges
0.295
0.317
0.339
Steel
2 free edges
1 free edges
no free edges
0.274
0.318
0.362
Titanium
2 free edges
1 free edges
no free edges
0.272
0.307
0.342

Table 3: Breakdown table including free edge
No
1
2
3
4
.
.
.
n-2
n-1
n
Free edge Ce bn tn bntn Fcc bntnFcc
If the values of Fccr ˃ Fcy, Then Fcy will be placed instead of Fcc
in the table shown above.bn is the longer distance of the element
and tn is the thickness of the elements. In this situation Fir ˃ Fcc
˃ Fcy and the inter-rivet failures will not occur.
b. Now, we calculate the Fc to compare with Fcc
- Pure compression for the flange on top
(5)
- Pure bending for the web
(6)
2. When this is completed, and the limits are places under
its assigned column, then the forces of each strip Pi are
calculated as the product of the area Ai and the
minimum of σi and Fcc or Fcr
3. The moment of each strip is calculated once again
4. The neutral axis is adjusted one more time, either
going upwards or downwards until the sum of the
element forces Pi equals the applied axial load Pax (or
zero if none)
5. The ultimate moment Mult is simply the sum of the
moment ΣM
TEST PREPARATION
Four-point bending provides values for the flexural stress,
strain, modulus of elasticity in bending and the flexural stress-
strain response of the material. In order to create a constant
moment at the center section of the beam we performed a four-
point bending testing. The moment in the center mortar joint is:
(7)
And the stress at mid span is found with
σ = (8)
Transverse vertical loads are applied to horizontal beams such
that a constant bending moment results between the two inner
load locations. Figure 4 shows the corresponding loading
diagrams, from free-body to bending moment.
Figure 4: Example of Shear, and Moment diagrams for the
laboratory experiment on four-point bending.
Stress-strain relationships are utilized to develop the theoretical
parametric relationship between strain, , bending moment, M,
tensile modulus, E, width, b and height, h. The relationship
between these parameters is shown in equation below.
 = 6M/(Eb ) (9)
To maintain a constant bending moment over much of the span
of the beams, variation in the modulus of elasticity and width is
minimized by cutting each set of test beams from a single piece
of aluminum plate stock. The relationship between flexural
stress and strain can then be reduced to the form depicted in the
equation below, where C denotes a constant that is based on a
single applied load at the load cell.
 = C * (10)
The fixture that loads the above beams in four-point bending
consists primarily of custom-design and fabricated elements
loaded into a standard universal testing machine. The upper
yoke fixture elements are designed to provide vertical tensile
loads. The beam specimens are tested with the fixture mounted
in a universal testing machine as shown in Figure 5.

Figure 5: Four point bending experimental setup
EQUIPMENTS
The test apparatus being used was a single hand pump operated
hydraulic ram that applies load from the top, with supporting
blocks on the underside. In this configuration, three-point
bending would be achieved, necessitating a fabricated adapter
to convert the three-point bending into a four-point bending.
This fabricated adapter was manufactured of 4130 Chromoly
steel and was load tested to 2,000 lbs. with 0.10” of deflection
to ensure the fixture itself would not deflect when under the
loads required to bring the specimen to crippling. As testing was
performed however, this adapter began to weaken as it was
repeatedly loaded and unloaded. During the final stages of
testing, a separate, more robust adapter, was fabricated to
replace it as it began to deform and fail where the hydraulic ram
applied force.
To measure the applied force on the system, a load cell, or
arrangement of load cells was needed; one high capacity load
cell at the ram or two moderate capacity load cells beneath the
roller supports. Due to time and budget constraints, purchasing
two moderate capacity load cells with a large platform provided
the best solution. This also allowed for the placement of the
beam to be verified, as discussed in the Test Procedure section.
For extrusions, 6061-T6 aluminum was used as the primary
material due to its ease of acquisition and relative low cost. All
samples were purchased from McMaster-Carr given the higher
quality control standards that they impose on materials, as
opposed to a general hardware and/or material store.
TEST SETUP
To induce crippling into the specimen, four-point bending
flexural testing must be performed. This type of loading is
shown below in Figure 6, in which force is applied through two
roller supports at the center of the test section. The test
specimen is then supported on either end at a set distance. All
supports are considered free floating, to allow the test specimen
to deflect in an unconstrained manner. Four-point bending in
this manner produces a larger volume that is under stress, in
comparison to simple three-point bending.
Figure 6: Test setup
The square 1” x 1” 1/6th
inch thick aluminum extrusion was
used as the primary test specimen due to its limited cost, and
limited force required to plastically deform it. It also was the
simplest specimen to setup within the apparatus, and appeared
to behave the most consistently amongst the other cross
sections. Because of these reasons, all apparatus setup testing
was performed with this extrusion.
Initial test setup was performed with a six-foot-long, square
section in an effort to keep the required loads to a minimum.
After two tests however, it had become evident that too much
deflection was occurring due to the section length, resulting in
the hydraulic ram reaching its extension limits. Due to this, the
specimen length was shortened to three feet. This also allowed
us to get two specimens for each aluminum section purchased.
For the Z and C cross section specimens, the load had to be
applied offset from the center of the section due its lack of
symmetry at the shear center. This was a different offset
distance for each cross section. The offset was calculated, and
the load was applied offset from the flange at that distance. This
prevented the specimens from rolling about the web of the
specimen when under load.
TEST PROCEDURES
To begin testing, the load cells and blocks were centered
between the hydraulic ram. Centering the load cells ensured that
the load cell would reflect the true load being applied, as it was
found that being off-center by six inches reflected a variation of
load readout of up to ten pounds. The proper stack of the outer
supports is as follows: Steel support – load cell – brick – ¾”
thick aluminum triangle (bottom to top). The beam was then
centered between the outer supports and temporarily loaded
with no more than 40 lbs. Doing so ensures that the beam is
properly centered; if the beam is not centered, the load cells
would read different values. After centering the beam, the
hydraulic ram was backed off so that no load was being applied,
and the load cells were zeroed/tared. After this was completed,
the 4-point bending adapter is moved into place between the
hydraulic ram and the beam and load was applied. The load
readout from each load cell was recorded, along with the
deflection of the hydraulic ram which was also recorded at 30
lbs intervals. Deflection and applied load were plotted at each
interval in real-time to monitor the region of the strain-strain

curve that we were operating in. At all times, the lab safety
procedures were followed.
ANALYSIS & ASSESSMENT
1. Rectangular Bar
Due to the symmetrical cross section, the rectangular bar was
chosen to test first. Before starting the test, an Excel
spreadsheet was created to calculate the deflection in different
load scenarios. Using superposition method to calculate the
deflection it allowed us to modify the load on each side in case
the applied load is not the same. Displacement equation:
(11)
A detail of displacement calculation is included in Appendix
and also in the USB contained all calculation.
2. T Bar
The same approach to calculate the displacement of rectangular
bar was applied to the T bar. However, due to the instability
from the web of the T bar when placing on the triangle support,
supports were attached to the T bar as Figure below to hold the
T bar stable when testing.
Figure 7: T bar with support to eliminate the instability
3. Channel Bar
We considered bending of a non-symmetrical beam with respect
to the longitudinal plane of bending, subjected to transverse
shear forces in addition to bending moments. The resultant of
the shear stresses produced by the transverse loads will act in a
plane that is parallel to, but offset from, the plane of loading.
Whenever the resultant shear forces do not act in the plane of
the applied loads, the beam will twist about its longitudinal axis
in addition to bending about its neutral axis figure (8).
Transverse loads applied through the shear center case no
torsion of the beam figure (10), if a beam twists as it bends,
torsional shear stresses will be developed in the cross section,
these shear stresses will be quite large in magnitude. For that
reason, it is important for the beam designer to ensure that all
loads are applied in a manner that eliminates twisting of the
beam. This can be accomplished when an external load is
applied through the shear center of the cross section. For cross
sections that are unsymmetrical about one axis, or both axes, the
shear center must be determined by computation or observation.
We will first assume that the beam cross section bends, but does
not twist. On this basis, the resultant internal shear forces in
thin-walled shape will be determined by considering of the
shear flow produced in the shape. Equilibrium between internal
and external resultant forces must be maintained. From this the
location of the external load necessary to satisfy equilibrium
can then be computed.
Figure 8: Effect of twisting on C Channel
To better understand what causes the channel shape to twist, it
is instructive to look at the internal shear flow produce in the
beam in response to the applied load P figure (9). The couple
formed by the flange forces Ff causes the channel to twist in a
counterclockwise direction. To counterbalance this twist, an
equal clockwise torsional moment is required. A torsional
moment can be produced by moving the external load P away
from the centroid. Because there is a moment equilibrium about
point B, the beam will no longer have a tendency to twist. The
distance e measured from the centerline of the channel web
defines the location of e shear center O figure (11).
(L
2
- b
2
- x
2
) for 0 ≤ x ≤ a

Figure 9: Shear flow in C Channel
As long as the external loads act through the shear center, the
beam will bend without twisting. When this requirement is met,
the stresses in the beam can be determined from the flexure
formula.
Figure 10: Applied load at the shear center
The shear center location only depends on the dimensions and
geometry of the cross section.
Figure 11: Shear center location
The eccentricity is given by
(12)
Where „e‟ is the distance from the centerline of the channel
web.
Shear Stress Distribution over a channel shape
The shear stress produced in each channel flange is linearly
distributed, only the maximum values, which occurs at point B
and D will need to be determined. The shear stress in the flange
is parabolically distributed, with its minimum values occurring
at points B and D and its maximum value occurring at point C
shown in figure 12.
Figure 12: Shear stress distribution on C channel
 Finite Element Analysis
Structural Analysis was done using FEMAP. For this it was
used 61 nodes, 1inch spacing from node to node, and 0.25in of

spacing in areas of interest such as the mid-section of each
beam, with a total length of the beam of 36 inches. The material
used in this analysis is Aluminum Alloy 6061. In figure (13) it
shows the material properties, using Isotropic material, with a
Young‟s Modulus, E of 10.1E6 and a Poisson‟s ration of 0.33
Figure 13: Define material - Isotropic
The next step in this analysis is to define our Cross Section in
our Property Card. a simple method of doing this is by placing
the dimensions of our structure shown in figures (14, 15).
1. Rectangular Bar
Figure 14: Cross Section definition for the rectangular bar
Figure 15: Property definition for the rectangular bar
2. T Bar
Figure 16: Cross section definition for the T bar

Figure 17: Property definition for the T bar
3. Channel Bar
Figure 18: Cross section definition for the channel bar
Figure 19: Property definition for the channel bar
After we have our Material, and Property card implemented, we
can start distributing the nodes, and creating the elements of the
beam. There is a need for careful attention when creating the
elements, since the unit vector has to always point
perpendicular to the beam, and parallel to bending. In this case,
our unit vector will point in the positive y axis
Next we create the constraints and loads on the beam. In this
case we are doing a four-point bending. Our constraints are 3
inches from the edges, using pin connections, and the loads are
3.75 inches apart from the center of the beam.
Lastly we analyze our program by using NX Nastran Editor for
full results of our structures.
4. Isometric view of tested beams
Figure 20: Finite Element Analysis on rectangular bar

Figure 21: Finite Element Analysis on T bar
Figure 22: Finite Element Analysis on channel bar
RESULTS
1. Rectangular Bar
We have created two graphs; the first graph is the actual graph
where it is shown with dotted lines which are the actual strain at
the top of the beam. The solid line represents the actual stress-
strain at the center of the flange. The solid triangle shows the
location of the cripple with a Fcc of 5,465 psi in graph (1).
Graph 1: Stress & Strain Curve – Actual Data
The stress strain curve was plotted by using theory data
shown if graph (2) During the testing, when the material
reached Fcy we changed in our calculations to tension Young‟s
Modulus, and we expected the creep to lower for the actual
data. These calculations are shown in the appendix for the
rectangular beam.
Graph 2: Stress & Strain Curve – Theory Data
During the testing, there was a noticeable dimple of the
rectangle beam in the center as shown in figure (23, 24). It was
also observed this happened in the plastic region of the stress-
strain curve as shown in the actual data of the beam.
Figure 23: Dimple formation
Figure 24: Dimple formation

2. T Bar
In the T Bar there are two graphs, one representing the actual
data, and the second one representing the theoretical data. In
this case it was observed the formation of two dimples during
testing. Also in the stress-strain curve with actual data shown in
graph (3) it also shows when this dimples happened.
Graph 3: Stress & Strain Curve – Actual Data
Graph 4: Stress & Strain Curve – Theory Data
In the testing of the tee beam, we observed the first dimple, and
then the weight dropped due to the beam being in the plastic
region of the stress-strain curve. Then the stress increased and
created another dimple shown in figure (25).
Figure 25: Formation of two dimples
When there was more load applied to the beam, there was a
sudden noise, and that‟s when the beam reached its ultimate
stress and created a noticeable crack at the bottom surface of
the web figure (26).
Figure 26: Fracture of T beam
3. Channel Bar
In the testing of the channel Bar figure (27), there was an
application of 443 pounds in each scale, there was no indication
of dimples and the beam returned to its original location figure
(28). One of the challenges in the testing of this beam was the
limitations of the lab not able to record data above certain
percentage load applied.

Figure 27: Channel Being tested
Figure 28: Channel after testing
It is important to notice that hand calculations and Finite
Element Analysis resulted in similar results. For Hand
calculations, computer simulations, and full details, it is
recommended to make reference in the appendix attached to
this document.
PROJECT CRITIQUE
While we believe the data that was recorded was indeed
accurate, the likelihood for error was drastically increased by
lack of quality lab equipment. On one occasion, the lab
temperature was higher than previous test days and the
hydraulic ram appeared to be leaking hydraulic fluid past the
internal O-ring seal. This resulted in an aborted test for the
remainder of that day until temperatures has dropped to a
reasonable level. Also, given that the hydraulic ram was
actuated by a hand pump, controlling exactly how much load
was being applied became nearly impossible. The objective was
to record deflection at each increment of 30 lbs., however the
actual interval varied between 25 and 50 lbs. Despite this, a
proper load-deflection curve was successfully generated.
Upon further investigation, it is believed that more accurate
results could be attained by utilizing proper material testing
equipment that records as a function of time, or at smaller load
intervals, while recording more accurately the total deflection of
the system at these much smaller interval.
CONCLUSION
In this paper, results of a study of Ultimate Strength of Aircraft
Structures are presented. Analysis and testing was conducted on
multiple beams with different shapes to check the local
crippling according to the theory from Dr. Todd Coburn‟s paper.
Finite element analysis was implemented to correlate the results
from hand calculations and beam testing. It was found that
computer modeling always had results between hand
calculations and actual data, whereas theory always had higher
values on the stress-strain curve, but similar shape as theory.
REFERENCES
[1] Todd Coburn. "Ultimate Strength of Aircraft Structures."
(2014): 1-9. IMECE2014-39986.
[2] Bruhn, E.F., June 1973, “Analysis & Design of Flight
Vehicle Structures”, S.R. Jacobs & Associates, IN.
[3] Philpot, Timothy A. Mechanics of Materials. 3e ed.
Danvers: Wiley, 2012. Print.
[4] Flabel, Jean-Claude. Practical Stress Analysis for Design
Engineers: Design and Analysis of Aerospace Vehicle
Structures. First Edition ed. Hayden Lake, ID: Lake City
Pub., 1997. Print.

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