AEROSPACE 305W STRUCTURES & DYNAMICS LABORATORY
Laboratory Experiment #2
Column Buckling
April 6, 2010
Chris CameronSection 18
Lab Partners:Joseph O’Leary
Zeljko RaicMichael YoungJonathan Hudak
Gregory Palencar
Course Instructor: Dr. Stephen ConlonLab TA: Mike Thiel
Abstract
Columns are commonly used in engineering and specifically in aerospace in situations
such as the ribbing in wings. They support a load but most often their critical load is determined
by when buckling occurs. This buckling is caused either by imperfections in the column or the
loading. This experiment was designed with the objectives of confirming the theoretical
predictions for when columns buckle and how to increase their critical load. It was assumed that
the longer columns would buckle sooner and also the simply supported vs. clamped end columns
would also buckle sooner by a factor of 4. Another assumption was made that the increasing
slenderness ratio of the columns would decrease their critical stress. The experiment was set up
by loading varying lengths of beams with both simply supported and clamped fixities. A load
was applied until the central displacement, measured using a linear variable differential
transformer, began to increase without increase in load. The data supported the assumptions
with an error being at all points below 20%.
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. I. Introduction
Columns are commonly used engineering structures that are used to carry compressive loads.
A common aerospace column is the ribbing found within the airfoils on a plane. However
instabilities cause columns to not only compress, but to buckle under loading. Buckling is a
disproportionate increase in displacement with an additional applied load. This buckling reduces
the columns ability to carry loads and must be understood in order to determine the maximum
load of a column.
The objective of this lab will be to determine if shorter or longer columns buckle under
different loads and if the method if fixing the ends also affects the buckling load. Also the
slenderness ratio effect on the critical stress will be examined. The experimental data will be
compared to theoretical data to find if the theory behind column buckling predicts the data
collected. Error between the theoretical and experimental data will give insight to improper
assumptions about boundary conditions, as well as other sources of error within the experiment.
Columns instabilities are due to both imperfections in the column as well as imperfections in
the loading. Columns imperfections can be due to imperfections in the material, as well as the
shape of the column being imperfect. The loading imperfections occur when loads are applied
that are not along the centerline of the beam, creating a moment on the end of the column.
Columns can buckle in different ways and this is mainly dependant on the method of fixing
the ends of the column. There are three common types being clamped, simply supported and
free. Each of these types of fixities corresponds to a set of boundary conditions at the end of the
beam. Free fixing allows for both displacement and rotation, simply supported will not allow
displacement, and clamped will not allow displacement or rotation. These boundary conditions
will be used to derive the governing equations for columns.
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Uniform columns are governed by the differential equation
EId4wd x4 +P d
2wd x2 =0 (1.1)
where E is the modulus of elasticity of the column, I is the cross-sectional area, and P is the load
applied to the column. EI together is the columns stiffness. Assuming that the stiffness and load
are constant, k can be defined as
k2= PEI (1.2)
Applying (1.2), (1.1) becomes
w ' ' ' '+k2w' '=0 (1.3)
The general solution to the differential equation (1.3) becomes
w (x )=c1sin (kx )+c2 cos (kx )+c3 x+c4 (1.4)
where c1, c2, c3, and c4 are constants of integration. The four boundary conditions corresponding
to a given columns displacement and rotation at the each end create four equations. The four
equations and five unknowns create an eigenvalue problem. Solving for the eigenvalues yields a
family of loads causing the column to buckle
P=c n2π2 EIL2 (1.5)
where c is a constant relating to how the ends are fixed, n is the mode shape, and L is the length
of the column. The first mode shape (n=1) gives the critical load of the column
Pcr=cπ2 EIL2 (1.6)
In this experiment columns were loaded until the critical load, corresponding to the lowest load
where buckling occurs, was reached for both columns simply supported at both ends and
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clamped at both ends. The eigenvalue problems yielded c values of c=1 for simply supported
and c=4 for clamped.
Six different columns were used in this experiment, each having the same cross section. This
allowed for one calculation to be made for I for all the beams. For a rectangular cross section the
moment of inertia is found using the equation
I=bh3
12(1.7)
Figure 1.1 shows the cross section of the columns with measurements for both base and height.
Figure 1.1- Cross section of columns
This orientation of the beam yields a smaller moment of inertia, which will cause the column to
buckle in the direction parallel to the short edge. The moment of inertia corresponding to this
orientation is 0.000122 in4.
By use of equation (1.6) and assuming that EI will be constant for all columns predictions
about the critical load can be made. For columns with the same end fixity the only variable
remaining is L2 in the denominator. This will cause the shortest beams to have the highest
critical loads. Also by comparing the values of c it is found that the clamped beams will have a
higher critical load.
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Another value of interest is the critical stress, σcr, of the column. This is defined in the
equation
σ cr=PcrA
(1.8)
The slenderness ratio, s, is the value of interest for determining the critical stress and is defined
as
s=L
r √c (1.9)
The radius of gyration, r, is defined as
r=√ IA (1.10)
Applying equations (1.6), (1.9), and (1.10) to equation (1.8) yields
σ cr=π2Es2
(1.11)
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II. Experimental Procedure
Data was acquired for this lab use of two different instruments. The first was a linear variable
differential transformer (LVDT), which was used to measure the deflection at the midpoint of the
column. The second was a load cell force gauge which was attached to a load wheel and the
horizontal beam. A converter box was used to convert the load from the load wheel to the load
being applied to the column specimen. All data was collected using Labview software, which
for averaged each acquired data point from a hundred instantaneous values. This average
corrected for vibrations. The set up of the experiment is shown in figure 2.1.
Figure 2.1- Experimental set up
Two different fixing methods were used for this experiment. The first, clamped, was created
by inserting the column into a bracket which was then tightened using a screw and hex key. The
simply supported case was created by flipping the clamping brackets so that the ends of the
column were placed into a notch in the bracket. The ends of the simply supported column were
angled into a point in order to have only one contact point in the bracket. These fixing methods
are shown in figure 2.2.
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Figure 2.2- End fixing methods
Once each column was fixed into the apparatus the horizontal beam was lowered into place.
Balance mass was used to counteract the moment of the beam about the fixed spring end. The
beam was leveled using the level adjust spring. The load cell was then attached to the load
wheel and the horizontal beam. The LVDT was then mounted onto the side of the specimen at
the center to measure the maximum displacement. Ensuring that the load was at zero the LVDT
was also zeroed.
Once the set up was complete data was then gathered. The load wheel was turned to increase
the load on the column. Data points were collected using the Labview software and were largely
spaced for the beginning loads. Once the load began to near the expected critical load, the data
points were measured at smaller load increments. Data was collected until the column began to
deflect greatly without a change in the load. The column was then unloaded and the test was
repeated for each of the six samples.
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III. Results and Discussion
The collected data was expected to show an asymptotic behavior as the applied load neared
the critical load. The critical load of the shorter specimens was expected to be much higher than
that of the longer specimens. Also due to boundary conditions the clamped specimens were
expected to have critical loads four times greater than their simply supported counterparts. The
theoretical values for the critical loads of the six experiments are shown in table 3.1.
Table 3.1- Theoretical critical buckling loads
Column Length (in) Clamped Pcr (lb) Simply Supported Pcr (lb)18 445.96 111.521 327.64 81.924 250.85 62.7
The horizontal asymptote of the load vs. displacement was found for each specimen and these
load values were used as the experimental critical buckling loads. The error using the asymptotic
method was low for this experiment. All values were under 15% error but also all values with
one exception were lower than the expected theoretical values, most likely caused by fatigue on
the specimens. These values are shown in table 3.2 while their error is shown in table 3.3. The
data is shown in figures 3.1 and 3.2.
Table 3.2- Experimental critical buckling loads
Column Length (in) Clamped Pcr (lb) Simply Supported Pcr (lb)18 401 10121 293 87.524 247 54.5
Table 3.3- Percent error, asymptotic vs. theoreticalLength (in) % Error Clamped % Error Simply Supported
18 10.082 9.41721 10.573 6.83824 1.535 13.078
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
50
100
150
200
250
300
350
400
450
Force vs. DisplacementClamped - Clamped
18"
21"
24"
18" Asymptote
21" Asymptote
24"AsymptoteDisplacement (in)
Forc
e (lb
)
293
401
247
Figure 3.1- Clamped-Clamped asymptotic graph
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
20
40
60
80
100
120
Force vs. DisplacementSimply Supported-Simply Supported
18"21"24"18" Asymptote21" Asymptote24" AsymptoteDisplacement (in)
Forc
e (lb
s)
54.5
78.5
101
Figure 3.2- Simply Supported asymptotic graph
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A second method of experimentally determining the critical buckling load was used. This
method of finding the slope between the displacement and the displacement-load ratio is referred
to as the imperfection accommodation method. Values for the critical load found using this
method were compared to both the theoretical values and the values found using the asymptotic
method. The percent error between the imperfection accommodation method and the theoretical
data was very similar to the percent error between the asymptotic method and the theoretical
data. However there was a large error between the two methods for the Clamped-Clamped
column but not for the Simply Supported column. The values for the critical load and percent
error are shown in tables 3.4 – 3.6 and the data for the critical load is found in figures 3.3 and
3.4.
Table 3.4- Imperfection accommodation critical loads
Column Length (in) Clamped Pcr (lb) Simply supported Pcr (lb)18 452.2 96.821 381.4 90.224 284 56.5
Table 3.5- Percent error imperfection accommodation vs. theoretical
Length (in) Percent error Clamped Percent error Simply Supported18 1.399 13.18421 16.408 10.13424 13.215 9.889
Table 3.6- Percent error imperfection accommodation vs. asymptotic
Length (in) Percent error Clamped Percent error Simply Supported18 12.768 4.15821 30.171 3.08624 14.98 3.67
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0 0.00005 0.0001 0.00015 0.0002 0.000250
0.01
0.02
0.03
0.04
0.05
0.06
f(x) = 283.969594782373 x − 0.0090920604652454
f(x) = NaN x + NaNf(x) = NaN x + NaN
Displacement vs. Displacement/LoadClamped-Clamped
18"Linear (18")21"Linear (21")24"Linear (24")
Displacement/Load (in/lb)
Disp
lace
men
t (in
)
Figure 3.3- Clamped imperfection accommodation graph
0 0.0002 0.0004 0.0006 0.0008 0.001 0.00120
0.01
0.02
0.03
0.04
0.05
0.06
0.07
f(x) = 56.4977135349544 x − 0.00485000237508319
f(x) = 90.1531676418197 x − 0.0163812197798266
f(x) = NaN x + NaN
Displacement vs. Displacement/LoadSimply Supported-Simply Supported
18"Linear (18")21"Linear (21")24"Linear (24")
Displacement/Load (in/lb)
Disp
lace
men
t (in
)
Figure 3.4- Simply supported imperfection accommodation graph
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The slenderness of the beams was also measured to find its effect on critical buckling stress.
The experimental values of the buckling stress were determined from both the asymptotic and
the imperfection accommodation methods. There was a reasonably small amount of error found
when comparing the theoretical data with the experimental data and all data supported the
assumption that the higher slenderness ratios would support a lower critical stress. Values for
the data and error can be found in tables 3.7 – 3.10 and graphs of the critical stress vs. the
slenderness ratio can be found in figures 3.5 and 3.6.
Table 3.7- Simply supported critical stress
Length (in.) Slenderness ratio Theoretical stress (psi) Asymptotic stress (psi) Imperfection stress (psi)18 498.974 1189.333 1077.333 1032.53321 582.137 873.6 933.333 962.13324 665.3 668.8 581.333 602.667
Table 3.8- Clamped critical stress
Length (in.) Slenderness ratio Theoretical stress (psi) Asymptotic stress (psi) Imperfection stress (psi)18 498.974 4756.907 4277.333 4823.46721 582.137 3494.827 3125.333 4068.26724 665.3 2675.733 2634.667 3029.333
Table 3.9- Simply supported critical stress percent error
Length (in.) Percent error asymptotic Percent error Imperfection18 0.094170404 0.13183856521 0.068376068 0.10134310124 0.130781499 0.098883573
Table 3.10- Clamped critical stress percent error
Length (in.) Percent error asymptotic Percent error Imperfection18 0.100816217 0.01399228621 0.105725797 0.1640825324 0.015347817 0.132150688
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450 500 550 600 650 7000
200
400
600
800
1000
1200
1400
Critical Stress vs. Slenderness RatioSimply Supported-Simply Supported
Theoretical AsymptoticImperfection Tech.
Slenderness Ratio
Criti
cal S
tres
s (ps
i)
Figure 3.5- Simply supported critical stress vs. slenderness ratio graph
450 500 550 600 650 7000
1000
2000
3000
4000
5000
6000
Critical Stress vs. Slenderness RatioClamped-Clamped
TheoreticalAsymptoticImperfection Tech.
Slenderness Ratio
Criti
cal S
tres
s (ps
i)
Figure 5.6- Clamped critical stress vs. slenderness ratio graph
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IV. Conclusions
The objectives of this lab were to determine how varying lengths of beam and end fixity
affected the critical loads. The experimental data supported the theoretical calculations that
the inverse square of the length of the column directly relates to the critical load. Also the
clamped method of fixing the ends was found to produce a critical load roughly four times
greater than the simply supported method. As for the slenderness ratio’s effect on the critical
stress, the theoretical data was again supported although with slightly more error than the
critical loads for the clamped condition. Each experiment did follow the trend that increased
slenderness ratio decreased the critical stress. This shows that when designed columns to
resist buckling they should be kept as short as possible, and also they should be clamped at
the ends. One other method not tested in this experiment would be to increase the stiffness of
the column either using geometry or material properties.
While the overall amount of error was small for each trial there was a common trend that
most buckled at a lower than expected applied load. There are two sources of error that most
likely are responsible for this. One is that the columns have been used repeatedly to repeat
this experiment and can be suffering from fatigue. Newly manufactured columns being used
for each experiment could reduce this error. Another source could be the sideways force
being applied by the LVDT. The spring that holds the device against the column applied a
force that could cause buckling to occur earlier than predicted. By mounting the LVDT with
a glue or tape instead of a spring the error here could be reduced.
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