AERSP 305 – AEROSPACE TECHNOLOGY LABORATORY
Experiments on Column Buckling
Author: Arthur Melo Cruz
Lab 2
Room: 047 Hammond
Lab Partners: Aaron Sheffield, Haralambos Zografos, Tyler Sheridan
Date of Experiment: February 6, 2017
TA: Shitiz Sehgal
Instructor: Benjamin Beck, PhD
Abstract
This report assesses the relationships between theoretical and experimental analyses of
buckling loads for columns with variable lengths, under different boundary conditions.
The purpose is to compare the performances of three specimens of columns of same
material/cross-section but different lengths, under simple support and clamped support,
determining which arrangement yields the highest structural stability in terms of
buckling. The motivation behind this experiment lies on the importance of stability for
aerospace structures, which are subject to high loads and require low factors of safety for
optimal design. This document first establishes a theoretical model, starting from the
governing differential equation for buckling and deriving critical buckling loads for two
different sets of boundary conditions. Furthermore, it lays the details of the experimental
procedure, informing about the physical setting used for conducting the experiment. This
report analyses the resulting experimental data, using three converging approaches:
asymptotic method, imperfect accommodation method and slenderness ratio study. All
methods compare experimental results with the theory, assessing accuracy of the
measurements, determining anomalies and evaluating reliability of the data. The report
concludes that the experiment offers an appropriate evaluation of the subject matter, due
to relatively low errors contained in the data acquired, offers hypotheses that explain
discrepancies between theory and reality, and suggests modifications for further tests to
be conducted within the subject matter.
Introduction
The purpose of this experiment is to analyze the performance of three specimens of
columns under compressive load, evaluating how the length of the columns and boundary
conditions affect their buckling behavior. The goal is to assess the correlations between
experimental data and the theoretical model, determining accuracy of the measurements
and evaluating the effects of variable loading profiles on lateral deflections. Compressive
loads are present in several aerospace applications, including deployable space structures
and honeycomb cores on wings, which illustrate the importance of assessing column
instability.
The three specimens are constituted of homogeneous type 304 stainless steel, with
Young’s Modulus of 27×106 𝑝𝑠𝑖 and constant rectangular cross-section of 0.75×0.125 𝑖𝑛,
yielding a second area moment of inertia of 1.2207×10−4 𝑖𝑛4. The specimens only differ
in their lengths, which are 18 𝑖𝑛, 21 𝑖𝑛 and 24 𝑖𝑛. With this specifications, it is possible to
determine a single relationship between length and critical buckling load for each set of
boundary conditions.
The experiment consists of applying variable compressive loads on the specimens under
two sets of end fixity, which simulate clamped-clamped and pinned-pinned boundary
conditions. A Linear Variable Differential Transformer (LVDT) is mounted at the
midpoint of each column to measure transverse displacements, and the loads are applied
through a screw-lever mechanism attached to a load cell. More details about the setting
can be found on the Experimental Procedure of this report.
Column Buckling Theory
Buckling theory follows Euler’s critical load theory, which use the Euler-Bernoulli’s
assumptions and free body diagram analysis on a small element of a deformed column to
derive a governing differential equation for column buckling. The governing equation is
presented below:
𝐸𝐼𝑑4𝑤
𝑑𝑥4 + 𝑃𝑑2𝑤
𝑑𝑥2 = 0 (Eq. 1)
Details on the derivation of this formula can be found in the appendix 1. In this case, 𝐸𝐼
represents flexural rigidity, w is the deflection of the column, P is the compressive load
and x is the position along the length of the column. For solving this equation to obtain
deflection 𝑤(𝑥), we define a constant 𝑘 as 𝑘2 =𝑃
𝐸𝐼 (Eq. 2). The solution for this linear
differential equation with constant 𝑘 is:
𝑤(𝑥) = 𝐶1 sin(𝑘𝑥) + 𝐶2 cos(𝑘𝑥) + 𝐶3𝑥 + 𝐶4 (Eq. 3)
The constants of integration will be determined through the application of boundary
conditions.
For the simply supported case, in which both ends of the column are pinned to the
supports, the boundary conditions are the following:
1. 𝑤(𝑥 = 0) = 0
2. 𝑤(𝑥 = 𝐿) = 0
3. 𝑤′′(𝑥 = 0) = 0
4. 𝑤′′(𝑥 = 𝐿) = 0
Applying these conditions to (Eq. 3) yields the following characteristic equation for a non-
trivial solution: sin(𝑘𝐿) = 0 (Eq. 4). For the simplest mode and lowest non-trivial value at
which the column buckles, 𝑘 =𝜋
𝐿. Equating with (Eq. 2), we obtain the following solution
for the simply supported buckling load: 𝑃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 𝜋2𝐸𝐼
𝐿2 (Eq. 5).
For the clamped case, at which both ends of the column are cantilevered, the boundary
conditions are:
1. 𝑤(𝑥 = 0) = 0
2. 𝑤(𝑥 = 𝐿) = 0
3. 𝑤′(𝑥 = 0) = 0
4. 𝑤′(𝑥 = 𝐿) = 0
The characteristic equation yielded is: cos(𝑘𝐿) = 1 (Eq. 6). For the first mode of buckling,
𝑘 =2𝜋
𝐿. Equating with (Eq. 2), we obtain the following solution for the clamped buckling
load: 𝑃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 4𝜋2𝐸𝐼
𝐿2 (Eq. 7). Note that the critical load for cantilevered condition is 4
times higher than the critical load for simple support condition.
Experimental Procedure
The experiment is conducted using a screw-lever apparatus made by TecQuipment, which
applies a uniform compressive load on the specimens under variable boundary
conditions. A schematic of the apparatus is shown below:
In the apparatus, the LVDT is mounted at the midpoint of the specimen, to measure
maximum deflections. The support blocks simulate the two boundary conditions tested.
The cantilevered condition is simulated with a clamp on the support blocks, and the
simple support is established with a dent that fits at the end of the specimens. The balance
mass levels the structure to ensure uniform vertical loading on the specimens. The load
wheel applies a force on the structure when turned, compressing the specimens, and the
level adjust controls the height of the compressing structure, also maintaining the
structure leveled.
For loading the specimens, a student controls the load wheel and the level adjust
simultaneously, maintaining vertical compression. The level adjust and load wheel are
turned in small increments, increasing compression and to acquire precise measurements
of force and deflection. Loading starts at zero and is maintained until it stabilizes
Level adjust
Load Cell
Specimen Load Wheel
LVDT
Balance mass
Support Blocks
Figure 1: The Experimental Apparatus
asymptotically, when the buckled state is achieved. Data from the load cell and LVDT are
transmitted to LabVIEW software, which correlates load with displacement. The
experiment is run a total of 6 times, for each specimen in a specific support configuration.
The results are discussed in the next section.
Results and Discussion
Applying the values of 𝐸 (27×106 𝑝𝑠𝑖), 𝐼 (1.2207×10−4 𝑖𝑛4) and 𝐿 (18 𝑖𝑛, 21 𝑖𝑛, 24 𝑖𝑛) to
(Eq. 5) and (Eq. 7) from the Column Buckling Theory section, we obtain the following
results for theoretical values of critical buckling load:
Now, it is worth analyzing the experimental results and comparing them with the
theoretical values from Table 1. Following are the plots of deflection at the midpoint vs.
load through the asymptotic method, for the three specimens, in which load converges to
the critical buckling value as deflection grows.
Length Support Type Critical Load (lb)
L = 18 in Simple Support 100.4
Clamped Support 401.59
L = 21 in Simple Support 73.76
Clamped Support 295.05
L = 24 in Simple Support 56.47
Clamped Support 225.9
Table 1: Theoretical values of critical load
In these plots, experimental data is represented by discrete points, while the asymptotes
represent the theoretical values to which the data should converge. The first aspect to be
noted is that the experimental values for the simple support (pin-pin condition) converge
better to their theoretical counterparts than the curves for clamped support, in all three
specimens. This indicates that the enforcement of boundary conditions for pin-pin cases
were stronger than for clamp-clamp cases. It is also worth noting that the loads for the
clamped support condition are stabilizing at a value much lower than the theoretical
asymptote, corroborating the hypothesis that the boundary conditions for that case were
not enforced properly. Besides that, the data follows the general behavior described by
the theory that higher lengths for the columns generate lower critical buckling loads, with
the 18 in specimen having the highest buckling loads for both sets of boundary conditions.
To further assess the differences between the models, following are the experimental data
using an imperfection accommodation method, in which the slopes of the graphs yield
the critical buckling loads for the experiments. This method accounts for the fact that the
columns start deflecting before they achieve their critical loads due to structural
imperfections, by displaying a relationship of deflection versus the ratio between
deflection and corresponding load.
Figure 2: Plots of Deflection vs. Load through Asymptotic Method
It is observed that the critical loads obtained by the fitted lines are relatively accurate for
the pin-pin conditions on all three specimens, but have lower accuracy on the clamped
conditions, agreeing with the results from the deflection vs. load plots. One clear
aberration noted relates to the critical load for the clamped conditions on the 21 in
specimen. Theoretically, its value should be higher than the one for 24 in, but the
experimental results show the opposite, with the buckling load for the 21 in being the
lowest among the 3 specimens. Again that result might also have been caused by poor
boundary conditions enforcement.
To finalize the experimental data analysis, it is worth looking at the data of buckling stress
versus effective slenderness ratio. This analysis allows the observation of how theory
Figure 3: Plots of Deflection/Load vs. Deflection through imperfection
accommodation method
matches with the experiments by comparing buckling performance with the slenderness
of the specimens, which relates to how effectively “thin” the specimens are. In theory,
slender subjects have higher slenderness ratios, being easier to buckle. Note that effective
slenderness ratio is also affected by boundary conditions, with clamped condition leading
to lower slenderness ratio than pinned condition. For details on the formulas for
slenderness ratio, see appendix 2.
From the plot, it is observable that experimental critical stress (load divided by area) for
pinned condition is again much closer to their theoretical parallels than the values for
clamped condition. The data displays large discrepancies for low effective slenderness
ratios, which is the case of clamped condition, agreeing with the previous two sets of plots.
The anomaly in the buckling load for the 21 in specimen in clamped condition is noted
again, with its critical stress posing to be the lowest among the data runs for the clamped
case.
In terms of percentage errors, the data for all 6 experiment runs are displayed below:
Length Support Type Theoretical
Critical Load (lb) Experimental
Data (lb) Error (%)
L = 18 in Simple Support 100.4 81.1996 19.1239
Clamped Support 401.59 197.0559 50.93107
L = 21 in Simple Support 73.76 58.0124 21.34978
Clamped Support 295.05 158.4843 46.28561
L = 24 in Simple Support 56.47 48.018 14.96724
Clamped Support 225.9 179.1683 20.6869
Figure 4: Critical Stress vs. Effective Slenderness Ratio plot
Table 2: Percentage Errors
The experimental data used was obtained from the slopes of the imperfection
accommodation plots, since they offer objective values for the critical loads. From this
assessment, it is concluded that the experimental data is validated as reasonable, since
the percentage errors are contained in a range roughly between 15% and 50%. As noted
from the previous analyses, the errors for simple support cases are among the lowest.
Even though the errors for clamped support are high, they are contained within a range
of acceptable values. One interesting fact from this analysis is that it does not make
evident the anomaly for the clamped condition on the 21 in specimen. In addition, the
error for the clamped condition on the 24 in specimen is similar to the errors for the pin-
pin condition, indicating better enforcement of boundary conditions on that test run than
on the 21 in and 18 in.
Conclusions and Recommendations
Given observed evidence on the experimental data, the theoretical model offers an
objective and relatively accurate assessment of the behavior of columns under
compressive loads. The theoretical calculations for buckling loads matched the
experimental results with a maximum error of 50.93107% on all test runs, and the general
behavior of the columns followed the relationships established by the theory, with longer
lengths resulting in lower buckling loads.
From this experiment and its correlation with the theory, it is concluded that clamped
columns with shorter lengths offer the best structural stability for aerospace applications,
with highest buckling loads. Thus, designers should prioritize structures that contain
lower effective slenderness ratios, reducing risk of instability. This conclusion comes from
both theoretical calculations and experimental evidence, with the 18 in specimen in
clamped condition having the highest buckling load.
This experiment offers a good initial assessment of buckling behavior of columns, but a
series of improvements must be made with the apparatus to reduce error and increase
reliability of the results. In this experiment, the main causes for inaccuracies are:
1. Poor enforcement of boundary conditions with the support blocks;
2. Failure in maintaining the structure leveled while loading the columns, causing
non-uniform loads;
3. Imperfections with the specimens, including pre-buckling;
From the causes listed above, the most critical is the first one, related to boundary
conditions. The poor clamping system in the support blocks might have created a non-
zero slope at the ends of the specimens during the compression, altering their boundary
values. For future experiments, a better clamping system should be implemented. In
addition, new specimens should be used, and an electronic apparatus that maintains the
columns leveled automatically without the need for human input should be introduced.
Appendix 1 – Derivations on the GDE
Consider the column under compressive loads displayed below:
Now, consider one element of the deformed column:
The sum of forces yields the following equation: ∆𝑉
∆𝑥+ 𝑞 = 0 (Eq. 8).
The sum of moments yields: ∆𝑀
∆𝑥+ 𝑉 +
∆𝑉
2+ 𝑃
∆𝑤
∆𝑥= 0 (Eq. 9).
Letting ∆𝑥, ∆𝑀, ∆𝑉, ∆𝑤 → 0, we obtain:
𝑑𝑉
𝑑𝑥+ 𝑞 = 0 (Eq. 10)
𝑑𝑀
𝑑𝑥+ 𝑉 + 𝑃
𝑑𝑤
𝑑𝑥= 0 (Eq. 11)
Now, recall the Moment-Curvature equation: 𝑀 = 𝐸𝐼𝑑2𝑤
𝑑𝑥2 (Eq. 12)
Assuming 𝑞 = 0 we obtain: 𝑑𝑉
𝑑𝑥= 0 (Eq. 13)
Taking the derivative of (Eq. 11) we obtain: 𝑑2𝑀
𝑑𝑥2+ 𝑃
𝑑2𝑤
𝑑𝑥2= 0 (Eq. 14)
Substituting (Eq. 12) on (Eq. 14) we obtain the GDE:
𝐸𝐼𝑑4𝑤
𝑑𝑥4+ 𝑃
𝑑2𝑤
𝑑𝑥2= 0 (Eq. 1)
P P
q(x)
Appendix 2 – Definition of Slenderness Ratio
Radius of Gyration is defined as: 𝑟 = √𝐼
𝐴 (Eq. 15), in which I is the second moment of
inertia and A is the area of the cross-section.
The effective slenderness ratio is defined as: 𝑠 =𝐿
𝑟√𝑐 (Eq. 16), in which L is the length of
the column, r is the radius of gyration and c serves as a factor defined upon boundary
conditions. For simple support 𝑐 = 1, and for clamped support 𝑐 = 4. The slenderness
ratio is a measure of how “thin” the column is, dividing the length by a function of the
area of the cross-section, boundary conditions and moment of inertia. Columns that have
higher slenderness ratio have lower buckling loads.