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.