Mechanics of Materials Laboratory

Column Buckling Test

David Clark

Group C:

David Clark

Jacob Parton

Zachary Tyler

Andrew Smith

10/27/2006

Abstract

Leonhard Euler first derived a series of equations that can successfully determine

the buckling behavior of columns. The following procedure attempts to verify one of

these equations. The maximum load, the highest load a column can support without

buckling, is correlated to the Young's modulus, moment of inertia, length of a beam, and

method of support. The practice of using a column buckling machine can be very

accurate with proper testing materials and procedure. The results within this experiment

exhibited up to 44% error.

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Table of Contents

1. Introduction & Background.............................................................................3

1.1. General Background................................................................................3

2. Equipment and Procedure................................................................................4

2.1. Equipment................................................................................................4

2.2. Experiment Setup.....................................................................................5

2.3. Procedure.................................................................................................5

3. Data, Analysis & Calculations.........................................................................5

3.1. Theoretical Calulcations..........................................................................6

4. Results..............................................................................................................9

5. Conclusions......................................................................................................9

6. References......................................................................................................10

7. Raw Notes......................................................................................................11

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1. Introduction & Background

1.1. General Background

The physicist and mathematician Leonhard Euler first derived a series of

equations to determine the deformation of columns under loads. The following procedure

attempts to verify one of these buckling equations for steel columns.

Euler determined the following expression for determining the critical load.

Equation 1

E is the elastic modulus of the specimen

I is the second moment of area (moment of inertia)

Le is the effective length.

The method of support determines the effective length. The figure below

demonstrates what these effective lengths are, as well as provide a visual explanation

why the effective length per setup is different.

Figure 1

3

When a critical load is applied, the buckling occurs in the plane perpendicular to

the corresponding principal axis of inertia. To utilize this, the radius of gyration is

introduced. "In structural engineering, the two-dimensional radius of gyration is used to

describe the distribution of cross-sectional area in a beam around its centroidal axis"

(Wikipedia). The radius of gyration is given by the following formula

Equation 2

Combining Equation 1 and 2, the expression for critial loading becomes

Equation 3

The critical load can be used to find the stress in the beam being loaded.

Equation 4

2. Equipment and Procedure

2.1. Equipment

1. Column Buckling Machine

2. Three Metal Beams: In this experiment, steel beams of known length

were used. The modulus of elasticity for the material tested was

predefined.

3. Calipers, a Dial Gage, and a Tape Measure: Calipers should be used to

measure the width and thickness of the beam. Dial gages will be used to

measure deflection along the length of the beam. The tape measure is used

to measure the length between supports

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4. Specimens to be Tested: The following procedure utilizes three

speciments: one specimen prepared to be fixed at both ends, one specimen

that models a pinning support at both ends, and a final specimen that has

one fixed end and another end acting as a pin support.

2.2. Experiment Setup

The specimen should be secured on the column buckling machine with each end

of the specimen being supported per case requirements. The effective length (the distance

between supports) should be measured and recorded.

A dial gage should be attached to the column buckling machine such that any

deflection of the beam can be easily measured and recorded.

2.3. Procedure

A load is then induced onto the beam by the column buckling machine. The

deflection should be recorded from the dial gage secured to the apparatus. The load is

measured by reading the load gage after balancing the beam applying the force. After

each applied load, record the force and deflection.

3. Data, Analysis & Calculations

In the following set of results, the scenarios are labeled as follows:

o Case 1 : A beam fixed on both ends

o Case 2: A beam fixed on one end with the second end acting as a

pin support

o Case 3: A beam with both ends acting as a pin support

The following table categorizes known dimensional data of the test specimens.

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The length, L, was the distance measured between supports. The area is the

product of the width multiplied by the thickness.

3.1. Theoretical Calulcations

The following calculations should serve as an example for the calculations used in

all three cases. The effective length, second moment of area, radius of gyration, and

critical load were found using Equations 1 through 3 and Figure 1.

Equation 5

Equation 6

Equation 7

Equation 8

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Table 1

Load vs Deflection - Case 1

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6

Deflection (in)

Lo

ad

(lb

)

Figure 2

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Load vs Deflection - Case 2

0

50

100

150

200

250

300

350

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Deflection (in)

Lo

ad

(lb

)

Figure 3

Load vs Deflection - Two Pinned Supports

0

50

100

150

200

250

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Deflection (in)

Lo

ad

(lb

)

Figure 4

The following figure combines Figures 2 through 4 to demonstrate how different

support conditions change the buckling load.

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Load vs Deflection

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6

Deflection (in)

Lo

ad

(lb

)

Case 1

Case 2

Case 3

Figure 5

4. Results

Table 2

5. Conclusions

The error within this experiment was grossly inaccurate and exhibited

unacceptable error. The main source of error was due to poor testing technique. Even

with proper technique, however, persistent and large error was still present. The

remaining error was due to ill conditioned test samples and poorly calibrated equipment.

The samples had experienced buckling many times previous to this test, and the clamps

were not aligned to place the force parallel along the beam.

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6. References

Gilbert, J. A and C. L. Carmen. "Chapter 4 – Column Buckling Test." MAE/CE 370 –

Mechanics of Materials Laboratory Manual. June 2000.

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7. Raw Notes

Figure 6

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Figure 7

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