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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.
1
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
2
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
4
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.
5
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
6
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
7
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.
8
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.
9
6. References
Gilbert, J. A and C. L. Carmen. "Chapter 4 – Column Buckling Test." MAE/CE 370 –
Mechanics of Materials Laboratory Manual. June 2000.
10
7. Raw Notes
Figure 6
11
Figure 7
12