(1).
MGJ 10 Nov 96
PURPOSE
The purpose of this exercise is to study the effects of end conditions,
column length, and material properties on compressive behaviour
and buckling in columns.
EQUIPMENT
Solid rods of various lengths of aluminum and steel
Universal test machine with grips, controller, and data acquisition
system
PROCEDURE
Repeat the following steps for each specimen.
Measure the diameter and lengths of each specimen to 0.02 mm.
Zero the load output (balance).
Activate load protect (~50 N) on the test machine to prevent overloading the specimen during installation.
Install the top end of the test specimen in the top grip of the test machine while the test machine is in displacement control.
Install the bottom end of the test specimen in the lower grip of the test machine.
In displacement control adjust the actuator position of the test machine to achieve nearly zero load on the specimen.
Deactivate load protect.
Initiate the data acquisition and control program.
Enter the correct file name and specimen information as required.
Initiate the test sequence via the computer program.
Continue the test until specimen buckling or compressive failure
Examine the load versus displacement trace for each test. Note
the load at the onset of buckling or compressive failure.
RESULTS
Structures and machines may fail in many ways depending on the
materials, kinds of loads, and conditions of support. Many machine
elements can be modeled as uniform members under uniaxial tension
or compression. For tensile loading, these members tend to self-align
and fail either by ductile deformation or brittle fracture depending
on the material. In compression, the failure mode is complicated
by the possibility of a geometric instability, called buckling,
in addition to ductile deformation.
Columns are structural members which support compressive loads.
Buckling occurs when the column has a tendency to deflect laterally,
out of the line of action of the load. Once buckling initiates,
the instability can lead to failure of the column because the
eccentric load acts as a moment causing greater stresses and deflections
due to the combination of the bending and axial loads.
The possibility of buckling increases for the following column
conditions: 1) longer, "thinner" columns, 2) pinned,
free, or non-fixed end conditions, 3) initial eccentricity of
the loading (e.g., bent columns) and/or 4) lower elastic modulus
of the column material.
In this exercise, two materials and two column lengths will be
studied. Anticipated buckling or compressive failure loads will
first be calculated for various length specimens and materials.
(1).
where 
is the compressive
failure load (yield), 
is proportional limit
stress (or yield strength), 
is the initial
area of the gage section, 
is the Euler critical
buckling load, I is the least moment
of inertia of the cross section, and 
is
the effective, unsupported length of the column.
The anticipated buckling or compressive failure loads will then
be compared to the actual measured loads at the onset of instability.
Observations will be made on the effects of end conditions, material
type, and column length.
Show all work and answers on the Worksheet, turning this in as
the In-class Lab report.
References:
"Mechanics of Materials," J.M. Gere and S.P. Timoshenko
"Mechanics of Materials," R.C. Hibbeler
MGJ 10 Noc 96
NAME______________________________________DATE______________
EQUIPMENT IDENTIFICATION______________________________________
1) Determine (look up) the following mechanical properties.
Table 1 Selected Properties for Test Materials
6061-T6 Aluminum 1018 HR Steel
Selected Mechanical Properties (R.T.) Selected Mechanical Properties (R.T.)
| E (GPa) | E (GPa) | |||
|
|
| |||
|
|
| |||
| % elongation | % elongation |
2) Measure and record the following dimensions.
Table 2 Pertinent column dimensions
Column Dimensions for Column Dimensions for
Aluminum Steel
| Diameter, d (mm) | Diameter, d (mm) | |||
| Length 1, L1 (mm) | Length 1, L1 (mm) | |||
| Length 2, L2 (mm) | Length 2, L2 (mm) |
3) For each column, determine the following geometric quantities.
Aluminum Steel
Moment of Inertia: 
________
Moment of Inertia: 
________
Cross sectional area: 
________
Cross sectional area: 
________
Radius of gyration squared: 
________
Radius of gyration squared: 
________
4) Buckling of columns with pinned ends is often called
the fundamental case of buckling. However, many other conditions
such as fixed ends, elastic supports, and free ends are encountered
in practice. The critical loads for buckling for each of these
end conditions can be determined by applying the appropriate boundary
conditions and solving the differential equations. These solutions
lead to the concept of an "effective length," 
,
appropriate for each end condition which is a multiple of the
actual length, L, of the column as shown in Table 3 and Figure
1.
 |
 |  |  |
5) In general, axially-loaded compression members may fail
by one of three modes: crushing; a combination of crushing or
buckling; or buckling alone. Columns can be placed into three
groups:
1) Short columns - the mode of failure is by crushing (simple compressive failure)
2) Intermediate columns - the mode of failure depends on simple compressive and/or bending
stress
3) Long columns - the mode of failure is primarily a function
of the bending stress (buckling).
A paramter which is employed to group these columns is the slenderness
ratio, 
. The minimum slenderness ratio
marks the nominal transition from crushing
to buckling. If the axial stress, 
, is
plotted as a function of slenderness ratio, then the minimum slenderness
ratio is the nominal transition from the constant stress for crushing,
, to the stress as function of 
for buckling, 
.
Aluminum Steel
Elastic modulus: 
=________
Elastic modulus: 
=________
Proportional limit stress: 
=________
Proportional limit stress: 
=________
Minimum slenderness ratio:
=______
Minimum slenderness ratio: 
=_______
On the following graphs, plot 
for 
and 
for 
.
6) Determine the following critical compressive loads for
the experimental columns
Aluminum Steel
For column length L1, the unsupported For column length L1, the unsupported
length if each grip end is 50 mm long length if each grip end is 50 mm long
such that L=L1-(2*50) = ________mm such that L=L1-(2*50)
= ________mm
Effective length, 
using Table 2 Effective length, 
using Table 2
for the Fixed/Fixed end condition_______mm for the
Fixed/Fixed end condition_______mm
For L1, slenderness ratio, 
=________
For L1, slenderness ratio, 
=________
Minimum slenderness ratio: 
=______Minimum
slenderness ratio: 
=_______
if 
. _________MPa 
if 
_________MPa
OR OR
if 
._________MPa
if 
._________MPa
Cross sectional area, A=___________
Cross sectional area, A=___________
Use the smaller of the stresses calculated above. Use the smaller of the stresses calculated above.
For L1, critical load, 
=
=___________N For L1, critical load, 
= 
=___________N
For column length L2, the unsupported For column length L2, the unsupported
length if each grip end is 50 mm long length if each grip end is 50 mm long
such that L=L2-(2*50) = ________mm such that L=L2-(2*50)
= ________mm
Effective length, 
using Table
2 Effective length, 
using Table 2
for the Fixed/Fixed end condition_______mm for the
Fixed/Fixed end condition_______mm
For L2 slenderness ratio, 
=________
For L2, slenderness ratio, 
=________
Minimum slenderness ratio: 
=______
Minimum slenderness ratio: 
=_______
if 
. _________MPa 
if 
_________MPa
OR OR
if 
._________MPa
if 
._________MPa
Cross sectional area, A_=__________
Cross sectional area, A=___________
Use the smaller of the stresses calculated above. Use the smaller of the stresses calculated above.
For L2, critical load, 
=
=___________N For L2, critical load, 
= 
=___________N
7) Measure the actual critical compressive loads for the
experimental columns.
For L1, Aluminum For L1, Steel
Measured critical compressive load,
=________N
Measured critical compressive load,
=________N
For L1, critical load, 
=
=___________N For L1, critical load, 
= 
=___________N
% diff __________ % diff __________
For L2, Aluminum For L2, Steel
Measured critical compressive load,
=________N
Measured critical compressive load,
=________N
For L2, critical load, 
=
=___________N For L2, critical load, 
= 
_=__________N
% diff __________ % diff __________
8) Comment on how well the equations predicted the actual
critical compression load? Were discrepancies reasonable? If
not, what could possible sources of error be attributed to? (Recall
that the assumptions for the buckling loads assume no initial
eccentricity and no off-axis loading).
9) As a designer, what steps can be taken to reduce the tendency to buckle, geometrically? material-wise?
