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Buckling of Columns ME 272 Experiment #6

Objective:

The objective of this experiment is to develop an experimental method utilizing the buckling of columns apparatus that will determine the load-deflection function and critical loads of a sample column. The developed method is then used to determine the load-deflection function and critical loads for a column with various end conditions (pinned-pinned, clamped-pinned, and clamped-clamped).

Reference: Sections 10.2 to 10.5 in Mechanics of Materials

; Beer, Johnston, and DeWolf

Apparatus:

You will be using the strut machine for this experiment. Once the test sample is installed into the machine, an axial load is then applied to the column through the hydraulic cylinder. This load is applied by turning the hand crank clockwise. Although the load is being applied axially, the dial indicator produces a slight side loading that causes deflection to occur almost immediately. As the column is loaded, the dial indicator will measure the deflection in the beam. The cross member that houses the indicator can be moved laterally to place the indicator at the location on the column with maximum deflection. This machine can simulate both pinned and clamped end conditions. For a clamped end condition, the end of the column is

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simply clamped into the fixture. For the pinned end condition, the clamp is tightened without the column in it. The column is then slid into the V-shaped groove. Although there will be some side loading from the V-shaped groove, it is negligible, allowing the assumption of a pinned end.

Theory:

Under compression above a critical load (Pcr), a column may experience a sudden change in its configuration. This phenomenon is called column buckling. A. Euler formula In one way, Euler’s formula (Eq. 6-1) gives the expression to determine this critical load.

2

2

ecr L

EIP π= (6-1)

Here, the value of effective length Le depends on the end conditions of columns.

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B. Secant Formula In another way, column buckling can be viewed as a result of eccentric compression load, which causes bending of the column.

As shown in the above figure, the maximum deflection of the column can expressed using Secant formula (Eq. 6-2).

−

= 1

2secmax

crPPey π (6-2)

Deliverables:

1. Document this experiment with a formal lab report following the report guidelines set forth in the ME 272 lab manual.

2. For theory section, present and discuss the equations and the derivation of the equations that were used to determine the calculated value of Pcr

3. For each end condition, create plots showing the load-deflection relationship using experimental data as well as the determined load-deflection function.

and the experimentally obtained value.

4. Be sure to compare the experimentally determine value for Pcr

with the value obtained using Euler’s formula.

Hints: The secant formula will still have 2 unknown values, e and Pcr, after you insert the experimentally obtained values. To solve this, you must use 2 data points to give you 2 equations. Once you have two equations, divide one of them by the other to eliminate e. Because the other unknown is within a secant function, it is easiest to set up the function in a spreadsheet using a range of test numbers to find the value of Pcr that satisfies the equation. Use several combinations of points to evaluate the function, returning several values of Pcr that can be averaged. To choose your various combinations of points, plot your experimental data and choose pairs of points that lie in the same linear portion of the curve.

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Buckling of Columns ME 272 Experiment #6 Pre-Lab Assignment

Find the critical load (in Newtons) using the Euler formula for each of three end conditions (pinned / pinned, pinned / clamped, clamped / clamped) for a rectangular aluminum (E = 610.5 10× psi) specimen of the following dimensions: L = 1.15 m w = 1.0 cm d = 1.5 mm