Mechanical and Aerospace Engineering Department

Polytechnic Institute of New York University

ME6213

Introduction to Solid Mechanics

1.Buckling of Columns

2.Deflection of Curved Beams

Date of Experiment:_______

Date of Lab Report Submission: _______

This lab report submission is approved by:

Amith Deshmukh| Signature:_________|

Bhavesh Joshi| Signature:_________|

Anoop Kumar| Signature:_________|

Sriniket Srinivas Achar| Signature:_________|

Experiment 1 – Buckling of Columns

Introduction

When a small compressive load is applied on an ideal column, it deflects laterally. And, when this load is removed, the column returns to its original position due to the elastic restoring forces. As the load is increased to a certain a value Pcr and then removed, the restoring forces are incapable to return the column to its original position but maintain equilibrium about the displaced lateral position. At this position there is displacement without the increase in the applied load. This critical load is termed as Buckling Load and this phenomenon is called as Buckling. The formula to calculate Buckling load is given by,

Where,

E – Young’s Modulus,

I – Area Moment of Inertia,

Le – Effective length of the column, which depends on the boundary conditions. For Pinned-Pinned condition, Le = L

For Fixed- Pinned condition, Le = 0.7L

For Fixed-Fixed condition, Le = 0.5L

Where, L – Length of the column

In this experiment an attempt is made to calculate Critical Buckling Load of a column experimentally and theoretically with different boundary conditions. These values will then be compared. Equipment and Procedure

Equipment

1. Column Buckling Machine

2. Test Specimen: Three Metal Beams. In this experiment, steel beams of known length were used. The modulus of elasticity for the material tested was predefined. The thickness and width of the beams were found to be 2mm and 20mm respectively 3. Calipers and Load Cells: Calipers are used to measure the width and thickness of the specimen. And Load cells measures the force.

Experiment Setup

The specimen should be secured on the column buckling machine with each end of the specimen being supported per case requirements.

Procedure

Load is gradually applied by twisting the knob present in the machine, till the column starts to buckle. The load cells display the load which is noted down when buckling starts. This procedure is repeated for different lengths. Then the boundary conditions are changed and the experiment is set up as required and this procedure is repeated. Effective lengths are calculated and from that the theoretical values are derived. The error is calculated and tabulated. Calculations

The experimental values were recorded in the Tabular Column. The theoretical values are calculated using (1). Specimen Calculation:

Condition: Pin-Pin

Length: 370mm

In (1),

PCR = (ԉ2 * 70e9 * 1.33e-11) / 0.372

PCR = 67.28 N

Condition| LengthL (mm)| Experimental LoadP (N)| Theoretical LoadPCR (N)| % Error| Pinned-Pinned(Le = L)| 370| 58| 67.28| 13.79|

| 320| 75| 89.9551| 16.62|

Fixed-Pinned(Le = 0.7L)| 350| 120| 153.4594| 21.8|

| 300| 198| 208.8753| 5.2|

Fixed-Fixed(Le = 0.5L)| 330| 303| 338.3434| 10.44|

| 280| 365| 469.9694| 22.33|

Discussion

The tabular column shows that the error ranges from 5% to 23%, with the error increasing as the length is decreased. This is because critical buckling load is inversely proportional to the square of the length of the column. Conclusion

Although there were errors in the outcome of the experiment, the values were fairly acceptable. The accuracy depends on the clamping of the machine and the level of calibration of the machine. The values also depend on condition of the test samples. The test samples were used a lot of times and were slightly deformed even before the conduction of the experiment. References

Advanced Mechanics of...

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