rephrasing

Abstract

The objectives of this experiment are to study the crushing of short structural members, and buckling behavior of long columns under axial compressive loading, and to determine the critical loads of the columns. We used aluminum alloy tubes on the MTS machine and we put them on fixed-fixed, pinned-pinned, pinned-fixed and fixed-pinned installations. Then we figured out the critical loads for the different length columns and the condition used. The theoretical and the experimental values percent error range from 5% to 30%.

Table of Contents

Abstract……………………………………………………………….…..2

Table of contents………………………………………………………….3

Introduction and theory……………………………………………….….4-7

Procedures…………………………………………………………..…….8-9

Summary of important results………………………………..…………..10-11

Sample calculations and error analysis…………………………..……....12

Discussion and conclusion……………………………………………….13

References………………………………………………………………..14

Appendix………………………………………………………...………15-16

Introduction and Theory

Columns are structural members loaded in compression. In the axial tension and beam tests, the load carrying ability of the structural members was directly related to the yield or ultimate strength of the material. This is also the case for a short column where the crushing strength of the column depends on the compressive yield and ultimate strength for the material. However, for long columns, another effect must be considered: that of static stability.

A tension member of any length is statically stable because the tensile forces tend to align the member along the axis of loading, and thus reduce any misalignment. A compression member, however, may be statically unstable because any misalignment of the compressive forces, or lack of straightness or uniformity of the member, tends to increase with increasing loads. Convince yourself of this effect by holding the two ends of a pencil eraser, and apply a compressive load at the two end points.

Consider a slender rod, free to rotate at the ends, and loaded axially with load P. If a lateral load Q is applied, the elastic rod will deflect by -y, and if P is not too large, it will return to its original shape when Q is removed. In this condition, the rod is stable with respect to P. If P is gradually increased, and Q is successively applied and released, it is expected that some value of P will be reached where the elastic forces in the rod are no longer able to bring the rod back to its original position when Q is removed; instead, it remains in its deflected position under the action of P. In this condition, the rod is neutrally stable, and the value of P is called the critical load, Pcr. Any slight further increase in P would cause the rod to become unstable, and to collapse.

To derive the equations needed for analyzing the above phenomenon, consider a rod in neutral

stability. From Fig. VI-1, it is seen that the moment at any section is M = −Pcry (using the same

sign convention as for the beam experiment). But, from beam theory, M = EId²y/dx²; thus, the

equation describing the deflected shape of the neutrally stable rod is

(V-1)

with the boundary conditions that y = 0 at x = 0 and x = L. The general solution of Eq. (V-1) is

(V-2)

where = EI/Pcr. The boundary condition at x = 0 can be met by taking B = 0, but the one at x = L can be met only for characteristic values of λ for which sin(L/λ) = 0; or for L/λ = nπ, n = 1, 2,3.... Thus, solving this equation, the smallest value of the critical load is obtained for n = 1:

(V-3)

Notice that this value depends only on the stiffness property, E, of the material, and not on the strength of the material. This result is known as Euler’s equation, and it is applicable to long slender elastic columns. In Eq. (V-3), I is the minimum moment of inertia of the cross-section of the rod, which may be replaced by Ak², where k is the corresponding “radius of gyration.” With this substitution, Euler’s equation takes the form

(V-4)

Where /A is the critical stress. The plot for Euler’s equation is shown by the solid curve

on the right side of the graph in Figure VI-2.

The ratio L/k is called the “slenderness ratio”. It provides the measure by which a compressive member is judged long or short. Euler’s equation is based on the assumption that the material of the column remains in the elastic range until buckling occurs. For this reason, the plot of Euler’s equation starts at a value of L/k , identified as (L/k)max, which corresponds to a critical stress

equal to the proportional limit of the material, σpl (see Fig. V-2). (L/k)max is calculated by using

Equation VI-4 and setting σcr = σpl:

The solid horizontal line on the left side of the graph in Figure V-2 belongs to short

compression members that fail by crushing with a critical stress equal to the yield point, σy.

Intermediate members that are neither “short” nor “long” may fail by a combination of crushing

and buckling. Their behavior is represented by the dashed curve in Figure V-2. Empirical equations are used to predict the critical stress of an intermediate column.

The length, L, in Euler’s equation is not necessarily the geometric length of the column. L is a function of the end conditions imposed on the column. It is the “effective length” of an equivalent column having zero moments at its ends.

In practice, most columns are of intermediate length, and various empirical formulas have been developed. For example, for structural steel, the American Institute of Steel Construction recommends the following parabolic type formula for the working stress in “PSI”.

For aluminum alloys, the Aluminum Company of America (Alcoa) recommends the following linear type formula (not including the safety factor) in “PSI”.

Procedure

In this experiment, the critical load will be determined for twelve various lengths of 6061-T6 aluminum alloy tubes (from 1in to 12in). First, we carefully measure and record the length and the dimensions of the tubes using a digital caliper and then record the measurement. After that we ensure that the emergency stop is released. After that we turned on the PC and the MTS then we double click on testworks 4 on the PC then open method drop down and highlight :Exp-3 4 point flex Mod 9-18A” the click OK. Then, we clicked on “Calibrate Device” icon and highlight the load cell serial number associated with the load cell that is installed on you MTS machine and clicked calibrate. We waited until the process in completed the clicked finish. After that we clicked ok on the calibrate box. Then we used the handset control to raise the cross head, making certain that the top fixture has enough clearance to fit the aluminum alloy tube column between free inserts of MTS fixture. We placed the aluminum tube column with fixed-fixed fittings in the testing machine as shown in the experiment manual. Then we used the cross head to hold the aluminum tube in place without causing any load on the MTS load cells. After that we zeroed the cross head indicator on the MTS display. After installing the column we identify a midpoint in the column and measure the distance between the reference bar and the aluminum tube column. We used a dial indicator to measure the deflection. The subsequent deflection is measured with respect to the undeflected position pf the specimen. While the dial indicator showing this undeflected position. We press the zero “button” on the calipers to initialize it. After that we increased the load gradually and slightly prod the column. When some deflection accrued, we rotated the column so that the dial indicator records the readings. We increased the load 10 lbs for the long columns, 50 lbs for the medium columns, and 75 lbs for the short columns until the specimen fails. After that we shared our group data with the other groups in our lab. We were unable to work on the pinned-fixed data because we did not receive it from the group that did the experiment.

Summary of Important Results

Sample Calculations and Error Analysis

%Error

Discussion and Conclusion

After finishing the experiment and calculating the data and graph them, we concluded that the difference between the theoretical values and the experimental values range between 5% to 30%. We also learned that as the length of the material increases the critical load decreases. Also, as the critical stress decreases the slenderness ratio increases. However, each condition (fixed-fixed, fixed-pinned, and pinned-pinned) has its own values of critical load and critical stress.

Errors in any experiment are unavoidable. One can only try to reduce them as much as possible. Therefore, the errors in this experiment could be due to wrong calculations, mistake in taking measurements, or a wrong setup for the sample in the machine. These are the errors that could happen by the one doing the experiment. However, there are errors that not related to the one doing the experiment. The machine itself could be old or hasn’t had the proper maintenance and that could cause errors. Not only the machine could cause errors, but also the samples could be not manufactured well.

REFERENCES

[1] CSUF EGME 306A Lab Manual

APPENDIX

Fixed-Fixed

Fixed-Pinned

Pinned-Pinned

Pinned-Pinned

Critical Stress vs Selenderness Ratio

Slenderness Ratio

Critical Stress

Critical Load vs Deflection for all Condition for each length

Deflection

Critical load

Fixed - Fixed

Critical Stress vs Slenderness Ratio

Slenderness ratio

Critical Stress (psi)

Fixed-Pinned

Critical Stress vs Slenderness Ratio

Slenderness Ratio

Critical Stress

2