The determination of critical buckling loads (Pcri) in supporting struts is crucial in statics calculations in ensuring the stability of a larger structure. The orientation and build-up of struts similar to those tested in this experiment is determined by the beam’s physical properties, crucially, the maximum load that can be applied to the beam before it begins to buckle and fail. Though this practical deals with struts on a relatively smaller scale, compared to larger structural struts that are used in buildings for example, the principle of determining this Pcri value is in essence the same (for more description of the methodology see section 3).
The values of this critical load can be calculated theoretically also; using a Euler formula (see section 2), as with all theoretical calculations, there is almost always a degree in error when the calculation is made without experimental values to compare and compliment said theoretical values. Therefore, this report will not only discuss the method and theory used to provide these experimental values and provide the theoretical values, but this report will also discuss comparisons between theoretically calculated values of Pcri, experimental values of Pcri, and the difference in error when comparing experimental data to theoretical calculation within all the different struts tested in this practical.
Therefore, demonstrating the differences in experimental values and theoretically calculated values, and furthermore, showing exactly where within the experimental data there is the greater amount of error in comparison to the raw calculated values of Pcri.
The primary aims of this practical are to test the understanding of static buckling and apply this knowledge into a practical setting, to compare the theoretical values of critical buckling load with the produced experimental values, to see where potential errors may occur within the data, why they occur and what can be done to make the data more reliable. Within the grander scheme of the project, a primary objective is to increase the knowledge of Southwell plots and how they are a very useful tool in dealing with static problems involving buckling.
In carrying out the theoretical analysis of the component some assumptions need to be made, these assumptions are that the strut is perfectly straight and perfectly homogeneous, even when the compressive load is zero. The strut is very unlikely to be perfectly straight and homogeneous even when initially being set up within the strut buckling rig (see section 3) where some buckling will still occur. However, the theoretical equation and principle proposes that the beam goes perfectly straight into the device and then immediately buckles, instead of the gradual buckling that actually occurs. Therefore, the theoretical calculations will need to be compared to some experimental data to corroborate the initial calculations done on an assumed to be perfect strut. The strut is faced with two equal compressive forces (P) measured in Newtons at both ends of the strut. This force acts along the x axis of the strut straight through its length. The displacement (δ) measured in mm, is the lateral displacement, perpendicular to the direction of force applied, that occurs in the centre of the strut relative to the struts own end points.
With the assumed perfect beam, a Euler equation for critical loading (equation 1) is used to give the theoretical perfect beams value for critical buckling load (Pcri). This theory and equation was adapted and applied into a series of integration techniques by Richard Southwell to account for the actual imperfections of the beam [01], a theory which was disputed but more conclusively backed up by P. Mandal and C. Calladine [02] with resolution of a suggested “paradox” that the Southwell plot could provide. In summary, the manipulation of the original Euler equation by Southwell suggests that the buckling of an ideal strut would take the form of a series of sine curves. Applying differential calculus and substituting the original Euler formula back into the new expression, then subsequently making the displacement δ the subject of the equations renders an expression comparable with that of a straight line (equation 2), with δ (in mm) being equivalent to the y axis and δ/P (ratio) being equivalent to the x axis. This expression shows that the gradient of this expression is equal to the critical buckling load Pcri, therefore plots were made of displacement against the ratio of displacement to force and the gradient of the line of best fit would be taken from this graph to provide an experimental value of Pcri suitable to compare to the theoretical values.
Fig 3. Close up of strut buckling rig taken from [03]
1. Measure the breadth, ‘b’, and thickness, ‘t’, of the beam with a caliper gauge and micrometer, respectively.
2. Calculate the theoretical buckling load of the strut being tested, with simply supported end conditions.
3. Calculate 80% of the critical load and decide upon a maximum load value in between that allows a sensible progression of load with at least 11 steps. Tabulate the load levels so calculated.
4. Make sure that the V groove blocks (one shown as Item A of Fig. 3) of the end fittings of the test machine (Items 1 in Fig. 1) are clamped tight at the end of their travel.
5. Adjust the cross beam of the moveable end fitting of the test machine (Item 2 of Fig. 1, at the opposite end to the force gauge, Item 3) so that the test strut can sit with its ends in the grooves of the two end fittings. Take care not to damage the deflection dial gauge probe (Item 4) when doing this.
6. Replace the pins that provide support to the movable end fitting beam (Items 5 of Fig. 1). Then turn the knob at the force gauge end of the test machine (Item 6) clockwise until the force gauge registers a small load. Adjust the knob so that the load gauge registers zero.
7. Adjust the position of the deflection dial gauge (Item 4 of Fig. 1, also Item C of Fig. 3) so that it is measuring the deflection of the mid-point of the strut.
8. Take the zero reading of the deflection dial gauge.
9. Turn the loading knob clockwise to apply the first load increment, as calculated in Step 3, and record the deflection dial gauge reading when the load has been reached.
10. Repeat Step 9 for all of the load steps tabulated in Step 3.
It should be noted that in the data provided from the module leader, the tables produced for each beams showed different values and steps. This is due to the nature of step three, as the critical load increases there will be a greater number of sensible steps taken to get from P=0 to P=max. for example, as shown in appendix A section 7, there are less data points in table 2 when compared to table 2. This is because the overall maximum value for P is greater for the 550 mm strut compared to the 600 mm strut. The overall maximum theoretical value for P does decrease with the increasing length of the strut, however, to effectively display the data in tables 3 and 4, there are more steps taken from P=0 to P=max. This is done within the results provided to strike a balance with the instructions of step 3, and the effectiveness of display the data in correlation with the overall decreasing size of P as the struts increase in length, thus explaining the different sizes of data tables in appendix A.
In conclusion it can be said that as the load applied to the beam increases the perpendicular displacement will also increase. It can also be simply stated that as the length of each strut increases, the critical buckling load will decrease by a factor of 4 due to proportionality being inverse to the square when comparing Pcri and length of strut.
Within the Southwell plots carried out in this practical it was more effective and accurate to remove the initial 30% of the data (30% and less of the critical buckling load) and focus on the more consistently linear sections of the graph. This then provided overall comparatively lower percentage errors in the value of Pcri when compared to the theoretically calculated value using the Euler equation based around the perfect strut assumptions, mentioned in section 2. Meaning that overall, the values for Pcri values produced by the graphs marked .03 are more accurate than the values of the same quantity calculated from the graphs marked .02.
Furthermore, it has been shown that fixing the beam will make a substantial difference to the amount of force needed to buckle the strut due to the additional force acting in the opposite direction to the applied load P with the critical buckling load being approximately 4 times as large as the average load required to buckle the different simply supported struts.
[01] Southwell Richard Vynne 1932On the analysis of experimental observations in problems of elastic stability Proc. R. Soc. Lond. A135601– 616
Link: On the analysis of experimental observations in problems of elastic stability | Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (royalsocietypublishing)
[02] P. Mandal and C. Calladine extract from the international journal of Mechanical sciences pages 2555-2571. Entitled “Lateral-torsional buckling of beams and the Southwell plot”. Taken from the science direct website.
Link: Lateral-torsional buckling of beams and the Southwell plot - ScienceDirect
[03] Source provided by the University of Salford’s school of Computing Science, Engineering and Environment. Entitled “FE & Aircraft Structures/FE & Structural Mechanics, L3, Laboratory Notes”. Written by Dr Walid Jouri.
Link: 10521362 (blackboardcdn)
Table 2 Table 3
