Solid Mechanics

02TTB204 - Mechanics of Solids

Part B Lab - Buckling of Struts

1. Introduction

The task was given to obtain the buckling stresses for pin-ended steel struts of various slenderness ratios and compare with theoretical predictions obtained using the Euler and Rankine-Gordon equations.

2. Theory

The method of obtaining the buckling stresses followed was to use data show in Appendix A. From the record of applied load, P, against deflection, δ, a Southwell plot of δ against δ/P can be drawn. The gradient of the Southwell plot yields the buckling load of the particular strut. The dimensions of each strut are given and therefore the experimental critical stresses can be obtained by division of the buckling load by the cross-sectional area. When taking data from the plot in Appendix A, it is only necessary to take values around the region where buckling occurs (These are highlighted red on the plot).

Ideally the experimental buckling stresses obtained should be closely linked to two theoretical methods for obtaining the same stresses. Eq.2.1 and Eq.2.2 both give methods of calculating the buckling stress in a strut.

Euler equation: - Eq.2.1

Rankine - Gordon:- Eq.2.2

Fig.2.1 shows a perfect pin-ended strut. This strut is assumed to undergo only axial loading and will remain in its elastic range prior to the critical load being reached. Deflection will only occur when the critical load is reached. Any deflections prior to this load are maintained inside the structure.

Fig.2.1 - a) Perfect pin-ended strut b) Deflection due to buckling

3. Discussion

3.1 Differences between two theories

Euler theory states that provided the critical buckling load is not exceeded a strut will not undergo any excessive deflection and will remain in equilibrium, i.e. the strut will only buckle if the applied load exceeds the critical load, it will not yield or fail in any other way before this point.

Experimentally real struts do not behave in this way, as immediately upon the addition of load, deflection will occur. At high deflections the Euler theory predictions and experimental results become very similar. Fig.3.1.1b shows how Euler and experimental results converge together as applied load and deflection increase.

Fig.3.1.1 - Load -deflection behaviour of ideal Euler and actual struts

At smaller slenderness ratios the Euler theory is not accurate to experimental results. This is because low slenderness ratio struts have high buckling stresses. In cases where the buckling stress is higher than the yield strength, the material of the strut will fail before the structure itself buckles. Therefore if the critical buckling stress calculated from the Euler equation is higher than that of the yield strength of the material, then buckling is not a limiting factor. Therefore it can be said that the Euler theory will not hold as a good approximation for experimental results in 'short, fat' struts, i.e. low slenderness ratios.

Fig.3.3.2 - Behaviour of ideal Euler and actual struts

Fig. 3.3.2 shows that for low slenderness ratios, between A and B, the Euler theory does not match experimental results. In practice many struts do not fall into the range of slenderness ratios relevant to the Euler theory, this requires further analysis and the use of other formulae. The Rankine - Gordon Theory covers the range A-B, where buckling is not a factor thus providing a closer approximation to buckling stresses which should occur. The R-G theory not only holds for range A-B but also for at higher slenderness ratios the theory is still valid. Therefore both theories should give accurate, close-together values at high slenderness ratios as they both are valid over this range.

Both theories take into account the dimensions of the struts but the R-G equation can be used for various materials and end conditions. Variable 'a', in Eq.2.2, changes depending on material of whether the strut is pinned or fixed. This would explain some difference between the two theories as the Euler Theory makes no consideration of such factors as end conditions. It only considers material selection. At low slenderness ratios the struts are short, therefore consideration of end conditions is important and hence the difference between the two. As length increases, the effect of end conditions upon buckling becomes less and less. Therefore as the effect of including end conditions in the analysis decreases the two theories will tend towards the same values. This is seen to happen in Appendix B.

3.1 Differences between experimental and theories themselves

From what has been discussed above, in 3.1, it would be anticipated that the two theoretical solutions would be very close together in accuracy at high slenderness ratios and would both be good approximations of the buckling stress in actual struts. From observation of Appendix B it is clear that neither theoretical solution are close to the actual results in all ranges of slenderness ratio, with the Euler theory producing the closest values. The credibility of the equations need not be tested so obviously there are sources of error in how the actual buckling stresses obtained and therefore these need to be evaluated as to why there is such a difference between theoretical and experimental values.

First and foremost there are reasons for differences between theoretical and experimental results that would exist no matter how small the difference actually was.

In arriving at the theoretical values it is assumed that the struts are perfect. In reality the struts are not perfect and have small but significant imperfections in their geometry or in the way they are loaded. For example the dimensions given on the lab sheet are probably not the exact dimensions of the actual struts used in experiment. Also the Young's modulus, E, of the actual specimens may not be equal to that used in the Euler equation.

As discussed in section 2, the theories expect axial loading. In practice this is not possible as the loading is not likely to be occurring directly on the strut's horizontal centroidal axis. Also the theories expect simple (infinitely small)