# Solid Mechanics

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...

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