☻ ☻ ☻ ☻ 2.0 Bending of Beams sx 2.1 Revision – Bending Moments 2.2 Stresses in Beams x sx Mxz P ☻ 2.3 Combined Bending and Axial Loading P1 P2 ☻ 2.4 Deflections in Beams 2.5 Buckling (Refer: B,C & A –Sec’s 10.1, 10.2)

2.5 Bars Under Axial Compression - Buckling (Refer: B, C & A–Sec 10.1, 10.2) Euler, Leonard (1741) For bars or columns under compressive loading, we may witness a sudden failure, known as buckling. This is a form of instability. P v Consider load, P, plotted against horizontal displacement, v. P P Point of Bifurcation Short, Thick Long, Slender

P P P Mxz=-P.v Consider a long slender bar under axial compression: y From the Engineering Beam Theory: Let Euler Equation

If A=0, v=0 for all values of x General Solution: (i) @ x=0 Boundary Conditions: (ii) @ x=L (Also, (iii) @x=L/2, ) From (i): From (ii): If A=0, v=0 for all values of x (i.e. The bar remains straight – NO BUCKLING) If Sin aL=0, then (possible buckling solutions)

P v The Lowest Real Value: Since Where PC is the Critical or Euler Buckling Load. For buckling (i.e. at point of bifurcation): P (Iz is the smallest 2nd Moment of Area) P v Points of Bifurcation v

sC Unstable sYield Stable S Let Let Kz2 (Radius of Gyration) and S (Slenderness Ratio) Short, Thick Long, Slender sC S Euler Stress Stable Unstable sYield Empirical Material Yielded

P L Free or Pinned Ends End Constraints: P L Fixed Ends L/2 P L One End Free & One End Fixed 2L P L One End Pinned & One End Fixed

Example: Find the shortest length L of a Pin-Ended strut, having a cross-section of 60mm x 100mm for which the Euler Equation applies. Assume E=200 Gpa and sYield=250 MPa Euler Eqn., i.e. Iz Min. value, sC S Euler Theory sYd =250 MPa Note: 90