Beam Bending and Euler Buckling

In slender-beam Euler–Bernoulli theory, cross-sections remain plane and perpendicular to the deflected centreline. A fibre farther from the neutral axis changes length more strongly under the same curvature. Within linear elasticity, its longitudinal stress is therefore proportional to its distance from that axis. Summing the stress moments over the area introduces the second moment of area. This is a geometric quantity measured in metres to the fourth power, not the mass moment of inertia used in rotational dynamics.

The product of Young’s modulus and second moment of area is flexural rigidity. Increasing it reduces curvature for a prescribed bending moment. For a rectangular section, depth enters cubically, so adding material far from the neutral axis is especially effective. This motivates efficient beam shapes, although local plate buckling, shear deformation, connection details, and torsion can impose other constraints. The simple theory assumes small strains, a slender member, and sufficiently small slopes when curvature is replaced by the second derivative of transverse deflection.

Now consider a perfectly straight, uniform, pin-ended column loaded through its centroid. A small lateral deflection creates a bending moment from the compressive force. Bending resists that deflection, but the load does work as the ends approach. Linearizing equilibrium produces a boundary-value problem: a nonzero lateral shape exists only at particular compression loads. Pins require zero lateral displacement at both ends while allowing rotation. The lowest mode is one half sine wave over the column length.

The first Euler load follows from that lowest mode. Higher eigenmodes have larger ideal critical loads and do not set the first instability. Support conditions enter through an effective length: a cantilever has twice its actual length in this formula, while an ideal column fixed against rotation and translation at both ends has half. A support label is a model of actual restraint, not a decorative choice. Frame sway, connection flexibility, or partial bracing can change the effective length and must be analysed rather than guessed.

Buckling and yielding are different limiting mechanisms. Euler theory can predict instability while the average compressive stress is still below the yield stress, especially for slender columns. A stocky column may yield first, and imperfections give real columns lateral deflection before the ideal critical load is reached. Eccentric loading, residual stress, and nonlinear material response further alter the load–deflection relation. The Euler value is therefore a mechanistic benchmark, not by itself a safe allowable load or a complete post-buckling prediction.

Use the function lab to inspect a small-slope shape with constant second derivative: its displayed curve is y = ax². Predict how its curvature approximation changes when a doubles, then check how the tangent slope at a fixed position changes. Both track the doubled coefficient. This supports the link between deflection derivatives and bending, but the plotted parabola is not the pin-ended buckling eigenmode. The lab has no axial load, supports, cross-section, or instability calculation. Use the equations and stated support conditions to calculate the Euler load separately.