Equation of the Elastic Curve

The concept of curvature in plane curves, crucial in structural engineering, defines how sharply a beam bends under load. This curvature is determined using the curve's first and second derivatives.

Consider a cantilever beam with a point load at its free end (for instance, a diving board). When analyzing beam deflection with small slopes, the shape of the beam's elastic curve becomes key. The governing equation for this analysis involves the bending moment and the beam's flexural rigidity, which is a product of the modulus of elasticity and the moment of inertia of the beam's cross-section.

For prismatic beams, where the cross-section remains constant, the analysis simplifies, making the flexural rigidity constant along the beam's length. Integrating the governing equation allows the calculation of the angle formed by the tangent to the curve at any point, which, upon further integration, yields the beam's deflection at that point.

Boundary conditions at the beam supports are vital for completing these calculations. Supported, overhanging, and cantilever are common types of beams, each with distinct boundary conditions. For example, the deflection and slope at a cantilever beam's support point are zero, which is essential for calculating the constants of the deflection equations.

Accurately predicting beam deflection is crucial for ensuring structural safety and functionality. Excessive deflection can cause structural failures or serviceability issues, underscoring the importance of understanding beam behavior under load.