The plane curve's curvature at a point on the curve can be expressed using an expression involving the curve's first and second derivatives.

The slope is insignificant for the beam's elastic curve, so the governing equation for the elastic curve is expressed as a second-order linear differential equation, considering flexural rigidity.

If flexural rigidity varies along the beam, it is expressed as a function of x. However, for a prismatic beam, flexural rigidity remains constant.

Integration of this equation provides the angle formed by the tangent to the elastic curve at a point with the horizontal.

This small angle, when integrated, gives us the deflection of the beam at any point. The beam supports' boundary conditions determine the constants in the equations.

Supported, overhanging, and cantilever are the three types of beams that are primarily considered. For supported and overhanging beams, the deflection at support points is zero.

Both deflection and slope at the support point are zero for cantilever beams. These conditions enable the calculation of the constants.