What is the equation of curvature expressed in relation to the radius of curvature?

The equation of curvature, particularly in the context of structural analysis, relates specifically to the bending of beams under load. The correct answer expresses curvature in terms of the bending moment (M), the modulus of elasticity (E), and the moment of inertia (I) of the beam's cross-section.

Curvature (often denoted as 'κ') is defined as the change in angle per unit length of the beam. In elastic theory, this curvature can be expressed mathematically as:

[ \kappa = \frac{M}{E \cdot I} ]

This relationship is critical because it establishes how much the beam bends under a given moment and allows engineers to calculate deflections and ensure that structures can sustain the loads applied to them without failing.

The terms in this equation play an important role:

• The bending moment (M) indicates the internal resistance force due to external loads.
• The modulus of elasticity (E) represents the material's stiffness, which influences how much it will deform under stress.
• The moment of inertia (I) reflects the beam's geometry and helps define its rigidity.

Together, these elements help establish a clear relationship between applied loads and resultant curvature, which is essential for safe and effective structural design. This understanding