25.6: Moment-Area Theorems
The Moment-Area Theorem is crucial in structural engineering for analyzing beam bending, particularly in applications like building floor supports. This theorem utilizes the geometric properties of the elastic curve, which depicts how a beam deforms under load, to simplify the calculations of deflections and slopes.
The theorem is divided into two parts. The first part connects the angle between tangents at any two points on the beam's elastic curve to the area under a curve derived by plotting the quantity M/EI (where M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia) against the beam's deflection along its length. The area under this curve directly corresponds to the total rotation occurring between these two points.
The second part of the theorem addresses the tangential deviation—or the vertical displacement—between any two points resulting from the beam's bending. It states that this deviation is equivalent to the first moment of the area under the M/EI curve about a vertical axis through one of these points, providing a measure of the beam segment's displacement from its original position. These theorems efficiently determine the slope and deflection at various points along a beam, essential for ensuring structural safety and performance under load.
