25.7: Beams with Symmetric Loadings
The moment-area method is an analytical tool used in structural engineering to determine the slope and deflection of beams under various loads. Consider a cantilever with a concentrated load and moment at the free end. The first step is constructing a free-body diagram to calculate the reactions at the fixed end. Next, the bending moment diagram is plotted to visualize how the bending moment varies along the beam's length, focusing on points where the bending moment equals zero.
The M/EI diagram is then drawn, where M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia of the beam's cross-section. This diagram identifies the areas under the curve for segments between points where the bending moment is zero, assigning positive or negative signs to the areas based on their position relative to the x-axis.
The first moment-area theorem is applied to calculate the slope at any point on the beam by integrating the area under the M/EI diagram between two points. The second moment-area theorem is then used to find the beam's deflection, equating it to the first moment of the area under the M/EI diagram about a vertical axis passing through the endpoint.
Finally, the deflected shape of the beam is sketched, providing a visual representation of the analytical findings and illustrating the beam's bending under the imposed load conditions. This method offers precise insights into the structural performance of cantilever beams.
