Consider a train moving on a bridge. Here, an unsymmetrical load is applied to a supported beam where the maximum deflection doesn't usually occur in the middle.
The maximum deflection of the beam is calculated by identifying the point O on the curve, where the beam's tangent is horizontal.
The slope of the tangent at point X is determined by calculating the tangential deviation between the supports and dividing it by their distance.
Since the slope at point O is zero, the slope between O and X equals the negative slope at X.
The first moment-area theorem is used to locate point O by measuring an area under the M/EI diagram equal to the negative slope at support X.
The maximum deflection equals the tangential deviation of support X about point O. This value can be obtained by calculating the first moment relating to the vertical axis through X of the area between X and O.
25.9:
Maximum Deflection
When analyzing beams under unsymmetrical loads, such as a train moving on a bridge, it is crucial to accurately determine the points of maximum stress and deflection. The process involves identifying the maximum deflection of the beam, which may not always occur at its midpoint due to the uneven distribution of the load.
The maximum deflection occurs at a specific point, known as point O, where the tangent to the deflection curve is horizontal. To find point O, the slope of the tangent at any given point X along the beam is examined. The slope at point X can be calculated by considering the tangential deviation between the supports and dividing it by their distance. This slope is zero at point O, indicating the maximum deflection location.
The First Moment-Area Theorem plays a key role in locating point O. According to this theorem, the area under the bending moment diagram between any two points along the beam corresponds to the change in the slope between these points. Point O can be identified by calculating this area up to the negative slope at support X.
Once point O is determined, the maximum deflection is calculated by analyzing the tangential deviation of support X about point O. This approach provides a systematic method to evaluate the structural behavior of beams under unsymmetrical loading, ensuring the safety and stability of structures such as railway bridges.