An improvement on Magnel's diagram, and an application

In designing the cross-section of a prestressed concrete beam that must withstand, over its history of loading, a known range of bending moments, the engineer must satisfy four inequalities which represent the allowable stress limits (+/–) at the upper and lower surfaces of the beam, respectively. Magnel described a graphical plot of these inequalities, which provided a two-dimensional ‘permissible region’ of prestressing force and thrust eccentricity in the form of a quadrilateral. Here a different way of plotting the same four inequalities is demonstrated, so that the ‘permissible region’ is always in the simpler form of a parallelogram. This ‘new’ plot is easier to deal with than Magnel's, and its advantages are listed. Also, an example is given of the use of the new plot in designing the thrust profile for a three-span bridge beam—an example originally given by Burgoyne to illustrate a point made first by Low.