The redistribution of internal forces and deflections in a uniformly loaded propped cantilever and a fixed-end beam caused by the insertion of a frictionless hinge is evaluated for an arbitrary position of the hinge. This is accomplished by an extended use of the method of discontinuity functions to incorporate the slope discontinuity at the hinge, without the separation of structures into their constituting parts, as commonly done in other methods of analysis. It is shown that the insertion of a hinge in the middle of a propped cantilever increases the reactive moment at the fixed end two times. A hinge in the middle of a fixed-end beam increases its reactive moments by 50%, while the maximum deflection increases three times. The maximum allowable load is determined for all considered structures by using the classical and the limit design criteria. If a hinge is placed in a propped cantilever at the distance from its fixed end smaller than one-fourth of its span, the classical design criterion predicts that a hinged propped cantilever can transmit a greater distributed load than a propped cantilever without a hinge. However, according to the limit design criterion, the insertion of a hinge in a propped cantilever decreases the ultimate load for any location of a hinge. The insertion of a hinge in a fixed-end beam decreases the maximum load according to both, the classical and the limit design criteria. For the rectangular cross section, the ratio of the maximum loads according to the limit and the classical design criterion is constant and equal to 3/2 in the case of a hinge-relaxed propped cantilever, while it varies with the position of the hinge in the case of a hinge-relaxed fixed-end beam. The presented analysis and the obtained results are of interest for undergraduate engineering education in the courses of mechanics of materials and structural design.
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