47 



beam symmetrical in form with material uniform throughout) 

 is restrained, the ends have no slope when the beam carries a load. 

 The case is shown in Fig. 41, where the ends are extended to 

 some distance, b, where a 

 load is placed which has suffi- 

 cient weight to hold the ends 

 in the original positions. 

 Tension under such condi- L .. 

 tions exists at A and C with 

 compression at B and D. At 

 the point where the beam 



Fig. 41 



Fig. 42 



ceases to be horizontal and bends down there is neither tension 

 nor compression, the only existing force being shear. This point 

 is termed the " point of contraflexure," the " point of reverse 

 moment," or the " point of inflection." 



In Fig. 42 the shaded tri- 

 angles represent the mo- 

 ments of the actual center 

 load and the two assumed 

 end loads. These loads, as 

 well as the length 6, may 

 actually exist, but the same 

 effect will be obtained by riveting or otherwise fastening the ends 

 of the beams to, or in, solid supports; therefore the loads and the 

 moment areas beyond the point of support are said to be imagi- 

 nary or assumed. The condition created is that of a simply sup- 

 ported beam, having a length measured between the points of 

 contraflexure, carried on the ends of 

 two short cantilevers. An expression 

 must be found for the force creating 

 such a condition and also for the 

 lengths of the cantilevers, that is, the 

 distance from the point of support to 

 the point of contraflexure. 



In Fig. 43 let the triangle ABC re- 

 present the moment area due to a 

 concentrated load at midspan of a 

 freely supported beam, AC. The two end triangles AGF and CDE 

 are the moment areas of the loads causing the restraint. An in- 

 spection will show that the combined areas of the two end triangles 



