Surface-Loading on the Flexure of Beams. 499 



problem of the equilibrium of an elastic solid, and investi- 

 gated the properties of what he termed " isostatic surfaces," 

 or surfaces where only normal " actions " are applied. 



In 1870 Saint- Venant * examined the differential equations 

 to which the subject of " isostatic surfaces " gave rise, and in 

 1872 Professor Boussinesqf gave a geometric method for 

 constructing isostatic lines passing through any given point. 

 This memoir was shortly followed by a second f, treating of 

 the integration of the equations involved. 



Rankine has examined the form of the curves of Principal 

 Stress, and given an expression from which the curves can be 

 drawn §. He neglects the surface- loading effect as "in most 

 cases practically of small intensity when compared with the 

 other elements of stress." On comparing his curves with 

 those in Plate II. it will be noticed how closely the curves of 

 tension agree, while the curves of compression are very 

 dissimilar. 



Sir George Airy has calculated and drawn the curves of 

 principal stress for several cases of flexure, including that of 

 a beam doubly supported and centrally loaded ||. He assumes 

 " that there is a neutral point in the centre of the depth ; that 

 on the upper side of this neutral point the forces are forces of 

 tension, and on the lower side are forces of compression, and 

 that these forces are proportional to the distances from the 

 neutral point; " but he says " These suppositions seem to imply 

 that the actual extensions or compressions correspond exactly 

 to the curvature of the edge of the lamina." The surface- 

 loading effect is not here taken into account ; and it would 

 have been interesting to compare the results as shown in 

 fig. 6, for a beam in which the span equals ten depths, with 

 the actual curve as found by experiment. This comparison, 

 however, would lead to erroneous conclusions, since it has 

 been shown TT that the results arrived at are not consistent 

 with the fundamental equations, and the form of the curves 

 can be accepted only as a very general approximation. 



Proposition VII. 



To determine the lines of Principal Stress in a glass beam 

 doubly supported and centrally loaded. 



Experiment 10. — A glass beam, 128 millim. x 19 millim. 



* Ibid. vol. lxx. t Ibid. vol. lxxiv. p. 242. % Ibid. vol. lxxiv. p. 318. 

 § ' Applied Mechanics,' §§ 310 and 311. 

 || Phil. Trans. 1863, part 1. 



•ff See criticism on Sir George Airy's solution in Ibbetson's ' Mathe- 

 matical Theory of Elasticity/ note on p. 358. 



2L2 



