482 Prof. C. A. Carus Wilson on the Influence of 



(2) The state of strain due to surface-loading only may be 

 found, with close approximation to truth, by resting the beam 

 on a flat plane instead of on two supports. 



(3) The strains due to bending only may be obtained from 

 the Bernoulli-Saint- Venant results ; viz. : — 



(a) The stretch for any cross section varies as the distance 

 from the neutral axis. 



(/3) The central axis is unstretched. 



(y) For the same point in different cross sections the 

 stretch varies as the bending-moment. 



Saint- Yenant has dealt with the shearing-strains at some 

 little distance from the load in the case of a beam doubly 

 supported and centrally loaded* ; and Professor Pearson has 

 shown f that, in the case of beams continuously loaded, the 

 results of the Bernoulli-Eulerian theory can only be con- 

 sidered as giving approximate formulae when the span of the 

 beam is not less than ten times its depth J. 



The mathematical determination of the state of strain pro- 

 duced by the loading of a beam as it rests on a flat plane is 

 one of considerable analytical difficulty. 



MM. Lame and Clapeyron have attempted the solution of 

 a more general problem in their " Memoire sur 1'equilibre 

 interieur des corps solides homogenes."" § The object of this 

 paper is stated to be " to investigate the way in which the 

 interior of a body is affected by the transmission through it 

 of the action of forces." Here they treat the problem of a 

 solid extending to infinity on one side of a plane, on which is 

 a given distribution of tractive load, and also of a solid con- 

 tained between two parallel infinite planes. They obtain as 

 a result a set of definite integrals giving the displacements, 

 introducing a function involving the distribution of tractive 

 load, from which the stresses may be deduced, but concerning 

 which they add : " Les formules precedentes, pour etre 

 obtenues en series numeriques et immediatement applicables, 

 exigent la connaissance des valeurs d'un genre particulier 

 d'integrales definies, dont il ne nous parait que les geometres 

 se soient encore occupes. - " 



The most successful attempt at a solution of this problem is 



* Pearson's ' Elastical Researches of Saint- Venant,' § § 69-99. 



t Pearson, " On the Flexure of Heavy Beams subject to continuous 

 systems of Load," Quarterly Journal of Mathematics, No. 93 (1889). 



X Rankine assumed that the surface-loading eifect might be neglected. 

 See his 'Applied Mechanics,' § 311. 



See also Resume des Legons &c. by Navier (Paris: Dunod, 1864), 

 vol. i. p. 41 : — " Observation sur le mode d'application et de distribution 

 des forces qui font fiechir," where the same assumption is made. 



§ Crelle's Journal, vol. vii. p. 145 et seq. 



