1902.] the Bending of a Beam of Rectangular Cross-section. 493 



solution for the most general system of applied stress of the above type 

 when a is finite. This is found to lead to infinite series of the form 



v/ , t x ("cosh") niry fcosl nirx 



2(a, + M{ sinh j^x{ s ; n j— (1), 



ti being a positive integer, and a n , b n arbitrary constants. Together 

 with these infinite series, there enter into the solutions a finite number 

 of terms of the form 



Cnrn^f 1 (2). 



These represent solutions for certain cases where the body stress 

 equations can be solved in finite terms, so as to give zero stress over 

 the boundaries y = ±b. For instance, a uniform tension parallel to x, 

 a uniform bending moment, and a uniform shear give rise to solutions 

 of this type. They can be superimposed upon the others without 

 affecting the stress distributions over y= ± b, and they are introduced 

 to satisfy the terminal " total " conditions. 



In the various cases considered, the length a of the beam is allowed 

 to tend to infinity. The series then degenerate into integrals. The 

 transformation and interpretation of these integrals are dealt with at 

 length. It is shown that they may be expanded in series of the form 



2 (d n + e n y) r n cos n<f> (3), 



r, cf> being polar co-ordinates about any point in the beam as origin, 

 n being an integer, and d n , e n being constants, which are determined. 



The form of these series varies with the origin chosen. When the 

 origin is a point where a concentrated load is applied, the series for 

 the stresses start with a negative value of n, giving terms which 

 become infinite when r = 0. 



In this case the corresponding series for the displacements contain 

 terms in log r and <£, which lead to discontinuities and infinities. 

 These of course could not occur in any actual problem, but in practice 

 the material immediately below a concentrated load would probably 

 become plastic, so that in the immediate neighbourhood of such loads 

 the solution will not apply. 



It is found that the terms involving infinities and discontinuities are 

 precisely those to which the solution reduces, when the height 2b is 

 made very large. They agree with the solutions given by Boussinesq 

 and Flamant* for two-dimensional strains in an infinite solid bounded 

 by a plane and subjected to load concentrated along a straight line. 



The series of terms involving positive powers of r represent therefore 

 the corrections to Boussinesq's expressions, when the finite height of 

 the beam is taken into account. 



* ' Cornptes Eendus,' vol. 114, pp. 1465—68 and pp. 1510—16. 



