] 26 Prof. S. P. Timoshenko on the Transverse 



In equation (1) 



EI denotes the flexural rigidity of the bar, 



O the area of the cross- section 



and - the density of the material. 

 9 



It was shown that the correction for shear, in a repre- 

 sentative example, was four times as important as the 

 correction for " rotatory inertia," and that both corrections 

 are unimportant if the wave-length of the transverse 

 vibrations is large in comparison with the dimensions of 

 the cross-section. 



In the present paper, an exact solution of the problem is 

 given in the case of a beam of rectangular section, of which 

 the breadth is great or small compared with the depth, so 

 that the problem is virtually one of plane strain or of 

 plane stress. The results ;ire compared with those of my 

 former paper, and confirm the conclusions which were there 

 obtained. 



§ 2. When the problem is one of plane strain (so that 

 w is constant), we have to solve the equations * 



(\+2/*)v 2 A + / o/A = 0, 

 and fjuy 2 ^-{-pp 2 ^T = 0, 



where A denotes the cubical dilatation f^ + ^- ), 



Vo# oyj 



v „ the rotation ^ ^"^j, 



V , B 2 



y (2) 



an 



d v 2 „ the operator ^ 2 + ^j. 



J 



•^Taking the #-axis in the direction of the central line, and 

 choosing for A an even and for w an uneven function of y, 

 we may write 



A = A sin ax sinh my Cos pt, 1 



2vr = B cos olx cosh ny cos pt 9 J v« J 



where A and B are undetermined coefficients, 



* A. E. H. Love, ' Theory of Elasticity/ §§ 14 (d) and 204. 



