744 ] 



LXVI. On the Correction for Shear of the Differential Equa- 

 tion for Transverse Vibrations of Prismatic Bars. By 



Prof. S. P. TlMOSHENKO *. 



IN studying the transverse vibrations of prismatic bars, 

 we usually start from the differential equation 



d.i' 4 g d«- 



(1) 



in which EI denotes the flexural rigidity of the bar, 

 12 the area of the cross-section, 



and " the density of the material. 

 9 

 When the " rotatory inertia " is taken into consideration, 

 the equation takes the form 



Ip BV p&~d 2 y 



g a* s a* 2 * g a* 8 



&ar 



0. 



00 



I now propose to show how the effect of the shear may be 

 taken into account in investio-atino- transverse vibrations, 

 and I shall deduce the general equation of vibration, from 

 which equations (1) and (2) may be obtained as special 

 cases. 



Fis-. 1. 



^Q + |Q dx 



Let abed (fig. 1) be an element bounded by two 

 adjacent cross-sections of a prismatic bar. M and Q denote 

 respectively the bending moment and the shearing force. 



* Communicated by Mr. R. V. Southwell, M.A. Translated from 

 the Russian by Prof. M. G. Yatsevitch. 



