19 ft 



Transverse Vibrations of Bars of Uniform Cross-Section. 125 



membrane would have some resemblance to the curve of a 

 catenary of uniform strength. 



If <j) is zero or negligible the equations in this paper 

 reduce to the usual Poisson equations for thin plates. Now 

 it follows from equation (12) that <f> can be zero provided 

 that 



te%) -§?•§? =°- • • • (106) 



But this is precisely the condition that the bent middle 

 surface should be a developable surface. When cf> is zero 

 the middle surface is .unstretched, and now we find from 

 our equations, what is quite obvious from geometry, that a 

 plane sheet can be bent into a developable surface without 

 the stretching or shrinking of any of its elements. 



If the expression on the left-hand side of equation (106) 

 is small but not zero, it is still possible that <£ be so small 

 that Poisson's equations are approximately true. The 

 examples we have worked out indicate that this condition 

 is satisfied if iv is everywhere small in comparison with the 

 thickness of the plate. Since the expression in (106) is of 

 the second order in 10 it is clear that <fi depends on ur rather 

 than on w. The condition that <f> should be neo-lio-ible will 

 still be satisfied if the deflexion of the middle surface, 

 measured from some developable surface, is small compared 

 with the thickness of the plate. The third problem worked 

 out above supplies an example of this. 



X. On the Transverse Vibrations of Bars of Uniform Cross- 

 Section. By Prof. S. P. TlMOSHENKO *. 



§ 1. TN a paper recently published in this Magazine f , 

 A I have dealt with the corrections which must 

 be introduced into the equation for transverse vibrations 

 of a prismatic bar, viz., 



m W + TW = > • • • • C 1 ) 



in order that the effects of " rotatory inertia " and of the 

 deflexion due to shear may be taken into account. 



* Communicated by Mr. R. V. Southwell, 31.A. 

 t Phil. Mag. vol. xli. pp. 744-746. 



