56 



For M and Q we have the well known expressions as follows 

 (b) M = -Ell^; Q = x'ßFG = x' (^_,^)fG. 



Here 



G denotes the modulus of rigidity and /.' — coefficient, the 

 value of which depends on the contour of the cross-section. 



The differential equation corresponding to the rotation of the 

 element ab cd will be 



or, after (b), 



Eip:^+^'m^AFG-'-^'^ = o (c) 



d X^ ' \dx . ) g dt 



The equation of motion of the element in the direction of y 

 will be 



dQ_^d^y 



dx ~ g d f 



^^-x'fö-fVG = ö Ld) 



g dt^ \dx^ dxl 



From (c) and (d) we obtain 

 ^^a^_ / ^x d^y .r^d^_l^o (3) 



dx* ' ^»/^ '^ V ^x'G/ ()jc^a/2 ^x'^a^^ 



where a and r are given by the equations 



vr r 



In oder to estimate the influence of shear on the frequency of 

 the Vibration we will consider the most simple case, the Vibration 

 of prismatical rods with supported ens. In such a case we can take 

 for the normal mode of Vibration 



>; =^ C sm — cospt, (e) 



where / — denotes the length of the span between the Supports. 



If we put (e) in the equation (3), we receive for the determi- 

 ning of frequency p the equation 



a-^l, -P--P'-J, .^G^" nr+^x'G^ =^--^^^ 



If we retain the two first terms of this equation. we receive 



p = a 



m^rJ' an- 



l' ~ \ 



2 ' 



where X denotes the length of waves. This result corresponds to the 

 equation (1). 



By the retaining of the three first terms in (4) we obtain 



OTT-' /, 1 ^V^ 



which corresponds to the equation (2). 



