I being the areal "moment of inertia" of the cross section and x denoting 

 the distance along the beam. 



In an actual beam there may well be resistance to variation of the 

 shear strains also. Inclusion of such an effect as an independent para- 

 meter, however, greatly complicates the theory. Accordingly, the 

 resistance of the shearing stresses to time variation is assumed to be 



such that the moment M due to variation with time of shear wairping along 

 s 



the beam retains its usual value as stated on p. 175 of Reference 11, 

 namely 



^s = - ^I i i^ 

 in terms of the shearing force P, the area of cross section A, a 



dimensionless constant K depending on the shape of the cross section, and 



the modulus of rigidity G where G ~ E/[2(1 + u) ]. 



The total moment (or "bending moment") M then equals M + M . As 



b s 



usual, P = - gM/ax, but the equation of motion now reads 



9 V sP % 9v 



^ _2 = 3x - =y 3t 

 9t 



Tn-os the basic equations for the damped \miform beam can be written as: 



P=-^ [23] 



3x 



M = EI 



12"^ 2/ KAG 2 



\3x 9t 3x / 3x 



^+ ^^il+ i!ll= [25] 



2 ^ 3t 2 



at 9x 



It is assumed that an external force F and an external moment G act 

 on the beam at one end, where x = 0, in the same principal plane with v. 



28 



