Beam theory assumes that the stresses a , ct^^, and a^^ vanish (a^^ is defined to be 

 the force per unit area acting on a face perpendicular to the i-axis and in the j-direction). 

 Thus the strain energy per unit length is given by (see Chapter 6 of Reference 7): 



-.2 „2 , „2 



w = — 



E 



G 



dA 



The stresses a^ , a , and o must be determined in terms of the beam forces V^^ , 

 . . . , M . Statics alone is not sufficient, and assumptions consistent with the theory of 

 elasticity must be made to solve for the stresses. If the distribution (except for a constant 

 factor) of the stresses is known, then statics can be used to find the stresses (find the 

 factor). Assume that the a^^ stresses are due to V^^, M , and M^; a^^ and a^^ stresses are 

 due to M , V , and V . 



x' y ' z 



For CT^ stresses, let Fj(y, z), F2(y, z) be three given functions and Kj, Kj, and Kg 

 be three unknown constants. As a simple example, the functions might be selected (see 

 Chapter 7 of Reference 8 or Chapter VI of Reference 9): 



1; 



y; 



This would duplicate the stresses existing due to tension, moment about z-axis, and moment 

 about y-axis if elementary beam theory is adopted, requiring tensile strain proportional to the 

 distance from the elastic axis when bending moment is carried; i.e., the basic assumption of 

 beam theory is that the longitudinal strain in the ship hull, deck, longitudinal members, etc., 

 varies linearly with the coordinates of a cross section. Hence assume* 



^xx(y' ^) = KiFi(y, z) + K2F2(y, z) + K3F3 (y, z) 

 Applying statics gives: 



Vx=/-xxdA =KiJFidA + K2jF2dA + K3|F3dA 



My =j za^^dA = KjzFjdA + K2 jzF2dA + K3 |'zF3dA 



M. =/(-yKxdA = K J(-y)F^dA + K2|(-y)F2dA + K3/(-y)F3dA 



Since F^, F2, and F3 are assumed to be known functions, the above three equations can be 

 solved for K,, K^, and K, as linear homogenous expressions in V , M , and M ; 



K. 



jF.dA / 



F^dA 

 fzFjdA 



-lyFidA 



F.dA 



zF.dA 



F,dA 



zF,dA 



2"^ 3 



ryF2dA - yF3dA 



-1 



V 

 M 



Km J 



*This will be shown to give rise to the bending parameters. 



46 



