K izz(y. -y)-iyz(^.-^) 



II -12 



yy zz ys 



M k' 



I (z. -z) - I (y. - y) 



yy\l / yz^'^l •'-' 



I I -P 



y y ZZ y 



For ships with a plane of symmetry, I = 0. 



If this expression is differentiated with respect to x, assuming the node locations and 



areas do not depend upon x, then the rate of change of tension is the same expression except 

 dV^ dMy dM^ dV^ dM^ 



that V^ is replaced by , M by , and M ' by . Assume that = 0, = V , 



dx y •^ dx ^ ■^ dx dx dx 



'^^'^ 

 and — — = - V (equilibrium of beam). Then, since rate of change of tension in a node is the 

 dx ^ 



sum of the shear flows, it follows that (note that shear flow is out of node, hence, force on 

 node is in +x direction; see Equations [16]-[20J of Appendix A. 2):* 



I (y. _ y) _ I (z. - i) 

 zz^-'i •' ' yz^ 1 ' 



I q. (out) = - V k^A. 



^1 ^ ' y 1 1 



y i i 

 V^k;A. 



Iyy(^i 



I I -l2 



yy zz yz 



-"^)-iyz(yi- 



-y) 





yy zz ~ yz 





This gives one equation involving the shear flows for each node. Usually there are more 

 plates than nodes, so additional equations are needed to solve for the shear flows. Any set 

 of shear flows which satisfies the above condition for the sum of q out of the nodes will 

 automatically have the correct resultant V and V^, thus no additional information is gained 

 by writing overall equilibrium equations; see page 70.** 



*2q. (out) is better termed (S q . ). or merely q » ■ , and is the algebraic sum of shear flows q on each plate 



connected to node i. Such a term q is positive if the force acting on the portion of the plate AA on the -x side 

 of the section shown in Figure 16 (as viewed from the +x side) is away from the node; another plate or portion 

 of a plate contiguous with side AA is assumed to exist to the left of AA. And (Sq ). is positive if the net 

 shear flow in all connecting plates is outward. 



**Overall equilibrium equates the total shear forces sustained by the section to the shear flows of the plates: 



V = iq. (y, . -y,.) ; V = X q. (z, . - z, .) 



y J ^J ^■'hj 'tj^ ' z 'ij "- hj tj-' 



Here, subscript j refers to all the plates, q. is the shear flow in a plate, and y, ■, y^., z, ., and z . locate the 

 head (h) and tail (t) of the plate. 



The point of the statement is that for shear flows q- based on a tree and values of (Sq ). at each node 

 given by the preceding equation, the above equations are automatically satisfied (and for loop shear flows they 

 give zero for V and V ); therefore, there is no point in invoking them. 



55 



