2. In order for the plate to be in equi- 

 librium in the x-direction, the product of r 

 times the thickness must not vary along 

 the plate [i.e., for a given plate, ( t ■ thickness) 

 is independent of y and z]. This is explained 

 as follows: Figure 7 shows the shear forces 

 acting on a plate. For equilibrium in the 

 x-direction, F^ = F2. But the thickness at 

 end 1 could be different from the thickness 

 at end 2: t^^t^. Then defining the shear 

 flow^'' q (force per unit length along the plate) 

 by q = T. thickness: 



qi Ax 

 q2 Ax 



Tjtj Ax 

 T2t2Ax 



5ut Fj = Fj , and, there- 



Figure 7 — Shear Forces Acting on a Plate 



Thus T, ^' T„ 



1 ^ 

 fore, q^ = q2. Also, by rotational equilib- 

 rium, q^ = qj at corner 1 and q^ = q2 at 

 corner 2. 



However, we could have selected 

 slices 1 and 2 at any points on the plate, 



not just at the nodes at the end. Consequently, no matter where one looks along edge 3, 

 q, = q^ = q^ is the same at any point on a single panel between nodes where tensile force 

 (in the x-direction) acts on the plate from an external source; i.e., the shear flow is a con- 

 stant for each plate. Thus the problem of finding the shear stresses has now been reduced 

 to finding one unknown (shear flow) for each plate. 



3. Each plate begins and ends at a node.* Also, each node has at least one plate 

 attached to it. The shear stress in a plate exerts an axial force on the node. This force is 

 q per unit length in the x-direction.^'^ Assign a positive direction to shear flow. When 

 looking at the cross section from the +x side, the shear stress acts upon the plate in one 

 direction. This is the direction of the flow. If the shear flows into a node, the plate exerts 

 a force on the node in the -x direction. Hence, the net force per unit length on a node by 

 all the plates which join it is the sum of the shear flows out of the node. (Hence, the name 

 "shear flow." For problems with no tension, the sum of the flows out of any node vanishes.) 

 From this study of the nodes, the tensile stress in a node is given in terms of the forces 

 V^, M , and M^ (see Equations [9J and [16] of Appendix A. 2 and the preceding development): 



*For additional detail on this section, see Figure 16 and the associated text in section Shear and Torsion 

 Appendix A. 2; also see footnote on page 55. 



54 



