A. 2 - ADDITIONAL THEORY USED IN EVALUATING 

 SECTION PROPERTIES 



ASSUMPTIONS 



Figure 10 illustrates the general class of structures to which this theory and digital 

 program are applicable, and it shows the idealizations incorporated into the representation of 

 the structure. In applying beam theory to a structure such as a beam hull, it is recognized 

 that the section properties will vary with position along the beam; however, the calculation 

 of the elastic parameters of the beam at a particular cross section is based on the assumption 

 that the structure is prismatic; that is, all sections are identical, at least in the immediate 

 vicinity of the section under consideration. Thus, Figure 10 shows the structure as a prism, 

 with all tension and shear members parallel to the x-axis. It is assumed that the section 

 lying in the plane x = is the section to be analyzed. For the purpose of establishing the 

 elastic properties of this section, the prismatic structure is assumed continuous, both in the 

 -X direction (shown) and in the +x direction (not shown). 



In Figure 10, the coordinate axis x, y, z locate points on the structure. The displace- 

 ments of points from their basic positions as in Appendix A.l is given by U^, U , and U^ in 

 translation and by 6 , 6 , and 6^ in rotation, with positive directions in the same sense as 

 the X-, y-, and z-axes (rotation established by the right-hand rule). Section forces are V , 

 V , and V ; section moments are M , M , and M . These forces and moments are positive 

 if the force (or moment) exerted by the portion of the structure not shown (x > 0) upon the 

 portion shown (x = 0) is in the direction of positive displacement. 



The figure shows that the structure has been idealized so that all the tensile stress 

 is carried by a finite number of axial elements, each located at a distinct node of the section 

 and having associated with it a finite area. A.. The remainder of the structure carries only 

 shear and consists of straight panels of constant thickness t. connecting pairs of nodes. For 

 the sake of simplicity, it is assumed here (although not in Appendix A.l) (1) that each 

 tensile element has full effectiveness and all are composed of the same material and 

 (2) that each shear element has full effectiveness and all are composed of the same material. 

 These assumptions do not really limit the generality of the theory. 



In this report the following subscripts are used: 



i to indicate the various nodes of the cross section 

 j to indicate the various plates of the cross section 

 / to indicate the independent loops formed by the plates 



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