APPENDIX A 



The author recognizes that reader interest will vary widely on the theoretical aspect 

 of the paper. A working knowledge of the physical meaning of the equations used in the 

 coding may be required only. Or, interest may also be centered upon all of the fundamental 

 ideas underlying the derivation and use of these equations. Therefore, this Appendix has 

 been subdivided into two parts; Appendix A.l, which describes the method for evaluating 

 the section properties of the ship (sufficient for understanding the general procedure), and 

 Appendix A. 2, which supplements this description with additional fundamental concepts and 

 mathematical detail. Thus, the theory can be pursued to the degree desired. 



A.l - METHOD FOR EVALUATING SECTION PROPERTIES 



The theory of beams may be considered as the limiting case of the general theory of 

 elasticity applied to slender objects. In the theory of elasticity, the displacements and 

 stresses are unknown functions of position. Strain displacement, stress-strain, and equilib- 

 rium laws are available to solve for the unknowns. Most engineers consider the strain- 

 displacement laws and the equilibrium laws as independent unrelated ideas; however, one is 

 obtainable from the other by using the stress-strain law and a minimum principle (minimum 

 potential energy theorem). In the theory of beams, instead of taking unknowns in three spatial 

 dimensions, quantities are defined only along one line, the "axis" of the beam. The unknowns 

 become six displacements (linear displacements in three directions and rotations about three 

 axes) of the cross section, and six forces* (tension, two bending moments, two shears, and a 

 torque). In the following, the elastic relations between the forces and displacements will be 

 found for beams constructed of stringers and plates. The equilibrium laws which come from 

 an application of Newton's Second Law (force = mass • acceleration) are not given, but they 

 may easily be found to complete the beam theory. 



Choose a rectangular cartesian coordinate system with the x-axis along the beam and 

 the y- and z-axis such as to form a right-hand coordinate system; see Figure 3. The dis- 

 placements of a cross section parallel and perpendicular to the x-axis will be given by U^^ 

 and U , U , respectively. The rotations of the cross section about these axes are 6^, d , 

 and d , where positive sense is given by the right-hand rule. The resultant force associated 

 with these motions acting on the positive side of the cross section (acting upon the body which 

 consists of those portions of the beam on the -x side of the section) will have three linear 

 components V , V , and V and three moments M , M , and M . All displacements, rotations, 

 moments, and forces are positive if the vector which represents them is in the positive 

 coordinate direction. In general, these 12 unknowns are functions of x (and possibly time). 

 Six equations for the unknowns come from equilibrium; the other six from elasticity. 



*Forces here is used in a generic sense in that it includes moments and torques. 



44 



