VARIATIONAL LAUNCH WINDOW STUDY 1129 mum launch azimuths, since for this launch time the optimum azimuth is due-east. The entire trajectory is seen to lie in the reference plane, and where the trajectory terminates, the reference plane is extended by the dashed curve. For the other two launch times, the reference plane is translated and drawn through the launch site in order to yield a visualization of two-dimensional motion. Some understanding of the variation of launch azimuth can be obtained from this figure. Consider, for example, the trajectories shown for Afo = +1 hr. For the due-east launch azimuth, the trajectory is "S" shaped. There are two "turnings" or bendings of the trajectory. Near the launch site, the flight plane is turned southerly to bring the motion toward the desired plane. Towards the injection point, a second turning is necessary to bring the velocity vector as well as the position vector into the plane.Optimization of the launch azimuth is seen to alleviate the necessity for the bending around the launch site. Thus, the bending of the trajectory, i.e., its three-dimensionality, can be equated roughly with energy expenditure.This would leave one with a final conjecture. The initial bending of the plane seen for the due-east launch azimuth is necessary to direct the motion toward the desired plane, and if this were not done, burnout would occur at some point relatively remote from the reference plane. But if a coast were introduced at a sufficiently high energy level to avoid excessive descent, and if the coast were of a sufficiently long duration to allow a displacement of the order of 180°, the vehicle would approach automatically the reference plane from the other side of the earth. This may improve performance in much the same manner as did optimization of the launch azimuth.Equations are derived for the slope, deflection, and maximum stresses in a cylindrical shell subjected to edge shears and moments, including the simultaneous action of axial loads. A sample problem is given in which the maximum stresses obtained by the refined equations are compared to those obtained by approximate methods currently used in industry in which the axial load restraints on shell rotations and deflections are neglected. A 100-in. radius cylinder with a wall thickness of 0.5 in. and an internal pressure of 1000 psi was selected arbitrarily as a basis for comparison. The maximum hoop stress was found to be 213,770 psi by the standard method and 200,000 psi by the method developed in this paper. The reduction in maximum hoop stress is due to the restraining influence of the axial loads on hoop displacements of the shell. Where warranted, the use of the refined method generally will give lower calculated maximum stresses and result in a lighter-weight shell design.
