Design for Deflection

Monograph VI of the Mechanical Design Engineering Series deals with the development and application of strain energy methods to determine the deflection of slender members subjected to bending, torsion and extension (or compression) loadings. Chapter 1 compares the use of classical mechanics techniques to strain energy methods of calculating deflection to illustrate the ease and convenience of using one over the other. Chapter 2 develops how the work done by simple normal and shear stresses along with their respective strains contributes to the strain energy stored in a deformed body. Hooke's Law relationships for an elastic material are introduced to eliminate shear strains and a state of plane strain is introduced for including slender member loading modes. In Chapter 3 the normal and shear stresses are replaced by the bending moments, torques and tensions in slender members that create the internal stresses. Castigliano's theorem is introduced as a means to determine the deflections and slopes that exist at applied moments, torques and axial tension or compression. Chapter 4 presents a number p special example applications of Castigliano's theorem: (1) finding deflections in beams where no external loads exist, (2), including the distributed weight of the beam, (3) determining deflection perpendicular to a force on a curved beam, (4) deflection of a curved quarter loop, (5) coil spring deflection, (6) a statically indeterminate beam, (7) beam built in at both ends, (8) two element structure analysis and (9) deflection of a planar truss. Two practical design engineering problems are analyzed in detail in Chapter 5. In Problem 1 a cantilever beam is employed in a unique design arrangement to produce a torsional spring. For a specified device geometry and spring constant the task is to determine appropriate cross sectional dimensions for the beam. Problem 2 deals with the lifting of a very long pipe as occurs in the installation of continuous pipelines. The task is to determine the magnitude of the required lifting force and the maximum bending stress generated as a function of the amount of lift.