The engineering community generally accepts that there exists only a small set of closed-form solutions for simple cases of bars, beams, columns, and plates. Despite the advances in powerful computing and advanced numerical techniques, closed-form solutions remain important for engineering; these include uses for preliminary design, for evaluation of the accuracy of approximate and numerical solutions, and for evaluating the role played by various geometric and loading parameters.
Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions offers the first new treatment of closed-form solutions since the works of Leonhard Euler over two centuries ago. It presents simple solutions for vibrating bars, beams, and plates, as well as solutions that can be used to verify finite element solutions. The closed solutions in this book not only have applications that allow for the design of tailored structures, but also transcend mechanical engineering to generalize into other fields of engineering. Also included are polynomial solutions, non-polynomial solutions, and discussions on axial variability of stiffness that offer the possibility of incorporating axial grading into functionally graded materials.
This single-package treatment of inhomogeneous structures presents the tools for optimization in many applications. Mechanical, aerospace, civil, and marine engineers will find this to be the most comprehensive book on the subject. In addition, senior undergraduate and graduate students and professors will find this to be a good supplement to other structural design texts, as it can be easily incorporated into the classroom.
Table of Contents
Foreword. Prologue. Introduction: Review of Direct, Semi-Inverse and Inverse Eigenvalue Problems. Unusual Closed-Form Solutions in Column Buckling. Unusual Closed-Form Solutions for Rod Vibrations. Unusual Closed-Form Solutions for Beam Vibrations. Beams and Columns with Higher-Order Polynomial Eigenfunctions. Influence of Boundary Conditions on Eigenvalues. Boundary Conditions Involving Guided Ends. Vibration of Beams in the Presence of an Axial Load. Unexpected Results for a Beam on an Elastic Foundation or with Elastic Support. Non-Polynomial Expressions for the Beam's Flexural Rigidity for Buckling or Vibration. Circular Plates. Epilogue. Appendices. References. Author Index. Subject Index.