Mechanics Of Materials

Draft DRAFT Lecture Notes Introduction to MECHANICS of MATERIALS II Fundamentals of Inelastic Analysis c VICTOR E. SAOUMA Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO 80309-0428 Draft ii Victor Saouma Mechanics of Materials II Draft iii PREFACE One of the most fundamental question that an Engineer has to ask him/herself is what is how does it deform, and when does it break. Ultimately, it its the answer to those two questions which would provide us with not only a proper safety assesment of a structure, but also how to properly design it. Ironically, botht he ACI and the AISC codes are based on limit state design, yet practically all design analyses are linear and elastic. On the other hand, the Engineer is often confronted with the task of determining the ultimate load carying capacity of a structure or to assess its progressive degradation (in the ontect of a forensic study, or the rehabilitation, or life extension of an existing structure). In those particular situations, the Engineer should be capable of going beyond the simple linear elastic analysis investigation. Whereas the Finite Element Method has proved to be a very powerful investigative tool, its proper (and correct) usage in the context of non-linear analysis requires a solid and thorough understanding of the fundamentals of Mechanics. Unfortunately, this is often forgotten as students rush into ever more advanced FEM classes without a proper solid background in Mechanics. In the humble opinion of the author, this understanding is best achieved in two stages. First, the student should be exposed to the basic principles of Continuum Mechanics. Detailed coverage of (3D) Stress, Strain, General Principles, and Constitutive Relations is essential. In here we shall go from the general to the specific. Then material models should be studied. Plasticity will provide a framework from where to determine the ultimate strength, Fracture Mechanics a framework to check both strength and stability of flawed structures, and finally Damage Mechanics will provide a framework to assess stiffness degradation under increased load. The course was originally offered to second year undergraduate Materials Science students at the Swiss Institute of Technology during the author’s sabbatical leave in French. The notes were developed with the following objectives in mind. First they must be complete and rigorous. At any time, a student should be able to trace back the development of an equation. Furthermore, by going through all the derivations, the student would understand the limitations and assumptions behind every model. Finally, the rigor adopted in the coverage of the subject should serve as an example to the students of the rigor expected from them in solving other scientific or engineering problems. This last aspect is often forgotten. The notes are broken down into a very hierarchical format. Each concept is broken down into a small section (a byte). This should not only facilitate comprehension, but also dialogue among the students or with the instructor. Whenever necessary, Mathematical preliminaries are introduced to make sure that the student is equipped with the appropriate tools. Illustrative problems are introduced whenever possible, and last but not least problem set using Mathematica is given in the Appendix. The author has no illusion as to the completeness or exactness of all these set of notes. They were entirely developed during a single academic year, and hence could greatly benefit from a thorough review. As such, corrections, criticisms and comments are welcome. Victor E. Saouma Boulder, January 2002 Victor Saouma Mechanics of Materials II Draft iv Victor Saouma Mechanics of Materials II