Course description

Matrix analysis of structures with material and geometric nonlinearities; nonlinear geometry response under large displacements, P-D approximation, stability of structures. Description of nonlinear material, section and element response. Nonlinear static and dynamic response of 2d frames and trusses.

Course organization

A midterm examination will cover concepts of the first half of the course. The comprehensive final examination will cover concepts from the entire course with some emphasis on the second half. Since the course is rather intensive in calculations, it is advisable to purchase a calculator that can handle basic operations on small matrices. A take home examination may also be given, if necessary.

Grading will be based on the following weights: class participation: 10%, homework: 20%,  midterm examination: 25%, final examination: 45%

Course objectives

The course extends the concepts presented in CE 220 to cover topics in nonlinear analysis of structures such as the response under nonlinear material and geometry, the stability of structures, and the nonlinear response under transient loads.

CE 220 covered two basic approaches to linear structural analysis: the displacement method and the force method of analysis. The objective of presenting these two methods was to enhance understanding of the duality relation between the compatibility and equilibrium relations and their virtual work equivalents, the principle of virtual forces and the principle of virtual displacements, respectively. These principles continue to apply to nonlinear material response under small displacements, but need to be suitably extended for the case of nonlinear geometry. This course makes almost exclusive use of the displacement method of structural analysis in the determination of the nonlinear response of  structural systems with few members. This contributes to better understanding of system response issues. On the other hand, the automatic computer analysis of structures makes use of the direct stiffness implementation of the displacement method. Since the assembly of the structure stiffness matrix is relatively straightforward from the procedural standpoint, the focus shifts on the determination of the element response under nonlinear geometry and material. To this end, both the force and displacement formulation will be used. At the same time, the development of suitable strategies for the solution of the nonlinear equilibrium equations needs to be addressed.

Emphasis will be placed on the rigorous definition of the analysis model, the formulation of the problem, and the logic of the computation process. Limited attention will be devoted to efficient computational methods, which are addressed in other courses. Since the solution of even the smallest nonlinear problem places a heavy calculation burden on the student, we will depend heavily on available computer tools, such as Mathcad and Matlab. Most homework assignments will contain a significant calculation and/or programming component, so that familiarity with computers is essential for this course. We will break down the solution process into several small steps and use a Matlab function or Mathcad calculations for each step. Examples of the organization of each task of the solution process will be provided. Fundamental concepts will be used in setting up the problem solution and checking the answers of an analysis. The midterm and final examinations will emphasize the conceptual aspects of the course but may also address calculations in the case of take home examinations.

The most suitable computational tools for this course are Mathcad and Matlab. Both have strengths and limitations. Mathcad is the better all around package, affords clear documentation of the solution process, but is clearly inferior in its programming capabilities. Matlab on the other hand is harder to master, but offers more programming power, better modularity and is easy to extend to more general cases once the right architecture is put in place. It is possible to develop an organized set of functions that can tackle gradually more complex problems. This is the intent of the toolbox FEDEASLab which was introduced in CE 220. We intend to make gradual extensions to the toolbox with the objective of studying history dependent nonlinear response under static and dynamic loads.