Syllabus

NONLINEAR ANALYSIS of STRUCTURES

Prof. Walter Lacarbonara

Academic year 2014-15, Spring Semester

Lecture 1. Introduction to nonlinearities in structures and notation.

Lecture 2. Nonlinear deformations in solids: the first step towards justification of reduced nonlinear theories

Lecture 3. Introduction to COMSOL Multiphysics for nonlinear problems

Lecture 4. Nonlinear models of elastic cables: theory of deformation and equations of motion; Total and Updated Lagrangian Formulations.

Lecture 5. Solution of special problems: funicular shape of hydrostatic pressure; catenary under self-weight; parabola of the suspension bridge problem.

Lecture 6. Discretization of nonlinear distributed-parameter systems via weighted residuals: Galerkin’s method.

Lecture 7. Linearization of the cable problem: geometric and elastic stiffness matrices. Straightforward incremental analysis with force control; incremental analysis with corrector based on Newton-Raphson’s method.

Lecture 8. Implementation of the Galerkin method in Mathematica for the cable problem.

Lecture 9. Path following methods: the pseudo-arclength method.

Lecture 10. Implementation of the pseudo-arclength method in Matlab and Mathematica.

Lecture 11. Nonlinear theory of 3D beams: deformation and equations of motion. Body-fixed component form of the equations of motion.

Lecture 12. Implementation of the 3D beam problem as a one-dimensional polar continuum or 3D continuum in COMSOL.

Lecture 13. Stability: concept and definition; variational equations and eigenvalues; static and dynamic instability; divergence and Hopf (flutter).

Lecture 14. Conservative and nonconservative problems; method of adjacent equilibrium and stationarity of the total potential energy for conservative systems.

Lecture 15. Stability and bifurcation of 1-degree-of-freedom systems: pre-critical, critical and post-critical states. Perfect and imperfect systems. Sensitivity to imperfections.

Lecture 16. Stability of n-degree-of-freedom systems: eigenvalues and eigenvectors; Rayleigh quotient; second-order effects; examples: von Mises structure and the snap-through phenomenon; flutter of a lifting surface.

Lecture 17. Examples in Mathematica of path following analyses with concurrent stability.

Lecture 18. Buckling instability of compressed beams. Instability of framed structures: elastic and geometric stiffness matrices.

Lecture 19. Solution of instability problems in multi-span beams and frames.

Lecture 20. Flexural-torsional instability of open thin-walled beams.

Lecture 21. Solution of instability problems for open thin-walled beams.

Lecture 22. Nonlinear theory of elastic istropic and laminated plates.

Lecture 23. Stability of plates subject to compressive and shear forces.

Lecture 24. Seminar: Static and dynamic stability of bridges

Lecture 25. On-site visit to Ponte della Musica on the Tiber river, Rome.

Lecture 26. Viscoelasticity: rheological models, Kelvin’s and Maxwell’s models, the 4-parameter model of standard visco-elastic solid; the creep and relaxation functions; the hereditary integrals.

Lecture 27. Rheological models of elasto-plasticity: elasto-perfectly plastic model, elasto-plastic model with hardening, visco-elasto-plastic model (Bingham’s model).

Lecture 28. Implementation of rheological models of elasto-plasticity in Matlab.

Lecture 29. The elasto-plastic constitutive response in solids. The associated phase laws of elasto-plasticity; the ultimate limit states due to elasto-plastic failure; the kinematic and static theorems for the limit states estimates.

Lecture 30. The elasto-plastic Saint-Venant problem for beam-like solids: bending and torsion. The moment-curvature law for steel beam sections and the interaction domains.

Lecture 31. The elasto-plastic Saint-Venant problem for beam-like solids: shear and bending; the plastic hinge model and the limit analysis. The ultimate load multiplier for statically determinate and indeterminate beams.

Lecture 32. Limit analysis of frames.

Lecture 33. Evolutive nonlinear analysis up to the ultimate limit state of beam-like structures using COMSOL Multiphysics.

Exams: The course offers traditional lectures, training sessions, a few seminars and on-site visits to important structures. The students will be assigned 3 homework and a project on a selected topic. The final exam consists of a discussion of the homeworks and a presentation of the project through slides.

Book:

W. Lacarbonara, Nonlinear Structural Mechanics: Theory, Dynamical Phenomena and Modeling. Springer, New York, 2013.

http://www.springer.com/materials/mechanics/book/978-1-4419-1275-6

http://www.amazon.com/Nonlinear-Structural-Mechanics-Dynamical-Phenomena/dp/1441912754