Analysis of beam columns. Stability functions. Behavior of ideal columns. Bifurcation buckling and limit point instability. Mechanical models of stability. Static and dynamic formulations. Energy methods. Finite element formulation. Buckling of frames. Imperfection sensitivity and post critical behavior. Buckling of beams on elastic foundations, arches and plates. Inelastic buckling. Dynamic analysis of stability. Parametric instabilities and stability under nonconservative forces. Divergence and flutter.
List of recommended books.
General theory and sources of exercise problems
S P Timoshenko and J M Gere, 1963, Theory of elastic stability, McGraw Hill, London.
A Chajes, 1974, Principles of elastic stability, Prentice Hall, NJ.
G J Simitses, 1976, An introduction to the elastic stability of structures, Prentice Hall, NJ.
Z P Bazant and L Cedolin, 1990, Stability of structures, Oxford University Press, Oxford.
Y G Panovko, 1965, Stability and oscillations of elastic systems, Consultants Bureau, NY.
3 Tests [Open book] (30 points)
4 Assignments (20 points)
1 Final examination [Open book] (50 points)
Schedule for tests: 14th February, 14th March and 14th April.
To study, through analytical means, class of static problems amenable for exact solutions. Learn, through this study, the basic notion of elastic stability.
Formulate problems of elastic stability using energy principles.
Develop an understanding of how finite element method can be used to investigate elastic stability of static systems.
To study, through analytical means, class of dynamic stability problems amenable for acceptable approximate solutions. Learn through this study, the basic notions of dynamic stability, divergence and flutter.
Details of lectures
Analysis of beam columns. Stability functions. Breakdown of traditional linear superposition principle. Modified version of principle of superpositioning. Single span and multi-span beam columns. Modified three moment equations. Eccentrically applied axial loads. Effect of initial imperfections. Effect of support flexibility. Southwell plot. Secant formula and Perry-Robertson formula.
Behavior of ideal columns. Concept of equilibrium states in the proximity. Buckling loads as eigenvalues of governing differential equation. Buckling mode shapes and their orthogonality. Tilt buckling. Beams with rigid segments. Concept of critical spring stiffness. Buckling under self-weight.
Mechanical models of stability. Static and dynamic formulations. Linear and nonlinear formulations. Lyapunov stability. Stability matrix. Singular points and their classification. Models for limit load buckling: snap through instability. Elements of Koiter’s theory.
Energy methods. Hamilton’s principle and Lagrange’s equation. Axioms for determining equilibrium and stability for static systems. Rayleigh-Ritz and Galerkin formulations. FE formulation of elastic stability problem.
Buckling of frames: exact solutions. FE solutions. Demonstrations on NISA/ANSYS softwares.
Topics in static stability: Lateral buckling of beams; Beams on elastic foundation. Buckling of perfect Arches. Von Mise’s truss. Snap through. Inelastic buckling. Double modulus theory.
Topics in dynamic stability: Introduction to parametric instability. Strutt’s diagram. Floquet’s coefficients. Nonconservative loads. Divergence and flutter. Concept of follower forces and discussion of conflicting views on its existence.