Sine Leergaard Wiggers

Buckling with Spring Supported BC

This chapter is initiated with a discussion of the concept of stability, because it is not well defined. The name stability covers broadly, and it is not limited to the subject of the present book. There is no unique definition but different subjective, problem-oriented definitions. The American mathematician Bellman characterizes by the statement “Stability is a heavy loaded concept with an unstable definition.”

Stability of 2D Frames

Then a solution procedure for these rather simple frames is presented as a kind of “cookbook.” A case (Roorda–Koiter frame) of critical imperfection analysis is shown with solution obtained by applying Berry functions. The necessary statics to formulate this problem is a good exercise.

Eigenfrequencies of Beam-Columns with Spring Supported BC

The general eigenvalue problem with three supporting parameters (non-dimensional (s_0), (s_1), and k) and also combined axial load and eigenfrequency is described with its general solution.

Routh–Hurwitz-Liénard–Chipart Criteria

The description of the Routh–Hurwitz criteria for stability of linear, free vibrations is presented in a number of text books, but seldom with more detailed analytical expressions. Furthermore, the simplifications by Lienard–Chipart (1914) are often not included. This is the background for the present chapter and for applying the name Routh–Hurwitz-Lienard–Chipart criteria.

Eigen Solutions for the Euler Cases

Specific cases with constant bending stiffness EI and non-elastic BC are named the Euler cases, and results for these are included in most textbooks on stability and vibration. Instability modes and eigenfrequency modes for bending of the Euler cases are presented in tables with corresponding critical instability load and corresponding eigenfrequencies for the first five eigenfrequencies.

Beam-Column Eigenfrequencies

In Sect. 3.3, the DE for eigenfrequencies is derived and discussed in relation to simply supported BC. Now including rotational end springs to get further BC, the analytical solutions complicate and generalized basic Berry functions with an added tilde notation must be introduced. For each case of no axial force, eigenfrequencies as a function of conservative axial force are obtained. Graphically results are shown for different combinations of rotational spring support. It is noted that the DE may describe the combination of harmonic vibration with continuous support (Winkler support) and with linear magnetic attraction. Relations to Kolousek functions are pointed out.

Structural Stability and Vibration: An Integrated Introduction by Analytical and Numerical Methods

From the Content: Introduction.- Beam-column Differential Equation.- Eigen Solutions for the Euler Cases.- Beam-columns and Applied Berry Functions.

Large Displacements, Pre-buckling Strains, and Snap-Through

From material nonlinearity to influence from geometrical nonlinearity. For Large displacements but still with linear strains, the solution to the Elastica is derived. The modified data for length and moment of inertia, just before buckling, also influence the determined buckling load. An example of a column, build up of truss bars, is presented with analytical result as well as a numerical 2D truss model. Linear elastic snap-through of von Mises truss is the final part of the chapter.

Discretized Stability Analysis

Discretized stability analysis for continua is described for two different versions, named linear elastic buckling analysis and nonlinear elastic buckling analysis. These approaches include eigenvalue problems with different formulations. Linear buckling analysis is often non-reliable and therefore nonlinear buckling analysis, based on Green-Lagrange strains, is described in more detail.

Dynamics of Discretized Linear Systems

Discretization of continuos systems as by the finite element (FE) method is here restricted to linear systems. The main focus is on mass matrices, damping matrices and stiffness matrices and on a linear eigenvalue problems. Derived knowledge on real, positive eigenvalues, on mutual orthogonal eigenvalues and on mode expansion for general linear systems. Finally vibrations with single degree of freedom (dof) give introductory information on the influence from damping.