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#### Related lectures

Concepts of stable and unstable elastic equilibrium

General criteria for elastic stability

Elastic instability modes

General methods for assessing critical loads

# Iterative methods for solving stability problems

##### OBJECTIVE

To present and illustrate the application of methods which can be used to solve stability problems iteratively.

##### SUMMARY

This lecture begins with an introduction which describes the reasons for using iterative methods to solve stability problems. Then the Vianello method is introduced. Next, the Newmark method for the calculation of internal forces and deflections in transversally loaded beams is reviewed as a preliminary step to the presentation of the Vianello-Newmark method. This method combines Vianello's method with Newmark's integration procedure.

### 1. INTRODUCTION

Even when deflections are assumed to be small, stability problems are always non-linear in the sense that the equilibrium equations and boundary conditions must be established for the deformed configuration of the structure. As a result, only in very simple cases, it is possible to obtain analytical solutions of the eigenvalue-eigenfunction problem leading to the determination of the critical buckling load and corresponding instability mode (see the Elastic instability modes lecture). In general, it is necessary to resort to approximate methods. One very important group of such methods - energy methods - was presented in the General method for assessing critical loads lecture. Basically, these methods consist of replacing the original continuous structure with a simpler discrete structure. This was achieved by constraining the real structure to deform in a manner which is the superposition of a set of defined shapes with unspecified amplitudes. The exact critical buckling load and mode of this simpler structure, which is the solution of an eigenvalue-eigenvector problem similar to the one addressed in the General criteria for elastic stability lecture, are approximate solutions for the original structure. Although the accuracy of these methods (and the effort involved) increases with the number of degrees of freedom considered, very satisfactory approximations can often be obtained using only a small number. One major drawback of energy methods is that they always lead to upper bounds of the critical buckling load, which is not convenient in design. The discretisation procedure of a continuous structure may also be achieved by dividing it into several rigid elements connected by springs that provide its stiffness. The deformation of the structure is a piecewise continuous function which is completely defined by the displacements of the nodes connecting the elements. The exact solution of this discretised structure was addressed in General criteria for elastic stability and is also an approximate solution of the original problem. However, in this case, nothing can be said concerning the amount or sign of the error. As before, the accuracy also increases with the number of elements.

Thus, the determination of the critical buckling load and mode of a structure requires the solution of a non-linear problem which is either a linear eigenvalue-eigenvector problem (discrete or discretised systems) or a linear eigenvalue-eigenfunction problem (continuous systems). In the first case, an analytical solution is always possible, but it requires the determination of the lowest root of the characteristic equation, which is often of a relatively high degree. In the second case, an analytical solution is possible only for simple problems. An alternative to either of these problems is provided by an iterative method first introduced by Vianello and, therefore, designated as Vianello's method. The basic idea consists of replacing the solution of the non-linear problem by the solution of a sequence of linear problems which can be shown to converge to the critical buckling mode and enable the calculation of the critical buckling load. A feature of Vianello's method, which is very convenient in the design and safety checking of structures, is that it is possible, after each iteration, to calculate upper and lower bounds of the critical buckling load and, therefore, to estimate the corresponding error.

Finally, the Vianello-Newmark method combines the concept of Vianello's method with Newmark's numerical integration technique. It is a very efficient alternative for the determination of critical buckling loads and modes of axially loaded columns, particularly if some non-standard features are present in the loads, the column, or its supporting conditions. This method can also be used to determine equilibrium configurations of columns acted on by specified axial loads and containing initial geometrical imperfections or transverse loads, such as beam-columns.