Dr.-Ing. Frank Schwabe
An information about the usability of structures with regard to failure is often more important than an exact knowledge about the stresses and deformations. This way for interpreting results in a save of material or the possibility to predict safety factors more exactly. The main aim of shakedown research ist the determination of the maximum limit load in such a way, that the structure will shakedown under variable loads and will not fail with respect to the time dependent evaluation of the load.
In the past several theories have been developed. For the simplicity in the most cases only a two-dimensional description of the problem were used. The extension to the general three-dimensional case and an experimental control of the numerical results were not done.
The aim of this work is now the software development for the three-dimensional case with respect to inhomogenous structures. This software should be used for the preparation of experimental investigations.
The Notion of Shakedown
The notion of shakedown can be simple explained with the example of a bending beam. The material of this beam should be elastic-perfect plastic, so that an elastic limit moment can be given with:
If the load is greater than this limit moment the beam will plastify about his cross section. The maximal carriable load is the limit moment:
By unloading the beam a residual stress come into existence. With a new loading up to the limit moment no new plastic deformation will occur. The behaviour of the beam is elastic and is called shakedown.
The general mathematical formulation was done by Melan (1938). With the existence of a factor a and a time-independent residual stress field in such a way, that the yield condition (e.g. Huber - von Mises) is fulfilled at every position the system can shakedown under the given loads. The residual stress field must fulfill the homogenous static equilibrium and the boundary conditions.
The aim is now to find a maximum factor under the given contraints. This factor ist called the shakedown-factor.
Finite-Element-Discretisation and solution
For the solution of the optimisation problem a discretisation of the structure can be done with the method of finite-elements. For the flexibility only isoparametric tetrahedron elements with linear and quadratic shapefunction are used.
With these elements it is possible to discretisize every structure (e.g. with the commercial software IDEAS).
The elastic stresses can be calculated with the known finite-element-methods. The calculation of the residual stresses in such is simple way is not possible. Starting with the principle of virtual work it is possible to formulate a system of equations with redundant information. So it is not possible to find a clear residual stress state.
With this system of equations a solution must be found in a space with large dimension, so that the shakedown factor will be maximized. The solution can be found with an optimisation-programm for non-linear problems.
Für groß dimensionierte Systeme ist die Lösung des zugehörigen Einspielproblems mit einem großen Speicher- und Rechenzeitaufwand verbunden. Da diese Probleme nicht hinnehmbar sind, wird derzeit versucht, eine Lösung effizient auf einer Workstation bzw. auf einem schnellen PC zu erzielen. Dies geschieht durch eine geschickte Wahl der Basisvektoren (Technik der reduzierten Basen). Durch eine zusätzliche plastische Berechnung wird ein Unterraum des ursprünglichen Problems erzeugt, in dem eine gute Näherung der Lösung gefunden werden kann.
In the absence of results for three-dimenional problems in the literature a well known two-dimensional problem was extended to a three-dimensional problem. A lot of authos use a plain plate with a hole to show their correctness of the numerical results. By calculating this plate with the software for three-dimensions good results were found.
To the end of 1998 a new big experimental equipment will be constructed. With this equipment we have the rare possibility to investigate three dimensional structures for shakedown and compare the numerical with the experimental results. The following picture should only give an impression about the dimension of the planed equipment.
- Prof. A. R. S. Ponter, Department of Engineering, Leicester University, Great Britain
- Prof. G. Maier, Department of Structural Engineering, Politecnico di Milano, Italy