The development and production of innovative products using new materials requires in-depth knowledge of analytical and numerical calculation methods for the hazard-free design of components and machines. The transfer of this knowledge is one of the essential tasks of Engineering Mechanics at LTM in the bachelor and master studies. With the coupling of training, modeling, performing experiments and practical application pursued at LTM, the prospective engineer is thus prepared in a multi-layered manner for the constantly increasing challenges in the calculation of mechanical engineering components in industry.

Illustrative object in teaching: The 4 Euler cases

An important tool for engineers are trusses, which are generally a structure of interconnected members. Therefore, trusses and their special features are a central element in the teaching of technical mechanics, in which the LTM makes a major contribution to the education of young prospective engineers in both the bachelor's and master's degree programs in mechanical engineering. The lecture "Technical Mechanics 2" deals, among other things, with various problems that arise when structures and bodies are subjected to compressive or tensile loads. This distinction plays an important role in the question of whether a component - e.g. a bar of a truss - can withstand the external force acting on it without damage or whether it will fail. If a member is loaded in compression, there may be a stability problem. The keyword at this point: the four Euler cases. In this theory, there are 4 materially as well as geometrically identical buckling bars, each with different bearings at both ends.

In addition to the cross-section of the bars and their length and the stiffness of the material, the upper and lower bearings have a considerable influence on the external force F that can be applied in the direction of the bar to the upper end of the still undeformed straight bar. Depending on the type of bearing, a sometimes quite small compressive force is sufficient here to cause the bars to buckle, which represents the failure.

In order to make these effects tangible for the students in addition to the teaching materials and to illustrate the buckling process to them, we have conceived, designed and manufactured our own Euler demonstrator as an illustrative object in a small project. On the demonstrator, which is made of aluminum and stainless steel, all 4 Euler cases are reproduced according to theory. The force acting on the rods is realized by a multitude of weights with masses between 6 and 285 grams in interaction with the acceleration due to gravity. For each of the cases shown, the weights can therefore be stacked "slice by slice" and the force acting on the rods thus increased until the critical force Fk is reached and the rods made of spaghetti buckle. This does not happen gradually, but suddenly. The deflection is limited in order to be able to represent the illustrations of buckled rods known from teaching in a non-destructive and yet as similar manner as possible.

In addition to a possible compressive loading of bars, there is, as already mentioned, depending on the application, the possibility of tensile loading and thus under certain circumstances a strength problem in which the bar length is irrelevant. In order to be able to reproduce this loading condition as well, the fixture was designed so that it can also be turned upside down and the weights cause a tensile load by clamping the fourth bar (right) on both sides.

Thus, in the example of the fourth case (on the right in the figure above), the bar previously loaded in compression is now loaded in tension. In our example, with a significantly higher acting force than before: The bar does not fail, although the weight has even been doubled.

The core observation, the buckling under compressive load and the holding under a considerably higher tensile load of one and the same rod, can thus be vividly conveyed to the students with our demonstrator and thus make teaching a bit more tangible.