Cremona diagram

The Cremonaplan used in statically determinate trusses, the graphic determination of the stresses for the design of the bars. It was developed in the 19th century by Antonio Luigi Gaudenzio Giuseppe Cremona and first published in the year 1865.

The Cremonaplan the underlying methods are very helpful in training. In the computer age, the determination of the stresses usually is faster and more convenient without the graphical representation of Kraftecks ​​.

Description

The basic principle is that at each node of a truss must prevail balance. When one perceives the rod forces the node to external forces, the sum of these forces is equal to zero. For each node can draw a closed polygon of forces. If you add together the individual power corner, gives the Cremonaplan.

Each bar force occurs at two nodes and two force corners. Therefore, the Cremonaplan yields a closed force polygon to scale, in which each bar force occurs only once.

Necessary equipment

Manually

You will need: A sheet of paper, ruler, Geodreiecke, pencil, colored pencils, sharpener and eraser. The scale on the ruler is used to read the input and output variables. All forces are translated to the graphic representation with an appropriate scale in length. The largest of the expected forces determined by the selected scale ( eg "1 cm corresponds to 1 kN " ) ultimately, the dimension of the paper. Preferably, the paper sizes A4, A3 used to A0. The larger the pattern, the more accurate the results. Advantage: bar forces can "locally", be determined quickly and reliably without electricity or expensive equipment. A disadvantage is may require a higher amount of time.

Computer-aided

A simple 2D CAD software ( with only a few basic functions such as: line, layer, parallel, measure, measurement) is sufficient for the creation of a Cremonaplanes. For expression is a (preferably color) printer or plotter required. Using different colors input and output forces can be easily distinguished, as well train and compression forces. If command macros for creating the forces plan to be present, thus accelerating this work considerably. The high accuracy ( 8-16 decimal places ) of a 2D CAD system are amazingly accurate calculations possible. Such precision is not necessary in practical structural analysis.

Method

First you have to release the forces on the object. To this end, the support reaction forces and the forces acting on the truss forces are determined in magnitude and direction. This is in a statically determinate system is not difficult, for example, drawing with the Seileckverfahren or by calculation according to the equilibrium conditions ΣMA = 0, ΣV = 0, ΣH = 0 ( see also statically determined ).

The bearing forces are following the example below calculated using the equilibrium conditions:

For the externally acting forces a bypass meaning must be determined (clockwise or counterclockwise ). The forces then entered in the correct order also result in a closed force polygon.

Next, you have to search a node at which only two rods are connected with unknown bar forces. This is set as the starting point. As a starting point is to i.d.R. one of the supports. From there you go through the series after all nodes. For each additional node in turn, a maximum of two new rods, rod whose power is not yet determined, connect. At each force in identifying to which staff they had heard and whether it is a tensile force or compressive force.

At all truss nodes must use the same bypass sense. Each of these members may be drawn only once. (Hint:. Helpful to highlight the already considered forces with a small flag is ) If you have done everything correctly, then the force polygon must be the " starting point " close. Since the force polygon is drawn to scale, the length of the lines drawn to the forces.

In all the directions of the power corners forces are irrelevant ( to the external forces). It's about the amounts and the determination of whether there is a train or a compressive force.

Below for the calculation of the magnitude and direction of the stresses is explained in the o a sample:

After the reaction forces were determined as described above and the force polygon of external forces was drawn to search for one node at which a maximum of two bar forces are unknown. In the example, the node B is selected with the two unknown Stäbkräften for U2 and D4. For a better overview the Einzelkrafteck has been drawn here again.

You do not draw your own force polygon for each node, but can, with a little practice, identify all the member forces in the Gesamtkrafteck. Here you have in the end, a control because the Gesamtkrafteck must close.

To obtain reliable results, the net weights of the bars of a truss must be considered as additional loads at the nodes. These loads can be appreciated in the first pass. They will later be checked and corrected if necessary. While this increases the computational complexity, but for example with larger dead weights, eg Bridges, essential.