Cremonaplan

The Cremona diagram used in statically determined trusses of the stresses to measure the rods of the drawing determination. It was developed by Antonio Luigi Gaudenzio Giuseppe Cremona in the 19th century and first published around 1865.

The methods on which the Cremonaplan is based are very helpful in training technicians or engineers and in understanding the flow of forces . In the computer age, the determination of the bar forces is usually faster and more convenient without the graphic representation of the force corner.

description

The basic principle is based on the fact that equilibrium must prevail at every junction of a framework. If you consider the bar forces at the nodes as external forces, the sum of these forces is zero. A closed force corner can be drawn for each node (see Culmann method ). If you put the individual strength corners together, the Cremonaplan is created.

Every bar force occurs at two nodes and in two force corners. Therefore, the Cremonaplan results in a closed, true to scale force corner in which each bar force occurs only once.

Necessary tools

Manually

triangle ruler

You need a sheet of paper, a ruler, a set square , pencil, colored pencils, a sharpener and an eraser. The scale on the ruler is used to read the input and output variables. All forces are converted into lengths using a suitable scale for graphical representation. The greatest of the forces to be expected ultimately determines the dimensions of the paper via the selected scale (e.g. " 1 cm corresponds to 1 kN "). In general, the larger the drawing, the more precise the relative accuracy and thus i. d. Usually also the result. Advantage: bar forces can be determined quickly and reliably "on site" without electricity and expensive equipment. One disadvantage is u. U. a higher expenditure of time.

Computer aided

A simple 2D CAD software (with only a few basic functions such as: line, plane, parallel, dimensioning, measuring) is sufficient to create a Cremona plan. If command macros are available for creating the force plan, this speeds up the work considerably. Due to the high accuracy (often 8 to 16 decimal places) of a 2D CAD system, amazingly accurate calculations are possible. In practical structural engineering, however, such accuracy is only necessary in special cases.

method

In the case of a statically determined (ideal) framework , you can (also in this case ) also determine the support forces with the Cremonaplan; The support forces are often calculated in advance in order to be able to check the construction again and because the construction of the Cremonaplan is also easier. The determination of the support forces in a statically determined system can generally be solved with the statics of rigid bodies and is included in the curriculum of high school schools . The determination can generally e.g. Drawing with the B. Seileckverfahren or by calculation according to the equilibrium conditions , , (see also determined statically ) take place. ${\ displaystyle {\ sum M_ {A}} = 0}$ ${\ displaystyle {\ sum V} = 0}$ ${\ displaystyle {\ sum H} = 0}$

Before determining the bar forces, it makes sense to determine any zero bars beforehand according to the applicable rules. In the example below there are no zero members.

The following are the support forces for the example below:

${\ displaystyle {\ sum M_ {A}} = 0 \ Rightarrow B = 12 {,} 5 \, \ mathrm {kN}}$ , , . ${\ displaystyle {\ sum V} = 0 \ Rightarrow A_ {V} = 12 {,} 5 \, \ mathrm {kN}}$ ${\ displaystyle {\ sum H} = 0 \ Rightarrow A_ {H} = 5 {,} 0 \, \ mathrm {kN}}$

For the forces acting from outside, a direction of bypassing must be defined (in mathematically positive direction or clockwise). The forces then entered in the correct order then result in a closed force corner.

If available, it is generally efficient to find a node that has a maximum of two unknown forces acting on it. The order of the nodes can be chosen arbitrarily, but it is advantageous to go through the nodes in order. In addition to the already known members, a maximum of two members whose member force has not yet been determined may connect to each further node. Each force is marked to which rod it belongs and whether it is a tensile or a compressive force.

The same circumferential direction must be used at all truss nodes. It automatically follows that each bar force only needs to be drawn once. Since the force corner is true to scale, the lengths can be measured and converted into the corresponding forces.

The directions of the forces are relevant in all force corners.

In the following, the o. A. Example for determining the size and direction of the bar forces explained:

If you determine the support forces arithmetically and not with the Cremonaplan, you draw the support forces in the force corner of the external forces. It is advantageous, but not necessary, to look for a node for which a maximum of two bar forces are unknown. In the example, node B was selected with the two unknown bar forces for U ₂ and D ₄ . For a better overview, the individual force corner has been drawn out here.

You mentally stand between the two unknown rods and turn (here to the right) around the knot. The first known force (and also the only one at this node) is the support force B, which - here in the individual force corner - is recorded to scale with its direction and magnitude.
Next you hit the unknown staff U ₂ . At the end of the previous force (B), the line of action (direction of force) is now applied in the force corner as a straight line, still without a size.
The force still remaining in rod D ₄ is also shown in the force corner with its line of action . Since it has to close the force corner, it goes through the starting point of the support force B.
Now you have the finished Krafteck B, U ₂ , D ₄ . Since the force corner must close, the direction arrows can be drawn accordingly. The size of the bar forces can be determined by measuring the individual lengths in the selected scale.
In the system sketch at the examined node one now also draws the determined arrows in the same direction. Here in node B the force U ₂ goes away from the node and the force D ₄ goes towards the node. Forces away from the node are always tensile forces in the member (here blue) and towards the node there are always compressive forces (here red). Now the magnitude and direction of the forces have been determined for the examined node.
Since the forces in the respective rod are constant in terms of direction and magnitude, the direction arrows can now be drawn on the opposite node of the two rods: tensile forces away from the node, compressive forces towards the node. The arrows at the ends of the rods therefore point in the opposite direction. Now node B is done.
Now you can determine the forces at the next node. In the example this would be node K ₃ with the known loads F ₂ , F ₃ and bar force D ₄ and the two unknown bars D ₃ and O ₁ . To do this, proceed as described in points 1 to 6.

You do not have to draw a separate force corner for each node, but can, with a little practice, determine all the bar forces in the total force corner. At the end of the day, you also have a control here, as the total force corner has to close.

literature

Luigi Cremona: Le figure reciproche nella statica grafica. Ulrico Hoepli, Milan 1872.
Gross, Hauger, Schröder, Wall: Technical Mechanics 1. Springer Verlag.