Ritter's cutting method

The Rittersche cutting process goes back to August Ritter . It is used for structural analysis to calculate internal forces in general for trusses .

In order for a system to be in equilibrium under the action of given forces, it is necessary and sufficient that every sub-system is in equilibrium of all force quantities acting on it. For statically determined structures, it is generally expedient to calculate all support forces before applying the cross-section . Before determining the bar forces, it makes sense to determine any zero bars beforehand according to the applicable rules.

Procedural principle

A part of the framework is cut out (thus creating two parts). In principle, one can cut arbitrarily to get a new equation.
However, it is advantageous to cut in such a way that an independent equation is obtained for each unknown. For this purpose, all lines of action of the remaining intersected, unknown bar forces must intersect at a point which i. A. is not cut by the unknown unknown (see round cut ). This point may also be at infinity. (If the point of intersection is at infinity, however, one does not make an equilibrium of moments in the actual sense, but an equilibrium of forces.)

If you cut three bars of a two-dimensional truss, with a maximum of two parallel to each other, the forces can always be calculated according to the equilibrium conditions, provided the parallel bars do not lie on a line of action, since you can set up three linearly independent equations, provided the outer ones Forces are known (e.g. support forces).

The equilibrium conditions in the 2-dimensional, in an xy-coordinate system, are z. B .:

• The sum of all torques that act at one point is zero: ${\displaystyle \sum M_{i}~=~0}$
• The sum of all forces acting in the X direction is zero: ${\displaystyle \sum F_{x,i}~=~0}$
• The sum of all forces acting in the Y direction is zero: ${\displaystyle \sum F_{y,i}~=~0}$

In the case of a three-dimensional framework, a total of six bar forces can be calculated by setting up a linear system of equations with six linearly independent equation components. ( and ) ${\displaystyle \sum {\vec {F}}_{i}={\vec {0}}}$ ${\displaystyle \sum {\vec {M}}_{i}={\vec {0}}}$

If you would then only cut out a single knot in the cross-cut, which in principle would not be the intention of the cross-cut, (since it is actually a structural cut that generally goes through the entire structure), one speaks of a round cut .

example

sketch

In this example the bar forces are determined for a roof truss. Red forces are given, green forces are determined from the equilibrium conditions. Note on the graphic: in the first step the green forces in the upper graphic (i.e. the support forces) are calculated so that they are known (red) and only then are the green arrows in the lower graphic calculated.

• The upper part of the picture shows the entire girder with fixed bearing on the left and floating bearing on the right. (The force F _Ax results in zero, F _Ay and F _B = F _By result in 1.5F.)
• In the lower part of the picture , the Rittersche section is shown, which takes place here after the support reactions have been determined. After cutting, the three bar forces (F ₁ , F ₂ , F ₃ ) can be calculated. By cleverly choosing the equilibrium equations, the solving effort of the linear equation system can be reduced, for example by

1. The sum of the moments around the point of intersection of the bar forces F ₂ and F ₃ gives the bar force F ₁ directly ,
2. The sum of the moments around the point of intersection of the bar forces F ₁ and F ₂ directly yields the bar force F ₃ ,
3. The sum of the moments around the point of intersection of the bar forces F ₁ and F ₃ provides the bar force F ₂ directly .

You can then control the determined bar forces via the equilibrium conditions and . ${\displaystyle \sum F_{x}=0}$ ${\displaystyle \sum F_{y}=0}$

See also

• Beam theory for bending bars
• Cremonaplan for a graphic solution

literature

• August Ritter: Elementary theory and calculation of iron roof and bridge constructions . C. Rümpler, Hanover 1863.
• Heinz Waller: Technical Mechanics in a nutshell . BI-Verlag, Mannheim et al. 1990, ISBN 3-411-76312-4 .
• C. Spura: Technical Mechanics 1. Stereostatics . Springer, 2016, ISBN 978-3-658-14984-0 .