# Cutting principle

The cutting principle ( English free-body principle, principle of intersection ) is a theoretical investigation method in mechanics , especially in statics to determine local stresses in a loaded object.

Part of the object is mentally cut off ( free cutting ). Forces and moments must then be applied to the cut surface of the remaining part, which must be in equilibrium with the forces and moments acting on it from further outside and via the bearings ( bearing reactions ). They are determined on the remaining part with the help of the equilibrium conditions that include these forces and moments.

The cutting reactions are the stresses at the interface. The stresses acting in the cut surface can be inferred directly from them .

The cut can also be made through a bearing or (mostly articulated ) connection point within the object if the forces and moments at such points are to be determined.

The object can be a solid , a liquid , a gas or a system of such.

## history

The realization of the benefit of decomposition is attributed to Francis Bacon

"The principle of dismantling [...] remained completely hidden from the ancients, only the great English natural philosopher Bacon of Verulam said it around 1600: 'You have to cut up nature.' As a whole, almost all natural phenomena would be incomprehensible to us, they would be so complicated and varied that man would soon have to give up any attempt at understanding. "

H. Bertram quotes Isaac Newton as saying :

"NEWTON also wrote as early as 1687 (p. 380):

'Furthermore, we learn from the phenomena that mutually touching particles can be separated.' "

- H. Bertram

This realization was reflected in Newton's principle of actio and reaction .

Based on this, Leonhard Euler developed the cutting principle in 1752 in order to be able to treat extended (also deformable or liquid) bodies: If any part of a body is cut out in thought, he follows Newton's laws of mechanics, whereby the forces acting on it are that the rest of the body exerts on it at the cut surfaces (plus any external forces such as weight, etc.).

Augustin-Louis Cauchy also successfully applied this cutting principle to mechanical stresses that are distributed over the cut surfaces. Accordingly, this cutting principle is also referred to as the Euler-Cauchy tension principle.

## General

The mechanical interactions always take place between material bodies or the particles they are made of. Press z. B. a particle T 1 on another T 2 with a force F , the force F occurs simultaneously at T 1 as an effect of T 2 , namely as a pressure force in the direction of the particle T 1 .

${\ displaystyle {\ stackrel {T_ {1}} {\ square}} \, {\ stackrel {F} {\ leftarrow}} \ quad {\ stackrel {F} {\ rightarrow}} \, {\ stackrel {T_ {2}} {\ square}}}$

According to the principle of "action and reaction", both forces are always oppositely equal.

In a system of masses that is demarcated from other masses, two types of forces can be distinguished: On the one hand, there are internal forces that act between two masses belonging to the system and therefore always occur in opposing pairs. On the other hand, there are the external forces that act between every system mass and a mass outside the system and therefore only occur once on the system. In sum, the internal forces always cancel each other out in pairs, so that only the external forces remain.

## Application in continuum mechanics

Fig. 2: A disc with blocks cut out

From the point of view of continuum mechanics , the entire universe can be viewed as a continuum with more or less solid components. The theories resulting from this, however, are formally complex and are not required for most local investigations. Instead, material bodies are generally introduced using the cutting principle. The body is cut out of the continuum by specifying a boundary, which divides the continuum into the body and its surroundings. The limitation is largely arbitrary and can be adapted to the respective task.

The internal and external interactions are always two-dimensional or spatially distributed in the continuum and, if necessary, are integrated into resultants over corresponding areas.

As in Fig. 2, (infinitesimal) small sub-bodies can also be extracted in order to derive equations of motion for the material points of the body, see for example rope statics and Cauchy-Euler's laws of motion . In particular, Cauchy's fundamental theorem shows the unlimited applicability of the intersection principle in continuum mechanics.

## Application in technology

In technical mechanics and structural engineering , the stresses at one point in a component or in a system of components that correspond to the cutting reactions to be applied at this point are of particular interest in free cutting .

### Location map

Fig. 3: Site plan of a mechanical system with incision (red) on a drum held by a rope on a ramp; the load on the rope as a result of the weight of the drum must be determined (indicated by the acceleration due to gravity g )

A site plan (see Fig. 3) shows the spatial arrangement of the body or system under consideration. The incision is drawn and marked in it.

The conceptual cut (s) separates the body under consideration (the upper rope in Fig. 3) from its surroundings or from a part of its surroundings (from the barrel in Fig. 3;). The dividing line encompasses the entire cut-away area (closed line), which is usually not drawn in completely (only the part through the upper rope in Fig. 3).

### Free body image, sectional reactions

Fig. 4: Free body image of the system from Fig. 3;
a) Cutting reaction F and inner rope force Z at the cutting point
b) The downhill force
F generated by the weight m · g (vector direction correct downhill!) is the cutting reaction

The site plan is abstracted to the so-called free body image. The symbolic representation of the interface (cut surface) of the cut body is sufficient. Of the cut-away environment, only that which is absolutely necessary for the determination of the cutting reactions emanating from it remains in hint (in Fig. 4 barrel, ramp surface and restraint rope from Fig. 3).

With the help of the free-body diagram, it is relatively easy to determine the cutting reactions (in Fig. 4 the tensile force F on the rope; Z is the internal force in the rope).

• Firmly connected bars, beams and other solid bodies transmit forces and moments.
• Ropes , chains , drive belts and similar limp structures only transfer tensile forces in the direction of the rope. Such rope forces are diverted by rollers , but their amount does not change if there is no friction.
• A pendulum rod , which is articulated at both ends, only transmits tensile and compressive forces along its axis.
• Joints do not transmit some forces and moments, depending on the type. However, depending on the type, other forces and moments can be transferred.
• Contact forces on smooth, frictionless surfaces always act perpendicular to the surfaces.
• In the same way, liquids and gases always exert their static pressure perpendicular to the wetted surfaces.
• Frictional forces always act tangentially to the contact surface opposite to the direction of movement of the body on which they act.

The drawing is given a coordinate system for orientation. In the various disciplines (structural engineering, mechanical engineering, point mechanics ) there are different ways of representing and interpreting the characters in the sketches.

#### Figure the body

Fig. 5: In dynamics, bodies can be represented as point masses

The free-body picture is a basic sketch that does not depend on a true-to-scale representation. The individual bodies are sketched, whereby arms and columns are reduced to lines and extended bodies - as in Fig. 5 - are drawn together to points. For complex components with many partial bodies, a free-body diagram is similar to an exploded view , and it is a matter of skill to make the free-body diagram clear.

#### Illustration of forces and moments

Fig. 6: Representation of a force F , moments M and T and a line  load q

Forces are shown as arrows, line loads as interconnected arrows and moments as double arrows or as simple arrows curved around an axis of rotation, see Fig. 6.Because the relative position of the force application points and the lines of action of the forces are important for a system of forces, the application points must and their relative positions can be seen from the free body image. When entering the forces, it must be considered whether it should only be determined whether the body is in equilibrium, whether the bodies are deformable or can be assumed to be rigid. With the latter and generally with questions of balance, moments may be shifted freely and forces may be shifted along their lines of action. When determining loads and deformations, the force application points must not be relocated.

#### Not to be shown

The free-body diagram is just one of several drawings that are created for solving mechanical systems.

The following system features should expressly not appear in their free-body images:

• Bodies that do not belong to the system under consideration, in particular "cut away" bodies (the forces they exert are already entered. There is a risk of forces being taken into account twice.),
• Ancillary, boundary and other constraints as well as bearings (the corresponding constraint forces must be entered instead),
• Forces and moments that the body exerts on its environment (Because these forces and moments are equal to those that are exerted on the body according to the principle of action and reaction, all forces and moments would be extinguished. There would be no imprinted forces more.),
• internal forces and moments (These only occur on cut surfaces. Internal forces entered in the body give rise to fear that they will be misunderstood as impressed forces.).

These details to be excluded can be shown in other, supplementary drawings if necessary.

### Concrete illustrations

Occasionally, loads, bearing reactions, stresses and also sectional reactions are entered in more specific illustrations (e.g. technical drawings) to simplify matters, as the pictures below show. The overview is easily lost compared to site plans and especially free-body images.

 The storage reactions are already entered as vectors. The storage symbols are dispensable. The system is not static, but dynamic (the two weight forces are not the same). The step to a free body picture is still considerable. The buoyancy forces B replace the hydrostatic pressure of the water, which can be dispensed with.

There is even the risk that some forces are taken into account twice as applied forces when applying the cutting principle, for example the bearing reactions in the left picture, the rope forces in the middle and the buoyancy forces in the right picture.

## Remarks

1. The cutting reactions now replace the forces and moments acting on the cut-away part from the outside and via the bearings.
2. ↑ You should refrain from cutting the body that is already being cut free. Errors can easily occur here, especially if simplifying assumptions have since been incorporated into it.
3. The short line between the two force vectors indicates that one sometimes speaks of the two banks of a cut, of which one (right in Fig. 4) is the cutting reaction, the other (left) the existing at the cut surface of the cut part assigns internal forces and moments or stresses.
4. The force arrows N and R represent the bearing reactions, which are compensated for by the forces on the bearing (ramp and counter rope) that are not shown and also result from the weight of the drum.

## Individual evidence

1. Haupt (2002), p. 75
2. Georg Hamel: Elementary Mechanics . A textbook. Teubner, Leipzig and Berlin 1912, DNB  580933865 , p. 5 ( archive.org [accessed December 1, 2016]).
3. a b H. Bertram: Axiomatic Introduction to Continuum Mechanics . Wissenschaftsverlag, 1989, ISBN 3-411-14031-3 , pp. 67 f .
4. ^ Otto Bruns, Theodor Lehmann: Elements of Mechanics I: Introduction, Statics . Vieweg, Braunschweig 1993, p. 92 ( google.de [accessed on December 1, 2016]).
5. Hans Georg Hahn: Theory of elasticity . Basics of linear theory and applications to one-dimensional, plane and spatial problems. Springer Fachmedien Wiesbaden, Berlin, Heidelberg 1985, DNB  850225965 , p. 16 ( google.de [accessed on December 1, 2016]).
6. H. Oertel (ed.): Prandtl guide through fluid mechanics. Fundamentals and phenomena . 13th edition. Springer Vieweg, 2012, ISBN 978-3-8348-1918-5 , p. 15 .
7. ^ H. Altenbach: Continuum Mechanics . Introduction to the material-independent and material-dependent equations. Springer, 2012, ISBN 3-642-24119-0 , pp. 71 .
8. Haupt (2002), p. 91

## literature

• Bruno Assmann, Peter Selke: Technical Mechanics . 18th edition. Oldenbourg Wissenschaftsverlag, 2006, ISBN 3-486-58010-8 , p. 99 ( google.com [accessed February 9, 2013]).
• István Szabó : History of Mechanical Principles . Springer, 2013, ISBN 978-3-0348-5301-9 ( google.de [accessed on December 3, 2016]).
• Alfred Böge: Technology / Technology - for technical high schools and technical colleges . 5th edition Friedr. Vieweg & Sohn, Braunschweig, Wiesbaden 1988, ISBN 978-3-528-44075-6 , pp. 5 f . ( google.de [accessed on January 5, 2017]).
• Dietmar Gross, Werner Hauger, Jörg Schröder, Wolfgang A. Wall: Technical Mechanics . tape 3 : kinetics . Springer DE, 2008, ISBN 978-3-540-68422-0 , pp. 191 f . ( google.com [accessed January 5, 2017]).
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2002, ISBN 978-3-642-07718-0 .