# Statics of rigid bodies

Beam construction with fixed and floating bearings and fulfilled equilibrium conditions (sum of all forces and moments equal to zero)

The statics of rigid bodies (also called rigid-body statics, stereostatics or stereo-statics ) is part of rigid body mechanics and statics . It deals with the balance of forces on rigid bodies , i.e. on non-deformable bodies. In statics, these bodies either move at a constant speed or they are at rest. Rigid body statics is important as a branch of technical mechanics for the calculation of forces that act in components. The rigid body statics deals with the fundamentals of technical mechanics; further areas are elastostatics , dynamics and structural engineering .

In the context of rigid body statics, many topics are dealt with that are also of importance in other areas of mechanics. The equilibrium conditions are valid in the entire statics, basic terms such as force, moment , area center of gravity and center of mass are common in all mechanics, while bearings , force systems , cutting principle and cutting reactions play a role in all technical mechanics. The model concept of the rigid body and the axiom of volatility of lines apply to all rigid body mechanics.

## Definition, classification, demarcation and relationship to adjacent areas

### definition

The static is defined as a branch of mechanics, in which the forces in equilibrium are and therefore cancel in their effect, or alternatively as "teaching of bodies that are at rest", ie non-moving bodies. Since bodies that move at constant speed are also in equilibrium, they can also be treated with the methods of statics.

A rigid body is a physical body that cannot be deformed. The statics of rigid bodies thus deals with the balance on these bodies.

### classification

The statics in technical mechanics
 Technical mechanics Statics dynamics Strength theory kinematics kinetics
Structuring the mechanics from the
point of view of the forces involved
 mechanics Kinematics Laws of motionwithout forces Dynamics effect offorces Statics Forces in equilibrium ofresting bodies Kinetic forces change the state ofmotion

Since the static is part of the mechanics, the rigid body statics both part of the rigid body mechanics as the general statics to the addition of the rigid body structure, among other things, the fluid statics (for gases and liquids) and the elastostatics (accommodate elastic body). The classification of statics is different in physics and engineering.

In physics , mechanics is divided into kinematics , which deals with the location, speed and acceleration of bodies, but not with forces or masses, as well as dynamics , which take forces and masses into account. The dynamics can be further subdivided into the statics for bodies in which the forces are in equilibrium, and in the kinetics , which deal with bodies in which the forces are not in equilibrium. In fact, statics is not treated separately in physics and is left to technical mechanics. The two physicists Joseph Honerkamp and Hartmann Römer write: The calculation of constraining forces is a technically very important problem; For example, engineering statics can be seen as a theory of the calculation of constraining forces.

Technical mechanics is divided into statics, strength theory and dynamics. In fact, textbooks and events under “statics” deal with some basic principles of general statics (e.g. equilibrium conditions ) as well as the statics of rigid bodies; areas such as elastostatics or structural engineering are explicitly referred to as such. Since there are separate disciplines for deformable bodies, statics in the narrower sense is often only understood as the statics of rigid bodies. In civil engineering , however, statics is often used to refer to static calculation or structural analysis, which also deals with deformable, solid bodies. In contrast, strength theory, as a further area of ​​technical mechanics, exclusively deals with deformable bodies.

### Delimitation and relationship to adjacent areas

The strength theory deals with all deformable, solid bodies that are in equilibrium. In the case of statically determined systems, it usually assumes (in the theory of the first order ) that the acting forces have been determined using the methods of rigid body statics and is therefore based on this. The known forces can then be used to investigate how much a component is deformed. The rigid body statics cannot make any statements about the spatial and areal distribution of these forces in the components ( mechanical tension ), which are of particular importance for strength theory. The bodies must then be regarded as deformable, also in the case of stability problems (buckling, bulges) and in the case of statically indeterminate systems.

In the kinetics the forces acting are not in equilibrium. However, these are in so-called dynamic equilibrium with the inertial forces (not forces in the sense of Newtonian mechanics) . The calculation of dynamic systems (rigid bodies) can then be carried out using methods of (rigid body) statics. In this respect, the kinetics are based on the statics. The statics, which get by with the methods of rigid body statics, is therefore practically always dealt with first in lectures and textbooks in technical mechanics.

### Areas of application

Example of a statically indeterminate system

The aim of statics as part of technical mechanics is essentially to calculate the forces that act in individual components. With the statics of rigid bodies, this is only possible with so-called statically determined systems . With these, the equilibrium conditions are sufficient to calculate the forces. For statically indeterminate systems, further conditions are required; the equilibrium conditions of the statics of rigid bodies remain valid. Statically indeterminate systems are dealt with in the field of strength theory and structural engineering .

## Topic overview

A framework made of six bars with two forces. The upper three small circles symbolize swivel joints, the lower ones are immovable (fixed) bearings in which the rods can also rotate.

In the textbooks of technical mechanics there is a great deal of agreement about which topics belong to the rigid body statics, only the order varies. The force as a central factor is often introduced at the beginning. Sometimes the closely related moment (force times lever arm) is introduced immediately afterwards , but often only when it is needed for the general force systems. Power systems make it possible to combine several forces to a resultant to split individual forces in several forces (especially those that are parallel to the coordinate axes), checking whether a number of forces are in equilibrium, and calculating unknown forces, if the forces in the Balance.

Bearings are components that connect bodies to their surroundings. Through them, forces act on the body under consideration. These forces only emerge when the bearings are mentally removed and replaced by these forces. This replacement is called free cutting .

Real machines and structures often consist of several parts that are put together. These can be cut freely at any point (e.g. at the joints) and broken down into several subsystems ( cutting principle ) in order to calculate the unknown forces. These systems include all structures . Forces or moments in the interior of bodies are referred to as internal forces , respectively. They are required in strength theory , but can be calculated using the statics of rigid bodies if the bodies are supported in a statically determined manner.

Further topics are friction and focal points ( geometric center of gravity and center of mass ). Friction also includes static friction for immobile bodies and sliding friction for moving bodies. The calculation of centers of gravity is required to determine the line of action of the resulting forces, especially if the forces are distributed across volumes, areas or lines, such as pressure forces (area) or weight forces (volume).

## Axioms of rigid body statics

The statics are based on axioms , i.e. assumptions that cannot be proven, but which are consistent with experiences with real bodies. In contrast to Newton's axioms, there is no consensus on the axioms of statics either in their order or in their number. The following axioms are mentioned in the literature; many are not only valid in rigid-body statics, but also in statics or mechanics in general, while others only apply to all rigid-body mechanics:

• Axiom of equilibrium: Two forces that act on a rigid body are in equilibrium if they have the same amount and line of action and point in opposite directions. It goes back to Pierre de Varignon .
For deformable bodies, this only applies if the two forces act at the same point. If they are only on the same line of action, you can compress or stretch it. If they are not on the same line of action, they rotate a freely moving body without shifting the center of gravity. They then form a so-called force couple .
• Axiom of interaction: If a force from one body acts on another body, then this exerts a force on the first that lies on the same line of action, has the same amount and is opposite to the first force.
This applies to all bodies, not only to rigid bodies and also to forces inside bodies.
The axiom of line volatility does not apply to a deformable body.
• Linienflüchtigkeitsaxiom (also displacement law, law of displacement or longitudinal displacement sentence ) a force acting on a rigid body, may be displaced along its line of action without the effect (acceleration) changes on the body.
This applies to all rigid body mechanics. With the same line of action and the same direction, it makes no difference for equilibrium and acceleration on a rigid body whether a force pulls it in front or pushes it back. However, this makes a difference for deformable bodies and also for the forces inside any (including rigid) bodies ( cutting forces ). It is therefore important that bodies are always cut free first and forces can only be shifted afterwards. A shift of a force on a parallel line of action, however, may only be carried out if the offset torque is taken into account.
• Axiom of the parallelogram of forces : Two forces that act at a common point can be replaced by a resulting force , which is the diagonal of the parallelogram that is spanned by the two forces.
This also applies to the special case that they are on the same line of action. The parallelogram then degenerates into a line.
• Superposition theorem: Two forces that are in equilibrium can be added to a group of forces without changing the effect on the rigid body.
The theorem applies to all of the mechanics of rigid bodies. Effect means the equilibrium and the acceleration of the body; internal forces generally change by adding equilibrium groups.

## Basic concepts

The most important basic concepts are the rigid body, the force, the moment and the degree of freedom.

### Rigid body

A rigid body is a theoretical body that does not deform under the action of forces. In reality there are no rigid bodies because every body deforms. In many practical problems, however, these deformations are so small that good results can be achieved with the model of the rigid body. The model is also used in the dynamics.

### force

The force is a physical quantity that tries to set a resting body in motion. Important forces in technical mechanics are the weight force and the frictional force including static friction . In mechanics, forces are mathematically modeled as vectors and represented by an arrow. They are defined by several determinants:

1. Your amount, represented as the length of the arrow.
2. Your point of attack, shown as the foot or starting point of the arrow.
3. Their line of action , usually not shown in drawings. It results as a straight line through the base and tip of the arrow.
4. Your sense of direction , i.e. the direction along the line of action

The sense of direction and the line of action are also combined to form the direction. In rigid body mechanics, the point of application can also be omitted.

In general, the force is a point-tied vector that relates to the point of application . In the statics of rigid bodies, however, forces can be shifted along their lines of action (as long as they act on the same rigid body); it is therefore a line- volatile vector.

In mechanics, forces are classified according to various criteria.

According to the spatial distribution one differentiates:

• Individual forces or point forces, individual load or individual torque acting on a single point. It is an idealization; real forces always act on surfaces or volumes. The weight force is often modeled as a single force that acts on the center of gravity.
• A linear force or line load is distributed on a line. It is also about idealizations.
• A surface force, surface load or surface tension acts on a surface, for example pressure forces from liquids act on container walls or winds on buildings. Surface forces are often constant in one or more dimensions. If only the cross-section of bodies is considered, then surface forces become distributed loads.
• Body forces act of a body on the whole volume. This primarily includes weight.

According to the cause, a distinction is made between:

• Applied force : It has a physical cause such as weight or wind pressure.
• Reaction force : It reacts to external influences, e.g. B. the force that counteracts its weight in the foundation of a building. The reaction forces also include, in particular, the bearing force and the constraining force , which limit the possibilities of movement of bodies.

According to their effect, a distinction is made between the

• Close force that only works when the two bodies between which it works are in direct contact and the
• Long- distance force , which can also act over a distance, such as gravitation or electrical and magnetic forces.

Local forces are usually surface forces that are transmitted via the contact surface. Remote forces are usually potential forces, which are applied as volume forces in continuum mechanics.

According to the place of activity, a distinction is made between

• external force that acts on a body from outside and which
• internal force that works inside a body. These include the so-called internal sizes .

### moment

Force around a reference point
Power couple

The moment is a physical quantity that tries to turn a body. In technical mechanics, a distinction is made between the moment of a (single) force with respect to a point and the moment of a force couple (without a reference point). The moment of a single force is only defined in relation to a freely selectable point . Its amount can be calculated from the vertical distance between the line of action of this force and the reference point and the amount of force ${\ displaystyle M ^ {(A)}}$${\ displaystyle F}$${\ displaystyle A}$${\ displaystyle a}$${\ displaystyle F}$

${\ displaystyle M ^ {(A)} = F \ cdot a}$.

A pair of forces consists of two forces that have the same amount , lie on parallel lines of action with the distance and point in opposite direction. A force couple cannot move a body like a force, but they try to turn it. The amount of the moment that is exerted by the force couple results from the distance and the amount of one of the two forces ${\ displaystyle F}$${\ displaystyle a}$${\ displaystyle M}$

${\ displaystyle M = F \ cdot a}$.

The moment of a pair of forces has no reference point and acts in the entire plane in which the two forces lie. In the mechanics of rigid bodies it can be replaced by its moment if no internal forces are of interest but not by a resulting force. A single force, on the other hand, cannot be replaced by its moment.

### Degrees of freedom of a rigid body

A body's degree of freedom is a possibility of movement that it has in principle. A rigid body that can only move within one plane can move in two dimensions and it can be rotated. He thus has three degrees of freedom. Rotations are also known as rotations and displacements are also known as translations . A rigid body that can move in three-dimensional space has a total of six degrees of freedom: rotation and translation are possible in each dimension. Deformable bodies also have deformation degrees of freedom, i.e. an infinite number.

## camp

Components that connect the bodies under consideration to their surroundings are called bearings . The way a body is stored is called storage. Bearings allow some movements and prevent others.

The bearings of a door, for example, which is idealized as a rigid body on a rigid bearing free of holes, only allow rotation and prevent all other movements. The bearings of most drawers, on the other hand, which are idealized as rigid bodies on an ideal immovable bearing free of holes, only allow displacement and prevent any rotation. The number of degrees of freedom that a bearing prevents is called its value . The bearings of the door and the drawer are thus pentavalent, since they prevent five of the six possible movements in principle.

Bearings can be permanently connected to the environment ( fixed bearings ), or they can be moved themselves ( floating bearings ). For example, a bridge that is only supported at its ends is usually equipped with a fixed bearing and several floating bearings. If it expands as a result of a change in temperature, it can move practically unhindered in the horizontal direction, both in the longitudinal and in the width direction, because the floating bearings generally allow the movement with negligible resistance.

For the numerous bearings there are corresponding symbols in technical mechanics. Bearings that allow rotations are particularly common, as these are usually easier to manufacture. These are represented by a small circle.

The three most common symbols in level support systems:

Bearings can be implemented using various components. For details, see warehouse (construction) and warehouse (machine element) .

## Cutting principle

Example of a wave
A straight bar (solid line), cut from two sides, together with the forces and moments cut free. The dashed line is the reference fiber and is used to indicate positive directions.

With the exception of remote forces, the forces acting on a body are introduced via adjacent bodies (especially bearings). In order to make these forces available for calculations, a method called the principle of intersection is used. The process is known as cutting free or freeing. Here, bodies are replaced in thoughts by the forces that they transmit. The cutting of the bearings, which are replaced by their bearing reactions , is of particular importance . Bearings that prevent displacement are replaced by forces (bearing force) and bearings that prevent rotation are replaced by moments ( clamping moment ).

In principle, any body can be cut at any point. The interfaces are illustrated in drawings by curved lines, unless the bearings are cut free. For example, if a body is hanging on a rope, the body can be replaced by its weight force and the rope can be cut in thought, whereby the rope force appears, which counteracts the weight force.

## Force systems

A force system or group of forces is a series of forces that act in a system: for example, all forces that act on a bridge, all of the forces that act on a vehicle or only those that act on the transmission. In the case of free-cut systems, the forces also include the cutting forces. Power systems allow multiple operations. This includes the combination of several forces into one resultant and the determination of unknown forces via the equilibrium conditions. This can be used to check whether two different force systems are statically equivalent , i.e. have the same effect on a body. You can also check whether a force system is in equilibrium. With the assumption that it is in equilibrium, the unknown forces can be calculated.

Force systems are classified according to two different criteria. According to the number of dimensions, a distinction is made between plane and spatial force systems. According to the occurrence of moments, a distinction is made between central force systems (without moments), in which the lines of action of all forces intersect at a single point, and general force systems (with moments), in which the forces do not intersect at a single point.

### Pooling and splitting of forces

Parallelogram of forces

Two forces with a common point of application can be combined into a resultant using the force parallelogram , which has the same effect as the individual forces . More than two forces with a common point of application can be summarized by first forming a resultant of two forces and then repeating the process. Conversely, a single force can be broken down into several components that point in predetermined directions (e.g. coordinate axes).

Forces at which the lines of action intersect at a common point can also be combined with the force parallelogram. To do this, they are first moved along their lines of action to the intersection and combined there. A decomposition also works accordingly.

If the lines of action do not intersect at a single point, the forces can be combined by moving them to one point. In the case of the parallel displacement to another line of action, an offset torque arises that must be taken into account. The system of the resulting force and the total moment is called the Dyname . The moment can be eliminated by shifting the resulting force in parallel. This means that the amount, direction and line of action of the resulting force are fixed.

A special case is the force couple . It cannot be summarized as a resultant force, but it can be replaced by its moment (without a resultant force).

### balance

A body is in equilibrium when the resulting force and moment with respect to any point are both zero. In a plane force system that spans exactly one horizontal and exactly one vertical direction, this means:

• The sum of all force components in the horizontal direction is zero.
• The sum of all force components in the vertical direction is zero.
• The sum of all moments in the plane with respect to any point is zero.

In the spatial force system, there is an equilibrium of forces and an equilibrium of moments for each dimension. The equilibria of forces apply in any direction.

These equilibrium conditions can be used to check for a number of known forces whether they are in equilibrium. If it is known that a body does not move and only some of the forces are known, the unknown forces can be calculated using the equilibrium conditions. Since only three independent equations can be set up in the plane case, only three unknowns can be calculated for a single body using the methods of rigid body statics. In the spatial case there are accordingly six equations and unknowns for a single body.

If further forces are unknown, further equations are required which then contain deformations and material properties. These are the subject of structural engineering and strength theory . In the case of several bodies that are connected to form a larger body (for example, individual parts that are assembled into assemblies and modules), a corresponding number of unknowns can be calculated for each rigid body (three in the plane).

## Rigid Body Systems

A tannery girder: It consists of three beams that are connected by two joints (small circles). On the left there is a fixed bearing (triangle), in the middle and on the right there are a total of three floating bearings (triangle with a horizontal line).

Rigid body systems consist of several rigid bodies that are connected to one another. The connections can also be rigid or pivotable or displaceable by means of joints. These systems are calculated by cutting the entire body. The overall system then breaks down into several subsystems, with the corresponding forces and moments being applied to the interfaces.

Ideal trusses are constructions that only consist of rods that are articulated at the connection points (nodes). They are connected to one another in a certain way so that the nodes each form a central force system, i.e. only tensile or compressive forces are transmitted in the bars, but no transverse forces or moments, which is why ideal trusses are particularly easy to calculate. Similar importance Gerber carrier as bridge construction. They are relatively easy to manufacture and are insensitive to deposits. In tanner girders, the loads are mainly carried by bending moments and lateral forces.

## Cutting sizes

Internal forces on a single-span beam with line load q and longitudinal force F, transverse force V, normal force N and bending moment M. The transverse force is greatest at the edges and has a linear profile. The torque curve here follows a quadratic function and has its maximum in the middle.

Forces are forces and moments acting in the interior of bodies. They can be calculated by cutting the bodies in mind. It is not the cutting force at a specific point that is of interest, but the course of the cutting forces or moments over the length of a rod. The internal forces are required to design the dimensions of the components or structures ( dimensioning or dimensioning ). To do this, however, the strength of the material must be known.

In the case of a rod-shaped body in the plane, there are three types of internal forces in first- order theory :

• The shear force that is perpendicular to the beam axis.
• The normal force that acts perpendicular (normal) to the section surface, i.e. in the direction of the member axis.
• The bending moment attempting to bend the beam.

For bars that are modeled in three-dimensional space, there are a total of six internal forces, one force and one moment per dimension. Compared to the plane system, there is an additional bending moment and a transverse force for bars as well as the torsional moment that tries to twist (twist) a bar. In "complex" structures such as a cube no cutting forces as normal force, shear force, bending moment and torsional moment can be defined, but only a stress tensor - field .

## main emphasis

Focal points are required to calculate the lines of action or points of application of distributed loads. This includes the weight force. A distinction is made between the purely geometric centers of gravity such as the line, area or volume center of gravity and the center of mass .

## Limits of statics and further areas

By definition, the field of rigid body statics is subject to two restrictions: on the one hand, only rigid bodies are dealt with, and on the other hand only bodies that are in equilibrium, which makes the mathematics necessary to describe static systems comparatively simple. On the other hand, deformations of components and structures as well as forces in statically indeterminate systems cannot be calculated using methods of rigid body statics.

If only the restriction to rigid bodies is lifted, this leads to the fields of strength theory and structural analysis. If you allow accelerated movements instead, this leads to (rigid body) dynamics. Both strength theory and dynamics are an integral part of technical mechanics, as found in many engineering courses, structural engineering only in civil engineering. The math required for these areas is at a higher level. For example, tensors and derivatives play a major role.

### Strength theory

In the Strength of non-rigid - - body treats deformable be. The forces acting on the body then lead to deformations. The strength is a material parameter that indicates how great the force ( mechanical tension ) related to the cross-sectional area may be so that no failure (breaking, excessive deformation) occurs. Since deformations are linked to the forces via material parameters, the strength theory can also be used to calculate statically indeterminate systems.

### Structural analysis

The structural analysis is a field of civil engineering. Like strength theory, it uses variables such as mechanical tension, deformation and strength. A special focus is on the various structures such as B. trusses, single-span girders, pane connections and frameworks. It deals with statically determinate as well as indeterminate systems. It also includes the equilibrium in the deformed position, which is realistic, since (loaded) systems are only in equilibrium in the deformed position.

### dynamics

Dynamics is an area that deals with accelerated movements in technical mechanics. The accelerations can be calculated with known forces and masses. Accelerated bodies are not in (static) equilibrium. If, in addition to the forces acting, the so-called D'Alembert inertial forces are also taken into account, then these are in dynamic equilibrium .

## history

Until the early 18th century, Archimedes' law of levers and the parallelogram of forces dominated the development of statics. Of Jordanus Nemorarius the first correct description of static weights on a native incline to 1250. In his description of the lever he used the first time the principle of virtual work .

During the Renaissance, the development was based primarily on observation and experimentation to solve construction problems on machines and structures (e.g. Leonardo da Vinci's construction of an arch bridge without connecting elements). His thought experiments and Galileo's work on strength theory led to the separation of elastostatics from rigid body statics. Galileo also specified the Archimedean concept of force and used the expression of the moment for forces directed at will.

Since the late 16th century, statics was no longer developed by architects and other practitioners, but by mathematicians and physicists. Simon Stevin was already using the parallelogram of forces and the virtual displacement to dissolve forces before 1600 . In 1669, Gilles Personne de Roberval built a scale with a parallelogram linkage that was always in balance. At the beginning of the 18th century, Pierre de Varignon traced the lever law, the parallelogram of forces and the principle of "virtual speed" formulated by Bernoulli back to the equilibrium conditions, which can also be derived from Newton's system, as demonstrated by Leonhard Euler in 1775. The kinematic and the geometric direction of the statics were summarized in 1788 by Joseph-Louis Lagrange in a synthesis based on the principle of virtual displacements. With his theory of the pairs of forces in 1803 Louis Poinsot made a further advance in the formulation of the statics of rigid bodies.

Commons : Statik  - collection of images, videos and audio files
Wikibooks: Stereostatics  - learning and teaching materials

## literature

• Bruno Assmann: Technical Mechanics 1 - Statics. 19th edition. Oldenbourg, Munich 2010, ISBN 978-3-486-59133-0 .
• Dietmar Gross, Werner Hauger, Jörg Schröder , Wolfgang Wall: Technical Mechanics 1 - Statics. 11th edition. Springer, Berlin / Heidelberg 2011, ISBN 978-3-642-13805-8 .
• Rolf Mahnken: Textbook of technical mechanics - statics. Springer, Berlin / Heidelberg 2012, ISBN 978-3-642-21711-1 .
• Martin Mayr: Technical mechanics: statics, kinematics, kinetics, vibrations, strength theory. 6th, revised edition. Hanser, Munich / Vienna 2008, ISBN 978-3-446-41690-1 .
• Hans Albert Richard, Manuela Sander: Technical mechanics - statics. 5th edition. Springer-Vieweg, Wiesbaden 2016, ISBN 978-3-658-14906-2 .
• Mahir Sayir, Jürg Dual, Stephan Kaufmann, Edoardo Mazza: Engineering Mechanics 1 - Statics. 3rd, revised edition. Springer-Vieweg, Wiesbaden 2015, ISBN 978-3-658-10046-9 .
• Christian Spura: Technical Mechanics 1 - Stereostatics. Springer-Vieweg, Wiesbaden 2016, ISBN 978-3-658-14985-7 .

### University publications

• Helmut Weyhmann: Development and methodical use of practice-oriented visual and motivational aids for the degree course in mechanical engineering at universities, colleges and technical schools for the section "Statics of rigid bodies". Dissertation. TU Dresden, 1992. DNB 930936442 .

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 This version was added to the list of articles worth reading on January 31, 2017 .