Archimedes (c. 287–c. 212 BC) did pioneering work in statics.[3]​[4]​

Later developments in the field of statics are found in works by Thábit (826-901).[5]​.

Until the beginning of the 19th century, Archimedes' law of the lever and the parallelogram of forces dominated the development of statics. The first correct description of the statics of weights on an inclined plane comes from Jordanus Nemorarius around 1250. In his description of the lever, he uses the principle of virtual work for the first time.

In the Renaissance, development was mainly based on observation and experimentation to solve problems in the construction of machines and structures (for example, Leonardo da Vinci's construction of an arch bridge without connecting elements). His and Galileo's thought experiments on the theory of force led to the separation of elastostatics from the statics of rigid bodies. Galileo also specified the Archimedean concept of force and used the term moment&action=edit&redlink=1 "Moment (technical mechanics) (not yet drafted)") for arbitrarily directed forces.

Starting at the end of the century, statics was no longer developed by architects and other professionals, but by mathematicians and physicists. Before 1600, Simon Stevin was already using the parallelogram of forces and virtual displacement to solve for forces. In 1669, Gilles Personne de Roberval built a balance with a parallelogram that was always in balance. At the beginning of the century, Pierre de Varignon introduced the law of levers, the parallelogram of forces and the parallelogram of forces developed by Bernoulli's principle of "virtual velocity" [7] to the equilibrium conditions that can also be derived from Newton's system, which Leonhard Euler demonstrated in 1775. The kinematic and geometric directions of statics were summarized in 1788 by Joseph-Louis Lagrange in a synthesis on the principle of displacements Virtuals. Louis Poinsot further advanced the formulation of the statics of rigid bodies with his theory of force couples in 1803.[8]​.

Statics provides, through the use of rigid solid mechanics, a solution to the so-called isostatic problems. In these problems, it is sufficient to state the basic equilibrium conditions, which are:

These two conditions, through linear algebra, are converted into a system of equations; The resolution of this system of equations is the solution of the equilibrium condition.

There are methods for solving this type of static problems using graphics, inherited from the times when the complexity of solving systems of equations was avoided through geometry, although currently there is a tendency towards computer calculation.

To solve hyperstatic problems (those in which equilibrium can be achieved with different combinations of efforts) it is necessary to consider compatibility equations. These additional compatibility equations are obtained by introducing deformations and internal stresses associated with deformations using the methods of deformable solid mechanics, which is an extension of rigid solid mechanics that also accounts for the deformability of solids and their internal effects.

There are several classical methods based on the mechanics of deformable solids, such as Castigliano's theorems or Navier-Bresse formulas.

Un sistema de fuerzas o grupo de fuerzas es una serie de fuerzas que actúan en un sistema: por ejemplo, todas las fuerzas que actúan sobre un puente, todas las fuerzas que actúan sobre un vehículo o sólo las que actúan sobre la transmisión. En el caso de los sistemas de corte libre, las fuerzas incluyen también las fuerzas de corte. Los sistemas de fuerzas permiten varias operaciones. Entre ellas, la combinación de varias fuerzas en una resultante y la determinación de fuerzas desconocidas mediante las condiciones de equilibrio. Esto permite comprobar si dos sistemas de fuerzas diferentes equivalencia estática son equivalentes, es decir, tienen el mismo efecto sobre un cuerpo. También es posible comprobar si un sistema de fuerzas está en equilibrio. Suponiendo que esté en equilibrio, se pueden calcular las fuerzas desconocidas.

Los sistemas de fuerzas se clasifican según dos criterios diferentes. Según el número de dimensiones, se distingue entre sistemas de fuerzas planos y espaciales. Según la aparición de momentos, se distingue entre sistemas de fuerzas centrales (sin momentos), en los que las líneas de acción de todas las fuerzas se cruzan en un único punto, y sistemas de fuerzas generales (con momentos), en los que las fuerzas no se cruzan en un único punto.[9]​.

Combination and division of forces

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

Forces whose lines of action intersect at a common point can also be summarized using the parallelogram of forces. To do this, they first move along their lines of action to the point of intersection and summarize there. Decomposition works the same way.

If the lines of action do not intersect at a single point, the forces can be summarized by moving them to a single point. Displacement parallel to another line of action gives rise to a "displacement moment" that must be taken into account. The system formed by the resultant force and the total moment is called a dynamo. The moment can be eliminated by parallel displacement of the resultant force. The magnitude, direction and line of action of the resultant force are then fixed.

A special case is the force pair. It cannot be summarized in a resultant force, but it can be replaced by its moment (without resultant force).[10]​.

Balance

A body is in equilibrium when the resultant force and the resultant moment about any point are both zero. In a planar system of forces spanning exactly one horizontal direction and exactly one vertical direction, this means:.

In the spatial force system, there is a force balance and a moment balance for each dimension. The balance of forces applies in any direction.

These equilibrium conditions can be used to check whether a series of known forces are in equilibrium. If a body is known not to move and only some of the forces are known, equilibrium conditions can be used to calculate the unknown forces. Since only three independent equations can be established 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, the result is six equations and unknowns for a single body.

If more forces are unknown, more equations are required, which then contain deformations and material parameters. These are the subject of structural analysis and material resistance. For several bodies that are connected to form a larger body (e.g., individual parts that are joined together to form assemblies and modules), a corresponding number of unknowns can be calculated for each rigid body (three in the plane).[11]​.

When several forces act on a body or mechanics of a rigid solid/rigid solid and are applied at the same point, the calculation of the resulting force is trivial: it is enough to add them vector-wise and apply the resulting vector to the common point of application.

However, when there are forces with different points of application, it is necessary to determine the point of application of the resulting force. For non-parallel forces this can be done by adding the forces two by two. To do this, two of the forces that draw straight lines are considered, prolonging the forces in both directions and looking for their intersection. That intersection will be a point of passage for the combined force of the two. The two forces are then replaced by a single vector force sum of the two previous ones applied at the point of intersection. This is repeated n-1 times for a system of n forces and the passing point of the resultant is obtained. In the limiting case in which there are n parallel forces, the funicular polygon can be used to find the point of passage of the resultant.

Por esta cuestión es que la estática resulta ser una materia indispensable en carreras y trabajos como los que llevan a cabo la ingeniería estructural, mecánica y de construcción, ya que siempre que se quiera construir una estructura fija, como ser, un edificio, en términos un poco más extendidos, los pilares de un rascacielos, o la viga de un puente, será necesario e indiscutible su participación y estudio para garantizar la seguridad de aquellos que luego transiten por las mencionadas estructuras.

La estática abarca el estudio del equilibrio tanto del conjunto como de sus partes constituyentes, incluyendo las porciones elementales de material.

Uno de los principales objetivos de la estática es la obtención de esfuerzos cortantes, fuerza normal, de torsión y momento flector a lo largo de una pieza, que puede ser desde una viga de un puente o los pilares de un rascacielos.

Su importancia reside en que una vez trazados los diagramas y obtenidas sus ecuaciones, se puede decidir el material con el que se construirá, las dimensiones que deberá tener, límites para un uso seguro, etcétera, mediante un análisis de materiales. Por tanto, resulta de aplicación en ingeniería estructural, ingeniería mecánica, construcción, siempre que se quiera construir una estructura fija. Para el análisis de una estructura en movimiento es necesario considerar la aceleración de las partes y las fuerzas resultantes.

El estudio de la Estática suele ser el primero dentro del área de la ingeniería mecánica, debido a que los procedimientos que se realizan suelen usarse a lo largo de los demás cursos de ingeniería mecánica.

Solids and structural analysis

Statics is used in the analysis of structures, for example in architecture and structural engineering and civil engineering. The strength of materials is a related field of mechanics that depends largely on the application of static equilibrium. A key concept is the center of gravity of a body at rest, which constitutes an imaginary point at which all of a body's mass resides. The position of the point relative to the foundations on which a body lies determines its stability to small movements. If the center of gravity is located outside the bases, then the body is unstable because there is a couple acting: any small disturbance will cause the body to fall. If the center of gravity falls within the bases, the body is stable, since it does not act on the body's net torque. If the center of gravity coincides with the foundations, then the body is said to be metastable.

In order to know the internal force or mechanical stress that some parts of a resistant structure are supporting, two means of calculation can frequently be used:

To obtain either of these two verifications, the sum of external forces in the structure (forces in x and y) must be taken into account, and then begin with the verification by nodes or by section. Although in practice it is not always possible to analyze a resistant structure exclusively through the equations of statics, and in those cases more general methods of resistance of materials, theory of elasticity "Elasticity (solid mechanics)"), mechanics of deformable solids and numerical techniques must be used to solve the equations to which these methods lead, such as the popular finite element method can be used in pulleys.

Archimedes (c. 287–c. 212 BC) did pioneering work in statics.[3]​[4]​

Later developments in the field of statics are found in works by Thábit (826-901).[5]​.

Until the beginning of the 19th century, Archimedes' law of the lever and the parallelogram of forces dominated the development of statics. The first correct description of the statics of weights on an inclined plane comes from Jordanus Nemorarius around 1250. In his description of the lever, he uses the principle of virtual work for the first time.

In the Renaissance, development was mainly based on observation and experimentation to solve problems in the construction of machines and structures (for example, Leonardo da Vinci's construction of an arch bridge without connecting elements). His and Galileo's thought experiments on the theory of force led to the separation of elastostatics from the statics of rigid bodies. Galileo also specified the Archimedean concept of force and used the term moment&action=edit&redlink=1 "Moment (technical mechanics) (not yet drafted)") for arbitrarily directed forces.

Starting at the end of the century, statics was no longer developed by architects and other professionals, but by mathematicians and physicists. Before 1600, Simon Stevin was already using the parallelogram of forces and virtual displacement to solve for forces. In 1669, Gilles Personne de Roberval built a balance with a parallelogram that was always in balance. At the beginning of the century, Pierre de Varignon introduced the law of levers, the parallelogram of forces and the parallelogram of forces developed by Bernoulli's principle of "virtual velocity" [7] to the equilibrium conditions that can also be derived from Newton's system, which Leonhard Euler demonstrated in 1775. The kinematic and geometric directions of statics were summarized in 1788 by Joseph-Louis Lagrange in a synthesis on the principle of displacements Virtuals. Louis Poinsot further advanced the formulation of the statics of rigid bodies with his theory of force couples in 1803.[8]​.

Statics provides, through the use of rigid solid mechanics, a solution to the so-called isostatic problems. In these problems, it is sufficient to state the basic equilibrium conditions, which are:

These two conditions, through linear algebra, are converted into a system of equations; The resolution of this system of equations is the solution of the equilibrium condition.

There are methods for solving this type of static problems using graphics, inherited from the times when the complexity of solving systems of equations was avoided through geometry, although currently there is a tendency towards computer calculation.

To solve hyperstatic problems (those in which equilibrium can be achieved with different combinations of efforts) it is necessary to consider compatibility equations. These additional compatibility equations are obtained by introducing deformations and internal stresses associated with deformations using the methods of deformable solid mechanics, which is an extension of rigid solid mechanics that also accounts for the deformability of solids and their internal effects.

There are several classical methods based on the mechanics of deformable solids, such as Castigliano's theorems or Navier-Bresse formulas.

Un sistema de fuerzas o grupo de fuerzas es una serie de fuerzas que actúan en un sistema: por ejemplo, todas las fuerzas que actúan sobre un puente, todas las fuerzas que actúan sobre un vehículo o sólo las que actúan sobre la transmisión. En el caso de los sistemas de corte libre, las fuerzas incluyen también las fuerzas de corte. Los sistemas de fuerzas permiten varias operaciones. Entre ellas, la combinación de varias fuerzas en una resultante y la determinación de fuerzas desconocidas mediante las condiciones de equilibrio. Esto permite comprobar si dos sistemas de fuerzas diferentes equivalencia estática son equivalentes, es decir, tienen el mismo efecto sobre un cuerpo. También es posible comprobar si un sistema de fuerzas está en equilibrio. Suponiendo que esté en equilibrio, se pueden calcular las fuerzas desconocidas.

Los sistemas de fuerzas se clasifican según dos criterios diferentes. Según el número de dimensiones, se distingue entre sistemas de fuerzas planos y espaciales. Según la aparición de momentos, se distingue entre sistemas de fuerzas centrales (sin momentos), en los que las líneas de acción de todas las fuerzas se cruzan en un único punto, y sistemas de fuerzas generales (con momentos), en los que las fuerzas no se cruzan en un único punto.[9]​.

Combination and division of forces

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

Forces whose lines of action intersect at a common point can also be summarized using the parallelogram of forces. To do this, they first move along their lines of action to the point of intersection and summarize there. Decomposition works the same way.

If the lines of action do not intersect at a single point, the forces can be summarized by moving them to a single point. Displacement parallel to another line of action gives rise to a "displacement moment" that must be taken into account. The system formed by the resultant force and the total moment is called a dynamo. The moment can be eliminated by parallel displacement of the resultant force. The magnitude, direction and line of action of the resultant force are then fixed.

A special case is the force pair. It cannot be summarized in a resultant force, but it can be replaced by its moment (without resultant force).[10]​.

Balance

A body is in equilibrium when the resultant force and the resultant moment about any point are both zero. In a planar system of forces spanning exactly one horizontal direction and exactly one vertical direction, this means:.

In the spatial force system, there is a force balance and a moment balance for each dimension. The balance of forces applies in any direction.

These equilibrium conditions can be used to check whether a series of known forces are in equilibrium. If a body is known not to move and only some of the forces are known, equilibrium conditions can be used to calculate the unknown forces. Since only three independent equations can be established 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, the result is six equations and unknowns for a single body.

If more forces are unknown, more equations are required, which then contain deformations and material parameters. These are the subject of structural analysis and material resistance. For several bodies that are connected to form a larger body (e.g., individual parts that are joined together to form assemblies and modules), a corresponding number of unknowns can be calculated for each rigid body (three in the plane).[11]​.

When several forces act on a body or mechanics of a rigid solid/rigid solid and are applied at the same point, the calculation of the resulting force is trivial: it is enough to add them vector-wise and apply the resulting vector to the common point of application.

However, when there are forces with different points of application, it is necessary to determine the point of application of the resulting force. For non-parallel forces this can be done by adding the forces two by two. To do this, two of the forces that draw straight lines are considered, prolonging the forces in both directions and looking for their intersection. That intersection will be a point of passage for the combined force of the two. The two forces are then replaced by a single vector force sum of the two previous ones applied at the point of intersection. This is repeated n-1 times for a system of n forces and the passing point of the resultant is obtained. In the limiting case in which there are n parallel forces, the funicular polygon can be used to find the point of passage of the resultant.

Por esta cuestión es que la estática resulta ser una materia indispensable en carreras y trabajos como los que llevan a cabo la ingeniería estructural, mecánica y de construcción, ya que siempre que se quiera construir una estructura fija, como ser, un edificio, en términos un poco más extendidos, los pilares de un rascacielos, o la viga de un puente, será necesario e indiscutible su participación y estudio para garantizar la seguridad de aquellos que luego transiten por las mencionadas estructuras.

La estática abarca el estudio del equilibrio tanto del conjunto como de sus partes constituyentes, incluyendo las porciones elementales de material.

Uno de los principales objetivos de la estática es la obtención de esfuerzos cortantes, fuerza normal, de torsión y momento flector a lo largo de una pieza, que puede ser desde una viga de un puente o los pilares de un rascacielos.

Su importancia reside en que una vez trazados los diagramas y obtenidas sus ecuaciones, se puede decidir el material con el que se construirá, las dimensiones que deberá tener, límites para un uso seguro, etcétera, mediante un análisis de materiales. Por tanto, resulta de aplicación en ingeniería estructural, ingeniería mecánica, construcción, siempre que se quiera construir una estructura fija. Para el análisis de una estructura en movimiento es necesario considerar la aceleración de las partes y las fuerzas resultantes.

El estudio de la Estática suele ser el primero dentro del área de la ingeniería mecánica, debido a que los procedimientos que se realizan suelen usarse a lo largo de los demás cursos de ingeniería mecánica.

Solids and structural analysis

Statics is used in the analysis of structures, for example in architecture and structural engineering and civil engineering. The strength of materials is a related field of mechanics that depends largely on the application of static equilibrium. A key concept is the center of gravity of a body at rest, which constitutes an imaginary point at which all of a body's mass resides. The position of the point relative to the foundations on which a body lies determines its stability to small movements. If the center of gravity is located outside the bases, then the body is unstable because there is a couple acting: any small disturbance will cause the body to fall. If the center of gravity falls within the bases, the body is stable, since it does not act on the body's net torque. If the center of gravity coincides with the foundations, then the body is said to be metastable.

In order to know the internal force or mechanical stress that some parts of a resistant structure are supporting, two means of calculation can frequently be used:

To obtain either of these two verifications, the sum of external forces in the structure (forces in x and y) must be taken into account, and then begin with the verification by nodes or by section. Although in practice it is not always possible to analyze a resistant structure exclusively through the equations of statics, and in those cases more general methods of resistance of materials, theory of elasticity "Elasticity (solid mechanics)"), mechanics of deformable solids and numerical techniques must be used to solve the equations to which these methods lead, such as the popular finite element method can be used in pulleys.