Betti's theorem
Encyclopedia
Betti's theorem, also known as Maxwell-Betti reciprocal work theorem, discovered by Enrico Betti
in 1872, states that for a linear elastic structure subject to two sets of forces {Pi} i=1,...,m and {Qj}, j=1,2,...,n, the work
done by the set P through the displacements produced by the set Q is equal to the work done by the set Q through the displacements produced by the set P. This theorem has applications in structural engineering
where it is used to define influence line
s and derive the boundary element method
.
Betti's theorem is used in the design of compliant mechanisms by topology optimization approach.
and 
. Consider that each force system causes a displacement fields, with the displacements measured at the external force's point of application referred to as 
and 
.
When the
force system is applied to the structure, the balance between the work performed by the external force system and the strain energy is:
The work-energy balance associated with the
force system is as follows:
Now, consider that with the
force system applied, the 
force system is applied subsequently. As the 
is already applied and therefore won't cause any extra displacement, the work-energy balance assumes the following expression:
Conversely, if we consider the
force system already applied and the 
external force system applied subsequently, the work-energy balance will assume the following expression:
If the work-energy balance for the cases where the external force systems are applied in isolation are respectively subtracted from the cases where the force systems are applied simultaneously, we arrive at the following equations:
If the solid body where the force systems are applied is formed by a linear elastic material
and if the force systems are such that only infinitesimal strains are observed in the body, then the body's constitutive equation
, which may follow Hooke's law
, can be expressed in the following manner:
Replacing this result in the previous set of equations leads us to the following result:
If we subtracting both equations then we obtain the following result:
on which two points have been defined: point 1 and point 2. First we apply a vertical force P at point 1 and measure the vertical displacement of point 2, denoted
. Next we remove force P and apply a vertical force Q at point 2, which produces the vertical displacement at point 1 of 
. Betti's reciprocity theorem states that:
Enrico Betti
-External links:...
in 1872, states that for a linear elastic structure subject to two sets of forces {Pi} i=1,...,m and {Qj}, j=1,2,...,n, the work
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...
done by the set P through the displacements produced by the set Q is equal to the work done by the set Q through the displacements produced by the set P. This theorem has applications in structural engineering
Structural engineering
Structural engineering is a field of engineering dealing with the analysis and design of structures that support or resist loads. Structural engineering is usually considered a specialty within civil engineering, but it can also be studied in its own right....
where it is used to define influence line
Influence line
In engineering, an influence line graphs the variation of a function at a specific point on a beam or truss caused by a unit load placed at any point along the structure...
s and derive the boundary element method
Boundary element method
The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations . It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture...
.
Betti's theorem is used in the design of compliant mechanisms by topology optimization approach.
Demonstration
Consider a solid body subjected to a pair of external force systems, referred to as and . Consider that each force system causes a displacement fields, with the displacements measured at the external force's point of application referred to as and .When the
force system is applied to the structure, the balance between the work performed by the external force system and the strain energy is:The work-energy balance associated with the
force system is as follows:Now, consider that with the
force system applied, the force system is applied subsequently. As the is already applied and therefore won't cause any extra displacement, the work-energy balance assumes the following expression:Conversely, if we consider the
force system already applied and the external force system applied subsequently, the work-energy balance will assume the following expression:If the work-energy balance for the cases where the external force systems are applied in isolation are respectively subtracted from the cases where the force systems are applied simultaneously, we arrive at the following equations:
If the solid body where the force systems are applied is formed by a linear elastic material
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
and if the force systems are such that only infinitesimal strains are observed in the body, then the body's constitutive equation
Constitutive equation
In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...
, which may follow Hooke's law
Hooke's law
In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law...
, can be expressed in the following manner:
Replacing this result in the previous set of equations leads us to the following result:
If we subtracting both equations then we obtain the following result:
Example
For a simple example let m=1 and n=1. Consider a horizontal beamBeam (structure)
A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.- Overview...
on which two points have been defined: point 1 and point 2. First we apply a vertical force P at point 1 and measure the vertical displacement of point 2, denoted
. Next we remove force P and apply a vertical force Q at point 2, which produces the vertical displacement at point 1 of . Betti's reciprocity theorem states that: