Let be a linear elastic body #Linear_Elasticity "Elasticity (solid mechanics)") on which two sets of forces act and applied to the points of the solid and , respectively. Sean:.

Similarly,

The work done by the loads does not depend on the order of their application, so it is enough to consider two cases:

As said at the beginning, the deformation energy does not depend on the order of application of the loads, so by equating the expressions obtained in the two cases considered:

This equality is what gives rise to the Maxwell-Betti Theorem, which can be stated as follows:

The main consequence of this result is that the reciprocal influence coefficients are equal. In fact, let's assume that both . Then the meaning of coincides with the definition of the influence coefficient, while it corresponds to the influence coefficient reciprocal to the previous one, that is, , and by the expression that defines the Maxwell-Betti Theorem, we have that:.

This theorem also applies in the case that the actions applied on the elastic solid are not forces but moments (or even forces and moments) "Elasticity (solid mechanics)"), in which case the displacements will be replaced by the corresponding angle of rotation.

Let be a deformable body whose behavior is linear-elastic. Two forces act on it: on any point and on a point, each of which generates a field of displacements that, due to the linearity of the problem, are mutually independent. Consider then that the joint effect on the deformable body is the sum of two systems:

Note that of the displacements at a point, only their value in the direction of the load applied at that point is considered. Likewise, it is noted that the notation corresponds to a displacement at the point due to the load applied at the point.

Due to the linearity of the problem, the principle of superposition is applicable. Therefore, the total work done by first applying system I and then system II (case 1) must be the same as if they are applied the other way around (case 2).

When system I is applied, the force does work when moving, which, taking into account that there is proportionality between displacements and forces, is the following:

When system II is applied, the force does work when moving, but it also does another work when moving a quantity. In the latter, it must also be considered that it acts at all times with its maximum value, so the coefficient of is not incorporated. With this, the work generated when applying system II is:

So the total work is the sum of both and is worth:

When system II is applied, the force does work when moving, which is as follows:

When system I is applied, the force does work when moving, but it also does another work when moving a quantity. With this, the work generated when applying system I is:

Adding the above, the total work in this new case is worth:

As has been said, the linearity of the problem requires that the total work of case 1 be equal to the total of case 2. Equating both expressions and simplifying:.

Which is a simplified case of the Maxwell-Betti theorem already presented, when the sets of charges and are each made up of a single charge. The demonstration of the general case can thus be approached in a simple way by simple induction based on what is explained in this section.

Maxwell's theorem

Prior to the Maxwell-Betti theorem, in 1864, James Clerk Maxwell published a theorem called "Method of Distortions or Reciprocal Displacements", to which the works of other scientists such as Mohr and Clapeyron contributed significantly. This theorem is considered to give rise to the first method of analysis for statically indeterminate structures. It is stated as follows:

As we see, this theorem is nothing more than a particular case of the general one, expressed through the Maxwell-Betti Theorem, since here only the performance of a force is considered and not a set of them.

Stiffness matrix symmetry

The Maxwell-Betti theorem applied to a linear structure characterizable by a stiffness matrix implies that the stiffness matrix must be symmetric. Since if it were not, the final elastic state would not be unique and would depend on the order and mode of application of the loads, which contradicts the linear character of the structure.

Let be a linear elastic body #Linear_Elasticity "Elasticity (solid mechanics)") on which two sets of forces act and applied to the points of the solid and , respectively. Sean:.

Similarly,

The work done by the loads does not depend on the order of their application, so it is enough to consider two cases:

As said at the beginning, the deformation energy does not depend on the order of application of the loads, so by equating the expressions obtained in the two cases considered:

This equality is what gives rise to the Maxwell-Betti Theorem, which can be stated as follows:

The main consequence of this result is that the reciprocal influence coefficients are equal. In fact, let's assume that both . Then the meaning of coincides with the definition of the influence coefficient, while it corresponds to the influence coefficient reciprocal to the previous one, that is, , and by the expression that defines the Maxwell-Betti Theorem, we have that:.

This theorem also applies in the case that the actions applied on the elastic solid are not forces but moments (or even forces and moments) "Elasticity (solid mechanics)"), in which case the displacements will be replaced by the corresponding angle of rotation.

Let be a deformable body whose behavior is linear-elastic. Two forces act on it: on any point and on a point, each of which generates a field of displacements that, due to the linearity of the problem, are mutually independent. Consider then that the joint effect on the deformable body is the sum of two systems:

Note that of the displacements at a point, only their value in the direction of the load applied at that point is considered. Likewise, it is noted that the notation corresponds to a displacement at the point due to the load applied at the point.

Due to the linearity of the problem, the principle of superposition is applicable. Therefore, the total work done by first applying system I and then system II (case 1) must be the same as if they are applied the other way around (case 2).

When system I is applied, the force does work when moving, which, taking into account that there is proportionality between displacements and forces, is the following:

When system II is applied, the force does work when moving, but it also does another work when moving a quantity. In the latter, it must also be considered that it acts at all times with its maximum value, so the coefficient of is not incorporated. With this, the work generated when applying system II is:

So the total work is the sum of both and is worth:

When system II is applied, the force does work when moving, which is as follows:

When system I is applied, the force does work when moving, but it also does another work when moving a quantity. With this, the work generated when applying system I is:

Adding the above, the total work in this new case is worth:

As has been said, the linearity of the problem requires that the total work of case 1 be equal to the total of case 2. Equating both expressions and simplifying:.

Which is a simplified case of the Maxwell-Betti theorem already presented, when the sets of charges and are each made up of a single charge. The demonstration of the general case can thus be approached in a simple way by simple induction based on what is explained in this section.

Maxwell's theorem

Prior to the Maxwell-Betti theorem, in 1864, James Clerk Maxwell published a theorem called "Method of Distortions or Reciprocal Displacements", to which the works of other scientists such as Mohr and Clapeyron contributed significantly. This theorem is considered to give rise to the first method of analysis for statically indeterminate structures. It is stated as follows:

As we see, this theorem is nothing more than a particular case of the general one, expressed through the Maxwell-Betti Theorem, since here only the performance of a force is considered and not a set of them.

Stiffness matrix symmetry

The Maxwell-Betti theorem applied to a linear structure characterizable by a stiffness matrix implies that the stiffness matrix must be symmetric. Since if it were not, the final elastic state would not be unique and would depend on the order and mode of application of the loads, which contradicts the linear character of the structure.