**Betti’s Theorem**, also known as the Maxwell-Betti Reciprocal Work Theorem, is a fundamental principle in structural analysis. *Betti’s theorem of reciprocal works states that in any elastic system, the work performed by a load of state 1 along displacement caused by a load of state 2 equals the work performed by a load of state 2 along displacement caused by a load of state 1*.

In other words, it states that for a linear elastic structure subjected to two sets of forces, the work done by the first set of forces in acting through the displacements produced by the second set of loads is equal to the work done by the second set of loads in acting through the displacements produced by the first set.

Reciprocal theorems reflect the fundamental properties of any linear statistically determinate or indeterminate elastic systems. These theorems find extensive application in the analysis of redundant structures.

**Proof of Betti’s theorem**

Let us consider an elastic structure subjected to loads P_{1} and P_{2} separately; let us call it the first and second states (Figure 1). Set of displacements Δ_{mn} for each state are shown below. The first index *m* indicates the direction of the displacement and the second index *n* denotes the load, which causes this displacement.

Thus, Δ_{11} and Δ_{12} are displacements in the direction of load P_{1} due to load P_{1} and P_{2}, respectively, Δ_{21} and Δ_{22} are displacements in the direction of load P_{2} due to load P_{1} and P_{2}, respectively.

Let us calculate the strain energy of the system by considering consequent applications of loads P_{1} and P_{2}, i.e., state 1 is additionally subjected to load P_{2}. The total work done by both of these loads consists of three parts:

- Work done by the force P
_{1}on the displacement Δ_{11}. Since load P_{1}is applied statically (from zero to P_{1}according to triangle law), then W_{1}= P_{1}Δ_{11}/2. - Work done by the force P
_{2}on the displacement Δ_{22}. Since load P_{2}is applied statically, then W_{2}= P_{2}Δ_{22}/2. - Work done by the force P
_{1 }on the displacement Δ_{12}; this displacement is caused by load P_{2}. The load P_{1}approached its maximum value P_{1}before the application of P_{2}. The corresponding P_{1}–Δ_{1}diagram is shown in Figure 1, so W_{3}= P_{1}Δ_{12}.

Since potential energy U equals to the total work, then;

U = ½P_{1}Δ_{11} + ½P_{2}Δ_{22} + P_{1}Δ_{12}

On the other hand, considering of application of load P_{2} first and then P_{1}, i.e., if state 2 is additionally subjected to load P_{1}, then potential energy *U* equals;

U = ½P_{2}Δ_{22} + ½P_{1}Δ_{11} + P_{2}Δ_{21}

Since strain energy does not depend on the order of loading, then the following fundamental relationship is obtained;

P_{1}Δ_{12} = P_{2}Δ_{21} or W_{12} = W_{21}

Work W_{12} can be positive or negative. It is only zero if and only if the displacement of the point of application of force P_{1} produced by force P_{2} is zero or perpendicular to the direction of P_{1}.

**A Simple Analogy**

Imagine two people pushing against each other. Person A pushes person B with a certain force, resulting in a displacement of person B. Simultaneously, person B pushes person A with an equal and opposite force, causing a displacement of person A. Betti’s Theorem states that the work done by person A on person B is equal to the work done by person B on person A.

**Implications and Applications**

At its core, Betti’s Theorem establishes a reciprocal relationship between loads and displacements in a linear elastic system. This principle has far-reaching implications in structural engineering:

**Influence Lines:**It is instrumental in constructing influence lines, which are essential for analyzing indeterminate structures under moving loads.**Boundary Element Method:**This numerical method, widely used in engineering, is based on Betti’s Theorem.**Analysis of statically indeterminate structures using force method:**It simplifies the calculation of influence coefficients when using the force method to analyse statically indeterminate structures.**Structural Optimization:**It contributes to the design of compliant mechanisms through topology optimization techniques.

**Limitations**

It is important to remember that Betti’s Theorem is applicable only to linear elastic structures. This means that the material of the structure must obey Hooke’s law, and the deformations must be small. In conclusion, Betti’s Theorem is a powerful tool for engineers and scientists, providing a foundation for understanding and analyzing the behaviour of structures under various loading conditions.