10.2: Maxwell-Betti Law of Reciprocal Deflections

The Maxwell-Betti law of reciprocal deflections establishes the fact that the displacements at two points in an elastic structure subjected to a unit load successively at those points are the same in magnitude. This law helps reduce the computational efforts required to obtain the flexibility coefficients for the compatibility equations when analyzing indeterminate structures with several redundant restraints by force method. The Maxwell-Betti law of reciprocal deflection states that the linear displacement at point \(A\) due to a unit load applied at \(B\) is equal in magnitude to the linear displacement at point \(B\) due to a unit load applied at \(A\) for a stable elastic structure.

To prove the Maxwell-Betti law of reciprocal deflections, consider a beam subjected to the loads \(P_{1}\) and \(P_{2}\) at point 1 and point 2, successively, as shown in Figure 10.2a and Figure 10.2b.

\(Fig. 10.2\). Beam subjected to loads.

Case 1:

Apply \(P_{1}\), followed by \(P_{2}\).

Work done at point 1 when \(P_{1}\) is applied: \[W_{1}=\frac{1}{2} P_{1} \delta_{11}\]

where

\(\delta_{11}\) = the deflection at point 1 due to the gradually applied load \(P_{1}\).

Work done at points 1 and 2 when \(P_{2}\) is applied and \(P_{1}\) is still in place: \[W_{2}=P_{1} \delta_{12}+\frac{1}{2} P_{2} \delta_{22}\]

where

\(\delta_{12}\) and \(\delta_{22}\) = the deflections at point 1 and point 2, respectively, when the load \(P_{2}\) is gradually at point 2.

Total work done \(W_{T}\): \[\begin{array}{c}
W_{T}=W_{1}+W_{2} \\
=\frac{1}{2} P_{1} \delta_{11}+\frac{1}{2} P_{2} \delta_{22}+P_{1} \delta_{12}
\end{array}\]

Case 2:

Apply \(P_{2}\), followed by \(P_{1}\).

Work done at point 1 when \(P_{1}\) is applied: \[W_{2}=\frac{1}{2} P_{2} \delta_{22}\]

Work done at points 1 and 2 when \(P_{1}\) is applied and \(P_{2}\) is still in place: \[W_{2}=P_{2} \delta_{21}+\frac{1}{2} P_{1} \delta_{11}\]

Total work done \(W_{T}\): \[\begin{array}{c}
W_{T}=W_{1}+W_{2} \\
=\frac{1}{2} P_{1} \delta_{11}+\frac{1}{2} P_{2} \delta_{22}+P_{2} \delta_{21}
\end{array}\]

Equate the total of both cases (from equations 3 and 6).

\[\begin{array}{c}
\frac{1}{2} P_{1} \delta_{11}+\frac{1}{2} P_{2} \delta_{22}+P_{1} \delta_{12}=\frac{1}{2} P_{1} \delta_{11}+\frac{1}{2} P_{2} \delta_{22}+P_{2} \delta_{21} \\
P_{1} \delta_{12}=P_{2} \delta_{21}
\end{array}\]

Substituting \(P_{1}=P_{2}=1\) into equation 7 suggests the following: \[\delta_{12}=\delta_{21}\]

The Maxwell-Betti law is also applicable for reciprocal rotation. The theorem for reciprocal rotation states that the rotation at point \(B\) due to a unit couple moment applied at point \(A\) is equal in magnitude to the rotation at \(A\) due to a unit couple moment applied at point \(B\). This is expressed as follows: \[\alpha_{A B}=\alpha_{B A}\]

where \(a_{A B}\) is the rotation at a point \(A\) due to a unit couple moment applied at \(B\) and \(a_{B A}\) is the rotation at a point \(B\) due to a unit couple moment applied at \(A\).