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Principle of Reciprocal Deflections explained

The Principle of Reciprocal Deflections is a fundamental concept in structural mechanics, often used in analysing deflections and verifying solutions in beam and structural problems.

It states:

If a unit load is applied at point A and the resulting deflection is measured at point B , the deflection at point B due to a unit load applied at A is the same as the deflection at point A due to a unit load applied at B .

This principle is based on the symmetry of the stiffness matrix in linear elastic systems.

Mathematical Representation

Let:

  • \delta_{AB} : Deflection at point B due to a unit load at A ,
  • \delta_{BA} : Deflection at point A due to a unit load at B .

The principle states: \delta_{AB} = \delta_{BA}

This relationship is valid for linear elastic systems that obey Hooke’s Law.

Illustrative Equations

  1. Work and Energy Argument:The principle can be derived from the work-energy theorem. The external work done by a force is stored as strain energy in the structure. For a structure under two forces P_A and P_B applied at points A and B , respectively: U = \frac{1}{2} P_A \delta_{AA} + \frac{1}{2} P_B \delta_{BB} + P_A \delta_{AB} Here:
    • \delta_{AA} : Deflection at A due to P_A ,
    • \delta_{BB} : Deflection at B due to P_B ,
    • \delta_{AB} : Deflection at B due to P_A .
    Interchange P_A and P_B , and equate terms to show \delta_{AB} = \delta_{BA} .
  2. Direct Integration for Beams:Consider a beam of flexural rigidity EI under distributed or point loads. The deflection v(x) at any point can be obtained by integrating the bending moment M(x) : \frac{d^2 v}{dx^2} = \frac{M(x)}{EI} Using the principle, for point loads at A and B , you can verify: \delta_{AB} = \int \frac{M_A(x) M_B(x)}{EI} , dx where M_A(x) and M_B(x) are the moment distributions due to unit loads at A and B , respectively.
  3. Matrix Formulation:In matrix structural analysis, the flexibility matrix [f] relates forces {P} to displacements {\delta} : {\delta} = [f] {P} The flexibility matrix [f] is symmetric, implying f_{ij} = f_{ji} , which corresponds to \delta_{AB} = \delta_{BA} .
  4. Example for Simply Supported Beam:For a simply supported beam of span L under a unit load at A (distance a from the left support), the deflection at B (distance b from the left support) is: \delta_{AB} = \frac{a b (L^2 - a^2 - b^2)}{6EI} If the load is placed at B , the deflection at A is: \delta_{BA} = \frac{a b (L^2 - a^2 - b^2)}{6EI} Clearly, \delta_{AB} = \delta_{BA} .

Applications

  1. Verification of Deflection Solutions: In structural analysis, reciprocal deflections provide a useful check for numerical or analytical solutions.
  2. Influence Line Analysis: The principle simplifies the computation of influence lines for deflections and internal forces in beams and trusses.
  3. Flexibility Method: In the flexibility (force) method of structural analysis, reciprocal relationships are used to relate displacements and forces.

This principle is especially powerful in symmetric systems and linear elastic analyses, where it ensures consistency between computed deflections at different points.

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