The method of consistent deformations (or method of compatibility) is a structural analysis technique used to solve statically indeterminate structures.

The method ensures equilibrium and compatibility of deformations by introducing unknown redundant forces or moments and solving for them using compatibility conditions.

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Steps in the Method of Consistent Deformations

1. Identify Redundants:
   Choose a set of redundant forces/moments to make the structure statically determinate by removing these redundants.
2. Analyse the Basic Determinate Structure:
   Solve the statically determinate structure under applied loads. This includes calculating deflections, slopes, or rotations caused by the loads.
3. Apply Compatibility Conditions:
   Write equations ensuring that the deformations at points where redundants were removed match those in the original structure (compatibility condition).
4. Account for Effects of Redundants:
   Include the deflections caused by the redundant forces/moments and solve the system of equations for redundants.

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Illustrative Example: Continuous Beam

Consider a continuous beam with a single redundant, such as a fixed-end support at BB. The steps involve:

1. Basic Determinate Structure:
   Convert the continuous beam to a simply supported beam by removing the redundant reaction at BB, denoted as .
2. Deflection of Basic Structure:
   Use beam theory (e.g., moment-area method or superposition) to calculate the deflection at due to the applied loads .
3. Effect of Redundant Force:
   Calculate the deflection at due to a unit redundant force .
4. Compatibility Equation:
   At point , the total deflection must be zero in the original structure:
5. Solve for Redundant Force:
   Rearrange the equation to find :
6. Final Analysis:
   Include in the analysis and calculate other internal forces and moments.

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Equations

For clarity, let us expand the equations:

1. Deflection due to Loads:
   (for a simply supported beam under a central point load ).
2. Deflection due to Redundant:
   (for a unit load applied at ).
3. Compatibility Condition:
   Substitute and into:
4. Solve for :
   

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This method can be generalised for multi-redundant systems by using matrix methods or superposition, where compatibility conditions lead to a system of simultaneous equations.