Flexibility method
In structural engineering, the flexibility method, also called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the members' flexibility matrices also has the name the matrix force method due to its use of member forces as the primary unknowns.
Flexibility is the inverse of stiffness. For example, consider a spring that has Q and q as, respectively, its force and deformation:
A typical member flexibility relation has the following general form:
where
For a system composed of many members interconnected at points called nodes, the members' flexibility relations can be put together into a single matrix equation, dropping the superscript m:
where M is the total number of members' characteristic deformations or forces in the system.
Unlike the matrix stiffness method, where the members' stiffness relations can be readily integrated via nodal equilibrium and compatibility conditions, the present flexibility form of equation (2) poses serious difficulty. With member forces as the primary unknowns, the number of nodal equilibrium equations is insufficient for solution, in general—unless the system is statically determinate.
To resolve this difficulty, first we make use of the nodal equilibrium equations in order to reduce the number of independent unknown member forces. The nodal equilibrium equation for the system has the form:
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