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: The spring stiffness relation is Q = k q where k is the spring stiffness. Its flexibility relation is q = f Q, where f is the spring flexibility. Hence, f = 1/k. A typical member flexibility relation has the following general form: where m = member number m. = vector of member's characteristic deformations. = member flexibility matrix which characterises the member's susceptibility to deform under forces. = vector of member's independent characteristic forces, which are unknown internal forces. These independent forces give rise to all member-end forces by member equilibrium. = vector of member's characteristic deformations caused by external effects (such as known forces and temperature changes) applied to the isolated, disconnected member (i.e. with ). 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 () 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.