Flexibility Method

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In structural engineering, the **flexibility method** is the *classical* consistent deformation 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.

## Member flexibility

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:

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- 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:

- <math>mathbf^m = mathbf^m mathbf^m + mathbf^ qquad qquad qquad mathrm</math>

*m*= member number*m*.- <math>mathbf^m </math> = vector of member's characteristic deformations.
- <math>mathbf^m </math> = member flexibility matrix which characterises the member's susceptibility to deform under forces.
- <math>mathbf^m </math> = 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.
- <math>mathbf^ </math> = 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....... ...

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