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

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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:

$\backslash mathbf\{q\}^m\; =\; \backslash mathbf\{f\}^m\; \backslash mathbf\{Q\}^m\; +\; \backslash mathbf\{q\}^\{om\}\; \backslash qquad\; \backslash qquad\; \backslash qquad\; \backslash mathrm\{(1)\}$

where

m = member number m.

$\backslash mathbf\{q\}^m$ = vector of member's characteristic deformations.

$\backslash mathbf\{f\}^m$ = member flexibility matrix which characterises the member's susceptibility to deform under forces.

$\backslash mathbf\{Q\}^m$ = 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.

$\backslash mathbf\{q\}^\{om\}$ = 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 $\backslash mathbf\{Q\}^m\; =\; 0$).

For a system composed of many members interconnected at points called nodes, the members' flexibility relations can be put together into one big matrix equation, dropping the superscript m:

$\backslash mathbf\{q\}\_\{M\; \backslash times\; 1\}\; =\; \backslash mathbf\{f\}\_\{M\; \backslash times\; M\}\; \backslash mathbf\{Q\}\_\{M\; \backslash times\; 1\}\; +\; \backslash mathbf\{q\}^\{o\}\_\{M\; \backslash times\; 1\}\; \backslash qquad\; \backslash qquad\; \backslash qquad\; \backslash mathrm\{(2)\}$

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 Eq.(2) poses serious difficulty. With member forces $\backslash mathbf\{Q\}\_\{M\; \backslash times\; 1\}$ as the primary unknowns, the number of nodal equilibrium equations is insufficient for solution, in general--unless the system is statically determinate.

Nodal equilibrium equations

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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:

$\backslash mathbf\{R\}\_\{N\; \backslash times\; 1\}\; =\; \backslash mathbf\{b\}\_\{N\; \backslash times\; M\}\; \backslash mathbf\{Q\}\_\{M\; \backslash times\; 1\}\; +\; \backslash mathbf\{W\}\_\{N\; \backslash times\; 1\}\; \backslash qquad\; \backslash qquad\; \backslash qquad\; \backslash mathrm\{(3)\}$

where

$\backslash mathbf\{R\}\_\{N\; \backslash times\; 1\}$: Vector of nodal forces at all N degrees of freedom of the system.

$\backslash mathbf\{b\}\_\{N\; \backslash times\; M\}$: The resulting nodal equilibrium matrix

$\backslash mathbf\{W\}\_\{N\; \backslash times\; 1\}$: The vector of forces arising from loading on the members.

In the case of determinate systems, matrix b is square and the solution for Q can be found immediately from (3) provided that the system is stable.

The primary system

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For indeterminate systems, M > N, and hence, we can augment (3) with I = M-N equations of the form:

$X\_i\; =\; \backslash alpha\; Q\_j\; +\; \backslash beta\; Q\_k\; +\; .\; .\; .\; \backslash qquad\; i=1,2,\; ...\; I\backslash qquad\; \backslash qquad\; \backslash mathrm\{(4)\}$

The vector X is the so-called vector of redundant forces and I is the degree of statical indeterminacy of the system. We usually choose j, k, ... , $\backslash alpha$, and $\backslash beta$ such that $X\_i$ is a support reaction or an internal member-end force. With suitable choices of redundant forces, the equation system (3) augmented by (4) can now be solved to obtain:

$\backslash mathbf\{Q\}\_\{M\; \backslash times\; 1\}\; =\; \backslash mathbf\{B\}\_R\; \backslash mathbf\{R\}\_\{N\; \backslash times\; 1\}\; +\; \backslash mathbf\{B\}\_X\; \backslash mathbf\{X\}\_\{I\; \backslash times\; 1\}\; +\; \backslash mathbf\{Q\}\_\{v\; \backslash cdot\; M\; \backslash times\; 1\}\; \backslash qquad\; \backslash qquad\; \backslash qquad\; \backslash mathrm\{(5)\}$

Substitution into (2) gives:

$\backslash mathbf\{q\}\_\{M\; \backslash times\; 1\}\; =\; \backslash mathbf\{f\}\_\{M\; \backslash times\; M\}$

\Big( \mathbf{B}_R \mathbf{R}_{N \times 1} + \mathbf{B}_X \mathbf{X}_{I \times 1} + \mathbf{Q}_{v \cdot M \times 1} \Big)

+ \mathbf{q}^{o}_{M \times 1} \qquad \qquad \qquad \mathrm{(6)}

Eqs.(5) and (6) are the solution for the primary system which is the original system that has been rendered statically determinate by cuts that expose the redundant forces $\backslash mathbf\{X\}$. Eq.(5) effectively reduces the set of unknown forces to $\backslash mathbf\{X\}$.

Compatibility equation and solution

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Next, we need to set up $I$ compatibility equations in order to find $\backslash mathbf\{X\}$. The compatibility equations restore the required continuity at the cut sections by setting the relative displacements $\backslash mathbf\{r\}\_\{X\}$ at the redundants X to zero. That is, using the unit dummy force method:

$\backslash mathbf\{r\}\_\{X\}\; =\; \backslash mathbf\{B\}\_X^T\; \backslash mathbf\{q\}\; =\; \backslash mathbf\{B\}\_X^T\; \backslash Big[\; \backslash mathbf\{f\}$

\Big( \mathbf{B}_R \mathbf{R} + \mathbf{B}_X \mathbf{X} + \mathbf{Q}_v \Big) + \mathbf{q}^{o} \Big] = 0 \qquad \qquad \qquad \mathrm{(7a)}

or $\backslash mathbf\{r\}\_\{X\}\; =\; \backslash mathbf\{F\}\_\{XX\}\; \backslash mathbf\{X\}\; +\; \backslash mathbf\{r\}^o\_X\; =\; 0\; \backslash qquad\; \backslash qquad\; \backslash qquad\; \backslash mathrm\{(7b)\}$

where

$\backslash mathbf\{F\}\_\{XX\}\; =\; \backslash mathbf\{B\}\_X^T\; \backslash mathbf\{f\}\; \backslash mathbf\{B\}\_X$

$\backslash mathbf\{r\}^o\_X\; =\; \backslash mathbf\{B\}\_X^T\; \backslash Big[\; \backslash mathbf\{f\}$

\Big( \mathbf{B}_R \mathbf{R} + \mathbf{Q}_v \Big) + \mathbf{q}^{o} \Big] Eq.(7b) can be solved for X, and the member forces are next found from (5) while the nodal displacements can be found by

$\backslash mathbf\{r\}\_\{R\}\; =\; \backslash mathbf\{B\}\_R^T\; \backslash mathbf\{q\}\; =\; \backslash mathbf\{F\}\_\{RR\}\; \backslash mathbf\{R\}\; +\; \backslash mathbf\{r\}^o\_R$

where

$\backslash mathbf\{F\}\_\{RR\}\; =\; \backslash mathbf\{B\}\_R^T\; \backslash mathbf\{f\}\; \backslash mathbf\{B\}\_R$ is the system flexibility matrix.

$\backslash mathbf\{r\}^o\_R\; =\; \backslash mathbf\{B\}\_R^T\; \backslash Big[\; \backslash mathbf\{f\}$

\Big( \mathbf{B}_X \mathbf{X} + \mathbf{Q}_v \Big) + \mathbf{q}^{o} \Big]

Supports' movements taking place at the redundants can be included in the righ-hand-side of Eq.(7), while supports' movements at other places must be included in $\backslash mathbf\{r\}^o\_X$ and $\backslash mathbf\{r\}^o\_R$ as well. For educational purposes, the above computations are best done using the computer with a suitable software package such as MATLAB or the freeware CMAP

Advantages and disadvantages

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While the choice of redundants in (4) appears to be arbitrary and troublesome for automatic computation, this objection can be overcome by proceeding from (3) directly to (5) using a modified Gauss-Jordan elimination process. This is a robust procedure that automatically selects a good set of redundant forces to ensure numerical stability.

It is apparent from the above process that the matrix stiffness method is easier to comprehend and to implement for automatic computation. It is also easier to extend for advanced applications such as non-linear analysis, stability, vibrations, etc.. For these reasons, the matrix stiffness method is the method of choice for use in general purpose structural analysis software packages. On the other hand, for linear systems with low degree of statical indeterminacy, the flexibility method has the advantage of being computationally less intensive. This advantage, however, is a moot point as personal computers are widely available and more powerful. The main redeeming factor in learning this method nowadays is its educational value in imparting the concepts of equilibrium and compatibility in addition to its historical value. In contrast, the procedure of the direct stiffness method is so mechanical that it risks being used without much understanding of the structural behaviours.

See also

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• Matrix stiffness method
• Finite element method
• Finite element method in structural mechanics
• Structural analysis

Structural engineering | Civil engineering