# Matrix stiffness method

Matrix stiffness method

In structural engineering, the matrix stiffness method (or simply "stiffness method", also known as Direct stiffness method) is a "matrix" method that makes use of the members' stiffness relations for computing member forces and displacements in structures. For example, if "k" is the stiffness of a spring that is subject to a force "Q", the spring's stiffness relation is:

:$Q = kq$

where "q" is the spring deformation. This relation gives "q = Q/k" as the resulting spring deformation.

Member stiffness relations

A typical member stiffness relation has the following general form:

:$mathbf\left\{Q\right\}^m = mathbf\left\{k\right\}^m mathbf\left\{q\right\}^m + mathbf\left\{Q\right\}^\left\{om\right\} qquad qquad qquad mathrm\left\{\left(1\right)\right\}$where:"m" = member number "m".:$mathbf\left\{Q\right\}^m$ = vector of member's characteristic forces, which are unknown internal forces.:$mathbf\left\{k\right\}^m$ = member stiffness matrix which characterises the member's resistance against deformations.:$mathbf\left\{q\right\}^m$ = vector of member's characteristic displacements or deformations.:$mathbf\left\{Q\right\}^\left\{om\right\}$ = vector of member's characteristic forces caused by external effects (such as known forces and temperature changes) applied to the member while $mathbf\left\{q\right\}^m = 0$).

If $mathbf\left\{q\right\}^m$ are member deformations rather than absolute displacements, then $mathbf\left\{Q\right\}^m$ are independent member forces, and in such case (1) can be inverted to yield the so-called "member flexibility matrix", which is used in the flexibility method.

ystem stiffness relation

For a system with many members interconnected at points called nodes, the members' stiffness relations such as Eq.(1) can be integrated by making use of the following observations:
* The member deformations $mathbf\left\{q\right\}^m$ can be expressed in terms of system nodal displacements r in order to ensure compatibility between members. This implies that r will be the primary unknowns.
* The member forces $mathbf\left\{Q\right\}^m$ help to the keep the nodes in equilibrium under the nodal forces R. This implies that the right-hand-side of (1) will be integrated into the right-hand-side of the following nodal equilibrium equations for the entire system:

:$mathbf\left\{R\right\} = mathbf\left\{Kr\right\} + mathbf\left\{R\right\}^o qquad qquad qquad mathrm\left\{\left(2\right)\right\}$where:$mathbf\left\{R\right\}$ = vector of nodal forces, representing external forces applied to the system's nodes.:$mathbf\left\{K\right\}$ = system stiffness matrix, which is established by "assembling" the members' stiffness matrices $mathbf\left\{k\right\}^m$.:$mathbf\left\{r\right\}$ = vector of system's nodal displacements that can define all possible deformed configurations of the system subject to arbitrary nodal forces R.:$mathbf\left\{R\right\}^o$ = vector of equivalent nodal forces, representing all external effects other than the nodal forces which are already included in the preceding nodal force vector R. This vector is established by assembling the members' $mathbf\left\{Q\right\}^\left\{om\right\}$.

olution

The system stiffness matrix K is square since the vectors R and r have the same size. In addition, it is symmetric because $mathbf\left\{k\right\}^m$ is symmetric. Once the supports' constraints are accounted for in (2), the nodal displacements are found by solving the system of linear equations (2), symbolically::$mathbf\left\{r\right\} = mathbf\left\{K\right\}^\left\{-1\right\} \left(mathbf\left\{R\right\}-mathbf\left\{R\right\}^o \right) qquad qquad qquad mathrm\left\{\left(3\right)\right\}$

Subsequently, the members' characteristic forces may be found from Eq.(1) where $mathbf\left\{q\right\}^m$ can be found from r by compatibility consideration.

The direct stiffness method

It is common to have Eq.(1) in a form where $mathbf\left\{q\right\}^m$ and $mathbf\left\{Q\right\}^\left\{om\right\}$ are, respectively, the member-end displacements and forces matching in direction with r and R. In such case, $mathbf\left\{K\right\}$ and $mathbf\left\{R\right\}^o$ can be obtained by direct summation of the members' matrices $mathbf\left\{k\right\}^m$ and $mathbf\left\{Q\right\}^\left\{om\right\}$. The method is then known as the direct stiffness method.

The advantages and disadvantages of the matrix stiffness method are compared and discussed in the flexibility method article.

ee also

*Finite element method
*Finite element method in structural mechanics
*Structural analysis
*Flexibility method

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