- Matrix stiffness method
In

structural engineering , the**matrix**(or simply "stiffness method", also known asstiffness methodDirect 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:

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

If $mathbf\{q\}^m$ are member deformations rather than absolute displacements, then $mathbf\{Q\}^m$ 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 $mathbf\{q\}^m$ 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 $mathbf\{Q\}^m$ 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::$mathbf\{R\}\; =\; mathbf\{Kr\}\; +\; mathbf\{R\}^o\; qquad\; qquad\; qquad\; mathrm\{(2)\}$where:$mathbf\{R\}$ = vector of nodal forces, representing external forces applied to the system's nodes.:$mathbf\{K\}$ = system stiffness matrix, which is established by "assembling" the members' stiffness matrices $mathbf\{k\}^m$.:$mathbf\{r\}$ = vector of system's nodal displacements that can define all possible deformed configurations of the system subject to arbitrary nodal forces

**R**.:$mathbf\{R\}^o$ = 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' $mathbf\{Q\}^\{om\}$.**olution**The system stiffness matrix

**K**is square since the vectors**R**and**r**have the same size. In addition, it is symmetric because $mathbf\{k\}^m$ is symmetric. Once the supports' constraints are accounted for in (2), the nodal displacements are found by solving thesystem of linear equations (2), symbolically::$mathbf\{r\}\; =\; mathbf\{K\}^\{-1\}\; (mathbf\{R\}-mathbf\{R\}^o\; )\; qquad\; qquad\; qquad\; mathrm\{(3)\}$Subsequently, the members' characteristic forces may be found from Eq.(1) where $mathbf\{q\}^m$ can be found from

**r**by compatibility consideration.**The direct stiffness method**It is common to have Eq.(1) in a form where $mathbf\{q\}^m$ and $mathbf\{Q\}^\{om\}$ are, respectively, the member-end displacements and forces matching in direction with

**r**and**R**. In such case, $mathbf\{K\}$ and $mathbf\{R\}^o$ can be obtained by direct summation of the members' matrices $mathbf\{k\}^m$ and $mathbf\{Q\}^\{om\}$. The method is then known as thedirect 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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