Matrix stiffness method

In Structural engineering, the Matrix stiffness method (or simply Stiffness method) is a matrix method that makes use of the members' stiffness relations for analysing the mechanical stresses and strains within a structure. For example, if k is the stiffness of a spring that is subject to a force Q, the spring's stiffness relation is

Q = k q

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

For complex systems, the 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 fixed-end forces caused by external effects (such as known forces and temperature changes) applied to the member.

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 tied to the system nodal displacements r in order to ensure compatibility between members.
The member forces $\mathbf {Q} ^{m}$ help to the keep the nodes in equilibrium under the nodal forces R.

These nodal equilibrium and compatibility conditions leads to the following equilibrium equation 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}

= sytem stiffness matrix, which can be 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}

.

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 the system 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.

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 the Direct Stiffness Method.

See also