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.

Using Eq.(1) together with 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}

.

Once the supports' constraints are accounted for, 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 $\mathbf {q} ^{m}$ is merely a subset of r pertaining to the member under consideration.

See also