Stiffness matrix

For the stiffness tensor in solid mechanics, see Hooke's law#Matrix representation (stiffness tensor).

In the finite element method and in analysis of spring systems, a stiffness matrix, K, is a symmetric positive-semidefinite matrix that generalizes the stiffness of Hooke's law to a matrix, describing the stiffness of between all of the degrees of freedom so that

$\mathbf{F}=-K\mathbf{x}$

where F and x are the force and the displacement vectors, and

$U=\frac{1}{2} \;\mathbf{x}^\top\! K \mathbf{x}$

is the system's total potential energy.

For a simple spring network, the stiffness matrix is a Laplacian matrix (in order to enforce Newton's third law) describing the connectivity graph between degrees of freedom. Off-diagonal entries contain $k_{ij}$, the negative stiffness of the spring connecting degree-of-freedom i to j. For example,

$K=\left(\begin{array}{rrrrrr} 13 & -1 & 0 & 0 & -12 & 0\\ -1 & 3 & -1 & 0 & -1 & 0\\ 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & -1 & 3 & -1 & -1\\ -12 & -1 & 0 & -1 & 14 & 0\\ 0 & 0 & 0 & -1 & 0 & 1\\ \end{array}\right)$

Truss Element Stiffness Matrix

The stiffness matrix of a horizontal prismatic truss element is  :

d.o.f. of the truss element

$K = \frac{EA}{L} \begin{align} \begin{bmatrix} 1 & -1 \\ -1 & 1 \\ \end{bmatrix} \begin{matrix} u1 \\ u2 \\ \end{matrix} \end{align}$

E: modules of elasticity

L: length

A: cross section area

Beam Element Stiffness Matrix

The stiffness matrix of a prismatic two dimensional horizontal beam element with negligible shear and axial deformation is :

d.o.f. of the frame element

$K = \frac{EI}{L} \begin{align} \begin{bmatrix} \frac{12}{L^2} & \frac{6}{L} & \frac{-12}{L^2}& \frac{6}{L}\\ \\ \frac{6}{L} & 4 & \frac{-6}{L} &2\\ \\ \frac{-12}{L^2}& \frac{-6}{L}&\frac{12}{L^2}& \frac{-6}{L}\\ \\ \frac{6}{L} & 2&\frac{-6}{L} &4 \end{bmatrix} \begin{matrix} v1 \\ \\ \Theta1 \\ \\ v2\\ \\ \Theta2 \\ \end{matrix} \end{align}$

The stiffness matrix of a prismatic two dimensional horizontal beam element with negligible shear deformation is :

$K = \begin{bmatrix} \frac{AE}{L} & 0 & 0& \frac{-AE}{L} & 0 & 0 \\ \\ 0 & \frac{12EI}{L^3} & \frac{6EI}{L^2} &0& \frac{-12EI}{L^3} &\frac{6EI}{L^2}\\ \\ 0 & \frac{6EI}{L^2} & \frac{4EI}{L} &0& \frac{-6EI}{L^2} &\frac{2EI}{L}\\ \\ \frac{-AE}{L} & 0 & 0& \frac{AE}{L} & 0 & 0 \\ \\ 0 & \frac{-12EI}{L^3} & \frac{-6EI}{L^2} &0& \frac{12EI}{L^3} &\frac{-6EI}{L^2}\\ \\ 0 & \frac{6EI}{L^2} & \frac{2EI}{L} &0& \frac{-6EI}{L^2} &\frac{4EI}{L}\\ \\ \end{bmatrix}$:

E: modules of elasticity
I: section moment of inertia perpendicular to page
L: length

See also

• Hooke's law
• Mass matrix