All courses › Finite Element Method › Boundary conditions and Ku = f

Boundary conditions and Ku = f

The global stiffness matrix is singular before boundary conditions are applied, because the structure can still move freely as a rigid body. By fixing known displacements (often zero) and removing the corresponding rows, the system becomes solvable for the remaining unknown displacements.

K\vec u = \vec f

the linear system that is solved

Symbols

$K$	global stiffness matrix (after boundary conditions)
$\vec u$	nodal displacements (unknown)	m
$\vec f$	nodal forces	N

Example

Node 1 fixed, force $F$ at node 2 of a single bar:

$u_2 = FL/(EA)$ after row/column 1 is removed.

Enough degrees of freedom must be fixed to prevent rigid-body motion, or

K

stays singular.

Practise bar elements and stiffness matrices for free →

← Assembly · Shape functions →

Part of Finite Element Method: Bar elements and stiffness matrices.