Element Stiffness Matrix Characteristics

An element stiffness matrix has many general characteristics that can be used to check the formulation of a particular stiffness matrix. An element stiffness matrix must have the following properties:

• Symmetric - This means that k_ij = k_ji. This is always the case when the displacements are directly proportional to the applied loads.

• Square - This implies that the number of rows are equal to the number of columns in the matrix.

• Singular - The element stiffness matrix is singular (the determinate of the matrix is equal to zero e.i., ) since no constraints (prescribed displacements and/or rotation) have been applied.

• Positive Diagonal Terms - All the terms in the main diagonal (upper left to lower right) must be positive. If k_ii is negative then the force and it's corresponding displacement would be oppositely directed, which is physically unreasonable. If k_ii = 0, then the displacement would produce no reaction force resisting it, which would imply that the structure is unstable.

The characteristics just mentioned can be seen in the beam element stiffness matrix shown below, although it should be noted that they apply to all element stiffness matrices.