Stiffness, Damping, or Mass Matrix

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MATRIX27 Element Description

MATRIX27 represents an arbitrary element whose geometry is undefined but whose elastic kinematic response can be specified by stiffness, damping, or mass coefficients in matrix form. The matrix is assumed to relate two nodes, each with six degrees of freedom per node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. See MATRIX27 in the Mechanical APDL Theory Reference for more details about this element. Other similar, but less general, elements are the spring-damper element (COMBIN14), and the mass element (MASS21).

Figure 27.1:  MATRIX27 Schematic

MATRIX27 Schematic

MATRIX27 Input Data

The node locations and the coordinate system for this element are shown in Figure 27.1: MATRIX27 Schematic. The element is defined by two nodes and the matrix coefficients. The stiffness, damping, or mass matrix constants are input as real constants. The units of the stiffness constants are Force/Length or Force*Length/Radian and the damping constants, Force*Time/Length and Force*Length*Time/Radian. The mass constants should have units of Force*Time2/Length or Force*Time2*Length/Radian.

All matrices generated by this element are 12 by 12. The degrees of freedom are ordered as UX, UY, UZ, ROTX, ROTY, ROTZ for node I followed by the same for node J. If one node is not used, simply let all rows and columns relating to that node default to zero.

A structural matrix that combines the effects of many elements is normally positive or zero definite, as are the element matrices that contribute to it. There may be unusual circumstances where an element matrix is negative definite, and this is okay if there are other matrices connected to the same nodes that are positive definite, resulting in a final system of equations is still positive or zero definite. A simple example of such a circumstance is a beam element loaded with half of the buckling load. The stress stiffness matrix is negative definite, but the combined regular and stress stiffness matrix is positive definite.

When using MATRIX27 with symmetric element matrices (KEYOPT(2) = 0), positive or zero definite matrices should be input using KEYOPT(1) = 0, the default. To input a negative definite element matrix, set KEYOPT(1) = 1 so that the negative definite checking will be bypassed. However, when using an unsymmetric or skew-symmetric element matrix, (KEYOPT(2) = 2 or 3), there are no limits on the form of the matrix, as the unsymmetric solvers are designed to solve any system of equations, as long as it is not singular.

The matrix constants should be input according to the matrix diagrams shown in "MATRIX27 Output Data". For example, if a simple spring of stiffness K in the nodal x direction is desired, the input constants would be C1 = C58 = K and C7 = -K for KEYOPT(2) = 0 and KEYOPT(3) = 4.

A summary of the element input is given in "MATRIX27 Input Summary". Element Input gives a general description of element input.

MATRIX27 Input Summary


I, J

Degrees of Freedom


Real Constants
C1, C2, ... C78 - Define the upper triangular portion of the symmetric matrix (if KEYOPT(2) = 0)
C1, C2, ... C144 - Define all terms of the unsymmetric matrix (if KEYOPT(2) = 2)
C1, C2, ... C66 - Define upper triangular portion (less diagonal terms) if skew symmetric (KEYOPT(2) = 3)
Material Properties

MP command: BETD, ALPD, DMPR

Surface Loads


Body Loads


Special Features
Birth and death
Linear perturbation

Matrix form (only works with KEYOPT(2) = 0):

0 -- 

Input positive or zero definite matrices only

1 -- 

Input positive, zero, or negative definite matrices


Matrix formulation:

0 -- 

Symmetric matrices (78 constants required)

2 -- 

Unsymmetric matrices(144 constants required)

3 -- 

Skew symmetric matrices (66 constants required)


Real constant input data:

2 -- 

Defines a 12 x 12 mass matrix

4 -- 

Defines a 12 x 12 stiffness matrix

5 -- 

Defines a 12 x 12 damping matrix


Element matrix output:

0 -- 

Do not print element matrix

1 -- 

Print element matrix at beginning of solution phase

MATRIX27 Output Data

The solution output associated with the element consists of node displacements included in the overall nodal solution. There is no element solution output associated with the element unless element reaction forces and/or energies are requested. KEYOPT(4) = 1 causes the element matrix to be printed (for the first substep of the first load step only). Solution Output gives a general description of solution output.

For KEYOPT(2) = 0, the symmetric matrix has the form: 

For KEYOPT(2) = 2, the unsymmetric matrix has the form: 

For KEYOPT(2) = 3, the skew symmetric matrix has the form: 

MATRIX27 Assumptions and Restrictions

  • Nodes may be coincident or noncoincident.

  • Since element matrices should normally not be negative definite, a note is printed for those cases where this can be easily detected.

  • With a lumped mass matrix [LUMPM,ON] all off-diagonal terms must be zero.

  • The matrix terms are associated with the nodal degrees of freedom and are assumed to act in the nodal coordinate directions (see Elements That Operate in the Nodal Coordinate System).

MATRIX27 Product Restrictions

There are not product-specific restrictions for this element.

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