Elements tested
Acoustic elements, beams, cohesive elements, elbows,
membranes, pipes, shells, trusses, continuum elements (except coupled pore
pressure-displacement and coupled temperature-displacement elements),
piezoelectric elements, springs, and
masses.
Problem description
The models consist of a single element. There are no boundary conditions,
except as required in spring-mass (see
SPRING, MASS, and JOINT2D elements)
and piezoelectric tests. For the piezoelectric element tests one electric
potential degree of freedom is constrained to remove singularities from the
dielectric portion of the structural stiffness.
Note:
There are no mass terms associated with potential degrees of freedom.
Results and discussion
The results presented in
Table 1
through
Table 7
show the number of zero-energy modes and the first nonzero eigenvalue. Some
elements have nonrigid-body zero-energy modes. Where two values are given in
the zero-energy modes column, the first is the number of zero-energy modes and
the second is the number of rigid-body zero-energy modes. When an assembly of
elements is tested, as in
Eigenvalue extraction for unconstrained patches of elements,
the nonrigid-body zero-energy modes disappear. The eigenvalue is shown only for
purposes of comparison. Elements with quadrilateral geometry can be degenerated
to triangular shape; these results are denoted by “(triangle)” in the tables.
Results for the piezoelectric elements are reported for Step 2.
Table 1. Acoustic elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| AC1D2
|
1
|
1.509 × 108
|
| AC1D3
|
1
|
4.527 × 108
|
| AC2D3
|
1
|
1.122 × 108
|
| AC2D4 (triangle)
|
1
|
1.122 × 108
|
| AC2D4
|
1
|
9.971 × 107
|
| AC2D6
|
1
|
4.116 × 108
|
| AC2D8 (triangle)
|
1
|
4.077 × 108
|
| AC2D8
|
1
|
4.447 × 108
|
| AC3D4
|
1
|
1.482 × 108
|
| AC3D5
|
1
|
9.94 × 107
|
| AC3D6
|
1
|
4.447 × 108
|
| AC3D8
|
1
|
3.743 × 107
|
| AC3D10
|
1
|
5.775 × 108
|
| AC3D15
|
1
|
4.447 × 108
|
| AC3D20
|
1
|
1.132 × 108
|
| ACAX3
|
1
|
1.218 × 108
|
| ACAX4 (triangle)
|
1
|
1.218 × 108
|
| ACAX4
|
1
|
9.331 × 107
|
| ACAX6
|
1
|
4.887 × 108
|
| ACAX8 (triangle)
|
1
|
4.870 × 108
|
| ACAX8
|
1
|
4.527 × 108
|
Table 2. Beam elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| B21
|
3
|
1.675 × 109
|
| B21H
|
3
|
1.675 × 109
|
| B22
|
3
|
4.621 × 109
|
| B22H
|
3
|
4.621 × 109
|
| B23
|
3
|
1.379 × 1010
|
| B23H
|
3
|
1.379 × 1010
|
| B31
|
6
|
3.127 × 109
|
| B31H
|
6
|
3.127 × 109
|
| B31OS
|
6
|
8.534 × 107
|
| B31OSH
|
6
|
8.534 × 107
|
| B32
|
6
|
7.170 × 109
|
| B32H
|
6
|
7.170 × 109
|
| B32OS
|
6
|
2.050 × 108
|
| B32OSH
|
6
|
2.050 × 108
|
| B33
|
6
|
1.714 × 1010
|
| B33H
|
6
|
1.714 × 1010
|
Table 3. Cohesive elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
Energy modes
|
eigenvalue
|
| COH2D4
|
5/3
|
1.0256 × 106
|
| COHAX4
|
5/1
|
1.0256 × 106
|
| COH3D6
|
12/6
|
1.2820 × 105
|
| COH3D8
|
16/6
|
5.1282 × 105
|
Table 4. Elbow and pipe elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| ELBOW31
|
6
|
5.481 × 107
|
| ELBOW31B
|
6
|
3.230 × 105
|
| ELBOW31C
|
6
|
3.230 × 105
|
| ELBOW32
|
6
|
1.065 × 108
|
| PIPE21
|
3
|
1.675 × 109
|
| PIPE21H
|
3
|
1.675 × 109
|
| PIPE22
|
3
|
4.621 × 109
|
| PIPE22H
|
3
|
4.621 × 109
|
| PIPE31
|
6
|
3.127 × 109
|
| PIPE31H
|
6
|
3.127 × 109
|
| PIPE32
|
6
|
9.321 × 109
|
| PIPE32H
|
6
|
9.321 × 109
|
The membrane elements have no bending stiffness, which accounts for the high
number of nonrigid-body zero-energy modes.
Table 5. Membrane elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| M3D3
|
6
|
2.350 × 108
|
| M3D4
|
7
|
1.615 × 108
|
| M3D4R
|
7
|
3.140 × 105
|
| M3D6
|
9
|
3.622 × 108
|
| M3D8
|
11
|
7.274 × 108
|
| M3D8R
|
12
|
7.274 × 108
|
| M3D9
|
12
|
7.274 × 108
|
| M3D9R
|
13
|
5.225 × 108
|
| MAX1
|
2
|
1.231 × 109
|
| MAX2
|
2
|
1.535 × 109
|
| MCL6
|
9
|
7.582 × 109
|
| MCL9
|
9
|
6.313 × 108
|
Table 6. Shell elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| S3/S3R
|
6
|
1.985 × 106
|
| S4
|
6
|
3.071 × 106
|
| S4R
|
6
|
3.071 × 106
|
| S4R5
|
6
|
3.074 × 106
|
| S8R
|
8/6
|
3.073 × 105
|
| S8R5
|
7/6
|
1.165 × 104
|
| S9R5
|
7/6
|
1.165 × 104
|
| STRI3
|
6
|
7.189 × 107
|
| STRI65
|
6
|
3.049 × 105
|
| SAXA11
|
4/3
|
1.228 × 105
|
| SAXA12
|
5/3
|
1.229 × 105
|
| SAXA13
|
6/3
|
1.229 × 105
|
| SAXA14
|
7/3
|
1.229 × 105
|
| SAXA21
|
3
|
2.636 × 106
|
| SAXA22
|
3
|
4.075 × 105
|
| SAXA23
|
3
|
4.075 × 105
|
| SAXA24
|
3
|
4.075 × 105
|
| SAX1
|
2/1
|
1.231 × 109
|
| SAX2
|
1
|
2.636 × 106
|
| SC6R
|
6
|
1.942 × 108
|
| SC8R
|
6
|
1.942 × 108
|
Table 7. Truss elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| T2D2
|
3
|
1.143 × 1010
|
| T2D2H
|
3
|
1.143 × 1010
|
| T2D3
|
4/3
|
3.429 × 1010
|
| T2D3H
|
4/3
|
3.429 × 1010
|
| T3D2
|
5
|
1.143 × 1010
|
| T3D2H
|
5
|
1.143 × 1010
|
| T3D3
|
7/6
|
3.429 × 1010
|
| T3D3H
|
7/6
|
3.429 × 1010
|
Table 8. Two-dimensional continuum elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| CPE3
|
3
|
2.488 × 108
|
| CPE3H
|
3
|
2.488 × 108
|
| CPE4
|
3
|
8.373 × 107
|
| CPE4H
|
3
|
8.373 × 107
|
| CPE4I
|
3
|
1.196 × 108
|
| CPE4IH
|
3
|
1.196 × 108
|
| CPE4R
|
3
|
3.140 × 105
|
| CPE4RH
|
3
|
3.140 × 105
|
| CPE6
|
3
|
3.868 × 108
|
| CPE6H
|
3
|
3.868 × 108
|
| CPE6M
|
3
|
1.289 × 108
|
| CPE6MH
|
3
|
1.289 × 108
|
| CPE8
|
3
|
7.535 × 108
|
| CPE8H
|
3
|
5.024 × 108
|
| CPE8R
|
4/3
|
7.535 × 108
|
| CPE8RH
|
4/3
|
7.535 × 108
|
| CPEG3
|
5/3
|
4.662 × 108
|
| CPEG3H
|
5/3
|
4.662 × 108
|
| CPEG4
|
5/3
|
8.373 × 107
|
| CPEG4H
|
5/3
|
8.373 × 107
|
| CPEG4I
|
3
|
1.086 × 108
|
| CPEG4IH
|
3
|
1.086 × 108
|
| CPEG4R
|
5/3
|
3.140 × 105
|
| CPEG4RH
|
5/3
|
3.140 × 105
|
| CPEG6
|
3
|
3.599 × 108
|
| CPEG6H
|
3
|
3.599 × 108
|
| CPEG8
|
3
|
7.168 × 108
|
| CPEG8H
|
3
|
5.024 × 108
|
| CPEG8R
|
4/3
|
7.168 × 108
|
| CPEG8RH
|
4/3
|
7.168 × 108
|
| CPS3
|
3
|
2.350 × 108
|
| CPS4
|
3
|
1.615 × 108
|
| CPS4I
|
3
|
1.088 × 108
|
| CPS4R
|
3
|
3.140 × 105
|
| CPS6
|
3
|
3.622 × 108
|
| CPS6M
|
3
|
1.206 × 108
|
| CPS8
|
3
|
7.274 × 108
|
| CPS8R
|
4/3
|
7.274 × 108
|
Table 9. Axisymmetric continuum elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| CAXA41
|
4/3
|
2.015 × 108
|
| CAXA42
|
4/3
|
4.887 × 107
|
| CAXA43
|
4/3
|
4.887 × 107
|
| CAXA44
|
4/3
|
4.887 × 107
|
| CAXA4H1
|
4/3
|
2.015 × 108
|
| CAXA4H2
|
4/3
|
4.887 × 107
|
| CAXA4H3
|
4/3
|
4.887 × 107
|
| CAXA4H4
|
4/3
|
4.887 × 107
|
| CAXA4R1
|
5/3
|
9.615 × 106
|
| CAXA4R2
|
8/3
|
9.615 × 106
|
| CAXA4R3
|
11/3
|
9.615 × 106
|
| CAXA4R4
|
14/3
|
9.615 × 106
|
| CAXA4RH1
|
5/3
|
9.615 × 106
|
| CAXA4RH2
|
8/3
|
9.615 × 106
|
| CAXA4RH3
|
11/3
|
9.615 × 106
|
| CAXA4RH4
|
14/3
|
9.615 × 106
|
| CAXA81
|
3
|
2.437 × 108
|
| CAXA82
|
3
|
8.526 × 107
|
| CAXA83
|
3
|
8.526 × 107
|
| CAXA84
|
3
|
8.526 × 107
|
| CAXA8H1
|
3
|
2.156 × 108
|
| CAXA8H2
|
3
|
8.461 × 107
|
| CAXA8H3
|
3
|
8.461 × 107
|
| CAXA8H4
|
3
|
8.461 × 107
|
| CAXA8R1
|
5/3
|
2.405 × 108
|
| CAXA8R2
|
6/3
|
8.457 × 107
|
| CAXA8R3
|
7/3
|
8.457 × 107
|
| CAXA8R4
|
8/3
|
8.457 × 107
|
| CAXA8RH1
|
5/3
|
2.099 × 108
|
| CAXA8RH2
|
6/3
|
8.384 × 107
|
| CAXA8RH3
|
7/3
|
8.384 × 107
|
| CAXA8RH4
|
8/3
|
8.348 × 107
|
| CAX3
|
2/1
|
7.402 × 108
|
| CAX3H
|
2/1
|
7.402 × 108
|
| CAX4
|
2/1
|
1.022 × 109
|
| CAX4H
|
2/1
|
1.022 × 109
|
| CAX4R
|
2/1
|
1.011 × 107
|
| CAX4RH
|
2/1
|
1.011 × 107
|
| CAX4I
|
1
|
7.711 × 107
|
| CAX4IH
|
1
|
7.456 × 107
|
| CAX6
|
1
|
1.448 × 108
|
| CAX6H
|
1
|
1.448 × 108
|
| CAX6M
|
1
|
8.949 × 107
|
| CAX6MH
|
1
|
8.949 × 107
|
| CAX8
|
1
|
2.437 × 108
|
| CAX8H
|
1
|
2.156 × 108
|
| CAX8R
|
2/1
|
2.405 × 108
|
| CAX8RH
|
2/1
|
2.099 × 108
|
Table 10. Three-dimensional continuum elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| C3D10
|
6
|
4.500 × 109
|
| C3D10H
|
6
|
4.500 × 109
|
| C3D10HS
|
6
|
4.500 × 109
|
| C3D10M
|
6
|
7.486 × 107
|
| C3D10MH
|
6
|
7.486 × 107
|
| C3D15
|
6
|
1.695 × 109
|
| C3D15H
|
6
|
1.967 × 109
|
| C3D15V
|
6
|
1.084 × 109
|
| C3D15VH
|
6
|
1.379 × 108
|
| C3D20
|
6
|
3.436 × 108
|
| C3D20H
|
6
|
2.213 × 108
|
| C3D20R
|
12/6
|
3.768 × 108
|
| C3D20RH
|
12/6
|
4.082 × 103
|
| C3D27 (21 nodes)
|
6
|
3.768 × 108
|
| C3D27 (22 nodes)
|
6
|
3.768 × 108
|
| C3D27 (23 nodes)
|
6
|
3.768 × 108
|
| C3D27 (24 nodes)
|
6
|
3.768 × 108
|
| C3D27 (25 nodes)
|
6
|
3.768 × 108
|
| C3D27 (26 nodes)
|
6
|
3.768 × 108
|
| C3D27 (27 nodes)
|
6
|
3.768 × 108
|
| C3D27H (21 nodes)
|
6
|
2.213 × 108
|
| C3D27H (22 nodes)
|
6
|
2.213 × 108
|
| C3D27H (23 nodes)
|
6
|
2.213 × 108
|
| C3D27H (24 nodes)
|
6
|
2.213 × 108
|
| C3D27H (25 nodes)
|
6
|
2.213 × 108
|
| C3D27H (26 nodes)
|
6
|
2.213 × 108
|
| C3D27H (27 nodes)
|
6
|
2.213 × 108
|
| C3D27R (21 nodes)
|
6
|
3.768 × 108
|
| C3D27R (22 nodes)
|
6
|
3.768 × 108
|
| C3D27R (23 nodes)
|
6
|
3.768 × 108
|
| C3D27R (24 nodes)
|
6
|
3.128 × 108
|
| C3D27R (25 nodes)
|
6
|
1.558 × 108
|
| C3D27R (26 nodes)
|
6
|
1.236 × 108
|
| C3D27R (27 nodes)
|
9/6
|
2.007 × 108
|
| C3D27RH (21 nodes)
|
6
|
2.213 × 108
|
| C3D27RH (22 nodes)
|
6
|
2.032 × 108
|
| C3D27RH (23 nodes)
|
6
|
1.467 × 108
|
| C3D27RH (24 nodes)
|
6
|
1.022 × 108
|
| C3D27RH (25 nodes)
|
6
|
2.767 × 107
|
| C3D27RH (26 nodes)
|
6
|
2.509 × 107
|
| C3D27RH (27 nodes)
|
9/6
|
3.069 × 107
|
| C3D4
|
6
|
3.623 × 109
|
| C3D4H
|
6
|
3.623 × 109
|
| C3D5
|
6
|
2.026 × 1012
|
| C3D5H
|
6
|
1.936 × 1012
|
| C3D6
|
7/6
|
3.846 × 108
|
| C3D6H
|
7/6
|
3.472 × 108
|
| C3D8
|
6
|
4.186 × 107
|
| C3D8H
|
6
|
4.186 × 107
|
| C3D8I
|
6
|
4.186 × 107
|
| C3D8IH
|
6
|
4.186 × 107
|
| C3D8R
|
6
|
1.184 × 106
|
| C3D8RH
|
6
|
1.184 × 106
|
| CCL9
|
9/6
|
1.410 × 105
|
| CCL9H
|
9/6
|
1.0572
|
| CCL12
|
6
|
3.1502 × 108
|
| CCL12H
|
6
|
3.1502 × 108
|
| CCL18
|
6
|
1.089 × 1010
|
| CCL18H
|
6
|
4.449 × 108
|
| CCL24
|
6
|
3.767 × 109
|
| CCL24R
|
9/6
|
3.394 × 109
|
| CCL24H
|
6
|
2.213 × 109
|
| CCL24RH
|
9/6
|
1.214 × 109
|
| CSS8
|
6
|
4.186 × 107
|
Table 11. Piezoelectric elements.
| Element
|
Number of zero-
|
First nonzero
|
| type
|
energy modes
|
eigenvalue
|
| C3D10E
|
6
|
4.825 × 109
|
| C3D15E
|
6
|
1.695 × 109
|
| C3D20E
|
6
|
3.768 × 108
|
| C3D20RE
|
12/6
|
3.768 × 108
|
| C3D4E
|
6
|
6.092 × 109
|
| C3D6E
|
7/6
|
3.846 × 108
|
| C3D8E
|
6
|
4.186 × 107
|
| CAX3E
|
2/1
|
8.828 × 108
|
| CAX4E
|
2/1
|
1.169 × 109
|
| CAX6E
|
1
|
1.604 × 108
|
| CAX8E
|
1
|
2.556 × 108
|
| CAX8RE
|
2/1
|
2.522 × 108
|
| CPE3E
|
3
|
6.567 × 108
|
| CPE4E
|
3
|
8.373 × 107
|
| CPE6E
|
3
|
6.006 × 108
|
| CPE8E
|
3
|
8.246 × 108
|
| CPE8RE
|
4/3
|
8.246 × 108
|
| CPS3E
|
3
|
5.024 × 108
|
| CPS4E
|
3
|
1.615 × 108
|
| CPS6E
|
3
|
5.265 × 108
|
| CPS8E
|
3
|
7.797 × 108
|
| CPS8RE
|
4/3
|
7.797 × 108
|
| T2D2E
|
3
|
1.476 × 1013
|
| T2D3E
|
4/3
|
1.714 × 1011
|
| T3D2E
|
5
|
1.476 × 1013
|
| T3D3E
|
7/6
|
1.714 × 1011
|
SPRING,
MASS, and JOINT2D elements
The models for the eigenvalue extraction tests for
SPRING and
MASS element types are slightly more complex
than the tests for the other elements.
Elements of type SPRINGA and MASS are tested together in files
exspame1.inp
and
exspame2.inp.
Three nodes lie along a straight line. One of the nodes is constrained, and
each of the other two nodes defines a point mass. SPRINGA elements are defined between each of the three possible pairs of
nodes. The spring-mass system acts in degree of freedom 1.
File
exspbue1.inp
tests element types SPRING1 and SPRING2 with a mass matrix defined by a user element. Two coincident
nodes are defined. These two nodes are used in the definition of the user
element. A SPRING2 element connects the nodes, and each node is also connected to a SPRING1 element. No boundary conditions are required since, by
definition, the other ends of the SPRING1 elements are connected to ground. The spring-mass system acts in
degree of freedom 1.
Results for both tests: =0.6340,
=2.3660.
File
exepxme1.inp
tests element type JOINT2D. One node of the JOINT2D element is fully constrained, and the other has MASS and ROTARYI elements applied to create a spring-mass system. The natural
frequencies and modes correspond to analytically calculated values.
|