Comparisons with equivalent beam MPC and
equivalent revolute and universal MPC problems
show that using coupling constraints yields identical behavior.
Figure 1. Geometry to test local orientation definitions.
In these tests the center node is the reference node, and the perimeter
nodes are the coupling nodes. Four separate coupling definitions that share the
same reference node are defined. Each coupling definition defines the local
coordinate system using a different orientation system: cylindrical,
rectangular, spherical, and, for the
Abaqus/Standard
analyses, a system defined by user subroutine
ORIENT. In all cases the resulting local constraint basis
directions coincide with the local directions of a cylindrical coordinate
system whose axis is normal to the plane containing the nodes and passes
through the reference node.
In problems
xcouplingk_std_orient_1.inp
and
xcouplingk_xpl_orient_1.inpthe
kinematic coupling constrains all but the radial degree of freedom at the
coupling nodes. Linear springs to ground (SPRING1) for the
Abaqus/Standard
analyses and connector elements to ground (CONN3D2) with linear elastic connector behavior for the
Abaqus/Explicit
analyses are attached to all coupling nodes and act in the
x- and y-directions. The reference
node is then rotated
radians about the z-axis.
In problems
xcouplingk_std_orient_2.inp
and
xcouplingk_xpl_orient_2.inpthe
kinematic coupling constrains the circumferential degree of freedom only.
Linear springs to ground (SPRING1) for the
Abaqus/Standard
analyses and connector elements to ground (CONN3D2) with linear elastic connector behavior for the
Abaqus/Explicit
analyses are attached to all coupling nodes and act in the
x-, y-, and
z-directions. The reference node is then rotated
about x-axis.
Results and discussion
These tests result in motion of the constrained nodes, under action of the
linear springs, as the reference node rotates. For tests
xcouplingk_std_orient_1.inp
and
xcouplingk_xpl_orient_1.inpthis
motion remains on the local radius passing through the node at all increments.
For tests
xcouplingk_std_orient_2.inp
and
xcouplingk_xpl_orient_2.inp
this motion remains in the plane defined by the original configuration local
radius and the global z-direction as this plane rotates
according to the motion prescribed at the reference node.
Test of local orientation and the release of two translational degrees of
freedom.
Internal sorting of kinematic coupling constraints
Features tested
The internal sorting of kinematic coupling constraints when used in
conjunction with MPC definitions is verified.
Problem description
The model consists of an axial arrangement of 20 shell elements. These
elements are tied together using a combination of kinematic coupling
constraints as well as MPCs. The constraints
are defined such that the kinematic coupling reference node appears after the
constraint definitions that are eliminated degrees of freedom on that node;
thus, constraint sorting is required. The structure is clamped on one end, and
a concentrated axial load is applied on the other end.
Results and discussion
The test results in an internal sorting of kinematic coupling definitions
and MPCs so that the proper elimination order
is achieved.
Test internal sorting of kinematic coupling constraints.
Distributing coupling constraints with user-specified weights
Features tested
The distributing coupling constraint is tested by using coupling and
distributing constraints with user-specified distributing weight factors.
Geometric linear and nonlinear tests are performed.
Problem description
Model:
The initial starting geometry for each test is shown in
Figure 2.
For the geometric linear test, for
Abaqus/Standard,
each coupling node is connected by a spring to ground (SPRING1) in each direction. In the geometrically nonlinear test in
Abaqus/Standard,
each coupling node is connected by a dashpot to ground (DASHPOT1) in each direction, and an axial spring element (SPRINGA) connects each pair of coupling nodes. In the geometrically
nonlinear test in
Abaqus/Explicit,
each coupling node is connected by a connector to ground (CONN3D2) with damping behavior specified in each direction, and a
connector element with specified elastic behavior connects each pair of
coupling nodes. The reference node for the coupling constraint is node 10.
Figure 2. Initial starting geometry.
Linear behavior
Properties:
The spring stiffnesses are 100, 200, and 300 for degrees of freedom 1, 2,
and 3, respectively, for the springs connected to all coupling nodes. The
distributing weight factors are 1, 2, and 3 for nodes 1, 2, and 3,
respectively.
Loading:
Step 1
The force at the reference node is 1.0 in the
x-direction. The moment at the reference node is 2.0 about
the z-axis.
Step 2
The force at the reference node is 1.0 in the
y-direction. The moment at the reference node is 2.0 about
the x-axis.
Step 3
The force at the reference node is 1.0 in the
z-direction. The moment at the reference node is 2.0 about
the y-axis.
Step 4
Frequency extraction.
Step 5
Transient modal dynamic step with a load,
1.0,
applied to the reference node.
Step 6
Mode-based steady-state dynamic step with a load,
1.0, applied to the reference node.
Nonlinear behavior
Properties:
The dashpot damping coefficients are 100, 200, and 300 for degrees of
freedom 1, 2, and 3, respectively, for the dashpots connected to all coupling
nodes. The axial springs connecting the coupling nodes each have a spring
constant of 1.0 × 108. The distributing weight factors are 1, 2, and
3 for nodes 1, 2, and 3, respectively.
Prescribed
reference node motion for
Abaqus/Standard:
Step 1
Total rotation of
about the z-axis. Translation .
Step 2
Total rotation of
about the y-axis. Translation .
Step 3
Total rotation of
about the x-axis. Translation .
Step 4
Direct-integration dynamic step with a total
rotation of
about the z-axis. Translation .
Prescribed
reference node motion for
Abaqus/Explicit:
Step 1
Total rotation of
about the z-axis. Translation .
Step 2
Total rotation of
about the y-axis. Translation .
Step 3
Total rotation of
about the x-axis. Translation .
Step 4
Total rotation of
about the z-axis. Translation .
Results and discussion
In all tests the load distribution among coupling nodes adheres to the
relation
where
is the force distribution at the coupling nodes,
and
are the force and moment at the reference node,
are the normalized distributing weight factors, is the coupling node
arrangement inertia tensor, and
and
are the positions of the reference and coupling nodes relative to the coupling
node arrangement centroid, respectively. See
Distributing coupling constraints
for a more detailed description of this load distribution.
Distributing coupling for geometric nonlinear case.
Default distributing weight factors
Elements tested
B21
B22
C3D8
C3D8R
C3D10M
C3D20
C3D27
CAX4
CAX4R
CAX8
CPE4
CPE4R
CPE8
S3R
S4
S8R
S9R5
CSS8
Features tested
The default distributing weight factors for a distributing coupling
constraint are verified. The weight factors are based on the nodal tributary
surface area at each coupling node.
Problem description
Various models consisting of either continuum, beam, or shell elements are
used in this test. In all models a uniform surface load is applied via a
reference node and a distributing coupling constraint. A nonuniform mesh
density is used to verify that the proper tributary area is calculated. The
reference node is located at the center of the loaded surface, offset in the
normal direction.
Results and discussion
The displacements are equal to the displacements obtained if the model were
loaded with a uniform pressure load, hence verifying that the proper
distributing weights are calculated at the coupling nodes.
The calculation of distributing weights as outlined in
Coupling Constraints
when the optional weighting method and influence region are specified is
verified. The use of coupling constraints at the part-instance level is also
illustrated.
Problem description
A part is defined consisting of two rows of 20 CPE4R elements. Each element is a unit square. The coupling nodes are
defined along the top surface. A reference node is created at the center of the
top surface. The part is then instanced three times in the assembly definition.
For each part instance a coupling constraint with a different influence region
is defined. The first part instance has an infinite influence radius; i.e., all
nodes defined on the surface will be included in the coupling definition. The
second part instance uses an influence radius of 5.5, and the third part
instance uses an influence radius of 0.5. A concentrated load is applied to
each reference node. Input files are provided for each weighting scheme:
uniform, linear, quadratic, and cubic.
Results and discussion
The distributing weight factor calculations are verified to be according to
the description provided in
Coupling Constraints.
For the first instance all nodes belonging to the facets are included in the
coupling definition. For the second instance the nodes of six facets adjacent
to the reference node are included in the coupling definition. In this case the
facet farthest away from the reference node (on either side) uses a facet
participation factor of 0.5, since only part of element surface facet is
included in the influence region. For the third case the nodes of the adjacent
facets to the reference node are included in the coupling definition. In this
case each facet has a participation factor of 0.5, since only part of the
element surface facet is included in the influence region.
Distributing coupling with a cubic weighting method.
Colinear coupling node arrangement
Features tested
A pathological situation in which all coupling nodes are colinear for a
distributing coupling constraint and the moment applied at the reference node
is not transmitted by the constraint is tested.
The distributing coupling constraint connects a single reference node that
has translational and rotational degrees of freedom to a collection of coupling
nodes that have only translational degrees of freedom. Thus, when the coupling
nodes are colinear in a three-dimensional analysis, a situation can arise where
the moments applied to the reference node are not transmitted. In such a case
Abaqus
will print a warning message specifying the axis about which the moments are
not transmitted.
Distributing coupling with colinear coupling nodes.
Moment release for distributing coupling
Features tested
A series of linear and nonlinear analyses are performed demonstrating the
ability of the distributing coupling constraints to release the rotation
constraints between the reference node and the coupling nodes about
user-specified axes.
Problem description
This example consists of both a two-dimensional and three-dimensional test.
In the two-dimensional test, two separate models are defined. Each model
consists of a single CPE4 element with one face coupled to a reference node with a
distributing constraint. The opposite face of the CPE4 element is fixed. Beam elements are attached to the reference
nodes for visualization purposes only. The first model uses the default
coupling in which the rotation degree of freedom of the reference node is
coupled to the solid surface (the displacement degrees of freedom of the
reference are always coupled to the surface with distributing constraints). The
second model releases the rotation constraint. A series of boundary conditions
are applied to the reference nodes simulating shear, tension, and bending (in
various linear and nonlinear steps).
In the three-dimensional test, eight separate models are defined. Each model
consists of a single C3D8 element with one face coupled to a reference node with a
distributing constraint. The opposite faces of the C3D8 elements are fixed. Beam elements are attached to the reference
nodes for visualization purposes only. The first model uses the default
coupling in which all three rotation degrees of freedom of the reference node
are coupled to the solid surface. The next three models respectively release
the rotation constraint in the 1, 2, and 3 directions. The final four models
are identical to the first four, except that the rotation constraint directions
are specified. A series of boundary conditions are applied to the reference
nodes simulating shear, tension, and bending (in linear and nonlinear steps).
Results and discussion
The results clearly show that both coupling definitions in both two and
three dimensions are being applied properly.
Three-dimensional examples of distributing coupling with the moment
constraints released.
Dimensional coupling
Features tested
A series of linear analyses are performed demonstrating the ability of the
distributing coupling constraints to provide accurate dimensional coupling of
beam elements to shell and solid elements.
Problem description
This example consists of two sets of tests in which a pipe is modeled with
beam and shell elements and with beam and continuum elements.
The pipe analyzed with beam and shell elements has a length of 0.8 m, an
outside radius of 0.1 m, and a thickness of 0.01 m. The material has a Young's
modulus of 200 GPa and a Poisson's ratio of 0.3. Half of the pipe is modeled
with beam elements and the other half is modeled with shell elements (see
Figure 4(a)).
The beam node closest to the shell model is defined as the reference node for
the distributing coupling constraint. An element-based edge surface is defined
on the shell model, which is coupled to the reference node. The coupled model
is subjected to four linear loading conditions simulating: (1) twist about the
pipe axis, (2) axial stretch along the pipe axis, (3) pure bending about the
x-axis, and (4) shear loading. The four load
conditions are applied in a single linear step as four load cases. Two models
are analyzed: one with linear beam and shell elements and one with quadratic
beam and shell elements.
The pipe analyzed with beam and continuum elements has a length of 0.8 m,
an outside radius of 0.1 m, and a thickness of 0.04 m. The material has a
Young's modulus of 200 GPa and a Poisson's ratio of 0.3. Half of the pipe is
modeled with beam elements and the other half is modeled with continuum
elements (see
Figure 4(b)).
The beam node closest to the continuum model is defined as the reference node
for the distributing coupling constraint. An element-based surface is defined
on the continuum model, which is coupled to the reference node. The coupled
model is subjected to four linear loading conditions simulating: (1) twist
about the pipe axis, (2) axial stretch along the pipe axis, (3) pure bending
about the x-axis, and (4) shear loading. The four
load conditions are applied in a single linear step. Two models are analyzed:
one with linear beam and continuum elements and one with quadratic beam and
continuum elements.
Results and discussion
The resulting stress fields in the shell and solid models show minimal
distortion at the coupling interface, indication that the dimensional coupling
is modeled accurately.
Coupling a beam model to a continuum model using quadratic beam and
continuum elements.
Structural coupling
Features tested
A series of analyses are performed demonstrating the structural coupling
capability of small distributing coupling constraints.
Problem description
Four different models, each with two small distributing couplings, are
analyzed. In the first model two small square plates are coupled together with
a BEAM connector. The connector nodes are coupled to the two small
surfaces using structural distributing couplings. One plate is kept fixed,
while the other is pulled upward (pried open) on one side. In the second model
the same plates are pulled upward from all sides. In the third model two
circular plates are fastened together by placing a BEAMMPC between the reference nodes of
two structural distributing couplings spanning two small patches on the two
plates. The plates are then subjected to relative shear motion. In the fourth
model two U-shaped shell specimens are connected in a fashion similar to that
in the second model. The lower specimen is fixed, while the upper specimen is
lifted and pried open simultaneously.
For comparison in
Abaqus/Explicit,
similar models are created to use continuum distributing coupling and
fasteners.
Results and discussion
The resulting deformed shapes match the expectations. More important, if
structural coupling is used, contact between the plates does not occur in the
area close to the fastener, as expected. By contrast, contact does occur if
continuum distributing couplings are used.