Tube support elements model the interaction between a tube and a
support that is not always in contact with the tube during dynamic events. The
tube is assumed to have a circular section and can interact with one of two
tube support geometries: a circular hole and an “egg-crate” support.
Two tube support interface elements are provided, one for each geometry, as
shown in
Figure 1
and
Figure 2.
As indicated in
Figure 2,
one cylindrical geometry interface is needed to model the interaction of the
tube with a circular hole, while
Figure 1
shows that several unidirectional geometry elements are needed to model the
interaction with an egg-crate—one element perpendicular to each pair of
egg-crate faces.
The interface elements themselves consist of a spring and friction link and
a dashpot, as shown in
Figure 3.
The spring is assumed to behave as shown in
Figure 4:
when there is no contact between the tube and the support, no force is
transmitted by the spring; when the tube is in contact with the support, the
force increases as the tube wall is deformed. This force can be modeled as a
linear or a nonlinear function of the relative displacement between the axis of
the tube and the center of the hole in the support.
The frictional part of the spring and friction link uses the Coulomb
friction model in
Abaqus:
that model is described in
Coulomb friction.
The dashpot is provided to model fluid effects in the annulus between the
tube and the support plate. Its behavior can be linear or nonlinear. The model
assumes that shear forces created by the fluid are negligible, so that the only
shear forces transmitted by one of these interface elements are the frictional
forces caused by direct contact between the tube and the support.
A major simplification in these elements that saves considerable
computational effort in dynamic applications is the assumption that impacts
between the tube and its support plates involve no instantaneous transfer of
momentum or energy loss: the standard impact algorithm of
Abaqus/Standard
used with gap and other interface elements (and described in
Intermittent contact/impact)
is not needed. This simplification derives from the assumption that these
elements will be used in conjunction with beam element models of the tube, so
the tube section is defined by the position and orientation of its axis and
local deformation of the cross-section of the tube is neglected. In reality,
when the tube hits a support, initially only a small part of the tube wall
loses momentum so that there is—instantaneously—only a small loss of kinetic
energy. This instantaneous energy loss is neglected when these elements are
used. The subsequent flattening of the tube wall is modeled by the spring link
in the element, acting between the node on the tube axis and the node
representing the center of the hole. Thus, the modeling of this local
flattening behavior as an equivalent spring provides the simplification that
instantaneous impact calculations are not needed. In cases where this approach
is not reasonable, gap elements can be used instead of these special interface
elements, at the cost of more computational effort.
The remainder of this section discusses the kinematic definitions used in
these elements and their contributions to the overall equilibrium equations and
to the Jacobian (stiffness) matrix needed in the Newton solution of those
equations.
Geometry and kinematics
Each tube support element has two nodes. One node represents the axis of the
tube, the other is the center of the hole in the support plate (or midway
between a pair of parallel sides of an “egg-crate”).
Let
be a unit vector along the axis of the tube, and let
be a unit vector along the axis of the interface element (that is,
perpendicular to the parallel sides of the support in the “unidirectional”
interface that is used with egg-crate supports, and parallel to the line
joining the two nodes of the element in the “cylindrical” interface that is
used with circular holes). It is assumed that
is in the cross-section of the tube and, hence, orthogonal to
.
We define a third basis vector as
Let
be the current position of node N of the element at any
point in time (here N is 1 or 2).
Relative displacements in the element are measured from the position when
the tube and the hole in the support plate are exactly aligned; that is, when
the nodes of the element are at the same location. They are defined as follows:
axial to the interface element,
axial to the tube,
and
tangential, in the plane of the tube's cross-section,
for the unidirectional case:
and, for the cylindrical case:
The basis vector—,
along the axis of the tube and of the hole in the support plate—is assumed to
be fixed. In the unidirectional element the
and
vectors are also fixed. In the cylindrical interface
is parallel to the line joining the nodes of the element, so
where
Therefore,
Thus, for this element
and
For simplicity we replace the integral with the backward difference
approximation
Forces in the element
The element generates an axial force—,
parallel to —and
two shear forces—,
parallel to
and ,
parallel to .
In addition, because the nodes of the element are at the center of the tube and
at the center of the hole in the support, while the interaction forces between
the tube and its support are transmitted at the point of contact of the tube
with the support, these forces also cause moments at the nodes of the element.
It is assumed that the moments caused by
and by
are not significant and can be neglected. The only moments considered are
at node 1, the center of the tube, and
at node 2, the center of the hole. Here
is the outside diameter of the tube and
is the diameter of the hole for the cylindrical interface or is the distance
between the parallel support plates for the uniaxial interface (see
Figure 5).
The virtual work contribution of the element is, then,
where
is the virtual rate of rotation about the -axis
at node N.
From this expression the contribution of the element to the Jacobian
(stiffness) matrix of the equilibrium equations is immediately available as
The “initial stress” terms,
are only nonzero for the cylindrical interface, for which
so that
and so
Also, for this element,
and so
This term is not symmetric.
The “initial stress” terms for the cylindrical interface are, therefore,
The other terms in the stiffness matrix are associated with changes in the
forces in the element, ,
,
and .
We assume
is made up of a spring force that is a function of
and a dashpot force that is a function of :
so that
For the implicit integration operator used for nonlinear dynamic analysis in
Abaqus/Standard,
where
where
is the time increment and
and
are the parameters of the integration operator.
Thus,
The values of
and
come from the friction theory and are defined from ,
,
and
by that theory (see
Coulomb friction).
In summary,
and
so the stiffness contribution of the element is
This matrix is not symmetric if ,
,
or
is nonzero. Without friction they are zero, and the terms in
and
are the only nonzero terms. With relatively small friction coefficients in
dynamic applications the terms
and—if the tube diameter is not very large—,
,
and
can be neglected and, thus, a symmetric approximation to the Jacobian matrix
used without serious degradation of the convergence rate of the Newton solution
of the equilibrium equations.