In CalculiX this is the general purpose beam element. The node numbering is shown in Figure 28.
In each node a local Carthesian system
is defined.
is the
normalized local tangential vector,
is a normalized
vector in the local 1-direction and
is a normalized
vector in the local 2-direction, also called the normal. The local
directions 1 and 2 are used to expand the beam element into a C3D20 or
C3D20R element according to Figure 29.
For each node of the beam element 8 new nodes are generated according to the scheme on the right of Figure 29. These new nodes are used in the definition of the brick element, and their position is defined by the local directions together with the thickness and offset in these directions.
The tangential direction follows from the geometry of the beam element. The normal direction (2-direction) can be defined in two ways:
In the latter case,
can be defined either
If a node belongs to more than one beam element, the tangent and the
normal is first calculated for all elements to which the node belongs.
Then, the element with the lowest element number in
this set for which the normal was defined explicitly using a *NORMAL
card is used as reference. Its normal and tangent are defined as
reference normal and reference tangent and the element is stored in a
new subset. All other elements of the same type in the set
for which the normal and tangent have an angle smaller than with the
reference normal and tangent and
which have the same local thicknesses, offsets and sections are also included in this
subset. All elements in the subset are considered to have the same
normal and tangent. The normal is defined as the normed mean of all normals
in the subset, the same applies to the tangent. Finally, the normal is
slightly modified within the tangent-normal plane such that it is
normal to the tangent. This procedure is repeated until no elements
are left with an explicitly defined normal. Then, the element with the
lowest element number left in the set is used as reference. Its normal and tangent are defined as
reference normal and reference tangent and the element is stored in a
new subset. All other elements of the same type in the set
for which the normal and tangent have an angle smaller than
with the
reference normal and tangent and
which have the same local thicknesses, offsets and sections are also included in this
subset. All elements in the subset are considered to have the same
normal and tangent. This normal is defined as the normed mean of all normals
in the subset, the same applies to the tangent. Finally, the normal is
slightly modified within the tangent-normal plane such that it is
normal to the tangent. This procedure is repeated until
a normal and tangent have been defined in each element. Finally, the 1-direction is defined by
.
If this procedure leads to more than one local coordinate system in one and the same node, all expanded nodes are considered to behave as a rigid body knot with the generating node as reference node. Graphically, the beam elements partially overlap (Figure 30).
Consequently, a node leads to a knot if
In addition, a knot is also generated if
Beam and shell elements are always connected in a stiff way if they share common nodes. This, however, does not apply to plane stress, plane strain and axisymmetric elements. Although any mixture of 1-D and 2-D elements generates a knot, the knot is modeled as a hinge for any plane stress, plane strain or axisymmetric elements involved in the knot. This is necessary to account for the special nature of these elements (the displacement normal to the symmetry plane and normal to the radial planes is zero for plane elements and axisymmetric elements, respectively).
The section of the beam must be specified on the *BEAM SECTION keyword card. It can be rectangular (SECTION=RECT) or elliptical (SECTION=CIRC). A circular cross section is a special case of elliptical section. For a rectangular cross section the local axes must be defined parallel to the sides of the section, for an elliptical section they are parallel to the minor and major axes of the section. The thickness of a section is the distance between the free surfaces, i.e. for a circular section it is the diameter.
The thicknesses of the beam element (in 1- and 2-direction) can be defined on the *BEAM SECTION keyword card. It applies to the complete element. Alternatively, the nodal thicknesses can be defined in each node separately using *NODAL THICKNESS. That way, a beam with variable thickness can be modeled. Thicknesses defined by a *NODAL THICKNESS card take precedence over thicknesses defined by a *BEAM SECTION card.
The offsets of a beam element (in 1- and 2-direction) can be set on the *BEAM SECTION card. Default is zero. The unit of the offset is the beam thickness in the appropriate direction. An offset of 0.5 means that the user-defined beam reference line lies in reality on the positive surface of the expanded beam (i.e. the surface with an external normal in direction of the local axis). The offset can take any real value. Consequently, it can be used to define composite structures, such as a plate supported by a beam, or a I cross section built up of rectangular cross sections.
The treatment of the boundary conditions for beam elements is straightforward. The user can independently fix any translational degree of freedom (DOF 1 through 3) or any rotational DOF (DOF 4 through 6). Here, DOF 4 is the rotation about the global x-axis, DOF 5 about the global y-axis and DOF 6 about the global z-axis. No local coordinate system should be defined in nodes with constrained rotational degrees of freedom. A hinge is defined by fixing the translational degrees of freedom only.
For an internal hinge between 1-D or 2-D elements the nodes must be doubled and connected with MPC's. The connection between 3-D elements and all other elements (1-D or 2-D) is always hinged.
Point forces defined in a beam node are not modified if a knot is generated (the reference node is the beam node). If no knot is generated, the point load is divided among the expanded nodes according to a 1/4-1/4-1/4-1/4 ratio for a beam midnode and a (-1/12)-(1/3)-(-1/12)-(1/3)-(-1/12)-(1/3)-(-1/12)-(1/3) ratio for a beam endnode. Concentrated bending moments or torques are defined as point loads (*CLOAD) acting on degree four to six in the node. Their use generates a knot in the node.
Distributed loading can be defined by the labels P1 and P2 in the *DLOAD card. A positive value corresponds to a pressure load in direction 1 and 2, respectively.
In addition to a temperature for the reference surface of the beam, a temperature gradient in 1-direction and in 2-direction can be specified on the *TEMPERATURE. Default is zero.