Beam added inertia

There are two sets of problems presented in this section. The first set includes four input files: b31_dyn_iso.inp, b31_dyn_exact.inp, b31_moddyn_iso.inp, and b31_moddyn_exact.inp. These analyses compare the dynamic response to an acceleration record on a single-element cantilever structure made of B31 elements using the isotropic or exact rotary inertia formulation. Comparisons are made between the direct-integration implicit dynamic and the modal dynamic procedures. To change the rotary inertia formulation for Timoshenko beams, isotropic rotary inertia or exact (default) rotary inertia is used for the beam or beam general sections.

The second set of problems verifies the beam with the additional inertia procedure. This procedure allows adding mass and rotary inertia properties per element length at specified locations on the beam cross-section. The beam's mass together with the added mass may combine to give an offset between the location of the node and the center of mass for the cross-section. That offset produces the coupling between the translational degrees of freedom and the rotational degrees of freedom in the mass matrix for the element. A pair of input files, xbeamaddinertia_std_lin3d.inp and xbeamaddinertia_std_quad3d.inp, shows the concept of the offset mass for the beam element that can also be modeled with MASS and ROTARYI elements with appropriate BEAM-type MPC definitions to accommodate the mass offset. The remaining single-element input files verify various cross-section types for transient dynamic and eigenvalue extraction procedures. Input files pmcp_pipe2d_bai.inp, pmcp_beam2d_bai.inp, pmcp_pipe3d_bai.inp, and pmcp_beam3d_bai.inp are collections of all pipe and all beam elements placed in a plane or space. The beam with the additional inertia procedure is used for all beam section definitions. These multiple step analyses verify the frequency, static (with mass-dependent loads), steady-state (mode-based and direct), modal dynamic, and direct-integration implicit dynamic procedures.

This problem verifies the use of beams with additional inertia in Abaqus/Explicit. Identical beam elements are assigned additional mass and rotary inertia in two ways: using the beams with the additional inertia procedure and by defining additional point mass and rotary inertia elements and rigidly constraining them to the beam nodes using BEAM-type MPCs. The solutions obtained using the two methods are compared. Four cases, each comprising one of the four beam element types available in Abaqus/Explicit, are considered.

For each case four beam elements with the same element length are defined. Two of the beam elements are assigned identical section properties using the beam section procedure, and the remaining two are assigned identical section properties using the general beam section procedure.

One of the elements with section properties given by the beam section procedure has additional mass and rotary inertia assigned to it using the beam with the additional inertia procedure. For the second beam element with a beam section, additional nodes are defined at locations offset from the element nodes and MASS and ROTARYI elements are defined at the offset nodes. BEAM-type MPCs connect each node of the second beam to its corresponding offset node. The offset node corresponding to each node of the second beam lies in the cross-section passing through the beam node and has the same local coordinates with respect to the beam node as the center of mass coordinates defined for the first beam. Similarly, the mass and inertia assigned to the offset nodes are exactly equivalent to those assigned to the first beam element using the beam with the additional inertia procedure.

The two beam elements with general beam sections are also subjected to the same test. One of them is assigned additional mass and inertia, while the other has BEAM-type MPCs connecting each node to nodal locations offset from it where MASS and ROTARYI elements with appropriate section properties are defined.

All four beams are cantilevered at one end and are subjected to the same concentrated load at the other end.