Frame Analysis using Beam Elements
Beam elements are widely used in engineering and the sciences to simulate complex physical problems. Computationally, the use of these elements can be effective because the analysis models are usually easily constructed, and the matrices in the finite element equations can have a small bandwidth. However, beam elements are formulated based on various kinematic and stress assumptions that in nonlinear analysis can be quite complex, for example considering closed and open sections in torsion. Of course, a specific beam element should only be used when, for the phenomena to be simulated, the underlying element assumptions are reasonable.
With ADINA* the analyst can use, for general nonlinear analysis, ‘the standard beam’ and the more powerful ‘warping beam’. Only two different beam elements are offered, both based on Hermitian displacement assumptions, in order to simplify the modeling of structures. The elements can have various cross-sections and can be used in large displacement elastic and elasto-plastic analyses. For open-section beams, the warping beam should be used in which the change in the angle of twist along the length of the beam is properly modeled. Of course, the beam element cannot represent local web or flange buckling.
The objective in this Brief is to illustrate the use of the ADINA beam elements in a typical structural analysis and compare the beam results with those obtained using a shell model of the structure.
We consider the frame structure shown in Figure 1. The frame consists of box beams for the columns and I-beams for the girders.
The member descriptions are briefly summarized in the Table 1 below.
Table 1 Section and material descriptions of the frame members
The objective of the analysis is to determine the lateral displacement of the frame for various load levels. In this type of analysis it is important to consider both geometric and material nonlinearities. The geometric nonlinearities are included by use of a large displacement formulation, and the material nonlinearities are included by use of elastic-plastic material models. In a frame analysis, the effects of the geometric nonlinearities lead to increased inelastic behavior, leading to additional displacements and additional geometric nonlinear effects.
We employ two frame models: a beam model and a shell model.
Beam model: For the columns, the box cross-section is used along with the standard (non-warping) beam element. For the girders, the I cross-section is used along with the warping beam element. Rigid links are employed to model the connections between the columns and girders.
Shell model: MITC4 shell elements are used throughout. Again, rigid links are used to model the connections between the columns and girders.
The results are as follows:
Frequency analysis: The first several frequencies/modes agree very well between the beam and shell models, as shown in Table 2 and Figure 2 below, and the movies above. However it is important to note that the 'warping beam' must be used for the I cross-section beam, otherwise wrong results would be obtained.
Table 2 Natural frequencies of frame models (Hz)
Figure 2 Mode 1 of the frame calculated using the beam model and the shell model
Figure 3 shows the force-deflection curves for the following analyses:
Linear elastic analysis: The stiffnesses of the two models are similar, with the shell model slightly softer than the beam model. The difference is primarily due to lack of shear deformation effects within the Hermitian beam elements.
Large displacement elastic analysis: Both models soften as the load is increased, due to the P-delta effect. Also, one of the girders buckles, as shown in Figure 4 below.
MNO elastic-plastic analysis: Both models exhibit a limit load, with a very little higher limit load obtained with the shell elements.
Large displacement elastic-plastic analysis: Both models exhibit softening after the limit load is reached. The limit load is smaller than the limit load obtained in the MNO analysis. The shell model softens more rapidly than the beam model. This additional softening is due to local buckling effects in the shell model, which cannot be captured in the beam model.
Figure 4 Large displacement elastic analysis results using the beam model and the shell model
Clearly the advantages of using the beam model are the relative simplicity, ease of model definition and solution speed, whereas the advantages of the shell model are the ability to include shear deformations and local buckling effects.
When local buckling is not important, the ADINA beam element, with the option of cross-section definition and an elastic-plastic material model, can be very effective in the geometrically and materially nonlinear analyses of frames and other structures.
The fact that these beam elements can be used with all other solution capabilities, including FSI and multiphysics, makes ADINA a powerful tool for many applications.
*ADINA version 9.0 is used