Vehicle Crashworthiness and Occupant Protection - Chapter 3

Vehicle Crashworthiness and Occupant Protection
The constitutive law is invoked in order to estimate hourglass stresses resulting
in a set of nodal forces that are supposed to correct missing stiffness components
of the element rather exactly. This approach is much more efficient than full- or
reduced-selective integration and requires only about 30 percent more computer
time then the original Belytschko and Tsay element. The element also passes the
Kichhoff patch test.
Next to the existence of zero-energy modes, a second potential problem with the
highly efficient Belytschko and Tsay shell element is its limitation to flat
geometries. This results from the use of the Mindlin plate theory where the fiber
direction is assumed to coincide with the normal to the plate surface. If this
assumption is used in warped elements, the bending strain can be underestimated
for two reasons. First rotations normal to the element will not cause any curvature
to be calculated although these rotations may not be parallel to the nodal drill
degrees of freedom (fiber direction) in all nodes of the element. Second, force
loads in the element plane should also cause bending strains in a warped element
and will fail to do so when the unmodified Belytschko and Tsay element is used.
Those effects will lead to the failure of the so-called twisted beam test by this
element. An efficient coupling of membrane and bending deformation is advocated
in [39]. These formulations allow accurate modeling of surfaces with double
curvature using warped elements with warp angles up to 15 to 20 degrees.
It should be emphasized strongly that none of the improvements described above
affect the original assumption that only small deformations exist within a single
shell element. These improvements are merely meant to correct some deficiencies
of the element formulations within this framework. Thus, the use of more
sophisticated elements does not allow the use of coarser meshes. All rules
developed earlier to determine mesh density of a crashworthiness model still
apply. A good crashworthiness simulation can only be obtained if the mesh is
able to smoothly represent the deformed geometry of the car body. In fact, this
condition may have become stronger then before since large element deformations
may lead to numerical problems and generate instabilities that were almost never
the case with the numerically ultra-robust Belytschko and Tsay element.
3.6 Applications
This section presents a summary of typical FE models used in frontal
crashworthiness analysis.
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