Layered shells exhibit localization when materials or components lose strength. This phenomenon is well-described in engineering literature, readily observed in reality, and implemented within

Software. It may be challenging to create a model such that localization occurs as it will in a real structure, though mathematical simulation does capture the mechanical attributes, and correlate with the underlying principles.

Localization in slabs

To demonstrate localization, we will observe the following case study:

Analytical model

Take a reinforced-concrete panel which is modeled using layered shell objects. The slab has an orthogonal grid of rebar, is fixed along its base, and is subjected to a vertical displacement along its top surface.

Linear behavior

Given linear response, these conditions will generate nearly uniform tensile membrane forces in the vertical direction. Distribution is not perfectly uniform because of mathematical round-off. Standard shell objects will report linear behavior beyond the cracking stress of concrete, and beyond the yield point of steel. Over-stressing is reported though deformation and stress distribution remain uniform. Given layered shells, however, this is not the case.

Nonlinear behavior

When the model uses nonlinear layered shells, stresses are nearly uniform until the concrete reaches its cracking strength, at which point nonuniform distributions may be observed, as shown in Figure 1:

As expected, this behavior is the result of localization. In both the analytical model and the real structure, localized cracks will form in distinct locations. Reinforced concrete will not lose strength uniformly, and the entire slab will not simultaneously crumble under tension. Instead, steel will carry the entire load across crack openings, and where cracking has not occurred, concrete will share the load with steel reinforcement. The example shown in Figure 1 is consistent with this explanation. Cracking is found to occur across the upper row of elements, vertical steel is found to carry all tension, and displacement is more pronounced in the region of cracking. Once this region begins to crack, stresses are relieved within the rest of the domain, which does not crack.

In reality, cracking is the result of stress concentrations which occur in locations of structural imperfection. Mathematically, cracking occurs because of round-off, and according to the geometry of the finite-element mesh.

For the cracking problem, calculated crack size will be equal to the mesh size, which should be decided upon beforehand. If detailed stress modeling is your objective, physical behavior will need to be considered when deciding upon the mesh size. If practical design information is your objective, such as with performance-based design, detailed modeling is not warranted, and may even be misleading. In this case, use the largest elements possible. Strain demand will be averaged over the larger element, and results will be more useful.

Localization in steel

Another physical example of localization is the necking seen in a steel tensile specimen. During loading, the strain is uniform over the central region of the specimen. Once strength loss occurs, it localizes and most of the straining occurs in a local region, the “neck”. The length of this region is controlled by the geometry of properties of the specimen, and is typically on the order of the width of the specimen. A finite element model of this behavior will also show localization, but the necking length will be controlled by the mesh size. This is not physically realistic, so care must be used to choose an appropriate mesh size.