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*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 {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} 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.


h1. Localization in slabs

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


h4. Analytical model

Take a reinforced-concrete panel which is modeled using [layered shell|kb:Layered shells] 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.


h4. Linear behavior

Given linear response, these conditions will produce nearly uniform tensile membrane forces in the vertical direction. Distribution is not perfectly uniform because of mathematical round-off. Standard [shell|kb:Shell] objects will report linear behavior beyond the cracking stress of concrete, and beyond the yield point of steel. When this occurs, an over-stressed condition is reported, and the distribution of deformation and stress remain uniform. Given layered shells, however, this is not the case.


h4. Nonlinear behavior

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

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!Figure 1.png|align=center,border=0!

{center-text}Figure 1 - Localization of material nonlinearity{center-text}

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Localization causes this behavior. 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 concentrated in the region of cracking. Once one region starts to crack, it relieves the stress on the rest of the region, and does not crack. This is localization.

In reality, structural imperfections cause stress concentrations, which may then advance into cracking behavior. Mathematically, cracking occurs because of round-off, and according to the geometry of the finite-element [mesh|kb:Meshing].

For the cracking problem, the calculated crack size will be equal to the mesh size. You should decide upon a reasonable size beforehand. If the goal is detailed stress modeling, the physical behavior will need to be considered to decide the mesh size. If the goal is practical design information, such as for performance-based design, detailed models are not warranted and may even be misleading. In the latter case, use the largest elements possible. The strain demand will be averaged over the larger element, and will be more useful.


h1. 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.