This is a classic case of localization, which can occur Nonlinear shells may exhibit strain localization when materials or components lose strength. This phenomenon is well-described in the engineering literature and easily , readily observed in nature. It is not always easy to make the behavior in a model follow that in nature, but the cause is the same.
If you look at the resultant force in the model, it is uniform, or very nearly so, throughout the loading. The stresses in the layers are also uniform until the concrete reaches cracking stress. It is not expected that the entire structure will lose strength uniformly. In fact, when cracking occurs in a real concrete structure, you see distinct localized cracks. The whole wall does not simultaneously disintegrate in tension. In the physical case, where the cracks occur, the steel is carrying the entire load. Where the wall has not cracked, the concrete shares the load with the steel.
This is also what is happening here. The top row of elements has cracked, and the steel is carrying all the load. For the rest of the rows of elements, the steel and concrete are sharing the load. Just before cracking, the stress is uniform. However, it is not perfectly so because of numerical round-off in the calculations. In a real structure imperfections would have the same effect. Once one region starts to crack, it relieves the stress on the rest of the region and it does not crack. That is called localization.
Another physical example of localization is the necking seen in reality, and implemented within CSI 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.
Strain localization in slabs
To demonstrate strain localization, we will observe the following case study:
Analytical model
Take a reinforced-concrete panel which is modeled using nonlinear 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. Linear 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 nonlinear 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:
Figure 1 - Localization of material nonlinearity
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.
Strain localization in steel
Another physical example of strain localization is the necking of 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 most strain occurs in a local region, or the “neck” neck. The length of this region is controlled by the specimen geometry of properties of the specimen, and is typically on the order of the specimen width of the specimen. A finite-element model of this behavior will also show demonstrate localization, but though mesh size will control the necking length will be controlled by the mesh sizeof the necking region. This is not physically realistic, so care must be used to choose an appropriate mesh size.For the cracking problem, the calculated crack size will also be equal to the mesh size. The user needs to decide what a reasonable size is for this. 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 usefulthe choice of an appropriate mesh size merits consideration.