# Non-linear calculations

# Uplift calculations

In the Wall and Plate modules of FEM-Design there is a possibility to define point, line and surface sup- ports resisting only compression. This is a problem with material non-linearity which can be solved with the iterative method. In this case the relationship between reaction forces and displacements can be interpreted by the following diagram.

The solution implemented in FEM-Design is very simple - in the first step, when supports also resist tension, it is checked if tension appeared in any support. If yes, and the support is defined to resist only compression, then the linear static analysis is repeated with setting the stiffness values in the tensioned elements to a very small value. We repeat this procedure until there is no tensioned support any more.

If the user defines this kind of supports, he has to be aware of the direction (local coordinate system) of them, furthermore to the fact that the structure can become kinematically undetermined.

# Crack analysis in FEM-Design Plate

In FEM-Design a crack analysis technique is applied, where an iteration mechanism is calculating the effect of the cracks. As the crack analysis is a non-linear calculation the principle of superposition is not true. By this fact the crack analysis is not applicable for load groups and the calculation has to be executed for every single combination.

Generally the iteration is loading the structure in load steps, and modifies the stiffness of it in every step as more and more cracks occur during the loading process. The stiffness of the plate will be decreased only in the direction that is perpendicular to the crack lines, in the direction of the crack lines the stiffness remains the same as for the uncracked state. The key of the calculation is the way the crack direction is calculated in a certain point. Dr. Ferenc Németh from the Technical University of Budapest has invented a method for this which is based on experiments. The cracked stiffness calculation is based on a conventional cross section modulus calculation of the second crack state which is combined with a Eurocode like crack distribution calculation (to consider the effect of un- cracked parts of the plate between two cracks).

The calculation for one combination is performed in the following steps:

- Loading the structure with the loads of the combination and performing a linear calculation of the internal forces.
- Calculating the moment that cause crack on the structure in every points of the plate. This value is calculated by the tensional strength (limit stress) of the plate’s concrete material, the reinforcements are not taken into account at this point.
- Searching for the place where the ratio of the crack moment and the actual (linear) moment has the smallest value. This value will describe the initial level of the load for the iteration. The size of a load step is calculated by user defined values.
- In the first step the initial load acts and is then increased by the calculated load steps.
- In every step is calculated weather the plate is cracked or not in a certain point (comparing the smallest principal moment to the crack moment of the plate). If the plate is cracked the direction of the crack is calculated and the stiffness of the cracked section. The element where the crack occurs then will have reduced stiffness. In the next load step it will change the behaviour of the plate as the crack does in the real structure.
- When the full load is applied on the structure the calculation is continued with full load level to consider cracks occurring in the last load step and to have a stable result. This phase is called final iteration. The final iteration is finished when the differences of the sum of the movements are less than a certain error percentage between two steps. The initial error percentage is 1% compared to the previous step, but this value could be adjusted.

Notes:

- It is possible that the plate is cracked in two directions in the same side. This is the case when the largest as well as the smallest principal value is over the crack limit. In this case the stiffness of the plate will be decreased in both directions (parallel and perpendicular to the crack line).
- It also can happen that the plate is cracked on both sides of it, but in this case the crack lines are nearly perpendicular to each other (depends on the reinforcement parameters).
- During the calculations the direction of the cracks and the stiffness of the cracked parts are recalculated in every step. This is because the cracks make changes in the behaviour of the structure, and depending on this the moment distribution is changed continuously along the structure. By numerical reasons the newly calculated directions and stiffnesses are not applied immediately with their full value but an intermediate value is used between the previous and the newly calculated values. This well known technique makes the calculation longer but the chance of success is increased. This technique is one reason why a final iteration is needed.
- As the numerical techniques are mandatory to get correct results and the- se techniques are affected by the structural conditions and by user defined values the user should be warned that a certain load step and final error value which is good for one structure perhaps is not suited for an- other structure. Smaller load steps means generally more accurate results, but the price is longer calculation time.
- The crack direction calculation is based on the least remaining moments method. This method suppose that the crack direction will be the same as the crack when the capacity of the plate is reached. In every investigated point the moments are increased virtually (multiplied with a certain value) until the yield state is reached. The method of Ferenc Németh can describe the crack line direction on this level.
- The stiffnesses of the cracked sections can be described in the directions of the reinforcements but the cracks occur in any direction. To calculate the stiffness perpendicular to the crack lines an average calculation method invented by Dr. Ferenc Németh is used. The technique is based on experiments.
- By the limitation of the finite element method the internal force distribution will not be as smooth as can be seen for uncracked structures, there would be small peaks on the border of two elements that have different crack direction and/or stiffness which is a normal state during crack analysis.