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# Non-linear analysis

In the field of analysis of building structures, we tend to use the linear calculation. What does that mean? It means the results (reactions, stresses, displacement and deformation) are proportional to data (actions).

So, if we get a 3 mm sag on a steel beam when we apply a load of 50 kN, no need to re-calculate to know what will be the sag of the beam in the case of applied load of 100 kN. It will be 6 mm. But there are cases in which this does not happen in this way. In these cases we use a non-linear analysis, also called in second order. And the tensile structures are one of these cases.

There are various types of non-linearity. We are going to comment on the four most common.

1. Geometric nonlinearity.
When the movement of a structure is very large (regardless whether or not they are the deformations), a balance must be sought in the final form, so that in each load case find a different final form, and the reactions are not proportional to the actions.
2. Mechanical non-linearity: elements.
It is possible that when displacements occur, certain elements of the structure are subjected to stresses that can not support. For example, a cable under compression. In this case, depending on the type of load, active elements will be more or less resistant. Again, the results will not be proportional to the actions.
3. Mechanical non-linearity: materials.
There are materials that present non-linear stress-strain behavior. If so, it is clear that the behavior of a structure consisting of elements of this type of material, will also be not linear.
4. Mechanical non-linearity: loads.
There are actions (such as wind or snow) that depend on the form of the structure. If big displacements exist, it changes the form and they change the actions. Obviously, the results are also variable and non-proportional actions.
How do we calculate these structures? The most common way is to do it through an iterative process:
1. We calculate the structure by a conventional method.
2. We check the displacements obtained:
• There are no elements subject to stresses that cannot withstand. If there are, we don’t take into account in this iteration.
• We check that the loads taken into account not have changed due to the new geometry of the structure.
• We calculate the stresses in the structural elements taking into account the mechanical characteristics of the material that forms the elements depending on the actual geometry of the structure.
• Finally, we calculate the balance of the structural elements: “the actions need to match to the reactions”.
3. If the structure is in balance (or imbalance is very small) we have already finished the analysis. If not, go back to point 1, change the initial loads by the current imbalances and the initial geometry by the actual geometry.
4. Repeat steps 1,2 and 3 to find the balance. We call each repeat as iteration.

If the structure is balanced quickly, as we were approaching in each iteration, we say that the method of analysis is convergent, or that there is a good convergence. If, on the contrary, we do not find the balance, with frequent approaches and deviates, we say that the calculation does not converge, or that there is no convergence. The quality of non-linear calculation program is measured according to the degree and speed of convergence.

Ramon SASTRE,
May 2012  