C13. More about non-linear calculations

This paragraph deals with non-linearities related to the laws of materials and the so-called geometric non-linearities.

It is common to associate nonlinear calculations with complex structures such as cable-stayed and suspended bridges, yet this subject appears in the daily life of any structural engineer, for example in the case of:

a partial detachment of foundation footings,

the inability of certain bars to work in compression (see § C.2.7 about bracing),

a buckling calculation in reinforced concrete,

cases beyond the field of beam theory (for example, the calculation of stresses in a bridge bracing under its own weight).

In general, for all non-linear calculations, it is important to perform a linear calculation before accounting for non-linearity to understand the behavior of the structure and the specific effect of non-linearity.

C.13.1 Theoretical Geometry and Imperfections

Most standards require that non-linear calculations incorporate an initial imperfection in the geometry of the structure or the elements implantation. Some software can directly integrate this imperfection. For others, it will be necessary either to apply a load case that creates the initial imperfection, or to define the geometry with the defect.

It can be observed that in a finite element calculation, the use of triangular elements always allows us to consider the pre-deformation of a flat surface.

C.13.2 Ropes and cables

The ropes and cables are essentially non-linear elements since they operate only in traction and because of the chain effect, an "apparent" Young's modulus must be considered. This modulus is a function of the tension, length, density, and gross Young's modulus of the cable.

In earlier phases of the project, it is not imperative to take these effects into account. It is possible to model the ropes using a bar, ideally bi-articulated, making sure to neglect the dead weight of the bar or to apply it manually directly to the edges. It should then be verified in the analysis that these bars are not compressed.

C.13.3 Zones with material non-linearity

A first linear calculation is used to identify the areas where non-linear behavior will appear. The calculation will continue by successive iterations, progressively integrating the non-linearities.

C.13.4 Buckling and large displacement calculations

Buckling - calculation of critical coefficients

Most of the software can determine the critical buckling loads of the compressed bars (i.e. the buckling lengths of the bars composing a structure) from a modal calculation, in small or even large displacements. The calculations must be performed for each combination. Many software also allow to carry out the normative verifications from this calculation of critical loads (or simply by manually entering the buckling lengths).

The calculation of critical coefficients is based on the search for αi values such as the determinant Det([Ko]+αi [Kσ])=0, where Ko is the stiffness matrix associated with small displacements and [Kσ] the stiffness matrix associated with initial stresses.

The objective here is not to develop all the possibilities offered by the software, but to insist (once again) on the fact that the modeler must understand what a given software does and what is the impact of the modification of the calculation parameters. A simple parameter may be the required subdivision of the bars to obtain the right results, as shown in the example below.

Illustration on the braced gantry in Chapter C.2. Link to the example of the calculation of critical buckling coefficients.

This small example confirms that it is necessary to master what the tool does. Moreover, when more challenging calculations are being performed (not linear elastic, or first-order), one should always refer to simple examples from the literature.

Calculations in large displacements:

These calculations require updating the stiffness matrices at each iteration, whether in reinforced concrete or steel structures. What was highlighted here before for the calculation of critical buckling coefficients with respect to the control of the software parameters, remains perfectly applicable.

We refer to two interesting articles on the subject:

- Calcul au flambement des arcs - Comparaison entre un calcul approché et un calcul en grands déplacements du Bulletin Ouvrages d'art n°32“ - Lien vers l'article.

- Instabilité par flambement des arcs (CTICM) - Lien vers l'article.