C16. More about dynamic and seismic calculations

The dynamic solicitations generate inertial and kinematic forces in the structure.

If Eurocode 8 is used for the design, a detailed explanation of the permissible simplifications is provided for the modeling and calculations of the structure, according to its uniformity. This notion of uniformity is explained in §4.2.3 of EN1998-1. For bridges, the guide "Ponts courants en zone sismique" also provides uniformity criteria and admissible simplifications for the calculations.

Depending on the refinement of the model and the objectives sought, several points should be considered.

C.16.1 Defining the general X and Y axes

Defining the general X and Y axes warrants particular attention. Indeed, seismic results can be erroneous if these axes are not close to the main axes of inertia of the structure.

2 examples illustrating the subject can be visualized below.

Example: Corner block of a stadium

Overview of the corner block model

It is necessary to adopt for the corner blocks different reference axes (X, Y) from those of the general project (XG, YG). The X-axis must be radial in the direction of the 1st vibration mode.

The study of the eigenmodes then shows that the fundamental modes of a classical building with well-defined modes are retrieved according to the X and Y directions as well as a torsion mode. This would not be the case if the general axes XG and YG were adopted because each mode would activate masses in both directions, which would disturb the Complete quadratic combinations (CQC) and Newmark combinations.

Example: Earthquake on a curved bridge

In the example below, two seismic calculations were performed for this TGV viaduct with an in-plan curvature.

The first computation depends on the general X and Y axes oriented according to the left abutment and the second with the axes oriented according to the right abutment.

It is possible to switch from one model to another by changing the coordinates of the model nodes. The comparison of the efforts at the base of the supports highlights very different results between the 2 models.

Graph n° foot support/effort (kN)

Therefore, it is advised to carry out seismic calculations for bridges on straight planar alignments if the curvature allows it (refer to the CEREMA guides). Otherwise, several calculations must be conducted by varying the axes for each support studied, it is a complex solution that should be avoided when possible.

C.16.2 Modeling of non-structural or secondary elements

See C.4 Modeling of non-structural elements or equipment.

Non-structural elements and equipment do not need to be modeled if their mass is small enough relative to the mass of the building. However, one must ensure that their stiffness do not influence the behavior of the structure. Otherwise, they must be considered. For more information, please refer to §2.4.5.2 of the ASN guide and the AFPS technical book 36 of the AFPS concerning the dimensioning of non-structural elements under seismic conditions.

When the mass of these non-structural elements is not negligible, it must be correctly defined in space. For example, the offset of the bridges’ lateral elements’ masses with respect to the center of the sections should be well defined so that the torsion modes are considered.

The distribution of the masses on the floors can generate many local modes during a modal analysis and make the model difficult to operate. In this case, it is recommended to apply punctual masses and to redo a local study if necessary.

C.16.3 Transformation of loads into masses

Most software calculate the weight of the structure directly or have an option to do so based on the density or volumetric weight of the materials.

For seismic and dynamic calculations, the weights, forces, and pressures must be converted into masses following the normative ponderation factors (it is the case for equipment and superstructures or part of the operational loads).

To reduce the number of eigenmodes that are not useful for a global calculation, one can focus on options that transform the distributed masses into masses at the nodes or introduce manually the masses at the nodes.

C.16.4 Pay attention to the units

The accuracy of seismic or dynamic calculations is particularly sensitive to the coherence of the units. Inertial forces involve the acceleration of gravity g, whose unit (generally defined by default) must be consistent with all others.

Trivial, but it is always useful to remember that the unit of mass is ... the kilogram.

Be careful when using old standards or regulations because they may use units such as kgf (kilogram-force). In general, it is advised to use exclusively the international system units, at the very least to control the results.

Taking the simple example of the self-weight calculation. For most software, the action of gravity is defined by the application of vertical acceleration applied to the whole structure. Internally, the software will calculate the mass of the structure by first calculating for each element its volume multiplied by the density of its material. If one wishes to obtain the self-weight in N and the geometrical dimensions and density have been defined respectively in mm and kg/mm3, the acceleration will have to be defined in ... m/s²:

For example, the mass equivalent to an operational accidental load of 20KN is equal to 20,000(N)/ 9.81(m/s2) = 2038 Kg or 2,038 tons.

C.16.5 Materials

The material laws as well as the partial coefficients depend on the type of analysis performed.

Taking concrete as an example, the instantaneous Young's modulus will be preferred.

To consider the cracking state of the elements, the EI module can be modified:

either by a minor coefficient applied to Young's modulus E,

or by modifying the net cross-section or the inertia of the section directly.

The applied reference frame can specify the Poisson's ratio to apply according to the type of calculation. it can be modified to consider the state of cracking, for exemple, equal to zero in the case of a cracked element or under ALS (Accidental Limit State) earthquake.

C.16.6 Modeling of bracing elements of steel structures

Bracing elements ensure the lateral stability of the structure. It is important to transcribe their actual behavior. For St. Andrew's Crosses, for instance, the bars only work in tension because they buckle instantaneously in compression. Therefore, they should not be modeled in their entirety if a linear calculation is planned. Otherwise, the capacity of the bracing would be overestimated by a factor of about 2. See C.2 Modelling of the main elements

C.16.7 Boundary Conditions

Depending on the models, the dynamic soil-structure interaction may need to be considered. If the springs are modeled this way, it is necessary to ensure that the structure do not uplift excessively to remain within the validity range of a linear study.

For earthquake studies, the engineer can calculate stiffnesses by referring to the following documents:

"Ponts en zone sismique" published by CEREMA, which proposes in chapter 4.3.3.2 fairly simple calculation formulas,

Seismic Design-Building - V. Davidovici §4.3.4.4 Modelling of the ground by a system of damped springs - Eurocode Collection - Afnor Editions,

Gazetas formulas: which can be consulted in Appendix D of "Fondations et procédés d’amélioration du sol de Davidovici" (or other references).

All these documents determine the stiffnesses from the shear moduli and Poisson's coefficients of the soils, but also from the characteristic dimensions of the foundation. These stiffnesses depend on the vibration frequencies of the studied structure.

The case of foundation slabs:

Modeling a foundation slab under dynamic loading is more complex because the springs will have to represent at the same time the vertical, horizontal, and rotational stiffnesses, as determined by the soil-structure interaction study.

One can refer to more specific documents for this type of study.

Several forms of modeling are possible:

using a punctual spring in the center of the invert affected by the 6 stiffnesses (and 6 dampings), with rigid connections on all the nodes of the foundation slab,

Advantage: an accurate representation of the SSI in dynamic calculations.

Disadvantage: it is not possible to determine the stresses in the foundation slab because of the presence of the rigid connections that artificially stiffen it. It is then necessary to carry out a local calculation of the foundation slab subjected to the soil pressures deduced from the forces in the central spring. More specifically, in the case of a foundation slab with large dimensions compared to its thickness, this method is not adapted.

using springs placed uniformly under the foundation slab (as for the static study);

In this case, each node of the invert is connected to 3 springs, one in each direction X, Y, Z.

The horizontal springs according to X or Y will be deducted directly from the global translational stiffnesses, whereas the stiffnesses of the vertical springs will represent either the global vertical stiffness or the global rotational stiffness in a given direction. This approach implies 3 computational models to analyze the 3 directions of the earthquake.

Advantage: simpler model, allowing to calculate the forces in the foundation slab.

Disadvantage: one of the 2 vertical or rotational stiffnesses is not represented in each of the calculation models. The torsional stiffness is not incorporated.

putting in place a spring mattress,

This type of modeling is mainly used in complex structures, a mattress of springs assigned to each node of the foundation slab allows to represent all the global stiffnesses.

Advantage: the SSI is modeled in detail.

Disadvantage: the modeling is complex and can only be applied using specific software with a complete understanding of the causes.

Figure: Diagram of the spring mattress, Tractebel image

C.16.8 Spectral modal analysis

Truncation - number of modes

The theoretical concept of truncation is defined in the 1st part of this guide. In practice, concerning the number of modes to be used for the calculation, we advise:

not to exceed 100 modes for classical works,

to go up to the cut-off frequency (generally 33 Hz),

to use a pseudo mode for the participating mass that is not being considered (EN 1998-2/§4.2.1.2),

not to be limited to the modes with the most participating masses, because the antisymmetric modes have classically a very low number of participating mass but induce non-zero forces,

to reflect on the relevance of retaining or not local modes in the analysis.

Behavioral coefficients

Since the coefficient, or rather the behavioral coefficients, can be different in each direction, they are incorporated in the definition of the calculation spectra. It is important to check that the calculated displacements are indeed re-multiplied by this same coefficient.

Modes signature

At the end of the combination of spectral responses, the sign of the efforts is lost (all values are positive). This can generate operating difficulties when one wishes to calculate a torsor or when one wishes to study concomitant forces (see D 7.4.5).

To reallocate a sign to the different calculated quantities, there are several possible approaches, including those described below:

Attribution of the sign of one of the modes. For structures having a predominant mode in each direction, it is possible to allocate the sign of the predominant mode to the calculated quantities. This is interesting for the overall behavior of the structure and is very efficient as long as the participation of this mode is greater than 60% of the modal mass of the structure. On the other hand, for elements responding on higher local modes, this may not be appropriate (see the example of thick floors in industrial sites),

Attribution of the sign following a uniform acceleration analysis. For each direction, a unit acceleration is applied, and the sign obtained is kept,

Ellipsis method analysis. When the justification of a structural element must consider several stress components, it is possible to establish the range of concomitance of these quantities, so as not to introduce conservatism in the calculation.

C.16.9 Damping

Within the framework of a structural study with a calculation spectrum including a behavioral coefficient, the latter already considers damping. Thus, it is not useful to introduce another one.

For dimensioning with an elastic response spectrum, the damping of the materials must be taken into account.

When setting up the data, it is important to ensure that the material damping included or taken by default by the software is consistent with the codes and the analyses carried out. For example, it is necessary to distinguish reinforced concrete from prestressed concrete, or welded structures from bolted structures in the definition of material damping.

C.16.10 Time discretization and integration scheme

The resolution of a dynamic solicitation requires the implementation of a specific integration method. Part 1 Chapter 2 provides details on these methods and guidance for choosing the time step and mesh size according to the problem. It is recommended to consult engineers specialized in this type of study.