D7. Further information specific to dynamic calculations

D7. Further information specific to dynamic calculations

D.7.1 Total mass verification

In the case of dynamic studies, one of the fundamental parameters is the mass of the structure which is used to determine its eigenfrequencies.

It is therefore very important to make sure that the entire mass of the structure is actually entered into the model. Indeed, in the case of using a model that has already been used for static calculations, it may happen that some permanent or variable loads, such as equipment, have been entered as loads (point loads, linear loads, surface loads, etc.) and not as a mass. Therefore, it can happen that the software doesn’t consider these loads as masses but only as overloads, and does not take them into account in its mass calculation. This may result in a reduction of the seismic forces.

It is therefore always necessary to make sure that the total mass of the structure is indeed the desired one. This information is generally available in the results of modal analysis or, even better, can be obtained by performing three static calculations, by applying a unit acceleration field in the 3 directions (X,Y,Z): only the elements with mass will therefore be taken into account, and the sum of the reactions will therefore make it possible to know the mass actually taken into account in the model, in each direction.

D.7.2 Verification of the participating masses

It should be verified that the modal analysis carried out takes into account enough eigenmodes. For this, it must be verified that the participating modal masses in the studied direction and cumulated for the different calculated modes, represent at least 90% of the total mass that can be set in motion, calculated from the unit cases of acceleration, otherwise the standards authorize the taking into account of a pseudo mode (per direction).

Trap: Some software indicates cumulative modal mass % which may be based on a wrong hypothesis of mobilized total mass: in fact, the parts of masses blocked in movement by supports will not be counted by the software, which will therefore overestimate the mobilized modal mass %. A trick to overcome this is to define elastic supports with high stiffness rather than fixed supports: the total mass will then be exact.

In general, it is preferable not to model mass associated with fixed supports.

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Example of the study of a skewer model with 5 degrees of freedom:

Three cases are studied:

• Case 1: Similar masses and stiffnesses at all levels;

• Case 2: Case 1 but with a stiffness 100 times higher in the height of the first floor;

• Case 3: Case 1 but with a stiffness 100 times higher in the height of the first 2 floors.

Assuming that all periods correspond to the spectrum plateau (identical spectral value for all periods):

It is therefore important to take into account all significant modes otherwise the calculation efforts could be significantly underestimated.

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 Trap: Symmetrical and anti-symmetrical modes.

Example of beam vibration

Depending on the type of calculation being carried out, modes that do not provide a % modal mass supplement in a given direction should not necessarily be considered irrelevant.

Simple case of the beam on two supports - the masses are mobilized only vertically. The table of modal results shows that all even modes do not add any additional % modal mass.

Looking at the modal deformations, we realize that these are modes with anti-symmetric deformations:

Modal deformation - mode 1

Modal deformation - mode 2

In the case of a spectral seismic calculation, these modes do not actually add anything new, whereas in the case of a beam or bridge vibration calculation, these modes have all their interest. It is indeed admitted that pedestrians, in their movements, can have actions in opposition and in the direction of the modal deformation. A harmonic calculation is indeed carried out from the loads positioned as below:

We will usefully refer to the SETRA (operating society for transport and automobile repairs)/CEREMA (center for studies and expertise on risks, environment, mobility and development) guide on pedestrian footbridges for more information.

Spectral analysis: finally, we give below the forces at the middle node of this beam, calculated by a spectral seismic analysis - it can be seen that even modes do not actually make any contribution.

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 Trap: Torsional modes

Example of a building in torsion

Generally, common buildings have a torsional mode. On the example below, we can observe:

• The 1st mode: with a preponderant mode according to UY (longitudinal),

• The 2nd mode: with a preponderant mode according to UX (transverse),

• The 3rd mode: with little modal participation while it significantly affects the structure. It is a torsional mode.

D.7.3 Verification of the main eigenfrequencies

In order to verify the effective behavior of the structure in dynamics, it must be ensured that the main eigenfrequencies of the structure have a coherent rough estimate.

These frequencies correspond to the main modes of bending, torsion, shear, and they generally correspond to modes with important participation factors and participating masses.

It should nonetheless be noted that limiting to modes with a significant participation factor is not an exhaustive guarantee of the modes that may be problematic under dynamic loading. Indeed, participation factors can be calculated by software based on signed modal displacements. Thus, it may happen that the participation factors taking into account the accumulation of values return low values when the mode is important.

This can happen, for example, for a continuous beam on three identical supports with two identical spans. The main mode of bending of this beam is the bending of one span in one direction and the bending of the other span in the other direction (wave form). The participation factor of this mode can be very small, as the displacements of one span counterbalance the displacements of the other span in the calculation, whereas this mode is the main bending mode of the structure, and can be the one giving the highest acceleration response if the structure is subjected to periodic excitation.

When the structure is complex, the dynamics formulas given in the literature do not allow to find precisely the eigenfrequency values obtained, since these formulas concern simple structures (eigenfrequency of an isostatic beam on two supports, of a cantilever beam, of a fixed end beam or an oscillator with a few degrees of freedom). However, these classical formulas allow to estimate the rough estimate of the main eigenfrequencies by estimating in a simplified way the behavior of the structure to reduce it to simple functioning for verification.

In the case of a beam type structure, we can thus estimate the eigenfrequencies of bending from the classical formulas. For example, and in a very general way, the eigenfrequency of bending of a beam on two supports with rotational flexibility will be between the eigenfrequency of bending of the same beam but on hinged supports and the eigenfrequency of bending of the same beam on fixed supports.

D.7.4 Modal/spectral dynamic seismic calculations

D.7.4.1 Verification of the first modes' relevance (instabilities, displacements)

The first 2 or 3 global modes visualize the functioning of the structure, which allows on the one hand to understand how it works, and on the other hand to identify modeling problems.

For a well-dimensioned building, the first 2 modes are always according to X and Y, the third mode is a torsional mode.

For common buildings, the fundamental period is of the order of 1/25 to 1/16 of the number of floors.

D.7.4.2 Verification of the global X and Y axes with respect to the first modes

It is necessary to verify that the seismic directions studied X and Y are aligned according to the first important modes. If not, the complete quadratic combination (CQC) calculations will be wrong.

The solutions are:

• in the case of a building, to rotate the model according to the main axes

• in the case of a curved bridge, to either calculate the earthquake on a straight model, or make several calculations by varying the axes according to the skew of the piles.

D.7.4.3 Validation of the number of modes - Complementary mode

Eurocode 8 specifies a minimum percentage of mass to be of interest in the spectral calculation.

If 90% of the mass is not interested, the software allows a complementary mode to be taken into account. It is a fictitious mode affected by the mass not excited by the studied modes and affected by the spectral acceleration associated with the last studied mode.

It has been shown that neglecting modes distorts the results (see 1st example in D.7.2).

D.7.4.4 Spectral Calculation and CQC Combinations

The spectral calculation allows to obtain the structural effects (efforts, displacements...) of each mode. Then, according to the statistical distribution of the earthquake according to the frequencies (defined by the regulatory spectrum), the effects are combined in order to obtain the statistical response of the structure to an earthquake.

The combination of the different unit modal calculations is done according to the CQC or SRSS mode, the theoretical definition of which is provided in Part 1 - Chapter 2.

It is important to differentiate between the regulatory spectra used in construction (which are generally dimensioning spectra) and those used for bridges (which are elastic spectra).

We go from the second to the first by dividing by a behavior coefficient (equal to or higher than 1.5) which takes into account the plasticity of the structure. The values of the behavior coefficients are defined by Eurocode 8.

In all spectral calculations, it is important to be sure that the damping of the structure is well defined. Depending on the software, the damping is assigned to modes or materials. In the second case, the mode damping will depend on the participation of each material in the deformation of the considered mode.

Caution, if we want to attribute a damping in the springs modeling the soil, this corresponds to a study of type soil-structure dynamic interaction, and it is not possible to use a dimensioning spectrum, only an elastic spectrum.

Finally, as indicated in C16.8, a different behavior coefficient can be assigned to each direction.

After the CQC or SRSS combinations (which combine the modes), the Newmark combinations must be made (seismic direction combinations), and then the action combinations.

D.7.4.5 Verification of support reactions under elementary cases

First of all, we evaluate the support reactions of the elementary seismic cases EX EY EZ and we compare it to the total masses.

Verification of support reactions can only be done by signing the modes.

For a building based on a base slab or a strip footing, it is important to limit the detachment of the supports. Indeed, the elastic calculation shows tensions in the supports that do not exist in reality because the foundations heave.

Negative support reactions (in red) cannot exist because in reality the foundations heave, so the actual stress distribution on the ground is different from the calculated one (cf. C16.8.3).

Support reactions of a building under superficial foundations

It is allowed to consider representative "elastic" models if the heave is limited: the limit is taken equal to 30% in the general case (10% in the case of nuclear buildings).

When detachment is important, much more complex non-linear seismic calculations are required. This verification of non-heave of the building must be done with care:

• it is necessary to give to the modes according to the main modes because CQC support reactions are always positive.

CQC Support reactions / Signed CQC Support reactions 

Seismic combinations (CP + E) make no sense if the seismic E efforts are all positive, while the CP dead loads are either positive or negative.

• It is necessary to study all Newmark combinations separately.

Example of the 24 Newmark combinations for the current frame:

• CP +/- 1.0 EX +/- 0.3 EY +/- 0.3 EZ

• CP +/- 0.3 EX +/- 1.0 EY +/- 0.3 EZ

• CP +/- 0.3 EX +/- 0.3 EY +/- 1.0 EZ

D.7.5 Dynamic calculations other than seismic calculations

This paragraph concerns dynamic calculations other than spectral modal calculations (e.g. vibration of a railway structure or a pedestrian bridge in order to verify comfort criteria), and corresponds to calculations in which the load and the structure are calculated with an evolution over time.

D.7.5.1 Verification of free vibration or resonance behavior

In the case of a comfort study of a railway structure or a footbridge, it is advisable to look for loads cases that can cause the structure to resonate and the consequences of such resonance. For this, it is therefore necessary to apply periodic loads that can cause these resonances.

As a reminder, the resonance of the structure occurs when the periodic loading is at a frequency identical to one of the structure’s eigenfrequencies.

To verify that the frequencies of the applied periodic loads are consistent with those of the structure, we can easily find them on the time graphs obtained a posteriori. This method is applied to the temporal evolution of the acceleration, velocity or displacement of a representative node (for example the middle of a beam).

To do this, we count the number of periods between two distant points in time and divide this number by the time separating these two points. This gives a good approximation of the vibration frequency of the node in question:

Structure’s first eigenmode   temporal acceleration of a representative node

6.36 Hz/6.25 Hz =1.018; the approximation giving 6.25 Hz gives the right rough estimate
=> the node is excited according to the first eigenmode of the structure.

If the curve shows a very marked periodic variation over the time period in which the loading is applied, this corresponds to a forced vibration of the structure, and the method described above ensures that the frequency of the loading is indeed that expected.

If the curve shows this periodic variation over a time period after loading (the calculation was continued after the loading was stopped), this corresponds to the free vibrations of the structure. The method described above allows, in this case, to estimate the main eigenfrequency of the structure and thus to make sure that the excited eigenmode is the right one.