B4. Specificities of the seismic analysis

B4. Specificities of the seismic analysis

Spectral response – Specific case of earthquakes

The principle of the method is, for a given seismic direction, to construct the maximum responses from the loading spectrum at all points, mode by mode, then, to accumulate all of them using various methods. 

When the seismic response is obtained for a given direction, the seismic directions are combined to get the global response. 

Let us be in a case where the modal vectors are normalized according to the mass matrix, which is the case for most of the FE computational models.

One can write the expression for a degree of freedom in a canonical form:

The term p_ik is called the weighting coefficient. It is an important concept in the case of seismic analysis because it restores the physical contribution of excitation in a given direction k. 

It is determined for the cases in which the modal shape vectors are normalized according to the mass matrix:

Moreover, for a seismic motion q¨_s(t) in a given direction Delta_k, the mean solutions greater than the maximum values of r¨_ik(t), r°_ik(t), and r_ik(t) in terms of pseudo-accelerations, pseudo-velocities, and displacements via the respective spectra of this motion, are known:

This response is constructed for a modal damping ξ_i that can be evaluated as a proportion of the modal strain energy (see §1.2.1 of Génie parasismique Tome 3, J. Betbeder-Matibet).

with:

• E_iT  is the total strain energy of the structure for mode i such that: E_iT=1/2 φ_i^TK φ_i

• E_{i material j}  is the internal strain energy of the structure with the damping of the material being ξ_j.

Then, from the simple oscillator response, one can construct the response in the structure for each mode, for the acceleration, the velocity, and the displacement respectively:

These modal values must then be combined for a given loading direction to obtain the global response of the structure. There exist several cumulation methods (from now on, a_k,p, for the sake of example, denotes the component p of the a_i,k vector, such as φ_i,p with φ_i) as shown below:

• The “Square Root of the Sum of the Square” (SRSS) – At all nodes p:

• The Total Quadratic Cumulation (TQC) – At all nodes p:

with P_ij denoting the quadratic coupling coefficient of the modes i and j, such that:

It is important to note that, in this case, it concerns a cumulation of algebraic terms, but the signs are all positive, placing it under the square root don’t cause any problem. 

Let us consider the modal damping values ξ_i and ξ_j equal to a variable such that x = {0.2, 4, 7, 20 et and 30%}, one can graph, for the natural frequency ratios ω_i/ω_j, the curve (figure 3) that highlights the cumulation of the neighboring modes. The cumulation coefficient decreases rapidly when the frequency ratio increases, and even more when the damping ratio is low. 

Graphic of the coupling coefficient for different damping ratios varying as a function of the natural frequency ratios.

This last observation shows that the TQC method is more appropriate than the SRSS because the latter does not consider the cumulation of modes close to one another. 

On the contrary, there are no advantages of using the TQC method as a comparison to the SRSS method when the nodes are far from one another. 

Be careful! The cumulated modal responses must not be used to calculate other quantities of interest. For instance, the efforts deduced from the total quadratic cumulation method cannot be calculated using: : this calculation is incorrect in comparison to the evaluation by the cumulation of the modal efforts. 

One must, therefore, for this special case, calculate as follows:

1.
2. For each term p of the fk vector, it can be stated that:

It is also important to keep in mind that criteria allowing to evaluate the state of cracks based on the invariants of the stress tensor, and thus, of the principal stresses, (cf. for instance the criterion proposed in the annex LL of the EN1992-2) cannot be considered after using a quadratic cumulation of any kind.

The spectral responses of each mode, after being cumulated for one seismic direction, must be combined to obtain the global response. In this case, it is called spatial cumulation. 

The spatial cumulations can be handled using different methods:

• By square root of the sum of the squares of the responses obtained in each direction. This method gets rid of the sign and all logical relations between solicitations. It provides only one scalar quantity for each quantity of interest:

• By algebraic cumulation of “Newmark” type. This approach relies on the hypothesis of independence of all the spatial responses. It introduces a weighting coefficient μ, with values ranging according to the different standards, which considers the two other unfavorable responses compared to the preferred direction. It takes into account the sign variability of all the quantities of interest. Thus, it leads to not one accumulated response, but a total of 24 as demonstrated in the following equations:

Truncation of modal bases – Seismic case

For the specific case of earthquakes, and normalizing the modes according to the mass matrix:

It corresponds to the mass driven by the mode i in the direction k. This mass is therefore related to a specific direction. For a mode i, there will be 3 modal masses as a function of the different loading directions (if we are in a 3D space, 2 if we are in 2D, and only one for a 1D problem). Figure 4 highlights a 2D example for a frame. 

The modal displacements multiplied by the weighting coefficient represent the deformation of the structure whose product at all points with the response of a simple oscillator gives the time-dependent response of a given model. This concept is detailed in the following chapter. 

Mode 1 and displacement in the ey direction                                          Mode 1 and displacement in the ex direction

Illustration of the mass displacements for the same nodes in 2 different directions

As explained earlier if the model contains N degrees of freedom, there will be at most N natural modes of vibration.

However, since the search for natural frequencies and vibration modes is conducted numerically, not all the modes are extracted. In theory, one must have:

In practice, the algorithm will stop running after a threshold value of the frequency fixed by the user is reached. Then, the user must verify that there are enough modes to restore the percentage of mass required by the engineering rules followed. Usually, the criterion is to restore 90% of the mass. 

Be careful! If at the threshold frequency, the percentage of mass targeted is not reached, it might be necessary to include a pseudo-mode. 

Conversely, if a significant percentage of mass is reached at a low frequency, way below the threshold frequency, it is necessary to incorporate a pseudo-mode or to complete the modal base. Indeed, the participating mass of a floor for a local flexural mode inside a sizable structure represents a very small percentage of the total mass. 

One should also be careful concerning the antisymmetric modes with a modal mass that can be equal to zero because the masses displaced around one axis of the structure will balance one another. 

Pseudo-mode or static correction

As formulated before, the selection criterion of the natural modes concerning the cumulation of the mass is equal to 90% for a frequency of the same order of magnitude as the seismic response’s threshold frequency i.e. 40Hz maximum. 

Whenever this value cannot be reached, the lowest percentage withheld will be completed with an additional mass associated with a “pseudo-mode” of vibration. 

Let us remember that for a seismic direction k, there is an accumulated response on n modes such that:

To be rigorous, if there are N degrees of freedom, the formulation becomes:

As seen previously, the modes after the threshold frequency are rigid body motion modes. The structure reacts in phase with the seismic loading that it is subjected to with a relative displacement equal to zero. 

To complete the modal base, one can construct the pseudo-modes considering that the total response is the sum of the “dynamic” response, taking into account the modal base of the n first modes withheld, and the term proportional to the seismic acceleration of the support q̈s(t). Let us remember the following equation:

with P_k the displacement of the structure in the direction k subjected to the static loading equivalent to the mass of the accelerated structure at the acceleration of the last mode n extracted from the natural frequency:

Since the relative acceleration response for the structure is equal to zero after the threshold frequency, one can simply write:

Reformulating the previous equation as a function of , it will be easier to link the pseudo-accelerations spectrum, considering the frame of spectrum analysis:

Thus, the following formulation can also be considered:

After evaluating these vectors, they must be cumulated.

Indeed, the temporal solution for a seismic direction is given using the summation of the components on the modal base:

However, this combination cannot be applied to the maximum values that were just calculated previously.