5. Problem data

5.1. Meshing

5.1.1. General advice

The discretization of the mesh must be adapted to the wavelength of the phenomena that one wishes to represent. A cutoff frequency and therefore an element size can be established by means of a prior modal analysis of the structure.

In many studies, nonlinear calculations are performed as a result of a linear study. In this case it is necessary to adapt the linear mesh by refining the zones that become non-linear, and also the areas where high levels of bending moment are likely to occur. In order to remail, the software HOMARD can be used via the MACR_ADAP_MAIL [U7.03.01] macro-command. In particular, it is possible to re-mesh part of a mesh by simply indicating the mesh group that you want to refine.

It is strongly recommended to carry out a study of the sensitivity of the results to the mesh. In practice, it is advisable to provide several mesh refinements when creating the mesh.

5.1.2. Massive local modeling and multi-layer shell

It is advisable to make the steel and concrete nodes coincide to reduce the size of the problem. The elements BARRE, GRILLE_MEMBRANE or GRILLE_EXCENTRE representing the reinforcements must have their nodes merged with those of the massive elements or concrete shells (see § 3.2.3 and 3.3.3).

For link modeling, refer to § 3.6.

5.2. Initial conditions

The preloading of the reinforced concrete structure (own weight and other static loads…) is necessary when using the DYNA_NON_LINE operator. It is done using the operator STAT_NON_LINE by imposing only the load due to the own weight and other loads. Dynamic loading is applied after the static calculation of the dead weight, while maintaining the latter load.

Moreover, it is possible to take into account the construction phase of the structure. Taking this phasing into account makes it possible to simulate a precrack appearing during construction. Moreover, it may be necessary, even in linear elasticity, in order to represent a state of non-zero initial stresses in the structure (obtained during construction by modification of the hyperstatisms).

These methods are currently not widely used for the calculation of building structures. However, it is recommended to exploit the work carried out on the problem of excavation [U2.04.06] and the construction by layers of a dike ([U2.04.07]).

5.3. Boundary conditions

Boundary conditions are to be defined classically as in any study.

However, the seismic analysis of a civil engineering structure placed on a flexible foundation may require taking into account the interaction between the ground and the structure (ISS). Refer to [U2.06.07] for the description of methods for taking ISS into account. In particular, ISS can be represented using foundation impedances adjusted around the linear behavior of the soil-based structure.

5.4. Seismic loading

In this paragraph we repeat the elements detailed in [U2.06.09].

It should be noted that the structure is said to be mono-supported if all the supports on which the structure is based are subjected to the same excitation. The structure is said to be multi-supported if there are at least two supports that are not subject to identical seismic excitation.

5.4.1. Mono-support

In this case, there are no constraints induced by the differential movements of the anchors. We therefore use a relative displacement approach and we apply either the accelerogram (case 1) or the displacement (case 2) to the structure (operator CALC_CHAR_SEISME, [U4.63.01]).

1st case

We have the seismic signal a (t), acceleration as a function of time in the three directions \(X\), \(Y\) and \(Z\) — Use of CALC_CHAR_SEISME.

The development of the « loading » concept used by AFFE_CHAR_MECA takes place as follows:

• assembling the assembled mass matrix \({M}_{\mathrm{ass}}\): operator ASSEMBLAGE;

• development of the assembled vectors (\({V}_{{\mathrm{ass}}_{X}}\), \({V}_{{\mathrm{ass}}_{Y}}\) and \({V}_{{\mathrm{ass}}_{Z}}\)), used as a basis for loading (1 per direction of the earthquake): operator CALC_CHAR_SEISME with the keyword MONO_APPUI =” OUI “;

• development of the « loading » concept (1 per direction of the earthquake): operator AFFE_CHAR_MECA with the keyword VECT_ASSE.

This case is the most common in studies of civil engineering structures under earthquakes.

2nd case

We have the seismic signal \(d(t)\), displacement as a function of time in the three directions \(X\), \(Y\) and \(Z\).

The development of the « loading » concept used by AFFE_CHAR_MECA is immediate: operator AFFE_CHAR_MECA with the keyword DDL_IMPO.

5.4.2. Multi-support

In this case, the differential movements of the anchors induce secondary stresses. It is no longer possible to use a relative displacement approach. We must therefore perform the resolution in the absolute coordinate system. The load must be imposed in the form of movements at the anchorages of the structure.

Compared to the single-support calculation, the first proposed method is no longer suitable. The use of CALC_CHAR_SEISME in multi-supports and in transient nonlinear (DYNA_NON_LINE) provides results whose validity is not assured because CALC_CHAR_SEISME operates by elasto-static lifting (static modes).

1st case

We have the seismic signal \(a(t)\), acceleration as a function of time in the three directions \(X\), \(Y\) and \(Z\) — Use of Fourier transforms.

It is necessary to build \(d(t)\), moving as a function of time starting with \(a(t)\). The proposed method consists in using Fourier transforms to integrate the signal:

• calculation of the acceleration \(a(\omega )\) as a function of the pulsation: realization of a Fourier transform of \(a(t)\) in each direction;

• calculation of the \(d(\omega )\) displacement as a function of the pulsation: multiplication of \(a(\omega )\) by \({\omega }^{2}\);

• calculation of the \(d(t)\) displacement as a function of time: realization of an inverse Fourier transform of \(d(\omega )\);

• development of the « loading » concept used by AFFE_CHAR_MECA: operator AFFE_CHAR_MECA with the keyword DDL_IMPO.

Note

Refer to [14] for the precautions to be taken in carrying out Fourier transforms: verification of the principle of causality (return to zero of the signal to perform FFT) and absence of displacement drift.

2nd case:

We have the seismic signal \(d(t)\), displacement as a function of time in the three directions \(X\), \(Y\) and \(Z\).

The development of the « load » concept used by AFFE_CHAR_MECA is immediate: operator AFFE_CHAR_MECA with the keyword DDL_IMPO.

5.4.3. General time signal precautions

The evolution of time as a function of time \(a(t)\) or \(d(t)\) used as input to the calculation must be as regular as possible. The time step that defines the sampling must therefore be sufficiently fine. If we call \({f}_{\mathrm{max}}\) the maximum frequency of the signal spectrum, we must have at least a sample rate 5 to 10 times greater than \({f}_{\mathrm{max}}\).

Initially, as the structure is considered to be subject only to permanent loads (weight,…), it is better to have zero displacement, speed and acceleration induced by the earthquake on the supports. If these conditions are not respected, there is a risk of observing initial oscillations in the response. In practice, to respect this, it is sufficient for the seismic signal imposed on the supports while moving to be zero over at least the first two steps of time. It will therefore be necessary to add null values manually at the beginning of the load files.

At the end of the calculation, it is recommended to also return to zero imposed travel conditions over a certain period of time. This ensures that the structure returns to rest. This is essential (causality condition) if we want to do a correct spectral analysis by FFT of the answers.

Note

The guide to ASN [15] recommends using a set of \(N\) accelerograms ( \(N=5\) minimum) representative of the seismic design movement for the design of civil engineering structures. We therefore carry out \(N\) nonlinear transient calculations in order to analyze the variability of the results according to the accelerograms used. We define the quantity to be used for the sizing according to the number of accelerograms used, the mean and the standard deviation of the absolute values of the results (for this purpose we use a Student-Fisher type confidence interval estimator, [15] , * of the mean type of nonlinear transient responses + k.sigma, with k depending on the number \(N\) of realizations (5, 10…).) .Some quantities of interest are more sensitive than others to variabilities; moreover the non- linearities can be localized and large parts of the structure can remain in a linear regime. Note that in the linear framework, * the guide to ASN [15] recommends using a simple average of the transient responses.

5.5. Depreciation

5.5.1. Definition

It is recommended to use Rayleigh damping [R5.05.04] and [R5.05.05] in direct transient resolution using DYNA_NON_LINE.

The elementary damping matrix \({C}_{\mathrm{elem}}\) is expressed as a linear combination of the elementary matrices of mass \({M}_{\mathrm{elem}}\) and stiffness \({K}_{\mathrm{elem}}\):

\({C}_{\mathrm{elem}}=\alpha {K}_{\mathrm{elem}}+\beta {M}_{\mathrm{elem}}\)

with \(\alpha\) and \(\beta\) the Rayleigh coefficients. Part \(\beta {M}_{\mathrm{elem}}\) corresponds to low frequency damping and part \(\alpha {K}_{\mathrm{elem}}\) to high frequency damping (Figure 5.5.1-a).

Damping factor \({\zeta }_{n}\) for the nth mode of the system under consideration is defined by:

\({\zeta }_{n}=\frac{\alpha {\omega }_{n}}{2}+\frac{\beta }{2{\omega }_{n}}\)

with the \({\omega }_{n}\) pulse associated with the nth mode.

This Rayleigh damping is based on two damping modes and factors \(({\omega }_{\mathrm{1,}}{\zeta }_{1})\) and \(({\omega }_{\mathrm{2,}}{\zeta }_{2})\) relevant for the analysis of the structure under study. When we know the damping values for these two frequencies, we can calculate \(\alpha\) and \(\beta\):

\(\{\begin{array}{ccc}\alpha & \text{=}& 2\frac{{\omega }_{1}}{{\omega }_{1}^{2}-{\omega }_{2}^{2}}{\zeta }_{1}-2\frac{{\omega }_{2}}{{\omega }_{1}^{2}-{\omega }_{2}^{2}}{\zeta }_{2}\\ \beta & \text{=}& 2\frac{{\omega }_{1}{\omega }_{2}^{2}}{{\omega }_{2}^{2}-{\omega }_{1}^{2}}{\zeta }_{1}-2\frac{{\omega }_{1}^{2}{\omega }_{2}}{{\omega }_{2}^{2}-{\omega }_{1}^{2}}{\zeta }_{2}\end{array}\)

In practice, we often consider the case where the two amortizations are equal \({\zeta }_{1}={\zeta }_{2}=\zeta\), we then obtain the formulas to be used for the calculation of \(\alpha\) and \(\beta\):

\(\{\begin{array}{ccc}\alpha & \text{=}& \frac{2}{{\omega }_{1}+{\omega }_{2}}\zeta \\ \beta & \text{=}& 2\frac{{\omega }_{1}{\omega }_{2}}{{\omega }_{1}+{\omega }_{2}}\zeta \end{array}\)

It is then necessary to enter the values \(\alpha\) and \(\beta\) using the command DEFI_MATERIAU, keywords AMOR_ALPHA/AMOR_BETA.

Figure 5.5.1-a shows the graphical description of Rayleigh damping for two frequencies \(({f}_{\mathrm{1,}}\zeta )\) and \(({f}_{\mathrm{2,}}\zeta )\) between which the damping is relatively uniform.

Figure 5.5.1-a : description of Rayleigh damping.

5.5.2. Nonlinear case

To calculate \({C}_{\mathrm{elem}}\), we take the stiffness matrix used in Newton’s method (RIGI_MECA). If we use the updated tangent matrix \({K}_{\mathrm{elem}}^{\mathrm{tan}}\) in DYNA_NON_LINE (NEWTON =_F (=_F (MATRICE =” TANGENTE “, REAC_ITER =1,),),), the damping matrix is then defined by:

\({C}_{\mathrm{elem}}=\alpha {K}_{\mathrm{elem}}^{\mathrm{tan}}+\beta {M}_{\mathrm{elem}}\)

The damping matrix \(C\) then does not remain constant during the calculation. In fact, when the structure becomes heavily damaged, \({K}^{\mathrm{tan}}\) decreases and may even become negative. As a result, the interpretation of the effect of proportional damping in non-linear terms is difficult (variation in the dissipation introduced over time).

The user sets the damping parameters to the range where the structure is elastic.

5.5.3. Other nonlinear tips

In non-linear mode, some of the dissipative phenomena are modelled using models of nonlinear behavior of concrete and steel. However, other dissipative phenomena are not modelled by the behaviors of the materials, which nevertheless dissipate the energy provided by the excitation throughout the transient and at every material point. Therefore, as in linear elastic analysis, a fixed viscous damping matrix is introduced into the calculation to represent these phenomena, cf. [33].

With Code_Aster, you can choose a Rayleigh damping matrix or a modal damping matrix, cf. u2.06.03.

It is recommended to use a maximum damping value of 2% on the first mode, and 5% at most for the cutoff frequency, to comply with the requirements of the ASN guide [15]). This value is much lower than that of 5 to 7% used linearly for dimensioning calculations. It generally leads to overestimating the peaks of the floor spectra, even when the load exceeds several times the design earthquake, see [18].

It should be noted that with multi-support (see 5.4.2) the problem is solved in the absolute frame of reference (\({U}_{\mathrm{absolu}}={U}_{\mathrm{relatif}}+{U}_{\mathrm{entraînement}}\)). In this case, it is essential not to introduce damping on the part corresponding to the drive speed. The damping matrix should be taken to be proportional to the stiffness matrix only (\({C}_{\mathrm{elem}}=\alpha {K}_{\mathrm{elem}}\) in the method recommended by ASN [15]). The choice of the modal damping matrix, by specifying a zero damping value on rigid modes, is also suitable. Another possibility consists in embedding one of the supports and in describing the movement of the other supports in relation to this one (this is equivalent to cancelling the driving movement). In this case, conventional Rayleigh damping can be used.

For the discrete elements DIS_T or DIS_TR [U3.11.02], the damping matrix is directly defined for each degree of freedom in AFFE_CARA_ELEM (keyword DISCRET [U4.42.01]). This remark is important when modeling added masses using discrete elements. It is then essential to associate a damping matrix with it otherwise these added masses are not amortized.

It is important to produce the balance of energies during the transition over the entire system: using the keyword ENERGIEde DYNA_NON_LINE. This makes it possible to verify the distribution of kinetic energy, of the deformation energy, of that provided to the structure by the stress at each moment, of that dissipated in viscous form, of that (called hysteretic) dissipated by the models of non-linear behavior. The remainder of the balance sheet is to be assigned to the temporal integration diagram, if this one is dissipative.

When using an explicit time integration scheme (DIFF_CENT or TCHAMWA), it is imperative to use damping that is proportional to the mass matrix only. The use of full Rayleigh damping causes a drop in the stability time step and therefore a very significant increase in the calculation time.