6. Soil-structure interaction#

The seismic behavior of a building depends on the characteristics of the ground on which it is placed since it depends on the seismic movement imposed on the ground and on the dynamic behavior of the building and its foundations in the ground. Soil-structure interaction most frequently contributes to reducing the response of the studied structure.

6.1. Impedance of a foundation#

Let it be a rigid superficial foundation without mass, subject to a harmonic pulsating force \(\omega\): \(P(t)={P}_{0}\text{.}{e}^{\mathrm{iwt}}\) It is therefore animated by a \(X(t)\) movement of the same frequency. We call foundation impedance, or dynamic stiffness, the complex number \(K(\omega )\), a function of frequency \(\omega\) such as: \(K(\omega )=\frac{P(t)}{X(t)}\).

Several analytical or numerical methods make it possible to calculate the impedance of a foundation according to the complexity of the foundation and the soil on which it is placed or partially buried. Among the most frequently used are:

  • the analytical methods of WOLF or DELEUZE springs where it is assumed that the raft is circular, rigid and placed on a homogeneous ground. The foundation should be superficial;

  • the numerical method of code CLASSI where it is assumed that the floor is of any shape, rigid and placed on a possibly laminated floor. The foundation should be superficial;

  • the numerical method of code MISS3D where the floor slab can be of any shape, possibly deformable and placed on a possibly laminated floor.

It is possible to treat the soil-foundation interaction by the frequency coupling method (taking into account the frequency dependence of the impedance matrix) by carrying out a coupled calculation MISS3D/Code_Aster. This type of calculation is not detailed in this reference material. Here we present only the most common case where the soil-foundation interaction is treated by the soil spring method (it is considered that the terms of the impedance matrix are independent of frequency).

In the case of a rigid superficial foundation, the impedance is calculated at the center of gravity of the surface in contact in a coordinate system linked to the main axes of inertia of this surface. For each frequency, it is expressed in the form of a matrix with dimension \((\mathrm{6,}6)\). The value of each term is then adjusted according to a particular mode specific to the building studied in a locked base:

  • frequency of the first balancing mode \({\omega }_{0}\) for horizontal stiffness \({K}_{x}({\omega }_{0}),{K}_{y}({\omega }_{0})\) and rotation stiffness \({K}_{\mathrm{rx}}({\omega }_{0}),{K}_{\mathrm{ry}}({\omega }_{0})\);

  • frequency of the first pumping mode \({\omega }_{1}\) for vertical stiffness \({K}_{z}({\omega }_{1})\) and for torsional stiffness \({K}_{\mathrm{rz}}({\omega }_{1})\).

As the natural frequencies of the building depend on ground stiffness, the calculation of the overall values of the six ground springs results from an iterative process illustrated in figure [Figure 6.1-a]. The first ground stiffness \({K}_{x}({\omega }_{0}),{K}_{y}({\omega }_{0}),{K}_{z}({\omega }_{1}),{K}_{\mathrm{rx}}({\omega }_{0}),{K}_{\mathrm{ry}}({\omega }_{0})\) and \({K}_{\mathrm{rz}}({\omega }_{1})\) are chosen as a function of the first natural frequencies of balancing \(({\omega }_{0})\) and pumping \(({\omega }_{1})\) of the locked base structure. The ground stiffness is then adjusted to the first significant natural frequencies of the spring structure until the frequencies at which the impedance functions are calculated correspond to the natural frequency values of the coupled soil-building system.

_images/Object_131.svg

Figure 6.1-a : Process for adjusting ground stiffness.

6.2. Taking into account modal depreciation calculated according to the RCC -G rule#

Damping due to the ground is broken down into a part of material origin and into a geometric part: damping due to the reflection of elastic waves in the ground.

The RCC -G rule consists in summing, for each mode, the amortizations of each substructure constituting the building in question and the structural and geometric depreciations of the ground weighted by their respective potential energy ratio in relation to the total potential energy:

\({\eta }_{i}\mathrm{=}\frac{\mathrm{\sum }_{k}{E}_{\text{ki}}\text{.}{\eta }_{k}+\mathrm{\sum }_{s}{E}_{\text{si}}\text{.}{\eta }_{\text{si}}}{\mathrm{\sum }_{k}{E}_{\text{ki}}+\mathrm{\sum }_{s}{E}_{\text{si}}}\)

with:

  • \({\eta }_{i}\), the average reduced damping of mode \(i\);

  • \({\eta }_{k}\), the reduced damping of the \(k\) th element of the structure;

  • \({\eta }_{\text{si}}\), the reduced damping of the ground spring s for mode \(i\);

  • \({E}_{\text{ki}}\), the potential energy of the \(k\) th element of the structure for the \(i\) mode;

  • and \({E}_{\text{si}}\), the potential energy of the \(s\) ground spring for the \(i\) mode.

In the regulation, modal depreciation is limited to a maximum value of 0.3.

The material portion of ground damping is calculated by weighting the damping of each substructure by the ratio: potential energy rate to total potential energy. As for the geometric part of the damping, it is calculated by distributing the damping values for each direction (three translations and three rotations) weighted by the potential energy rate in the ground of the direction. The directional damping values are obtained by interpolating, for each calculated natural frequency, the directional damping functions resulting from a soil-structure interaction code (PARASOL, CLASSI or MISS3D). The ratio of the imaginary part to twice the real part of the impedance matrix: \(\frac{\text{Im}(K(\omega ))}{2\text{.}\text{Re}(K(\omega ))}\), provides the values of this radiative damping.

The procedure to follow is as follows:

  • Calculation of the potential energy dissipated in the studied structure: POST_ELEM [U4.81.22]

\({E}_{k}\) = POST_ELEM (ENER_POT =_F (TOUT = “OUI”));

  • Calculation of modal depreciation using the RCC -G rule: CALC_AMOR_MODAL [U4.52.13]

l_love = CALC_AMOR_MODAL (

ENER_SOL =_F (MODE_MECA = modal_base, GROUP_NO_RADIER = … ,

KX= \({K}_{x}({\omega }_{0})\), KY= \({K}_{y}({\omega }_{0})\), KZ= \({K}_{z}({\omega }_{1})\),

KRX = \({K}_{\mathrm{rx}}({\omega }_{0})\), KRY = \({K}_{\mathrm{ry}}({\omega }_{0})\), KRZ: \({K}_{\mathrm{rz}}({\omega }_{1})\)));

AMOR_INTERNE =_F (GROUP_MA =…, ENER_POT = \({E}_{k}\), AMOR_REDUIT = \({\eta }_{k}\))

AMOR_SOL =F (FONC_AMOR_GEO = \(\frac{\text{Im}(K(\omega ))}{2\text{.}\text{Re}(K(\omega ))}\))

);

The calculation of the soil’s contribution to potential energy \({E}_{s}\) (keyword factor ENER_SOL) is calculated from the ground impedance values determined previously (cf. [§6.1]). It can be calculated using two different methods depending on whether the modal forces are averaged (keyword RIGI_PARASOL) or the modal displacements at the nodes of the raft.

The reduced damping of ground spring \({\eta }_{s}\) (keyword factor AMOR_SOL) is calculated from radiative damping values.

6.3. Stiffness distribution and ground damping#

If we want to study the effect of an earthquake on the possible detachment of the foundation, for example, we may have to model the ground no longer by a single spring at the center of gravity of the soil-building interface but by a carpet of springs. This is possible thanks to the command AFFE_CARA_ELEM [U4.42.01] option RIGI_PARASOL.

The approach consists in calculating at each node of the grid of the slab the elementary stiffness \(({k}_{x},{k}_{y},{k}_{z},{\text{kr}}_{x},{\text{kr}}_{y},{\text{kr}}_{z})\) to be applied from the global values of the three translation springs: \(\text{kx},\text{ky},\text{kz}\) and of the three rotation springs: \(\text{krx},\text{kry},\text{krz}\) resulting from a soil-structure interaction code (or calculated analytically).

It is assumed that the elementary translation stiffness is proportional to the surface \(S(P)\) represented by the node \(P\) and to a distribution function \(f(r)\) depending on the distance \(r\) from the node \(P\) to the center of gravity of the raft \(O\):

\(\mathrm{\{}\begin{array}{c}{K}_{x}\mathrm{=}\mathrm{\sum }_{P}{k}_{x}(P)\mathrm{=}{k}_{x}\text{.}\mathrm{\sum }_{P}S(P)\text{.}f(\mathit{OP})\\ {K}_{y}\mathrm{=}\mathrm{\sum }_{P}{k}_{y}(P)\mathrm{=}{k}_{y}\text{.}\mathrm{\sum }_{P}S(P)\text{.}f(\mathit{OP})\\ {K}_{z}\mathrm{=}\mathrm{\sum }_{P}{k}_{z}(P)\mathrm{=}{k}_{z}\text{.}\mathrm{\sum }_{P}S(P)\text{.}f(\mathit{OP})\end{array}\)

We then deduce \({k}_{x}\) then \({k}_{x}(P)\) from the calculation:

\({k}_{x}(P)\mathrm{=}{k}_{x}\text{.}S(P)\text{.}f(\mathit{OP})\mathrm{=}{K}_{x}\text{.}\frac{S(P)\text{.}f(\mathit{OP})}{\mathrm{\sum }_{P}S(P)\text{.}f(\mathit{OP})}\).

We also deduce \({k}_{y}(P)\) and \({k}_{z}(P)\).

For the elementary stiffness of rotation, what is left after removing the contributions due to translations is distributed in the same way as the translations:

\(\{\begin{array}{c}{K}_{\text{rx}}=\sum _{P}{k}_{\text{rx}}(P)+\sum _{P}\left[{k}_{y}(P)\text{.}{z}_{\mathrm{OP}}^{2}+{k}_{z}(P)\text{.}{y}_{\mathrm{OP}}^{2}\right]={k}_{\text{rx}}\text{.}\sum _{P}S(P)\text{.}f(\mathrm{OP})+\sum _{P}\left[{k}_{y}(P)\text{.}{z}_{\mathrm{OP}}^{2}+{k}_{z}(P)\text{.}{y}_{\mathrm{OP}}^{2}\right]\\ {K}_{\text{ry}}=\sum _{P}{k}_{\text{ry}}(P)+\sum _{P}\left[{k}_{x}(P)\text{.}{z}_{\mathrm{OP}}^{2}+{k}_{z}(P)\text{.}{x}_{\mathrm{OP}}^{2}\right]={k}_{\text{ry}}\text{.}\sum _{P}S(P)\text{.}f(\mathrm{OP})+\sum _{P}\left[{k}_{x}(P)\text{.}{z}_{\mathrm{OP}}^{2}+{k}_{z}(P)\text{.}{x}_{\mathrm{OP}}^{2}\right]\\ {K}_{\text{rz}}=\sum _{P}{k}_{\text{rz}}(P)+\sum _{P}\left[{k}_{x}(P)\text{.}{y}_{\mathrm{OP}}^{2}+{k}_{y}(P)\text{.}{x}_{\mathrm{OP}}^{2}\right]={k}_{\text{rz}}\text{.}\sum _{P}S(P)\text{.}f(\mathrm{OP})+\sum _{P}\left[{k}_{x}(P)\text{.}{y}_{\mathrm{OP}}^{2}+{k}_{y}(P)\text{.}{x}_{\mathrm{OP}}^{2}\right]\end{array}\)

We then deduce \({k}_{\text{rx}}\) then \({k}_{\text{rx}}(P)\) from the calculation:

\(\begin{array}{c}{k}_{\text{rx}}(P)\mathrm{=}{k}_{\text{rx}}\text{.}S(P)\text{.}f(\mathit{OP})\\ \mathrm{=}({K}_{\text{rx}}\mathrm{-}\mathrm{\sum }_{P}\left[{k}_{y}(P)\text{.}{z}_{\mathit{OP}}^{2}+{k}_{z}(P)\text{.}{y}_{\mathit{OP}}^{2}\right])\text{.}\frac{S(P)\text{.}f(\mathit{OP})}{\mathrm{\sum }_{P}S(P)\text{.}f(\mathit{OP})}\end{array}\)

We also deduce \({k}_{\text{ry}}(P)\) and \({k}_{\text{rz}}(P)\).

Note:

By default, the distribution function is considered to be constant and unitary, i.e. each surface is affected by the same weight.

Six global damping values can be distributed in the same way, analytical or calculated by a soil-structure interaction code.

6.4. Taking into account an absorbent border#

If we want to calculate the seismic response of a dam, for example, we must, among other things, be able to take into account the non-reflection of waves on the arbitrary border of the finite element model within the ground or reservoir. This feature is not detailed in this document. It is the subject of the documentation [r4.02.05].