Frequency Dependent LPM for Soil-Structure Interaction under Induced Seismicity

Table of Content

Introduction

The extraction of geothermal heat can induce repeated micro-seismic events. The effects of such events on nearby buildings can vary widely depending on the underlying soil and the building properties. The aim of this project is, to create a ANSYS FEM model, which can quickly be adjusted for any type of building, soil and seismic loading.

This will be especially useful for future expansions of geothermal heat extraction within big cities like Munich. Due to its geography, Munich has a rich supply of geothermal energy. However, the effect of induced seismicity can become quite problematic due to the high density of buildings in the area. Most likely, because Munich lies far from typical earthquake regions, only very few buildings can be expected to be built with seismicity in mind. Therefore, this project aims in predicting possible serviceability issues in advance, in order to prevent a shutdown of the extraction activities.

Geothermal Energy

Geothermal energy is heat, stored in the earth's crust. In Germany, this could cover over 50% of the heat demand. In addition, also power can be generated [1]. The richest areas for geothermal energy in Germany are the North German Basin, the Upper Rhine valley, and the Molasse Basin in Southern Germany. Despite all of these advantages, thus far only a very small percentage of the geothermal energy in Germany, but also around the world is tapped upon. This is partly due to the high risk of induced seismicity. Subsequently, nearby residents often object against the construction of geothermal heat extractions [2].

Shallow Geothermal

Shallow geothermal reach depths of up to 400 meters. Heat is usually extracted with help of heat transfer fluids. On the surface level (usually in resident basements), heat exchangers are utilized to heat buildings. Unfortunately, the heat from shallow geothermal is not enough to heat larger building arrays or suburbs [1].

Deep Geothermal

Deep geothermal energy is generally extracted from depths up to 5000 meters. Unlike shallow geothermal, it possesses far more energy, thus enabling the generation of power or the heating of entire suburbs [1]. Deep geothermal is further distinguished between hydrothermal and petrothermal heat extractions [2].

Hydrothermal Geothermal

Hydrothermal geothermal heat extraction uses thermal water to extract the heat. Due to the extreme depths of the water sources, simple heat transfer fluids can no longer be utilized. Thus, the thermal water itself has to be pumped to the surface, where heat exchangers extract the heat from the water. The cooled water is then injected back below the surface through a second borehole [1].

Petrothermal Geothermal

If thermal water is not abundant underground, heat extraction is still possible by taking advantage of the heat in the rocks itself. In this case, the stones act as heat exchangers. To improve the efficiency of this method, the surface area of the rocks at the bottom of the drill site needs to be increased. This can be achieved via hydraulic stimulation, a process, where cold water is injected down the borehole to break up the rock into smaller pieces. Due to hydraulic stimulation, petrothermal geothermal heat extraction is most prone to induced seismic events [2].

Induced Seismicity

Induced seismicity is earthquakes that occur due to human activity. However, with growing intensity, the percentage of human-caused earthquakes reduces dramatically. While below a magnitude of 3, roughly half of the registered earthquakes are induced, above a magnitude of 3, only around one-quarter of the earthquakes are induced [2]. Usually, these events are not perceptible for humans (i.e. magnitude < 3). Induced seismicity occurs most commonly during mining processes, however, also geothermal heat extraction can cause these tremors [2].

In the case of hydrothermal geothermal heat extraction, the seismic activities happen through a change in the rock's stress. By pumping out the hot thermal water and replacing it with cooled water, the rocks perceive a change in the state of stress. If in succession, the shear strength is surpassed, the rocks will break apart and thus cause a seismic event.

Even higher are the risks for induced seismicity in petrothermal geothermal heat extractions. Due to injecting water into the ground (hydraulic stimulation), the stress state of the rocks is not only changed but in addition, also the shear strength is reduced (due to pore water pressure) [2]. This usually results in constant seismic events. However, also chemical processes or cooling of water (in the case of hydrothermal geothermal), can cause a reduction in the shear strength. Usually, a seismic response to an activity caused by heat extraction is expected to come with a delay, ranging from days, up to months (dependent on the permeability of the rock, i.e. how quickly the water can saturate the ground) [2].

Compared to other induced seismic events, geothermal heat extraction causes only very subtle vibrations. According to [2], no earthquakes above magnitude 3 have been registered so far in Germany with a direct correlation to geothermal activity. This response actually isn't even stronger than the vibrations caused by a truck on a bumpy road [1]. However, many factors can play into the impact of a seismic event. In order to quickly assess the consequences, a FEM-Analysis can be used to evaluate the response of a system with a given input response.

Theoretical Base

Lumped-parameter model of rigid massless foundation (LPM)

In this project, since frequency-dependent soil stiffness and damping are introduced, Lysmer's analog LPM is not valid anymore. The frequency-dependent model has to be used instead.[5]

Model of rigid massless foundation on the surface of a homogeneous half-space

Equation of motion:

(1)	$\begin{array}{l}m_{f} \ddot{x}(t) +r(t) = f(t)\\\end{array}$

(2)	$\begin{array}{l}\displaystyle r(t) = c(\omega) \dot{x}(t) + k(\omega) x(t)\end{array}$

where $\begin{array}{l}M_f\end{array}$ is the mass of the foundation and $\begin{array}{l}R(t)\end{array}$ is the reaction force in between the foundation and the soil.

Substitute $\begin{array}{l}R(t)\end{array}$ into equation(1):

(3)	$\begin{array}{l}\displaystyle M_{f} \ddot{X}(t) + C(\omega) \dot{X}(t) + K(\omega) X(t) = F(t)\end{array}$

Since the stiffness $\begin{array}{l}K(\omega)\end{array}$ and damping $\begin{array}{l}C(\omega)\end{array}$ are frequency dependent, the displacement $\begin{array}{l}X(t)\end{array}$ and force $\begin{array}{l}F(t)\end{array}$ above will be transferred into the frequency domain as $\begin{array}{l}X(\omega)\end{array}$ and $\begin{array}{l}F(\omega)\end{array}$ respectively. The derivatives of $\begin{array}{l}X(\omega)\end{array}$ in the frequency domain are:

(4)	$\begin{array}{l}\dot{X}(\omega) = i \omega X(\omega) \\\ddot{X}(\omega) = -\omega^{2} X(\omega)\end{array}$

Substituting(4) into the equation of motion(3), we receive the following equation:

(5)	$\begin{array}{l}\displaystyle -M_{f} \omega^{2} {X}(\omega) + i \omega C(\omega) X(\omega) + K(\omega) X(\omega) = F(\omega)\end{array}$

The dynamic stiffness part from eq.(5) denotes the following term:

(6)	$\begin{array}{l}\displaystyle \hat{K}_{dyn}(\omega) = i \omega C(\omega) + K(\omega)\end{array}$

Therefore, the equation of motion in the frequency domain can also be written as:

(7)	$\begin{array}{l}\displaystyle (-M_{f}\omega^{2} + \hat{K}_{dyn}(\omega))X(\omega) = F(\omega)\end{array}$

In engineering convention, the dynamic stiffness function $\begin{array}{l}\hat{K}_{dyn}(\omega)\end{array}$ usually is represented as (8), where $\begin{array}{l}k_{i}\end{array}$ and $\begin{array}{l}c_{i}\end{array}$ are the soil's stiffness and damping coefficients. Those coefficients are normalized with respect to the static stiffness $\begin{array}{l}{K}_{stat}\end{array}$ .

(8)	$\begin{array}{l}\displaystyle \hat{K}_{dyn}(\omega) = {K}_{stat}(k_{i}+i \omega c_{i})\end{array}$

It can also be described under the dimensionless frequency $\begin{array}{l}a_{0} = \dfrac {\omega \cdot r_{equ}} {v_{s}}\end{array}$ , where $\begin{array}{l}r_{equ}\end{array}$ is the equivalent radius of the foundation and $\begin{array}{l}v_{s}\end{array}$ is shear wave velocity. The dynamics stiffness function will be re-written as

(9)	$\begin{array}{l}\displaystyle \hat{K}_{dyn}(a_{0}) = {K}_{stat}(k_{i}+i a_{0} c_{i})\end{array}$

In LPM, there is no physical meaning of the dynamics stiffness function $\begin{array}{l}\hat{K}_{dyn}(a_{0})\end{array}$ . It is usually generated by the curve fitting technique. It approximates the stiffness and damping which are calculated or measured based on the more complex or realistic model of the foundation.

It is worth noticing that the mass of the foundation $\begin{array}{l}M_{f}\end{array}$ also has no physical meaning, they are used as a mathematical technique for better matching with the exact response of the system. By adding $\begin{array}{l}M_{f}\end{array}$ , a more accurate response might be estimated especially in the low- and medium-frequency ranges.[6]

Excitation and transfer function

When the excitation, Dirac function $\begin{array}{l}\delta(t)\end{array}$ , is inputted, its Fourier transform is:

(10)	$\begin{array}{l}\displaystyle \delta(t) \xrightarrow{\mathcal{F}} 1\end{array}$

which is unity in a whole frequency domain. Its response in the frequency domain is called the transfer function $\begin{array}{l}H(\omega)\end{array}$ .

The transfer function implies the amplification ratio as the function of frequency between response $\begin{array}{l}{Y}(\omega)\end{array}$ and excitation $\begin{array}{l}{X}(\omega)\end{array}$ . The use of it makes the analysis simpler because one can obtain the response easily by multiplication of the particular excitation with the transfer function:

(11)	$\begin{array}{l}\displaystyle Y(\omega) = H(\omega) \cdot X(\omega)\end{array}$

In this project, all the responses will be presented as the transfer function, meaning all the input excitation for the simulation is set as 1 under the frequency domain in Harmonic analysis.

Implementation of the impedance model

The motion of the equation is extended now by adding the foundation input motion $\begin{array}{l}U^{*}\end{array}$ for the seismic excitation and ignoring the effect of the mass $\begin{array}{l}M_{f}\end{array}$ . This is described in matrix form according to Wong and Luco [4].

(12)	$\begin{array}{l}\displaystyle \textbf{F} = \textbf{K}(\textbf{U} - \textbf{U$^*$})\end{array}$

where $\begin{array}{l}\textbf{F}\end{array}$ is the vector of the resultant force, $\begin{array}{l}\textbf{U}\end{array}$ is the total motion, and $\begin{array}{l}\textbf{U$^*$}\end{array}$ is the foundation input motion (seismic excitation) [4].

(13)	$\begin{array}{l}\displaystyle \textbf{F} =( F_{1},F_{2},F_{3},\dfrac{M_{1}}{r_{equ}} , \dfrac{M_{2}}{r_{equ}} ,\dfrac{M_{3}}{r_{equ}})^T\end{array}$

(14)	$\begin{array}{l}\displaystyle \textbf{U} = (U_{1}, U_{2} ,U_{3} ,r_{equ}\theta_{1} ,r_{equ}\theta_{2} ,r_{equ}\theta_{3} )^T\end{array}$

(15)	$\begin{array}{l}\displaystyle \textbf{U$^$} = (U_{1}^{}, U_{2}^{} ,U_{3}^{} ,r_{equ}\theta_{1}^{} ,r_{equ}\theta_{2}^{} ,r_{equ}\theta_{3}^{*} )^T\end{array}$

$\begin{array}{l}\textbf{K}\end{array}$ is the foundation impedance matrix. Here we assumed the use of a rectangular foundation, and $\begin{array}{l}r_{equ}=(\dfrac{4l_{f}w_{f}}{\pi})^{0.5}\end{array}$ , where $\begin{array}{l}l_{f}\end{array}$ and $\begin{array}{l}w_{f}\end{array}$ are the length and width of the foundation were measured from the center point. Therefore, the foundation impedance matrix $\begin{array}{l}\textbf{K}\end{array}$ can be simplified as a symmetric matrix, with the coupling terms $\begin{array}{l}k_{15} = k_{51}\end{array}$ and $\begin{array}{l}k_{24} = k_{42}\end{array}$ .

(16)	$\begin{array}{l}\textbf{K}=G\cdot r_{equ} \cdot \[ \left[ {\begin{array}{ccccc} k_{11} & 0 &0 & 0 & k_{15}&0\\ 0 & k_{22} & 0 & k_{24} & 0&0\\ 0 & 0 & k_{33} & 0 & 0&0\\ 0 & k_{42} & 0 & k_{44} & 0&0\\ k_{51} & 0 & 0 & 0 & k_{55}&0\\ 0 &0 & 0 &0 & 0&k_{66}\\ \end{array} } \right] \]\end{array}$

Modeling

FEM model of building with footing

All the elements used for modeling are selected from the ANSYS library. The slabs are modeled by the SHELL169 element and all of the columns and beams are modeled by the BEAM element.

The foundations are modeled by a combination of the MASS21 and COMBIN14 or MATRIX27. A brief introduction to the use of elements is following.

MASS21 element

MASS21 element is a point element with 6 degrees of freedom. The user can assign the mass or inertia quantities for each direction.

COMBIN14 element

COMBIN14 element can be modeled as a 1-D longitudinal spring-damper element. This element is defined by two nodes, regardless of the distance between them. Since it is only in 1-D, If one wants to model the stiffness in three directions, the elements have to be added three times based on different axes.

Pros:

The frequency-dependent stiffness and damping can be inputted directly by using tabular parameters.
This element is easy to implement and can access stiffness and damping at the same time.

Con:

It is very difficult to describe the coupling terms, such as $\begin{array}{l}k_{15}\end{array}$ , $\begin{array}{l}k_{51}\end{array}$ , $\begin{array}{l}k_{24}\end{array}$ and $\begin{array}{l}k_{42}\end{array}$ from eq.(16)

MATRIX27 element

$\begin{array}{l}\textbf{K}_{i j , mn}= \[ \left[ {\begin{array}{cccccccccccc} k_{11,11} & \cdots & k_{11,16} &k_{12,11} & \cdots & k_{12,16} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \\ k_{11,61} & \cdots & k_{11,66} &k_{12,11} & \cdots & k_{12,66} \\ k_{21,11} & \cdots & k_{21,16} &k_{22,11} & \cdots & k_{22,16} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \\ k_{21,61} & \cdots & k_{21,66} &k_{22,11} & \cdots & k_{22,66} \\ \end{array} } \right] \] \\ \\ \text{where} ~i,j \text{ = index of node and} ~m,n \text{ = degree of freedom}\end{array}$

MATRIX27 element is a general stiffness matrix element that is also defined by two nodes. The size of this matrix element is 12 by 12, where the first 6 indexes denote the 6 degrees of freedom for the first node and the remaining are for the second node.

Pro:

It is the general stiffness matrix that can account for the coupling effect from equation(16). In our cases, it is applied to deal with the impedance function from Wong and Luco [4] and the impedance function of the embedded function from Gazetas [3]

Cons:

The implementation of MATRIX27 is a lot more complicated than the COMBIN14 element.
Each element can only access one type of elastic kinematic response such as stiffness or damping. The implementation of the spring-damper model has to be treated as the superposition of the stiffness element and the damping element.
Tabular parameters cannot be assigned to the element, therefore frequency-dependent model can only be fulfilled by using for-loop.

Impedance Function

Finding the correct entries for the foundation impedance matrix (eq.(16)) is essential for an accurate flow of forces into the upper floors of the FEM building. Due to the uniform shape and size of most foundations, it makes sense to take into account tabulated values for the impedance function. Commonly used are the impedance tables from Wong and Luco [4] and the impedance charts and formulas from Gezetas [3]. Due to the large dataset in the literature, most foundation types can be directly input into the FEM model without the need of evaluating manually the impedance function for the foundation.

In our project, the rectangular foundation with an aspect ratio $\begin{array}{l}\dfrac{l_{f}}{w_{f}}=1\end{array}$ is chosen.

Impedance Charts and Formulas from Gazetas [3]

The impedance values from Gazetas [3] have been specifically designed for the use in dynamic analysis. For accurate results, a seismic or machine-type inertial loading is expected [3]. The charts and formulas from Gazetas [3] are only useable for rectangular foundations. However, according to Gazetas [3], it is possible to circumscribe arbitrary foundation shapes into rectangular shapes to evaluate the impedance functions. An example of this method can be seen in the image below. The foundation parameters of the rectangular foundation will later be needed in Tables 1 and 2.

Surface Foundation (left), Embedded Foundation (right)

Following the naming convention from Gazetas [3], the equations(6) and(12) can be rewritten as:

(17)	$\begin{array}{l}\displaystyle \textbf{S}=i \omega C(\omega) + K(\omega) = \bar{K}(\omega) + i \omega C(\omega)\end{array}$

(18)	$\begin{array}{l}\displaystyle \textbf{R} = \textbf{K}(\textbf{U} - \textbf{U$^$}) = \textbf{S}(\textbf{U} - \textbf{U$^$})\end{array}$

Analogously, the impdeance matrix can be written as follows:

(19)

$\begin{array}{l}\textbf{S}= \[ \left[ {\begin{array}{ccccc} S_{x} & 0 &0 & 0 & \color{red} S_{x-ry} &0\\ 0 & S_{y} & 0 & \color{red} S_{y-rx} & 0&0\\ 0 & 0 & S_{z} & 0 & 0&0\\ 0 & \color{red} S_{y-rx} & 0 & S_{rx}& 0&0\\ \color{red} S_{x-ry} & 0 & 0 & 0 &S_{ry}&0\\ 0 &0 & 0 &0 & 0&S_{t}\\ \end{array} } \right] \]\end{array}$

Where $\begin{array}{l}S_{x}\end{array}$ = longitudinal swaying, $\begin{array}{l}S_{y}\end{array}$ = lateral swaying, $\begin{array}{l}S_{z}\end{array}$ = vertical, $\begin{array}{l}S_{rx}\end{array}$ and $\begin{array}{l}S_{ry}\end{array}$ = rocking and $\begin{array}{l}S_{t}\end{array}$ = torsional impedance. In the case of embedded foundations (red only), swaying-rocking impedance has to be considered.

As can be seen from eq.(17), each impedance function consists of a "real" and "imaginary" part. While the "real" part refers to the dynamic stiffness $\begin{array}{l}\bar{K}\end{array}$ , the "imaginary" part refers to the product of the dashpot coefficient $\begin{array}{l}C\end{array}$ with the circular frequency $\begin{array}{l}\omega\end{array}$ . The dynamic stiffness ("real" part) can be evaluated with the following formula:

(20)	$\begin{array}{l}\displaystyle \bar{K}(\omega) = K\cdot k(\omega)\end{array}$

The values for the static stiffness $\begin{array}{l}K\end{array}$ and the dynamic stiffness coefficient $\begin{array}{l}k(\omega)\end{array}$ can be evaluated with help of table 1-2. The "imaginary" part can be split into radiation damping and hysteretic damping:

(21)	$\begin{array}{l}\displaystyle total~C = radiation~C + \frac{2\bar{K}}{\omega}\beta\end{array}$

While the radiation dampening coefficient $\begin{array}{l}radiation~C\end{array}$ and the dynamic stiffness $\begin{array}{l}\bar{K}\end{array}$ are also evaluated with help of table 1-2, the hysteretic damping $\begin{array}{l}\beta\end{array}$ and the circular frequency $\begin{array}{l}\omega\end{array}$ can freely be chosen dependent on the project's boundary conditions.

Surface Foundations

In surface foundations, the coupling terms from the swaying-rocking impedance become negligibly small. Due to this, the coupling terms do not have to be considered in the modeling of the impedance function. Therefore, it is possible to entirely model the impedance function using multiple sets of COMBIN14 elements. Using the following table, the impedance for the foundation can be found.

Table 1: Dynamic Stiffnesses and Dashpot Coefficients

Additionally, dimensionless graphs for the dynamic impedance coefficients have to be used to find all frequency dependent values in Table 1. Shown in the image below are the graphs used in this project. In the literature of Gazetas [3], more graphs can be found for different impedances or foundation types.

For small values of A0, the dynamic stiffness equates to roughly 1. It thus can be assumed, that a frequency-independent approach (dynamic stiffness = static stiffness), is sufficient enough for finding the final displacements of a given system. Since $\begin{array}{l}a_{0} = \dfrac {\omega \cdot B} {v_{s}}\end{array}$ *, it becomes clear, that with higher shear wave velocities $\begin{array}{l}v_s\end{array}$ or smaller foundations ( $\begin{array}{l}B\end{array}$ ), the frequency-independent result becomes more accurate.

*A0 is often also expressed with the following formula: $\begin{array}{l}a_{0} = \dfrac {\omega \cdot r_{equ}} {v_{s}}\end{array}$ , where $\begin{array}{l}r_{equ}\end{array}$ is the equivalent radius of the foundation.

Embedded Foundations

The approach for the embedded foundations is roughly identical to surface foundations. However, additionally, swaying-rocking impedance has to be considered. With the following table, the impedance for the foundation can be evaluated.

Table 2: Dynamic Stiffnesses and Dashpot Coefficients

Just like with surface foundations, dynamic impedance coefficients have to be evaluated to find all impedance values from table 2. New to embedded foundations are coefficients that define the impact of the embedment depth. An example of such graphs can be seen in the image below. The graph is only valid for square foundations.

Impedance Tables from Wong and Luco [4]

In Wong and Luco's literature, the foundation is assumed to be the surface type. Each of the impedance functions is recorded as 21 values under the dimensionless frequency $\begin{array}{l}a_{0}\end{array}$ from 0 to 10 with an interval of 0.5.

The figure shown uses the Poisson ratio $\begin{array}{l}\nu = 0.33\end{array}$ and hysteretic damping ratio $\begin{array}{l}\xi = 0.02\end{array}$ .

Flowchart of simulation

/PREP7, /SOL, and /POST26 are the three main processes of ANSYS simulation, which are pre-processor, solver, and post-processor respectively. Notice that before the solver, the stiffness type has to be chosen. If the stiffness type is "decoupled", the COMBIN14 element is used. Since the COMBIN14 element can accept the "tabular parameter", meaning the frequency-dependent stiffness can be inputted as the array or table, the simulation will be straightforward. The stiffness value stored in the COMBIN14 element can vary automatically when the frequency is updated. When the "coupled" type is chosen, then the MATRIX27 element will be used. This element cannot access the tabular parameter, therefore the MATRIX27 element has to be created and deleted manually when the frequency is updated each time.

Results - Harmonic Analysis

Unit definitions

Soil parameters:

Density $\begin{array}{l}\rho = 1400~ [kg/m^3]\end{array}$
The Poisson ratio $\begin{array}{l}\nu =0.33\end{array}$
Shear wave velocity $\begin{array}{l}v_{s} ~[m/s]\end{array}$

Foundation parameters:

Foundation size $\begin{array}{l}l_{f} ,w_{f} ~[m]\end{array}$
If an embedded foundation is applied, total depth $\begin{array}{l}D_{emb} ~[m]\end{array}$ and effective depth $\begin{array}{l}d_{emb} ~[m]\end{array}$ will be included.

Geometric definitions

Location of recorded responses

Node definitions

Node 1 is located at the bottom end of the column (BEAM element). In order to add base constraints for the column, node 1 can be used. However, since both COMBIN14 and MATRIX27 require two nodes as input, the additional nodes 2-x, 2-y and 2-z have to be introduced.

Simplification for MATRIX27

MATRIX27 elements are used for:

a surface foundation by Wong and Luco [4]
an embedded foundation from Gazetas [3]

A simplification is done for the use of MATRIX27 that we treated the model as the 2-D structure, and the BCs are set as $\begin{array}{l}U_{node1,i} = 0 ~,i=2,4,6\end{array}$ . Therefore the dynamics stiffness model could be reduced which only $\begin{array}{l}k_{11},~k_{15},~k_{33},~k_{55}\end{array}$ remained.

Result comparison: frequency independent and dependent impedances

Results with respect to the impedance function from Gazetas [3]

Excitation in the vertical direction:

Foundation size: $\begin{array}{l}l_{f} = w_{f} = 0.75m\end{array}$
Shear wave velocity: $\begin{array}{l}v_{s} = 100,~200,~400~[m/s]\end{array}$
COMBIN14 elements between nodes 1 and 2-x/2-y/2-z
Excitation: $\begin{array}{l}U^{*}_{node2-z} = [~0,0,1,0,0,0~]^T\end{array}$
Constraints: $\begin{array}{l}U_{node2-x,i}= 0 ,~for~i=1\end{array}$
$\begin{array}{l}U_{node2-y,i}=0,~for~i=2\end{array}$

Story 0

Story 1

Because the dynamic stiffness coefficient is roughly 1 at lower frequencies, the solution of the frequency-dependent model is roughly the same as for the frequency-independent model for small frequencies. Additionally, the distribution of the dynamic impedance coefficients depends on the shear wave velocity and the size of the foundation ( $\begin{array}{l}a_{0} = \dfrac {\omega \cdot r_{equ}} {v_{s}}\end{array}$ ). Therefore, with smaller foundation sizes, or alternatively, with higher shear wave velocities (as can be seen in the results above), the frequency-independent model is sufficient enough for an accurate result. Analyzing these results further, a rather interesting observation can be made. It is quite obvious, that the eigenmode of the plate is around 15-18Hz, depending on the dynamic stiffness of the system. However, right after this peak, a sudden drop in the displacements can be observed. It is likely, that the mass of the plate acts similarly to a tuned mass-damping system at this frequency.

Impact of the shear wave velocity on the structure's response

Foundation size : $\begin{array}{l}l_{f} = w_{f} = 0.75m\end{array}$
Shear wave velocity: $\begin{array}{l}v_{s} = 100,~200,~400~[m/s]\end{array}$

Results with respect to the impedance function from Gazetas [3]

Excitation in vertical direction:

COMBIN14 elements between nodes 1 and 2-x/2-y/2-z
Excitation: $\begin{array}{l}U^{*}_{node2-z} = [~0,0,1,0,0,0~]^T\end{array}$
Constraints: $\begin{array}{l}U_{node2-x,i}= 0 ,~for~i=1\end{array}$
$\begin{array}{l}U_{node2-y,i}=0,~for~i=2\end{array}$

Story 0

Story 1

Excitation in vertical + horizontal direction:

COMBIN14 elements between nodes 1 and 2-x/2-y/2-z
Excitation: $\begin{array}{l}U^{*}_{node2-x} = [~1,0,0,0,0,0~]^T\end{array}$
$\begin{array}{l}U^{*}_{node2-y} = [~0,1,0,0,0,0~]^T\end{array}$
$\begin{array}{l}U^{*}_{node2-z} = [~0,0,1,0,0,0~]^T\end{array}$

Story 0

Story 1

As already stated in "Result comparison: frequency independent and dependent impedances", the dynamic impedance coefficients are dependent on the shear wave velocity. The coefficients from Gazetas [3] only range from $\begin{array}{l}( 0 \leq a_{0} \leq 2 )\end{array}$ . This is a big limitation compared to Wong and Luco [4], which has documented impedance values up to $\begin{array}{l}a_0 = 10\end{array}$ . Due to this limitation, soils with a low shear wave velocity cannot evaluate frequency-dependent results above roughly 40Hz. However, the dynamic stiffness generally decreases, while the dynamic damping increases with higher frequencies. Therefore, as a simplification, a mainly damped system can be used for the analysis at high frequencies.

Impact of the foundation size on the structure's response

Foundation size : $\begin{array}{l}l_{f} = w_{f} = 0.5,~0.75,~1.0~[m]\end{array}$
Shear wave velocity: $\begin{array}{l}v_{s} = 200m/s\end{array}$

Results with respect to the impedance function from Gazetas [3]

Excitation in vertical direction:

COMBIN14 elements between nodes 1 and 2-x/2-y/2-z
Excitation: $\begin{array}{l}U^{*}_{node2-z} = [~0,0,1,0,0,0~]^T\end{array}$
Constraints: $\begin{array}{l}U_{node2-x,i}= 0 ,~for~i=1\end{array}$
$\begin{array}{l}U_{node2-y,i}=0,~for~i=2\end{array}$

Story 0

Story 1

Excitation in vertical + horizontal direction:

COMBIN14 elements between nodes 1 and 2-x/2-y/2-z
Excitation: $\begin{array}{l}U^{*}_{node2-x} = [~1,0,0,0,0,0~]^T\end{array}$
$\begin{array}{l}U^{*}_{node2-y} = [~0,1,0,0,0,0~]^T\end{array}$
$\begin{array}{l}U^{*}_{node2-z} = [~0,0,1,0,0,0~]^T\end{array}$

Story 0

Story 1

Varying the foundation size has a similar effect as varying the shear wave velocity. Unlike with the shear wave velocity, an increase of the parameter in this case will lead to a decreased analyzable range in the frequency domain ( $\begin{array}{l}( 0 \leq a_{0} \leq 2 )\end{array}$ with $\begin{array}{l}a_{0} = \dfrac {\omega \cdot r_{equ}} {v_{s}}\end{array}$ ). Additionally, the foundation size will also affect the static stiffness of the foundation (Table 1).

Impact of hysteretic damping on the structure's response

Foundation size : $\begin{array}{l}l_{f} = w_{f} =0.75m\end{array}$
Shear wave velocity: $\begin{array}{l}v_{s} = 200m/s\end{array}$

Excitation in vertical + horizontal direction:

COMBIN14 elements between nodes 1 and 2-x/2-y/2-z
Excitation: $\begin{array}{l}U^{*}_{node2-x} = [~1,0,0,0,0,0~]^T\end{array}$
$\begin{array}{l}U^{*}_{node2-y} = [~0,1,0,0,0,0~]^T\end{array}$
$\begin{array}{l}U^{*}_{node2-z} = [~0,0,1,0,0,0~]^T\end{array}$

Story 0

Story 1

Hysteretic damping can also be considered during the harmonic analysis of the structure. Following eq.(21), an increase in damping is to be expected. With increased damping, the structure's response to excitations at the eigenfrequency will be reduced. This expected effect can also be seen in the results shown above. In this case, 5% ( $\begin{array}{l}\beta=0.05\end{array}$ ) hysteretic damping was compared with 0% ( $\begin{array}{l}\beta=0\end{array}$ ) hysteretic damping.

Result comparison: impedance function from Gazetas[3] and Wong and Luco [4]

Foundation size : $\begin{array}{l}l_{f} = w_{f} = 0.75\end{array}$
Shear wave velocity: $\begin{array}{l}v_{s} = 200\end{array}$
Constraints at the ground: $\begin{array}{l}U_{node1,i} = 0 ~,i=2,4,6\end{array}$

Load case of Gazetas[3]

$\begin{array}{l}U^{*}_{node2-x} = [~1,0,0,0,0,0~]^T\end{array}$
$\begin{array}{l}U^{*}_{node2-z} = [~0,0,1,0,0,0~]^T\end{array}$

Load case of Wong and Luco[4]

Excitation: $\begin{array}{l}U^{*}_{node2} = [~1,0,1,0,0,0~]^T\end{array}$

Story 0

Story 1

In order to compare the results, we consider it as the 2-D model and input both the vertical and horizontal excitation simultaneously.

The use of the impedance function provided by the two different literature [3],[4] leads us to similar results. Although they have slight differences in the higher frequency ranges, they align pretty well in the lower frequency range.

Result comparison: Surface foundation (Wong and Luco [4]) and embedded foundation (Gazetas[3])

Foundation size : $\begin{array}{l}l_{f} = w_{f} = 0.75\end{array}$
Shear wave velocity: $\begin{array}{l}v_{s} = 200\end{array}$
Constraints at the ground: $\begin{array}{l}U_{node1,i} = 0 ~,i=2,4,6\end{array}$
Comparison of the depth ratio w.r.t the size of the foundation

Load case of Gazetas[3]

$\begin{array}{l}U^{*}_{node2-x} = [~1,0,0,0,0,0~]^T\end{array}$
$\begin{array}{l}U^{*}_{node2-z} = [~0,0,1,0,0,0~]^T\end{array}$

Load case of Wong and Luco[4]

Excitation: $\begin{array}{l}U^{*}_{node2} = [~1,0,1,0,0,0~]^T\end{array}$

Story 0

Story 1

For interpreting the result of the use of an embedded foundation, let us focus on the result of the ground floor (Story 0). The results on the first floor usually depend more based on the modal shape of the structure, i.e. the eigenmodes of the slab, however, the one at the level of the ground is connected directly with the impedance model, which means it will be affected by the foundation model mainly.

When using the surface foundation, there are not only vertical and horizontal responses but also a response in rocking, which is the rotation along the y-axis. When it is replaced by the embedded foundation, we could expect that all the responses will be reduced especially the rocking, since the foundation is not laid on the surface but built "into" the ground, and if the depth ratio gets higher, the response should be decreased more.

The results in story 0 match our expectations, We can observe that compared to the use of a surface foundation, all the responses are reduced and the rocking responses were reduced significantly when using an embedded foundation.

References

[1] J. Ponstingl, "Süddeutsche Zeitung," 15.11.2017. [Online]. Available: https://www.sueddeutsche.de/wissen/geothermie-unter-unseren-fuessen-1.3747216. [Accessed 30.01.2023].

[2] M. Pilger, T. Plenefisch, U. Wegler and L. Ceranna, "BINE Informationsdienst," 2018. [Online]. Available: https://api.deutsche-digitale-bibliothek.de/binary/b318c12c-fe85-48ac-911f-efb3a56145f1.pdf. [Accessed 30.01.2023].

[3] Gazetas, "Formulas and Charts for Impedances of Surfaces and Embedded Foundations," Journal of Geotechnical Engineering, vol. 117, no. 9, pp. 1363-1381, 1991.

[4] Wong and Luco, "Tables of Impedance Functions and Input Motions for Rectangular Foundations," University of Southern California, Los Angeles, 1978.

[5] G. Muller, "Soil Vibrations - Emission, Propagation, Immission, Abatement," University of Munich, Germany, 2022.

[6] John P. Wolf, "Spring‐dashpot‐mass models for foundation vibrations,” Earthquake Engineering and Structural Dynamics, vol.26, 931–949, 1997.

$\begin{array}{l}\end{array}$

Seitenhierarchie

Frequency Dependent LPM for Soil-Structure Interaction under Induced Seismicity

Table of Content

Introduction

Geothermal Energy

Shallow Geothermal

Deep Geothermal

Hydrothermal Geothermal

Petrothermal Geothermal

Induced Seismicity

Theoretical Base

Lumped-parameter model of rigid massless foundation (LPM)

Excitation and transfer function

Implementation of the impedance model

Modeling

FEM model of building with footing

MASS21 element

COMBIN14 element

MATRIX27 element

Impedance Function

Impedance Charts and Formulas from Gazetas [3]

Surface Foundations

Embedded Foundations

Impedance Tables from Wong and Luco [4]

Flowchart of simulation

Results - Harmonic Analysis

Unit definitions

Soil parameters:

Foundation parameters:

Geometric definitions

Location of recorded responses

Node definitions

Simplification for MATRIX27

Result comparison: frequency independent and dependent impedances

Results with respect to the impedance function from Gazetas [3]

Excitation in the vertical direction:

Impact of the shear wave velocity on the structure's response

Results with respect to the impedance function from Gazetas [3]

Excitation in vertical direction:

Excitation in vertical + horizontal direction:

Impact of the foundation size on the structure's response

Results with respect to the impedance function from Gazetas [3]

Excitation in vertical direction:

Excitation in vertical + horizontal direction:

Impact of hysteretic damping on the structure's response

Excitation in vertical + horizontal direction:

Result comparison: impedance function from Gazetas[3] and Wong and Luco [4]

Result comparison: Surface foundation (Wong and Luco [4]) and embedded foundation (Gazetas[3])

References