# Effects of Rayleigh waves on seismic response of liquid storage tanks

Yaël Perraud, Charisis T Chatzigogos, Kristel Carolina Meza Fajardo

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Yaël Perraud, Charisis T Chatzigogos, Kristel Carolina Meza Fajardo. Effects of Rayleigh waves on seismic response of liquid storage tanks. 3rd International Conference on Natural Hazards & Infrastructure (ICONHIC2022), Jul 2022, Athènes, Greece. hal-04050006

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# Effects of Rayleigh waves on seismic response of liquid storage tanks

Yaël Perraud<fnref n="1" />, Charisis T. Chatzigogos

*Géodynamique & Structure (Montrouge, France)*

Kristel C. Meza-Fajardo

*Bureau de Recherches Géologiques et Minières (Orléans, France)*

## ABSTRACT

The present study focuses on the effect of Rayleigh waves on the seismic response of liquid storage tanks. Rayleigh waves generated at geological basins are characterized by low-frequency content and can thus excite the sloshing modes of liquid storage tanks, which are also low-frequency modes. The effects of Rayleigh waves are studied with synthetic signals generated using the Specific Barrier Model, enriched with basin-generated surface waves. Two loading cases are considered to study the effect of Rayleigh waves: a first one, in which the seismic excitation is devoid of Rayleigh waves, and a second one, including a Rayleigh wave component. We perform a parametric study to identify the parameters controlling the response of the tanks and the amplification of the response due to Rayleigh waves (tank geometry, soil properties, etc.). For this purpose, we implement an analytical model of the fluid-tank-soil system, which is based on the resolution of Laplace potential flow equation. Results of this work show that Rayleigh waves can amplify significantly the seismic response of a liquid storage tank (up to 3 times for some signals and some Engineering Demand Parameters) and that the most important amplification is obtained for storage tanks characterized by low aspect ratios (height/radius).

*Keywords: Liquid storage tanks, Rayleigh waves, parametric study, Specific Barrier Model*

## INTRODUCTION

Several recent seismic events have led to non-direct, far from source and with important delay, damage on liquid-storage tanks. One flagrant example is the Tokachi-Oki earthquake (Japan, 2003, $M_w$ 8.3), which led to extensive damage in an oil refinery in Tomakomai (Coffin & Hirata, 2003). This city is located in a sedimentary basin (Yufutsu basin) and the earthquake motion recorded during this event was shown to be rich in surface waves (Koketsu et al., 2003). Post-seismic analyses have shown that fire accidents were induced by damage on tank roofs caused by liquid sloshing. Since liquid sloshing modes inside the tank are characterized by low frequencies that can be excited by surface waves, liquid-storage tanks appear as a type of structure particularly vulnerable to the incidence of surface waves. This paper is dedicated to the study of the effects of Rayleigh waves on the seismic response of rigid, cylindrical, ground-supported, liquid storage tanks.

<footnote n="1">¹ Corresponding author: Yaël Perraud, <mailto:yael.perraud@geodynamique.com>
Géodynamique & Structure, Immeuble Cap Sud, 106 Avenue Marx Dormoy, 92120, Montrouge, France.</footnote>
LIQUID STORAGE TANK MODEL

Previous studies on the effects of surface waves on long-period civil structures, such as high-rise buildings (Meza-Fajardo & Papageorgiou 2018, 2019) and bridge pylons (Chatzigogos & Meza-Fajardo, 2020), (Perraud et al., 2022) used semi-analytical methods and/or simplified finite element models for quantifying the additional discharge in these structures caused by the incidence of Rayleigh waves. For the study of liquid storage tanks, we present in the following, an analytical method based on the works of Yang (Yang, 1976). The model is based on the resolution of Laplace potential flow equation for rigid, cylindrical, laterally excited, liquid-storage-tanks (Equation (1)), which reads as follows:

$$ \nabla^2 \phi = \frac{\partial^2 \phi}{\partial r^2} + \frac{1}{r} \frac{\partial \phi}{\partial r} + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0 \quad (1) $$

The approach proposed in (Yang, 1976) is based on the common decomposition of the global solution as the sum of two solutions: an impulsive one, assuming that part of the liquid mass that moves in unison with the motion of the structure (Jacobsen & Ayre, 1949) and a convective one, assuming that part of the fluid mass that vertically oscillates at the free surface of the liquid (Housner, 1957). This model uses also the subsequent developments presented in (Veletsos & Tang, 1990) to take into account soil-structure interaction effects in the study of liquid-storage tanks.

The hydrodynamic pressure can be obtained via the time derivative of the velocity potential $\phi$ (cf. Equation (2)); the global base shear $Q$, the moment due to pressure on walls $M$ and the moment due to pressure on tank base $\Delta M$ can then be deduced by a spatial integration of pressure field at the liquid-tank contact surface (cf. Equations (3), (4) and (5)):

$$ p(r, \theta, z, t) = \rho_{\ell} \frac{\partial \phi}{\partial t} \quad (2) $$

$$ Q(t) = \int_{0}^{H} \int_{0}^{2\pi} p(r, \theta, z, t)|_{r=a} a \cos \theta \, d\theta \, dz \quad (3) $$

$$ M(t) = \int_{0}^{H} \int_{0}^{2\pi} p(r, \theta, z, t)|_{r=a} az \cos \theta \, d\theta \, dz \quad (4) $$

$$ \Delta M(t) = \int_{0}^{2\pi} \int_{0}^{a} p(r, \theta, z, t)|_{z=0} r^2 \cos \theta \, dr \, d\theta \quad (5) $$

The height of the sloshing wave, $\eta$, can also be obtained by integrating the vertical velocity component of the fluid, $q$, at the free surface:

$$ q(r, \theta, z, t)|_{z=H} = \left. \frac{\partial \phi}{\partial z} \right|_{z=H} \quad (6) $$

$$ \eta(r, \theta, t) = \int q(r, \theta, t)|_{z=H} dt \quad (7) $$

Following the definition of appropriate boundary conditions associated to the impulsive and convective parts of the response, the above differential equations (1) to (7) can be solved. The impulsive component can be represented as a rigidly connected mass $m_0$ on the tank walls (cf. Figure 1). Because the tank is assumed to be rigid, the different engineering demand parameters (EDPs) of the impulsive response are proportional to the input signals. The convective component can be represented as a series of masses elastically connected to
tank walls (cf. Figure 1) with each mass corresponding to the sloshing modes of the liquid. Some seismic design standards such as the Eurocode 8 propose this simplified approach as long as it is physically based on robust description of liquid motion and since it has been validated against numerous experimental tests (e.g. Clough, 1977).

Because the tank is assumed to be rigid and under the assumption of linear behavior, the different EDPs due to the convective component can be computed as the sum of a number of Single Degree Of Freedom (SDOF) responses corresponding to each sloshing mode. In this study, these responses are computed in the frequency domain via multiplication of the Fourier transform of the input signals with the transfer functions of the tank-soil system. The resolution of the equations arising in this model have been coded in MATLAB® environment.

<figure><img src="image_1.png" /><figcaption>**Figure 1.** Housner equivalent model of liquid storage tank</figcaption></figure>

Input parameters of the proposed model are listed hereafter in a vector **a**, and are schematically represented on Figure 2:

$$ \mathbf{a}^{\mathrm{T}} = \{a, H, L, \rho_{\ell}, \xi, e_t, \rho_t, e_f, \rho_f, \rho_s, c_s, v_s\} \qquad (8) $$

<figure><img src="image_2.png" /><figcaption>**Figure 2.** Liquid-storage tank input parameters</figcaption></figure>

<table><thead><tr><th>Parameter</th><th>Symbol</th></tr></thead><tbody><tr><td>radius of the tank</td><td>a</td></tr><tr><td>height of the liquid inside the tank</td><td>H</td></tr><tr><td>height of the tank</td><td>L</td></tr><tr><td>density of the liquid inside the tank</td><td>ρ<sub>l</sub></td></tr><tr><td>liquid damping ratio</td><td>ξ</td></tr><tr><td>thickness of the tank walls</td><td>e<sub>t</sub></td></tr><tr><td>density of tank wall material</td><td>ρ<sub>t</sub></td></tr><tr><td>thickness of foundation</td><td>e<sub>f</sub></td></tr><tr><td>density of foundation material</td><td>ρ<sub>f</sub></td></tr><tr><td>density of soil material</td><td>ρ<sub>s</sub></td></tr><tr><td>soil shear wave velocity</td><td>c<sub>s</sub></td></tr><tr><td>soil Poisson ratio</td><td>ν<sub>s</sub></td></tr></tbody></table>
## PARAMETRIC STUDY

The previously described model focuses on the hydrodynamic effects of the fluid inside the tank, while neglecting tank flexibility. Because hydrodynamic effects are generally much larger and because of the important mass of fluid compared to tank mass itself it has been decided to keep constant values for the parameters related to tank structural mass: $e_t, \rho_t, e_f, \rho_f$. Hydrodynamic forces generated by fluid motion are mainly controlled by the geometry and parameters of the fluid inside the tank: tank inner radius $a$, liquid height $H$, tank height $L$, liquid density $\rho_l$ and liquid damping ratio $\xi$ for sloshing modes. These variables are parametrized in our study. Soil-Structure Interaction (SSI) is also taken into account (Veletsos & Tang, 1990) and soil variables are parametrized on the basis of soil shear wave velocity $c_s$ (Table 2).

<table><caption>**Table 1.** Fluid-tank input parameters</caption><thead><tr><th>N°</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th></tr></thead><tbody><tr><th>H (m)</th><td>2.00</td><td>4.33</td><td>6.67</td><td>9.00</td><td>2.00</td><td>8.00</td><td>14.00</td><td>20.00</td><td>5.00</td><td>11.67</td><td>18.33</td></tr><tr><th>a (m)</th><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>5.00</td><td>5.00</td><td>5.00</td><td>5.00</td><td>12.00</td><td>12.00</td><td>12.00</td></tr><tr><th>H/a</th><td>2.00</td><td>4.33</td><td>6.67</td><td>9.00</td><td>0.40</td><td>1.60</td><td>2.80</td><td>4.00</td><td>0.42</td><td>0.97</td><td>1.53</td></tr></tbody></table>

(a)

<table><thead><tr><th>N°</th><th>12</th><th>13</th><th>14</th><th>15</th><th>16</th><th>17</th><th>18</th><th>19</th><th>20</th><th>21</th><th>22</th></tr></thead><tbody><tr><th>H (m)</th><td>25.00</td><td>5.00</td><td>13.33</td><td>21.67</td><td>30.00</td><td>10.00</td><td>20.00</td><td>30.00</td><td>10.00</td><td>20.00</td><td>30.00</td></tr><tr><th>a (m)</th><td>12.00</td><td>23.00</td><td>23.00</td><td>23.00</td><td>23.00</td><td>34.00</td><td>34.00</td><td>34.00</td><td>45.00</td><td>45.00</td><td>45.00</td></tr><tr><th>H/a</th><td>2.08</td><td>0.22</td><td>0.58</td><td>0.94</td><td>1.30</td><td>0.29</td><td>0.59</td><td>0.88</td><td>0.22</td><td>0.44</td><td>0.67</td></tr></tbody></table>

(b)

The tested ranges of parameters for tank geometry are listed in Table 1. Selected values are in agreement with the database compiled by the American Lifeline Alliance in the context of post-seismic inspections on liquid storage tanks (American Lifelines Alliance (ALA), 2001). Fixed parameters related to tank/foundation mass are deduced from the followings expressions: $e_f = 0.50m, \rho_f = 2.40, e_t = 0.002a, \rho_t = \rho_l/0.127$. We have also decided to consider constant values for the fill ratio $H/L$ (full tank), fluid density $\rho_l$ (1.0), liquid damping $\xi$ (0.75%), soil density (2.0t/m³) and soil Poisson ratio (1/3). In total 88 tank-fluid-soil configurations are tested.

<table><caption>**Table 2.** Soil input parameters</caption><thead><tr><th>Soil type</th><th>Soil 1</th><th>Soil 2</th><th>Soil 3</th><th>Soil 4</th></tr></thead><tbody><tr><th>c<sub>s</sub> (m/s)</th><td>150</td><td>200</td><td>250</td><td>300</td></tr></tbody></table>

## INPUT GROUND MOTIONS

Previous studies on the effect of Rayleigh waves (and of the rocking seismic component) on high-rise buildings (Meza Fajardo & Papageorgiou 2018, 2019) and bridge pylons (Chatzigogos & Meza Fajardo, 2020), (Perraud *et al.*, 2022) were performed with a reduced number of recorded signals that were processed in order to extract the Rayleigh waves component (Meza Fajardo *et al.*, 2015) and estimate the rocking seismic component. In the case of liquid storage tanks, synthetic signals generated using the Specific Barrier Model (Papageorgiou & Aki, 1983a, b), (Halldorsson & Papageorgiou, 2005) are used for the definition of input seismic excitation.
Recent developments intend to integrate a better understanding of wave generation/propagation mechanisms, with path effects and local site effects, within the Specific Barrier Model. The model is adapted to generate signals that are rich in surface waves, and is here integrated in a stochastic modelling approach allowing to make multiple realizations (i.e., generate multiple signals with a single set of parameters). Signals are generated in a parametric way. Values of parameters selected within this parametrization are listed hereafter:

* Magnitude Mw: 6.0, 6.5, 7.0, 7.5

* Eurocode soil class: C, D

* Joyner-Boore distance: 30, 50, 100, 150km

* Basin edge - site distance: 10, 20km

* Surface waves amplification factor: 1.5, 2.0

* Surface waves central frequencies: 0.15, 0.25, 0.40, 0.50Hz

Three signal realizations were generated for each set of parameters. A total number of 1536 signals were thus obtained. One example of signal acceleration time-history is presented in Figure 3. This figure also displays the associated Rayleigh wave time-history extracted from the total signal.

<figure><img src="image_3.png" /><figcaption>**Figure 3.** Synthetic signal generated within the Specific Barrier Model: (a) total signal, (b) extracted Rayleigh waves signal</figcaption></figure>

If the Rayleigh wave signal is subtracted from the total signal, it can be considered that the resulting signal is devoid of any surface wave component. For each signal, two loading cases have accordingly been defined:

* Case 1: Signals devoid of Rayleigh wave component

* Case 2: Total signals

## RESULTS

The main objective of the proposed tank model is to reproduce the dynamic behavior of the studied system while keeping a reasonable computational cost and also for obtaining meaningful EDPs to characterize the different damage states/failure modes of the structure. In the case of liquid-storage tanks, failure modes can be of several types: settlement, liquefaction, “diamond buckling” (elastic buckling), “elephant foot buckling” (post-elastic buckling), damage induced by sloshing waves (roof damage), anchoring failure or tear of tank base (depending on whether the tank is anchored or not). Soil liquefaction will not be considered hereafter. In this context, and given the fact that the model is linear (consequently, no failure mechanisms can be directly obtained within the model), the considered EDPs of this study are:

* The maximum hydrodynamic pressure at the base of tank walls: $p_{tot,max}$

* The maximum base shear at the base of the tank: $Q_{tot,max}$

* The maximum overturning moment due to the hydrodynamic pressure on tank walls: $M_{tot,max}$

* The maximum overturning moment due to the hydrodynamic pressure on tank base: $\Delta M_{tot,max}$

* The total maximum overturning moment at the foundation: $M'_{tot,max}$

* The maximum height of sloshing wave: $\eta_{tot,max}$
These EDPs are normalized, respectively, by the hydrostatic pressure, the liquid weight, the product of liquid height times liquid weight and the height of liquid inside the tank for the height of the sloshing wave.

$$ \tilde{p} = \frac{|p_{\text{tot,max}}|}{\rho_{\ell} g L} \qquad (9) $$

$$ \tilde{Q} = \frac{|Q_{\text{tot,max}}|}{m_{\ell} g} \qquad (10) $$

$$ \tilde{M} = \frac{|M_{\text{tot,max}}|}{m_{\ell} g L}; \tilde{\Delta M} = \frac{|\Delta M_{\text{tot,max}}|}{m_{\ell} g L}; \tilde{M}' = \frac{|M'_{\text{tot,max}}|}{m_{\ell} g L} \qquad (11) $$

$$ \tilde{\eta} = \frac{|\eta_{\text{tot,max}}|}{L} \qquad (12) $$

In the above equations, the normalized EDPs are designated with a tilde (~).

The results displayed in this part are plotted against the following parameters: tank aspect ratio $H/a$, and a period ratio between fundamental period of first sloshing mode and Rayleigh waves central period $T_1/T_{c,RW}$.

It is well known that aspect ratio $H/a$ is a controlling parameter of hydrodynamic effects inside the tank. It controls the influence of the impulsive and convective components on the overall response of the system (impulsive/convective equivalent masses and heights), and the frequencies of the sloshing modes, which, in this case study, are the basic parameters of interest because the main focus is on potential amplification of convective components due to Rayleigh waves (low frequency waves). The ranges of tested tank configurations lead to periods of first sloshing modes as large as 12 seconds (large, broad tanks). The surfaces of the first and second sloshing periods, plotted against the values of aspect ratio $H/a$ and liquid height $H$ for the studied tank configurations, are displayed on Figure 4.

<figure><img src="image_4.png" /><figcaption>Figure 4. Fundamental period of first (a) and second (b) sloshing modes against aspect ratio $H/a$ and liquid height $H$</figcaption></figure>

Ratio $T_1/T_{c,RW}$ is used to highlight potential resonance phenomena between Rayleigh wave frequency content and tank response amplification due to this component. The central periods of Rayleigh wave signals vary between 2.00 seconds and 6.67 seconds.
The number of performed analyses is important (88 tank-liquid-soil configurations, 1536 signals), and in order to make the results more easily encompassed, Figures 5 and 6 are plotted as 2D histograms with a color convention representing the density of configurations yielding a particular result obtained for a given range of EDP values (plotted in logarithmic scale).

Figures 5 and 6 display the response of liquid storage tanks when they are subjected to signals devoid of Rayleigh waves (Figures 5a, 6a) and with signals incorporating the Rayleigh wave component (Figures 5b, 6b). The amplification ratio (EDP of load case 2 divided by EDP of load case 1) (amplification associated to Rayleigh waves component) is displayed on Figures 5c and 6c. The results are either plotted against tank aspect ratio *H/a* (Figure 5) or period ratio *T*₁/*T*<sub>C,RW</sub> (Figure 6).

<figure><img src="image_5.png" /><figcaption>**Figure 5.** 2D histograms of liquid storage tank response: (a) load case 1, (b) load case 2, and (c) response amplification ratios (load case 2/load case 1) against aspect ratio *H/a*</figcaption></figure>
An initial observation that deserves to be made is that among the large number of performed analyses, comparison between figures 5a and 5b (and respectively 6a and 6b) does not clearly highlight a flagrant qualitative difference due to the surface wave component. The trends for the two load cases are qualitatively similar, with the normalized hydrodynamic pressure $\tilde{p}$ and the normalized moment due to pressure on tank base $\Delta\tilde{M}$ to decrease with the increase of parameter $H/a$. On the contrary, the normalized base shear $\tilde{Q}$ and the normalized moment due to pressure on tank walls $\tilde{M}$ increase with the increase of $H/a$. As a result, the global normalized overturning moment $\tilde{M}'$ (sum of moment due to pressure on tank walls and moment due to pressure on base) is not monotonically related to ratio $H/a$. Finally, the height of the sloshing wave $\tilde{\eta}$ does not exhibit either a clear correlation with aspect ratio $H/a$.

<figure><img src="image_6.png" /><figcaption>**Figure 6.** 2D histograms of liquid storage tank response: (a) load case 1, (b) load case 2, and (c) response amplification ratios (load case 2/load case 1) plotted against period ratio $T_1/T_{c,RW}$</figcaption></figure>
Despite the large amount of plotted data points in Figures 5c and 6c, it can be understood that there is indeed an amplification in the EDPs due to the presence of Rayleigh waves. These amplifications can go up to values as high as 2.5 or 3.0 for certain isolated cases. The most pronounced amplifications are obtained for low aspect ratios (broad tanks: large, with small height), typically close to 0.5, except for the normalized height of sloshing wave which clearly amplified for period ratios $T_1/T_{c,RW}$ close to 1.

Figures 5 and 6 present trends and peaks that allow us to interpret the response amplification associated to Rayleigh waves as controlled mainly by two parameters:

* Period ratios $T_1/T_{c,RW}$ close to 1: this situation corresponds to a resonance between Rayleigh wave central period and the fundamental convective mode;

* Low aspect ratios ($H/a < 1$): this situation corresponds to wide tanks with small height which indeed, develop more significant convective masses and exhibit larger sloshing heights.

Furthermore, Figures 5 and 6 allow us to quantify the additional demand associated to Rayleigh waves on the most critical configurations: this can reach up to 2.5 (hydrodynamic pressure, base shear, overturning moments) and even 3.0 (height of sloshing wave).

Figure 7 displays the cumulative distribution of the amplification factors obtained among all tested soil-fluid-tank configurations. This type of plot allows to judge whether high or very high amplification factors can be obtained for several tank configurations or not. The results reveal that for the most important part of the analyses (e.g., 97% of the results) the amplification factors are below 1.3 for every EDPs, except for the height of sloshing wave, which reach a value of 2.5. This can be explained by the fact that sloshing wave is almost exclusively controlled by the convective (low frequency) part of the response.

<figure><img src="image_7.png" /><figcaption>**Figure 7.** Normalized cumulative distribution of response amplification due to RAYLEIGH waves.</figcaption></figure>

The obtained results in terms of response amplification associated to Rayleigh waves are in agreement with previous works performed on high-rise buildings via linear analyses (Meza Fajardo & Papageorgiou, 2018).
Finally, it is worth noting that nonlinear analyses on high-rise buildings (Meza Fajardo & Papageorgiou, 2019) and on bridge pylons (Perraud et al., 2022) have shown that for some special cases and EDPs, the response amplification due to Rayleigh waves can reach up to approximately an order of magnitude (~10), especially for EDPs of cumulative character (accumulated settlement, amount of dissipated energy, etc.). This difference in amplification factors obtained via linear and nonlinear analyses is explained by multiple facts as the reduced stiffness beyond linear/nonlinear threshold, which can conduce to rapidly growing displacement EDPs, and can be explained by the fact that surface waves are generally characterized by signals of long duration exhibiting a significant number of equivalent load cycles. This is why surface waves can appear more detrimental for EDPs of cumulative character.

## CONCLUSIONS

A large parametric study has been presented to characterize the effects of Rayleigh waves on liquid storage tank response. An analytical linear model for the seismic response of such structures has been considered, based on the decomposition of fluid response on a convective and an impulsive component. The resolution of the model is performed in the frequency domain.

The study is primarily parametrized on the basis of tank-fluid-soil parameters with different tank geometry, soil types, etc. and is conducted for a large number of synthetic signals (generated within the Specific Barrier Model) corresponding to a large range of seismic scenarios. Two loading cases are defined to study the additional demand associated to Rayleigh waves component: case 1 considering earthquakes exempt of the effect of Rayleigh waves; case 2 considering total signals (Rayleigh waves and body waves).

Main controlling parameter of the response of the structure is the tank aspect ratio (height of stored liquid divided by tank radius). This parameter controls the sloshing frequencies, the definition of impulsive/convective components, and consequently the hydrodynamic forces on the tank. The correlation between aspect ratio and tank response/amplification associated to Rayleigh waves/rocking component is investigated. The results of this study also investigate the influence of potential resonance phenomena between Rayleigh frequency content and the sloshing frequencies on tank global response amplification. Results highlight that broad tanks (i.e. exhibiting low aspect ratios) are the most impacted by the Rayleigh waves. This is true for all of the computed EDPs.

A quantification of the additional demand associated to Rayleigh waves and the distribution of this additional demand is also studied through the large number of results generated in this study. Results displayed amplification factors that can reach up to values of approximately 2.5 to 3 when Rayleigh waves are taken into account. However, this amplification factor is below 1.3 for the large majority of tested configurations and EDPs.

Future works aim at exploring more sophisticated, nonlinear models, using different sources of signals (i.e. recorded, simulated, etc.). The results will eventually be incorporated into a global approach for probing the effect of Rayleigh waves on the seismic fragility of liquid storage tanks.

## ACKNOWLEDGMENTS

This work has received funding from French National Research agency within the framework of project ANR Modulate (2018-2022).
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