The Impact of Differential Settlement on Sloshing Dynamics in Coastal Zone Storage Tanks Under External Excitation: Implications for Sustainable Development

Abstract

Large storage tanks situated in coastal areas are vulnerable to environmental hazards, with earthquakes being one of the most destructive forces threatening their structural safety. Additionally, differential settlement can significantly alter conditions in the tank, including the inclination, thereby changing the direction of external applied excitation forces and affecting the liquid sloshing response. To investigate the coupled effects of structural settlement and external excitation, model tests were conducted in series to analyze liquid sloshing behavior in a tilted tank subjected to harmonic excitation. The results revealed that the liquid response under combined environmental loads displayed distinct characteristics compared with that under single excitation. While the inclination angle had minimal influence during the unstable sloshing stage, it became crucial during the stable stage, particularly for third-order resonant responses, leading to intensified sloshing. More specifically, as the tilt angle of the storage tank from 0° to 8°, the steady-state wave height at third-order resonance increased by approximately 69%. This highlights the amplified risks to the structural stability and safety posed by differential settlement. Furthermore, variations in steady-state wave heights due to differential settlement conditions were investigated. The water level elevation along the tank walls varies as the inclination angles increase, which leads to potential risks to the stability of liquid storage under forced motion, especially under symmetric structural designs, and increases the likelihood of structural instability, oil spills, and other coastal disasters. These results provide valuable insights into the safety risks and sustainable utilization of coastal infrastructure, serving a basis for assessing and mitigating the risks associated with structural settlement and seismic excitations.

1. Introduction

External excitations, including strong winds [1] and ground motion [2], induce liquid sloshing, a critical factor affecting the stability of liquid containers. When the frequency of external excitation nears the natural frequency of the liquid tank, the internal sloshing can lead to unforeseen impacts on the tank, thereby compromising the operational safety [3,4]. Meanwhile, the effects of external excitation on liquid sloshing, such as an increase in the wave climbing height, can be further amplified by differential settlement when the structure is tilted, which subsequently affects the stability of the structure and the wall stress threshold. This may easily cause tank leaking, splashing, oil spillage, and other disasters, which pose significant threats to the coastal environment and the safety of people. Therefore, it is necessary to study liquid sloshing in storage tanks under multiple environmental loads.

Since 1966, researchers have conducted physical experiments to explore the liquid sloshing phenomenon [5]. Over the past few decades, numerous researchers have examined the sloshing characteristics in liquid storage tanks [6,7,8]. Sakai et al. [9] investigated the sloshing characteristics of floating-roof storage tanks through theoretical analyses and model tests. Faltinsen and colleagues proposed approximate analytical solutions for natural sloshing modes in liquid tanks of various geometries and extensively discussed the sloshing modes [10,11].

Numerous researchers have conducted studies on liquid sloshing using experimental approaches [12]. The liquid responses in tanks of various shapes under external excitation, such as 2D rectangular elastic tanks [13], 2D U-shaped tanks [14], cylindrical tanks [15], and horizontal Cassini tanks [16] have been investigated. Cui et al. [17] investigated the sloshing response under seismic excitation and revealed the distribution of wave peaks within the tank. Nonlinear sloshing responses in shallow water conditions with large amplitudes were investigated by Gurusamy and Kumar [18]. They noted that the resonant frequency of tanks with large aspect ratios increased by approximately 45% compared with the linear sloshing frequency under high excitation amplitude. The type of tank has an effect on the motion of the sloshing liquid [19]. Therefore, it is possible that different wall forms in liquid tanks may result in different deformations and propagation of the free liquid. This can be seen in the effects on the sloshing amplitude and modal characteristics associated with curved walls, sloped bulkheads, and beveled walls in spherical and hexagonal liquid tanks [20].

Meanwhile, the initial field characteristics of the liquid inside the tank are affected by settlement when the tank is tilted; additionally, the angle of interaction between the liquid and the wall changes and the direction of motion of the liquid along the wall changes. The outward manifestation of this phenomenon is a shift in the center of gravity and a change in the direction of the force. Nan et al. [21] noted that the lateral component of gravity in a low-gravity environment can lead to violent liquid sloshing in the tank. Subsequently, Li et al. [22] found that the level of residual gravity had a significant influence on the dynamic characteristics of free surfaces. With a reduction to low-gravity conditions, the dynamic behavior of the free surface remained qualitatively the same. However, the residual gravitational underdamping resulted in significant differences not only in the equilibrium state of the ullage but also the temporal evolution of oscillation frequencies and magnitudes. Liu et al. [23] further explored the effect of gravitational acceleration on sloshing. Furthermore, many scholars have found that the response of the liquid in the tank can also be affected by non-excitation-induced changes in the state. Lu et al. [24] developed a three-dimensional full-scale numerical model for analyzing tank sloshing response under leakage conditions. It was confirmed that the degree of sloshing had a huge impact on oil leakage. Zhang and Wan [25] analyzed the liquid sloshing response in an elastic tank and observed a fluctuating characteristic in the pressure signal due to the elasticity of the tank. Jiang et al. [26] investigated liquid sloshing loads and flow characteristics in liquid tanks of variable mass. They reported that fuel filling increased the mass of the fluid, suppressing sloshing and increasing the tank load. Cao et al. [27] investigated the sloshing response of a two-layer liquid. It was found that the inter-face displacement was mainly affected by the lower fluid depth, while the free surface displacement was mainly dependent on the total fluid depth. The above studies have shown that some static changes can also lead to alterations in the sloshing fluid response. Consequently, this aspect warrants further research attention.

Liquid response is different in various types of tanks under various external excitations. The oil storage tank, a classical liquid tank, is a pivotal component for industrial fuel and chemicals storage, integral to numerous aspects of industrial development [28,29]. Oil storage tanks are selected as the focus of this study, as they may be subjected to multiple excitations simultaneously in practical engineering applications. Considering the convenience and safety of industrial liquid loading and unloading, large storage tanks are usually built in the nearshore area of coastal cities (Figure 1). The landform and terrain can be affected by the usage of underground water, heavy rainfall, and the construction of buildings, potentially leading to ground settlement [30]. When this settlement occurs at the bottom of the tank, it may threaten the safety of the tank [31] (Figure 2). Tank settlement problems may induce many potential hazards, which can cause structural damage or instability, thereby reducing the durability and seismic resistance of the storage tank [32,33]. Currently, the research on tank settlement mainly focuses on the effect on the tank structure. Godoy and Sosa [30] studied the effect of support settlement on the out-of-plane displacement of a cylindrical tank with a fixed top cover. The results of the study indicated that the equilibrium path has highly nonlinear characteristics. Zhao et al. [34] investigated the nonlinear response and stability behavior of a floating roof steel tank under two types of settlement (global differential settlement and local differential settlement). They pointed out that very small differential settlements (about 0.3 times the wall thickness) can lead to local buckling of the eave ring or tank wall. Hotala and Ignatowicz [35] examined tanks with different foundations and showed the corresponding settlement modes. The numerical analysis method was also used to demonstrate that the condition of soil settlement and the deformation of the bottom slab can be evaluated through simple measurements of the lower part of the tank. As a special container with liquid inside, the liquid in the settled storage tank may exhibit different response modes to external excitation, and sloshing may occur. For soft foundations, the main form of settlement is planar [36]. After the planar settlement problem arises during the operation of the storage tank, the tilted bottom of the tank can cause the redistribution of the fluid in the tank, thereby affecting the safety of the storage tank. In addition, researchers investigated liquid sloshing in a full scale storage tank due to external excitation and found that the liquid sloshing excited by ground motion can cause serious structural failure [2].

Moreover, earthquakes are one of the most important dynamic loads that can damage storage tanks and have garnered considerable attention [38,39]. The study of liquid sloshing induced by seismic excitation can be dated back to the 1930s [40]. Several scholars have theoretically analyzed the response of storage tanks and developed various mechanical models [40,41,42]. In recent years, with the development of the computer technology, computational fluid dynamic (CFD) methods have been developed. Zhang et al. [43] investigated the effect of the direction of seismic excitation on the sloshing fluid and found that the effect of vertical seismic waves was small, accounting for only 5.72% of the maximum sloshing wave height under resonance conditions. Jin et al. [44] analyzed the nonlinear sloshing characteristics induced by 13 earthquakes and discovered that the sloshing height was closely related to the peak ground velocity (PGV) and frequency components. Additionally, Jin et al. [45] demonstrated that two-dimensional profiles could be employed to estimate the maximum response to three-dimensional liquid sloshing under nonlinear ground motion using the CFD method.

The above studies primarily concentrated on the liquid response of tanks in the initial horizontal state or the structural stability of settled tanks. Therefore, in the present work, the sloshing liquid response in a settled tank is reported. Consequently, factors such as the filled liquid depths, resonant modes, excitation amplitudes, and settled angles are considered in the experiments. The study contributes to a further understanding of the effects of settlement on the liquid response in storage tanks and provides guidance on how to prevent the hazards that may arise from settlement-affected storage tanks. Furthermore, it can aid in a more accurate evaluation of the safety of storage tanks to ensure environmental development in the coastal areas.

2. Materials and Methods

In this study, an experimental system was set up, including a liquid tank, a six degrees of freedom (6DoF) vibration platform, and a wave probe. A Plexiglas rectangular tank (L–0.58 m length, H–0.6 m height, and B–0.1 m breadth) was installed on a 6DoF vibration platform (Figure 3). The tank was a typical 2D rectangular tank, which effectively displays the relevant features of the actual 3D tank sloshing [45]. The problem was simplified to liquid sloshing in a rectangular tank in the sway direction. The motion displacement is shown in Equation (1), where A₀ and ω are the amplitude and angular frequency of excitation, respectively.

The natural frequencies of the tank are calculated according to Lamb [46]-Equation (2).

$$S=A_0\sin (\omega t)$$

(1)

$${\omega }_i=\sqrt{\mathrm{g}{\displaystyle \frac{\mathsf{\pi }i}{L}}\tanh \left({\displaystyle \frac{\mathsf{\pi }i}{L}}d\right)},i=1,2,3\dots n,$$

(2)

where S represents the displacement of the tank, g represents the gravitational acceleration (g = 9.81 m/s²), L represents the length of the tank, d represents the fluid depth in the tank, and i represents the order of the natural frequency.

The excitation displacement was generated by the 6DoF vibration platform. The wave probe W1 was installed on the sidewall of the tank, positioned 0.01 m (l) away from the sidewall (Figure 3b), and the results from the tilted side were used for analysis in the present research. According to API [47], the allowable settling angle for actual tanks is tiny. Therefore, to better observe the effect of settlement on liquid sloshing under the experimental conditions, the settlement angle of the liquid tank was selected as 0–10° at 1° intervals. The time series of free surface elevations were recorded at a sampling rate of 100 Hz over 180 s to collect sufficient data. The water depth in the test was selected based on the study by Faltinsen [48]. The finite liquid depth and critical depth are shown in

Table 1. Differentiating standards and test values for the water depths.

States	Standards	Values
Critical depth	d/L = 0.3368	d/L = 0.34
Finite liquid depth	0.2–0.25 < d/L < 1.0	d/L = 0.60

d is the water depth.

.

The diagram of the 6DoF vibration platform and the wave probe in the test is shown in Figure 4. The parameters of the 6DoF vibration platform and the wave probe are listed in

Table 2. Main parameters of WIN06-010A six-degree-of-freedom vibration platform.

Motion	Displacement (Angle)	Velocity	Acceleration	Position Accuracy	Repeat Positioning Accuracy
Rotary	±25°	±60°/s	120°/s2	≤0.05°	≤0.04°
Linear	±250 mm	500 mm/s	4900 mm/s2	≤0.05 mm	≤0.02 mm

Loadable weight 250 kg.

and

Table 3. The parameters of the wave probe.

Parameters	Value
Sampling frequency	100 Hz
Operating range	500 mm
Working temperature	0–40 °C

, respectively. Therefore, the experimental equipment met the test requirements and enabled us to obtain the required test data.

To better describe the experimental results, the time histories of the wave climbing height are presented in Figure 5. The time history for the 0° and 8° states (d/L = 0.34, A₀ = 0.002 m) is shown, and they are divided into resting, unstable, and stable stages. For the unstable and stable stages, H_max and H_stable are used to represent the characteristics in the two stages, respectively. The extreme peak H_max of the sloshing liquid in the unstable stage can have a significant impact on the tank, while in the stable stage, it exhibits a repeated stable waveform; therefore, it is more reasonable to use H_stable, as shown in Equation (3). Figure 6 depicts the time history of the liquid redistribution in the resting stage. When the tank is suddenly tilted to its side, the internal liquid is redistributed and eventually reaches a new state of rest in the midst of the fluctuations observed in the resting stage.

$$H_{\mathrm{stable}}=H_{\mathrm{stable}\_\mathrm{peak}}-H_{\mathrm{stable}\_\mathrm{valley}},$$

(3)

where H_{stable_peak} is the mean peak value in the stable stage, and H_{stable_valley} is the mean valley value in the stable stage.

The experimental arrangement is detailed in

Table 4. Experimental arrangements.

d/L [-]	ω/ω1 [-]	A0 [m]	θ [°]
0.34	1.00	0.0003	0–10 (interval 1)
		0.0005
		0.0020
	1.90	0.0003
		0.0005
		0.0020
0.60	1.00	0.0003
		0.0005
		0.0020
	1.79	0.0003
		0.0005
		0.0020

, where the liquid sloshing response of two different water-depth states of liquid tanks under 0–10° of lateral inclination subjected to first-order and third-order resonance frequencies and three distinct excitation amplitudes of external excitation are investigated, respectively. In addition, the flow chart illustrating the operation procedure in the test is presented in Figure 7.

3. Results and Discussion

3.1. Validation of Experimental Results

The validation of the accuracy of the test model is described in this section. The experimental results obtained with an excitation amplitude A₀ = 0.002 m were compared with the results of the potential flow theory for the water depth states of d/L = 0.34 and d/L = 0.60. The sloshing modes covered the first–and third–order resonant frequencies of the liquid tank.

The analytical solution of H_{stable_peak}/A₀ for a clean tank was calculated by Linton and McIver [49], as shown in Equations (4)–(8).

$${\displaystyle \frac{H_{\mathrm{stable}\_\mathrm{peak}}}{A_0}}={\displaystyle \frac{\omega }{g{A}_0}}\left|A_0\omega b+{\displaystyle \sum _{\mathrm{m}=0}^{\infty }\frac{2K}{{\mu }_{\mathrm{m}}b(K_{\mathrm{m}}-k)}}\right|$$

(4)

$$b=L/2$$

(5)

$${\mu }_{\mathrm{m}}=\left(\mathrm{m}+0.5\right)\pi /b$$

(6)

$$K_{\mathrm{m}}={\mu }_{\mathrm{m}}\tan \mathrm{h}\left(d{\mu }_{\mathrm{m}}\right)$$

(7)

$$K={\omega }^2/g$$

(8)

Figure 8 and Figure 9 illustrate the experimental results and theoretical predictions in the two liquid depth states. During the process of liquid sloshing, before the sloshing liquid reaches a stable state, there is a transient extreme value of the response amplitude H_max, which has the potential to cause damage to the structure of the storage vessel and affect the stability of the vessel. It is difficult for the potential flow theory to predict the amplitude of the extreme response before the stable stage, as can be seen in Figure 8 and Figure 9. Therefore, the complex nonlinear phenomena, which contain high energies, necessitate further investigation. Liquids in the resonant state can also exhibit significant liquid response, and the verified first-order frequency ω₁ and third-order frequency ω₃ in the stable state were utilized in the following tests, as shown in

Table 5. Experimentally obtained first-order (ω₁) and third-order (ω₃) resonant frequencies for different water depths.

d/L [-]	ω1 [rad/s]	ω3 [rad/s]
0.34	6.53	12.41 (1.90 ω1)
0.60	6.94	12.43 (1.79 ω1)

.

3.2. The Time Histories of Sloshing Liquid Under Horizontal and Tilted Conditions

Normally, storage tanks are filled to a high-level liquid depth to maximize the tank utilization. Meanwhile, a high liquid depth can enhance the robustness of the tank against potential extreme external loads. It also helps to reduce the shell vibration and the slamming of the shallow fluid. Considering the case of a partial filling liquid depth, many excitation conditions ae discussed in this section. Figure 10 illustrates that the water depth had little effect on the development trend of the wave climbing height at the sidewall of the liquid tank. The appearance of a tilt angle can cause a positive offset in the liquid elevation on the tilted side of the storage tank. Similarly, variations in the liquid depth had no significant impact on this non-dimensional value. The H_max and H_stable/A₀ for 0° and 8° tilted liquid tanks at first- and third-order frequencies in two liquid depths are presented in

Table 6. H_max and H_stable for 0° and 8° tilted liquid tanks at first- and third-order frequencies in two liquid depths.

d/L [-]	0.34				0.60
ω/ω1 [-]	1.00		1.90		1.00		1.79
α [°]	Hmax [m]	Hstable/A0 [-]	Hmax [m]	Hstable/A0 [-]	Hmax [m]	Hstable/A0 [-]	Hmax [m]	Hstable/A0 [-]
0	0.101	65.00	0.044	28.50	0.159	100.50	0.039	21.00
8	0.103	46.80	0.085	27.10	0.174	85.70	0.083	25.70

and Figure 11.

The initial inclination of the liquid tank has different impacts on sloshing under the first-order and third-order modes. Additionally, the maximum wave climbing height during the transient stage and the steady-state amplitude exhibit different responses across various modes. Under the same tilted angle, variations in the water depth resulted in an increase of approximately 57% in the maximum value of the first-order sloshing wave climbing height during the unstable stage, compared to the original value. Conversely, for the resonance in the third-order modes, the trend was opposite. A similar trend was observed in the stable stage, and the wave climbing height increased to 69% at the first-order resonance frequency.

The results indicate that the H_max increases with the inclination angle, a phenomenon particularly pronounced in the third-order mode. This may be attributed to the higher excitation energy in higher-order modes, where the additional forces induced by the inclination exacerbate the liquid surface motion response during the transient stage. However, for H_stable, the wave climbing height exhibits an opposite phenomenon for both depths. The H_stable at the first-order mode decreases, as the inclination increases, while it increases at the third-order mode. This may be explained by the fact that, in a settled liquid tank, the liquid climbs along the two tank walls in the first-order sloshing mode. Under this condition, the inclination reduces the climbing height on one side while increasing the height on the opposite side. The original symmetric resonant wave pattern is disrupted. For the third-order mode, the wave nodes of the free surface are located at 1/3 and 2/3 of the tank length. The inclination of the tank alters the direction of the external excitation forces, causing the oscillating free surface at the wave nodes to be pushed toward the lower side. This phenomenon results in an increase in both the maximum wave height and the steady-state wave height, and this characteristic becomes more pronounced with the increasing water depth.

3.3. The Effect of Excitation Amplitude on Sloshing Liquid Response

When considering the effect of excitation amplitude, two different tilt angles of 0° and 8° were also taken into account. In the horizontal state (0°), the response amplitude in the first-order frequency is larger than that in the third-order frequency. Furthermore, the effect of the excitation amplitude on the wave climbing height becomes negligible through dimensionless processing (Figure 12). When the liquid tank is tilted, due to settlement or other reasons, variations in the excitation amplitude alter the developmental processing of wave climbing height, modifying the motions of the sloshing liquid in the first- and third-order frequencies (Figure 13). In addition, this leads to the difference between the first- and third-order responses of the sloshing fluid at different tilt angles.

It should be noted that, at any angle, the excitation amplitude will have an effect on the time for the liquid to reach stable stage. The fastest time to reach a stable stage occurs at amplitude A₀ = 0.00050 m. In addition, it can be observed from Figure 13 that when the amplitude is very small (A₀ = 0.00030 m), the liquid surface grows slowly in the third-order resonance response of the 8° tilted liquid tank. The excitation time reaches 350 s before the rate of variation of the wave crest is below 0.001 m. Moreover, as the excitation amplitude increases, a clear phenomenon can be observed from Figure 12 and Figure 13 that the waveform of H_max in the unstable stage is steeper and sharper, which causes a more violent pressure impact on the sidewall.

An increasing excitation amplitude leads to a proportional increase in the H_max during the unstable stage (

Table 7. H_max and H_stable for 0° and 8° liquid tanks at first- and third-order frequencies in three excitation amplitudes.

α [°]	0				8
ω/ω1 [-]	1.00		1.90		1.00		1.90
A0 [m]	Hmax [m]	Hstable/A0 [-]	Hmax [m]	Hstable/A0 [-]	Hmax [m]	Hstable/A0 [-]	Hmax [m]	Hstable/A0 [-]
0.0003	0.0178	83.50	0.0040	23.20	0.0504	51.80	0.0482	48.10
0.0005	0.0228	67.70	0.0091	32.20	0.0553	42.00	0.0604	65.60
0.0020	0.1009	65.40	0.0444	28.60	0.1029	46.80	0.0852	27.10

and Figure 14). In the horizontal state, the response of the unstable stage in the first-order frequency increases to 1.28 and 5.66 times when the excitation amplitude is increased to 1.67 and 6.67 times, respectively. A similar trend is observed at the third-order frequency, with increases to 2.275 and 11.1 times, respectively. For the 8° tilted state, this tendency decreases, with 1.06 and 2.04 times, respectively, at first-order resonance and 1.25 and 1.77 times at third-order resonance. In addition, a more significant point is that the third-order resonance state corresponding to a medium excitation amplitude (A₀ = 0.0005 m) has a wider range of fluctuations in the stable stage. This implies that medium amplitude should also be considered in practical applications.

3.4. The Free Surface Elevation Affected by Different Tilted Angles

The tilt angle of the tank is also an important factor that can induce a more violent sloshing response within the tank. This section explores two main aspects: the unstable stage and the stable stage. In particular, the response frequency and response amplitude in the unstable and stable stages are investigated. A ratio of d/L = 0.34 was selected for this study, as previous findings have shown that the water depth has little influence on the development of the time history of the wave climbing heights. Figure 15 and Figure 16 present the FFT responses of the experimental results for the case of A₀ = 0.002 m, respectively. The maximum response amplitude for each condition is demonstrated in Figure 17 and Figure 18.

3.4.1. The Unstable Stage

In the unstable stage, for both the first- and third-order resonant states, the response frequency is related to the excitation frequency and consists of the excitation frequency and its multiples (Figure 15). This indicates that the excitation frequency of the dynamic load is the main factor influencing the response frequency of the liquid. As the excitation amplitude varies, the first-order response and the third-order response display different trends. The maximum response amplitude (A_max) of the first-order resonance decreases with the increase in the tilt angle in all excitation amplitudes. However, the maximum response amplitude of the third-order resonance shows a steady trend with the increasing tilt angle, as the excitation amplitude increases (Figure 17). This can be attributed to the position of the energy concentration of the resonant modes of the sloshing liquid. In the first-order resonant state, the energy of the liquid is concentrated at the sidewalls, so that part of the wave energy is used to resist the effect of the sideways tilt, as the angle of the sideways tilt increases. In contrast, for the third-order resonance, the liquid energy is not only concentrated at the sidewalls, there is also a concentration of energy in the liquid tank. Therefore, this observation may explain the different trends in the resulting amplitudes. Furthermore, it can be observed from Figure 17 that the maximum value of the response amplitude in the unstable stage appears in the first-order resonance response at the horizontal state. The third-order resonant amplitude is smaller than the first-order resonant amplitude in all states. This suggests that the effect of the tilt angle on the response amplitude in the unstable stage may be ignored in real engineering applications. Meanwhile, from Figure 17, it can be found that an increase in the excitation amplitude does not significantly and continuously increase the liquid climbing height.

3.4.2. The Stable Stage

In the stable stage, for both the first- and third-order resonant states, the response frequency is related to the excitation frequency and consists of the excitation frequency and its multiples (Figure 16). The trends of the response frequency and amplitude were consistent with those observed in the unstable stage (Figure 18). It should be noted that the amplitude of the third-order response in the stable stage was larger than that in the unstable stage. Moreover, at small excitation amplitudes, increasing the tilt angle gradually increased the amplitude of the third-order response and even exceeded the first-order response (Figure 18a,b). This suggests that in the stable stage, continuous liquid impacts necessitate considering not only first-order resonances but also third-order resonances, which are equally important. Additionally, it is demonstrated that this phenomenon may be related to the distribution of the energy concentration points within the liquid. As settling occurs, the energy is reduced at the sidewalls and transferred to the interior of the liquid. Unlike the maximum value, the steady-state third-order sloshing response is more significantly influenced by the inclined angle, and the amplitude of the third-order sloshing increases rapidly with the increase in the angle and exceeds that of the first-order response at the corresponding angle.

3.4.3. The Comparison Between the Unstable Stage and the Stable Stage

The comparative results between the stable and unstable stages for the first-order and third-order resonance states are displayed in Figure 19. In the first-order resonance condition, both the unstable and stable stages have the same trend: the response amplitude is decreasing as the tilted angle increases. This rule is the same as mentioned above. In the third-order resonant state, the difference between the stable and unstable stages decreases with an increase in the excitation amplitude. This is attributed to the response process of the third-order excitation, which is influenced by the excitation amplitude. The third-order resonance response increases with an increasing angle under all conditions, except for the large excitation amplitude case, which needs to be paid more attention in the engineering process.

The observed results can be attributed to the initial stage of excitation, during which a sloshing initiation process takes place in the liquid. During this process, the propagating modes of the fluid undergo gradual attenuation until they vanish. Consequently, the fluid response frequency spectrum transitions from a complex broadband spectrum to a combination of the external excitation frequency and the natural frequencies of the liquid tank. However, in this study, the inclination of the liquid tank significantly influenced the propagation and deformation of free-surface waves in the liquid tank under sway excitation. A strong nonlinear response emerged in the third-order resonance mode when subjected to the combined effects of horizontal and vertical accelerations. However, due to the differing sloshing fluid motion patterns across various modes, the effect of the gravitational component caused by inclination on the fluid motion in the first-order mode was not significant. As a result, the evolution patterns of the free surface response differed under the first- and third-order resonance modes.

For the first-order response shown in Figure 18a, the time series curve captures the intense sloshing stage observed during the initial phase, during which the liquid surface is excited under the propagating mode. During this stage, the liquid exhibits significant climbing along the sidewalls, resembling the behavior observed between 180 and 220 s in Figure 12. As the liquid sloshing transitions to a stable phase, the motion enters a resonant steady state. At this point, the liquid reciprocally climbs along the sidewalls with a stabilized sloshing amplitude, characteristic of the non-propagating mode. For the third-order response presented in Figure 12, the amplitude of the sloshing liquid increases gradually, and the liquid sloshing can be divided into several wave knots. Additionally, the inclination of the tank alters the direction of the external excitation forces, causing the oscillating free surface at the wave nodes to be pushed toward the lower side.

4. Conclusions

The settlement induced by earthquakes poses such an aggressive excitation of the liquid sloshing in storage tanks that the study of their physical design and safety precautions is both significant and essential. This work contributes to the design and precautionary measures for storage tanks in coastal areas subjected to multiple environmental excitations, thereby supporting sustainable industrialization. With the primary focus on the differential settlement, liquid sloshing in the inclined tanks under external excitation was investigated. The present work aimed to explore the mechanism of the hydrodynamics influenced by the inclination of the tank, with particular attention to the characteristics of the wave height and sloshing mode. Furthermore, the resonant sloshing wave may climb higher and slam stronger along the inclined sidewall, posing a threat to the structure’s strength, while regular sloshing affects the fatigue strength. The specific conclusions are summarized as follows:

(a) The tilt of the storage tank does not alter the resonance frequencies of the sloshing modes, but it does affect the liquid elevation along the tank walls. In the first-order mode, the wave height along the sidewalls decreases as the tilt angles increase, whereas in the third-order mode, it exhibits a notable increase. This observation demonstrates that tank inclination disrupts the symmetric sloshing pattern, thereby altering the force distribution and response characteristics of the liquid along the walls.

(b) As the tilt angle of the storage tank increases from 0° to 8°, the steady-state wave height at third-order resonance increases by approximately 69%. This significant amplification of the sloshing response in higher-order modes highlights the elevated risks of increased wall stresses and structural instability, particularly when the tank has a high liquid fill level. This phenomenon can be attributed to the changes in the force direction and the interaction between the liquid motion and the tank walls resulting from the inclination of the storage tank after settlement. These changes influence the propagation and deformation patterns of the free liquid surface under first- and third-order sloshing modes during forced vibration.

(c) The liquid elevation along the sidewalls of the tank was significantly influenced by the excitation amplitude. At third-order resonance, increasing the excitation amplitude from 0.0003 m to 0.0020 m resulted in a 10 times increase in the maximum wave height during the unstable stage. A similar trend was observed in the steady-state wave height, indicating that larger excitation amplitudes can considerably amplify the impact forces exerted on the tank walls, posing potential threats to structural safety.

The findings of this study show that the liquid response of multiple environmental loads acting on storage tanks is specific and warrants further attention. Further studies will focus on the combination of different loads and a more in-depth discussion on real 3D tanks.