Review of evaluation of tsunami-induced debris collision force

ABSTRACT

The evaluation of the impact of tsunami-borne debris collisions is important in designing tsunami-resistant structures and quantifying the risk for regional and industrial facilities located along the coast. Consequently, research on this topic is being actively conducted. Methods that evaluate debris behavior during tsunamis, including collision probability, velocity, and force, as well as the necessary parameters for these evaluations, are invaluable from a practical perspective and are being implemented through their incorporation into guidelines. Although research continues and evaluation techniques are maturing, identifying future challenges remains essential. This review article covers a wide range of technologies, both those already in use and those under development, and the physical processes that should be considered in these evaluations. It summarizes the applicability of evaluation methods based on accuracy, computational cost, and practicality, and clarifies the technologies currently available. The review identifies issues for future research, particularly the development of a probabilistic evaluation framework, which is highlighted as an area requiring further research. This review serves not only as a collection of practical findings but also as a basis for developing advanced evaluation techniques for tsunami-induced debris collisions, thereby contributing to the advancement of future structural design and the enhancement of local disaster preparedness against tsunamis.

KEYWORDS:

• Tsunami
• tsunami debris
• debris motion
• debris collision

1. Introduction

When a tsunami reaches the coast, it impacts both social and economic activities. For example, the 2011 Tohoku earthquake and tsunami (Mori et al., 2011) caused significant loss of urban functions due to inundation and structural damage from the tsunami impact (Takahashi et al., 2011). Between 1998 and 2017, tsunamis were responsible for approximately 251,770 casualties and US$ 280 billion in damages (UNISDR, 2018, Imamura et al., 2019). The destruction of houses and other infrastructure by the tsunami delayed recovery efforts in the affected areas (Takahashi et al., 2011). The tsunami caused by the Tohoku Pacific Offshore Earthquake also significantly damaged vital electric power facilities, which are an important social infrastructure (Shoji et al., 2012), leading to long-term power shortages in the East Japan region and significantly impacting society (Ministry of Economy, Trade and Industry (METI, 2011). The restoration of heavily damaged power plants took approximately two years to complete (Shiratori 2012, Yuyama and Kajitani, 2014).

Tsunami debris has had varied impacts in past disasters, including the Indian Ocean Tsunami on December 26 2004 (Lay et al., 2005, Kanamori, 2006), the Tohoku Earthquake on March 11 2011 (Mori et al., 2011), and the Indonesian tsunami on September 28 2018 (Stolle et al., 2020a). Various kinds of debris such as wood, vehicles, containers, ships, concrete pieces from collapsed buildings, and asphalt slabs, collided with structures, causing significant damage (EEFIT 2006, Rossetto et al. 2007, Takahashi et al., 2011, Suppasri et al., 2012, Mohammad et al., 2020, Stolle et al., 2020a). The generation and impact of debris from the collapse of structures have been experimentally investigated, shedding light on the mechanisms involved (Krauswald et al. 2022). In addition, the accumulation of large amounts of debris on land and in the sea complicates the post-disaster rescue and recovery efforts (Takahashi et al., 2011, Okal et al., 2006). A tsunami caused a large number of containers to arrive at the port, resulting in a loss of logistical functions (Takahashi et al., 2011). There is also a concern that the movement of debris at high densities may increase the risk of fire (Tomita and Chida, 2016, Imai et al., 2022). Furthermore, when structures cause the damming of debris objects, they not only affect the hydraulic field, but also increase the hydrodynamic forces acting on the structure (Stolle et al., 2017, 2018).

Predicting the damage caused by tsunami-borne debris and mitigating its impact are important engineering issues that have been the subject of extensive research. This review focuses on evaluating the collision force of tsunami-borne debris. The accumulation of such knowledge has influenced the improvement of design guidelines (Japan Society of Civil Engineers JSCE, 2016, JEA, 2021, FEMA 2008, FEMA, 2012, FEMA, 2019, ASCE 2016, ASCE, 2022) and led to the inclusion of tsunami load evaluations in design guidelines of ASCE (2016). These guidelines serve as practical references for evaluating debris collision forces. A periodic review of studies related to tsunami-induced debris is important because a lot of research is published each year. Efforts should be made to apply this evolving body of knowledge in practical settings.

Nistor et al. (2017) critically examined existing research on the motion of tsunami-borne debris, the evaluation of debris collision forces, and debris-tracking simulations, highlighting their importance in assessing the impacts of tsunami-borne debris. They identified a gap in techniques for evaluating the effects of multiple collisions against a target, such as a seawall, noting the tendency of tsunami debris to form agglomerations. They also pointed out the necessity of addressing the probabilistic nature of tsunami-borne debris issues and refining the scaling laws used in model experiments. Kaida and Kihara, (2017a) conducted a detailed review of debris-tracking simulations and collision force estimation methods, summarizing concepts for using debris-tracking simulations for various purposes and the applicability of collision force evaluation equations. The substitution parameters representing the physical properties of debris, essential for applying the evaluation equations for collision forces, have been compiled based on current knowledge. It was noted that numerous types of debris lack corresponding substitution parameters, highlighting the need for a more comprehensive set of parameters for these estimation equations. Furthermore, the development of a probabilistic evaluation method for debris collision requires further studies on the uncertainties related to various evaluation factors.

Previous studies (e.g. Haehnel and Daly 2004, Arikawa et al. 2003, Mizutani et al. 2004, 2005, Takabatake et al. 2015b, 2015a, Kaida et al. 2017b) and reviews (Nistor et al. 2017, Kaida and Kihara, 2017a) have both experimentally and theoretically established a relationship between debris collision force and debris collision speed, demonstrating that higher collision speeds result in greater forces. Therefore, accurately setting the collision speed is critical in the evaluation of collision forces for tsunami-driven debris. During tsunami propagation and run-up, various physical processes occur, such as wave-breaking along the coast and the creation of complex and turbulent flow fields around structures impacted by the on-rush of waves (Krauswald et al. 2022; Kaida and Kihara, 2016). These phenomena significantly influence the behavior of tsunami-driven debris, as well as the collision speed and probability. Accordingly, evaluations of debris collision speeds in coastal and tsunami run-up areas should incorporate these physical processes. The collision force of tsunami-driven debris is also determined by the physics of its interaction with structures. Important factors in this evaluation include the physical properties of both the debris and the impacted structures, the debris collision angle, and added-mass effects (Haehnel and Daly 2004, Ikeno et al. 2016). Experimental studies indicate that the impact force increases with the number of debris objects impacting simultaneously (Stolle et al., 2020b), suggesting the need for evaluation methods that appropriately reflect the physics of tsunami-driven debris motion and collision forces. Since the publication of reviews by Nistor et al. (2017) and Kaida and Kihara (2016), significant research has been conducted that could enhance the assessment of debris collision forces. However, some of these findings, particularly those published in Japanese, may not be widely recognized internationally. A comprehensive review of the latest research on the physics of tsunami-driven debris motion and collision, encompassing both English-language international journals and Japanese literature, would greatly benefit engineering applications.

This paper provides an overview of the existing literature on the physical processes essential for evaluating debris collision and the methods for assessing debris collision force and collision speed that incorporate these physical processes. The aim is to promote the further advancement of evaluation methods for tsunami debris collision forces, contributing to more accurate assessments in practice through the establishment of a standardized method. This review assesses the suitability of various evaluation methods in terms of accuracy, computational cost, and practicality, while also highlighting the technologies that are currently available. The review results also help in identification of issues to be studied in the future.

The structure of this paper is as follows: Section 2 offers a detailed review of research related to the evaluation of collision forces. It begins by outlining the essential physics that must be considered when determining collision speed – a key factor in the evaluation of collision forces – and in calculating these forces. It also explores the methods related to these aspects. Section 3 provides a summary of the evaluations presented in existing guidelines and discusses the issues encountered in practical evaluations. Section 4 identifies and discusses the issues that must be overcome to improve the methods for evaluating forces resulting from collisions with tsunami-driven debris. The paper concludes with Section 5, which summarizes the findings of this study.

2. Literature review

2.1. Evaluation of debris collision speed

2.1.1. Movement process of tsunami-borne debris

Tsunami propagation from offshore to coastal areas involves floating debris to drift with the induced currents and waves, and in some cases, causing damage, as described in Section 1. During this process, the behavior of tsunami-borne debris is affected by the dynamics of tsunami propagation and inundation, including wave-breaking and the formation of tsunami bores, which differs significantly from behavior in steady currents (Yao et al. 2014, Goseberg et al. 2016, Nistor et al. 2016, Oda et al. 2019, Oda et al. 2020, Ohashi et al. 2021, Kaida et al. 2023). The acceleration of debris motion at the early stages and the overall behavior of debris objects are notably affected by tsunami-induced bores and breaking waves (Yao et al. 2014; Goseberg et al. 2016; Nistor et al., 2016; Oda et al. 2020; Ohashi et al. 2021; Kaida et al. 2023), as further explained in the referenced studies. In addition, fluid turbulence and the initial direction and placement of the debris contribute to variations in its behavior (Suga et al. 2012, Tomita et al. 2020, Kimoto and Tomita, 2021). The friction between the debris and the bottom floor effects the initial phase of debris motion and how far it travels in the flow direction (Stolle et al. 2019, Kihara and Kaida, 2020). Debris that is not fully sealed and contains air pockets, such as cars and containers, can become submerged during tsunamis (Kumagai et al. 2008, Takeuchi et al. 2013). Studies indicate that containers can take 15–24 h to submerge (Kumagai et al. 2008), and vehicles, 5–20 min (Takeuchi et al. 2013). Nojima et al. (2014) showed that considering whether debris becomes submerged significantly impacts the evaluation of collision speed and probability for debris with voids. The specific gravity of debris also plays a significant role in its behavior during tsunamis (Kaida and Kihara, 2016). The initial response of debris and its mechanisms of movement vary substantially based on its buoyancy (Stolle et al., 2020a, Krautwald et al. 2021). As debris nears a structure, the complex flow field generated around the structure affects its behavior (Kaida and Kihara, 2016). The arrival of a tsunami at a structure creates a turbulent flow field with reflected waves around it (Kaida and Kihara, 2016, Krauswald et al. 2022), potentially reducing the collision speed and collision probability debris subsequently reaching the structure (Matsutomi, 1999, Ikeda and Arikawa, 2014, Kaida and Kihara, 2016, Sano et al. 2018, Murase et al. 2019). Experimental studies have shown that the interactions among numerous debris objects can affect their behavior (Stolle et al. 2016, Tomita et al. 2020), as can interactions between debris and the flow field (Rueben et al. 2014). Moreover, there is considerable uncertainty regarding the collision angle of debris, which effects to the collision forces (Kimoto and Tomita, 2021, Morita et al. 2021). Accurately analyzing debris behavior or evaluating its collision speed and probability necessitates careful consideration of these physical processes. When applying the evaluation methods described in the following section, it is critical to ensure these processes are adequately accounted for to maintain evaluation accuracy.

2.1.2. Evaluation method for debris collision speed

This section offers a comprehensive review of both simple and technically advanced methods for evaluating debris collision speed, supplementing existing reviews with the latest insights (Nistor et al. 2017;Kaida and Kihara, 2017a).

The simplest approach to determining collision speed involves setting it to the maximum flow velocity around the structure under evaluation or at its location. Despite its relative inaccuracy, this method is widely used in numerous guidelines due to its practicality (e.g. ASCE 2022; Ministry of Land, 2013). Experiments on debris behavior during tsunami run-up have shown that the peak speed of debris movement is often comparable to the maximum measured flow velocity (Kaida and Kihara, 2016). Moreover, when debris is affected by wave-breaking, especially if located near the wave-breaking zone, its maximum speed can exceed the maximum flow velocity, approximating the wave speed (Kaida et al. 2023). These findings indicate the necessity of accurately determining collision speed. However, such simplified methods do not account for modifications to the flow field by structures (Kaida and Kihara, 2016, Derschum et al. 2018, Stolle et al., 2018), water depth, flow velocity, etc., affecting debris behavior. The drawback of some methods is their inability to physically solve for debris trajectories, leading to collision speed evaluations that are typically conservative but may lack accuracy. Therefore, more sophisticated methods are necessary for obtaining realistic results. However, a significant practical advantage of these simpler approaches is their minimal computational cost compared to more complex methods. Therefore, it is widely used because of its practicality (See Section 3).

For more accurate assessments of collision speed, debris-tracking simulations are a valuable tool. The development of such simulations began in 1980, leading to a variety of modeling methods. These debris-tracking simulations cater to the physical processes discussed earlier, offering different levels of computational complexity and accuracy. Depending on the specific scenario, these methods can accurately depict debris trajectories and collision speeds.

T. Goto, Sasaki, and Shuto (1982) proposed a method to model debris as a mass point, drawing on hydraulic experiment outcomes. The equations of motion for debris are derived by calculating the hydrodynamic forces acting on it, using the Morison equation (e.g. Fujii et al., 2005a; Heo et al. 2015; Kihara and Kaida, 2020). This approach describes the behavior of tsunami-borne debris by considering factors such as inertial forces, pressure gradients, added-mass, drag, and diffusion, with stochastic elements accounted for only in diffusion. The Morison equation has been widely employed to construct the equations of motion for debris, enabling the consideration of inertial forces and facilitating the evaluation of collision speed. This modeling method accounts for forces other than rotation acting on the debris and is less computationally intensive than other methods discussed below.

Fujii, Oomori, Ikeya, Asakura, Iriya, et al. (2005a) suggested a method using the discrete element method (DEM), which simplifies the determination of contact between tsunami-borne debris, structures, and other debris objects. External forces exerted on the debris are calculated using the Morison equation and applied to the elements that make up the debris. Debris behavior is then calculated by solving the equations of motion for a system with six degrees of freedom, covering both translation and rotation. The accuracy of this method was verified through comparisons with hydraulic experiment results, which examined debris motion in complex flow fields within ports and harbors (Fujii, Oomori, Ikeya, Asakura, Takeda, et al. 2005b). This method successfully replicated the characteristic floating behavior of vessels in a harbor, including rotation due to eddies. A key advantage of DEM-based debris-tracking simulations is that debris is represented by interconnected elements, allowing for precise application of fluid forces to each element and accurate calculation of rotational motions. Furthermore, the debris shape can be more accurately modeled by increasing the number of interconnected elements, though this also increases computational demands.

Honda et al. (2009) proposed a model that treats tsunami-borne debris as a rectangular rigid body, calculating translational and rotational motions around the vertical axis using a moving coordinate system centered at the center of gravity of debris. This approach utilizes the method of Ikeya et al. (2006), which extends the Morison equation to include drag and inertial forces for evaluating hydrodynamic forces acting on the debris. This model was incorporated into the software T-STOC (Tsunami Storm surge and Tsunami simulator in Oceans and Coastal areas), developed by Tomita et al. (2016), with the source code made publicly available in 2016 (Tomita et al. 2016). The concept of modeling debris as rigid bodies has been the focus of active research (e.g. Heo et al. 2015).

These analytical models have been extensively used to investigate the behavior of tsunami-driven debris. The movement of debris in agglomerations is affected by interactions between the debris and the flow field (Rueben et al. 2014), and debris behavior is also affected by collisions among debris objects (Stolle et al. 2016, Tomita et al. 2020). Tajima et al. (2016) developed a model incorporating the interaction between numerous wood chips and the fluid, enhancing the nonlinear dispersive wave theory model with the reaction of fluid forces due to debris influence. Their simulations aimed to reproduce hydraulic experiments, with results showing a significant improvement in reproducibility when these interactions were accounted for. This interaction also modified the water-level fluctuation characteristics within the flow field. Many models incorporate the effects of debris – debris and debris – flow field interactions, which can significantly modify debris behavior (Nojima et al. 2014, Tomita and Chida, 2016, Shigihara et al. 2016, Kihara and Kaida, 2020).

Simplified methods for calculating the motion of tsunami-driven vessels with minimal computational effort have been proposed. Kobayashi et al. (2005) proposed a method using a hull translation coordinate system centered on the vessel’s center of gravity to calculate the equations of motion. Hashimoto et al. (2009) suggested dividing the hull into several elements to calculate the external forces on each, enabling the calculation of moments around the hull. This approach is particularly useful when the vessel’s size exceeds the horizontal resolution of the tsunami flow field calculation, allowing for the inclusion of the complex flow field’s effects on vessel behavior.

Various physical characteristics of the debris drifting process can be incorporated into the analysis of debris behavior. Previous study that considered debris submersion effects showed how submergence influences debris behavior. Nojima et al. (2014) reported that accounting for the submergence of debris, given the change in mass, allows for a more accurate analysis, including submergence effects. In simulations tracking containers in a port, the area affected by the tsunami was significantly smaller when container submersion was considered (Nojima et al. 2014), highlighting the importance of considering submergence in evaluating collision speed and probability for debris with voids, such as vehicles and containers. The behavior of debris is also influenced by turbulence in the fluid and the initial direction and placement of debris. Turbulence effects have been modeled as diffusion effects in debris-tracking simulations (T. Goto, Sasaki, and Shuto 1982, Kihara and Kaida, 2020), while the initial direction and location of debris are set as initial conditions for behavior calculations. Kihara and Kaida (2019) proposed a turbulent bore model for reflected waves formed in front of structures, enabling the determination of the vertical structure of reflected waves from two-dimensional flow field simulations generated by tsunamis. Incorporating this turbulent bore model into debris-tracking simulations, Kihara and Kaida (2020) achieved more realistic evaluations of debris collision speed and probability when considering the effect of reflected waves in front of structures.

Enhancing the accuracy of debris-tracking simulations crucially depends on the accuracy reproduction of the flow field (Takabatake et al. 2021). Takabatake et al. (2021) benchmarked hydraulic experiments on debris initially located on land (Stolle et al. 2016) and compared the computational results from four debris-tracking models (Heo et al. 2015, Kihara and Kaida, 2019, Nojima et al. 2017, Tomita et al. 2016). The benchmark test showed that the trajectory and velocity of debris obtained from the numerical debris-tracking simulation depend on the model’s coefficient settings, such as drag and mass coefficients. Additionally, it was found that accurate calculation of debris behavior is contingent upon the correct simulation of the flow field. When using the debris tracking simulation reviewed so far in this paper, the water level and flow velocity calculated by the two-dimensional shallow water equation model are used in the calculation of external forces acting on debris objects. Therefore, the accuracy of the reproduction of the vertical motion of debris is not physically correct. To accurately reflect the complex three-dimensional flow field around a structure during debris behavior evaluation, employing a method that uses flow fields obtained from three-dimensional hydraulic calculations is desirable. Yoneyama and Nagashima (2009) proposed calculating debris behavior using a six-degree-of-freedom system within a flow field from three-dimensional calculations. The particle method, a fully Lagrangian approach, offers accurate determination of debris behavior (Amicarelli et al. 2015, H. Goto et al. 2009, Gomez-Gesteira et al. 2012). The Lagrangian nature of the smoothed particle hydrodynamics method makes it particularly suitable for scenarios involving large deformations and distorted free surfaces, offering advantages in modeling highly nonlinear phenomena such as breaking waves (Colagrosshi and Landrini 2003, Monaghan et al. 2003). Despite its benefits, the high computational cost poses a challenge for practical uses, leading to proposals for coupling with planar two-dimensional calculations and other methods to solve this problem (H. Goto et al. 2009). The release of the open-source software DualSPHysics (Crespo et al. 2015) promises broader application possibilities, although the substantial computational resources required for these 3D numerical analyzes remain a significant hurdle for practical application.

The drag and inertial forces, crucial components of the Morison equation, are commonly included when deriving the equations of motion for debris. The Morison equation with drag and inertia forces is generally expressed as follows (Kihara and Kaida, 2020):

(1) $\mathit{F}=\frac{1}{2}{\rho }_w{C}_D{\displaystyle \int \left(\mathit{U}-\frac{d{x}_p}{\mathit{dt}}\right)}\left|\mathit{U}-\frac{d{x}_p}{\mathit{dt}}\right|\mathit{dA}+{\rho }_w\mathit{V}{C}_M\left(\frac{\mathit{DU}}{\mathit{Dt}}-\frac{d^2{x}_p}{\mathit{d}{t}^2}\right)$(1)

where x_p, C_D, C_M, A, V, ρ_w, and U represent the center position, drag coefficient, mass (or inertia) coefficient, projected area of the submerged part in the flow direction (m²), volume of the submerged debris fraction (m³), density of water (kg/m³), and velocity vectors (m/s), respectively. The chosen values for the inertia and drag coefficients are influenced by the flow conditions and the shape and draft of the debris. Fujii et al. (2005b) calculated C_D and the coefficient of inertia to be 0.74 and 0.41, respectively, for scenarios where wave direction aligns with the vessel’s bow, based on ship model hydraulic experiments. These coefficients produced hydrodynamic forces on the debris that aligned with experimental results. C_D and C_M are dependent on the shape of the debris and its direction relative to the flow, with a rectangular shape often assumed in studies presented below. In a benchmark study, C_D was set between 0.75 to 3.0 and C_M between 1.75 to 3.0 (Takabatake et al. 2021). Kihara and Kaida (2020), referencingYeh (2007), assigned a C_D of 3.0 for the initial seconds post-tsunami front collision with debris. Ikesue et al. (2017) suggested a model for C_M that varies based on the width and draft of the debris, with the projected aspect ratio in the flow direction also influencing C_M. These coefficients significantly impact debris behavior (Nojima et al. 2017, Takabatake et al. 2021). Moreover, Froude-Krylov forces play a role in the movement mechanism of debris at the tsunami front (Ikesue et al. 2017, Kaida et al. 2023). In the case of a tsunami breaking wave, floating debris in the vicinity of the wave breaking zone is temporarily trapped at the tsunami tip and transported toward the shore. One of the transport mechanisms is the effect of the Froude-Krylov force. Therefore, in scenarios requiring consideration of breaking wave and tsunami bore effects on debris behavior, Froude-Krylov forces should also be accurately incorporated into the model.

2.1.3. Applicability of evaluation method for debris collision speed

The characteristics of various evaluation methods for debris collision speeds are concisely captured in Table 1. A method that does not analyze debris trajectories (Naito et al. 2014), is limited to evaluating the spread area of debris. This approach tended to overestimate the dispersion area of debris compared to results from deterministic debris-tracking simulations (Koh et al. 2024, Nistor et al. 2017). The evaluation method of diffusion area proposed by Naito et al. (2014) derives from post-tsunami field survey data on the diffusion of shipping containers, influenced by factors such as inundation characteristics and the presence of buildings, potentially leading to the differences observed between experimental and numerical analysis outcomes. Given the need for further accuracy verification, this method is rated as “△” in the table. Such methods may help identify structures at risk from debris collisions, but since they do not calculate collision speed, they are marked as “×” for this parameter in the table. If these methods determine that it is necessary to set the debris collision speed in the area where debris collision possibly occurs, the maximum flow velocity around the impacted object may be used as the debris collision speed.

Table 1. Evaluation methods for debris spreading area, trajectory, and collision speed. ○ : The method is physically solved by each method, and a certain level of accuracy can be expected; △: the resulting output is obtained when each method is used; however, the accuracy may not be considered sufficient; and ×: the accuracy is not expected to be sufficient for practical use.

	Evaluation accuracy			Required resource
	Spreading area of the debris	Debris trajectory	Debris collision speed
Method using inundation depth information without debris-tracking simulation Naito et al. (2014)	△	×	×	Very low
Methods that model tsunami debris as concentrations and passive tracer ASCE (2022); Kobayashi et al. (2007); Kawamura et al. (2015)	〇	△	×	Low
Methods to model debris as mass points or rigid bodies T. Goto, Sasaki, and Shuto (1982); Fujii et al. (2005); Honda et al. (2009); Tomita and Chida, (2016) etc.	〇	〇	〇	Normal
Full-3D model Yoneyama and Nagashima (2009); Goto et al. (2009); Gomez-Gesteira et al. (2012); Amicarelli et al. (2015)	〇	〇	〇	High

Additionally, methods that require less computational effort have been developed, including those based on the massless tracer model (ASCE 2022 and the method of Naito et al. (2014). A simplified evaluation can be conducted using site-specific tsunami inundation analyzes, as conducted by the Federal Emergency Management Agency (FEMA 2012) and the American Society of Civil Engineers (ASCE 2022). These methods, not accounting for inertial forces, are not suited for evaluating debris collision speed but may offer a simple way to estimate the spreading area of debris. The primary advantage of this method is its potential for a broad estimation of debris behavior with minimal computational demand, a prospect that warrants future validation. In this case, the maximum flow velocity around the impacted object may also be used as the collision speed of the tsunami-induced debris object.

The above methods are effective when the debris collision speed that is necessary to determine the collision force of the debris needs to be set on the simple and safe side. As a more realistic way to set the impact velocity, modeling the tsunami-borne debris as a mass or rigid body and analyzing its behavior is a practical and well-balanced approach. It accounts for the inertial forces of debris objects and accurately resolves their trajectories, allowing for a comprehensive evaluation across all criteria listed in Table 1. As a result, this method received a “○” rating for all items, with the computational resources required for evaluation deemed moderate. Methods capable of representing effects such as turbulence in the flow, complex flow fields around structures, and the submersion of debris objects enhance the realism of debris behavior predictions. Integrating these advanced methods into debris-tracking simulations, depending on the specific evaluation context, will yield more realistic results reflective of the unique characteristics of each situation. It should be noted that the vertical motion of debris cannot be properly taken into account unless the water level and velocity in the flow field calculated using a two-dimensional shallow water equation model are used to calculate the external forces acting on the debris.

If a more detailed assessment of collision forces is required, it is important to set collision velocities with high accuracy. The use of 3D debris tracking simulations (Amicarelli et al. 2015, Gomez-Gesteira et al. 2012, H. Goto et al. 2009, Yoneyama and Nagashima, 2009), together with water levels and velocities of the flow field calculated in 3D, is expected to provide accurate analyzes of debris behavior, meriting a “○” rating for all criteria in Table 1. However, the practical application of such detailed models is limited by their high computational demands.

There exists a tradeoff between the desired accuracy of debris-tracking simulations and the computational resources available. In practical applications, it is important to select a method that best matches the specific model characteristics and to find a balance between computational expenditure and expected evaluation accuracy.

2.2. Evaluation of debris collision force

2.2.1. Dynamic process of debris collision

Several methods are available for evaluating the collision force of debris, including collision experiments, collision analysis, and collision force estimation equations. The most practical among these uses collision force estimation equations, which are widely referenced in numerous guidelines and standards. For these equations, it is essential to accurately account for the dynamic processes involved in debris collisions with structures. This section discusses these requirements and describes the estimation equations based on the presented theories. Although debris collision is inherently a three-dimensional phenomenon, for simplicity, this paper focuses on collision phenomena within a one-dimensional vector field.

For a debris collision, the impulse generated by the collision is considered equivalent to the change in momentum of the debris before and after the collision. This relationship can be expressed as follows:

(2) $\begin{array}{c}\mathit{m}\left(\mathit{u}{ }^{\mathrm{\prime }}-\mathit{u}\right)={\int }_0^{\mathrm{\Delta }t}\mathit{Fdt}\end{array}$(2)

where m, u, u’, Δt, and F represent the mass of the debris (kg), collision speed of the debris (m/s), rebound speed of the debris (m/s), impact duration (s), and collision force (N), respectively. Δt is defined as the time from when the collision force begins to increase to when it returns to its original level. Building on this theory, Haehnel and Daly (2004) proposed the following equation to evaluate the maximum impact force, assuming the collision’s time history is sinusoidal:

(3) $\begin{array}{c}{F}_{\mathit{max}}=\frac{\pi }{2}\frac{\mathit{um}}{\mathrm{\Delta }\mathit{t}/2}\end{array}$(3)

The work exerted on the structure by debris collisions and the change in the kinetic energy of the debris until its collision speed reduces to zero are considered equal. The work W corresponding to the change in momentum, is expressed as follows:

(4) $\begin{array}{c}\mathit{W}=\int \mathit{F}\left(\mathit{x}\right)\mathit{dx}=\int \mathit{d}\left(\frac{1}{2}\mathit{m}{u}^2\right)\end{array}$(4)

Let S (m) be the distance from the point where the debris contacts an object to where it comes to a complete stop. Representing elastic deformation with the spring coefficient k (N/m), the following equation is derived as follows:

(5) ${\displaystyle {\int }_0^S\mathit{k}}\mathit{xdx}=\frac{1}{2}\mathit{m}{u}^2$(5)

From the above equation, kS² = mu². Using F = kS, the collision force can be calculated as follows:

(6) $\begin{array}{c}\mathit{F}=\frac{m{u}^2}{\mathit{S}}\end{array}$(6)

Generally, S decreases as the stiffness of the object collided with increases. Marine Bridge Research Committee (1978) provided an equation reflecting this principle.

As debris collides with structures, both the debris and the structures undergo deformation. Haehnel and Daly (2004) conceptualized the debris collision process using springs and mass points, deriving an estimation equation to account for debris deformation during collision. This equation is based on the equation of motion for a mass point in a one-degree-of-freedom (1-DOF) system:

(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7)

where C is the added-mass coefficient, m_f is the mass of the fluid displaced by debris objects (kg), and $\hat{k}$ is effective contact stiffness of the collision. For collisions involving rigid objects, $\hat{k}$ can be replaced by the rigidity k of the debris. In cases where the collided object is not rigid, the 2-DOF system model proposed by Stolle et al. (2019) is applicable.

Drawing on Hertz’s theory of contact deformation between elastic bodies (Hertz, 1881) and the results of collision experiments between concrete tetrapods and steel plates, Arikawa et al. (2003) proposed an evaluation for collision forces as follows:

(8) $\begin{array}{c}\mathit{F}={\gamma }_p{\chi }^{\frac{2}{5}}{\left(\frac{5}{4}\mathit{m}\right)}^{\frac{3}{5}}{u}^{\frac{6}{5}}\end{array}$(8)

(9) $\begin{array}{c}\mathit{\chi }=\frac{4\sqrt{\mathit{a}}}{3}\frac{E}{1-{\nu }^2}\end{array}$(9)

where γ_p, a, E, and ν represent the experimental coefficient, radius of a sphere (m), modulus of longitudinal elasticity (N/m²), and Poisson’s ratio, respectively.

The dynamic processes involved in debris collisions can be described by several theories, leading to the proposition of various theory-based collision force estimation formulas. Given that actual collision phenomena are more complex than theoretical models might suggest, these equations might not accurately represent the physics of collisions. When considering debris consists of multiple components, as described below, it is essential to select parameters that accurately represent the physical properties of the debris in each equation. Parameters such as stiffness and collision duration can significantly vary depending on the debris type, as shown by experimental studies (Kaida and Kihara, 2017b). These variations should be carefully considered when estimating collision forces in practical applications.

Takabatake et al. (2015b) explored the collision forces during full-scale vehicle collisions in both air and water, seeking to refine methods for evaluating these forces. They found that in car crashes, the vehicle’s members and crash box initially absorb axial compressive loads. As these components buckle axially, other parts such as the rear engine and front-side extension take over the load-bearing role. This shift highlights the need to account for changing physical properties of components throughout the collision process. The maximum collision speed in the experiments of Takabatake et al. (2015b) of approximately 5 m/s, with similar characteristics observed in NHTSA experiments for impacts approximately 15 m/s.

Toyoda et al. (2022) and Kuriyama et al. (2024) conducted detailed analyzes of full-scale fiber-reinforced plastic (FRP) vessel collisions, initiated by a free fall from the bow. The collision force time histories showed several large peaks, each associated with impacts on sturdy members such as bulkheads. This indicates, as with vehicles, that the primary load-resisting member shifts as damage progresses. Accordingly, for complex structures composed of multiple components, employing a method that accommodates the progress of damage and consequent changes in component properties is recommended. This approach is also applicable to evaluating collision forces for containers (Madurapperuma and Wijeyewickrema, 2013, P. Aghl, Naito, and Riggs 2014, P. P. Aghl, Naito, and Riggs 2015). For single-component structures such as wood logs, changes in stiffness during collision are less of a concern, though effects of plasticization should be considered (P. P. Aghl, Naito, and Riggs 2015).

The collision force for debris increases with collision speed within the elastic deformation range. Should the debris undergo plastic deformation, the collision force peaks at the speed inducing such deformation. Similar to vehicles and ships, if one component undergoes plastic deformation and subsequent components absorb further deformation, the collision force continues to increase. For wood logs, the peak collision force corresponds to the speed at which plastic deformation begins (P. Aghl, Naito, and Riggs 2014). The Federal Highway Administration (FHMA 2007) proposed an elastoplastic model for wood, incorporating variables such as moisture content and mechanical strengths, implemented via the crash analysis code, LS-DYNA. Therefore, understanding the plasticization of debris is key for accurate collision force estimation.

Experimental studies have shed light on how the collision angle and added-mass of debris influence collision force. Specifically, for wood logs, Haehnel and Daly (2004) and Ikeno et al. (2016) experimentally showed that the collision angle significantly affects collision force. In their findings, collisions oriented along the long axis of the wood, serving as the collision axis, resulted in the highest load, whereas collisions along the wood’s short axis halved the maximum load value. Ikeno et al. (2016) proposed that this collision angle effect arises from the conversion of the debris kinetic energy into rotational energy before the collision. The effect of collision angle on the collision force for debris involving complex structures, such as vehicles and ships, cannot be evaluated in the same manner as for wood logs. This complexity arises because, as previously noted, subsequent structural members continue to resist the collision force even as one member experiences plastic deformation. The deformation process is considered to vary with the collision’s direction, necessitating the acquisition of specific parameters such as axial stiffness and collision duration relative to the collision direction. These parameters are essential for considering the effect of the collision angle. Additionally, when debris is propelled by a tsunami, the water surrounding the debris also moves, influencing the collision force for a given speed, depending on whether the collision occurs in air or water. Ko et al. (2014) found that at a given collision speed, the force of a collision in water can be up to 17% greater than in air, highlighting the need for an accurate hydrodynamic mass coefficient to obtain the collision force of waterborne debris accurately.

2.2.2. Evaluation equations for debris collision force

In the previous section, a theoretical study on the dynamic processes occurring during debris collisions is represented, along with theoretical collision force estimation equations derived from these studies. This section introduces additional evaluation equations for debris collision forces, considerations for applying these equations, and the necessary parameters for each equation’s use.

Mizutani et al. (2004) and Haehnel and Daly (2004) explored the relationship between the change in momentum of debris before and after collision and the resulting impulse. In their experiment, a container was propelled along an apron by tsunami-induced floodwaters, leading to the proposal of the following equation:

(10) $\mathit{F}=2{\rho }_w{\eta }_m{B}_C{u}^2+\left(\frac{\mathit{Wu}}{\mathit{g\Delta t}/2}\right)$(10)

where η_m, W, B_c, and g are the maximum water-level (m), weight of the container (N), width of the container (m), and gravitational acceleration (m/s²), respectively. Mizutani et al. (2005) introduced an additional term to account for the added-mass effect on the right-hand side of the container equation.

When estimating debris collision force using equations from Haehnel and Daly (2004) or Mizutani et al. (2004), setting the correct collision duration time, Δt, is important. Δt can be readily determined and incorporated into these equations, with values available from existing collision experiments and numerical analyzes. Values for Δt across various debris types are summarized in Table 2, an update from Kaida and Kihara, (2017a). This table also includes input parameter values from literature not previously mentioned (Sogabe et al. 1981).

Table 2. Collision duration for debris (u: collision speed of debris (m/s)).

Type of debris		Impact duration (s)	References
Wood logs	Pine (L = 2 m, m = 179 kg, D = 0.42 m)	0.007～0.025 (airborne, |u| = 0.2～1.4) 0.007～0.018 (waterborne, |u | = 2.0～5.2)	Takabatake et al. (2015b)
	Cedar (L = 2 m, m = 156 kg, D = 0.38 m)	0.007～0.025 (airborne, |u | = 0.2～1.4) 0.007～0.018 (waterborne, |u | = 2.0～5.2)
	Cedar (L = 1 m, m = 72 kg, D = 0.38 m)	0.007～0.025 (airborne, |u| = 0.2～1.4) 0.007～0.018 (waterborne, |u| = 2.0～5.2)
	Pine (L = 6.0 m, m = 204 kg, D = 0.24 m)	0.004～0.008 (airborne, |u| = 0.25～3.0) Decreases linearly with increasing speed	Aghl et al. (2015)
	Lawang (L = 0.4～1.6 m, m = 0.3～86 kg, D = 0.2～0.3 m)	About 0.01 (airborne) (|u| = 0.5～2.5 m/s)	Matsutomi (1999)
	Pine (L = 8.05 m, m = 1,030 kg)	0.007～0.01(airborne): Decreases linearly with increasing speed 0.01s(waterborne): Independent of collision speed	Sogabe et al. (1981) (|u| = 0.5～4 m/s)
	Lawang (L = 5.97 m, m = 2,020 kg)	0.005～0.01 (airborne): Decreases linearly with increasing speed 0.009s (waterborne): Independent of collision speed
Containers	20-ft standard shipping container oriented longitudinally (m = 2,200 kg)	0.010～0.017 (airborne, |u| = 0.5～2.5 m/s) 0.010～0.014 (waterborne, |u| = 0.25～2.0 m/s)	Aghl et al. (2015)
	20-ft standard shipping container (m = 2,200 kg)	If target structure is prism; 0.0127 + 0.0017|u| If target structure is cylinder; 0.0164 + 0.0031|u|	Madurapperuma and Wijeyewickrema (2013)
	40-ft standard shipping container (m = 3,800 kg)	If target structure is prism; 0.0207 + 0.0025|u| If target structure is cylinder; 0.0241 + 0.0033|u|
Vehicle	L = 3.295 m, H = 1.475 m, W = 1.395 m, m = 320 kg (Mass with glass, engine, etc. removed)	0.2 ± 0.05 (airborne, |u| < 0.1 m/s) About 0.05 (waterborne)	Takabatake et al. (2015a)
	Sedan type vehicle	Mean 0.07	Results analyzed in this study based on Kaida and Kihara (2017)
	SUV type vehicle	Mean 0.06
	Minivan type vehicle	Mean 0.05
	Pickup track	Mean 0.07
Small fishing boat	m = 2,760 kg, L = 8.1 m	0.125 (Collision speed is approximately 10 m/s.)	Toyoda et al. (2022)

In a wood-log collision experiment by Takabatake et al. (2015b), low collision speeds in air resulted in a wide range of collision durations, with Δt ranging from 0.007 to 0.025 s. As collision speed increased to approximately 1 m/s, the variation in collision duration decreased, stabilizing at approximately 0.012 s. In waterborne collision experiments, collision duration varied from 0.007 to 0.018 s, showing no significant change or correlation with debris collision speed. Vehicle collision durations in air, reported by Takabatake et al (2015b, 2015a). were longer than those summarized by Kaida et al. (2018) from NHTSA experimental results, attributed to the nature of airborne collision experiments at low speeds by Takabatake et al. (2015b), where the collision force was primarily absorbed by the vehicle’s less rigid front part.

Toyoda et al. (2022) conducted air-drop experiments to determine the collision duration for small fishing vessels, with a collision speed of approximately 10 m/s. The collision duration ranged from 0.125 to 0.165 s, longer than that observed for other debris types, indicating the need for more data on this debris category.

In applying Eq. (6)(6) $\begin{array}{c}\mathit{F}=\frac{m{u}^2}{\mathit{S}}\end{array}$(6) , the focus is on the work exerted on the structure by debris collision and the change in kinetic energy of the debris during the collision process. Here, the distance S (m), from the point where the debris contacts the structure to where it stops completely, is crucial for estimating the impact force. However, accurately determining S is challenging due to limited practical knowledge about its value, making Eq. (6)(6) $\begin{array}{c}\mathit{F}=\frac{m{u}^2}{\mathit{S}}\end{array}$(6) difficult to apply in many cases.

When utilizing Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) , based on the equation of motion for a mass point in a one-degree-of-freedom system as proposed by Haehnel and Daly (2004), setting the axial stiffness of the debris, k₁, becomes essential. The collision force is directly proportional to the collision speed, making the axial stiffness the constant of proportionality obtained from this relationship. This parameter, along with the impact duration Δt, can be obtained from collision experiments and analysis. The axial stiffness is calculated using the relationship between collision force and displacement, with data on axial stiffness compiled for various debris types from numerous studies. It is important to note that in the derivation process of Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) the stiffness of the collided object is assumed to be significantly greater than that of the debris, a condition necessary for accurately evaluating the interaction between colliding objects and debris.

The axial stiffness (k) of the debris object can also be calculated as follows, based on the principle that stiffness is directly proportional to the debris’ cross-sectional area and inversely proportional to its length:

(11) $\begin{array}{c}\mathit{k}=\frac{\mathit{EA}}{L},\end{array}$(11)

where A and L are the cross-sectional area (m²) and length (m) of debris, respectively.

For vehicles and ships, axial stiffness exhibits a nonlinear relationship with collision speed, necessitating caution in its determination. Figure 1 shows the collision force – displacement relationship from a vehicle collision experiment by Takabatake et al. (2015b). The slope changes with displacement due to the shift in load-resisting members as deformation progresses during the collision. Based on this observation, Takabatake et al. (2015b) proposed a collision force estimation equation that accounts for changes in axial stiffness in multiple steps, under the premise that the kinetic energy of the debris before collision equals the energy absorbed by deformation:

(12) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{k_n}\sqrt{1+\frac{1}{m{u}^2}\left(\sum _i^{\mathit{n}-1}{F}_i^2\left(-\frac{1}{k_i}-\frac{1}{{\mathit{k}}_{\mathit{i}+1}}\right)\right)}\end{array}$(12)

Figure 1. Collision force versus deformation measured in experiment (Takabatake et al., 2015b).

This equation allows for the calculation of collision force at a given collision speed (u(n-1) < u < u(n)) when stiffness changes in n steps. Experimental comparisons indicate that Eq. (12)(12) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{k_n}\sqrt{1+\frac{1}{m{u}^2}\left(\sum _i^{\mathit{n}-1}{F}_i^2\left(-\frac{1}{k_i}-\frac{1}{{\mathit{k}}_{\mathit{i}+1}}\right)\right)}\end{array}$(12) accurately estimates the collision force for debris experiencing changes in load-resisting members throughout the collision (Takabatake et al. 2015b). Thus, accurate evaluations of debris with complex structures are feasible with the appropriate axial stiffness values. However, determining multi-step axial stiffness values for practical debris collision force evaluations poses a challenge. Using the maximum slope in the collision force – displacement relationship as the axial stiffness in Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) enable a conservative estimation of the collision force; however, sufficient caution should be taken to prevent significant overestimation.

Another method for calculating the axial stiffness of debris k₁, involves using the equivalent stiffness derived from the maximum collision force and the area under the force – displacement curve up to the maximum collision force point. Miyagawa and Kaida (2019) calculated the equivalent stiffness as follows:

(13) $\mathit{k}=\frac{F_{max}^2}{2E_a}$(13)

where F_max² is the maximum collision force (N) and E_a represents the energy absorbed by the deformation of the debris during collision (Nm), effectively corresponding to the historical area leading to the maximum collision force point.

The axial stiffness values obtained from previous experiments and computational analyzes are compiled in Table 3, an update from Kaida and Kihara (2016). This table also incorporates input parameter values from literature not previously mentioned, such as Arita (1988). Axial stiffness values (N/m) for common tsunami debris range from O (10⁵) to O (10⁷), with containers being relatively stiffer compared to vehicles and timber. As shown in Table 3 and Figure 2, stiffness values for a specific type of debris exhibit variability. Kaida et al. (2017) categorized axial stiffness values for 161 vehicles based on collision experiments conducted by the National Highway Traffic Safety Administration. Based on these data, the variation of the axial stiffness of the vehicles was calculated; the results are shown in Figure 2. The analysis revealed that vehicle axial stiffness values are broadly distributed between O (10⁵) and O (10⁷), with 88.8% of vehicles displaying values from 1 × 10⁶ to 5 × 10⁶. A minor percentage of vehicles exhibited values exceeding 1 × 10⁷ N/m.

Figure 2. Histogram of equivalent stiffness of vehicles calculated by Kaida et al. (2018).

Table 3. Axial stiffness of debris (u: collision speed of debris).

Type of debris		Setting of the stiffness (×10^6 N/m)	References
Wood logs	Pine (m = 450 kg)	2.4	Haehnel and Daly (2004)
	Pine (L = 2 m, m = 179 kg, D = 0.42 m)	Mean: 9.55, Median: 9.19	Takabatake et al. (2015); Kaida and Kihara (2016)
	Cedar (L = 2 m, m = 156 kg, D = 0.38 m)	Mean: 14.4, Median: 14.3
	Cedar (L = 1 m, m = 72 kg, D = 0.38 m)	Mean: 11.9, Median: 10.0
	Pine (L = 6 m，m = 204 kg，D = 0.242 m)	73	Aghl et al. (2014)）
Container	20-ft standard shipping container oriented longitudinally (m = 2,200 kg)	24	FEMA (2012)
	20-ft standard shipping container oriented longitudinally (m = 2,270 kg)	42.9	ASCE (2022)
	20-ft standard shipping container-oriented transverse to flow (m = 2,200 kg)	80	FEMA (2012)
	20-ft heavy shipping container oriented longitudinally (m = 2,400 kg)	93
	20-ft heavy shipping container-oriented transverse to flow (m = 2,400 kg)	87
	40-ft standard shipping container oriented longitudinally (m = 3,800 kg)	60
	40-ft standard shipping container oriented longitudinally (m = 3,810 kg)	29.8	ASCE (2022)
	40-ft standard shipping container-oriented transverse to flow (m = 3,800 kg)	40	FEMA (2012)
	20-ft standard shipping container (m = 2,200 kg)	If target structure is prism; (22.85–7.66ln(|u|+0.7292)) If target structure is cylinder; (8.95–3.16ln(|u|-1.1195))	Madurapperuma and Wijeyewickrema (2013)
	40-ft standard shipping container (m = 3,800 kg)	If target structure is prism; (14.26–4.24ln(|u|-1.6485)) If target structure is cylinder; (10.20–3.39ln(|u|-1.7131))
Vehicle	L = 3.295 m, H = 1.475 m, W = 1.395 m, m = 320 kg (Mass with glass, engine, etc. removed)	|u| < 0.09; 6.25 × 10^−3 0.09 < |u| < 2.04; 1.18 × 10^−1 2.04 < |u| < 3.90; 2.04	Takabatake et al. (2015a)
	Track	2.5～6.9 (7.1 < |u| < 9.4)	Kubota and Kokubu (1995)
	Large size vehicle	1.3～2.8 (5.1 < |u| < 8.4)
	Off-road type vehicle	1.5～4.3 (9.8 < |u| < 12.4)
	Sedan type vehicle	9.010^−1～2.4 (5.6 < |u| < 12.2)
	Sedan type vehicle	Mean: 2.21	Results analyzed in this study based on Kaida and Kihara (2017)
	SUV type vehicle	Mean: 2.91
	Minivan type vehicle	Mean: 2.74
	Pickup track	Mean: 5.23
Ship	400GT	5.1（|u| < 2.30）	Arita (1988)
	1000GT	6.4（|u| < 2.30）
	2000GT	8.2（|u| < 2.31）
	4000GT	11（|u| < 2.34）
Small fishing boat	m = 2,760 kg, L = 8.1 m	0.54	Toyoda et al. (2022)
	m = 1,690 kg, L = 8.2 m	0.35

The axial stiffness of wood logs differs depending on the species and length. Research by Takabatake et al. (2015a) and Kaida and Kihara (2016) showed that for cedar logs, axial stiffness increases with length. This result contrasts with Eq. (3)(3) $\begin{array}{c}{F}_{\mathit{max}}=\frac{\pi }{2}\frac{\mathit{um}}{\mathrm{\Delta }\mathit{t}/2}\end{array}$(3) , possibly due to variations in cross-sectional area along the log, wood heterogeneity, moisture content, and other factors affecting physical properties. Kubota and Kokubu (1995) calculated axial stiffness for vehicles (referenced in Table 2) by utilizing the collision force – displacement relationship from vehicle collision experiments. Specifically, stiffness was calculated as F_max/x_Fmax from the maximum collision force F_max and the displacement x_Fmax at the first peak. The collision speed was then obtained by substituting the obtained axial stiffness and maximum collision force into Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) proposed by Haehnel and Daly (2004), establishing an upper limit for the applicable range. Note that the vehicle mass was assumed to be 1,500 kg. Stiffness values for small fishing vessels, obtained from air-drop experiments by Toyoda et al. (2022), indicate a collision speed of approximately 10 m/s and a stiffness O (10⁵), which is relatively low compared to other debris types, resulting in a longer collision duration. This suggests that, on average, small vessels are more flexible than wood and other materials. Given the limited data available for this type of debris, further research is necessary.

Equation (8)(8) $\begin{array}{c}\mathit{F}={\gamma }_p{\chi }^{\frac{2}{5}}{\left(\frac{5}{4}\mathit{m}\right)}^{\frac{3}{5}}{u}^{\frac{6}{5}}\end{array}$(8) by Arikawa et al. (2003), which presumes the debris and the collided object to be elastic, incorporates the coefficient γ_p to align with experimental results. In their study, Arikawa et al. (2003) observed that calculation results significantly overestimated experimental results when applying in Eq. (8)(8) $\begin{array}{c}\mathit{F}={\gamma }_p{\chi }^{\frac{2}{5}}{\left(\frac{5}{4}\mathit{m}\right)}^{\frac{3}{5}}{u}^{\frac{6}{5}}\end{array}$(8) without incorporating γ_p. The authors suggested that this difference might be stem from energy loss associated with the plastic deformation of the collided object. Arikawa et al. (2003) set γ_p at 0.25 based on collision experiments with concrete wave blocks, and this value proved consistent in subsequent experiments involving containers (Arikawa et al. 2007) and wood (Arikawa and Washizaki 2010), showing no significant deviation. Caution is advised when setting γ_p in Eq. (8)(8) $\begin{array}{c}\mathit{F}={\gamma }_p{\chi }^{\frac{2}{5}}{\left(\frac{5}{4}\mathit{m}\right)}^{\frac{3}{5}}{u}^{\frac{6}{5}}\end{array}$(8) , and its appropriateness should be validated through various collision experiments. Given that Eq. (8)(8) $\begin{array}{c}\mathit{F}={\gamma }_p{\chi }^{\frac{2}{5}}{\left(\frac{5}{4}\mathit{m}\right)}^{\frac{3}{5}}{u}^{\frac{6}{5}}\end{array}$(8) is grounded in Hertz’s (1881) theory, it assumes that deformation remains within the elastic range for both the debris and the collided objects. Therefore, cases where wood undergoes plastic deformation due to high-speed impacts (P. Aghl, Naito, and Riggs 2014) might fall outside its applicable range.

Beyond theory-derived equations, experimental results have also inspired equations based on dimensional analysis. Matsutomi (1999) conducted hydraulic experiments with a wood model and air tests with full-scale wood logs, leading to the proposal of the following collision force estimation equation from dimensional analysis of the air tests:

(14) $\begin{array}{c}\frac{F}{\gamma D^2\mathit{L}}=1.6C_{\mathit{MA}}{\left\{\frac{\mathit{u}}{{\left(\mathit{gD}\right)}^2}\right\}}^2{\left(\frac{{\sigma }_f}{\mathit{\gamma L}}\right)}^{0.4}\end{array}$(14)

where D, L, γ, C_MA, and σ_f are the diameter (m), length (m), unit weight (N/m³), hydrodynamic mass coefficient, and yield stress (N/m²) of the wood-log, respectively. The exponents in this power-law equation were determined through dimensional and regression analysis of the air-impact test results. The equation considers the wood’s physical properties, such as yield stress σ_f, dimensions, and weight. A linear relationship between the yield stress σ_f and the modulus of elasticity E was inferred from experiments on pine and lauan, indicating that Eq. (14)(14) $\begin{array}{c}\frac{F}{\gamma D^2\mathit{L}}=1.6C_{\mathit{MA}}{\left\{\frac{\mathit{u}}{{\left(\mathit{gD}\right)}^2}\right\}}^2{\left(\frac{{\sigma }_f}{\mathit{\gamma L}}\right)}^{0.4}\end{array}$(14) was specifically developed to evaluate wood collision forces.

When evaluating collision forces, it is necessary to account for the effects of hydrodynamic added-mass and collision angle. The methods for determining hydrodynamic mass coefficients are summarized in Table 4, with a range of proposed values for the coefficient. Ikesue et al. (2017) suggested that a larger aspect ratio leads to a lower added-mass coefficient, supporting Haehnel and Daly’s (2004) notion that the added-mass coefficient depending on the collision direction of the debris. Therefore, setting the appropriate coefficient requires consideration of the debris type and expected collision angle. Currently, there is a lack of guidance on setting the hydrodynamic mass coefficient for vehicles, often considered debris in many situations. Therefore, it may be necessary to adopt values from other debris types with similar shapes to vehicles.

Table 4. Hydrodynamic mass coefficient for debris.

Target debris	Setting of hydraulic mass coefficient	References
Wood log（oriented longitudinally）	Bore or surge; CMA = 1.7 Steady flow; CMA = 1.9 in Eq. (12)(12) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{k_n}\sqrt{1+\frac{1}{m{u}^2}\left(\sum _i^{\mathit{n}-1}{F}_i^2\left(-\frac{1}{k_i}-\frac{1}{{\mathit{k}}_{\mathit{i}+1}}\right)\right)}\end{array}$(12)	Matsutomi (1999)
	• C = 0; short rectangular timber oriented longitudinally • C = 1; long cylindrical logs oriented transverse to flow • C = 0.22; cylindrical logs oriented transverse to flow • C = 2.4; square timbers oriented transverse to flow • C = 3.5; rectangular timbers oriented transverse to flow in Eq. (5)(5) ${\displaystyle {\int }_0^S\mathit{k}}\mathit{xdx}=\frac{1}{2}\mathit{m}{u}^2$(5) .	Haehnel and Daly (2004)
Limber or wood log – oriented longitudinally	C = 1.3	FEMA (2012)
20-ft standard shipping container oriented longitudinally (m = 2,200 kg)	C = 1.30
20-ft standard shipping container-oriented transverse to flow (m = 2,200 kg)	C = 2.00
20-ft heavy shipping container oriented longitudinally (m = 2,400 kg)	C = 1.30
20-ft heavy shipping container-oriented transverse to flow (m = 2,400 kg)	C = 2.00
40-ft standard shipping container oriented longitudinally (m = 3,800 kg)	C = 1.20
40-ft standard shipping container-oriented transverse to flow (m = 3,800 kg)	C = 2.00
	If the draft of the debris (d (m)) is less than the width of the debris (B (m)): $\mathit{c}=\frac{\pi K_1\mathit{d}}{2\mathit{L}}$ Where K1 is coefficient described in JSME (2004). If d < B: $\mathit{c}=\frac{\pi K_1\mathit{B}}{4\mathit{L}}$	Ikesue et al. (2017)

The effect of the debris collision angle has been addressed previously. Drawing on large-scale collision experiments with wood, Ikeno et al. (2016) proposed a reduction factor γ for the collision force in cases of oblique collisions with wood:

(15) $\begin{array}{c}\mathit{\gamma }=\frac{1+{\left(\frac{{\epsilon }_0}{\mathit{r}}\right)}^2\mathit{co}{s}^2\mathit{\theta }}{1+{\left(\frac{{\epsilon }_0}{\mathit{r}}\right)}^2}\end{array}$(15)

where θ (°) represents the collision angle between the debris and the structure it collides with.

2.2.3. Applicability of evaluation equations

In addition to confirming the theoretical background of collision force estimation equations, it is crucial to assess their practical utility. Kaida and Kihara (2016) applied various previously proposed evaluation equations (Arikawa et al. 2003, Haehnel and Daly, 2004, Japan Road Association, 2012, Matsutomi, 1999) to different types of debris, such as vehicles, wood logs, containers, and ships. They highlighted the necessity of choosing appropriate substitution parameters to accurately estimate debris collision forces. The applicability of each equation was evaluated, revealing that results from guidelines and technical standards might not always be conservative. Yamamoto et al. (2021) also tested the efficacy of previously proposed equations (Arikawa et al. 2003; ASCE, 2016; Bridge Engineering Association, 1978; FEMA 2012; Ikeno and Tanaka, 2003; Ikeno et al., 2003; Japan Road Association, 2012; Matsutomi, 1999), assessing their applicability through comparative analysis of their magnitude relationships. However, detailed discussion on the specific conditions suitable for each equation was limited, focusing mainly on the comparison of relative magnitudes.

Recent research (Kaida and Kihara, 2017a, Kaida et al. 2017b, ASCE 2022) has broadened the range of collision forces estimable for vehicles, containers, and wood logs by introducing new substitution parameters to the collision force estimation equations. This review applies these updated parameters to assess the impact forces of vehicles, containers, and wood logs. As mentioned earlier, debris collision forces have an upper limit due to plastic deformation, yet this evaluation was performed within the elastic deformation range. The effects of hydrodynamic added mass and collision angle on collision force evaluation were not considered. This section delves into the applicability of equations by Matsutomi (1999), Arikawa et al. 2003), and Haehnel and Daly (2004). The equation by Mizutani et al. (2005) for containers is not included in this discussion as it aligns with Eq. (3)(3) $\begin{array}{c}{F}_{\mathit{max}}=\frac{\pi }{2}\frac{\mathit{um}}{\mathrm{\Delta }\mathit{t}/2}\end{array}$(3) , tailored for air collisions.

First, the accuracy of the collision force estimation equation for wood logs is verified using the results of two different experiments (P. Aghl, Naito, and Riggs 2014, Kaida et al. 2017b). The results of applying conditions from Table 5 to three different equations (Matsutomi, 1999, Arikawa et al. 2003, Haehnel and Daly, 2004) are shown in Figure 3. Consistent with prior research, the coefficients for the equations by Matsutomi (1999) and Arikawa et al. (2003) were set to ν = 0.4, γ_p = 0.25, and σ_f = 0.0044E. For Pine 1, Eq. (3)(3) $\begin{array}{c}{F}_{\mathit{max}}=\frac{\pi }{2}\frac{\mathit{um}}{\mathrm{\Delta }\mathit{t}/2}\end{array}$(3) , which incorporates collision duration, closely aligns with the experimental results (represented by gray plots). Using Eq. (11)(11) $\begin{array}{c}\mathit{k}=\frac{\mathit{EA}}{L},\end{array}$(11) to calculate axial stiffness and subsequently substituting it into Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) with the standard Young’s modulus (shown in blue plot) slightly underestimates the experimental results, yet generally maintains good accuracy. The other equations tend to underestimate the experimental results. Adjusting the experimental data, a precise fit was obtained with an axial stiffness k = 7.3 × 10⁷ N/m for Pine 1 (indicated by the dashed blue plot). For Pine 2, Cedar 1, and Cedar 2, calculations that employed axial stiffness values obtained from Eq. (11)(11) $\begin{array}{c}\mathit{k}=\frac{\mathit{EA}}{L},\end{array}$(11) and substituted into Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) (Haehnel and Daly, 2004) overestimated the experimental outcomes. The other equations produced more accurate estimations. The discrepancy suggests that axial stiffness values calculated via Eq. (11)(11) $\begin{array}{c}\mathit{k}=\frac{\mathit{EA}}{L},\end{array}$(11) do not match those obtained from experimental collision data, potentially due to variations in wood cross-section along the longitudinal axis, wood heterogeneity, moisture content, and other factors influencing physical properties.

Figure 3. Estimated collision force for (a) Cedar 1, (b) Cedar 2, (c) Pine 1, and (d) Pine 2 (see Table 5). White circles are experimental results reported by Takabatake et al. (2015) and Aghl et al. (2014). In the calculation using the equations of Haehnel and Daly (2004) (Eqs. (3) and (7)), indicated by the gray line and dashed blue line, the collision duration Δt and stiffnesses k listed in Table 5 were used.

Table 5. Types and specifications of debris used in example calculations of collision forces.

	Specifications	Input parameters	References
Pine 1	M = 204 kg L = 6 m D = 0.216 m	E = 7584MPa Δt = 0.005s	Aghl et al. (2014)
Pine 2	m = 179 kg L = 2 m D = 0.42 m	E = 1.1GPa Δt = 0.011 s k = 9.7e +6 N/m	Takabatake et al. (2015) Kaida et al. (2017)
Cedar 1	m = 156 kg L = 2 m D = 0.38 m	E = 7100MPa Δt = 0.007 s k = 15.70e +6 N/m	Takabatake et al. (2015) Kaida et al. (2017)
Cedar 2	m = 72 kg L = 1 m D = 0.38 m	E = 7100MPa Δt = 0.009 s k = 7.73e +6 N/m	Takabatake et al. (2015) Kaida et al. (2017)
Container	m = 2,300 kg L = 6 m	Δt = 0.011 s E = 207GPa	Aghl et al. 2015)
	–	k = 42.9e +6 N/m	ASCE (2022)
	–	k = 85e +6 N/m	FEMA (2012)
Vehicles	m = 1,118～3,448 (kg, mean is 1,880 kg)	Δt = 0.011～0.11 s k = 6.3e + 6～3.07e +7 N/m	See Table 2 See Table 3

When evaluating the collision forces for containers, substituting the axial stiffness values recommended by ASCE (2022) into Haehnel and Daly’s equation showed good alignment with experimental results (P. P. Aghl, Naito, and Riggs 2015) as depicted in Figure 4. Other equations tended to overestimate the experimental results. Eq. (3)(3) $\begin{array}{c}{F}_{\mathit{max}}=\frac{\pi }{2}\frac{\mathit{um}}{\mathrm{\Delta }\mathit{t}/2}\end{array}$(3) , which accounts for collision duration, overestimated the results. A possible reason for this is that the dependency of collision duration on the collision angle (P. Aghl, Naito, and Riggs 2014. Arikawa et al. 2003) equation also overestimated the experimental results, indicating the need for future studies to carefully consider the setting of the collision area and coefficients.

Figure 4. Estimated crash loads for 20-ft (6.1-m) containers. White circles are experimental results reported by Aghl et al. 2015).

In experiments involving wooden logs and containers, the maximum collision speed was below 5 m/s. However, during the Tohoku tsunami, triggered by the Mw 9.0 Tohoku-oki earthquake, maximum tsunami velocities in Kesennuma Bay hit 11 m/s (Fritz et al. 2012), and tsunami currents reached 8 m/s upstream of the Sendai Plain (Hayashi and Koshimura, 2012). Hydraulic experiments suggest that drifting objects may collide at or near the flow field’s maximum velocity (Kaida and Kihara, 2016), indicating that actual tsunami-driven debris impact velocities might exceed those used in the experiments. Therefore, the validity of the collision force evaluation equations for higher speed impacts needs verification. Yamamoto et al. (2021) tested various equations at collision speeds over 10 m/s, calculating the average estimated collision force from each equation. Equations yielding values close to the average were deemed applicable. However, as collision speed significantly influences the physical properties of debris materials affecting collision force – and considering the upper limit of collision force due to plasticization – evaluating these equations’ applicability to high-speed collisions requires an alternative approach. Future research, including collision analysis and high-speed collision experiment analysis, is vital to address this issue.

For vehicle collision forces, this review used equivalent stiffness values (referenced in Table 3 and Figure 2). By incorporating these stiffness values into Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) , a slightly overestimated maximum collision force was calculated, as illustrated in Figure 5. With Eq. (3)(3) $\begin{array}{c}{F}_{\mathit{max}}=\frac{\pi }{2}\frac{\mathit{um}}{\mathrm{\Delta }\mathit{t}/2}\end{array}$(3) , factoring in collision duration, collision forces were overestimated, likely because the collision force time series deviates from the sinusoidal waveform assumed by Haehnel and Daly (2004). The collision force – displacement relationship offers a means to deduce the collision speed (Kubota and Kokubu, 1995). Assuming the kinetic energy at collision equals the energy absorbed, as depicted by the collision force – displacement relationship, collision forces at speeds of 5 and 10 m/s were determined. On average, the review’s summarized equivalent stiffness values suggest conservative estimates for collision forces at speeds lower than those in the experiments.

Figure 5. Results of evaluation of collision forces on vehicles. (a) Comparison between Eq. (3) and Eq. (7). (b) Collision force evaluated by Eq. (7) proposed by Haehnel and Daly (2004), white, black and red circle show the results at collision speeds of 15.6 m/s, respectively.

The results of the mentioned investigations are summarized in Table 6, categorized according to three criteria: accuracy, theoretical foundation, and applicability. Each equation’s performance is evaluated against these criteria. Arikawa et al. 2003) derived an equation grounded in Hertz’s theory of elastic contact, requiring both the debris and the collided object to remain within the elastic deformation range. The equation acknowledges the plasticization effect of drifting objects through the coefficient γ_p; however, Arikawa et al. 2003) set γ_p based on a limited set of experimental data, and the coefficient lacks a physical basis. Matsutomi’s (1999) equation, derived from dimensional analysis of low-speed debris collisions, establishes the power-law exponent through dimensional analysis. It assumes a linear relationship between yield stress and modulus of elasticity, a simplification that does not fully align with collision physics theories. Due to these limitations, both equations receive a “△” rating for all criteria in Table 6. The Haehnel and Daly’s (2004) equation offers a stable and accurate evaluation of collision forces if axial stiffness or collision duration is correctly determined. However, setting these parameters accurately for all debris conditions is challenging. This equation benefits from a clear theoretical background and simple formulation, making it practically applicable. Incorporating axial stiffness and collision duration allows for the consideration of dynamic collision processes. Representing the impact plasticization of a drifting object can be achieved in Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) by setting multiple levels of axial stiffness, with information on these parameters growing increasingly detailed. Therefore, Eq. (3)(3) $\begin{array}{c}{F}_{\mathit{max}}=\frac{\pi }{2}\frac{\mathit{um}}{\mathrm{\Delta }\mathit{t}/2}\end{array}$(3) by Haehnel and Daly (2004) is rated as “○” for accuracy and ”△“for the other criteria. Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) receives an “○” rating for all criteria in Table 6.

Table 6. Summary of evaluation equations for collision force.

Equations	Target debris	Consideration of dynamic processes related to collisions	Evaluation accuracy	Versatility
Eq. (14)(14) $\begin{array}{c}\frac{F}{\gamma D^2\mathit{L}}=1.6C_{\mathit{MA}}{\left\{\frac{\mathit{u}}{{\left(\mathit{gD}\right)}^2}\right\}}^2{\left(\frac{{\sigma }_f}{\mathit{\gamma L}}\right)}^{0.4}\end{array}$(14) proposed by Matsutomi (1999)	Wood logs	△	△	△
Eq. (8)(8) $\begin{array}{c}\mathit{F}={\gamma }_p{\chi }^{\frac{2}{5}}{\left(\frac{5}{4}\mathit{m}\right)}^{\frac{3}{5}}{u}^{\frac{6}{5}}\end{array}$(8) proposed by Arikawa et al. (2003)	Wood log	△	△	△
Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) proposed by Haehnel and Daly (2004)	Debris for which input parameters can be set	○	○	○
Eq. (3)(3) $\begin{array}{c}{F}_{\mathit{max}}=\frac{\pi }{2}\frac{\mathit{um}}{\mathrm{\Delta }\mathit{t}/2}\end{array}$(3) proposed by Haehnel and Daly (2004)		△	○	△

Research to date, including this review, underscores the diverse applicability of evaluation equations for debris collision forces. They also highlight the need for developing high-quality, comprehensive substitution parameters for various debris types to address uncertainties in collision force evaluations. Despite considerable efforts, perfectly replicating specific conditions in practice remains a challenge. Incorrect application of substitution parameters or using evaluation equations beyond their intended conditions can lead to inaccurate results. Therefore, a thorough understanding of each evaluation equation’s derivation process and the rationale behind setting substitution parameters is essential for accurately estimating debris collision forces.

2.2.4. Collision experiments and numerical analysis of debris collision

When existing collision force estimation equations are not applicable, collision experiments and numerical analyzes become indispensable. Collision experiments provide crucial information, such as damage modes during a collision, which aid in setting input parameters for collision force estimation equations and in validating collision numerical analyzes. However, these experiments are costly and labor-intensive. Therefore, a thorough review of relevant literature is recommended before undertaking such experiments.

Collision analysis serves to evaluate the force of debris collisions and to closely examine the damage inflicted during the collision process. Previous studies have successfully replicated the collision process and the time series of collision force by accurately modeling the structure and physical properties of tsunami-driven debris (Madurapperuma et al., 2013; Toyoda et al., 2022; Kuriyama et al. 2024). This indicates that collision analysis, when the debris structure and physical properties are accurately modeled, offers a viable alternative to physical experiments. It allows for the simulation of collisions involving non-rigid structures and, by incorporating fluid-structure interactions, can couple with complex flow fields around structures to simulate advanced collision physics (Obi, 2016). However, conducting a collision analysis requires significant technical expertise to ensure accurate results. In addition, the resources required for collision analysis – such as computational power and time – far exceed those required for evaluations using collision force evaluation equations. While the need for numerical collision analysis should be carefully considered due to its high accuracy and the difficulties in setting appropriate substitution parameters in some cases, it remains a feasible method for evaluating collision forces.

3. Design guidelines for evaluation of debris collision force

Section 2 presents a comprehensive review of the literature concerning the establishment of collision speed and the formulation of collision force evaluation equations. Various organizations, including (ASCE 2022; FEMA 2008; FEMA 2012; FEMA 2019) have published guidelines offering valuable recommendations for assessing debris collision forces in a practical context. These guidelines, which inform the design of port structures and industrial facilities in flood-prone areas, encapsulate methods for evaluating debris collision forces. While numerous methods exist for this purpose, certain guidelines advocate for the application of slightly modified methods. This section reviews how debris collision evaluations are described across various guidelines, highlighting the assumptions and characteristics unique to each. Given that practical evaluations of debris collision force often rely on these guidelines, this section also delves into potential improvements of practical evaluations.

In 2008, FEMA introduced guidelines aimed at designing structures for vertical evacuation from tsunamis (FEMA 2008), which have since been updated in 2012 and 2019 (FEMA 2012; FEMA 2019). The collision force estimation equations in the first and second editions of FEMA P646 (FEMA, 2008; FEMA 2012) are based on Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) (Haehnel and Daly, 2004). To ensure conservativeness, these editions recommend using the maximum flow velocity at the site as the debris collision speed, as opposed to speeds obtained from debris-tracking simulations or other investigative methods. In addition, a safety factor is applied to ensure conservative estimation results. The most recent version of the FEMA guidelines (FEMA, 2019) incorporates references to a new chapter on tsunami load assessment found in ASCE (2016).

In the 2016 revision of ASCE 7 (ASCE, 2016), a chapter dedicated to tsunami loads was introduced. This addition specifies that in areas where the inundation depth is 0.914 m or less, the effects of tsunami-driven debris collisions need not be considered. Similarly, it is deemed unnecessary to combine the debris collision force with other tsunami-induced loads, such as wave forces. A simplified method for determining tsunami debris hazard regions, based on inundation depth and suggested by Naito et al. (2014), has been adopted to determine areas requiring debris collision force evaluation in design considerations (Figure 6). This method involves drawing a perpendicular line from the geometric center of potential debris accumulation sites (e.g. container yards, lumber yards) toward the land, along with two additional lines at ± 22.5° from this central line. The area enclosed by the landward limit line (defining the maximum reach of debris) and these angled lines is defined as susceptible to debris impact. To account for tsunami-induced drawback effects, two additional lines at ± 22.5° from the landward limit of debris run-up are drawn toward the shoreline, delineating areas at risk from retreating tsunami waves. This estimation method by Naito et al. (2014), which leverages existing hazard maps without the need for complex numerical inundation analyzes or debris-tracking simulations, offers a more straightforward and less time-consuming method for assessing debris collision risks. However, as it is grounded in the results of field surveys on the final resting places of tsunami debris, it does not account for debris trajectories, potentially underestimating the extent of areas at risk (Naito et al. 2014). In addition, this method does not facilitate the evaluation of collision speeds, which are influenced by debris movement behavior.

Figure 6. Conceptual diagram of Naito et al. (2014) for the evaluation of potential area for debris collision.

ASCE (2022) introduces several equations for evaluating collision forces, based on Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) , aiming to enhance engineering convenience albeit at the expense of accuracy. The maximum instantaneous debris collision force F_ni and the design instantaneous debris collision force F_i for various objects, including wooden poles, logs, shipping containers, large vessels, and barges, are calculated as follows:

(16) $\begin{array}{c}{F}_{\mathit{ni}}=u_{\mathit{max}}\sqrt{\mathit{km}}\end{array}$(16)

(17) $\begin{array}{c}{F}_i=I_{\mathit{tsu}}{C}_0{F}_{\mathit{ni}}\end{array}$(17)

where I_tsu, C_O, and u_max represents the importance coefficient, the rotational coefficient (set at 0.65), and the maximum flow velocity (m/s) at the site, specifically at the water depth (m) where debris floats sufficiently. The impact of debris collisions is significantly affected by the angle of impact, making it important to compensate for this effect in evaluations (Haehnel and Daly, 2004, Ikeno et al. 2016). However, the application of the C_O setting in ASCE (2022) does not lead to a conservative estimation of collision forces. Contrary to this, other guidelines, as discussed below, do not consider factors related to the debris collision angle, reasoning that including such effects might reduce the calculated collision force. This approach marks a significant difference between ASCE (2022) and other guidelines, which prioritize establishing a minimum design load for consideration. The duration of the collision is calculated as follows:

(18) $\begin{array}{c}{t}_d=\frac{2\mathit{m}{\mathit{u}}_{\mathit{max}}}{F_{\mathit{ni}}}\end{array}$(18)

In conducting an equivalent elastic-static analysis with the designed instantaneous debris collision force, the impact force is adjusted by multiplying it by a dynamic response factor. ASCE (2022) includes a table that allows for the determination of this dynamic response factor based on the ratio of the structural element’s natural period to the collision duration. For assessing floating vehicles, the collision force is calculated as follows:

(19) $\begin{array}{c}{F}_{\mathit{ni}}=130I_{\mathit{tsu}}\left(\mathit{kN}\right)\end{array}$(19)

Regarding the evaluation of large submerged, rolling debris, and concrete fragments, the impact force on vertical structural elements located from 2 ft (0.61 m) above ground level to the maximum water depth is calculated in areas where the maximum inundation depth exceeds 6 ft (1.83 m). The impact force calculation method is outlined as follows:

(20) $\begin{array}{c}{F}_{\mathit{ni}}=36I_{\mathit{tsu}}\left(\mathit{kN}\right)\end{array}$(20)

Equations (19)(19) $\begin{array}{c}{F}_{\mathit{ni}}=130I_{\mathit{tsu}}\left(\mathit{kN}\right)\end{array}$(19) and (20(20) $\begin{array}{c}{F}_{\mathit{ni}}=36I_{\mathit{tsu}}\left(\mathit{kN}\right)\end{array}$(20) ) exclude collision speed as a parameter because they incorporate the importance factor and the collision force for the debris at a given collision speed, as calculated using Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) (Haehnel and Daly, 2004). For example, taking a vehicle with a mass m = 1,090 kg and k = 1 × 10⁶ N/m and substituting u_max = 4 m/s into Eq. (7)(7) $\begin{array}{c}\mathit{F}=\mathit{u}\sqrt{\hat{k}\left(\mathit{m}+\mathit{C}{m}_f\right)}\end{array}$(7) , results in an impact force of 133 kN. Consequently, Eq. (19)(19) $\begin{array}{c}{F}_{\mathit{ni}}=130I_{\mathit{tsu}}\left(\mathit{kN}\right)\end{array}$(19) is approached as an engineering configuration and warrants cautious application. The axial stiffness magnitude used aligns with those depicted in Figure 2 and Table 3, albeit slightly lower. A detailed review of the collision conditions is essential for a conservative assessment of collision forces.

To assess the impact force of a shipping container, Eqs. (16)(16) $\begin{array}{c}{F}_{\mathit{ni}}=u_{\mathit{max}}\sqrt{\mathit{km}}\end{array}$(16) and (17(17) $\begin{array}{c}{F}_i=I_{\mathit{tsu}}{C}_0{F}_{\mathit{ni}}\end{array}$(17) ) are used to calculate the instantaneous debris collision force, with the mass of the empty container as the input. The point of impact is assumed to be the container’s bottom corner, and its stiffness is calculated using Eq. (13)(13) $\mathit{k}=\frac{F_{max}^2}{2E_a}$(13) . The axial stiffness and mass of the empty container are summarized in Table 3.

ASCE (2022) offers an alternative approach for evaluating the static force from debris collisions involving wooden poles, logs, vehicles, large gravel, concrete pieces, and containers, using the equation below:

(21) $\begin{array}{c}{F}_i=1470C_0{I}_{\mathit{tsu}}\left(\mathit{kN}\right)\end{array}$(21)

In scenarios unlikely to involve collisions with shipping containers, vessels, or barges, the calculated force from the above equation may be halved.

ASCE (2022) stands out among guidelines for providing a specific method to evaluate the collision forces of common tsunami debris, offering substantial engineering utility. However, it is crucial to note that ASCE (2022) focuses on defining the minimum design forces for consideration. Therefore, this guideline should be applied judiciously, particularly in the design of facilities and structures that necessitate high safety margins.

Since the Great East Japan Earthquake (Mori et al., 2011), Japan has established and continually updated various guidelines for tsunami risk assessment, covering a broad spectrum from general and fishing ports to nuclear power plants. The MLIT (2013) developed guidelines for designing tsunami evacuation facilities at ports and harbors, stipulating that debris collision forces must be accounted for in addition to other external tsunami forces. The collision speed is determined based on the maximum flow speed at the evaluation point, with methods introduced in Table 7 for evaluating collision forces, acknowledging the challenge of quantitatively accurate impact force estimation. Fisheries Agency (2016) developed design guidelines for tsunami debris management at fishing ports, aligning closely with MLIT’s (2013) recommendations. These guidelines are notable for incorporating debris-tracking simulations and evaluating collision speeds using models from Goto et al. 1982) and Shigihara et al. (2016) for specific ports, comparing speeds from simulations with those derived from maximum flow velocity to highlight differences in debris trajectories and collision speeds. Technical references for nuclear power plants were published by the JSCE (2016) and the JEA (2021), advising that collision speed estimations be based on hydraulic data from tsunami propagation and inundation simulations, with debris-tracking simulations used upon validation of their applicability. For nuclear power plant sites, evaluations must reflect the facility layout through tsunami inundation simulations. MLIT (2013) and the Fisheries Agency of Japan (2016) introduced several collision force estimation equations, the importance of considering debris types and collision conditions. JSCE (2016) illustrated the application of these equations in collision experiments with vehicles and wood logs, underscoring the necessity of setting appropriate parameters. Furthermore, the combination of debris collision forces with other tsunami-induced loads in evaluations was highlighted, alongside measures to prevent potential debris, beneficial not just for nuclear sites but also for general disaster prevention. The Coastal Development Institute of Technology (CDIT) and the Cold Regions Air and Sea Ports Engineering Research Center (CDIA and CPC, 2014) proposed a design for tsunami debris mitigation facilities using steel pipe struts and wire ropes, requiring the integration of collision forces with tsunami flow forces. Given that facilities are designed to absorb debris collisions through deformation, the collision force estimation equations shown in Section 2.2 are not used in these instances. Instead, a design method that accounts for the energy absorbed through the deformation of the impacted object is proposed (Coastal Development Institute of Technology 2014). These guidelines require setting the collision speeds based on the maximum flow velocity at the site intended for protective measures; however, this requirement is appropriate only under limited conditions. Specifically, it is only appropriate in the case wherein a tsunami bore, characterized by a steep front and high flow velocity, impacts debris close to the structure. Otherwise, the evaluated collision force may not accurately reflect the actual physical dynamics of the situation. The guidelines acknowledge that complex topography or significant structural impacts at the proposed site must be considered to ensure a realistic design. Consequently, setting collision speeds that align with the real physics of collisions is essential. For this purpose, debris-tracking simulations are permitted by various guidelines (Fisheries Agency of JEA, 2021; Japan 2016; JSCE, 2016).

Table 7. Summary of evaluation methods described in technical guidelines for tsunami debris.

	Superposition with other loads	Structures and facilities to be designed	Evaluation
			Debris collision speed	Debris collision force
MLIT(2013)	○	Tsunami evacuation facilities in ports and harbors	Setting based on the maximum flow speed at the proposed site	Matsutomi (1999) Ikeno and Tanaka (2003) Arikawa et al. 2003) Mizutani et al. (2005) FEMA (2012)
CDIT and CPC (2014)	○	Tsunami debris measurement facilities consisting of steel pipe structures and wire ropes	Setting based on the maximum flow speed at the proposed site	A method based on the energy due to deformation of the collided object
Fisheries Agency (2016)	○	Tsunami debris measurement facilities at fishing ports	Setting based on the maximum flow speed at the proposed site The debris-tracking simulation	Matsutomi (1999) Ikeno and Tanaka (2003) Arikawa et al. (2003) Mizutani et al. (2005) FEMA (2012)
JSCE (2016)	○	Nuclear power plant	Setting based on hydraulic quantities obtained from tsunami propagation and inundation simulations The debris-tracking simulation (if applicability is confirmed)	Matsutomi (1999) Ikeno and Tanaka (2003) Arikawa et al. (2003) Mizutani et al. (2005) FEMA (2012)
JEA (2021)	○			Matsutomi (1999) Ikeno and Tanaka (2003) Arikawa et al. (2003) Mizutani et al. (2005) FEMA (2012)
ASCE (2022)	×	Buildings and other structures	Maximum flow speed at the relevant site	Equation introducing various assumptions in Haehnel and Daly (2004)

The characteristics of the guidelines reviewed are summarized in Table 7. When evaluating the effects of debris collisions, it is essential to ensure that the assessment’s objectives and targets align with those specified in the chosen guidelines. ASCE (2022) offers a standardized method for evaluating debris collision forces, but it is important to carefully consider the conditions under which this method is applied, indicating that the ASCE (2022) approach is not universally applicable. Conversely, guidelines issued by Japanese organizations introduce a range of equations for the evaluation of collision forces, instead of focusing on a single equation, and have not yet achieved standardization. This diversity of methods underscores the complexity of accurately evaluating debris impact forces. One reason for this is that the physical processes involved vary widely depending on the particular situation being evaluated; this is related to the difficulty in developing a universally applicable standard method.

4. Discussion on future research

Methods for evaluating the forces of debris collisions have advanced significantly. Debris-tracking simulations, which assess collision speeds, enable detailed analysis of the behavior of tsunami-borne debris. Accurate evaluation of debris collision forces can be achieved using an equation with precisely substituted parameters and correct collision speeds. However, the accuracy of these speeds and forces varies significantly, depending on the evaluation methods and the values of the parameters used.

Conducting debris-tracking simulations introduces several sources of uncertainty, particularly in evaluating debris collision speeds. The drag and added-mass coefficients in the Morison equation, as shown in Eq. (1)(1) $\mathit{F}=\frac{1}{2}{\rho }_w{C}_D{\displaystyle \int \left(\mathit{U}-\frac{d{x}_p}{\mathit{dt}}\right)}\left|\mathit{U}-\frac{d{x}_p}{\mathit{dt}}\right|\mathit{dA}+{\rho }_w\mathit{V}{C}_M\left(\frac{\mathit{DU}}{\mathit{Dt}}-\frac{d^2{x}_p}{\mathit{d}{t}^2}\right)$(1) , must be carefully chosen, yet their values vary depending on the shape of the debris and flow conditions. Although numerous studies have reported on these coefficients, their values differ significantly. Additionally, factors such as the submersion time of debris, which is relevant for debris with voids, and the coefficient of bottom friction, must be considered. Consequently, there are no standardized values for these coefficients. Altering these coefficients in the simulations can change the predicted debris trajectory, as well as the collision speed, position, and angle. Moreover, the initial position and direction of the debris, coupled with flow turbulence, introduce randomness to its trajectory. It is difficult to set a single specific value for the various conditions mentioned above, and in practice, certain values are often set for convenience. However, an engineering approach to obtain conservative results involves bypassing debris-tracking simulations and assuming the maximum flow velocity as the collision speed. This approach, though, overlooks the physics of the drifting process.

The evaluation of debris collision force necessitates the setting of various parameters, introducing a significant source of uncertainty into the results. These substitution parameters, which represent the physical properties of the debris, significantly impact the results of a collision force evaluation. Their values can vary significantly across various types of debris (Tables 2 and 3). Even within the same type of debris, values depend on specific debris specifications (Table 4). Furthermore, the effects of debris shape and collision angle on collision forces can vary substantially under different conditions. While it is possible to account for these effects in the evaluation of collision force, assigning deterministic values to these variables is challenging.

This inherent uncertainty complicates the practical application of these evaluation techniques. This issue likely contributes to the absence of a standardized method for evaluating the force of collisions caused by tsunami-induced debris.

The development of deterministic methods for accurately determining debris collision force is nearing its limit. To achieve more realistic designs and risk assessments, establishing a methodology that includes probabilistic evaluation is essential. The need for a probabilistic evaluation of tsunami-driven debris collisions has been mentioned in previous studies (Kaida and Kihara, 2017; Nistor et al. 2017; Koh et al., 2024). Although several studies (Kihara and Kaida, 2019; Stolle et al., 2020a; Stolle et al., 2020c; Kaida and Kihara, 2020; Kaida et al. 2022) have contributed to the development of probabilistic evaluations, a standard methodology for the probabilistic evaluation of debris collision forces has yet to be established. Kihara and Kaida (2019) explored the use of debris-tracking simulations in the probabilistic evaluation of debris collision speed and probability. They performed reproduction numerical simulations under conditions that encompassed the uncertainties inherent in debris-tracking simulations, based on an experiment that assessed the collision probability and speed of land-placed debris (Kaida and Kihara, 2016). Their approach modeled the debris collision as an exceedance probability, considering a range of uncertainties, with experimental values falling within this uncertainty range. Unlike deterministic evaluations, which engineer various conditions to derive a single solution, probabilistic evaluations provide solutions that encompasses the results of the deterministic evaluations. Stolle et al. (2020c) proposed a framework for the probabilistic evaluation of tsunami-induced debris collision probabilities, illustrating its application under simplified conditions. Implementing this approach in real-world scenarios, which involve more complex structural arrangements and flow fields, will undoubtedly complicate debris behavior. Although the applicability of this method to such complex situations remains to be verified, its high engineering utility suggests it has significant potential for future development. The proposed probabilistic approaches are considered reasonable and could serve as a basis for future probabilistic evaluations.

Kaida and Kihara (2020, 2022) proposed a method for evaluating the probability of structural damage due to debris collision, building upon their previous study (Kihara and Kaida, 2019). They tested the practicality of this method by applying it to a hypothetical scenario involving a nuclear power plant protected by a seawall and surrounded by various structures. The evaluation of collision speed and probability for this nuclear power plant considered the altered flow field caused by the presence of these structures, a factor not considered in ideal hydraulic experiments. In their method, the exceedance probability of debris collision speed, obtained from debris-tracking simulations, was used. Parameters without standardized values (e.g. drag coefficient, added-mass, submersion time) and elements of random physics (e.g. turbulence) were represented probabilistically. For parameters such as drag coefficient and added-mass, a range of possible values was assigned, and a combination of all these values was used to capture the uncertainty surrounding debris behavior. Random physics, such as turbulence, were modeled using Monte Carlo simulations with the random walk model. The probability of structural damage from debris collisions was calculated by correlating the evaluated debris collision speed with the design collision speed of the structure. For various tsunami conditions, this method was used to establish a relationship between the probability of structural damage due to debris collision P_D and tsunami intensity h, taking uncertainties into account. The obtained h-P_D relationships varied, as illustrated in Figure 7, with tsunami height at a specific position used as a measure of tsunami intensity. The range of uncertainty in these relationships stemmed from the different parameter settings in the debris-tracking simulation. It was anticipated that the probability of structural damage from both tsunami pressure and debris collision force would increase with tsunami intensity. This hypothesis was supported by a linear h-P_D relationship observed in several cases in Kaida et al. (2022), as indicated by a black line in Figure 7. However, a nonlinear h-P_D relationship was also noted under certain tsunami conditions (red line in Figure 7), attributed to the complex flow fields around industrial and urban areas, which house multiple structures (Chida and Mori 2023; Kaida et al. 2022,). The complexity of the flow field, which increases with the tsunami height, may affect debris motion. Specifically, the complex and strongly turbulent flow field, including reflected waves formed in the vicinity of the structure, may have disturbed the collision of debris. It has been suggested that both the collision speed and probability may be reduced by these factors (Kaida et al. 2022). The cause of this problem has not been verified, and further verification is required.

Figure 7. Conceptual diagram showing relationship between damage probability of structures due to debris collision P_D and hazard, expressed as tsunami height h at a given point. Cases in which the damage probability increases linearly with the hazard and an example of the nonlinear relationship between h and P_D are indicated by black and red lines, respectively. The vertical bars indicate the range of uncertainty.

Methods for the probabilistic assessment of debris collision speed exist, yet the development of a probabilistic evaluation method for debris collision encompasses several challenges. When determining the damage probability based on collision force, it is crucial to consider the uncertainties related to the evaluation of collision force. Consequently, a broader range of uncertainty than that associated with the exceedance probability of collision speed might emerge. In the probabilistic evaluation of debris collision force, the true values of collision speed, collision force, and damage probability due to debris collision lie within an uncertainty range, as shown in Figure 7. The accurate evaluation of collision forces, regarded as true values, is important. Whether applying probabilistic or deterministic evaluations in practice, refining evaluation techniques and parameters for the most reliable estimation of collision forces is important. In probabilistic evaluations, leveraging both the best estimated value and the uncertainty in the evaluation results enables risk-informed decisions and designs. Probabilistic methods do not need to be applied to every step of evaluating debris collision force; meaningful probabilistic evaluation results can still be obtained by applying probabilistic methods to specific parts of the process. For example, if the uncertainty range associated with a parameter in the debris collision force equation has been quantified, and this parameter is treated probabilistically, then collision speed might be determined deterministically on the conservative side, but the resulting collision force would carry uncertainty due to the probabilistic treatment of the substituting parameter. Collision probability is an essential measure in probabilistic evaluations, with debris-tracking simulation techniques showing promise for this evaluation. However, simplifying the evaluation by assuming a collision probability of 1.0 allows for a straightforward safety assumption. Beyond integrating probabilistic and deterministic approaches as described, it is essential to continue identifying and quantifying uncertainties, referencing prior research (e.g. AESJ 2019; Kaida and Kihara 2017). Understanding the statistical properties of parameters, such as those used in debris-tracking simulations for evaluating collision speed and in equations for evaluating collision forces (e.g. collision time, debris stiffness), will improve the accuracy of evaluations and reduce uncertainties. Recent large-scale collision experiments aimed at determining the physical properties of small FRP vessels (Toyoda et al. 2022) exemplify efforts to refine parameter values. Further experimental and numerical analysis can provide more precise parameter values in guidelines, similarly, improving the setting of collision speeds.

As Kaida et al. (2022) demonstrated, comprehensively accounting for the uncertainties involved in evaluating debris collision force can significantly increase computational demands. Minimizing computational load is crucial for the practical implementation of probabilistic evaluations. Although Kaida et al. (2022) investigated methods to lessen this computational burden in their methodology that uses debris-tracking simulations (Kaida and Kihara, 2020), their efforts did not obtain satisfactory results. Addressing these issues is critical for enabling the practical use of probabilistic approaches.

Developing a standardized method for evaluating debris collision force would greatly benefit practical applications. To achieve this, it is important to advance the construction of a probabilistic evaluation method that demands a realistic amount of effort. This includes establishing and quantifying various uncertainty factors and ensuring a comprehensive understanding of the parameters required for evaluating debris collision force and speed. Selecting potential methodologies from the array available and reflecting them into guidelines as standard methods requires in-depth discussion among a wide range of academic experts. For both evaluating collision forces and speeds, it is necessary to periodically review the latest research findings and engage in discussions and research aimed at standardization.

This review focused on evaluating debris collision speed and force, detailing a process that includes (i) setting up both the tsunami-generated debris and the object it collides with, (ii) establishing the collision conditions, (iii) configuring the debris collision force, and (iv) evaluating the response of the object impacted (Kaida et al. 2023). While this study did not cover the methodologies used in steps (i) and (iv), a comprehensive examination of these areas would be beneficial, especially considering the issues related to step (iv). For example, in scenarios where the stiffness of the objects collided with is on par with that of the debris, the application of methods reviewed in Section 2 would be unsuitable. In addition, the phenomenon where multiple pieces of debris collide and potentially increase the collision force has been acknowledged, yet there is a lack of research on addressing this effect probabilistically, highlighting an important gap in practical applications. As mentioned in Section 2.1.1, the initial and subsequent behavior of debris is influenced by its buoyancy, a factor that should be incorporated into debris-tracking simulations. The mechanism of debris movement by tsunamis has been studied mainly for floating buoyant debris. Research and technological advancements in debris-tracking simulations have mainly focused on floating buoyant debris. However, there is a pressing need for research on non-buoyant debris objects. Such studies would significantly enhance understanding of debris movement mechanisms and improve the validation of debris-tracking simulation techniques.

5. Conclusions

This paper provides a detailed review of the methods for assessing the collision speed and force of debris generated by tsunamis, alongside the essential physical processes involved in these evaluations, including the state-of-the-art findings. It outlines the effectiveness of these evaluation methods in terms of accuracy, computational effort, and practicality, providing useful insights for practitioners in the field. In addition, it identifies the evaluation methods currently available for use. A critical analysis of existing guidelines reveals a lack of standardization in assessing the force of collisions caused by tsunami debris. Building on this observation, the paper engages in a discussion on improving the evaluation methods for the impact of tsunami-induced debris and the need for developing a standardized method.

The paper identifies critical issues that require attention. First, there is a need for a comprehensive probabilistic framework to evaluate the debris collisions. This framework should be capable of assessing multiple debris collisions simultaneously. By combining deterministic and probabilistic approaches, evaluations can achieve a balance between rationality and necessary conservatism. However, probabilistic evaluations tend to increase both the time and cost of numerical analysis, necessitating the development of a practical and feasible method. Essential to probabilistic evaluations is the systematic organization and quantification of uncertainties. Addressing various parameters, such as drag and inertial force coefficients used in debris-tracking simulations and parameters representing the physical properties of the debris in collision force equations, as previously done, will importantly aid this process. Advancements in understanding should narrow the uncertainty margin in debris collision force evaluations. Furthermore, these improvements will enhance the applicability of deterministic assessments, which are crucial for future research directions.

While the method for evaluating debris collision force remains unstandardized, addressing the issues outlined is crucial for its standardization. For this purpose, it is imperative to engage academic experts in meaningful and in-depth discussions. These experts should have the capability to assess a variety of evaluation technologies, such as those highlighted in this review.

As mentioned, developing a probabilistic evaluation framework, enhancing understanding of parameters to reduce the uncertainty of such evaluations, and standardizing methods for assessing the impact force of tsunami-induced debris collisions stands out as important issues future research. Moreover, exploring the development of evaluation methods for multiple debris objects and verifying the accuracy of simulation techniques for tracking buoyant, non-floating debris are also desirable research directions. Advancements in the evaluation method for debris collision force, achieving results that closely approximate real-world values, will significantly contribute to the rational design of coastal structures and the enhancement of local tsunami resilience.

Acknowledgments

We would like to thank the members of the Tsunami Review Group in the Seismic Design Subcommittee of the Nuclear Standards Board of the JEA for their contributions to the discussion on the prediction for the tsunami debris collision force.

Disclosure statement

No potential conflict of interest was reported by the author(s).