Nonlinear Response of Four-Tier Container Multiple Stacks with Different Lashing Approach Under Dynamic Excitation

Abstract

This study delves into the nonlinear dynamic response of four-tier dual stacks subjected to dynamic excitation and employs internal and external lashing methods to identify causes of container damage and lashing failure. Researchers constructed a scaled model based on Froude scaling laws and used a shaking table to simulate the dynamic excitation. By integrating numerical simulations with model experiments, the study systematically analyzed the displacement and acceleration responses of the container stacks, with particular focus on nonlinear factors such as sliding and collision. The results reveal that exceeding specific critical points in excitation amplitude and frequency leads to the gradual overcoming of friction between adjacent container corner castings, resulting in noticeable relative sliding and collision. Twist lock gaps significantly worsen collision behavior, highlighting their critical impact on the stacking system’s nonlinear collision dynamics. Additionally, under conditions of high-amplitude and high-frequency excitation, external lashing schemes proved more stable and resistant to collisions than internal ones. The study also emphasizes that collisions between adjacent stacks can trigger load redistribution, thereby altering the stack’s load transfer path and impacting the stability of the entire stacking and lashing system.

1. Introduction

With the rapid expansion of global trade, container shipping now accounts for about 90% of non-bulk maritime transport [1]. Every year, lashing failures and stack collapses result in the loss of thousands of containers, leading to economic losses totaling billions of dollars [2]. On ultra-large container ships, stacking heights on decks have surpassed 12 tiers, which significantly increases dynamic accelerations caused by vessel movements. Current lashing system design standards [3,4,5] still rely on static or quasi-static assumptions, failing to consider the nonlinear dynamic effects resulting from ship motion. About 70% of container loss incidents are directly linked to the failure of lashing systems under extreme dynamic loads [6,7,8], highlighting the urgent need to study the coupled response of container stacks and lashing systems when subjected to dynamic excitation.

In recent years, academia has focused extensively on the dynamics of container stacks. Scaled model tests with Froude similarity laws have become a prevalent method for predicting the behavior of prototype structures [9,10,11]. Kirkayak et al. [12] uncovered the nonlinear mechanical properties of twist locks in double-tier stacks through shaking table experiments and developed a mathematical model to study the impact of vertical gaps on dynamic response. Aguiar et al. [2] investigated the relationship between stack structure deformation and twist lock loads under heave excitation by employing a scaled model of a seven-tier container stack. They found that increased stack height amplifies the nonlinear characteristics of dynamic responses. Li and colleagues designed a scaled distortion model of a 20 ft container and established single stack test systems for seven-tier [13] configurations. These setups were used to explore variables such as rolling and pitching excitations, twist lock gaps, different lashing methods, lashing pretension, and stiffness on lashing force, twist lock load, and container stack displacement response. Their research revealed coupling effects between various lashing methods and twist lock gaps. However, most current research predominantly focuses on the lashing system within a single stack, lacking a deep exploration of collision effects among multiple stacks and the dynamic characteristics of external lashing methods.

It is noteworthy that ultra-large container ships are increasingly adopting external and hybrid lashing schemes to better optimize deck space [14]. However, current standards have not systematically evaluated their coupling effects with hull elastic deformation [6]. Li et al. [15] designed a scaled distortion model for the lashing bridge of a 20,000 TEU container ship to study the coupling effects between the transverse bulkhead and the lashing bridge using model tests and numerical simulations. They proposed methods to determine stiffness values for the lashing bridge and components of these large container ships. In laboratory settings, sinusoidal excitations mimicking rolling and pitching motions were applied to perform dynamic experiments and numerical simulations on the lashing bridge and an eleven-tier single container stack [16]. The experimental results revealed significant discrepancies between the loads and stack deformations under dynamic excitation compared to the calculated values in existing standards. These discrepancies become even more pronounced when adjacent high-tier stacks collide due to dynamic excitation. Additionally, the impact mechanisms of nonlinear factors such as twist lock gaps and friction between container corner fittings on system dynamic stability are still unclear [17,18], which limits advancements in optimized lashing designs.

To tackle these issues, this study concentrates on a four-tier dual stack of 20 ft ISO containers, and constructs a 1:10 scaled model test system (Figure 1) based on Froude similarity laws. The model incorporates both internal and external lashing methods, with dynamic excitations of controllable amplitude and frequency applied via a shaking table. During the experiment, a twist lock gap quantification module (gap range 0–1.5 mm) and a series of sensors were implemented to monitor stack displacement and acceleration responses in real time. Additionally, a nonlinear finite element model was used to simulate stack collisions, contact, and friction effects. The research focuses include: (1) uncovering the influence patterns of multiple stack collisions on container deformation, lashing forces, and twist lock loads; (2) comparing the structural response differences between internal and external lashing systems under dynamic excitation; and (3) exploring the contributions of stack collision, twist lock gaps, and other factors to the overall system’s nonlinear dynamic response. These findings can provide theoretical support for revising dynamic design standards for lashing systems and optimizing lashing schemes, offering significant value in reducing the risk of cargo loss during maritime transport.

2. Experimental and Numerical Simulation Details

2.1. 20-ft Container Scaled Model

This study employed Froude scaling laws [2] to create a test model of a four-tier dual stack of 20-ft ISO freight containers and their securing components. This approach ensures similarity in geometric dimensions, mass, stiffness, kinematic, and dynamic characteristics. The scaled model aims to simulate the nonlinear dynamic response caused by collisions between adjacent container stacks during transport, thus investigating potential causes of container damage and securing failure. The model’s length, width, height, and mass are scaled at a 1:10 ratio (similarity parameter: λ = 10). Standard carbon steel Q235 was selected for its consistency in elastic modulus, Poisson’s ratio, and density with actual container materials, ensuring that the model accurately reflects the prototype’s mechanical behavior.

Loads generated by stack collisions are transmitted through the lashing system to the interior of the stacks and the deck structure. Therefore, the scaled model design follows two dimensionless numbers: the ratio of gravitational to inertial forces and the ratio of elastic to inertial forces [19]. Consequently, it becomes essential to omit parameters that have minimal impact on structural mechanical behavior to conduct effective research. For a comprehensive overview of the structural features and material parameters of 20-ft standard containers, refer to the work of Zha and Zuo [20]. The design process of the container scaled model adheres to the scaling laws summarized in

Table 1. Similarity parameters (λ = 10). Data from [16].

20-ft Container Parameter		Full-Scale	Target	Scaled Model
External dimensions (m)	Length	6.058/λ	0.606	0.606
	Width	2.438/λ	0.244	0.244
	Height	2.591/λ	0.259	0.259
Mass (kg)		2330/λ3	2.330	2.450
Moment of Inertia	Ixx (kg/m2)	3765/λ5	0.038	0.040
	Iyy (kg/m2)	11,830/λ5	0.118	0.120
	Izz (kg/m2)	11,486/λ5	0.115	0.135
Transversal stiffness (MN/m)	Closed	58/λ2	0.580	0.600
	Open	3.400/λ2	0.034	0.040
Lateral stiffness (MN/m)	Closed	86.910/λ2	0.870	0.850
	Open	61.090/λ2	0.610	0.630
Longitudinal stiffness (MN/m)		58/λ2	0.580	0.630
Torsional stiffness (kNm/rad)		3200/λ4	0.320	0.350
1st mode frequency (Hz)		20λ1/2	63.250	65.380

. For more detailed information on the mechanical properties of containers, scaled model design, and numerical simulation validation, consult the literature by Li et al. [15].

2.2. Experimental Details

The experiment utilized a high-precision electro-hydraulic servo horizontal shaking table capable of applying sinusoidal excitation in the horizontal direction. Amplitude and frequency served as input variables to primarily simulate the unidirectional inertial forces caused by collisions of container stacks on ships. Although the actual ship motion involves multi-axial degrees of freedom, the lateral inertial force is the dominant factor leading to the racking and sliding of container stacks. Therefore, this experiment employs unidirectional horizontal excitation to simulate the critical lateral acceleration component acting on the stack. The dynamic response of four-tier dual container stacks under different lashing methods (internal and external) was simulated, with a particular focus on the nonlinear dynamic responses resulting from collisions between adjacent stacks. Experimental parameters included excitation amplitudes of 100 mm, 150 mm, and 200 mm, frequencies of 0.6 Hz, 0.7 Hz, and 0.8 Hz, twist lock gaps of 0.0 mm and 1.5 mm, and counterweight loads of the container stacks. The excitation amplitudes and frequencies were derived using Froude scaling laws to represent realistic ship roll periods.

The overall structure of the scaled model is illustrated in

Table 2. Main geometric parameters of container scale model. Data from [16].

	Color	Cross Section	Dimension	Thickness
	Magenta	Square	20 × 20 mm	1.5 mm
	Red	Square	20 × 20 mm	1.5 mm
	Cornflower	Pipe	5 mm (radius)	1 mm
	Green	Pipe	5 mm	2 mm
	Yellow	Pipe	9 mm (radius)	2 mm
	Blue	L section	20 × 20 mm	2 mm
	Smoke Gray	Plate	-	2 mm
	Orange	Square	30 × 30 mm	2 mm

. All dynamic tests were conducted via a hull motion simulation device at the Structural Mechanics Laboratory of Shanghai Jiao Tong University, China. The experimental system consisted of four-tier dual-row container stacks, a lashing system, and a multi-type sensor network. Based on the cargo securing manual of a 3400 TEU container ship, experimental parameters such as lashing methods, counterweight arrangements, and stack spacing were set. Adjacent containers in different tiers were bolted together through corner castings, and lashings used 3 mm diameter steel cables (with lashing points located at the corners of the second-tier containers) to simulate actual maritime securing methods and stiffness (Figure 2). For more detailed information about parameter design, refer to references [15,16]. To simulate full-load conditions, stack loading was achieved by placing steel weights inside each tier’s containers, with the first to fourth tiers containing weights of 24 kg, 24 kg, 22 kg, and 22 kg, respectively.

To ensure the accuracy of experimental data, high-precision force sensors, accelerometers, and displacement sensors were employed. Force sensors were installed at key locations on the lashing components to measure changes in lashing forces, maintaining an initial pretension force of 5 kg for each trial. Displacement sensors were placed at the upper casting locations of the top layer’s open end (door end) and closed end (wall end) to record the lateral dynamic response of the container stacks. Accelerometers were positioned at the upper casting locations on the open and closed ends of the third and fourth layers. This multidimensional sensor arrangement enabled a comprehensive capture of the system’s dynamic behavior, providing detailed data support for subsequent analysis. No fewer than five repeated tests were conducted for each working condition to ensure the statistical reliability of the results. Data processing methods included time history curve analysis and comparisons of maximum and root mean square (RMS) values to comprehensively evaluate the structure’s dynamic response. RMS value is defined as the root mean square value of the signal during the steady-state response phase (i.e., within the time window from 10 s after the start of excitation to 5 s before its end).

2.3. Numerical Model Construction

This article uses the commercial finite element software ABAQUS 6.14 to analyze the dynamic response of a four-tier double-stack system. Figure 3 shows the numerical simulation model of the system using both external and internal lashing systems. Figure 3a shows the complete external lashing model, Figure 3b shows the complete internal lashing model. In the simulation models, Q235 is selected as the material, with an elastic modulus of 210 GPa and a Poisson’s ratio of 0.3. The container is primarily meshed using quadrilateral shell elements, with a mesh size of 5 mm, and the total number of elements in a single container is 6294. In these models, the test bench is simplified as a rigid body, and the bottom container is fixedly connected to the test bench through its corner castings. The lashing positions for the four-tier double-stack system are at the lower corner castings of the second-tier containers. The force between the lashing component and the container is only transmitted as tension. Thus, nonlinear connectors are used in the numerical simulation model. The nonlinear behavior of the lashing component is shown in Figure 4, similar as Li et al. [13].

The containers are interconnected through twist locks, and due to the gap between the twist locks and the containers, the force between them is nonlinear. Considering only the vertical clearance between the twist lock and the container, a nonlinear translator connector is used to simulate their interaction, with a vertical clearance set to 1.5 mm. This mechanical behavior is shown in Figure 5, proposed by Kirkayak and Suzuki [11].

Due to the use of bolt-simulated twist locks in the experimental model, contact forces arise between the corners of adjacent containers when they approach. Therefore, the numerical simulation model includes face-to-face contact between adjacent corners with a friction coefficient of 0.25, same as references [2,13]. Under the excitation of the test bench, the double-stack container system may experience collisions between containers. Thus, this model also defines face-to-face contact with a friction coefficient of 0.25 for the locations where collisions may occur. The cargo in containers is simulated using mass points, positioned at 45% of the container height. The mass points are coupled to the container using coupling constraints, as shown in Figure 3.

3. Numerical and Test Results

3.1. Single Stack with Different Lashing Methods

Classification societies such as CCS [3], BV [4], and ABS [5] emphasize the importance of monitoring the maximum displacement response of container stacks as a key measure for evaluating transport safety. When subjected to dynamic forces, the displacement observed at the nodes of the container framework highlights the load distribution within both the frame structure and the lashing system. Figure 6 illustrates the results from an experiment measuring the displacement response of a four-tier single container stack under frequencies of 0.7 Hz and 0.8 Hz. The data clearly shows that the open end of the structure experiences a greater response than the closed end. Furthermore, gaps in the twist lock significantly increase the displacement response of the stack structure, a conclusion supported by previous studies conducted by Li et al. [15]. During the experiment, several noteworthy phenomena were detected. With a twist lock gap of 0 mm, the greatest displacement response occurred at the container’s open end, which aligned with the sensor placed at the upper corner casting of this end. In contrast, when the twist lock gap increased to 1.5 mm, the maximum displacement response shifted to the closed end of the stack structure, as indicated by the sensor at the upper corner casting of the closed end. This pattern was consistently observed under various conditions. The main reason may relate to differences in stiffness between the open end and closed end of the container. Nonetheless, this explanation does not seem comprehensive enough to fully account for the phenomenon observed.

To better understand how different lashing methods influence the displacement response of stack structures, Figure 7 contrasts the displacement responses at the open end under full-load conditions. The findings show that with a full-load and a twist lock gap of 0 mm, the displacement values at the open end are consistently higher than those at the closed end. An analysis of the data reveals that the application of external lashing scheme results in lower maximum and effective structural displacement responses compared to an internal lashing scheme. Therefore, the external lashing method is more advantageous than the internal approach, particularly for securing high-tier container stacks.

During the experiments, another intriguing phenomenon was observed when twist lock gaps were present. In dynamic tests with an excitation amplitude of 100 mm, noticeable slippage started at the corner castings of the top-tier container as the frequency increased to 0.8 Hz. This indicates that relative motion overcame the friction between the corners of the upper and lower containers. When the excitation amplitude reached 150 mm, significant slippage was accompanied by vertical separation at the twist lock connections between the first and second tiers once the frequency hit 0.8 Hz. Throughout these tests, slippage occurred more frequently at the closed end than at the open end. A comparison with the data from Figure 8 makes clear that the displacement values due to slippage and separation exceed those caused by structural loading alone. Figure 8 presents time history curves derived from numerical simulations and model tests, which demonstrates that systems with gap configurations show more complex time history curves than those with a 0 mm gap. For systems with twist lock gaps, the total displacement response of the stack structure comprises not just structural deformation under load but also includes relative slippage and separation at the corner castings. This phenomenon is connected to the friction at adjacent corners, as well as the inconsistencies in stiffness between the container’s open and closed ends.

To further corroborate these observations, data from accelerometers were analyzed, as detailed in

Table 3. Maximum acceleration of single stack system (m/s²).

Excitation Amplitude and Frequency	External Lashing, Fourth Tier (Gap: 0 mm)		Internal Lashing, Fourth Tier (Gap: 0 mm)		Internal Lashing, Fourth Tier (Gap: 1.5 mm)		Internal Lashing, Third Tier (Gap: 1.5 mm)
	Closed End	Open End	Closed End	Open End	Closed End	Open End	Closed End	Open End
100 mm, 0.6 Hz	1.4211	1.4610	1.4262	1.4769	1.4176	1.4662	1.4045	1.4327
100 mm, 0.7 Hz	1.9265	1.9691	1.9361	1.9686	1.3398	1.3570	1.3264	1.3536
100 mm, 0.8 Hz	2.54039	2.6026	2.5321	2.6146	3.8743	3.5217	3.4957	3.4516
150 mm, 0.6 Hz	2.1843	2.2389	2.1048	2.1506	2.9925	2.814	2.5816	2.6837
150 mm, 0.7 Hz	2.8887	2.9405	2.9357	2.9803	4.7771	4.0001	4.1976	4.0504
150 mm, 0.8 Hz	3.8780	3.9404	3.8244	3.8959	6.4639	5.5190	5.8092	5.3449
200 mm, 0.6 Hz	2.8096	2.8568	2.8550	2.8939	4.5598	4.3269	3.9496	4.1745
200 mm, 0.7 Hz	3.9952	4.0560	3.9534	4.0062	6.5702	5.3012	5.4634	5.5781
200 mm, 0.8 Hz	5.0722	5.1666	5.0530	5.1360	7.6363	6.4837	7.2282	7.1176

. In systems without twist lock gaps, there is no relative slippage between adjacent containers. The acceleration response at the open end is slightly higher than at the closed end, with minimal differences observed between internal and external lashing schemes. However, the presence of twist lock gaps significantly increases acceleration responses. When the excitation amplitude and frequency are set at 200 mm and 0.8 Hz, respectively, the acceleration response at the closed and open ends of the internal lashing system rises from 5.0530 m/s² and 5.1360 m/s² to 7.6363 m/s² and 6.4837 m/s². These changes represent increases of 51.12% and 26.23%, respectively. Furthermore, as the excitation amplitude and frequency increase, the discrepancy in acceleration responses between the third and fourth tiers shows a clear upward trend. This can be attributed to the significant rise in acceleration responses due to relative slippage caused by horizontal gaps and to the separation of adjacent corner castings due to vertical gaps in the twist locks. These factors collectively exacerbate the complexity of the system’s response.

3.2. Collision of Multiple Stacks with Different Lashing Methods

Under fully loaded dual stack test conditions with a twist lock gap of 0 mm, Figure 9 presents the effective value of the acceleration response for both single and dual stacks. The acceleration response data for the single stack closely matches that of the dual stack system, indicating no collision occurred between the dual stacks. This suggests that the relative deformation of the two stack structures did not exceed the inter-stack spacing. In both internal and external lashing systems, the acceleration response at the open end is slightly higher than at the closed end, though the difference remains minimal. Compared to Figure 6, despite the consistent acceleration responses, the displacement response under the internal lashing scheme is higher than that under the external lashing scheme. This indicates that the external lashing scheme can help reduce the displacement response of the top-tier container in systems without twist lock gaps.

For conditions with a twist lock gap of 1.5 mm and an excitation amplitude of 100 mm at a frequency of 0.6 Hz, there is no significant relative slippage between adjacent containers. However, as the frequency increases to 0.7 Hz, slippage begins to emerge within the stack. By 0.8 Hz, there is noticeable relative slippage within the stack, though no collision between adjacent stacks occurs. In the internal lashing system, when the excitation amplitude increases to 150 mm, collisions due to slippage first occur at the bottom corner casting of the fourth-tier container and subsequently at the top corner of the same tier. Data from

Table 4. Acceleration maximum value (double stack, unit: m/s²).

Excitation Amplitude and Frequency	Internal Lashing, Gap 1.5 mm				External Lashing, Gap 1.5 mm
	Third Tier		Fourth Tier		Third Tier		Fourth Tier
	Closed End	Open End	Closed End	Open End	Closed End	Open End	Closed End	Open End
100 mm, 0.6 Hz	1.4301	1.4887	1.4282	1.4998	1.6616	1.8733	1.7046	1.9231
100 mm, 0.7 Hz	2.3127	2.3126	2.3231	2.3392	2.0269	2.1836	2.0399	2.2109
100 mm, 0.8 Hz	4.7911	6.0964	5.4506	5.4364	5.6212	5.7380	5.8868	6.1477
150 mm, 0.6 Hz	2.2844	2.4251	3.8034	3.7218	4.0100	3.8944	4.0657	4.2533
150 mm, 0.7 Hz	6.1715	6.4118	6.0854	6.3984	6.4994	6.3994	6.4971	7.0544
150 mm, 0.8 Hz	7.3342	8.9275	7.2022	9.7729	7.9060	8.0229	8.6675	9.5740
200 mm, 0.6 Hz	6.7107	7.2807	7.3049	7.4613	6.4490	5.4730	7.6771	7.8434
200 mm, 0.7 Hz	7.6408	8.5872	7.3959	8.4723	7.6127	7.8730	7.6606	8.7150
200 mm, 0.8 Hz	8.5774	10.3178	9.6922	10.7695	9.5166	10.8477	10.2583	11.2783

show that the differences between the open and closed ends are more pronounced at excitation frequencies of 0.6 Hz and 0.8 Hz than at 0.7 Hz.

Compared to the internal lashing system, the external lashing system exhibits smaller differences in acceleration responses between the third and fourth tiers and experiences less relative slippage. When the excitation amplitude increases to 200 mm, noticeable slippage and collisions occur at the corner castings of each tier under both lashing systems. These collisions result in a significant increase in displacement response, indicating that the system is subjected to greater internal loads. Observations from the experimental process reveal that as the excitation frequency changes, the method of load transfer within the system also evolves. Initially, during low-intensity excitation, no relative slippage occurs within the system due to the frictional force at the corner castings. As the excitation intensity rises, local relative slippage begins within the system and gradually extends until slippage occurs between every corner casting. This demonstrates that with increasing excitation levels, the physical behavior and load transfer paths within the entire system undergo a significant transformation.

Figure 10 and Figure 11 illustrate the displacement responses of stack structures using internal and external lashing methods under various amplitudes and excitation frequencies. In the dual-stack test system with internal lashing, the displacement response at the open end is greater than that at the closed end. Conversely, in the dual-stack system employing external lashing, the opposite trend occurs. During the experiment, it was noted that for the internal lashing system, when the excitation amplitude reaches 150 mm, noticeable collisions occur at the top corners of the third and fourth-tier containers. At an excitation amplitude of 200 mm, significant collisions take place across all top-tier containers, with a consistent collision pattern at both ends. In the external lashing system, significant collisions at the top corners of the third and fourth-tier containers are also observed at excitation amplitudes of 150 mm and 200 mm. Notably, all collisions happen at the open end, while the closed end remains free from collisions. Within the external lashing scheme, the difference in displacement response between the open and closed ends is minimal, and no significant collisions occur at the top end of the second-tier container. It is clear that the displacement response values for internal lashing are higher than those for external lashing. Specifically, under the conditions of 200 mm amplitude and 0.8 Hz frequency, the maximum displacements at the closed and open ends for the internal lashing scheme are 20.7025 mm and 24.0756 mm, respectively. In contrast, for the external lashing scheme, they are 18.3376 mm and 18.1286 mm. These results demonstrate that the external lashing scheme effectively reduces the displacement at the open end by approximately 24.7% compared to the internal lashing scheme.

3.3. Analysis of Collision Effects Between Adjacent Stacks

This analysis reveals that the internal lashing scheme results in more pronounced displacement responses in stack structures. Under conditions with 150 mm and 200 mm excitation amplitudes, depicted in Figure 6d and Figure 10a, the displacement responses for both single and dual stacks are particularly noticeable. Figure 12 illustrates the comparative differences in displacement responses between the closed and open ends. Under a 150 mm and 0.6 Hz excitation, the displacement differences at the top containers of both single and dual stacks are minimal. However, as dynamic excitation intensifies, significant differences in structural displacement responses emerge between the closed and open ends, reaching up to 40.26% and 86.61%, respectively.

A comparison of Table 3 and Table 4 reveals that the presence of twist lock gaps significantly increases acceleration responses. For instance, in the internal lashing system, when compared to a single stack, the acceleration responses at the closed and open ends rise from 6.4639 m/s² and 5.5190 m/s² to 7.2022 m/s² and 9.7729 m/s², respectively, under a driving excitation amplitude and frequency of 150 mm and 0.8 Hz. This represents increases of 11.42% and 77.08%, as depicted in Figure 13. When the driving excitation amplitude and frequency increase to 200 mm and 0.8 Hz, the acceleration responses further rise from 7.6363 m/s² and 6.4837 m/s² to 9.6922 m/s² and 10.7695 m/s², marking increases of 26.94% and 66.10%. Additionally, acceleration responses for the third-tier container within the stack were analyzed, as shown in Figure 13b. At a driving excitation amplitude and frequency of 100 mm and 0.8 Hz, the acceleration responses at the closed and open ends increase from 1.3264 m/s² and 1.3536 m/s² to 2.3127 m/s² and 2.3126 m/s², with growth rates of 74.36% and 70.85%. When the driving excitation amplitude and frequency are 200 mm and 0.8 Hz, the closed and open end acceleration responses increase from 7.2282 m/s² and 7.1176 m/s² to 8.5774 m/s² and 10.3178 m/s², with growth rates of 18.67% and 44.96%. It is evident that stack collisions have a more significant impact on the open end, indicating that in the presence of twist lock gaps, the stack structure becomes more sensitive to dynamic excitation, particularly at the open end.

4. Discussion

In the context of booming global trade, container shipping volumes continue to rise. However, losses due to lashing failures and stack collapses during maritime transport pose significant economic and safety risks to the industry. This study focuses on the nonlinear dynamic response of four-tier dual-stacked containers under dynamic excitation with various lashing methods, providing critical support for optimizing lashing schemes and reducing the risk of cargo loss at sea.

While in principle, the mechanical behavior of adjacent dual stacks should remain consistent, factors such as friction, slippage, and minor structural differences can lead to changes in the relative positions of containers within the stacks during dynamic excitation, resulting in collisions. Specifically, under minor dynamic excitation, internal friction within the container stack can maintain relative stability, preventing slippage. However, as the excitation intensity increases and friction reaches its limit, slippage between corner castings begins, which disrupts the initial geometric configuration of the stack and alters the positions of the containers. Moreover, due to errors in manufacturing and installation, along with slight variations in material properties, container stacks are not entirely symmetrical or consistent. These subtle structural differences become magnified under dynamic excitation, further exacerbating changes in container positions and increasing the likelihood of collisions.

Under high-intensity dynamic excitation, the system’s nonlinear behaviors dominate the structural response, such as lashing relaxation-tension switching, friction between corner castings, twist lock gaps, and stack collisions. These nonlinear behaviors cause changes in internal load transfer, making the structure’s dynamic characteristics complex and variable. Traditional linear dynamic analysis methods cannot accurately capture the impact of these nonlinear behaviors on structural responses. As a result, conventional assessment methods used in current standards may underestimate peak loads, highlighting the need for nonlinear dynamic models to improve prediction accuracy. Nonlinear behaviors in the system can lead to localized deformation and damage of containers, potentially causing load redistribution that results in abrupt changes in the forces on lashings. This increases the risk of lashing system failures.

This research aims to effectively predict the nonlinear interactions of container stacks, twist lock devices, and securing arrangements through nonlinear numerical simulation methods, thereby providing a robust tool for the accurate assessment of the response of container stacks under dynamic excitation. However, the lashing system and container framework constitute a flexible coupled system, and the setting of different numerical simulation parameters can significantly influence the overall simulation results. For instance, stack collision simulation is not an easily controllable factor. Collisions under various excitation conditions may occur due to factors such as overcoming friction between adjacent corner castings, separation of adjacent corners leading to stack tilt, or a combination of multiple factors. This study also sought to use a unified nonlinear mathematical model to describe collisions between stacks, but this proved challenging. Additionally, these factors can vary with differences in internal mass distribution, stack height, and lashing methods.

Under dynamic excitation, noticeable differences emerge in the displacement response of stack structures and the acceleration response distribution of adjacent layers. These differences indicate that contact collisions between adjacent stacks lead to load redistribution. When collisions occur, significant impact forces are generated at the collision points and are transmitted through the container structure to the lashing system, causing abrupt changes in the force conditions on lashing rods. Such sudden load changes can result in uneven stress distribution within the lashing rods, leading to localized stress concentrations and increased risk of lashing rod failure. Furthermore, this phenomenon alters the internal stress transmission path within the stack, transferring loads originally borne by certain containers to others, thereby affecting the entire stack’s stability. Thus, understanding this dynamic coupling mechanism is crucial for accurately assessing the performance and reliability of the lashing system.

Differences in structural response between internal and external lashing systems are mainly observed in several areas. First, the external lashing system secures the lifting side of the container directly rather than the compression side, thereby reducing stress on both sides. Due to smaller deformation at the closed end, the container structure must bear more load, which may cause the lower-tier container corner to bear higher compressive loads. Second, when using an external lashing system, peak lashing force becomes a critical concern. The lashing force may be very high due to the presence of vertical gaps, depending on the locking design used. The internal lashing system can provide more favorable results because the lifting process does not overload the lashing rods. Moreover, comparative studies underscore the importance of the interaction between container structure, twist lock gaps, and securing equipment. Minor inconsistencies can create significant differences in predicting stack responses. Therefore, it is necessary to understand the interaction among these factors, which is crucial for accurately evaluating the performance of lashing systems.

During testing, it was found that the lashing locks sometimes disengaged. This mainly occurred because the pretension in the lashing system could not be maintained when subjected to dynamic forces, leading to loosening of the locks. If the lashing be-comes too loose, it will fail, but if it is overtightened, it might place excessive stress on the stack structure. Gaps can also lead to friction and collisions at the container corner castings, potentially causing fatigue and damage to the lashing components. Therefore, it is critical to inspect lashing components before severe weather, with special attention given to any loosening of turnbuckles caused by vibration. Improved monitoring of pretension and the use of automatic tensioning devices can lower the risk of lashing failure, an approach that is particularly effective in long-term vibrational environments. By continuously tracking the lashing system’s pretension and adjusting the tension as necessary, the system can be kept in optimal working order. In actual maritime transport, the design of hatch covers is affected by ship bending and flexing, while container movement and temperature variations can affect lashing tension. This makes regular inspections crucial to enhance the reliability and safety of the lashing system.

5. Conclusions

This study conducted a comprehensive experimental investigation of multi-stack systems, uncovering the impact of collisions on container deformation and acceleration distribution. It also examined how structural responses differ between internal and external lashing systems when exposed to dynamic forces and explored the influence of factors such as stack collisions and twist lock gaps on the system’s nonlinear dynamic behavior. The findings of this research are highly valuable for improving the economic efficiency and safety of ship operations. The main conclusions are as follows:

(1)

Collisions between adjacent stacks under dynamic excitation are significant and affected by various factors. The study reveals that when the amplitude and frequency of excitation reach certain thresholds, such as a 150 mm amplitude and 0.7 Hz or higher, friction between corner castings in both internal and external lashing systems is overcome. This results in considerable relative slippage and collisions. Twist lock gaps further intensify these collision behaviors, highlighting their crucial role in influencing the nonlinear collision dynamics of stack systems.

(2)

Collisions lead to load redistribution across container stacks and lashing systems, raising the risk of lashing system failure. They change the stress transmission paths within stacks, affect the overall stability, and lead to significant changes in the distribution of displacement and acceleration responses. This underscores the importance of considering dynamic load changes caused by collisions when assessing the structural safety and performance of lashing systems and containers.

(3)

Internal and external lashing systems demonstrate notable differences in collision behavior and structural response. In internal lashing systems, collisions frequently occur at the top corners of third and fourth-tier containers with significant displacement response differences between the open and closed ends. Conversely, external lashing systems mainly experience collisions at the open end, showing minimal differences in displacement responses between ends. By fixing the lifting side of the container, external lashing systems reduce stress on both sides and improve the stability and anti-collision performance, providing a valuable reference for optimizing lashing strategies.

(4)

Collisions cause abrupt changes in lashing rod forces and load redistribution, leading to localized stress concentrations and increased probabilities of failure. Dynamic excitations can also loosen lashing locks, thus diminishing system reliability. Collisions exacerbate fatigue and damage to lashing components and emphasize the necessity for diligent inspection and maintenance under dynamic conditions in maritime transport to boost system reliability and structural safety.

Although this study has produced some notable results, there are limitations that need to be considered. One issue is the experimental model, which was constructed based on Froude scaling laws. While efforts were made to ensure similarity in geometric dimensions, mass, and stiffness, it is challenging to achieve an exact match across all parameters. This may impact the accuracy of the experimental results. Another limitation is the focus on only two lashing methods: internal and external lashing. Other potential lashing schemes and more complex real-world conditions, such as multi-directional excitations and different types of containers, have not been thoroughly examined. Future research could broaden its scope to provide a more comprehensive understanding and prediction of the dynamic responses of container stacks under various actual working conditions. Additionally, while displacement and acceleration responses can shed light on the complexity of load transfer within the system, the specific changes in lashing and bottom lock loads require deeper investigation. These factors are directly linked to lashing failures and stack load-bearing capacity. By integrating advanced monitoring technologies and data analysis methods, it is possible to monitor and assess the dynamic responses and safety status of container stacks in real-time during actual transport. Such measures will offer stronger support for developing navigation safety strategies.