Resistance of Plane Lock Gates Subjected to Ship Impact

Abstract

This paper presents the analysis of lock gates submitted to ship impacts. This problem is generally approached in two different ways: firstly, by the use of an equivalent static method, in which the impact is modeled by a quasi-static force, or by the use of dynamic numerical simulations, during which the progress of the ship and the temporal evolution of the impact force is taken into account. The second approach requires extensive calculation and modeling efforts, which are generally prohibitive in the early design stage.

A simplified analytical method is presented to evaluate the resistance of such structures when impacted by a ship. The principle is based in the super-element method, firstly evaluating the resistance of the lock gate in local deformation mode, assuming crushing only of certain structural elements in a limited zone, located in the close vicinity of the impact. Secondly, the entire resistance of the lock gate is calculated, considering the global deformation mode, assuming a bending of the entire structure.

The scientific objective is to extend this approach to lock gates which cannot currently be treated with this method, in particular those where the layout of the stiffening elements can be highly irregular. This implies to study thoroughly the global deformation modes of the structure and the plastic mechanisms involved in the energy dissipation, in order to predict a correct displacement field and resistance force.

1 Introduction

The design of lock gates requires to account for accidental limit states, in which ship impacts shall be considered. Such analyses are not always straight forward. Dynamic numerical simulations in which the temporal evolution of the impact force is taken into account, require extensive calculation and modeling efforts, which are generally prohibitive in the early design stage. The objective of this research is to develop a calculation tool based on the super-element method to circumvent these difficulties and allow the consideration of ship collisions in the early design phase of lock gates.

The analytical approach of the super-element method assumes that the failure modes of the gate subjected to the impact are known. But it is obviously difficult to imagine in a generic and exhaustive way all the failure modes which can cover all conceivable lock gates.

In order to identify the failure modes of the plane lock gates, it appeared necessary to quickly carry out finite element analyzes using the LS-DYNA software, and compare the results of the analytical method based on the use of super-elements. Such preliminary simulations allowed to model the lock gates in a simplified manner, with only vertical and horizontal frames and girders (minimizing the structural details in some parts, such as the openings in the horizontal girders, who don’t contribute to the collision resistance). These simplifications can reduce the number of elements, while reducing significantly the computational time.

In addition, the stresses caused by a ship impact on a lock gate depend strongly on the type of ship (more precisely the shape of the bow). It was observed that the most dangerous situation (in terms of deformation of the gate) appears during an impact with a raked bow (in contrast with a collision with a barge). Indeed, in this case, it was observed that the plastic deformations of the gate remain mainly concentrated around the point of impact, the rest of the gate being little mobilized. This reflects a high stress on only a few structural elements, so that the impact resistance will be relatively low. This is why in this article only the cases of collision with a raked bow are highlighted.

This paper presents an analytical method to evaluate the resistance of plane lock gates impacted by a ship. Such approach is an approximation, as the goal is not to replace finite element analyses, but to have a complementary tool that could be used at the pre-design stage. At this point several collision cases can be considered in order to find an optimal configuration of the gate or to identify the most damaging scenario. This latter to be further analyzed in detail with a finite element simulation for example.

Some studies have already been presented in this matter, as for instance Le Sourne (2007), who integrated developments to evaluate the crushing resistance of ship components in a program to evaluate the resistance of two colliding ships. On the other hand, Buldgen et al. (2015, 2013 and 2012) achieved the developments for analysis of plane and miter gates impacted by a raked bow and also by a barge. Other authors have advanced in the developments of super-elements to analyze crashworthiness of other type of structures, such as offshore wind turbines, namely Pire (2018) for jacket structures, Echeverry (2021) for a spar buoy floating wind turbine and Marquez et al. (2021) for a concrete floating wind turbine.

2 Description of the Problem

The collision scenario considered in this research is depicted in Fig. 1, where the ship advances with an initial velocity V₀ in the X direction of the global coordinate system. For simplicity of the model, only the bow is modelled but the entire mass of the ship is taken into consideration in the energy balance. The transversal girders are simply supported at each side of the gate, therefore it is free to rotate once the collision happens. It is assumed that the ship is colliding the gate from the upstream, impacting directly the plate of the downstream gate of the lock.

Fig. 1.

Collision scenario, FEM model

The general principles of the problem and methodologies used in this research are presented more in detail in the subsequent sections. The dimensions of the gate and each of the stiffening members are not presented for confidential reasons.

2.1 General Principles

As the ship is characterized by its velocity V₀ and mass (including its added mass M_∞ in the direction of collision) M₀ = M_ship + M_∞. The initial kinetic energy to be dissipated by the collision is assumed to be M₀V₀²/2.

The gate is constituted of steel, therefore all the simulations and analytical calculations in this paper deal with this material, represented by a bilinear elastoplastic material law. The elastic phase is characterized by a Young modulus E of 210 GPa and a yield stress σ₀ of 255 MPa. A linear strain hardening is considered by a tangent modulus obtained as E_T = 0,5%E.

For the type of lock gate analyzed, it is assumed that a ship of mass 4500 tons is sufficiently large to induce important damage to the structure. The ship speed is then chosen according to the energy limits to study. For instance, a collision in the order of 2 m/s for such a vessel can, a priori, induce high deformations because the energy to be dissipated is 9 MJ. Having a relation between the collision resistance P_t and the ship penetration δ, the energy dissipated by the gate can then be calculated by simple integration.

The theoretical basis for the methodology used in this research is the so-called upper-bound theorem, which states that a system of applied loads on a structure will cause collapse of it, if the work rate of the system of applied loads equal to the corresponding internal energy dissipation rate. In the case of lock gate crashworthiness, it is then obvious that the external energy rate ${\dot{{\varvec{E}}}}_{{\varvec{e}}{\varvec{x}}{\varvec{t}}}$ is produced by the force P_t applied by the ship on the gate. This way, we can obtain:

$${\dot{E}}_{ext}={P}_{t}{V}_{0}={P}_{t}\dot{\delta }$$

(1)

If we neglect the dynamic effects in the structure and assuming an infinitely rigid ship, all this external energy is to be dissipated by deformation of the gate, this is ${\dot{{\varvec{E}}}}_{{\varvec{e}}{\varvec{x}}{\varvec{t}}}={\dot{{\varvec{E}}}}_{{\varvec{i}}{\varvec{n}}{\varvec{t}}}$. By integration over the total volume of the structure, it is possible to obtain a relation between the stress and strain rate tensors. Finally, using the Green-Lagrange tensor within the upper-bound theorem, it is possible to find a link between the deformation of the gate and the ship penetration δ in a kinematically admissible displacement field. More details in this theoretical approach are described by Buldgen et al. (2012).

2.2 Super-Element Method

The integration of the force P_t over the entire volume of the gate is quite a laborious task to be performed analytically, this is why it is assumed that the structure can be split into super-elements. Pre-existing analytical models, i.e. Buldgen (2014) and Lützen et al. (2000), are used as a basis for analyzing plane lock gates impacted by a ship. In order to obtain an analytical expression of the local resistance, the gate structure is modeled with a limited number of nodes and some large structural components (girders and frames).

Three types of super-elements (namely SE1, SE2 and SE3) are defined to form the structure of the gate and allow to evaluate its local resistance. These three components (SE1, SE2, and SE3) are shown in Fig. 2. Each super-element achieves a different failure mode, and the combination of all failure modes defines the total strength of the structure. These failure modes may include bending or buckling mechanisms after local deformation by crushing of the bow.

When the structure is struck by a perfectly rigid ship, the ship continues to move forward until the total initial kinetic energy has been fully dissipated (either as internal energy or kinetic energy). At the start of the collision, for low values of penetration δ, the damage caused to the structure remains mainly localized in a rather restricted zone confined around the initial point of contact. This region is subject to significant plastic deformations which can sometimes lead to the rupture of certain structural elements. Simultaneously, an elastoplastic global bending movement of the whole structure is superimposed on this localized indentation and is responsible for quite small out-of-plane displacements affecting the whole structure. In such a case, the structure is said to be resistant by a local mode of deformation.

Fig. 2.

Super-elements developed for studying lock gates (Buldgen 2014)

On the other hand, the impact can lead to large out-of-plane displacements affecting the entire structure. In this situation, a generalized rigid-plastic mechanism develops over the whole grid and this second process is called global deformation mode.

In order to analytically derive the internal energy E_int, the structure is first assumed to resist through a local deformation mode. Nevertheless, with increasing values of the ship displacement δ, it becomes more and more difficult for the ship to continue moving. Therefore, for a given penetration δ, the collision force reaches a sufficient level to activate a global plastic mechanism and a switchover is assumed to occur from the local mode to the global mode. An example of transition from local to global deformation modes of a lock gate is shown in Fig. 3.

As it is hard to take the coupling between local and global modes into account, it is assumed that a sudden switch between the two modes happens at certain penetration δ_t. At the beginning, when the ship first impacts the gate, only the local mode is activated and provides a resistance force P_t. This is true as long as the penetration does not exceed the value δ_t, which activates then the global mode. This means that at this penetration, the force P_t applied by the ship is sufficient to cause an overall bending of the whole structure.

In the local mode, each super-element is evaluated separately as they might present different dissipation energy mechanisms. For instance, the plate element (SE1) presents a deformation shape which is influenced in high manner by the shape of the bow (or the bulbous bow) striking directly into it. The beam elements (SE2) present a localized folding mechanism and a global bending that contribute to the resistance of the element. And the cross elements (SE3) present a folding mechanism that can be represented as the combination of folding mechanisms for each of the wings connected.

A representation of such deformation patterns is depicted in Fig. 4. More details in the calculation of P_t for the local modes (for each activated super-elements) and the global mode, are presented by Buldgen et al. (2012, 2014).

Fig. 3.

Example of deformation modes (local and global) of a lock gate (Buldgen 2014)

Fig. 4.

Deformation patterns of the super-elements of a lock gate (Buldgen 2014)

2.3 Finite Element Simulations

Several numerical simulations have been performed using the finite element software LS-Dyna. A preliminary analysis has shown that a raked bow can cause important damage to the gate, presenting a more prominent localized deformation of the impacted area in comparison with a barge.

For the FEM simulations, the gate is entirely modeled with Belytschko- Tsay shell elements of size 12 cm, which is sufficient to see the folding mechanisms of some of the impacted elements at the collision speeds analyzed. The material law used is a stainless steel already described in Sect. 2.1. The bow of the ship is modelled with same elements but with an infinitely rigid material, in order to save some computational time and focus on the deformation of the gate at this stage of the research. The mass of the ship is considered in the simulations for the sake of energy conservation. The models were presented previously in Fig. 1.

In preliminary simulations, the importance of some parameters of the simulation were observed. Such analyses concerned the material law, collision speed and selection of boundary conditions of the gate. Also the rupture and size of elements can play an important role if the collision happens near a vertical or horizontal reinforcement element, causing folding of the web. These results are not presented in this paper, but it is important to have in mind that such parametric analyses would lead to a more or less reliable result of the FEM simulations, which are crucial for the validation and verification of the analytical calculations.

Several collision cases were analyzed, but four of them are highlighted as specific stiffening elements are activated in each case: a collision on an intersection, a horizontal girder, a vertical frame or a plate. These collision scenarios are presented in Fig. 5. In all of the cases, the most stressed element is the plate, as well as the horizontal girder close to the point of impact. These stresses obviously appear depending on the phase of the collision, since the plating is activated in the local deformation phase in each case.

Fig. 5.

Collision scenarios for specific super-element activation

It was observed, that it is important to analyze the gate crashworthiness in a plastic regime in order to correctly identify the failure modes. Such failure modes are directly related to the deformation patterns described before and depicted in Fig. 3, where a transition from local to global mode can be assumed in the analytical calculation.

Choosing adequate mass and speed parameters is not always easy. The mass must be chosen according to the vessels currently navigating on the waterway where the lock in question is located. Regarding the speed, there is no particular recommendation on how it should be chosen if it is not contractually specified. It is obvious that at a lower speed and for a lower mass, the energy to be dissipated also decreases, which can certainly reduce the effects due to plasticity, which is an important limit to consider in an analysis of impact.

Finally, it is noted in Fig. 6 a) that a collision of a ship of 4500 tons (added mass included) at a speed of 0.5 m/s is not sufficient to observe an overall deformation of the gate, because the structural response is clearly in the local deformation regime (only the horizontal girder is activated locally), therefore the energy dissipated is much lower compared to a higher velocity collision (for instance 2 m/s in Fig. 6 b).

Fig. 6.

Effective plastic Strain for a collision on a horizontal girder

The vertical deformation modes observed for the four collision cases at the impact point are presented in Fig. 7, where it is observed in all cases a localized deformation due to the shape of the bow and a global vertical deformation of the gate, the latter tending to be almost linear. This first approach of linear displacement field of the gate is of vital importance for the analytical developments, where the energy dissipation depends on such known displacement field. This is a different assumption as the one from Buldgen et al. (2012, 2014), who considered a bilinear displacement field.

It is important to note that for each case of collision (on an intersection, horizontal girder, vertical frame or plate), the most stressed element is the plate, as well as the horizontal girder close to the point of impact. These stresses obviously appear depending on the phase of the collision, as mentioned.

Fig. 7.

Vertical displacement field of the impacted area for four collision cases

Another method for designing lock gates against collisions mentioned by Buldgen (2014), could be to consider an equivalent static force to be applied to the structure. For example, in the European standards (EN 1991-1-7), the design values of the force are given according to the class of waterways. They were derived for a collision speed of about 2 m/s and assuming that the energy dissipated by the structure was negligible. This approach is not very realistic for lock gates, therefore extreme care should be taken before applying an equivalent static force method.

Using other methods and/or analytical theories (such as a stiffness matrix or an orthotropic plate analysis) would also permit to predict the displacement field of the gate in a quasi-static-elastic regime, including simple static simulations where a constant (known) force is applied. These studies are not in the scope of this paper but were also considered during this research for further analyses in this topic.

3 Results and Discussion

From the preliminary analyses and using the linear displacement field observed for the global deformation of the gate, an analytical calculation was performed, including the local and global deformation of the super-elements activated in each case. Main results for the collision force with respect to the ship penetration are presented in Fig. 8, where the blue line depicts the gate resistance calculated with FE simulation (LS-Dyna) and the orange line presents the analytical calculation. The collision cases concern a ship of 4500 tons (added mass included) at a speed of 2 m/s.

Some differences are observed due to assumptions regarding to the membrane effects in the case of a collision on a horizontal girder, which makes the analytical model overestimate the gate resistance after 0.2 m penetration. Similarly, for the case of collision on a vertical frame or an intersection, at the beginning of the collision, the force calculated with the LS-Dyna simulation presents an interesting hump, that might be related to the element size or collision speed. Considering this, the analytical method underestimates the force in global bending of the gate, which needs to be further analyzed as this might be directly related to the displacement field chosen.

Despite these differences observed, the model is promising to calculate the resistance of a plane lock gate in more general cases such as collision on the plate. In further analyses for the current case study, other methodologies to predict the vertical displacement field can be evaluated, looking forward to predict the displacement field for each collision scenario in a prior analytical calculation (in the elastic regime). Moreover, the effects of membrane strains should be analyzed in detail for some of the scenarios, where the gate resistance force might be overestimated due to a miss conception of the membrane effects.

Similar analyses are to be performed in lock gates with a more irregular stiffness system, in order to see the applicability of the method in such cases.

Fig. 8.

Force vs displacement, results for four collision cases

4 Conclusion

This paper exposes a simplified method for assessing the crashworthiness of plane lock gates, based on the super-element method proposed by Buldgen (2014). At this stage of the project a plane gate with simple plating and classical orthogonal stiffness system has been considered. The deformation of the impacted structure is assumed to happen in two phases: a local indentation of the impacted elements and a global bending of the whole gate after a certain penetration of the ship inside the gate.

It is important to remember that the upper bound energy theorem systematically provides an estimate by excess of the resistance to impact. Then, the accuracy of the analytical results depends on the adequacy of the displacement field of the gate assumed prior any calculation. It is therefore crucial that this displacement field is chosen as carefully as possible. For this, the use of prior numerical simulations is currently the only option allowing reasonable assumptions to be made.

It was found, compared to the situations studied in the context of Buldgen (2014), that the lock gate configuration considered in this project presents significant differences. An important modification of the displacement field, in particular the introduction of a linear displacement field g(Y,δ) in the analytical model (and not bilinear as the hypothesis presented in Fig. 3) made it possible to obtain more accurate results. This confirms that the assumed displacement field remains a rather important condition to know before making the crashworthiness analyses using the super element method.

For the prediction of displacement field of different lock gates, a proposal of using other theories (such as the stiffness matrix method or the orthotropic plate) in the elastic regime, is a relevant option for future implementation and discussion of the current method. Furthermore, for a better understanding of the plastic global deformation of the gate, the membrane effects (related to the lateral supports of the gate) should be analyzed in detail, in order to predict a correct gate resistance for some specific collision cases. Finally, gates with an irregular stiffness system could be analyzed to validate and improve the current method.