Keywords

1 Introduction

Inland navigation locks are mostly located between sections of impounded rivers or canals, at connection points within the waterway network or at tidal rivers. They enable ships to overcome water level differences. The safe and efficient operation of these locks is essential for the transport of goods on the waterways.

Inside the navigation locks ships are connected to the structure by mooring lines to avoid a collision of the ship with the lock structure. During the leveling process the ship in a lock chamber is exposed to forces acting on the hull. The mooring lines must absorb these forces to avoid any unwanted displacement of the ship. The lines can be strained up to a certain breaking load. If a mooring line fails (breaks) a snap-back of the mooring line ends can produce great forces and the mooring staff is at risk for major injuries. Furthermore, the ship will start to move and can damage the lock or the ship itself. Thus, the mooring line force must be limited during the locking process.

The forces acting on the ship depend on the hydraulic effects during the filling process. A major factor is the gradient of the filling discharge. A faster opening of the filling facilities (e.g. valves) will cause higher forces but reduce the filling time and vice versa. Another relevant factor for the forces is the effect of jets on the ship. These jets can be reduced by adding energy dissipation objects (i.g. breakers) at the culvert outlets, but these hinder the flow. The hydrodynamics of the lock filling process are described in Belzner et al. (2018).

Thus, to reach a short filling time and to reduce the forces acting on a ship are contrary objectives. During the planning process of a lock a compromise between the filling time and acceptable forces must be found. The aim is to fill the lock as fast as possible and as slowly as needed.

2 Force Criteria

The force acting on the ship is the sum of external forces acting the hull. It depends on the hydrodynamics in the space between lock chamber floor and walls and the ship hull. During the hydraulic design process analytical, numerical or physical models are used to reproducibly determine this force. The mooring line force, on the other hand, depends on the temporal variation of the amplitude of the ship force and the configuration of the mooring line system: number, length, diameter, material and pretention of the mooring lines. Thus, it can only be reproduced for exactly the same setup, which is unlikely to occur in reality.

Figure 1 shows a principle sketch of a ship in a lock chamber. The force acting on the hull can be decomposed in longitudinal and transversal components. For through the head filling systems the longitudinal force component usually dominates. The mooring line force is the force that acts in the mooring lines, which are illustrated with dashed lines in Fig. 1.

Fig. 1.
figure 1

Principle sketch of a ship (yellow) in a lock chamber with mooring lines (dashed lines)

Full size image

Today the ship force is a commonly used criteria to assess the safety of the locking process: in practice it is assumed, that the mooring lines will not break if the ship force does not exceed a certain level. In the literature this certain level is often called the “ship force criteria” or the “hawser force criteria” and is mostly based on a mixture of semi-deterministic considerations and experience.

The ship force criterion used in Germany is based on estimations of Partenscky (1986) combined with practical considerations of the Federal Waterways Research and Engineering Institute (BAW). Partenscky (1986) analytically analyzed one specific (worst) case and generalized his results to the fleet of the 1980s. For a ship with a length of 110 m (CEMT Va) a force of 23.5 kN is considered acceptable (PIANC 2015), which is approximately 0.80 ‰ of the weight force of the displaced volume. In the Netherlands this criterion is defined as 0.85 ‰ of the weight force of the displaced volume and 1.10 ‰ of the weight force of the displaced volume for locks with floating bollards (Rijkswaterstaat 2000). These criteria are based on considerations of Vrijburcht (1994), who analyzed a range of configurations which fitted to the Dutch fleet in those times. Experience shows that almost no accidents with breaking mooring lines occur in locks due to the locking process itself. Thus, the approaches of Partenscky (1986) and Vrijburcht (1994) seem to be on the safe side.

The requirements for the mooring line equipment on board of the ships are specified for the European Union in the transnational ES-TRIN regulation (European committee for drawing up standards in the field of inland navigation 2021) and depend on the size of the ship (Eq. (1)):

$$ R_{{\text{S}}} = \left\{ {\begin{array}{*{20}c} {60 + \frac{{L_{{\text{V}}} \,\cdot\, W_{{\text{V}}} \,\cdot\, {\text{y}}_{{\text{V}}} }}{10}\;{\text{for}}\;L_{{\text{V}}} \,\cdot\, W_{{\text{V}}} \,\cdot\, {\text{y}}_{{\text{V}}} \le 1000\,{\text{m}}^{3} } \\ {150 + \frac{{L_{{\text{V}}} \,\cdot\, W_{{\text{V}}} \,\cdot\, {\text{y}}_{{\text{V}}} }}{100}\;{\text{for}}\;L_{{\text{V}}} \,\cdot\, W_{{\text{V}}} \,\cdot\, {\text{y}}_{{\text{V}}} > 1000\,{\text{m}}^{3} } \\ \end{array} } \right. $$
(1)

where,

RS:

minimum breaking load [kN]

LV:

length [m]

WV:

width [m]

yV:

draught [m].

3 Physics of the Mooring Line System

In a simplified way, the ship can be considered as a mass and the mooring lines as springs. Thus, the interaction between ship force, the reaction of the ship and the mooring lines can be described by the mass-spring equation (Eq. (2)):

$$ m_{{\text{V}}} \cdot a + k \cdot \Delta x = F_{V} $$
(2)

where,

mv:

mass of the ship [kg]

a:

acceleration of the ship [m/s2]

k:

spring constant [N/m], k = 0 for \(\Delta x \le 0\)

\(\Delta x\):

elongation of the spring [m].

Angular effects, the number and mass of mooring lines and damping effects also influence the behavior of the system. For convenience these parameters are neglected in Eq. (2). Later the model is enhanced with all these parameters to get a more realistic approximation.

The mooring line force Fm.l. can be derived from the elongation by Hook’s law:

$$ F_{\text{m.l.}} = k \cdot \Delta x $$
(3)

where the spring constant has to be calculated from the mooring line’s material properties:

$$ k = \frac{E \cdot A}{{l_{0} }} $$
(4)

where,

E:

Young’s modulus [N/m2]

A:

cross sectional area [m2]

l0:

original length of the mooring line [m].

The mooring line force is calculated by Hook’s law (Eq. (3)) from the elongation, which depends on the dynamic behavior of the mooring system and the spring constant. The ES-TRIN regulation requires a minimum breaking load for the mooring lines but there are no requirements concerning the stress-strain response of the lines, expressed by Young’s modulus. The spring constant is calculated from the mooring lines’ material and geometric properties, which are not regulated as long as the mooring fulfills the requirements for the minimum breaking load. A study from the Netherlands shows that the ES-TRIN requirements are over-fulfilled by most of the currently circulating mooring lines (PIANC 2015), indicating that these lines will also be more stiff. The considerations of Partenscky (1986) are based on mooring lines made of steel. Today it can be observed that the predominant part of the mooring lines is made of fiber-synthetic materials like polyester or polypropylene, which shows significantly different material properties compared to steel. Under certain circumstances, however, tankers must moor with steel hawsers. Furthermore, the spring constant scales inversely proportional with the length of the mooring line, which is also hard to predict.

In summary, the properties of the mooring line system cover a wide spectrum between rather “soft” and noticeably “stiffer” systems. Figure 2 illustrates the reactions of two mooring systems with significantly different characteristic. The forces are illustrated on the vertical axis in a normalized form where the forces are divided by the maximum amplitude of the ship force. The blue line shows a ship force acting on a ship with a mass of about mv = 4200 t in longitudinal direction. Following the thoughts of Partenscky (1986) the acting external force is applied in shape of a ramp. Its peak amplitude is at about 1 ‰ of the ship’s weight force. The mooring line force is calculated by Eq. (2) considering that even pretensioned lines will have a certain slack allowing the ship to accelerate before the ship force and the ship’s kinetic energy is absorbed by the mooring line. Two cases are illustrated: a rather stiff system (short steel rope) and a rather soft system (long synthetic rope).

Fig. 2.
figure 2

Reaction force in the mooring lines in comparison with the ship force for a soft system and a stiff system

Full size image

At the beginning the ship force acts on the ship, which accelerates until the line slack is reduced and the horizontal line force due to dead load and the ship force are in equilibrium, respectively. When the line is strained, the ship’s kinetic energy and the ship force are converted into line tension. The mooring line force is the result of the elongation of the line and the spring constant and acts as a reset force accelerating the ship in the opposite direction. Depending on the configuration of the mooring line system, the line will slack again or remain strained. The example illustrated in Fig. 2 shows the difference in mooring line forces depending on the so called “mooring system configuration”: the peak forces in the stiff and the soft system differ significantly. Softer or stiffer systems result from a combination of the material of the lines, the length and the diameter. One may assume that the stiffer system will be able to absorb higher forces without failing, but the only reliable comparative value is still given in the ES-TRIN standard (Eq. (1)).

This example shows in a very simplified way, that the mooring line force cannot be deterministically predicted on the base of the ship force alone. It depends on the magnitude and the temporal development of the ship force and, furthermore, on the configuration of the mooring line system. Even if the timeline of the ship force can be a known parameter, the configuration of the mooring system must be considered as a random variable.

Figure 2 shows an oversimplified example where only one mooring line is considered. A more realistic case is the application of two mooring lines: a bow line and a stern line. This amplifies the shown effect due to the interaction of two slacking and straining lines.

4 Probabilistic Approach

4.1 Amplification Factor

For a given ship, we can assume that the ship force depends on the characteristics of the lock filling system in a deterministic way. Then for this ship, the mooring line force depends on the ship force and the characteristic of the mooring line system, which is an unknown random variable. With considerations like shown in Fig. 2 an amplification factor can be defined, which is the ratio between the maximum of the mooring line force and the maximum of the ship force:

$$ V = \frac{{F_{{{\text{V}},\,\max }} }}{{F_{{{\text{m.l.}},\,\max}}}} $$
(5)

where,

V:

amplification factor [-]

FV, max:

maximum of ship force [N]

Fm.l., max:

maximum of mooring line force [N]

The amplification factor can be interpreted as an extremal value of every experiment.

4.2 Semi-probabilistic Approach

The current generation of the standards for structural engineering (e.g. ISO 22111:2019 2019) uses semi-probabilistic design approaches to determine impacts on structures. Structural loads and resistance are treated as probabilistic distributions from which partial safety factors can be calculated. This allows the planning engineers to guarantee a certain failure probability by considering a distributed quantity without having detailed knowledge of its statistical distribution. For example: the snow load on the rooftop of a building depends on its geographical position. It follows a statistical distribution based on long-standing observations. To avoid overloading and as consequence failure of the roof, a sufficient snow load must be used for the structural calculation. Using the maximum value of the observation will be uneconomic, using the mean value will underestimate the load with a probability of 50%. Absolute safety does not exist but a certain level of safety with a known failure probability (e.g. 10–6) can be guaranteed. Therefore, the structure must be designed to sustain a certain quantile of the snow load with a chosen reliability.

4.3 Monte Carlo Simulations

For wind or snow loads long-standing observations exist, allowing statistical analysis with the aim to choose a “safe” load for the structural calculation. For mooring line forces observations also exist, but the quantity is not sufficient for statistical analysis. That might be the reason why Partenscky (1986) did a kind of “worst case” calculation and Vrijburcht (1994) considered a limited range of situations, which covers the situation in the Netherlands and probably also on the waterway network in Belgium and Germany.

Another approach to overcome this lack of information is to perform Monte Carlo simulations with a high variety of randomly chosen parameters for the configuration of the mooring line systems. With this approach a large sample set of mooring line loads is produced where the amplification factor V can be calculated from.

The main influence factors like length of the mooring lines, diameter, Young’s modulus and pretention were randomly chosen from a range of uniform distributed values between certain limits that seem reasonably from an engineering point of view. Equation 2 was enhanced to consider more than one spring, angular effects, damping, rope characteristics and the fact that mooring lines cannot absorb pressure. As a consequence, Eq. 2 becomes a discontinuous differential equation, which cannot be solved analytically. Thus, the solution was performed numerically with a leap-frog algorithm (Skeel 1993).

After a sufficient number of simulations to obtain a statistically significant result the amplification factor will follow a Gumbel distribution, also called LogWeibull or Fisher-Tippetts distribution. An example is given in Fig. 3(left) where the results are shown compared to a fitted Gumbel probability density function (pdf). With this distribution function a chosen probability of exceedance P can be assigned to a corresponding amplification factor Vp. An example is given in Fig. 3(right) where a chosen probability on the vertical axis is assigned to an amplification by the cumulative distribution function (cdf). Note: this example is showing an exceedance probability of 1:10, which is very unsecure. This example was chosen to improve the readability. In reality a probability in the order of 1:106 would be chosen.

Fig. 3.
figure 3

Qualitative example for the probability density function (pdf, left) and cumulative distribution function (cdf, right) of the amplification factor

Full size image

Now the correlation between the maximum ship force and the mooring line force in one experiment or locking process with a defined probability is known. Additionally, the minimum breaking load of the mooring lines Rs is known from the ES-TRIN standard or other regulation. From this relationship, the maximum allowable ship force can be calculated:

$$ F_{{{\text{V,}}\,{\text{max}}}} = \frac{{R_{S} }}{{V_{p} }} $$
(6)

where,

VP:

amplification factor with the probability P [-]

FV, max:

maximum of ship force [N]

RS:

minimum breaking load of the mooring line [N]

The amplification factor Vp is the result from a probabilistic approach. For further usage, the design of the filling system and the valve schedules depend only this factor or furthermore the resulting maximum acceptable ship force are used without doing any statistics. This approach is based on probabilistic considerations but due to its reduction to one factor representing the probability, it is called a semi-probabilistic approach.

5 Outlook

Here, the ship in the lock chamber and the mooring line system are treated as a spring-mass system (Eq. (2)). Even if this approach is enhanced by several influence factors to make it more realistic, it does not consider the interaction between the ship and the water in the lock chamber. The authors perform numerical simulations with the CFD toolbox OpenFOAM® like presented in Thorenz and Schulze (2021) and additionally physical experiments with a scale model to investigate the coupling between the ship and the water in the lock chamber. As a result, the coupling will either be neglected or must be considered in the model.

Most semi-probabilistic design concepts consider both, the impact and the resistance as distributed values. The safety is in-between, where the impact does not exceed a certain value and the resistance does not fall below a certain value. Currently, the resistance (minimum breaking load) is typically treated as a deterministic value. In the future, the breaking load of mooring lines has to be described statistically to be able to assess probabilities on the resistance side.

An acceptable probability of exceedance of the mooring line force must be chosen. This will be a measure of how many locking processes can (statistically) be conducted without failing of the mooring lines. This may be a political decision regarding also other disciplines. It must be taken into account, that not every mooring line failure will have lethal consequences but it cannot be ruled out.

6 Conclusions

The authors showed a probabilistic approach that aims at an updated concept for planning engineers to determine the maximum acceptable ship forces on the base of a chosen exceedance probability.

The main findings of this paper in a short summary:

  • During the design process of a lock or the lock filling process a criterion for a maximum acceptable ship force is needed.

  • The ship force can reproducibly be determined with physical or numerical models. It must be distinguished from the mooring line force. Actually, the mooring line force is relevant for the safety but depends furthermore on the parameters of the ship and the mooring line configuration, which are random variables and cannot be forecasted. Thus, an approach to determine a maximum acceptable ship force considering various mooring line configurations is needed.

  • Existing approaches are ether based on outdated assumptions or do not enable to consider a chosen reliability. Nevertheless, these approaches seem to be rather conservative.

  • Existing observations of ship forces and mooring line forces are not sufficient for statistical analyses and forecast.

  • Monte Carlo simulations with a high variety of randomly chosen parameters for the configuration of the mooring line systems were made to produce a sample set which allows statistical analysis of the expectable mooring line forces depending on a given ship force.

  • Bases on statistical analysis, an amplification factor can be defined, which allows to determine the maximum acceptable ship force depending on the required mooring line equipment.

In this paper the procedure was summarized without giving hard values for target probabilities or acceptable forces. At the present time the procedure is in a proof-of-concept state. The German Federal Waterways Engineering and Research Institute and the Helmut Schmidt University are currently further developing this approach to update the criteria for the hydraulic design of navigation locks based on a scientific base.