The wave profile evolution before wave reflection is presented in Fig. 4. At t = 5.17 s, the surface sarcomport appeared at the top of the inlet of the water saving basin. At t = 10.12 s, the surface sarcomport region extended. At t = 15.1 s, a nonlinear long wave propagated to the left impoundment with the maximum water surface slope of the wave front of 0.022. At t = 20.11 s, the maximum slope of water surface slope reached 0.045. At t = 25.1 s, a series of well-formed undulations appeared at the wave front of the long wave. At t = 30.11–35.1, the propagation of the secondary wave was characterized by a considerable wave amplifications. At t = 45.2–55.2 s, the secondary wave showed some wave breaking and eventually evolved into a breaking bore. At this moment, the wave front of the long wave was characterized by a steep water wall. At t = 60.2s, the wave front arrived at the upstream impoundment.

Figure 5 exhibits the development of wave profiles after the first wave reflection. The secondary wave located at the wave front of the long wave showed an undular appearance immediately after wave reflection. At t = 95.2 s, the secondary wave exhibited some wave breaking immediately after its wave amplitude reached the maximum value. At t = 115.2 s, the free-surface undulations disappeared. The wave profiles after the long wave arrived at the downstream symmetrical boundary are presented in Fig. 6. Notably, the collisions of long waves facilitated the occurrence of undular secondary waves. In this context, the wave length of the secondary wave increased and the slope of the wave front decreased. The evolution of these waves after the second wall reflection is presented in Fig. 7. It can be seen that the wave amplitude of the secondary wave during wave propagation showed obvious wave attenuation due to the combined effect of wall friction and dispersion (Chanson 2010b).

Fig. 4. Wave profile evolution before wall reflection.

Fig. 5. Wave profile evolution after the first wall reflection.

Fig. 6. Wave profile evolution after wave collision.

Fig. 7. Wave profile evolution after the second wall reflection.

#### 3.2 Wave Celerity

The leading edge of a long wave propagated along the water saving basin with a celerity which varied with time and space. The evolution of long wave celerity in the water saving basin is shown in Fig. 8. C1 referred to the wave celerity of the long wave propagating in still water. C2 denoted the wave celerity of the long wave propagating from the upstream impoundment to the symmetric boundary immediately after the first wall reflection. C3 was the wave celerity of the long wave advancing from the symmetric boundary to the upstream impoundment. C4 was defined as the wave celerity after the second wall reflection. It can be observed that two distinct phases existed in the propagation process of the long wave in still water: a rapid acceleration phase characterized by a sharp increase in the celerity with propagation distance up to x = 16.78 m, and a gentle acceleration phase where the long wave propagated in a more gradual manner with a maximum wave celerity equal to 3.5 m/s. It is worth noting that the wave celerity at the phase transition was 2.5 m/s, which was close to the shallow wave celerity (i.e., 2.4 m/s). A sharp drop in the wave celerity was observed shortly immediately after the first wall reflection, i.e., C2 = 2.6. After that, the wave celerity showed some fluctuations during the subsequent wave propagation. C3 and C4 exhibited some irregular fluctuations over propagation distance with average values of 5.6 m/s and 5.2 m/s, respectively.

Fig. 8. Wave celerity evolution during wave propagation.

#### 3.3 Wave Force

The temporary evolution of wall force at the upstream impoundment is present in Fig. 9. Considering the first wall reflection, the wave force profile exhibited a double maximum structure due to the nonhydrostatic effects resulted from wave nonlinearity (Cooker et al. 1997). The wall force increased significantly with the water level and reached the first maximum value before the maximum wave run up at the impoundment. Subsequently, the force profile presented a local minimum value near the maximum wave run up and the second maximum value. For the second wall reflection, the wall force tended to oscillated in phase with the free-surface undulations. In this situation, the wave nonlinearity was rather weak and only single peak in force distribution was exhibited (Chen et al. 2015).

Fig. 9. Time evolution of free surface and wall force.

#### 4 Conclusions

The basin in a water saving lock is an impounded channel where long waves generated by the filling and emptying operation of lock chambers exist. During the propagation of the waves along the basin, several complicated hydraulic phenomena can be observed, such as the appearance of secondary waves over the body of the long waves, the wave type transition of secondary waves, and the wave reflection at the upstream and downstream impoundments. In this study, the propagation and evolution of long waves induced by ship-lock operation in a water saving basin was numerically investigated. The results showed the leading edge of a long wave propagated along the water saving basin with a celerity which varied with time and space. Two distinct stages were observed in the propagation process of the long wave in still water: a rapid acceleration phase characterized by a sharp increase in the celerity with propagation distance up to 16.78 m, and a gentle acceleration phase where the long wave propagated in a more gradual manner. It was found secondary waves occurred on the wave front of the long wave when its slop exceeded 0.045. In addition, the present results demonstrated that the impact of water waves in a water saving basin on impoundments was strongly dependent on wave nonlinearity. The present results have the potential to provide some guidance for the optimum design of water saving basins in integrated lock layout.

Acknowledgements. This study was supported by the China Postdoctoral Fund (Grant No. 2021M700620); Natural Science Foundation of Chongqing, China (Grant No. cstc2021jcyjbshX0049; cstc2020jcyj-bshX0043); Natural Science Funds of Chongqing (No. cstc2019jcyjmsxmX0759).

### References

Zheng F, Wang P, Wang M, Zhang J (2021) The evolution and runup of nonbreaking undular surges. Mar Georesour Geotechnol 40(7):774–781

Zheng F, Li X (2021) Undular surges interaction with a vertical wall. Mar Georesour Geotechnol

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Reliability Based Rehabilitation of Existing Hydraulic Structures

Arslan Tahir(&) and Claus Kunz

Department of Structural Engineering, Bundesanstalt für Wasserbau, Karlsruhe, Germany<EMAIL_ADDRESS>
Abstract. Existing hydraulic structures may show damage with increasing age and operation, so structural verification is crucial. In case of structural deficits, repair measures must be planned, and their effectiveness demonstrated. The advent of improved structural analysis methods and subsequent standardization processes facilitate the verification of existing structures to ensure sufficient reliability of infrastructure. Among the existing inland navigation hydraulic structures, older ship locks had been constructed with primitive construction materials such as damped plain concrete. At times, the structure exhibited neither any severe damages nor an indication of failure but failed to satisfy the limit states prescribed by the latest standards. This contribution considers a similar ship lock built in 1922 as a case study. The ship lock has a half-frame structural system with plain concrete gravity walls and a lightly transverse reinforced base slab. Cross-section based static verification revealed that the structure does not provide sufficient resistance in case of sliding and overturning limit states which could be attributed to crack and pore-water pressures in the cross-section. Consequently, rehabilitation of the lock walls with a vertical anchoring system was proposed to conform to required standards. Similar problems are expected for other existing locks in the German waterway system. Therefore, a methodology was developed to verify and to optimize the structural reliability of similar structures using full probabilistic methods while considering standardbased limit state functions. This involved uncertainty quantification of parameters for relevant loads (self-weight, water pressure, earth/ groundwater pressure, temperature, etc.) and materials (concrete, steel). To calculate the probability of failure and reliability indexes First Order Reliability Methods (FORM) was applied, considering its computational efficiency and more suitable for the presented Reliability-Based Design Optimization (RBDO) scheme. The contribution provides a probabilistic framework to study the influence of three aspects on the reliability of existing hydraulic structures, crack and pore-water pressures, operational conditions and lastly, the effect and optimization of rehabilitation in the form of anchoring.

Keywords: Existing hydraulic structures Structural reliability Rehabilitation Optimization Crack/pore-water pressures

#### 1 Introduction

Hydraulic structures are critical elements of the inland navigation network, and the safe functioning of these elements directly influences the system's reliability. A considerable number of these structures are passing their planned service lifetime, i.e. more than 100 years, and based on safety reassessment, require minor repairs, rehabilitation or replacement. Since constructing these structures, several advancements have been made in codes, standards and limited state functions for verification. Three issues need to be reviewed during the safety reanalysis of these structures.

Firstly, construction-based limit state functions, i.e. gravity-based unreinforced concrete hydraulic structures, differ significantly from slender reinforced concrete structures DIN 19702 (2013) and BAW (2016), a guideline for existing structures, indicate several of these differences and provide recommendations to deal with them.

Secondly, compared to ancient approaches, Crack and Pore-Water Pressures (CPWP) have to be considered at a section level and hydraulic structure-specific resistance models developed. (Eugene and Victor 1995a; 1995b) concluded through a series of experimentations that CPWP exists and water penetration into a concrete crack is time-dependent, and water under pressure penetrates the fracture process zone of a concrete crack. This creates additional uplifting pressures against the gravity principle and destabilizes the structure. During reassessment of existing hydraulic structures, it was found that the stresses and forces due to CPWP were neither considered in the original design BAW (2016) and Kiesel (2018). Therefore, future safety assessments require an appropriate CPWP analysis method for solid hydraulic structures. Thirdly, in case of insufficient safety, the selection, modelling and application of efficient rehabilitation methods for the extension of service life (Brown 2015; USACE 2016).These methods include pre-stressed anchoring, local bolting anchors or passive anchoring.

We adapted the probabilistic methods rather than the code-based partial safety factor method to evaluate the safety. Employing a full probabilistic approach provide two significant advantages firstly, parametric uncertainty can be considered through probabilistic modelling of field data and secondly, its flexibility to adapt the design to a designer's desired target reliability. We adapted the First Order Reliability Method for our framework, primarily due to its computational efficiency.

Not much research or recommendations exist that consider these three aspects discussed earlier, and almost none exist that consider the reliability-based design under parametric uncertainty associated with hydraulic structures. The current study provides a practical solution with a primary focus on unreinforced concrete gravity-based systems experiencing CPWP and requiring rehabilitation. We developed a probabilistic framework that provides reliability-based optimized rehabilitation considering hydraulic structure-specific failure modes. Additionally, the framework allows the designer to design rehabilitation based on his desired extension in service life and risk acceptance rather than code dictated fixed service design life, i.e.100 years.

We provide a framework for the analysis model at the section level, which incorporates the CPWP and models the steel reinforcement based on the anchoring principle.

Section 2 of this paper presents the methodology for evaluating crack and porewater pressures, followed by hydraulic structure-specific limit state functions in Sect. 3. The formulation of a reliability-based rehabilitation optimization problem is discussed in Sect. 4 in combination to target reliability for existing hydraulic structures. All presented methods are sewed together, and their application is presented through a case study of a typical 100-year-old structure in Sect. 5. Finally, the conclusion of the current study and discussion of the possible future outlook of work is presented.

### 2 Crack and Pore-Water Pressures (CPWP) in Rehabilitated Sections

DIN 19702 (2013) advises the analyst to consider crack and porewater pressures (CPWP) for hydraulic structures but provides a basic crack length and stress calculation method. The standard was further extended by BAW (2016) for existing hydraulic structures. The algorithm extends to cases with water on both sides and incorporates its effect on the effective normal forces and bending moment. Additionally, it provides calculations for the compressive stresses. Neither DIN 19702 (2013) nor BAW (2016) recommend the consideration of the tensile strength of concrete. Therefore, a section under tensile stresses is considered cracked and effective CPWP is applicable. Eventually, in cases where the resultant force acts outside the section base, the structure is considered unsafe. This is indicated in Fig. 1, part A, where the evolution of cracking is shown for an unreinforced section with increased tensile stresses (bending moment) and concrete tensile strength is ignored. Figure 1, part B, indicates the cases where the destabilizing tensile stresses are transferred through steel anchor and provide safety even in cases where the section is completely cracked. Figure 1, part C, shows all the sectional stress distribution and balance of forces for all possible scenarios from parts A and B. The essential contribution of this study is the extension of the BAW (2016) based analytical method for cases where anchoring is provided and evaluating the influence of CPWP in cases where the system is unstable.

This study considers the anchor as a point load contributing with its full strength to counter the destabilizing bending moments and sliding forces. Other designs and analyses to model steel reinforcement and CPWP are possible. The anchoring principle's major advantage is that no significant difference is seen in limit state functions and its probabilistic parameters in cases with/without CPWP and with/without rehabilitation, hence making the comparison considerably transparent.

### 3 Limit State Functions for Concrete Section Under CPWP

The stress distribution on a section level considering CPWP is further used in limit state functions to verify the safety of the structure. We consider three limits state function for the analysis of section considering different sections and operational load conditions. These include compression failure limit state function, eccentricity verification and sliding in construction joint. The last two are Ultimate Limit States.

#### 3.1 Compression Failure

Most of the existing structures were constructed using non-standardized plain concrete, which has considerably variable and low compressive strengths compared to modern concrete. Additionally, the CPWP compounds the compressive forces increasing the probability of failure due to compressive failure. For all load combinations in the ultimate limit state (ULS) and in according to Code of Practice BAW (2016), the concrete compressive stresses shall be limited as follows:

$$
\sigma\_{cd}^\* = \frac{3}{4} \cdot \sigma\_{cd} < f\_{cd} \tag{1}
$$

with: rcd Maximum stress value of the triangular compressive stress distribution.

r cd - Stress value of the simplified rectangular compressive stress distribution.

In case the entire section in under compression and no crack is present the stress in the system is calculated using the following equations

$$
\sigma\_{\rm cd} = \frac{\mathbf{N\_d}}{\mathbf{h}} + \frac{\mathbf{6} \cdot \mathbf{N\_d} \cdot \mathbf{e\_d}}{\mathbf{h}^2} - H\_2 \cdot \boldsymbol{\gamma}\_{\rm w} \tag{2}
$$

With water pressure on both sides, the effective concrete compressive stress rcd can be determined according to BAW (2016), Eq. (8), as follows:

$$
\sigma\_{cd} = \sigma\_{abs} - H\_2 \cdot \gamma\_w \tag{3}
$$

$$
\sigma\_{abs} = H\_2 \cdot \gamma\_w + \frac{4}{3} \cdot \frac{\left(N\_{Ed} - h \cdot H\_1 \cdot \gamma\_w\right)^2}{\left[h \cdot \left(N\_{Ed} - h \cdot H\_1 \cdot \gamma\_w\right) - 2 \cdot N\_{Ed} \cdot e\_d\right]} \tag{4}
$$

with: H1 - Water level on the side with compressive stresses in [m]. H2 - opposite water level height from H1 in [m]. cw - Weight of water in [kN/m3 ]. ed - load eccentricity under the design internal forces MEd und NEd in [m].

The final limit state function for reliability evaluation becomes

$$\mathbf{G(x)} = \theta\_{\mathbb{R}} \cdot (\mathbf{z}\_{\text{cr}} \cdot f\_{\text{cm}}) - \theta\_{\mathbb{E}} \cdot \sigma\_{\text{cd}}^{\*} \tag{5}$$

where acc the coefficient to account for long-term effects on the concrete compressive strength.

hR is the coefficient of uncertainty in the resistance model and hE is the coefficient of uncertainty in the action model.

Fig. 1. Different possible states with CPWP in a rehabilitated concrete section

#### 3.2 Eccentricity Based Design Check

Although not a standard-based verification, eccentricity determines the section's cracked state, which is vital in the calculation of CPWP and is also a geometrical indicator of the system stability. This limit state only includes loads with no consideration for material resistance. However, in case concrete tensile strength is not ignored, further modifications of the Limit State Function (LSF) can be transformed into a material based LSF. Furthermore, CPWP is a direct function of eccentricity, and in case the limit of h/2 as the half of the cross-section is breached, the section can no longer be verified by the methods discussed above.

The resultant force appears in three limit states corresponding to different tensile states and cracked based. Limit-1 is when the applied load eccentricity(ed) to the section length (h) is ed/h < 1/6, considering the case where a full section is under compression. Limit-2 represents cracked sections with 1/6 < ed/h < 1/2 and Limit-3 is when the section is fully cracked with ed/h > ½. Furthermore, this aspect directly influences the effective area, an important aspect for calculation of CPWP and consequently Nwd, Mwd (revised normal/moments with CPWP).

Therefore, the equation for rabs for calculation of compression states is valid only if the following application limit is met:

$$\frac{e\_{wd}}{h} = \frac{e\_d/h}{1 - \overline{\sigma}\_{wd}} < 0.5\tag{6}$$

At an eccentricity ewd=h[ 0:5 there is a stability problem (no compression zone any more, load resultant NEd outside the cross section). Thus the limit state function for reliability evaluation becomes

$$\mathbf{G}(\mathbf{x}) = 0.5 \cdot \theta\_{\mathbb{R}} - \theta\_{\mathbb{E}} \cdot \frac{e\_d/h}{1 - \overline{\sigma}\_{\mathbb{w}d}} \tag{7}$$

#### 3.3 Sliding Failure in Construction Joints

As it is expected that the solid concrete hssydraulic structures were constructed in concreting steps. It can be assumed that construction joints were provided while construction. For these joints, the following verification must be performed with regard to ensuring the sliding force transmission:

$$
\Sigma(V\_{\rm E}) \le A\_{\rm cc} \cdot v\_{\rm Rd} \tag{8}
$$

In this case, the design value of the sliding capacity in the joint vRd has to be determined following DIN EN 1992–1-1, by

$$\mathbf{v}\_{Rd} = \mathbf{c} \cdot f\_{cdd} + \boldsymbol{\mu} \cdot \boldsymbol{\sigma}\_n \le \mathbf{0}, \mathbf{S} \cdot \boldsymbol{\nu} \cdot f\_{cd} \tag{9}$$

where, c, l and v are coefficients as a function of the joint roughness.

For usual cross sections the verification is performed assuming a "rough" joint (c ¼ 0; 4 / l ¼ 0; 7=v ¼ 0; 5Þ according to the standard. But for the investigated lock lab tests for the evaluation of the joint roughness has been performed which lead to higher than the expected values. In the case of unreinforced concrete cross-sections, the load-bearing component of cohesion bond strength is completely omitted. The shear resistance is thus exclusively composed of the static friction (joint roughness) and normal stress components. Tensile strength of concrete is ignored in this contribution. Addition of the anchor leads to a modification of the above presented set of equations, since anchor is considered as a point load hence sliding resistance it provides is

$$\mathbf{V}\_{\text{rd,anchor}} = \mathbf{0}.\mathbf{5} \cdot \mathbf{A}\_{\text{anchor}} \cdot \mathbf{f}\_{\text{y}} \tag{10}$$

where fy is the strength of the anchor and Aanchor is the cross sectional area of anchor.

The final limit state function for reliability evaluation becomes

$$\mathbf{G(x)} = \theta\_\mathbf{R} \cdot \left( A\_{\rm cc} \cdot \mu \cdot \sigma\_n + 0.5 \cdot A\_{\rm anchor} \cdot \mathbf{f\_y} \right) - \theta\_\mathbf{E} \cdot \Sigma(V\_\mathbf{E}) \tag{11}$$

where hR is the coefficient of uncertainty in the resistance model and hE is the coefficient of uncertainty in action model. It must be noted that the anchor is expected to provide simultaneously sliding/shearing resistance and normal resistance to bending stresses.

#### 4 Reliability Based Design Optimization

This phase has three components: reliability analysis of a structural system; secondly, computationally efficient optimization scheme for design variables; and thirdly, integration of the two aspects. Several methods exist for the first two components for reliability analysis (first/second-order reliability methods, subset simulations, importance sampling). Since an analytical set of equations for limits state functions has been derived, FORM due to its computation efficiency can be used. The probability of a failure and reliability analysis is essentially solving the following integral.

$$P\_f = \Pr[g(\underline{\mathbf{X}}) \le 0] = \int\_{\mathcal{S}(\underline{\mathbf{X}})} f\_{\underline{\mathbf{X}}}(\underline{\mathbf{x}}) d\underline{\mathbf{x}} \tag{12}$$

where X is a random vector of input parameters with joint probability density function fX (x) and g(X) is the limit state function (LSF).

Furthermore, Reliability Based Design Optimization (RBDO) model can be articulated as:

$$\min\_{d, \mathbf{X}} F(d, \mathbf{X}) \tag{13}$$

$$\text{s.t.} \begin{cases} \Pr[L\_i(d, X) \le 0] \le P\_f^{\text{Target}}(i = 1, 2, \dots, n) \\ C\_j(d) \le 0(j = 1, 2, \dots, m) \end{cases} \tag{14}$$

where d is the deterministic design vector, X random vector is uncertain design vector, F dð Þ ; X is the optimization function, Lið Þ d; X 0 is the limit state function for failure, PTarget f is the target failure (allowable), Pr½ Lið Þ d; X 0 is the probability of failure under the given operational conditions and state, n is the number of uncertain constraints,Cjð Þ d 0 are deterministic constraint with m number of deterministic constraint.

Dissociated double loop approach was used. Where reliability analysis is conducted within the inner loop, we employed First Order Reliability Method FORM (Rackwitz and Fiessler 1978). Whereas the exterior loop, we used the golden search optimization algorithm to find the optimal point for design parameters. The approach was adopted for two reasons; firstly, the optimization problems can be reduced to one essential parameter, i.e. the area of steel, since we fixed the steel cover based on some dimensional requirements. Secondly, implementing the golden search algorithm with the FORM algorithm is computationally efficient in combination to multi limit state multi-operational conditions-based system reliability.

### 5 Target Reliability

The entire process depended on the P>arget f ,i.e. the allowable failure probability, which can be also translated into the reliability index b through the relation Pf Target ¼ U bTarget where U is the cumulative density function (CDF) of the standard normal distribution. Several aspects and approaches influence the evaluation of the target reliability. (DIN EN 1990, 2010) provides a target reliability index of 3.8 for 50 years service lifetime for new constructions. It has been adopted for solid hydraulic structures and 100 years service lifetime by DIN 19702 (2013). The discussion on modified reliability target becomes even more important as several of the existing structures do not fulfill partial safety factor design in codes for new structures. In BAW (2016), for example, modified partial safety factors for existing hydraulic structures are already considered.

#### 6 Case Study

The considered structure is representative of a broad portfolio of similar structures in the German inland water navigation network. Lock Oldenburg, built in 1922, has a clear width of 12.0 m and a usable length of 105 m to accommodate ships with 1350 tons bearing capacity. The lock walls are constructed as a gravity wall from unreinforced rammed concrete. On the ground side, the walls are stepped upwards so that the chamber walls are 4.50 m thick at the bottom and reduced like stairs to a thickness of 1.50 m as shown in the Fig. 2. The geotechnical assessment carried out in 2007 indicated two layers of soil. The first layer from NN + 5.90 m (ground level) to NN −2.00 m has fine/medium sands of very low strength. The second layer from NN −2.00 to NN −4.00 m has fine/medium sands of low to medium strength.

The bearing capacity deficits in the northern chamber wall and the bottom, even using the favorable design approach EQU according to DIN EN 1990 /NA (2010). The lack of safety is primarily due to the crack and pore water pressure, usually not considered during construction. Therefore, a rehabilitation of the structure was necessary in this case through steel reinforcement. Figure 2 shows the old structure without rehabilitation and the designed rehabilitation configuration.

Fig. 2. Structural configuration of a component pre and post rehabilitation

#### 6.1 Probabilistic Design Parameters

Information regarding parameters of the models of CPWP and limit state function was gathered through different sources. For collected data, probabilistic modelling was conducted and appropriate distributions and their parameters were calculated. The following table indicates the selected distributions and parameters for the limit state functions and CPWP evaluation system.

Table 1. Probabilistic models for parameters of different limit state functions

#### 6.2 Results

Reliability assessment was conducted for lock Oldenburg northern wall considering the CPWP and stress distributions shown in Fig. 1 and load models shown in Fig. 2, although with the probabilistic parameters indicated in Table 1. Table 2 indicates the reliability assessment conducted with three classifications. Class 1 is based on the significant operational conditions in a typical ship such as Oldenburg, i.e. revision with no water in the chamber, tail water level and head water level under operational conditions. Class 2 considers limit state functions, i.e. eccentricity, compression failure and sliding failure. Class 3 indicates the three different states of section analysis, i.e. State-1 represents the original design condition of the structure, which neither considered CPWP nor anchoring in the structure. This represents the safety level at the time of construction in the 1920ies. State-2 is when the structure is reanalyzed as an unreinforced section with CPWP; this state validates if the rehabilitation is required or not. State-3 represents cases where the structure is rehabilitated with anchor under CPWP. It must be noted that only state 3 is further used to optimize rehabilitation, whereas other states and their corresponding reliabilities are only used for comparative analysis. The anchor steel area for rehabilitation was designed according BAW (2016) with as = 6 cm2 /m and cover of 0.9 m from the earth side.

Table 2. Reliability levels achieved for different levels and anchor steel area of 6 cm2 /m

From the above analysis, it can be concluded that water level operations significantly influence the reliability of the structure. The most critical load case is case 2, which represents the chamber's minimum operational water level. In cases where eccentricity and compressive failure without anchors and considering CPWP was considered, no results were possible. The primary reason being the entire section is under tensile stresses, and the entire section is cracked. Consequently, the method presented in Fig. 1 and their corresponding equations provide numerical instabilities. This is indicated by the state of cracking in the last column in Table 2, where the percentage of base crack is shown for each load case, using the deterministic mean values. This further supports why eccentricity or overturning should be considered as a limit state function for structures under CPWP.

The results presented in Table 2 indicate results for only one steel design dimensions. We use the optimization routine discussed in the section earlier and conclude that for a supposed target reliability index b = 3.8 the optimized anchor steel required for eccentricity LSF is 6.25 cm2 /m, compression LSF is 4.75 cm2 /m, and sliding LSF is 4.92 cm2 /m, and consequently, therefore for the entire system 6.25 cm2 /m.

The following graph in Fig. 3 indicates the minimum reliability and probability of failure achieved for different operational conditions and limits state functions and systems for a variable steel content in the cross section. At the lower steel content, the difference between the different limit state functions is low. A lower steel content makes compression failure a dominant failure condition, and an anchor area of 3 cm2 /m and lesser has no significant influence on the reduction of reliability.

Because of a code-based ignorance of tensile strength eccentricity check remains the most critical limit state function hence the dominant failure mode for system reliability. A lower steel content structure is safer in sliding than compression failure. This could be attributed to the resistance doubling effect of the anchoring to the system. Firstly, due to the reduction of crack, CPWP and increased effective normal forces. Secondly, the anchor provides shear resistance to the horizontal applied forces. As the steel content increases in the cross section, the crack in the section closes, and applied compression is reduced exponentially, making compression the least probable failure mode.

Fig. 3. Evolution of reliability of different limit state functions with area of steel in anchor

#### 7 Conclusions

This contribution presents a practical solution through a framework with its field application for the impending problem of deficits in the reliability of existing concrete hydraulic infrastructure. The framework provides an agile reliability-based approach which provides the flexibility on an extension of service life and simultaneously provides a stepping stone for risk-based planning of assets at the network level. An example of lock Oldenburg represents structures that are a significant part of the existing inland navigation infrastructure throughout the world. These structures are nearing their planned service lifetime, and with good performance, reanalysis and reliability based minor rehabilitation they can extend their lifetime hence saving financial and environmental costs. The novelty of the contribution is the solution it provides to address the emerging issue of crack and porewater pressures in solid hydraulic structures, considering structure-specific uncertainties and integrating the anchoring system into the framework. Furthermore, the optimization scheme momentarily only for a single parameter since the primary target of the study was validation of the developed concepts and application of the framework was presented. However, the number of optimization parameters can be increased for complex problems and different structural components and systems. The authors believe that three aspects are of high importance and require further considerations. However, concrete split tensile tests indicate that the concrete has some tensile strength, but following the standards and codes, tensile strength is ignored, leading to a conservative verification. The presented method can be further modified to the inclusion of tensile strength of concrete. Sliding friction coefficients achieved during the testing of this exemplary lock structure was considerably higher than the literature-based values. Therefore, further investigating of the appropriate evaluation of the coefficient, which might significantly influence the results of the presented work is needed. Lastly, the anchoring principle presented in this contribution is one of the methods to model and assess the reliability of rehabilitated sections under crack and porewater pressures. The authors dealt with two further methods for the reliability assessment which will be presented in the future.

### References

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Research Developments in Hydrodynamics of Ships Entering and Leaving the Tank of a Ship Lift

Luzhidan Fu(&) , Yaan Hu, and Zhonghua Li

Key Laboratory of Navigation Structure Construction Technology, Nanjing Hydraulic Research Institute, Nanjing, China<EMAIL_ADDRESS>
Abstract. In the design process of a ship lift, the size of the tank and the depth of water inside the tank are always minimized under the premise of satisfying the standard of ship size and freight volume, in order to reduce the power of electrical driving system, the construction cost of the project and the difficulty of equipment manufacturing, as a result, the cross-section coefficient of the tank is small. Therefore, the up-and downstream docking between the tank and the reaches of canal involves many related hydraulic problems, for instance, opening and closing of the vertical lift gates, entering and leaving of the tank, are very complicated. The docking process is an important link in the whole operations of the ship lift. At the same time, the change of unsteady flow of the hub will also affect the safety and efficiency of the docking process. This paper first briefly introduces the docking process of different types of ship lift, and systematically summarizes the scientific problems involved in the docking process when the ship enters and leaves the tank. Secondly, combined with the research results obtained from a variety of methods such as theoretical analysis, physical model test, mathematical simulation calculation and real ship test of previous researchers, the hydrodynamic research progress of ships entering and leaving the tank is introduced. In summary, the design criteria for the water depth of the ship lift's tank obtained on the basis of the progress of the hydrodynamic research of the ship entering and leaving the tank are introduced above.

Keywords: Ship lift Research development Hydrodynamics

### 1 Introduction

The ship lifts play an important role in river channelization and communication of different water systems in inland water transportation. Compared with ship locks, ship lifts have obvious advantages, especially suitable for high dam navigation, canal navigation between different water systems. The process of ship entering and exiting the ship lift chamber is one of the core links of the whole process of the ship lift operation. Due to the high blockage ratios of the ship lift chamber, the hydraulics problem becomes a complex three-dimensional water flow problem coupling with ship operation. To prevent the ship from touching bottom it is necessary to formulate reasonable ship draught control standards and ship navigation methods based on the ship squat and the water fluctuation of the ship lift chamber, so as to ensure the safety of ship navigation and ship lift docking.

Some scholars proposed the practical methods for calculating the squat with the general ship characteristics (Tuck 1973; Husska 1976; Eryzulu 1978; Icorels 1980; Barrass 1981; Norrbin 1985; Millward 1990; Millward 1992; Eryzulu, 1994; Gourlay 2008;). Bao (Bao 1991) presented the formula of maximum squat in ship exiting the chamber according to the previous studies in China. The German Federal Waterway Design and Research Institute carried out the field investigations and the numerical calculation on the extra-large ships entering and exiting the Lüneburg ship lift. NHRI (Hu et al. 2011) advanced the formula of Bao based on the experiments of the Three Gorges ship lift.

The current studies at home and abroad mainly combine with the concrete engineering lack of the universality. In addition, most of them uses the formulas of ordinary restricted channels to make calculations, without considering the plane dimension influence of narrow navigation, enclosure and so forth.

In conclusion, this paper introduces the docking process of different types of ship lift, and systematically summarizes the scientific problems involved in the docking process when the ship enters and leaves the tank. Explain in detail the variation law of the ship's sailing resistance characteristics during the process of entering and leaving the tank. Describe the variation law of water surface fluctuation in the tank during the process of ships entering and leaving, analyze the relationship between the fluctuation characteristics and the ship speed, cross-section coefficient, entering or leaving, the water level in up-and downstream docking process. Summarizes the variation characteristics of water surface in the tank. Introduce the main influencing factors of ship squat, including ship size, ship draught, ship speed, tank size, tank water depth, etc. Research results and the relevant empirical formulas are summarized in chronological order according to different research methods. Outline the impact of the ship's entering and leaving of the tank on the characteristics of tank's load change and the force characteristics of the ship lock mechanism.

### 2 The Docking Process of Ship Lifts

Ship lifts come in a variety of classifications. According to whether the ship tank going into the navigation pool or not, it can be divided into two types: tank-launching ship lift and non-tank-launching ship lift. For the non-tank-launching ship lift, the upstream and downstream docking processes are consistent. For the tank-launching ship lift, the upstream docking process is the same as that of the non-tank-launching ship lift, but the downstream docking process is special.

The tank of the tank-launching ship lift enters the downstream approach channel directly, eliminating the need of the lower lock heads and its corresponding equipment, which not only saves the quantity of work, but also eliminates some auxiliary equipment such as retaining and sealing mechanism; meanwhile, the procedure of the process is greatly reduced, the time ships take to pass the dam is shortened, the operational reliability, passing capacity and operational efficiency are all increased a lot, compare to the non-tank-launching ship lift. The main factors affecting the hydrodynamic characteristics of tank-launching ship lift's process which tank going in and out the navigation pool is: the shape of the ship lift's tank, the size of the navigation pool, the speed of the tank going in and out, and the size of the approach channel.

The docking process between the ship and the upstream and downstream is an important part of the whole process of the ship lift operation. It involves problems like the hydrodynamic characteristics of the ship lift's tank when the tank's door is opened and closed; the ship navigation conditions and hydrodynamic characteristics in the tank during the process of ships entering and leaving; the change affections of downstream unsteady flow on the operation of the ship lift and other related hydrodynamic problems.

### 3 The Process of Ships Entering Leaving the Tank of a Ship Lift and its Hydraulic Problems

During the ship entering the tank, the bow kept pushing the water flow into the tank, which leads to the water level of tank gradually rises, and the water level of the stern lowers, forming a fore and aft water level difference, causing reverse flow of water on the ship's side and the bottom of the ship. The ship trim and sink then appears. When the ship completes the entrance and parks in the tank, the water surface decreases due to the inertia of the flow in the tank, yet the stern water is connected to the channel, with the supplement by the channel water, the water level drops less, and the stern sinks. During the ship leaving the tank, the bow water is connected to the channel, the bow kept pushing the water flow into the channel and the water level in the tank gradually decreases. Although the approach channel replenishes the water in the tank, but because of the ships blocking while the water in the stern is small, water supply insufficient, the water level drops more obviously, and the stern sinks is also larger.

The safety of the ship in the tank is one of the most concerned issues in the design and research of the ship lift as the cross-section coefficient of the tank is small. It is necessary to determine the reasonable water depth of the tank and the way of ships' navigation to prevent the bottoming and the drastically changing of the tank' load when the ships entering and leaving the tank, and to ensure the safety of the ship and the tank. Using the combination method of mathematical model and physical model and real ship test, the hydrodynamic characteristics of the ship entering and leaving the ship lift at different speeds are studied to determine the reasonable water depth of the tank and provide a basis for reasonable navigation.

Using the combination method of mathematical model and physical model and real ship test, the hydrodynamic characteristics of the ship entering and leaving the ship lift at different speeds are studied to determine the reasonable water depth of the tank and provide a basis for reasonable navigation (Figs. 1 and 2).

Fig. 1. Schematic diagram of three typical moments of ship entering the tank

Fig. 2. Schematic diagram of three typical moments of ship leaving the tank

### 4 Hydrodynamics Study of Ships Entering and Leaving the Tank

#### 4.1 Characteristics of Ship's Sailing Resistance

During the process of ship entering and leaving the tank, the ship's navigation environment is narrow and shallow, which lead to the flow field around the hull presents obvious bank effect and shallow water effect. The total resistance, lateral force, pitching moment and turning moment of the ship increase as the speed of the ship increases, and the speed of the ship has a significant impact on the force of the ship. Therefore, from a safety perspective, it is recommended that the ships enter and leave the tank at a lower speed. Meanwhile, when the ship enters the approach channel from the open water, the asymmetry of the flow field on both sides of the ship increases due to the narrowing of the water on both sides of the ship, which result in the rapid increase of lateral force and the turning moment. Therefore, it is more likely hit the wall or turn when ship sailing at this moment. Anti-collision measures can be considered in practical work.

#### 4.2 Characteristics of Water Surface Fluctuation in the Tank During the Process of Ships Entering and Leaving

When the ship leaves the tank, the water pushed out by the ship in the rear of the stern can only be replenished by the narrow space around the ship and the tank, resulting in a drop in the depth of the tank and the channel behind the ship. Obviously, the closer to the approach channel, the greater the maximum drop in the water surface, and the maximum drop in the water surface in the tank basically occurs before the bow completely leaves the tank. When the ship enters the tank, the distribution of the fluctuation of the water surface in the tank is opposite to that of the ship's leaving. The closer to the closed end of the tank, the greater the fluctuation of the water level, and the maximum drop in the water surface of the tank appears after the ship enters the tank. The ship speed in and out of the tank has a greater impact on the maximum reduction of the tank's water surface. The maximum decrease in the water surface increases as the speed of ship increases.

#### 4.3 Characteristics of Ship Squat During the Process of Ships Entering and Leaving

The ship's speed has a significant impact on the ship squat. At the same speed, the squat of the ship when leaving the tank is larger than that when entering the tank. Therefore, the ship squat when sailing out of the tank is the safe control condition for the ship to enter and leave the tank without the bottoming. As the tank's water depth decreases, the ship squat also decreases, the influence of the tank's water depth is less than the impact of the ship's speed on the squat. Under the same depth of the tank and the speed of the ship in and out, the larger the ship's tonnage, the smaller the section coefficient of the tank, and the greater the squat of the ship (Fig. 3).

Fig. 3. Schematic diagram of ships leaving chamber

The relationship between the maximum squat of the ship (d), entering and leaving the tank, and the water depth (H), section coefficient (n), and speed of the ship (V) can be expressed by two dimensionless quantities:

$$P = \frac{\delta}{H}, \ K = \frac{V^2}{2gH} \times \left[ \left( \frac{n}{n-1} \right)^2 - 1 \right] \tag{1}$$

in the range of 0.3–1.0 m/s, the maximum squat of the ship is basically linear with the square of the ship speed out of the tank (Fig. 4).

Fig. 4. Relationship of ship squat of different ship lifts

#### 4.4 Force Characteristics of the Ship Lock Mechanism

During the process of upstream and downstream docking, ship's entering and leaving the tank, the two groups of locks at the same end of the tank are subject to fluctuations in force and amplitude, and the amplitudes of the lock mechanisms at the upper and lower ends are greatly different, the fluctuation near the closed end of the tank is large, and on the other end is small. It is necessary to pay attention to the impact of downstream water level fluctuation on the load of the ship. Because the downstream water level changes too fast, when the ship is docked with the downstream, the water level between the tank and the downstream waterway may form a difference, resulting in the water loss or full water of the tank. At this time, the four locking mechanisms are significantly more stressed than the docking state of the upstream side, and may even exceed its acceptable range. Therefore, before opening the door of tank, it is necessary to confirm whether there is a water level difference between the downstream and the tank.

#### 4.5 The Design Criteria for the Water Depth of the Ship Lift's Tank

China's "Inland Navigation Standards" (GB50139-2004) stipulates the standard water depth (H) of the tank as calculated by the following formula:

$$H = T + \Delta H \tag{2}$$

In the formula: T- the standard draught of the ship; DH- the additional depth, which is the sum of the ship squat and safety margin without bottoming.

In the "Inland Navigation Standards", the value of the safety margin without bottoming is not clearly stated. Factors that should be considered for the additional depth of the ship: the ship will increase the draught when sailing in shallow water; the unevenness of the tank bottom; the sinking of the stern when the ship leaves the tank. Therefore, the safety margin of the tank should generally be not less than 0.5–0.6 m.

For the 500t class ship lift (designed ship draught 1.6 m) and the 1000 t class ship lift (designed ship draught 2.0 m) to ensure that the minimum normal design water depth of the corresponding tank is 2.5 m or 3.0 m, the safety margin of the tank is 0.5– 0.6 m, the design speed of the ship out of the tank should be less than 0.6–0.8 m/s. Considering the misloaded water depth of the tank, which the limit should not exceed −0.2 m, the speed of the ship out of the tank should be controlled at 0.6–0.7 m/s.

After the ship's speed exceeds 0.7 m/s, the ship's tilting moment caused by the fluctuation of the water surface in the tank increases rapidly. When the ship's entering speed is around 0.8–0.9 m/s, the ship's tilting moment peaks. Therefore, considering the safety margin, the longitudinal tilting moment of the tank, the maximum speed of the ship entering the tank should not exceed 0.7 m/s.

### 5 Conclusion

After years of research, the general formula of calculating the maximum ship squat when ships entering and leaving the tank is established by NHRI. Meanwhile, the mooring force in ship lift tank when the reclining door is opened and closed is established which reveals the unsteady flow influence on safe operation of ship lift mechanism. Putting forward ship lift docking control method to assure the safety of ships and the ship lift which include the reclining door opening and closing speed and water depth control standards, solved the hydrodynamic safety issues when ship entering and leaving the tank of a ship lift.

#### References

Tuck EO (1973) Sinkage and trim in shallow water of finite width. Schiffstechnik 1973:4

Icorels (1980) Report of Working Group IV, No. 35. Bulletin

Barrass CB (1981) Squat-a reply to a comparison of 2 prediction methods. Nav Archit 6:268–271

Norrbin N (1985) Bank clearance and optimal section shape for ship canals. In: 26th PIANC international navigation congress, Belgium

Millward A (1990) A preliminary design method for the prediction of squat in shallow water. Mar Technol SNAME News 27(1):10–19

Millward A (1992) Comparison of the theoretical and empirical prediction of squat in shallow water. Int Shipbuild Prog 39(417):69–78

Eryuzlu NE (1994) Under keel requirements for large vessels in shallow waterways. In: 28th international navigation congress, PIANC, Paper, vol 2, pp 17–25

Gourlay T (2008) Slender-body methods for predicting ship squat. Ocean Eng 35(2):191–200

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# Research on Influence from Ship Navigating in the Intermediate Channel Between Ship Lifts on Hydraulic Characteristics

Yingying Chen1,2(&) , Yaan Hu1,2, and Zhonghua Li1,2

1 Nanjing Hydraulic Research Institute, Nanjing, People's Republic of China

<EMAIL_ADDRESS>2 Key Laboratory of Navigation Structure Construction Technology, Ministry of Transport, Nanjing, People's Republic of China

Abstract. Through the method of physical model test, tests of ship navigating in the intermediate channel of the navigation structure of Longtan Hydropower Station were carried out to study the hydraulic characteristics, such as the water surface fluctuation in the ship chamber and the intermediate channel, the heave of ship bow and stern, and the mooring force of the berthing ship under different navigational conditions. The results indicate that from the perspective of ship lift operational efficiency, docking safety, ship navigating safety and berthing safety, the design navigable water depth of 3.6 m in the intermediate channel is reasonable and the navigational speed of the ship in the intermediate channel should not exceed 1.5 m/s. The research results can provide scientific basis and technical reference for the design and operation of the navigation structure of Longtan Hydropower Station.

Keywords: Longtan hydropower station Ship lift Intermediate channel Hydraulic characteristics Physical model

### 1 Introduction

Navigation structures of high dams mainly include ship lift and multistage ship lock (Li et al. 2014). In the construction of navigation structures for high dams and ultra-high dams, ship lift has been more and more applied in China because of its characteristics of fast operation speed, low investment and water saving (Hu et al. 2016). So far, several different types of vertical ship lifts have been built and operated in Geheyan, Yantan, Shuikou, Silin, Shatuo, Three Gorges, Jinghong, Xiangjiaba and other projects. When the working head of the hydropower station is higher, the multistage ship lift with intermediate channels can be adopted, for instance, the navigation structure of Goupitan Hydropower Station adopted three vertical ship lifts linked by two intermediate channels which has been put into trial operation (Chen et al. 2018).

The intermediate channel between ship lifts is an artificial channel that links multistage ship lifts and is used for the navigation of upstream and downstream ships. Its two ends are both closed, and the channel flow velocity is almost zero. Due to the limitation of engineering terrain conditions, the cross section scale of the intermediate channel is usually smaller than the requirements of restricted channel scale in navigation standard of inland waterway. It is a special restrictive channel. At present, there is no specification or standard for its scale design.

When ship navigates in the narrow closed channel in shallow water like the intermediate channel, its stern may sink obviously and even rub the bottom due to the blocking effect of the ship. And meanwhile, it produces a large surge wave. The surge wave will be reflected after being transmitted to the closed end, superimposed with the advancing surge wave, and oscillate back and forth in the intermediate channel and the ship chambers to form a complex long wave motion, which takes a long time to make the water surface stable. This kind of long wave motion of water flow has certain harm to ships in the intermediate channel, which will cause the ships to sway and shake, and has an adverse effect on the docking of the ship lift.

The navigation structure of Longtan Hydropower Station is a two-stage vertical ship lift with intermediate channel. The maximum lifting height of the two-stage ship lift is 62.4 m and 93.6 m respectively. The two-stage ship lift adopts fully balanced winch lifting system (Chen et al. 2020). The structure of the ship chamber is totally the same, the total length of the ship chamber is 88 m, and the design effective size is 73.0 12.2 3.5 m (length width water depth). The design navigable water depth of the intermediate channel is 3.6 m, and the size of the design ship is 68.0 m 11.0 m 2.4 m (total length shaped width design draft). In this paper, through the method of physical model test, the influence from the ship's navigation in the intermediate channel between ship lifts on the hydraulic characteristics is studied, and the control standards such as the limit speed of the ship's navigation in the intermediate channel is proposed.

### 2 Test Methods and Research Conditions

#### 2.1 Physical Model Design

This paper mainly studies the characteristics of the water surface fluctuation in the intermediate channel and the ship chambers, the heave of the navigating ship, and the mooring force of the berthing ship, etc. Therefore, the model design does not consider the lifting equipment and the itinerary of the ship chamber, and only considers the actual water area between the two-stage ship lift of Longtan Hydropower Station. The actual water area between the two-stage ship lift includes the two ship chambers and the upper head of the first stage ship lift, aqueduct 3 one-way channel and other sections of the intermediate channel between the two-stage ship lift. The plane layout of the two-stage ship lift with intermediate channel of the navigation structure of Longtan Hydropower Station is shown in Fig. 1.

The prototype of the test ship is a 1000t-class single ship. One test ship is the navigating ship and the other one is the berthing ship. The ship model is mainly designed according to the gravity similarity criterion, and the geometric scale is consistent with the hydraulic physical model. In addition to meeting the size and linear geometry similarity of the ship model, the speed of ship model should be similar to that of the real ship, and the displacement of the ship model and the real ship should also be similar.

Fig. 1. Plane layout of the two-stage ship lift with intermediate channel of the navigation structure of longtan hydropower station and the ship navigation starting point and direction

#### 2.2 Measuring Point Arrangement and Measuring Method

In the test, the capacitance liquid level meter was used to measure the water surface fluctuation of the two ship chambers and the intermediate channel between the twostage ship lift, and 24 measuring points were arranged, as shown in Fig. 1. The ultrasonic range finders ware fixed on the front and rear traction device to measure the heave of the navigating ship. The total station instrument measured the speed and distance of the ship navigation. The pressure sensors measured the mooring force of the berthing ship. The test data were automatically collected and preliminarily processed by a multi-channel high-speed data acquisition system.

#### 2.3 Research Conditions

The navigation method of ships in the intermediate channel should ensure the safety of the ships and navigation buildings. On this premise, the navigating time should be shortened as much as possible to improve the passing efficiency. Therefore, the more reasonable navigation method is to navigate along the route with variable speed.

According to previous research, when the ship navigates out of the ship chamber, a large amount of water in the ship chamber is pushed out, which may cause the ship to rub the bottom, affecting the safety of the ship and the ship chamber, and also cause water surface fluctuations in the ship chamber and the intermediate channel, affecting the docking safety of the ship lift and the berthing safety of ship in the intermediate channel (Li et al. 2016). In addition, considering the factors of preventing the ship from collision with the side wall and stabilizing the course, the navigational speed of the ship entering and leaving the upper and lower heads and the aqueduct 3 one-way channel (the narrow section of the intermediate channel) should not be too high. It is therefore decided that the ship should navigate out of the ship chamber of the second stage ship lift (hereinafter referred to as "ship chamber 2"), the upper head and the aqueduct 3 one-way channel at the maximum speed of 0.6 m/s in the test to ensure safety. After the ship leaves the aqueduct 3 one-way channel, the ship navigates in the aqueduct 3 twoway channel, open channel 3 and other intermediate channel sections at the maximum navigational speed accelerating to 1.0, 1.5, 2.0, 2.5 and 3.0 m/s (high speed in the intermediate channel, hereinafter referred to as "high speed").

There were two ship models in the test. One was towed by the traction system and navigated with variable speed in the ship chamber and the intermediate channel, and the other was berthed in the right ship berthing section on the upstream side of the second-stage ship lift (as shown in Fig. 1), and its mooring force was measured to analyze the influence from ship navigating in the ship chamber and the intermediate channel on the berthing conditions of the ship berthing in the intermediate channel.

At the same time, in the test, it is necessary to study the influence from ship navigation on the flow in the intermediate channel and the berthing conditions of the ship when the lower chamber gate of ship chamber of the first-stage ship lift (hereinafter referred to as "ship chamber 1") is open (docking with the intermediate channel) or closed. In particular, it is necessary to measure the water level change outside the downstream gate of ship chamber 1, and analyze whether the water surface fluctuation affects the smooth docking of the first stage ship lift and its downstream intermediate channel. The test groups are shown in Table 1.

Table 1. Test groups

### 3 Water Surface Fluctuation in Intermediate Channel and Ship Chamber

The water level change at typical measuring points and the ship navigation distance with time under the condition that ship navigates at a high speed of 1.5 m/s in the intermediate channel in test group LZ1 are shown in Fig. 2. Table 2 shows the maximum water surface fluctuation at typical measuring points in different test groups.

Combining with the Fig. 1 and Table 2, due to the blocking effect of the ship, when the ship starts, a large amount of water in ship chamber 2 is pushed to the intermediate channel and ship chamber 1 during the ship navigation out of ship chamber 2, upper head of ship chamber 2 and aqueduct 3 one-way channel. The water level of ship chamber 2, ship chamber one-way channel, upper head of ship chamber 2 and aqueduct 3 behind the ship stern decreases, the ship stern sinks obviously and the ship may rub the bottom. The water level of aqueduct 3 two-way channel and its upstream intermediate channel and ship chamber 1 rises, but the amplitude of water surface rise is basically less than that of the water surface decline in upper head of ship chamber 2 and ship chamber 2. When the stern of the navigating ship navigates out of aqueduct 3 oneway channel and enters aqueduct 3 two-way channel, a large amount of water suddenly enters ship chamber 2, upper head of ship chamber 2 and aqueduct 3 one-way channel from the upstream side intermediate channel of aqueduct 3 one-way channel, and forms a relatively large surge wave. This kind of surge wave is reflected after being transmitted from the upstream end of aqueduct 3 one-way channel to the downstream end of ship chamber 2 and superimposed with the surge wave advancing from the upstream. Then the complex long wave motion is formed in the intermediate channel and the ship chambers, which takes a long time for the water surface to restore stability. In the process of ship navigation, aqueduct 3, open channel 2 and aqueduct1 have relatively small water surface fluctuation due to wide section size and large section coefficient.

The navigation structure of Longtan Hydropower Station is a two-stage vertical ship lift with intermediate channel. It is necessary to consider the influence of ship navigating from chamber 2 to the intermediate channel on the downstream docking of the first stage ship lift and the upstream docking of the second stage ship lift. If the water surface at lower head of the first stage ship lift fluctuates greatly and the water level difference between the inside and outside of the ship chamber gate is too large, it may be difficult to open the ship chamber gate for docking. When the water level outside the ship chamber gate is higher than that inside the ship chamber, a large amount of water in the intermediate channel will flow into the ship chamber after the ship chamber gate is opened. When the water depth outside the ship chamber gate is lower than that inside the ship chamber, a large amount of water in the ship chamber will flow out. The water flowing into or out of the ship chamber causes large changes in the weight of water and the longitudinal tilt torque in the ship chamber in a short period of time, which not only affects the operation efficiency of the ship lift, but also threatens the locking mechanism of the ship lift and the safety of navigation or berthing of ships in the ship chamber and the intermediate channel.

Referring to the design requirements of the Three Gorges ship lift, when the ship chamber gate is opened, the water level difference between the inside and outside of the ship chamber is required to be no more than 0.1m. During the docking process of the ship lift, the water surface fluctuation in the approach channel should be no more than 0.1 m too. Therefore, during the docking process of the Longtan ship lift, the water surface fluctuation at lower head of ship chamber 1 and upper head of ship chamber 2 can also be controlled by not more than 0.1 m.

From the process line of water level change and the maximum statistical value of the measuring point at the lower head of ship chamber 1, it can be seen that when the high speed of the navigating ship reaches 2.0 m/s or above, the maximum value of the water surface fluctuation at lower head of ship chamber1 exceeds 0.1 m which will adversely affect the downstream docking safety of the first stage ship lift. Therefore, in order to ensure the downstream safe docking of the first stage ship lift, the navigational speed of the ship in the intermediate channel should not exceed 1.5 m/s.

In the process of ship navigating from ship chamber 2 to the intermediate channel, the water surface decline value is greater than the rise value. The maximum water surface fluctuation at upper head of ship chamber 2 is greater than 0.25m. When the high speed is 1.0–1.5 m/s, the water surface fluctuation at upper head of ship chamber 2 is relatively small which is basically less than 0.1 m after about 15 min. When the high speed is greater than 2.0 m/s, the water surface fluctuation at upper head of ship chamber 2 needs to wait more than 40 min until it is less than 0.1 m. The greater the high speed is, the longer the waiting time for water surface to stabilize is, and the lower the operation efficiency of the ship lift is. Therefore, from the perspective of improving the safety and operation efficiency of the upstream docking of the second stage ship lift, the navigational speed of the ship in the intermediate channel should be lower than or equal to 1.5 m/s.

Fig. 2. Process line of water level change and ship navigation distance with time

Table 2. The maximum water surface fluctuation at typical measuring points (Unit: m)

#### 4 Heave of Ship Bow and Stern

When the ship navigates out of the ship chamber and navigates in the intermediate channel, the water flow pushed away by the forward process of the ship produces a blocking effect under the influence of boundary conditions. The water in front of the ship bow moves to the stern, forming a movement of water around the ship. This backflow movement of water is accompanied by a loss of velocity, resulting in a water level difference. The drawdown of the water level will cause the ship to sink. When the ship sinks a large amount and the water depth of the channel is insufficient, it may cause the ship to rub the bottom, which will have an adverse impact on the safety of the ship and the ship lift. Accordingly, it is necessary study the heave of the ship bow and stern during the ship navigation, so as to determine the reasonable channel water depth and ship speed limits.

The heave of ship bow and stern with ship navigation distance under the condition that ship navigates at a high speed of 1.5 m/s and 2.5 m/s in the intermediate channel in test group LZ1 are shown in Fig. 3. The maximum sinkage and minimum safety margin of the navigating ship and the corresponding ship navigation distance and position in different test groups are listed in Table 3.

The figure and table show that in the process of ship navigation, the ship sinkage value is greater than the rising value. When the high speed of the navigating ship is 1.0–2.0 m/s, the maximum sinkage of the ship occurs when the stern is still in the ship chamber 2, and the maximum sinkage is mainly affected by the speed of the navigating ship out of the ship chamber 2. In the test, the maximum navigational speed of the ship out of the ship chamber is 0.6 m/s, so the maximum sinkage of the ship is 0.27–0.29 m, and the minimum safety margin is 0.93–0.91 m, which is relatively close. When the speed of the sailing ship is 2.5–3.0 m/s, the maximum sinkage occurs when the ship stern is located in aqueduct 2, and the maximum sinkage is mainly affected by the high speed of the navigating ship. When the high speed is 2.5 m/s, the maximum sinkage is 0.43–0.53 m, the minimum safety margin is 0.78–0.67 m. And when the high speed is 3.0 m/s, the maximum sinkage is 1.08 m, the minimum safety margin is only 0.12 m.

In order to ensure the safety of ship navigation, the safety margin at the bottom of the 1000 t-class design ship with a draft of 2.4 m for the Longtan ship lift should not be less than 0.7 m. Therefore, considering that the minimum safety margin of the ship is 0.7 m, the navigational speed of the ship in the intermediate channel should not be greater than 2.0 m/s.

Fig. 3. Process line of heave of ship bow and stern with ship navigation distance

Table 3. The maximum sinkage and the minimum safety margin of the navigatiing ship

Note: The maximum sinkage of the navigating ship is represented by negative value, and the minimum safety margin is represented by positive value, the ship navigation starting point is represented by X = 0 and the X value gradually increases along the ship navigation distance.

#### 5 Ship Berthing Condition

The complex long wave motion oscillates back and forth in the intermediate channel and ship chambers will cause the ships to sway and shake and may easily cause the ship's mooring force to exceed the standard, which will do certain harm to the ships berthing in the intermediate channel and produce potential safety hazards.

The mooring force with time and ship navigation distance under the condition that ship navigates at a high speed of 1.5 m/s in the intermediate channel in test group LZ1 are shown in Fig. 4 and Fig. 5. The maximum mooring force of the berthing ship in different test groups are listed in Table 4.

From the analysis of the figures and table, it can be seen that the maximum longitudinal mooring force of the berthing ship is greater than the maximum lateral mooring force of the corresponding test condition. The maximum mooring force of the

Fig. 4. Process line of mooring force and ship navigational distance with time

Fig. 5. Process line of mooring force with ship navigation distance

berthing ship basically increases with the increase of the high speed of the navigating ship, and the increase is more obvious when the high speed is greater than or equal to 2.0 m/s.

The design ship of the navigation structure of Longtan Hydropower Station is 1000t-class. The maximum limit value of longitudinal mooring force allowed by the code is 32 kN (GB 51177-2016, China), and the maximum limit value of lateral mooring force is 16 kN. Therefore, starting from controlling the mooring force of the berthing ship to meet the requirements of the code, the navigational speed of the ship in the intermediate channel should not exceed 1.5 m/s to ensure the safety of berthing ship in the intermediate channel.

Table 4. The maximum mooring force of the berthing ship (Unit: kN)

Note: The longitudinal mooring force of the berthing ship points to the upstream side as " +" and points to the downstream end as "–", the front lateral and the rear lateral mooring force point to the left as " +" and point to the right as "–".

### 6 Discussion on Ship Navigational Speed

According to the analysis of the above test results, the navigating process of the ship causes the water surface fluctuation in ship chambers and the intermediate channel, the heave of the navigating ship, and the mooring force change of the berthing ship in the intermediate channel. Considering these factors, the navigational speed control standard of the ship in the intermediate channel is discussed.

The process of ship navigation causes the water surface fluctuation in the ship chambers and the intermediate channel, which affects the safety and efficiency of the ship lift docking. From the perspective of docking safety, the water surface fluctuation of lower head of ship chamber 1 shall be less than 0.1m, and the high speed of the navigating ship in the intermediate channel should not be greater than 1.5 m/s. From the perspective of improving the efficiency of the upstream docking operation of the second stage ship lift, the high speed of the navigating ship in the intermediate channel should not exceed 1.5 m/s.

During the navigation of the ship, the stern of the navigating ship sinks obviously. In order to prevent the ship from rubbing the bottom, it is necessary to ensure that there is a certain safety margin at the bottom of the ship. From the perspective of ship navigation safety, the minimum safety margin at the bottom of the ship is 0.7 m, and the high speed of the navigating ship in the intermediate channel should not exceed 2.0 m/s.

The process of ship navigation causes the long wave motion in the intermediate channel and ship chambers which will do certain harm to the ship berthing in the intermediate. Considering the requirement that the mooring force of the berthing ship does not exceed the allowable limit of the code (the maximum allowable limit of longitudinal mooring force is 32 kN and the maximum allowable limit of lateral mooring force is 16 kN), the navigating speed of the ship in the intermediate channel should not be greater than 1.5 m/s.

In addition, after the navigating ship entering the intermediate channel, its acceleration also needs to be controlled. The speed increase should not be too fast, especially when there are berthed ships in the channel of the speed increase section.

### 7 Conclusions

Through the ship navigation test in the intermediate channel of the Longtan ship lift, this paper analyzes the variation characteristics of the water surface fluctuation, the heave of the bow and stern of the navigating ship, and the mooring force of the berthing ship in the intermediate channel. The influence of ship navigation on the docking of the ship lift, the navigating ship and the berthing ship is studied, and the navigational speed control standard of the ship in the intermediate channel is discussed.

Considering the operation safety and efficiency of the ship lift, the design navigable water depth of 3.6m in the intermediate channel is reasonable and the navigational speed of the ship in the intermediate channel should not exceed 1.5 m/s.

Under the conditions of the navigable water depth of 3.6 m in the intermediate channel, the maximum navigational speed of 0.6 m/s of the ship out of the ship chamber, and the high speed of 1.5 m/s of the ship in the intermediate channel, the maximum water surface fluctuation of the lower head of the first stage ship lift is 0.1m, and the water surface fluctuation of the upper head of the second stage ship lift is also less than 0.1 m after about 15 min. The maximum sinkage of the navigating ship is 0.27 m and the minimum safety margin is 0.94 m. The maximum longitudinal mooring force of the berthing ship in the intermediate channel is 28.4 kN and the maximum lateral mooring force is 6.4 kN. The research results can provide scientific basis and technical reference for the design and operation of Longtan ship lift.

### References

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Resistance of Plane Lock Gates Subjected to Ship Impact

Sara Echeverry Jaramillo1(&) , Marine Geers2 , Loïc Buldgen3 , Jean-Philippe Pecquet4 , and Philippe Rigo5

1 Centre de Recherches Des Instituts Groupés de La Haute École Libre Mosane (CRIG-HELMo), Liège, Belgium<EMAIL_ADDRESS>2 University of Liège, Liège, Belgium<EMAIL_ADDRESS>3 Haute École Libre Mosane (HELMo), Liège, Belgium<EMAIL_ADDRESS>4 SBE Engineering, Namur, Belgium 5 University of Liège, Department ARGENCO-ANAST, Liège, Belgium<EMAIL_ADDRESS>
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.

Keywords: Lock gate Super elements Elastic-plastic structural analysis Ship collision

#### 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 = Mship + M∞. The initial kinetic energy to be dissipated by the collision is assumed to be M0V0 2 /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 r0 of 255 MPa. A linear strain hardening is considered by a tangent modulus obtained as ET = 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 Pt and the ship penetration d, 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 E\_ ext is produced by the force Pt applied by the ship on the gate. This way, we can obtain:

$$
\dot{E}\_{\text{ext}} = P\_t V\_0 = P\_t \dot{\delta} \tag{1}
$$

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 E\_ ext ¼ E\_ int. 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 d 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 Pt 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 d, 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-ofplane 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 Eint, the structure is first assumed to resist through a local deformation mode. Nevertheless, with increasing values of the ship displacement d, it becomes more and more difficult for the ship to continue moving. Therefore, for a given penetration d, 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 dt. At the beginning, when the ship first impacts the gate, only the local mode is activated and provides a resistance force Pt. This is true as long as the penetration does not exceed the value dt, which activates then the global mode. This means that at this penetration, the force Pt 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 Pt for the local modes (for each activated super-elements) and the global mode, are presented by Buldgen et al. (2012, 2014).

#### *(a) Local and global deformation modes (b) Collision resistance and internal energy*

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.

c) Collision on a vertical frame d) Collision on a plate

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,d) 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.

Acknowledgements. This research is included within the framework of the Belgian COLLFOWT project from the Walloon Region, plan Marshall- GreenWin-Belgium, 2021– 2023, in partnership with the University of Liège and SBE.

#### References

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Risk-Based Maintenance of Lock Gates Based on Multiple Critical Welded Joints

Thuong Van Dang1(&) and Philippe Rigo2

1 Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam<EMAIL_ADDRESS>
2 Department ArGEnCo, ANAST, University of Liege, Liege, Belgium<EMAIL_ADDRESS>
Abstract. In many countries, inland waterway transport plays a significant role in the overall transport system. Navigation locks and dams are critical for inland waterways which regulate water and allow vessels to navigate. These infrastructures normally utilize large hydraulic steel structures, which are primarily of welded steel structures. Many of the critical welded joints of navigation lock gates have been in service for decades and are experiencing varying degrees of deterioration, mainly from fatigue. Inspection and maintenance of lock gates are expensive, generally requiring the complete closure of locks. Therefore, innovative strategies for the inspection and maintenance of lock gates are required. Fatigue cracks reflect the inherently poor fatigue performance of welded joints. Due to the cyclic loading nature of lock gates the fatigue of critical details requires assessment methods based on the reliability methods. Risk-based inspection planning are used for marine structures but seldom applied to inland navigation lock gates. This paper presents methods to update the failure probability of welded joints considering crack inspection data by using Dynamic Bayesian Network. Optimal inspection and repair plans can be evaluated by risk analysis, combining failure probabilities and associated expected costs for different events. In this study, a numerical example of the procedure of risk-based maintenance of a lock gate based on multiple critical welded joints is described. This reference case is a lock gate fabricated with five critical welded joints corresponding to different equivalent stress ranges. The conclusion is that risk-based maintenance of lock gates based on the optimization of the total expected cost is recommended.

Keywords: Fatigue Lock gates Risk-based inspection planning Dynamic Bayesian networks Welded joints

### 1 Introduction

Hydraulic steel structures are very diverse. Depending on the type of structures and environmental conditions, their components and structures may be suffered from different degradations (fatigue damage). Fatigue damage usually occurs in stressconcentrated areas where localized stress is high. Lock gates are large hydraulic steel structures fabricated by welded vertically and/or horizontally members and steel plates. Due to complex geometry, there are imperfections and residual stresses during the manufacturing process. Welded components are particularly prone to fatigue rather than the base metal. Therefore, the fatigue strength analysis of welded structures at the design stage and operation stage as well as is of high practical interest. In this paper, a numerical example of the procedure of risk-based maintenance of a lock gate with multiple critical welded joints is described. This reference case is a lock gate fabricated with five critical welded joints corresponding to different equivalent stress ranges. These components are vulnerable to fatigue and are designed according to the "category E" of AASHTO.

### 2 Deterioration Modeling

Navigation lock gates, are subjected to cyclic loads. The primary fatigue load is the differential water head between the sides of a lock gate (called water head). It stands for the difference between water levels on the upstream and downstream sides of the lock gate. Since the water head is not exactly the same for each lockage due to seasonal flows of the river, the different stress ranges can be represented by an unique "equivalent stress range" (EUROCODE 2005). In the United States, the standard method for fatigue design and assessment of hydraulic steel structure is the nominal stress approach. The nominal stress approach is based on S–N curves when the capacity of welded steel joints related to the fatigue strength is represented by these S-N curves. There is no specific S-N curve available for hydraulic steel structures. The United States Army Corps of Engineers has recommended the code ETL 1100-2-584 (USACE-ETL-1110-2-584 2014) to perform fatigue assessment with the American Association of State Highway and Transportation Officials (AASHTO) code. Table 1 summarizes the parameters of the S-N.

Table 1. Input data of welded joints for S-N model

The fatigue crack growth model is utilized for incorporating crack inspection results in assessing the failure probability. The most widely used model is the Paris law (Paris & Erdogan 1963). Table 2 presents the parameters of the fatigue crack growth model. Assuming that the geometry function (Y) is constant, crack size at time t can be calculated as shown in Eq. (1).

$$a\_t = \left[a\_{t-1}^{1-\frac{m}{2}} + \left(1 - \frac{m}{2}\right)CB\_s^m \Delta \sigma^m Y^m n\pi^{m/2}\right]^{2/(2-m)}\tag{1}$$

where a0 (t = 1) is the initial crack size, C and m are material parameters. Dr is the equivalent nominal stress range used for fatigue analysis, n is the number of cycles per year and Bs is the load uncertainty.

Table 2. Input data of critical welded joints for the FM model

Figure 1 shows the fatigue reliability curves for the five critical welded joints.

Fig. 1. Fatigue reliability of critical welded joints

#### 3 Dynamic Bayesian Network (DBN)

A Dynamic Bayesian Network (DBN) is a special class of Bayesian network, that represents the temporal evolution of variables over time. DBN was developed in the early 1990s by extending static belief-network models to more general dynamic forecasting models (Dagum et al. 1992). A DBN framework for stochastic modeling of deterioration process and updating the failure probability is proposed by Straub (Straub 2010). DBN is used to update the failure probability for critical welded joints of the lock gate. For a conservative way, it is assumed the lock gate fails when a critical welded joint fails.

The variable q = 1 m 2 CBm s DrmYmnpm=2 (see Eq. (1)) is defined in order to reduce the dimension of the joint distribution and consequently the computational time. The crack depth at the end of each year can be expressed recursively as a function of the crack depth in the previous year as shown in Eq. (2).

$$a\_t = \left[a\_{t-1}^{1-\frac{\mathbf{w}}{2}} + \mathbf{q}\right]^{2/(2-m)}\tag{2}$$

By instantiating the inspection variables It in the DBN with the observed events at the times of inspection, the failure probability is updated considering the inspection outcomes.

The DBN representation including inspection results It for critical welded joints are shown in Fig. 2.

Fig. 2. DBN of multiple components

### 4 Risk-Based Decision Analysis

Risk-Based Inspection (RBI) is an attractive issue for hydraulic steel structures due to the increase of aging structures and that many failures are detected. The aim of RBI is to find a balance between the benefit of inspection, repair schedule versus failure cost. The failure probabilities using DBN are combined with the cost model.

Risk-based maintenance of the lock gate is performed according to three scenarios:

The different unitary costs utilized in this numerical example are shown in Table 3. Because there is a difference of maintenance cost between a single welded joint and multiple welded joints, it is assumed that inspection cost and repair cost for an individual welded joint is equal to 50% of the inspection cost and repair cost that are used for multiple welded joints.

Table 3. Relative cost characteristics of critical welded joints

In the first scenario, the optimal schedule is performed for each welded joint independently.

As result, the expected cost for the lock gate is calculated by Eq. (3).

$$E\_{\text{totgate}} = E\_{\text{totA}} + E\_{\text{totB}} + E\_{\text{totC}} + E\_{\text{totD}} + E\_{\text{totE}} \tag{3}$$

Figure 3 shows the result of the optimization of the total expected cost for the critical welded joints, A to E, independently.

The result of the optimization cost for each critical welded joint (A to E) is given in Table 4 and Fig. 3.

Fig. 3. Results of optimizing the total expected cost of single joint cost considered separately

Table 4. Result of optimizing for each critical welded joint

In the second scenario (combined critical welded joints), the inspection and repair of the five critical welded joints are performed with a similar time period to figure out the optimal schedule. Repair cost for each component is equal to the repair cost of the entire gate divided by the number of investigated critical welded joints. The failure probability of the lock gate Pfgate is calculated by Eq. (4).

$$\begin{aligned} P\_{\text{fyx}} = P\left(\sum\_{i=1}^{v} A\_i\right) &= \sum\_{i=1}^{v} P(A\_i) - \sum\_{l$$

The total expected cost for the lock gate of the second approach is then calculated by Eq. (5)

$$E\_{\text{totgate}} = \sum\_{t=1}^{T} C\_f P\_{f\text{gate}}(t) \frac{1}{\left(1 + \mathbf{z}\_r\right)^t} + \sum\_{i=1}^{T\_{\text{imp}}} \frac{\sum\_{j=1}^{v} \left(C\_{rj} P\_{\eta j}\right)(T\_i) + C\_{\text{insp}}}{\left(1 + \mathbf{z}\_r\right)^{T\_i}} \tag{5}$$

where:

Cinsp inspection cost,

Crj repair cost for a critical welded joint,

Cf failure cost,

ar annual discounting rate,

Pf probability of failure,

Prj probability of repair for a critical welded joint,

Ti year of inspection.

v number of critical welded joint.

Figure 4 shows the outcome of the optimal schedule the lock gate with multiple critical welded joints. The optimal expected cost of the gate is equal 2.38104 . The inspection and repair interval is of 6 years.

Fig. 4. Results of risk-based maintenance the lock gate with multiple critical welded joints

Thirdly, we consider two groups of combined critical welded joints. Critical welded joints having simultaneously inspection and repair time are combined together.

In this example, five critical welded joints are divided into two groups:

Depending on the groups, the cost of inspection and repair are different. It is assumed that inspection and repair cost for group 1 and group 2 are equal to 80% and 90% of the inspection and repair cost that are used for the multiple welded joints (Table 3).

The procedure of the calculation in the third scenario is then similar to the second scenario (Eq. (4) and Eq. (5)) and the result is provided in Table 5.

Table 5. Total expected cost of the lock gate (100 years)

In the comparison of the three scenarios, we see that, the optimized total expected cost for the lock gate based on multiple critical welded joints gets better result (Table 5), with smaller cost (50% and 60%) than the optimized total expected cost based on the failure probability of independently critical welded joints and the groups of multiple joints.

### 5 Conclusions

This paper presents an innovative methodology to establish inspection and maintenance of lock gates considering multiple critical welded joints. The risk-based maintenance of a lock gate performed based on the optimization of the total expected cost of the combined critical welded joints is recommended.

Acknowledgements. The authors acknowledge the financial support provided by the Wallonie-Bruxelles International (WBI), Belgium for this research.

### References

Dagum P, Horvitz E, S U (1992) Dynamic network models for forecasting section on medical informatics stanford university school of medicine abstract we have developed a probabilistic forecasting methodology through a synthesis of belief network models and classical timeseries analys. pp 41–48

EUROCODE (2005) European standard EN 1993.1.9. vol 7, no 2006

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# Ship Impact for Suederelbe Bridge Crossing in Hamburg

Claus Kunz1(&) and Jan Schülke2

1 Bundesanstalt fuer Wasserbau (BAW), Karlsruhe, Germany<EMAIL_ADDRESS>2 Deutsche Einheit Fernstrassenplanung- und bau GmbH (DEGES), Berlin, Germany

Abstract. In the harbour of Hamburg a new bridge crossing the Suederelbe at Elbe km 620 has been planned. The bridge is part of the BAB26 motorway. Contracted by DEGES, a German company for highway planning and design, BAW determined the ship impact loads on the basis and methodology of EN 1991-1-7 (2006) on a site - specific basis, BAW (2016). The Suederelbe Bridge crossing will have a length of 695,60 m with main span bridging the fairway over 350 m. The pylon height of the main piers is planned to be 150 m. The clearance height of the Suederelbe crossing should be 53 m above sea level. The two pylons and main piers of the Suederelbe bridge crossing, which are to be positioned close to the bank, are at risk of ship impact. About 6,000 seagoing vessels up to 33,000 dwt are passing the future bridge. Fleet structure, ship passages, speeds, accident rates and nautical conditions were analysed and ship impact loads for the piers and protective structures were determined using a load model on one hand and a collision model on the other hand. Load and collision model are probabilistic and based on corresponding distributions of the decisive influencing parameters, whereby this approach represents state-of-the-art technology. Based on the collision model, the average time that the eastern pylon is being impacted is expected to be 470 years, while the average time for the western pylon is expected to be 1,240 years.

The impact loads for the east pylon are dynamic loads with FFdyn = 17.5 MN for frontal impact and FLdyn = 4.0 MN for lateral impact. Out of the models the impact load can be determined as dependent on the distance of the piers (pylons) to the fairway center line. The variation of the span distance of the pylons show the plausibility of the developed and used modelling.

Keywords: Ship collision Seagoing vessels Load model Collision model Probabilistic ship impact

#### 1 Introduction

DEGES GmbH is planning the new Suederelbe Bridge crossing Hamburg-Moorburg as part of the BAB26 motorway at Elbe km 620 for the Hanseatic City of Hamburg. For the planned bridge, bridge piers close to the bank are planned on both sides of the fairway. Ship impact loads for the service lifetime condition cannot be derived directly from the introduced relevant code EN 1991-1-7 (2006) and DIN EN 1991-1-7/NA (2010) from a technical-economic point of view. This is due, on the one hand, to the insufficiently classified and blanket ship impact loads for seagoing vessels specified in EN 1991-1-7 and, on the other hand, to the pier positions provided in the bank area, so that a case-by-case investigation has to been carried out with regard to ship impact loads.

On behalf of DEGES, Bundesanstalt fuer Wasserbau (BAW) determined the ship impact loads within the scope of a case-by-case consideration on the basis (methodology) of EN 1991-1-7 and adjacent German National Annex for the final condition and specified object-specific requirements regarding the proof for ship impact (e.g. geometries, e.g. deformation proofs,…). Traffic pattern, ship passages, speeds, accident occurrence, nautical conditions have been analysed and the impact loads for piers or protective bank structures are determined by a link of a load model with a collision model. In the absence of data, comparative observations have been made.

### 2 Bridge and Waterway Specifications for Suederelbe

#### 2.1 Bridge Crossing Suederelbe

The new Suederelbe Bridge, Figs. 1 and 2, will be part of the BAB26, which in the eastern section will serve as a cross-connection between the motorways BAB7 and the BAB1 to accommodate supra-regional east-west traffic, improve the accessibility to Hamburg port, bundle long-range port traffic and thus relieve inner-city residential quarters in the south of Hamburg. Currently, work on the construction of the BAB26 from the west to the BAB7 is in full swing. In preparation for the work up to the BAB1, the approval procedures have been initiated in three sections and the designs are being planned. The total length of the new bridge infrastructure amounts to about 5 kms.

Fig. 1. Animation Suederelbe Bridge crossing, view from west to east; Source: DEGES

Fig. 2. Top view of the planned Suederelbe Bridge crossing; Source: DEGES

The new Süderelbe crossing in Hamburg will be designed as a cable-stayed bridge with two high piers (pylons), Figs. 1 and 3. The winning design of the competition comes from a German-Danish planning consortium. The construction is scheduled to begin in 2026.

Fig. 3. Longitudinal section/view of the planned Suederelbe crossing; Source: DEGES

The current bridge of the new Suederelbe crossing is a five-span symmetrical cablestayed bridge, with centrally arranged double cable levels in fan shape. The structure is designed as a semi-integral construction with spans of 86.05–86.75 - 350.00 - 86.75– 86.05 m, total length is 695,6 m. The clearance under the bridge has a height of 53 m and thus approximately 51 m above the mean high tide, with a width of 150 m. The planned construction depth is - 17 m + NN corresponding to a water depth of 15 m plus 2 m for tolerances and scouring. The cable-stayed bridge is supported by pylon stems arranged centrally in the route axis with a height of about 150 m next to the Suederelbe and piers in the foreland areas. The superstructure consists of a one-piece cross-section consisting of two partial cross-sections coupled by cross girders, separated by a centrally arranged light gap, Fig. 4. The anchoring of the cables takes place within the inner cells of the superstructure cross-section adjacent to the light gap.

The western pier (pylon) stands on the edge of the Suederelbe and is protected by a bank wall parallel to the course of the river. On the east side, the pier is located directly at the entrance to the Hohe Schaar harbour a harbor for inland vessels, roughly in line with the eastern course of the river bank. To protect the piers additionally to the impact

Fig. 4. Cross section of the planned Suederelbe crossing; Source: DEGES

design, bank walls and impact protection measures will be provided in front of the deep foundations.

The Suederelbe is tidally influenced, so that water levels of up to 6.9 m + NN can be expected at the bridge site. The planned heights of the ground surfaces in the area of the pier cap are 6.0 m + NN on the west and east sides. A bank wall in front is also being considered for the west side.

The locations of the planned piers (pylons) are in the bank or embankment area, so that direct ship impacts are not very likely to occur. However, an impact can be indirectly transmitted to the bridge pier foundations via the bank walls, which must be taken into account in the planning. Due to the nearby Kattwyk Bridge with a main opening of 108.5 m, which is directly adjacent to the planned Suederelbe Bridge crossing, the vehicle flows from the north are bundled to a lane width of B = 94 m.

Endangered by ship impact are therefore:

#### 2.2 Suederelbe Waterway

The Suederelbe is an approximately 16 km long section of the Lower Elbe in the area of the city of Hamburg, Fig. 5. The Suederelbe bypasses the historic port areas of Hamburg and instead passes Harburg and some newer port areas. Like the entire Lower Elbe, the Suederelbe is also subjected to the tides. The mean tidal range is about 3.60 m and leads to strong currents, Faltboot (2016).

On average, about 725 seagoing vessels sail per year and direction. The ships come repeatedly or regularly. The displacement ranges from approx. 200 m3 to 107,000 m3 , the average value is 10,550 m3 ; 98% of the transits for seagoing vessels remain below 50,000 m3 , Table 1. The most significant shares in the transit volume are accounted for by the general cargo ship with approx. 42% (length < 120 m), the chemical tanker with approx. 36% (length < 180 m), the tanker with approx. 13% (length < 201 m) and the

Fig. 5. Overview of the port of Hamburg and the Suederelbe; Source: Hamburg Port Authority

bulk carrier with approx. 8% (length < 204 m). Tug assistance is provided to vessels with a length of more than 200 m, in addition to wind forces of more than 8 Beaufort and visibility of less than 3 ship lengths. Every year, about 70 coal ships with up to 100,000 tdw for the Moorburg power plant station with a maximum discharge T = 12.8 m have been towed aft ahead.

There are about three times as many inland vessels as seagoing vessels, so that these are additionally taken into account with a vessel number in the size of 17,400 inland vessels in the smallest displacement class <5,500 m3 . For the planning horizon, which is the design service life of the new bridge from the start of construction, the updated vessel passages and distribution were retained after discussion with the client. The updated size distribution, which was divided into m3 -classes with a class width of 5,500 m3 , is shown in Table 1.

Table 1. Class frequencies of ship displacements for the Suederelbe crossing; Source: BAW (2016)

Table 1 shows an annual passage rate of about 3,000 ships per direction and year, a total of about 6,000 ships per year. Approx. 90% of all ships, including the inland waterway vessels, have a displacement of less than 11,000 m3 , approx. 95% less than 33,000 m3 .

#### 3 Determination of Ship Impact

#### 3.1 General

In general, the impact action is characterised by two load directions which do not occur simultaneously and which are assumed to act perpendicularly to the surface of the structural member:

The point of application of the lateral impact should be set in such a way that the most unfavourable effect is achieved (moveable action).

According to EN 1991-1-7 (2006), C.4.4(2), probabilistic models of the basic variables determining the deformation energy or the impact behaviour of the ship can be used for the determination of the impact loads. A collision model according to EN 1991-1-7 (2006), B.9.3.3, has been used to determine the collision rate. For the treatment of the ship impact the impact load model was linked with the collision model in order to get a distribution of probable impact loads, with which the design value is determined using acceptable risk criteria.

#### 3.2 Impact Mechanics