Reliability Based Rehabilitation of Existing Hydraulic Structures

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 standard-based 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.

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:

$$ \sigma_{cd}^{*} = \frac{3}{4} \cdot \sigma_{cd} < f_{cd} { } $$

(1)

with: ${ }\sigma_{cd} -$ Maximum stress value of the triangular compressive stress distribution.

$\sigma_{cd}^{*}$ - 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

$$ \sigma_{{{\rm{cd}}}} = \frac{{{\rm{N}}_{{\rm{d}}} }}{{\rm{h}}} + \frac{{6 \cdot {\rm{N}}_{{\rm{d}}} \cdot {\rm{e}}_{{\rm{d}}} }}{{{\rm{h}}^{2} }} - H_{2} \cdot \gamma_{w} $$

(2)

With water pressure on both sides, the effective concrete compressive stress $\sigma_{cd}$ can be determined according to BAW (2016), Eq. (8), as follows:

$$ \sigma_{cd} { } = \sigma_{abs} - H_{2} \cdot \gamma_{w} $$

(3)

$$ \sigma_{abs} = H_{2} \cdot \gamma_{w} + \frac{4}{3} \cdot \frac{{\left( {N_{Ed} - h \cdot H_{1} \cdot \gamma_{w} } \right)^{2} }}{{\left[ {h \cdot \left( {N_{Ed} - h \cdot H_{1} \cdot \gamma_{w} } \right) - 2 \cdot N_{Ed} \cdot e_{d} } \right]}} $$

(4)

with: ${H}_{1}$ - Water level on the side with compressive stresses in [m].

${H}_{2}$ - opposite water level height from ${H}_{1}$ in [m].

${\gamma }_{w}$ - Weight of water in [kN/m³].

${e}_{d}$ - load eccentricity under the design internal forces ${M}_{Ed}$ und ${N}_{Ed}$ in [m].

The final limit state function for reliability evaluation becomes

$$ {\rm{G}}\left( {\rm{x}} \right) = \theta_{{\rm{R}}} \cdot \left( {\alpha_{cc} \cdot f_{{c{\rm{m}}}} } \right) - { }\theta_{{\rm{E}}} \cdot \sigma_{cd}^{*} $$

(5)

where ${\alpha }_{cc}$ the coefficient to account for long-term effects on the concrete compressive strength.

${\theta }_{\mathrm{R}}$ is the coefficient of uncertainty in the resistance model and ${\theta }_{\mathrm{E}}$ 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(e_d) to the section length (h) is e_d/h < 1/6, considering the case where a full section is under compression. Limit-2 represents cracked sections with 1/6 < e_d/h < 1/2 and Limit-3 is when the section is fully cracked with e_d/h > ½. Furthermore, this aspect directly influences the effective area, an important aspect for calculation of CPWP and consequently N_wd, M_wd (revised normal/moments with CPWP).

Therefore, the equation for $\sigma_{abs}$ for calculation of compression states is valid only if the following application limit is met:

$$ \frac{{e_{wd} }}{h} = \frac{{e_{d} /h}}{{1 - \overline{\sigma }_{wd} }}\, < \,0.5 $$

(6)

At an eccentricity $e_{wd} /h > 0.5$ there is a stability problem (no compression zone any more, load resultant $N_{Ed}$ outside the cross section). Thus the limit state function for reliability evaluation becomes

$$ {\rm{G}}\left( {\rm{x}} \right) = 0.5 \cdot \theta_{{\rm{R}}} { } - { }\theta_{{\rm{E}}} \cdot \frac{{e_{d} /h}}{{1 - \overline{\sigma }_{wd} }} $$

(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:

$$ {\Sigma }\left( {V_{{\rm{E}}} } \right)\,\, \le \,\,A_{cc} \cdot v_{Rd} $$

(8)

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

$$ { }v_{Rd} = c \cdot f_{ctd} + \mu \cdot \sigma_{n} \le 0,5 \cdot v \cdot f_{cd} $$

(9)

where, c, $\mu$ and $v$ are coefficients as a function of the joint roughness.

For usual cross sections the verification is performed assuming a “rough” joint ($c = 0,4$ / $\mu = 0,7/v = 0,5)$ 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

$$ {\rm{V}}_{{{\rm{rd}},{\rm{ anchor}}}} = 0.5 \cdot A_{{{\rm{anchor}}}} \cdot {\rm{f}}_{{\rm{y}}} $$

(10)

where ${\rm{f}}_{{\rm{y}}}$ is the strength of the anchor and $A_{{{\rm{anchor}}}}$ is the cross sectional area of anchor.

The final limit state function for reliability evaluation becomes

$$ {\rm{G}}\left( {\rm{x}} \right) = \theta_{{\rm{R}}} \cdot \left( {A_{cc} \cdot \mu \cdot \sigma_{n} + 0.5 \cdot A_{{{\rm{anchor}}}} \cdot {\rm{f}}_{{\rm{y}}} } \right) - { }\theta_{{\rm{E}}} \cdot {\Sigma }\left( {V_{{\rm{E}}} } \right) $$

(11)

where ${\theta }_{\mathrm{R}}$ is the coefficient of uncertainty in the resistance model and ${\theta }_{\mathrm{E}}$ 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.

$$ P_{f} = {\rm{Pr}}\left[ {g\left( {{\mathbf{\underline {X} }}} \right) \le 0} \right] = \int_{{g\left( {{\mathbf{\underline {X} }}} \right) \le 0}}^{{}} {f_{{{\mathbf{\underline {X} }}}} \left( {{\mathbf{\underline {x} }}} \right)d{\mathbf{\underline {x} }}} $$

(12)

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

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

$$ \mathop {{\rm{min}}}\limits_{d,X} \,F\left( {d,{\rm{X}}} \right) $$

(13)

$$ {\rm{s}}{\rm{.t}}{. }\left\{ {\begin{array}{*{20}l} {{\rm{Pr}}\left[ {L_{i} \left( {d,X} \right) \le 0} \right] \le P_{{\rm{f}}}^{{{\rm{Target}}}} \left( {i = 1,2, \cdots ,n} \right)} \hfill \\ {C_{j} \left( d \right) \le 0{ }\left( {j = 1,2, \cdots ,m} \right)} \hfill \\ \end{array} } \right. $$

(14)

where $d$ is the deterministic design vector, ${\varvec{X}}$ random vector is uncertain design vector, $F\left( {d,{\rm{X}}} \right)$ is the optimization function, $L_{i} \left( {d,X} \right) \le 0$ is the limit state function for failure, $P_{f}^{{{\rm{Target}}}}$ is the target failure (allowable), ${\rm{Pr}}\left[ {L_{i} \left( {d,X} \right) \le 0} \right]$ is the probability of failure under the given operational conditions and state, $n$ is the number of uncertain constraints,$C_{j} \left( {\varvec{d}} \right) \le 0$ 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 ${P}_{f}^{\mathrm{\top arget}}$,i.e. the allowable failure probability, which can be also translated into the reliability index β through the relation ${P}_{{f}_{\mathrm{Target}}}=\Phi \left(-{\beta }_{\mathrm{Target}}\right)$ where $\Phi$ 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 a_s = 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 β = 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.