**Rüd, Sophie; Müller, Hilmar; Fleischer, Helmut; Stephan, Christoph**

**Development of a verification procedure of partial loading** **on existing solid hydraulic structures - probabilistic**

**assessment for 3D material variations**

### Lecture Notes in Civil Engineering

### Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/110889 Vorgeschlagene Zitierweise/Suggested citation:

### Rüd, Sophie; Müller, Hilmar; Fleischer, Helmut; Stephan, Christoph (2023): Development of a verification procedure of partial loading on existing solid hydraulic structures - probabilistic assessment for 3D material variations. In: Li, Y.; Hu, Y.; Rigo, P.; Lefler, F.E.; Zhao, G. (Hg.):

### Proceedings of PIANC Smart Rivers 2022. Lecture Notes in Civil Engineering 264.

### Singapore: Springer. S. 372-383. https://doi.org/10.1007/978-981-19-6138-0_33.

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### of Partial Loading on Existing Solid Hydraulic Structures - Probabilistic Assessment for 3D

### Material Variations

Sophie Rüd^{(}^{&}^{)}, Hilmar Müller, Helmut Fleischer,
and Christoph Stephan

Federal Waterways Engineering and Research Institute, Karlsruhe, Germany sophie.rued@baw.de

Abstract. One of the challenges in assessing the load-bearing capacity of existing solid hydraulic structures is the formal veriﬁcation of concentrated loads for plain concrete. Due to the age bandwidth of such structures in Ger- many, this applies to hundreds of cases and especially to older structures of rammed concrete. Typical examples of components subjected to partial loads are found at weir pillars: e.g. support niches of inspection closures. Although they cannot be formally veriﬁed using the current regulations, the BAW Code of Practice “Evaluation of the load bearing capacity of existing solid hydraulic structures”(TbW) allows more detailed investigation methods to be applied, e.g.

the use of non-linear probabilistic calculations. The principle research motiva- tions are a higher loading capacity by numerical simulations with a more real- istic material model compared to the usual linear calculations and a higher loading capacity by reproducing a “natural” bandwidth of material character- istics in these simulations. The aim of the current research project is the development and standardisation of the numerical simulations for such a veri- ﬁcation procedure and its underlying safety concept by a classiﬁcation of structural markers. As a result, the necessity of complex reinforcements for such structures could be assessed. The paper introduces the research concept and addresses the investigation steps regarding measured and generic 3D material distributions and FEM representation speciﬁcs as the material model. Further- more, the preparation of the stochastic analysis is introduced by a demonstration model: The resulting hundreds to thousands of simulations of individual cases enable the stochastic analysis of metamodels to deduce general probabilistic results. Prospectively, the demonstration model will be transferred to further component measures and compressive strength classes.

Keywords: Veriﬁcation

^{FEM}3D material distributionProbabilistic

Existing solid hydraulic structures

©The Author(s) 2023

Y. Li et al. (Eds.): PIANC 2022, LNCE 264, pp. 372–383, 2023.

https://doi.org/10.1007/978-981-19-6138-0_33

### 1 Introduction

Assessing the load-bearing capacity of existing solid hydraulic structures is generally based upon stylized assumptions in the calculation methods, including a severely simpliﬁed model of the material behaviour. Additionally, the assessment assumes an at least per section homogeneous construction material and therefore constant material parameters. Such simpliﬁcations are balanced in the assessment process, e.g. by safety factors and characteristic values as material parameters. For example, the broad scat- tering of the composition, processing and mixing of the building materials is addressed in the hypothesis of a normed material distribution and a 5% quantile approach of the characteristic material parameters. Furthermore, the standard-compliant zeroing of the tensile strength under bending stress prevents formal veriﬁcation of the load-bearing capacity of plain concrete structures components under partial loading. This may result in complex reinforcements such as anchoring or additional constructions as sheathing of the loading areas. Figure1illustrates this kind of loading case for a weir pillar and the corresponding anchor reinforcements constructed at a weir pillar at the Donau- barrage Kachlet. In Germany, there are currently circa 700 weirs and locks in the portfolio of the Federal Waterways and Shipping Administration (WSV), of which almost 200 structures are over 100 years old and about 300 between 50 and 100 years, therefore this applies to hundreds of cases and especially to older structures build of rammed concrete. To address this challenge not on an individual case basis but to develop a veriﬁcation procedure, the numerical simulations require a scientiﬁc foun- dation and standardisation in their simulation scenarios as well as their analysis in accordance to an underlying safety concept, e. g. by a classiﬁcation of structural markers. As a result, the appraisal of subsequent and complex reinforcements for such structures could be supported in terms of necessity and efﬁciency. The classiﬁcation of revision recesses and their loads in a decision matrix, according to their geometric and material-speciﬁc/construction time period, corresponds with a reduction in the inves- tigation effort per individual case. The BAW Code of Practice“Evaluation of the load bearing capacity of existing solid hydraulic structures” (TbW) offers an assessment process set in three stages A, B and C. The stages mirror a rising accuracy in the assessment and the corresponding exploration of possible reserves in loading capacity:

In the calculations of the ﬁrst two stages, characteristic material values are either proposed according to the relevant material or construction period (stage A) or result from measurement campaigns (stage B) as 5% quantile values. In the third stage in accordance with TbW, FEM simulations are used to represent the non-linear material behaviour as realistically as possible, while probabilistic methods account for the anticipated range of the building material.

Regarding the FEM simulations, the main motivations are twofold. First, a higher loading capacity compared to the linear calculations by numerical simulations with a more realistic material model. The appeal of this aspect includes not only an extension of the linear material behaviour stipulated in typical veriﬁcation calculation, but notably the load case independent inclusion of tensile strength. Second, reproducing a heterogeneous realistic 3D bandwidth of material characteristics averages to a higher overall loading capacity than the homogenous applied 5% quantile. However, the 5%

quantile is part of the safety concept in EC0 (2002) and based upon model assumptions as well as material uncertainties, e.g. the broad spread of compositions and 3D localisation of the materials used in concrete construction. Furthermore, the material distribution of the loading areas is not known in such a detail as the simulations require.

A representative appraisal therefore requires an appropriate probabilistic analysis of FEM simulations with 3D distributionﬁelds of the material characteristics to account for the influence of material distribution and spatial permutations. The resulting hun- dreds to thousands of individual case simulations can be merged in parameterised result analysis and thereby enable a stochastic analysis. By building and sampling meta- models, general probabilistic results can be deduced at a deﬁned safety level. Figure2 introduces the FEM simulation preparation by illustrated process steps. The 3D dis- tributionﬁelds are spatial layout variations of generically produced distributions based on literature research and evaluation of measurements. Mapped on a section of a structural component and infused with a nonlinear material model, they constitute variations in otherwise constant FEM simulation scenarios. Each scenario yields one set of result values, e.g. a maximum force or number and failed elements prior to the maximum force. Approximating a sufﬁcient amount of result sets via metamodels enables a probabilistic analysis of further scenarios. In the next sections these process steps are discussed and the preparation of the stochastic analysis is showcased by a demonstration model.

Fig. 1. Left: loading scenario partial loading, middle: loading scenario revision recess, right:

reinforcements realized at Kachlet. Drawings modiﬁed from KW011A-SG01-4 (2016)

### 2 3D Distribution Fields of Material Parameters

2.1 Measurements

The influence of the spatial variation of material properties on the load-bearing behaviour of a structure depends not only on the distribution ensemble of the examined material properties, but also on the distribution pattern and its characteristics.

Depending on its properties, the 3D variation can furnish decisive insights into the load-bearing capacity of the component in question and thus constitute a design parameter itself. Generating 3D distributionﬁelds requires choices about the type and parameters of the 3D arrangement, e.g. random, structured or partially structured.

A key attribute is the (self-) correlation length, which describes the distance of influ- ence for a speciﬁc property. Regarding the correlation lengths of concrete, current studies point to a range of meters. The Joint Committee on Structural Safety (2002) for example invokes a correlation length of 5 m, based on Kersken-Bradley (1985) with 5 m for slim and 1 m for massive structures. Bouhjiti (2018) concurringly stipulates a correlation length of 1 m based onﬁndings of De Larrard (2010), whose experiments and analysis however cautioned regarding scattering and the dependence per investi- gated parameter. Correlation lengths less than 1 m were obtained in a non-destructive measurement campaign by Borosnyói (2015) conducting spatial analysis with different rebound hammers. Rebound measurements offer a higher resolution than typical destructive testing methods and yield a velocity based rebound quotient (Q value), correlating to the compressive strength. Turning to older plain concrete, and especially to rammed concrete, the construction process per layer and successive condensing by manual or mechanical rammed implies a vertically dominated layout of the concrete density and composition within each layer and across layers. This layout is visually detectable in cut-outs and the height per layer usually ranges from 0.15 to 0.5 m (Rehm 2019; DIN 1045:1925-09). However, in view of the recess area size, the numerical simulations aim for a spatial and materialﬁeld discretising in the range of centimetres.

In order to obtain realistic distribution ﬁelds of correspondingly high resolution of Fig. 2. Creation process of metamodels for reliability analysis based on FEM simulations of recesses with three-dimensionally distributed material parameters.

rammed concrete, three series of rebound measurements were carried out on recent cut-
out sections of two existing hydraulic structures and a test wall in 2020/21. Theﬁrst
structure is a double lock at the barrage Kachlet, situated at Donau-km 2.230 and in
operation since 1928. Measurements were taken inside the massive lower head of the
lock in the shaft partition during a repair measure. As the shaft excavations exposed
relatively smooth vertical sections of quasi-unreinforced concrete, two measurement
areas of about 2 and 3 m^{2}at different height levels inside a massive hydraulic structure
were available. The measurements at the Koblenz weir (Mosel-km 1.944, 1951) formed
an edge area counterpart, offered by the construction of an inspection recess in vicinity
to theﬁrst weir pillar. The rammed concrete test wall, courtesy of another research
project from 2015 at the BAW headquarters in Karlsruhe, represents a relatively slim
construction under laboratory conditions and encompasses three measurement areas.

Although their horizontal extent is limited to less than half a meter, as for the recess, the cut-out section of the test wall enabled a horizontal comparison over a meter (cut-out length) as well as for the decimetre range (homogenisation depth per respective cut-out face). Due to a signiﬁcant amount of non-valid measurements at theﬁrst measurements in Kachlet, the rebound hammer was switched and, contrary to DIN EN 12504-2:2012- 12, the presented tests have been carried out with a rebound hammer of low impact energy to include low compressive strength values of less than 10 MPa. In order to record several individual layers and layer transition areas, measuring grid sizes of 0.3 m (Koblenz, Karlsruhe) to 0.5 m (Kachlet) match the spatial discretisation requirements of the numerical simulations in accordance with the minimal distance requirement (EN 12504-2:2012) of 2.5 cm between the individual measuring points.

Due to the grid measurement, there was no pre-selection according to DIN EN 12504- 2:2012-12, so areas were tested that would normally have been avoided due to rock grain, roughness, porosity, etc. Equally, test results were included even if the test caused subsiding of the test point. All in all, the measurements amounted to*4000 single point values, each visually categorised (unannotated, stone, subsided on testing etc.) to form subsets per measurement area. For each area, the analysis differentiated ensembles by these subsets and their combinations. An example of the rebound results mapped on the measurement area is given in Fig.3on the left side with the face colour corresponding to the measured Q-values of the rebound hammer. Their edge colouring and line style indicate the visual category.

The main investigation objectives are the distribution characteristics, correlation length in vertical as well as horizontal direction respectively patterns or trends tailored to layers induced by the construction process. Furthermore, differences between the measurement objects are discussed. Analysing eventual layer patterns included the line wise comparison of median, mean and standard deviation and horizontal and vertical direction per (sub-)sets as well as semi-variograms, yielding a vertical correlation length in the decimetre range per area. Apart from outliers at rock grains and subsided points, the measurement areas support a horizontal layering as they yield a rather more homogeneous curve of mean values, standard deviations and medians in the vertical direction than per horizontal grouping. The repetition of a vertically sloped layer pattern is most pronounced in the measurements of the test wall, for which it is not only possible to match the measurement positions to the current component pictures, but also to validate the stipulated layers via matchup to the actual ones of the construction

process as documented in the previous research project. The least pronounced layering was found at Kachlet, where the visual overlay in Fig.3 suggests a division into an upper and lower subarea. Each subarea encompasses local clusters as well as distinctive counter layers of higher or lower means, highlighted in the slopes or stretched C-shapes in the respective horizontal means on the right side of Fig.3.

Histograms and distributionﬁts of the (sub-)sets formed a starting point regarding the distribution shape, whereas Anderson-Darling-Tests and probability plots assessed whether the (sub-)sets follow rather a normal or lognormal distribution (types as sup- posed by Eurocode). Additionally, they visualised the ranges and the extent of deviations.

Mostly, a lognormal distribution type similarity could only be attributed to unannotated ensembles and in case of the Kachlet measurements would classify only in rather log- normal than normal, especially due to higher values. The regularity within the mea- surement areas was approached by varying the sample ensembles for the median based classiﬁcation in compressive strength classes detailed in DIN EN 13791/A20:2020-02.

Figure4 compares histograms per subsets for the recess area at Koblenz with an inside area at Kachlet and a measurement area of the test wall, suggesting a logarithmic- like distribution for the functional values in the former case and a less obvious one at the latter. In all cases, the majority of the tests (Kachlet:*80%, Koblenz:*64%, test wall

*92%) displayed no visual peculiarities. The contra intuitive differences between the visual markers and the test values can be ascribed to the remaining homogenisation in depth, e.g. if a rock grain was covered, it was not recorded as unannotated, even if the level of the rebound number suggests a rock grain directly underneath.

Fig. 3. Q-values of the rebound tests at Kachlet, left: colour-coded results with subsets indicated by line style (no line: unannotated, dotted line: stone, continuous line: subsided), right: horizontal mean values

2.2 Generic Distributions and FEM Implementation

The modelling of the spatial distribution of the material properties can be decisive for the calculated load-bearing capacity by prearranging probable load paths, and conse- quently resistance extent to speciﬁc loading conditions, or by inducing material weaknesses. For example, nests of gravel in the concrete can be represented by a cluster of higher values with reduced connectivity of the elements within and construction joints by a localisation in levels. The selection, implementation and evaluation of the distribution functions and parameters are therefore a focus of the research project. By generating random realisations of a distribution type with speciﬁc design parameters (log-normal distribution, characteristic value, standard deviation and population size), so-called populations are obtained as ensemble sets of material characteristics with different value compositions. In accordance with normative assumptions and the pro- cedure assumptions to obtaining material characteristics in TbW, the base population of the compressive strength per structure is set as a logarithmic distribution. Furthermore, the strength classes of DIN EN 1992-1-1 imply a mean value corresponding to its 5%

characteristic value plus 8 MPa. Each base population ensemble yields a large number of 3D distributionﬁelds for the FEM model by random or structured mapping of the material property to the modelled component. These combinations of spatial allocation and material properties represent sub-ensembles per material distribution. The creation of these (sub-)ensembles as input ﬁles for the FEM calculation was automated in Matlab for random mapping and, based on literature and the rebound measurement analysis, a yet more realistically structured set-up is proposed as visualised in Fig.5:

The compressive strength population of the analysed component (e.g. loading area) follows the lognormal distribution of the left side, with the red bar indicating the characteristic value. The mapping divides into two population subsets. Firstly, the main pattern, constituted of the layers and layer transition areas, and secondly the strength outliers as deviations. As a simpliﬁed example, one possible division of the population is shown on the right side by color-coding. The main pattern without the deviations consists of two aspects: a vertical gradient in groups per cluster level and a horizontally uncorrelated distribution of these clusters per level.

Fig. 4. Q-value histograms of the rebound tests at Koblenz (left), test wall (middle) and Kachlet (right) by subsets of functional, subsided and stone

### 3 Numerical Simulation Set-Up

3.1 Non-linear Material Model

Various material models for geotechnical or concrete applications are already imple- mented in LS-Dyna (LSTC2020). Aﬁrst plausibility check regarding their suitability for the simulation of plain concrete behaviour included their capacity to reproduce speciﬁc damage for tensile and compressive loading. In further benchmarks, the material model“Continuous Surface Cap Model”(CSCM) was selected according to criteria relating to the tensile-compression behaviour as well as fracture patterns. The former behaviour was studied for single element tests and multi-element tests of cylinder and cube specimens, and the latter by comparing the damage and failure of the multi-element compound tests to general fracture patterns required for testing (DIN EN 12390-3:2009). CSCM offers a material parameterisation to be generated from com- pressive strength and maximum aggregate size. However, this primarily addresses the compressive strength range from 20 to 58 MPa, whereas hydraulic engineering and in particular its historical solid hydraulic structures also addresses signiﬁcantly lower values. Parameter studies with the optimisation program LS-Opt facilitated the derivation of a correlation of the CSCM-generated compressive strengths for values down to and below 1 MPa. In numerical aspects, speciﬁcs as hourglassing parameters have been determined by sensitivity studies. Furthermore, series of simulations were carried out on cylinder models, beam models and a simpliﬁed model of a generic recess for the investigation of the 3D distribution influence. In these benchmarks, the selected material model, variations of the spatial distribution and distribution characteristics are evaluated regarding their damage behaviour and compared with standard calculations.

Figure6depicts results of such a benchmark for a 22 10 m beam under bending load by symmetrical single forces, represented in the simulation counterparts as two slowly advancing rigid cylinders. The dotted lines represent the analytical maximal force obtained by formula with the material values as in DIN EN 1992-1-1, whereas the histograms and their respective logarithmic ﬁts visualise the simulation results of the matching 3D distributions. Although the number of simulations is rather low (50 per compressive strength), their bandwidth matches well in between of their encompassing quantile-based values.

Fig. 5. Spatial mapping structogram for rammed concrete simulation, visualised by color-coded partitions of a histogram of one distribution

3.2 FEM Simulations and Stochastic Analysis

For the example of a speciﬁc characteristic compressive strength, the FEM steps start with a distribution population generated with speciﬁc design parameters as shown in Fig.5. This population contains the total amount of compressive strength values to be mapped as part properties with the respective generic material model parameters on the component’s geometry in hundreds of spatially different variations, as visualised for a recess section in the left side of Fig.7for two variations. Each of these variations then experiences an identical loading scenario in LS-Dyna and fails somewhat differently due to their spatial strength conﬁguration, with the differences encompassing damage patterns and scope of the failing elements as well as the failure mode itself in case of changing load and damage paths as depicted in Fig.7(middle). The scenario includes the embedding of the analysed part into the homogenous component with boundary conditions and gravity loading on the modelled structure as well as due to overlying structure parts. The loading itself models a continuously rising, corresponding to a rise inﬁctional water level, normal force application as depicted in Fig.1. Key histories (time or value-based value curves, e.g. force-time, stress-strain) and virtual sensors (e.g. number of failed elements in a speciﬁc area) are deﬁned as part of the FEM input for numerical dependencies and in preparation of the postprocessing. So far, the pro- cess described above would require the conﬁguration of hundreds to thousands of scenarios and individual result analysis. As for the case of the material benchmarks simulations, the optimisation software LS-Opt facilitates the conﬁguration and result analyses in order to evaluate all these variations in a parameterised manner. Such a set- up in LS-Opt includes on the pre-processing side the deﬁnition of the design space (parameters, their type and range), the sampling procedure, the parameter based spatial conﬁguration and the LS-Dyna simulation control. On the post-processing side, his- tories are deﬁned and extracted from the simulation runs for subsequent processing, as visualised for force trajectories per failed element number in Fig.7on the right. As a further step in this processing, single value responses are obtained, e.g. maximal value at a speciﬁed time or value. The approximation of such resulting value sets by meta- modelling methods enable a propagation between the individual simulation cases, e.g.

between two recess depths or loading positions, and thereby the evaluation of a higher number of scenarios than actual simulations could deliver due to cpu and time limits.

Fig. 6. Left: Calculated maximal vertical force of beams under bending for fcpas in DIN EN 1992-1-1 compared to simulation result histograms and logarithmic ﬁts of 3D distributions.

Right: FEM model example.

These approximation methods span from cpu-friendly sequential approaches with linear basis functions up to Radial Basis Functions and Feedforward Neural Networks and yield quantitative and qualitative descriptions of the scenarios. In combination with failure classiﬁcation, e.g. by response values, the probability of failure and the level of accuracy and reliability of the load cases can be determined in the overall context of the safety concept.

3.3 Failure Classiﬁcation

Contrary to the usually determined utilisation ratio or to individual case assessment, the proposed simulations require speciﬁc and automatable failure classiﬁcation. For example, a partial damage per element could be acceptable, as it implies a change in bearing capacity without equalling a loss of bearing capacity. Due to robustness aspects as possible load path redundancies in the massive structures, even failure and thereby loss of bearing capacity of a few elements leads not unconditionally to the failure of the structural system. However, tipping points may be breached and transfer the structural system into a non-acceptable susceptibility. The requirements of failure classiﬁcation are therefore divided in two parts: ﬁrst the deﬁnition of qualitative indicators and secondly their combined quantitative description into an automated classiﬁcation scheme, e.g. as the classiﬁer approach implemented in LS-Opt. Indicators encompass testing related types as cracks (cohesion, element failure), movement at deﬁned points etc. as well as numerically typical aspects as changes in energy types or element damage accumulation. The current simulations serve to validate the modelling approaches and to estimate the applicability of indicators for evaluation and cate- gorisation for the recess simulations. Furthermore, the demonstrator runs constitute a base of the overall set-up. For example, their results enable to identify combinations of the failure indicators, which are functional for constructing classiﬁer-based sampling to reduce simulations scenarios in less susceptible design areas. This addresses the Fig. 7. Left to right: Color-coded compressive strength visualisations of two random 3D conﬁgurations of the same population. Failure variations differing only due to their 3D conﬁgurations. Section force per failed (deleted) elements with color-coding per 3D conﬁguration.

amount of simulation cases necessary to deduct the structural response over the design space depending on sensitivity analysis per parameter of the design space.

### 4 Conclusion and Outlook

Failure type and resistance capacity are both influenced by the material values ensembles and their 3D distributions. Reserves in bearing capacity are afﬁrmative, however, simulations mirror the limitations of the 3D ﬁelds assumptions and the approximations due to material model and simulation set-up. This emphasises the relevance of the literature and measurement-based generation of 3D material parameter ﬁelds and of tailoring theseﬁelds for the rammed concrete typical for the existing solid hydraulic structures. Based upon showcase simulation set-ups, direct response values were successfully used as failure indicators for failure probability analysis. The next step in the project addresses the classiﬁcation based upon combined failure indicators for the analysis of extended recess models reproducing component measures and compressive strength classes.

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