Keywords

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 scattering 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. Figure 1 illustrates 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 foundation 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 investigation 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.

Fig. 1.
figure 1

Left: loading scenario partial loading, middle: loading scenario revision recess, right: reinforcements realized at Kachlet. Drawings modified from KW011A-SG01-4 (2016)

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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 distribution fields of the material characteristics to account for the influence of material distribution and spatial permutations. The resulting hundreds to thousands of individual case simulations can be merged in parameterised result analysis and thereby enable a stochastic analysis. By building and sampling metamodels, general probabilistic results can be deduced at a defined safety level. Figure 2 introduces the FEM simulation preparation by illustrated process steps. The 3D distribution fields 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. 2.
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Creation process of metamodels for reliability analysis based on FEM simulations of recesses with three-dimensionally distributed material parameters.

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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 distribution fields 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 influence 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 on findings of De Larrard (2010), whose experiments and analysis however cautioned regarding scattering and the dependence per investigated 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 material field discretising in the range of centimetres. In order to obtain realistic distribution fields of correspondingly high resolution of 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. The first 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 m2 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 the first 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 the first 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. 3 on 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.

Fig. 3.
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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

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Histograms and distribution fits 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 supposed 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 lognormal than normal, especially due to higher values. The regularity within the measurement 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.

Figure 4 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. 4.
figure 4

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

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

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3 Numerical Simulation Set-Up

3.1 Non-linear Material Model

Various material models for geotechnical or concrete applications are already implemented in LS-Dyna (LSTC 2020). A first 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 compressive 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. Figure 6 depicts results of such a benchmark for a 2 × 2 × 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. 6.
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Left: Calculated maximal vertical force of beams under bending for fcp as in DIN EN 1992-1-1 compared to simulation result histograms and logarithmic fits of 3D distributions. Right: FEM model example.

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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. 7 for 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 in fictional 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 process 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, histories are defined and extracted from the simulation runs for subsequent processing, as visualised for force trajectories per failed element number in Fig. 7 on 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. 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.

Fig. 7.
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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.

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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 categorisation 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 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 these fields 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.