Theoretical Probabilistic Nonlinear Analysis of Post-Tensioned Bridge Cross-Sections with the Application of Random Fields

Numerous studies on the application of random fields in the analyses of civil engineering structures and building materials can be found in the current research literature. In the study [1] random fields were applied for modelling complicated mesoscopic fractures in quasi brittle materials (including structural concrete). In [2] random field simulation was used as a benchmark for developing a simplified method for risk assessment of surface damage of marine reinforced concrete structures. Within the framework of stochastic damage mechanics, the spatial variability of concrete was represented by two-scale stationary random fields in the study [3]. The Karhunen-Loève expansion random field discretization was used in the study [4] assessing the quasibrittle damage of a concrete beam under four-point bending in the context of non-linear finite element problems. The problem of uncertain concrete cracking was also tackled in the study [5] where random fields were used in conjunction with the neural network performance function approximation. Paper [6] extended these considerations to the problem of reliability of concrete members subjected to corrosion, with a reference to static strain data. Corrosion modelling – using random fields to generate an artificial steel corrosion distribution with the support of machine learning algorithms - was also used in the study [7] for the estimation of the structural capacity of corroded reinforced concrete members. General reliability updating framework was introduced to these issues in the publication [8].

The application of random fields in structural engineering is not limited to the proper representation of the issues related to concrete. In [9] a novel, practical methodology to determine the safety of suspension bridge main cables was proposed. Three-dimensional random field was applied to simulate the reduced strength of the cable due to friction and broken wires. In the study [10] a two-dimensional random field was developed where load effects and time-dependent structural resistance were calculated for each segment in the field, in the aspect of spatial time-dependent reliability analysis of slab bridges.

The encapsulation of the uncertain spatial variability of civil engineering structures and materials in the form of random fields is gaining popularity also in engineering practice. It can be observed that this approach is more and more present in commercial finite element (FE) engineering solutions. It has a place in both the geotechnical [11] and structural analysis domain [12]. It is widely recognized that random field application can enhance the accuracy and reliability of probabilistic computations of civil engineering structures, especially in comparison to the adoption of a single (global) random variable approach [13].

Based on the overview of the literature it can be stated that the academia is dominated by the approaches of direct application of random fields to advanced types of finite elements, including spatial elements. While in the engineering practice, especially in the design of bridges, bar models (“e1s3” [14]) are often used. On the basis of processing those models, advanced cross-sectional analysis takes place, but as a separate, second stage (based on internal forces from the numerical model). Due to this, and the fact that the parameters of concrete may differ significantly along the height of the bridge cross-section - because of the characteristics of the concrete pouring process and various other technological aspects – this study focuses on the non-linear analysis of cross-sections taking into account these probabilistic variabilities represented with random fields. This study is limited to consideration of one source of uncertainty, laying in the stress-strain relationships only in a vertical direction (1d, vertical random fields application). Thus, the aim of the paper is: (I)

to present the probabilistic, nonlinear analysis (nonlinear material parameters) of post-tensioned (PT) cross-sections with the application of one-dimensional random fields,

Deterministic procedure and computational code for the non-linear PT sections analysis wERE developed by the Author as part of his work for the Aspekt Laboratrium sp. z.o.o. company [15]. Thus, the presented algorithm is a commercially used, proprietary software. The procedure implemented in the software can be simplified to the following steps.

For reinforcing steel and tendon steel an elastic-perfectly-plastic material model was assumed with function parameters e.g. modulus of elasticity adopted accordingly to the material class given by the user.

As can be noted, the abovementioned method provides detailed insight into the levels of strains and stresses in all components of the analyzed cross-section, which can be then assessed in relation to the design strength of given materials. This approach has therefore a significant advantage over the envelope-only-based procedures which are commonly used in civil engineering.

On the basis of deterministic procedure and the computer code, the additional probabilistic Python modules were developed to expand the capabilities of the developed software. The cross-sections under consideration are presented in Fig. 1. They are the simplified versions of real post-tensioned bridge cross-sections that are used in typical road bridges [17, 18], [19]. The first three cross-sections are often present in double girder, double or triple-span viaducts crossing highways [20]. The section marked as “ILB” was inspired by members used in incrementally launched bridges (ILB) with vertical webs (omitting the deviators and external tendons) [21]. The IFC models of both solutions are presented in Fig. 2. In Tab. 1 the assumed values of key parameters of concrete, rebar steel and tendons are presented.

Figure 1.

Figure 2.

The pair of values M_Ed and N_Ed should be treated as purely theoretical and were chosen arbitrarily for each cross-section. They are collected in Tab. 2. These values were adjusted in such a manner to induce each time (for each cross-section) the same highest normal stresses in the cross-sectional concrete fibres in deterministic calculations of 24.2 MPa. Tt is worth emphasizing that this arbitrary value does not represent the design strength of concrete.

It must be emphasized that the values collected in Tab. 2 do not represent the actual values from specific bridges although the selected values were roughly inspired by the values of the cross-sectional forces for the cross-section in the mid-span (for T-cross-sections) and for the support cross-section for the ILB [22, 23]. Thus, the sole purpose of these theoretically selected values is to enable comparative sensitivity and probabilistic calculations, which are the focus of this paper.

Regarding PT concrete members, one of the key factors influencing the uncertainty in stress analysis lies in the strain-stress relationship of given concrete and given realization (job) [24, 25]. This factor was used as the source of uncertainty in this study. This relationship in deterministic procedure was given in formula (1). In order to assign to it a probabilistic characteristic, the value of the design strength of concrete was assumed in several probabilistic ways. Thus, the only purpose of introducing random properties into f_cd at this stage of analysis is to introduce appropriate randomness into equation 1 (stress-strain relationships) and not to model the random strength of concrete (direct modelling of concrete strength in terms of reliability index estimation is explained in further sections of the paper). In the first approach (I), the randomness to eq. 1 was intruded by defining the f_cd as a single “global”, lognormal, random variable over the entire height of the section with the mean value equal to the deterministic assumption f_cd = 25 MPa [16]. Due to the theoretical, comparative nature of the study, this value was adopted with the assumption of a concrete partial material safety factor of 1.4 and the long-term effects coefficient equal to 1.00. The coefficient of variation CV was assumed as equal to 0.20. Although the laboratory research indicates lower levels of CV [26, 27] it is worth emphasizing that the real in-situ variability in bridge cross-sections can be significantly higher. Thus, the CV values found in the relevant literature range from 0.10 to 0.20 [28, 29], [13, 30, 6, 31]. The upper value of this range was chosen arbitrarily to emphasize probabilistic relationships in comparative analyses.

The other three approaches (II, III, IV) to the introduction of the uncertainties to the strain-stress relationship rely on the use of suitably correlated random variables in the form of Gaussian, one-dimensional random fields which change over the height of the cross-section. Namely, in approach (II) the correlation in the field was defined as an exponential function given in formula (2). (2) ρΔx=e−ΔXLc \rho \left( {\Delta x} \right) = {e^{-{{\Delta X} \over {{L_c}}}}} Where: ρ(Δx) – the correlation value, ΔX – the distance between random variables under consideration (so-called “lag” distance), L_c assumed correlation length (or “radius”).

In approach (III) squared exponential function was used (formula (3)). (3) ρΔx=e−ΔX2Lc2 \rho \left( {\Delta x} \right) = {e^{-{{\Delta {X^2}} \over {L_c^2}}}} Contrary to previous solutions, in approach (IV) the asymptote of the correlation function (formula (4)) is not the horizontal zero axis but the threshold value C₁. (4) ρΔx=C1+1−C1⋅e−ΔX2Lc2 \rho \left( {\Delta x} \right) = {C_1} + \left( {1 - {C_1}} \right) \cdot {e^{-{{\Delta {X^2}} \over {L_c^2}}}} Where: C₁ is the threshold value – in this study adopted in accordance with the JCSS [13] guidelines, as equal to 0.5. The L_c value was also adopted according to JCSS as equal to 5.0 m. Those value is independent of the geometrical parameters of member length or cross-section height. Thus the L_c, is related to material properties and not the geometrical properties of the member that the concrete mix is poured in [13].

The functions from approaches (II) to (IV) are graphically shown in Fig. 3.

Figure 3.

In order to demonstrate the influence of the applied correlation functions (II, III, IV) in the form of appropriate random fields, exemplary realizations of 1d random fields were computed (Fig. 4).

Figure 4.

It should be clarified that all three types of random field types were applied vertically to the cross section (the ΔX in Fig. 3 and the abscissa axis in Fig. 4 were applied vertically). Thus, the number of random variables (RVs) was equal to the number of the horizontal computational bands (5 mm thick) that “fitted” into a given cross-section. Therefore, the total number of RVs in each field varied depending on the height of the given cross-section, and ranged from 200 for the “T1” cross-section up to 600 for the ILB cross-section.

There are multiple random field generation techniques [32], which can be divided into two main groups. The first group encompasses spatial correlated variables generators with the application of chosen discretization method. Those generators include, for example, the Covariance Matrix Decomposition method (CMD), the Moving Average method (MA), the Fast Fourier Transform method (FFT), the Turing Bands method (TB) and others. The second group includes series expansion generators, for example, the Karhunen-Loève expansion-based generator (KLE) or Polynomial Chaos Expansion-based generator (PC) [31]. Due to a relatively small number of random variables (max. 600 – one random variable per one computational band of the cross-section), in this study, the Covariance Matrix Decomposition (CMD) was used. It is described as a procedure in [28] and as an open-source computational simple example coded by the author in the repository [33]. Thanks to this, it was possible to directly obtain appropriate sets of random variables representing the discretized vertical 1d random field of uncertain strain-stress relationships along the cross-section height in all computational bands.

In order to obtain cumulative histograms and probabilistic ranges of the maximum edge stresses, the generations of RVs in each of four approaches [(I) constant value of RV over the entire height, and (II, III, IV) three types of random fields] were simulated 3000 times in each cross-section under consideration. This allowed for proper estimation of the parameters of response random variables, namely the expected values of stress ranges. On this basis, in order to properly estimate the simulation-based reliability index, further Monte Carlo simulation was possible and applied with 106 random trails for each approach and each cross-section.

As stated in the assumptions section, the theoretical values of N_Ed and M_Ed were adjusted in such a manner to induce each time (for each cross-section) the same maximum normal stresses in the cross-sectional concrete fibres in the deterministic calculations of 24.2 MPa. Thus, the results of deterministic procedure for each section are equal to this value. The deterministic results of strains and stresses in all components (in other concrete fibres, tendons and reinforcement) also did not exceed the design limit values for all sections. On the other hand – as expected – when approaches I–IV were applied, for some simulations the maximum values of the edge stress results exceeded the limit design value. An example of such results in relation to the ILB section is shown in Fig. 5, where the design strength of concrete was equal to 25 MPa.

Figure 5.

As can be noted from Fig. 5, in the case of the ILB, the concrete compressive zone x_eff [m] covered the whole section and thus in this case the x_eff = 3.0 [m]. In contrast to the ILB cross-section, in the case of the other cross-sections, the height of the concrete compressive zone x_eff [m] did not cover whole cross-sections’ respective heights. In the case of T1, x_eff was equal to 0.32 m, in the case of T2, x_eff = 0.66 m, and in the case of T3, x_eff = 1.29 m. Similarly, in sections T1-T3, the ratio of the compressive zone to the section height increased, and was equal to: 32%, 44% and 52%, respectively (100% in ILB).

Before discussing the final results of all aggregated simulations for all cross-sections, it is worth highlighting the exemplary results of cumulative histograms of the maximum edge stresses for a chosen instance of analyses (Fig. 6).

Figure 6.

Based on this data it was then possible to compute the corresponding stress range values expected with 95% probability. E.g. according to Fig. 6 (left part), it was equal to: 28.33 – 18.90 = 9.43 MPa. Analogical histograms, and thus the probable ranges of the highest stresses, were determined for all four (I–IV) approaches in relation to all four analyzed cross-sections. This gave a total of 16 × 3000 simulations. On this basis, the parameters of random variables representing the maximum probable stresses in each case were determined. This, in turn, allowed for further simulations in the context of reliability index estimation, which required a total of 16 × 10⁶ Monte-Carlo random trails.

In order to properly run those simulations, it was also necessary not only to introduce the randomness in a stress-strain relationship (“demand” side of the reliability equation) but also on the capacity (i.e., resistance) side. It was done by representing the design strength of concrete as an random lognormal variable with a mean value of 43 MPa and coefficient of variation CV equal to 0.20 according to the review of [16, 28, 29, 13, 30, 6, 31].

With these computations, it was possible to collect the key results in the form of collective comparative charts (Fig. 7) in accordance with the previously defined research goals of this study.

Figure 7.

Based on Fig. 7 key observations can be made. The probable (expected) range of the maximum edge stresses decreases with the increase of the cross-section height and the concrete compressive zone. The highest values of the range are approx. 13 MPa (section T1), and the smallest are approx. 8 MPa, in the ILB section. Regarding the reliability index, the tendencies are reversed: The highest values of the simulation-based reliability index are observed in the cross-sections with the greatest height and compressive zone (ILB, and T3). Those values range from 2.75 (ILB) down to 2.45 for the T1 section. However, it is worth emphasizing, that the most important conclusions concern the sensitivity of individual types of cross-sections in the context of the selected method of representing uncertainty (random field type and parameters). Namely, a key tendency can be noticed here in which both in terms of the simulation reliability index and the expected stress ranges, the sensitivity to the type of the random field increases with the increase of the compressive zone and the height of the cross-section. Therefore, the T0 cross-section exhibits the lowest sensitivity, and the ILB cross-section shows the highest sensitivity. There are also consistent, additional observations for all sections. In all cross-sections, the largest relative change in the quantities sought (index and range) takes place when using a random field with a correlation function of the exponential type (approach II). In turn, the largest relative difference in the expected ranges, between the JCSS field (approach IV) and one random variable (approach I) can be observed in cross-sections T2 and T3.

This paper presents the method and results of probabilistic analyses and simulations of PT members, based on the original proprietary computer algorithm. The research was aimed at finding the dependency between the input data in the form of (a) characteristics of selected, simplified cross-sections of typical bridges, and (b) ways of representing uncertainty in stress-strain relationships (various types and parameters of random fields), on the output data in the form of (a’) expected, probable values of stress ranges and (b’) simulation-based reliability indexes. Based on the literature overview concerning the application of probabilistic random fields in civil engineering, the presented scope and purpose of the study can be identified as valuable and novel. The computations showed that the random field using the exponential correlation function had the greatest impact on increasing the range of expected stresses and reducing the value of the reliability index. The highest sensitivity to the type of random field was visible in the cross-sections with the greatest height and compressive zone (ILB and T3 sections). The lowest uncertainties in the output data were recorded for the relatively lowest sections with the lowest compressive zone (T1 and T2). The application of the JCSS random fields (approach IV) gave closest (but not identical) results to the approach with the adoption of a single, global lognormal random variable in the majority of the analysed configurations.

Despite the interesting and rich conclusions resulting from the simulations, further studies on this topic are planned by the author. Further works will include the implementation of an additional module in the algorithm which would enable automated, systematic, parametric search for dependencies between the individual input variables (material, geometrical, mechanical, probabilistic) and the mechanical and probabilistic features of the response of cross-sections.