Lifetime predictions of prestressed concrete bridges—Evaluating parameters of relevance using Sobol' indices

The residual structural lifetime of concrete bridges is often limited by fatigue of the prestressing steel. In practice, numerical calculations of the structure are combined with Miner's rule—based on load frequencies and stress amplitudes—to estimate the damage. Thereby, various parameters must be accounted for whereof a few are actually relevant. Structural monitoring is valuable to increase the accuracy of lifetime predictions, but usually experts choose the right parameters on experience only. A more objective assessment can be reached by sensitivity analysis. A powerful method is provided by Sobol's variance‐based indices. They quantify the influence of a single parameter's variance on the model's total variance and account for interactions in nonlinear models, too. Exemplified on a reference structure, sensitivity indices are determined. Meanwhile specific characteristics like the nonlinear behavior of concrete after cracking are considered. In this case, the key elements of lifetime prediction turn out to be the time‐dependent losses of prestress and the traffic loads.

methods can be categorized into quantitative 18 and qualitative 19 ones.
While the latter come along with a reduced computational cost, they should preferably be used to identify less relevant parameters, as it is possible with the elementary effect-method by Morris. 19,20 An application of this to engineering structures and their lifetime predictions is given in Reference 21. In contrast, variance-based global sensitivity analyses are a powerful tool to quantify the individual impact of a parameter on the model output. 18 One of the most sophisticated types are the variance-based sensitivity indices by Sobol', 16

| Variance-based sensitivity indices
Variance-based sensitivity indices aim for analyzing complex models like nonlinear, nonmonotonic discontinuous systems. 22 The impact of a model parameter X i on the variability of the output Y can be quantified by analysis of variance (ANOVA) techniques. Therefore, the total variance of the output is decomposed into the variances of single parameters and into (co-)variances induced by correlation between the parameters. To assess the impact of the parameters, single parameters are fixed temporarily.
Let the model be a simple function of q input parameters X i at first and assume all X i are square-integrable. Thus, mathematically each parameter possesses a variance V X i ð Þ< ∞ The model spans a q-dimensional unit hyperspace Ω q and can be split into 2 q components by ANOVA high-dimensional model representation. 23,24 It comprises a single constant term of order zero f 0 , q linear terms f i and q over 2 quadratic components f ij , etc.
Indeed, decomposition is not unique, but Sobol' 16 has proven all terms being orthogonal assuming all means but f 0 being zero. Thus, f 0 equals the expectation of Y Moreover, the higher order components are obtained from conditional expectations 23 Therein, the index-i denotes all dimensions but i, analogously-{i,j} all dimensions but i and j The total variance is gained by integration of the squared function Due to orthogonality, the variance of Y can be decomposed analo- Þand con- Employing Equation (4), the variance due to a single parameter Þ . Related to the total variance, this directly leads to the first-order sensitivity indices S i , also called direct effects, according to Sobol' 16 Higher order sensitivity indices (S ij up to S 1,2,…q ) can be obtained analogously if the parameter interaction f ij X i , X j À Á is also considered.
From Equations (7) and (8), it follows that the sum of all 2 q-1 sensitivity indices (all first and higher order ones) is always ΣS i = 1. Moreover, in case of additive models, all first-order sensitivity indices always sum up to 1 while in nonadditive models, the sum might be smaller than 1, too. Then, the difference 1 À ΣS i indicates the amount of interaction in the model.
The model's total variance V(Y) can be expressed as the sum of conditional variances analog to Equation (7) and further variances.
Again, the conditional variances are obtained fixing all parameters but X i , while the further ones result from Y if X i is fixed to x * i simulation-wise. The result in nonlinear models might depend on the choice of x * i , too. For example, this is true for S-N-curves in fatigue lifetime estimations. So, the expected value of the conditional vari- Dividing Equation (9) by V(Y) yields again the first-order index. Additionally, a second term for the variance due to interaction with other parameters is obtained. The latter one subtracted from 1 gives the total sensitivity index, also called total effect S Ti Thus, the total sensitivity index S Ti additionally comprises all parameter interactions. Moreover, S Ti is always greater than S i . If S Ti is close to S i , interaction is of minor importance. For purely additive models, the sum of all parameters total sensitivity indices is 1, otherwise a value greater than 1.

| Numerical determination of variance-based sensitivity indices
Originally, Saltelli et al. 14 From both, q further matrices C i (i = 1,2,…,q) are generated by substituting columns. While the ith column comes from A, all others columns come from B 27 : ð12Þ For the parameter sets in A, B, and C i , the output by means of ndimensional vectors a, b, and c i is obtained from model simulation.
Altogether, nÁ(q + 2) simulation runs are necessary to gain the output.
The first-order index S i follows as Pearson's correlation coefficient between a and c i when all results Y of the model for the matrices A and B are used to get enhanced means μ Y and variances σ 2 Interpretation of an index determined from correlation is straightfor- ) results if the impact of X i is small 3 | NUMERICAL MODEL FOR LIFETIME PREDICTIONS OF CONCRETE BRIDGES At the joints, only a part of all tendons is coupled, which is nowadays standard but rarely done in those years. Due to threaded connections, coupling-joints are generally prone to fatigue and the more connections exist in a section, the higher the danger of fatigue is. Additionally, these construction joints often exhibit lowered tensile concrete strength, little reinforcement amounts, and tend to nonlinear stress distributions. 30 3.2 | Structure, loads, and stress calculation In this way, uncertainty can be considered by scaling of the internal forces according to the variability of the loads.

| Prediction model
On cross-sectional level, the nonlinear distribution of stresses has been computed on a layer model. Based on Bernoulli's hypothesis (plane-remains-plane) the strain gradient is iterated until equilibrium of internal and external forces is gained. To find the equilibrium, the Simplex-Algorithm 31 is applied. Once knowing the strains, axial (N R ) and bending resistances (M R ) follow from Equations (15) and (16) by integration over the cross-sectional area A. Therein, the tendons resistance (index p) reflects the determinate part of the prestress The reinforcement is incorporated by five single rebar-layers (index s).
Each one idealizes the reinforcement as a smeared layer in distinct position. For both, the prestressed tendons and the rebars, a linearelastic behavior is assumed. This is true in case of high-cycle fatigue loads in typical frequencies n ) 10 6 , where plastic deformation of rebar and tendons is generally disregarded. 28 In case of plastic deformation, fatigue failure would occur after only a few load-cycles (low cycle fatigue).

Resistance of concrete (index
. The stresses are evaluated at the representative coupling joint (3B in Figure 2).

| Time-dependent material behavior
Material properties of the steel practically do not change over time.
Concrete exhibits posthardening that increases its strength and stiff-   Figure 3 as the mean.

| Lifetime prediction by damage accumulation
Damage assessment relies on the linear accumulation hypotheses according to Pålmgren 35 Dependent on the stress amplitude, two branches are defined by an inclination parameter k 1/2 Partial damage D i is determined for different load levels i of Δσ and accumulated according to Miner's rule Accumulation of all partial damages from erection at T 0 to the times associated with structural failure at D ¼ 1 yields the structural life- During accumulation and due to nonlinearly evolving gradients of stress amplitudes and frequencies with time, the damage function D(t) itself increases nonlinearly. To reduce computational efforts, the timedependent model parameters are evaluated at discrete time instants.  The histogram of accumulated damages obtained from simulation is shown in Figure 5. The empirical cumulative density function is given in Figure 6. The region of fatigue failure (D ≥ 1) is gray-shaded.

| Traffic frequencies
Due to logarithmic scaling and supported by the higher order moments distribution's skewness (g 1 > 0) and kurtosis (g 2 > 0) become obvious. Kurtosis indicates a comparatively flatter distribution with heavy tails.
Goodness of fit testing employing Kolmogorov-Smirnov's statistic supports a log-normal distribution of data (gray in Figure 6) on a significance level α = .05% while Anderson-Darling's more powerful test rejects the hypothesis. Other distribution functions like the Weibull's one cannot be established at all. Thus, due to lack of evidence, the total damage is assumed log-normally distributed in the remainder.

| General remarks
Sensitivity indices based on correlation coefficients follow from Equations (13) and (14). While the first-order index S i quantifies just the direct impact of a parameter's variance V(X i ) on the variance of the output V(Y), the total index S Ti also includes covariances regarding parameter interaction (e.g., V(X i ,X j ) or V(X i ,X j ,X k )).

| Sampling
The sample sets for model simulation are generated by drawing sam- The fundamental idea of all sampling methods is the same and independent from the distribution being sampled. By inversion of the A mathematically invertible distribution function is prerequisite (which is practically not an issue), otherwise special procedures must be employed that are not addressed here.
Computation of sensitivity indices rests on two independent matrices A and B, both containing n realizations of all q parameters. As already discussed in greater detail in Section 2.2, the indices are obtained from result vectors of combined matrices C i and the original matrices A and B. Fundamental is the independency of A and B, since spurious correlation might occur and impair the sensitivity indices otherwise. 25 Saltelli et al. 14 propose to generate a joint matrix of random numbers of the size (2n Â q) by using a random sampling, see Figure 7A).
Afterward, it is split in half and realizations for A and B are generated. So, both matrices are independent. Sobol' 16 specifically proposes to use a random sampling based on quasi-random numbers.
The proposed modification involves a sampling with correlation control and iterative adaption among the input. It aims at zero correlation as granted by the unity matrix. The whole procedure of a sampling with correlation control has been introduced by Iman and Conover. 44 Its principal idea is to change the entries within each column of the sampling matrix (here: size [n Â 2q], Figure 7B) until the target correlation is reached. Therefore, Cholesky decomposition, the inverse standard-normal distribution and an auxiliary matrix are necessary to iteratively approximate the target correlation. A detailed description is presented by Helton and Davis. 45 The proposed sampling approach is briefly summarized in its key elements in the remainder.
All q columns of matrix A contain realizations of the basic model parameters X i by Latin Hypercube sampling. Next, the realizations in B are added as q further but independent basic parameters X 0 i as shown in light gray in Figure 7B). This ends up with a matrix (n Â 2Áq) and an associated correlation matrix with a doubled number of entries (2Áq Â 2Áq); cf. Figure 7C). Here, the new correlation matrix still contains the two original correlation matrices at the upper left and lower right quadrants along its main diagonal. Added superscripts a and b on the symbol of the correlation coefficient ρ ij index the associated matrices A and B, respectively. The dark-gray-shaded quadrants aside the main diagonal of the correlation matrix include the correlation coefficients between the two matrices A and B. Entries along the secondary diagonals capture the correlation between the basic variables X i and X 0 i from A and B. They quantify the de facto dependency of the two matrices which is at best zero to grant unaffected result vectors a and b.
The sampling procedure with correlation control is applied to the matrix in Figure 7B). A target correlation matrix is defined, with the previously described entries to be zero for independence. By repeating the procedure, the correlation was reduced to values <0.01 for independent input.

| Results of sensitivity analyses
All sensitivity indices for the 19 model parameters according to Table 1  In practice, the uncertainty of the governing parameters can be reduced by on-site measurements. In particular, the following monitoring measures are recommended to be monitored.
1. Traffic loads obviously have a considerable potential for increasing the accuracy, since they are incorporated into the calculation by means of conservative load models. Real loads can be measured for the current condition-for example, by weigh in motion systems or direct strain measurements, cf. Reference 30.
2. The frequency of the transition of the cross section to the cracked state due to decreasing prestress and increasing loads can be determined by crack widths measurement. Combined with corresponding temperature or traffic loads, the model can be calibrated.
3. Regarding the load frequency, accurate load histories often can be estimated from traffic counts on-site or nearby. So, even retrospective evaluations are possible. They can also serve as a basis for predicting load frequencies.
4. The accuracy of the structural model can be significantly increased with comparatively little effort by a 3D digital survey of the structure-for example, by means of laser scans, cf. Reference 8.
Finally, it can be concluded that especially measurements of the right elements can reduce uncertainty in the prediction of structural lifetime significantly. The most relevant parameters were identified in this paper for a real reference structure. The results serve as a basis for experts' decisions on the right elements for a structural monitoring, some elements are given. The quantitative evaluation of this gain in accuracy through measurements is part of further investigations. 28 An approach by the authors will be published soon.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.