 Research
 Open Access
Characterization of steel buildings by means of nondestructive testing methods
 Markus Sebastian Doktor^{1, 2}Email author,
 Christian Fox^{2},
 Wolfgang Kurz^{2} and
 JeanPierre Stockis^{1}
https://doi.org/10.1186/s1336201800525
© The Author(s) 2018
 Received: 16 February 2018
 Accepted: 8 October 2018
 Published: 16 October 2018
Abstract
Nondestructive testing methods became popular within the last few years. For steel beams incorporated in buildings there are currently only destructive ways for testing the yield limit as well as for determination of the current stress level. Rise of ultrasonic and micromagnetic tools for (nondestructive) measurements allows the characterization of the inbuilt material especially of old steel bridges as economical maintenance of the infrastructure. It is possible to determine the reserve of residuence of bridges or of any other existing steel buildings in order to upgrade them competitively for future usage by the possibility of a simple way of strengthening by welding or using bolds. This is done using modern devices for ultrasonic and micro magnetic data recording on the one hand and modern techniques from nonparametric statistics such as sieve, partition and semirecursive estimators on the other hand.
Keywords
 Nonparametric regression
 Robust regression
 Mathematical and mechanical modeling
 Dependency modeling
 Civil engineering
 Nondestructive testing
 Material characterization
1 Introduction
The load bearing capacity in existing buildings is classically determined by means of load tests. If a calculation model gives no sufficient results, highly complex test loadings have to be done to determine the load bearing capacity. For verification in existing buildings material characteristics as the yield strength and the existing stresses have to be known, determining them nondestructively is an obvious advantage. Furthermore, the internal forces could be estimated, which includes second order theory as well as e.g. imperfections and signal denoising (outlier detection). This is crucial for the verification of (sufficient) load bearing capacity. The appointed lifetime of a new building is 50 to 100 years. Especially for steel buildings, with high variations but well known material characteristics, extensions of the lifetime in the sense of sustainability might be possible, cf. [1]. The actual standards for condition monitoring are in [2, 3] with [4, 5] explaining how to localise fatigue effects before crack initiation starts. Therewith mechanical stresses can be obtained via electromagnetic induced ultrasonic measurements and acustoelastic effects, cf. [6, 7]. Thus, mathematical modeling is necessary for the micromagnetic records to determine material characteristics, the ultrasonic records to determine the current state of load and finally generalising the mechanical model to estimate the internal forces. This work links the nondestructive records with modeling and the generally accepted engineering standards.
2 Nondestructive testing methods
To determine the residual carrying capacity of a construction without risk of collapse is just possible with nondestructive testing methods. The determination of the yield strength by micromagnetic measurements and the determination of the current load state by ultrasonic measurements are described below.
2.1 Unknown material characteristics
The yield strength is the most important material characteristic of the steel to determine the loadbearing capacity. It is defined via regulations to characterize different steel qualities. If the yield limit is determined by measurements, the ultimate limit state design can be proven by real material characteristics without model uncertainties and not by guaranteed minimum values. For determination of the yield strength of an investigated beam the ferromagnetic properties of steel are used. There is a causal relation between magnetism and the yield limit to be determined, see [8] for details. For calibration means, the magnetic properties of the different steel types are determined and assigned to their yield limit measured in a classical tensile test.
2.2 Determination of the current load state
3 Construction & measurements in existing buildings
3.1 Construction
Higher static loads can easily be supported by steel constructions with simple strengthening measures. But the used steel and its material characteristics, especially his bearing capacity, are not known. Furthermore, the current state of load is typically unknown. Modifications of buildings in their previous service lifes caused changed load transfers, which are not incorporated in current construction plans. In many cases, construction plans of the building to be investigated do not exist anymore. Thus, it is impossible to make a serious statement about the load bearing capacity still available in the beams.
3.2 Recording data
4 Mathematical modeling and applications
4.1 Mathematical and mechanical models
The stresses existing in a steal beam can be decomposed in different sources of stress via classical/technical mechanics, see e.g. [9]. For security and safety concepts, we have to decompose them in their single parts: normal (N) and residual (E) stress, stress due to bending around the y or z axis (\(M_{y}\) and \(M_{z}\), respectively) and stress due to warping torsion (\(M_{w}\)). The concept of measuring points uses symmetry as far as possible and looks, locally in every cross section of the steel beam along the xaxis, as follows:
4.2 Regression model for segmented stress estimation and dependency modeling
Due to the statical system we expect the stress curve to show intervalwise different behaviour. The points of change of the (local) regression function, driven by the stress curve, are well known through the statical system.
4.2.1 The regression model
4.2.2 Statistical tests for outlier detection
4.2.3 Dependency modeling
 (i)
Continue with the model with corrected dependency structure
 (ii)
Restart the estimation procedure with reduced influence due to dependency.
4.3 Local regression estimates for internal forces
Algorithm
First, normalise all observations, i.e. remove the constants \(c= ( c_{1},\ldots,c_{7} ) \), and keep them for estimation of the occurring constants \(\alpha_{0}\), \(\beta_{0}\), \(\gamma_{0}\). The part left is the sum of normal and residual stress. Note, that the separation in residual and normal stress (i.e. splitting the constant) is not possible in general unless further information, e.g. from micro magnetics, see Sect. 4.5, are available.
Second, estimate the internal force \(M_{y}\) from traces 3 and 5 using (robust) Least Squares (cf. [20]) under the constraint \(\vert \tilde{\alpha}_{0}^{1} \vert \leq \vert c_{3} \vert , \vert c_{5} \vert \). The estimate obtained is called \(\theta^{2} = ( \alpha_{0}^{2},\ldots ,\alpha_{n_{1}}^{2},\beta_{0}^{0},\ldots,\beta_{n_{2}}^{0},\gamma_{0} ^{0},\gamma_{1}^{0},\gamma_{2}^{0} ) \) (analogously without mentioning in the sequel).
Third, estimate the internal forces \(M_{z} + M_{w}\) (note, that they are either both trivial/constant or linearly independent, thus, there is a unique solution to this estimation problem) in traces \(1, 2, 3, 5, 6, 7\) under knowledge of \(M_{y}\) (i.e. subtraction) from the previous step and keep the constraints \(\vert \tilde{\beta}_{0}^{3} + \tilde{\gamma}_{0}^{3} + \tilde{\alpha}_{0}^{2} \vert \leq \vert c_{1} \vert , \vert c_{2} \vert , \vert c_{6} \vert , \vert c_{7} \vert \).
Sixth, check whether \(\Vert \theta^{4}\theta^{5} \Vert <\kappa\) for a previously fixed constant \(\kappa>0\) (i.e. a Cauchy sequence criterion) or repeat updating the estimates under all information available in random and nonrepeating order at most M times, checking the Cauchy criterion after each iteration.

Weaker dependence structures are crucial for the application of the estimation techniques, therefore, the dependency reduction was necessary.

The constants \(\alpha_{0}\), \(\beta_{0}\), \(\gamma_{0}\) can not be estimated simultaneously in steps 3 and 5, thus, a fixedconstant approach has to be chosen including several trials.

A deterioration of the local estimation is not possible due the construction of the LevenbergMarquardt least squares, cf. [21, 22].

Using the geometry constants from equations (2) to (8) and the estimates \(\alpha_{0}\), \(\beta_{0}\), \(\gamma _{0}\) from the final estimation, the remaining parts of c, c̃ is the estimate for the sum of residual and normal stress and with knowledge of the normal stress, a decomposition in residual and normal stress can be done, see [18].

An absolute value bound (or in some cases: estimate) for the normal stress could be obtained viaan estimate for the residual stresses is given via$$ N = \min_{i=1,\ldots,7} \bigl\{ \bigl\vert c_{i}  ( g_{y_{i}}\cdot \alpha_{0} + g_{z_{i}}\cdot \beta_{0} + g_{w_{i}}\cdot\gamma_{0} ) \bigr\vert \bigr\} , $$Note, that those estimates are not necessarily consistent ones.$$ E_{i} = c_{i} \pm N ( g_{y_{i}}\cdot \alpha_{0} + g_{z_{i}} \cdot\beta_{0} + g_{w_{i}}\cdot\gamma_{0} ) . $$

If there is no feasible solution to the estimation problem, plasticising areas might have been identified/detected. A local decrease of residual stress (internal forces devour residual stress, cf. [18]), i.e. searching in double of the yield limit (see Sect. 4.5) for the constants is the approach of choice in this case.
4.4 Properties of the estimates
The estimates in this procedure are partition, sieve and semirecursive estimates, unless the procedure to estimate within this steps locally is a (robust) version of the LevenbergMarquardt Algorithm, compare [17, 20–23].
Theorem 1
(Doktor, Stockis)
Let \(M_{y}\), \(M_{z}\) be polynomials of degree at most \(n\in\mathbb{N}\). Further, let \(M_{w}\) be a hyperbolic function (i. e. the mechanical model is stated properly). Furthermore, let the tuple \((x_{i}, \varsigma_{i})_{i=1,\ldots,N}\in\mathbb{R}^{8}\) be pairwise uncorrelated with finite expectation \(\lvert\mu\rvert<\infty\) and uniformly bounded variance \(0<\sigma^{2}<\infty\).
Then the estimates used in every step are consistent. Furthermore, θ is a consistent estimate for the coefficients of the internal forces.
Proof
The statement follows mainly from the Theorems 10.3, 20.3 and 24.1 in [23] and Slutsky’s Lemma. □
Theorem 2
(Doktor, Stockis)
Consider the setup of Theorem 1. Then the estimates obtained for the internal forces are asymptotically normally distributed.
Proof
The statement for sieve and partition estimates follows from consistency and Theorems 11.4 and 21.1 in [23], the final statement uses additionally the rules of calculus for the multivariate normal distribution. □
Note, that this statement can be generalized in a robust setting according to [17] unless the rate of convergence decreases which is of high importance in practical applications as recording data is timeconsuming and might be expansive.
4.5 Further improvement with micromagnetics
The determination of valid confidence bands is a necessity for proper safety concepts in civil engineering. Further, the classification of residuals observed in ultrasonics has to be done properly to avoid economical and ecological disadvantages. Based on [18], additional micromagnetic measurements are used to link techniques and concepts.
4.5.1 Reduction of complexity
The devices used to record micromagnetic quantities measure 42 different quantities. They use four different implemented sensors which are rather expensive. To make the device handable in practice, a goal is the reduction of the number of quantities necessary without significant loss of quality. For this purpose we applied multiple statistical tests of independence, based on \(\chi^{2}\)tests, for details, we refer to [15]. With a (combined) level of 10%, 36 quantities are identified to be stochastically dependent. The quantities left are driven by two sensors only: the Barkhausen effect and incremental permeability, which halves the amount of sensors required, see [8] for details.
4.5.2 Combining ultrasonic waves and micromagnetism
5 Discussion and results
5.1 A simulation example
A MonteCarlo study with 1,000,000 independent repetitions leads, for 69 equidistant noisy (additive independent zeromean normally distributed error terms, standard deviation σ) stresses, to the following estimated residual means, standard deviations and repetitions of step 6 (upper limit 100 was never reached):
σ  5  10  25  50  75  100 
Residual mean  −0.089  −0.104  −0.054  0.107  0.224  0.0189 
Residual standard deviation  0.821  1.158  2.491  4.848  7.269  9.711 
Mean number of iterations  2  5  7  11  15  21 
5.2 A real data example
6 Conclusion and further developments
Nondestructive testing is always in competition with destructive testing. Using a combination of micromagnetic and ultrasonic measurements it has been shown that the reserve of resistance of existing steel buildings can be determined nondestructively. The occurring dependencies in the steal beam were modeled and approximated to obtain a weaker dependency structure. For the estimation of internal forces the traces have been chosen to contain all necessary informations but keep the mathematical model handable. The combination of different nonparametric estimation techniques and stepwise robust least square estimation leads to surprisingly good results, even for small sample sizes. Furthermore, these estimation techniques can be further generalised in a future work to a robust setting (e.g. optimally bias robust estimators) to keep applicability for e.g. coated beams. This ends up in new and improved security, usage and sustainability concepts for existing buildings including all relevant internal forces.
Declarations
Acknowledgements
Special thanks to Claudia Redenbach and Peter Ruckdeschel for several fruitful discussions about mathematics and its implementation. Further, thanks to Claudia Seck and Nicole Schmeckebier for fruitful discussions regarding structural analysis.
Availability of data and materials
Please contact author for data requests.
Funding
This contribution uses results obtained in the research project Bestandsbewertung von Stahlbauwerken mithilfe zerstörungsfreier Prüfverfahren supported by the Forschungsvereinigung Stahlbau in the context of IGF (IGFVorhaben 466 ZN) based on a resolution of the German Bundestag as well as the AiFZIMProject Bestandsbewertung von Stahlbauwerken mithilfe zerstörungsfreier Prüfverfahren, FZ: ZF4163502LT6, by the Federal Ministry of Economic Affairs and Energy.
Authors’ contributions
The main idea of this paper was proposed by MD and the implementation was also done by MD. JPS supported MD in mathematical modeling as well as in validation. The general engineering context was proposed by WK and CF. CF and WK assisted MD in mechanical modeling and JPS helped preparing the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 BeuthVerlag: DIN 1055100: Einwirkungen Auf Tragwerke  Teil 100: Grundlagen der Tragwerksplanung  Sicherheitskonzept und Bemessungsregeln. Berlin. 200103. Google Scholar
 Normenausschuss Bauwesen (NABau) im DIN Deutsches Institut für Normung e.V.: DIN 1076: Ingenieurbauwerke Im Zuge Von Straßen und Wegen. Überwachung und Prüfung. Berlin. 1999. Google Scholar
 Deutsche Bahn AG: Richtline DS 805, Tragsicherheit Bestehender Brückenbauwerke. Berlin. 2002. Google Scholar
 Boller C, Altpeter I, Dobmann G, Rabung M, Schreiber J, Szielasko K, Tschunky R. Electromagnetism as a means for understanding material mechanics phenomena in magnetic materials. Materialwissenschaft und Werkstofftechnik. 2011;42:269–77. View ArticleGoogle Scholar
 Boller C, Starke P. Enhanced assessment of ageing phenomena in steel structures based on material data and nondestructive testing. Mater Sci Eng Technol. 2016;47:876–86. Google Scholar
 Schneider E, Bindseil P, Boller C, Kurz W. Stand der entwicklung zur zerstörungsfreien bestimmung der längsspannung in bewehrungsstäben in betonbauwerken. Beton und Stahlbetonbau. 2012;107(4):244–54. View ArticleGoogle Scholar
 Fox C, Doktor M, Schneider E, Kurz W. Beitrag zur bewertung von stahlbauwerken mithilfe zerstörungsfreier prüfverfahren. Stahlbau. 2016;85(1):1–15. View ArticleGoogle Scholar
 Mayer JP. Aufbau und Kalibrierung Eines Magnetischen HallSensors zur Detektion und Bewertung Von Schädigung an Stahlbauwerken. Seminar project. Institute of Materials Science and Engineering, University of Kaiserslautern; 2016. Google Scholar
 Kurz W, Fox C, Doktor M, Hanke R, Kopp M, Schwender T, Nüsse G. Bestandsbewertung Von Stahlbauwerken Mithilfe Zerstörungsfreier Prüfverfahren (P 859). Düsseldorf: Forschungsvereinigung Stahlanwendung e.V. (FOSTA); 2016. Google Scholar
 Kurz W, Fox C. Evaluation of steel buildings by means of nondestructive testing methods. Schriftenreihe des Studiengangs Bauingenieurwesen der TU Kaiserslautern, Band 18; 2014. Google Scholar
 Lerman PM. Fitting segmented regression models by grid search. J R Stat Soc, Ser C, Appl Stat. 1980;29(1):77–84. Google Scholar
 The Mathworks Inc. MATLAB and statistics toolbox release 2017. 2017. Natick: The Mathworks Inc. Google Scholar
 R Developement Core Team. R: a language and environment for statistical computing. R Foundation for Statistical Computing; 2017. Google Scholar
 Francke W, Friemann H. Schub und Torsion in Geraden Stäben. Wiesbaden: Vieweg; 2005. View ArticleGoogle Scholar
 Roy SN, Bargmann RE. Tests of multiple independence and the associated confidence bounds. Ann Math Stat. 1958;29(2):491–503. MathSciNetView ArticleGoogle Scholar
 van der Vaart AW. Asymptotic statistics. Cambridge: Cambridge University Press; 2012. Google Scholar
 Hampel FR, Ronchetti EM, Roussew PJ, Stahel WA. Robust statistics, the approach based on influence functions. Wiley series in probability and statistics. Indianapolis; 2005. View ArticleGoogle Scholar
 Ackermann J. Die barkhausenrauschanalyse zur ermittlung von eigenspannungen im stahlbau [PhD thesis]. Darmstadt University, Department of Civil Engineering; 2008. Google Scholar
 Brockwell PJ, Davis RA. Introduction to time series and forecasting. New York: Springer; 2002. View ArticleGoogle Scholar
 Holland PW, Welsch RE. Robust regression using iteratively reweighted leastsquares. Commun Stat, Theory Methods. 1977;6(9):813–25. View ArticleGoogle Scholar
 Levenberg K. A method for the solution of certain nonlinear problems in least squares. Q Appl Math. 1944;2(2):164–8. MathSciNetMATHGoogle Scholar
 Marquardt DW. An algorithm for leastsquares estimation of nonlinear parameters. J Soc Ind Appl Math. 1963;11(2):431–41. MathSciNetView ArticleGoogle Scholar
 Györfi L, Kohler M, Krzyzak A, Walk A. A distributionfree theory of nonparametric regression. New York: Springer; 2010. MATHGoogle Scholar
 Thein C. Sicherheitskonzept Eines Zerstörungsfrei Geführten Tragfähigkeitsnachweises [BSc thesis]. Department of Civil Engeneering, University of Kaiserslautern; 2017. Google Scholar