Predictive Modeling Methods for Estimating the Residual Strength of Wooden Structures Based on Experimental Data

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1. Introduction It is well known that the construction industry is the main wood consumer [1; 2]. Any building can be considered as a system, each element of which has its own operational lifespan. According to the Russian building code SP 64.13330[27], the operational life of wooden elements and structures of buildings of large-scale construction under normal operating conditions (residential and industrial buildings) is at least 50 years. Scientific literature review shows that wooden structures are often operated outside the service standards [3; 4]. The available operational life of wood allows it to be used in the elements of wooden structures of unique buildings [4; 5]. From the analysis of the studies on the strength of operating wooden structures, it can be noted that there are practically no data and recommendations for estimating and predicting their strength capacity, in particular beyond the limits of the service life standards [7]. To ensure the safety and durability of the structure, it is necessary to consider the factors affecting the reliability of wooden structures: internal stresses in structures that do not correspond to their design values; external effects; maintenance system (preventive and systematic); technical qualifications of maintenance and repair personnel [7; 8]. During operation, the physical and mechanical properties of wood change, which leads to the emergence of imperfections [10]. Many imperfections are gradual in character: the system parameters gradually deteriorate during operation and at some point in time they reach values, at which further operation becomes impossible or impractical [9]. The study and development of recommendations to substantiate the remaining life for wooden structures will allow to correctly predict the necessary material resources for repairs, as well as to justify the feasibility of constructing a new building or structure to replace an existing one, if necessary. Thus, the creation of a reliable model for predicting the residual strength of wooden structures is a key task. The purpose of this methodology is to perform predictive modeling to estimate the residual strength of wooden structures using the example of a rafter system (spar) based solely on the results of experimental studies of samples, which is the main novelty of this study. 2. Methods For establishing the experimental study program, it is necessary to identify the factors that affect the reliability of the obtained results and the number of performed experiments [12]. An insufficient number of the affecting factors will lead to mediocre results that cannot be accepted as reliable without additional investigation [13]. An increased number of influencing factors will increase the accuracy of the work results, however, this will simultaneously lead to an increase in the number of experiments and a decrease in the economic effect of the study. The following factors are considered optimal for estimating the residual strength of load-bearing wooden structures beyond the limits of the service life: - service class; - operating conditions; - service life of construction objects; - availability of information on current/total repairs; - physical deterioration of structural elements; - load-bearing structure type based on strength; - cross-section design features; - loading mode of structural elements. Taking into account the above factors made it possible to prepare the necessary experimental research program and ensure the reliability of the obtained results [14]. Residential buildings built in Vladimir, Russia, in the 50s and early 60s were selected as the sites where standard wood samples were taken for the study of the physical and mechanical properties (Table 1). Table 1 Results of experimental studies of wood samples under compression along the fibers Type of structure Rafter spar Purpose of the building Residential building Age of the structure (t) New 49 60 64 65 Year of construction - 1974 1963 1959 1958 Test year 2023 2023 2023 2023 2023 Short-term resistance of pure wood Rn, MPA 30 29.6 27.87 25.38 24.13 Physical deterioration, % - 37 43 46 46 S o u r c e: made by A.V. Lukina The samples were selected from structural elements in the same stress-strain state - rafter spars. Blanks for the manufacture of standard samples were selected from the structural elements that had no visible defects or damage. Then, small reference samples were made from blanks cut from structures in laboratory conditions to determine the ultimate strength of the wood under compression along the fibers according to the interstate standard GOST 16483.10-73[28]. Statistical data processing was carried out using the test results. The physical deterioration of wooden structures of residential buildings was determined by the Delphi method [15]. The obtained experimental results of short-term resistance of pure wood (Table 1) served as the initial data for predictive modeling to estimate the residual strength of wooden structures (Table 2). Table 2 Experimental results of short-term resistance of pure wood for interpolation and extrapolation № 1 2 3 4 5 t 0 49 60 64 65 Rn 30 29.6 27.87 25.38 24.13 S o u r c e: made by A.V. Lukina For more accurate predictive modeling of the residual strength of wooden structures, the design resistance values for the service life of 75 and 100 years according to SP 64.13330 are determined as: . Then, the design resistance for the grade II wood Rdes (bending, compression and crushing along the fibers) for the service life of 75 years is: , for 100 years and more: Thus, the minimum value of the design resistance cannot be less than 10.3 MPa. This value takes into account the decrease in the strength characteristics of wood over time due to prolonged exposure to loads and environmental factors. The experimental results of short-term resistance of pure wood are recorded in tabular form in Table 2 for interpolation and extrapolation. The experimental values of the short-term resistance of wood are presented in the form of nodal points on the graph (Figure 1). To represent patterns, as well as in the process of scientific and engineering calculations, relationships in the form y(x) are often used, and the number of specified points (hereinafter nodal) of these relationships is limited. The task of approximate computation of the function values in the intervals between the nodal points (interpolation) and beyond (extrapolation) inevitably arises. This task is solved by replacing it with some fairly simple function. Thus, it is necessary to use the methods of interpolation and extrapolation to estimate the value of short-term resistance of pure wood of wooden structures over the observed, historical, time period (0-65 years) and predict their future behavior (forecast period 66-200 years). A decrease in the strength of wooden structures can occur as a result of bioerosion or physical deterioration [16]. It should be noted that the task of predictive modeling was not to identify the causes of the decrease in strength. The authors perform the simulation, assuming that the wooden structures were in normal operating conditions. Age of the structure, years Figure 1. Nodal points of experimental data S o u r c e: made by A.V. Lukina 2.1. Interpolation of Experimental Data Interpolation allows to estimate resistance values for conditions, for which there are no direct measurements, but the interpolated values will accurately match the available experimental data. The results of interpolation beyond the limits of the range of available experimental data are not considered. The most widely used interpolation methods are: - piecewise-linear interpolation; - polynomial interpolation; - spline interpolation. ¡ Piecewise-linear Interpolation. Piecewise-linear interpolation is a simple method that assumes that the parameter value changes linearly between two known points in the experimental data. Piecewise-linear interpolation involves representing a tabulated function on each interval between the horizontal coordinates of the nodal points by a linear relationship: R(t) = a1+ a2t. Coefficients a1 and a2 are determined for each interval [ti -1, ti] separately according to conditions: As a result, the piecewise-linear approximating function on the interval of [ti -1, ti] has the form: , and is continuous. For the experimental data presented in Table 2, the piecewise-linear function will take the form: (1) The graph of the piecewise-linear interpolation is shown in Figure 2. The interpolated values of short-term resistance of pure wood obtained over a historical time interval with piecewise-linear interpolation will be used to predict their future behavior during extrapolation. Age of the structure, years Age of the structure, years Experimental data Short-term resistance of pure wood (Rn), MPa Figure 2. Piecewise linear interpolation of short-term resistance of pure wood S o u r c e: made by S.I. Abrakhin ¡ Polynomial Interpolation Short-term resistance of pure wood (Rn), MPa. Polynomial interpolation uses power polynomials to interpolate the relationships between points. A more accurate method than linear interpolation, especially when the relationship is nonlinear. Such polynomials are the Newton and Lagrange polynomials. They are the same polynomial, but its coefficients are calculated in different ways [17]. The interpretation of experimental data by the Lagrange polynomial is considered below. The Lagrange interpolation polynomial is a method of constructing a power polynomial that passes exactly through the given set of points (interpolation nodes) (t₁, R₁), ..., (t5, R5). The Lagrange interpolation polynomial P(t) is such that P(ti)=Ri for all i = 1, ..., 5. This method belongs to a class of interpolation methods that allow to reconstruct (approximate) function values at points that are not explicitly specified, based on a known set of function values at other points. The Lagrange interpolation polynomial has the following form: , (2) where is the interpolation polynomial of degree no higher than the number of interpolation nodes; is the function value at the interpolation node tᵢ. is the fundamental Lagrange polynomial, which has the form of: , (3) where t is the variable, for which the polynomial is evaluated; tᵢ is the horizontal coordinate of the i-th interpolation node; tj is the horizontal coordinate of the j-th interpolation node. As a result of substituting the experimental data and performing calculations in the Mathcad software, the Lagrange polynomial will take the form: The graph of the Lagrange polynomial is presented in Figure 3. The interpolated data of short-term resistance values of pure wood obtained over a historical time interval when interpolated by the Lagrange polynomial cannot be used to predict their future behavior by extrapolation methods because the interpolated values over a time interval from 0 to 49 years reach more than 100 MPa, which they do not correspond to the physical conduct of the material. Short-term resistance of pure wood (Rn), MPa Interpolation data Experimental data Age of the structure, years Figure 3. Lagrange polynomial interpolation of short-term resistance of pure wood S o u r c e: made by S.I. Abrakhin ¡ Spline Interpolation. Spline interpolation is a method of constructing a smooth function passing through specified nodal points of experimental data. The function is constructed from segments, each of which is a separate cubic function, selected so that at the nodal points not only the function itself, but also its first and second derivatives are continuous. At the boundaries of the experimental data, where it is impossible to achieve a complete convergence of the derivatives, additional conditions are applied: linear boundary conditions that define zero second derivatives at the boundaries; parabolic conditions ensure that the extreme segments of the spline are parabolas, the coefficient for the cubic term is zero; and cubic conditions ensure that the spline behaves like a cubic polynomial at the boundaries [18]. In the Figure 4 shows the results of spline interpolation of experimental data performed in MathCad, where Rl, Rp, and Rc are spline interpolation under linear, parabolic, and cubic boundary conditions, respectively. Short-term resistance of pure wood (Rn), MPa Experimental data Age of the structure, years Figure 4. Spline interpolation of short-term resistance of pure wood S o u r c e: made by S.I. Abrakhin The results of spline interpolation using cubic conditions at the boundaries of experimental data (Rc) are not suitable for predicting their further behavior using the extrapolation method. The reason is that the interpolated values show a significant drop in strength to ~25 MPa over a time interval from 0 to 49 years, and then an unexpected increase, which contradicts the nature of the degradation of wood properties. On the contrary, the interpolated data obtained using linear (Rl) and parabolic (Rp) boundary conditions at the boundaries of the experimental data for coupling the first and second derivatives show a moderate increase in the values of short-term resistance of pure wood over a time interval from 0 to 49 years to ~30.5 MPa. The data obtained can be used to predict future behavior by extrapolation, as they correspond to the physical behavior of wood. 2.2. Extrapolation of Experimental Data Extrapolation is a method of estimating the values of short-term resistance of wood outside the range of available experimental data. It is used to predict the resistance values of pure wood in the future or under other unexplored conditions. Extrapolation is riskier than interpolation, as it assumes that the relationship identified within the experimental data will persist beyond its boundaries. Experimental data extrapolation is the process of selecting a mathematical function or model that best describes a set of experimental nodal points. The goal is to find a relationship between variables that minimizes the error between the predicted model values and the actual experimental data. Extrapolation by a power polynomial of power n: and a linear function, which is its particular case when n = 1, is not considered, since these results will correspond to polynomial interpolation. The following extrapolation options for the available experimental data are considered: - exponential extrapolation: - logistic extrapolation: - sinusoidal extrapolation: - power extrapolation: - logarithmic extrapolation: Based on the results of minimizing the error between the model values and the real experimental data, coefficients a, b, and c are calculated, which, when substituted, will allow to estimate the resistance value outside the range of the experimental data. The results of extrapolation of the experimental data performed in the Mathcad software are shown in Figure 5. It can be concluded from the obtained results that sinusoidal extrapolation gives the resistance values varying from » 25 to » 31 MPa with a period of around 7 years. Sigmoidal and logarithmic extrapolation give a constant value (30 and 0 respectively). Exponential and power extrapolation show good results in the historical range of experimental data with a coefficient of determination of 0.999 and 0.998, respectively, but outside the historical period, the resistance values become negative by the age of 75, which is not consistent with reality. Therefore, these extrapolation methods cannot be used to predict the residual strength of wooden structures. There remains the option of extrapolating experimental data using the autoregression method. The autoregression (AR) method is a statistical time series analysis method that is used to predict future values of a series based on its previous observations. The underlying concept of the autoregression method is that the current value of a time series can be represented as a linear combination of several previous values of the same series and a random error. The formula of the autoregression model is as follows: (4) where Rt is the current value of the time series; at-1, at-2, …, at-p are the regression coefficients; Rt-1, Rt-2, …, Rt-p are the time series values from the previous periods; p is the order of the model (the amount of previous values of the time series considered); e(p) is the prediction error. Experimental data Age of the structure, years Short-term resistance of pure wood (Rn), MPa Figure 5. Approximation of short-term resistance values of pure wood S o u r c e: made by S.I. Abrakhin One of the autoregression methods of is the Burg method. Its main task is to find the coefficients at-1, at-2, …, at-p that minimize the prediction error e(p). The main difference between Burg method and other algorithms is that it minimizes the prediction error, both in the forward and backward directions. This leads to more stable and reliable estimates of the coefficients, especially when working with short time series. The results of the extrapolation carried out in the MathCad software using the Burg method of interpolation of the experimental data obtained as a result of piecewise-linear interpolation are shown in Figure 6. Age of the structure, years Experimental data Extrapolation (forecast) Piecewise-linear approximation Design resistance of wood (100 years) Short-term resistance of pure wood (Rn), MPa Figure 6. Results of extrapolation of piecewise linear interpolation of experimental data S o u r c e: made by S.I. Abrakhin and A.V. Lukina 3. Results and Discussion The developed model (Figure 6) should be verified by comparison with theoretical and experimental studies [19; 20]. As seen from the graph in Fig. 6, the system proposed as a predictive model for estimating the residual strength of wooden structures is consistent with the curve of long-term resistance of wood developed by F.P. Belyankin and U.V. Ivanov and the Weibull exponential law [21; 22]. The asymptotic nature of the curve of long-term resistance of wood shows that, although the ultimate strength decreases with increasing duration of the applied load, it is not unlimited - it tends to a certain constant value , equal to the vertical coordinate of the asymptote of the curve [23; 24]. In the context of the reliability of wooden structures, the use of the Weibull exponential law allows to take into account the variability of the mechanical properties of wood and estimate the likelihood of failure under prolonged loading. In applied calculations, the reliability of a system is expressed by an exponential law [25]. A model based on the Weibull distribution is also often used. The probability of fail-safe operation is defined as: (5) where tc and β are positive parameters. For β > 1, formula (5) describes the behavior of “aging” objects, in which the failure rate increases over time. The failure rate is usually relatively high at the beginning. Then it decreases and remains approximately constant over a long period of operation, increasing towards the end due to aging or deterioration. The developed predictive modeling graph for estimating the residual strength of wooden structures (rafter spars) based on experimental data (Figure 6) has a nonlinear relationship, where the initial strength of wood decreases with increasing time of being subjected to the load. An important aspect of this curve is that it allows to take into account the time factor when assessing the strength of wood, which is critical for long-term structural design. In [26], the values of the ultimate strength of rafter spars in residential buildings built in the 1930s were determined: 15.25 and 15.36 MPa. At the time of the experimental studies, the age of the structures was just over 90 years. Analyzing the predictive modeling for estimating the residual strength of wooden structures based on the Burg method, it can be found that the experimental data of the author of [27] correlate very well with the graph in Figure 6. The predictive modeling curve can be interpreted as a reflection of the change in the strength parameter over time. As the load exposure time increases, the strength parameter decreases, which leads to an increased probability of failure (destruction). Thus, the developed predictive modeling graph can be used to quantify the residual strength of wood over time. 4. Conclusion Interpolation and extrapolation of experimental data on the short-term resistance of pure wood are important tools for estimating the residual life of wooden structures. They allow using limited data to predict changes in the strength characteristics of wood over time, which is essential for the safe and efficient operation of wooden structures. Extrapolation, being a riskier process, requires the use of mathematical models of wood aging and confirmation of the results by additional experimental studies. 1. An interpretation of the processes of deformation and degradation of the strength of wooden structures is proposed based on modeling the values of the residual strength of wooden structures obtained experimentally. 2. A predictive modeling algorithm has been developed and implemented to estimate the residual strength of wooden structures based on experimental data. 3. The developed predictive modeling graph for estimating the residual strength of wooden structures has good convergence with the theoretical assumptions and experimental results of other studies. 4. The use of the described approaches will help engineers to manage the operational life of wooden structures more effectively, ensuring their reliability and durability. The use of the results of the work in updating the regulatory framework and the practice of construction (repair) will ensure the adaptability of wooden load-bearing structures to prevent and eliminate failures through maintenance and timely repairs.

About the authors

Sergey I. Abrakhin

Vladimir State University named after Alexander and Nikolay Stoletovs

Email: abrahin_s@vlsu.ru
ORCID iD: 0009-0002-8589-4826
SPIN-code: 2121-2007

Candidate of Technical Sciences, Associate Professor of the Department of Building Structures, Institute of Аrchitecture, Civil Engineering and Energy

87 Gorky St, Vladimir, 600000, Russian Federation

Anastasiya V. Lukina

Moscow State University of Civil Engineering (National Research University)

Author for correspondence.
Email: pismo.33@yandex.ru
ORCID iD: 0000-0001-6065-678X
SPIN-code: 8745-0004

Candidate of Technical Sciences, Associate Professor of the Department of Architectural and Construction Design and Environmental Physics, Institute of Architecture and Urban Planning

26 Yaroslavskoye highway, Moscow, 129337, Russian Federation

Mikhail S. Lisyatnikov

Vladimir State University named after Alexander and Nikolay Stoletovs

Email: mlisyatnikov@mail.ru
ORCID iD: 0000-0002-5262-6609
SPIN-code: 4089-7216

Candidate of Technical Sciences, Associate Professor of the Department of Building Structures, Institute of Аrchitecture, Civil Engineering and Energy

87 Gorky St, Vladimir, 600000, Russian Federation

Danila A. Chibrikin

Vladimir State University named after Alexander and Nikolay Stoletovs

Email: dachibrikin@outlook.com
ORCID iD: 0000-0001-9278-4559
SPIN-code: 1809-6997

Candidate of Technical Sciences, Associate Professor of the Department of Building Structures, Institute of Аrchitecture, Civil Engineering and Energy

87 Gorky St, Vladimir, 600000, Russian Federation

References

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