Assessment of Knot-Induced Degradation in Timber Beams: Probabilistic Modeling and Data-Driven Prediction of Load Capacity Loss

Abstract

Timber structural performance is significantly influenced by natural knots, which serve as critical indicators in ancient architectural heritage preservation and modern sustainable building design. However, existing studies lack a comprehensive quantitative analysis of how the randomness of timber knot parameters relates to load-bearing capacity degradation. This study introduces a multiscale evaluation framework that integrates physical testing, probabilistic modeling, and data-driven techniques. Firstly, static tests on full-scale timber beams with artificially introduced knots reveal the failure mechanisms and load capacity reduction associated with knots in the tension zone. Subsequently, a three-dimensional Monte Carlo simulation, modeling random distributions of knot position and size, demonstrates that the midspan region is most sensitive to knot effects, with load capacity loss being more pronounced on the tension side than on the compression side. Finally, a predictive model based on a fully connected neural network is developed; feature analysis indicates that the longitudinal position of knots exerts a stronger nonlinear influence on load capacity than radial depth or diameter. The results establish a mapping between knot characteristics, stress field distortion, and ultimate load capacity, providing a theoretical basis for safety evaluation of historic timber structures and the design of defect-tolerant timber beams in modern engineering.

1. Introduction

Timber is an integral building material celebrated for its sustainability and historical significance, playing a crucial role in the preservation of ancient architecture and modern structural engineering. However, inherent natural defects, particularly the distribution and characteristics of knots, pose substantial challenges to the mechanical reliability of timber beams. Knots disrupt fiber continuity and create localized stress concentrations, which can reduce the bearing capacity of timber elements by 30% or more [1]. Historical inspection data from historical structures indicate that many instances of beam failure are directly associated with the presence of knots in tension zones, underscoring the critical importance of understanding knot effects.

Research into the impact mechanisms of timber knots has progressed. During the initial experimental phase, researchers established foundational insights: physical testing [2] revealed that knots could alter damage modes from ductile flexural failure to brittle fracture, thereby highlighting their detrimental mechanical influence. Chun-Wei Chang and Far-Ching Lin [3] further advanced this understanding using digital image correlation (DIC), observing strain distributions in Japanese cedar boards with knots under tensile load, which vividly demonstrated strain concentration effects at knot regions, which was an experimental breakthrough that quantified how knots locally amplify deformation. Complementing such mechanical analyses, Erik Johansson [4] et al. and F. Longeteaud [5] et al. developed automated knot detection algorithms for high-speed CT and X-ray CT images, respectively. These techniques enabled precise extraction of knot geometry from timber samples, resolving critical data acquisition challenges for subsequent mechanical modeling. Zhong Yong [6] et al. developed empirical formulas illustrating the inverse relationship between knot diameter and bending strength, while Winandy [7] et al. demonstrated that a mere 5% mass loss due to decay could lead to a 20% reduction in strength, emphasizing the importance of biological degradation effects.

The subsequent numerical simulation marked a breakthrough in modeling knot effects at various scales. Pablo Guindos and Manuel Guaita employed finite element analysis (FEA) to investigate how knot geometry influences wood bending, validating a 3D knot model and revealing seven characteristic positions that dominate mechanical response [8]. Building on this, Khaled Sand and András Lengyel used parametric FEA to simulate knot-bearing spruce beams, quantifying how knot parameters affect flexural behavior, which is an approach that bridged experimental observations with numerical prediction [1]. Georg Kandler and colleagues further refined geometric modeling: by leveraging tracheid effects and pith position estimation; they reconstructed knot geometry in timber boards, verifying 3D fiber angles via simulated annealing optimization. This work provided a robust geometric framework for timber grading and FEA, with applications in both material characterization and structural simulation [9]. Baño et al. constructed finite element models of three types of knots, achieving prediction errors within 9.7% using simplified hole models, thus laying a foundation for further quantitative analysis [10]. Shi Bainan [11] et al. found that live knots could enhance load-bearing capacity through stress redistribution, while Ding Xing [12] et al. introduced a critical area threshold to quantitatively assess knot impact.

Recently, research has shifted toward multiscale modeling approaches. Lukacevic et al. developed a 3D fiber deviation model with stiffness prediction errors below 8%, capturing how knot-induced fiber misalignment affects macroscopic mechanical properties [13]. Vanessa Baño et al. explored the effects of knots on the load-bearing capacity and stress distribution in Scots pine (*Pinus sylvestris*) timber beams using finite element models: simulating multiple working conditions, analyzing the model adaptability of knot positions, and correlating knots with stress distribution and bending strength [14].

In the interdisciplinary applications of timber structure and computer science, Johannes Belz and Benjamin Kromoser developed a structural optimization framework for timber building components by integrating Python programming, genetic algorithms, and ABAQUS finite element analysis (FEA) [15]. This framework optimizes the structure of the oriented strand board (OSB) through multi-technology integration and verifies its performance. Angela Balzano and other researchers utilized cutting-edge technologies such as machine learning to analyze wood’s adaptation to environmental stress and anatomical changes, providing interdisciplinary support for forest management and timber technology [16]. Rana Ehtisham and colleagues achieved knot classification and defect quantification in timber structures through image processing and convolutional neural network (CNN) models, demonstrating the potential of AI in health monitoring of timber structures [17].

However, there are three shortcomings in the current research. First, the existing deterministic analysis framework struggles to characterize the randomness of knot parameters. Most studies conduct mechanical analyses based on knots with fixed positions and sizes, ignoring the random variability of knot positions and geometric dimensions in actual timber. For example, the European Code EN 1995 [18] classifies knots using simplified assumptions, neither distinguishing the sensitivity differences between different positions, such as the mid-span and ends of beams, nor incorporating the probabilistic impact of random knot distribution on load-bearing capacity. This deterministic approach leads to deviations between safety assessment results and the actual structural performance, especially in dealing with the complexity of knot distribution in ancient buildings. Second, there is a lack of cross-scale correlation between mesoscopic damage mechanisms and macroscopic mechanical responses. Existing studies mostly analyze micro-fiber fracture or macroscopic load-bearing capacity degradation in isolation, failing to establish a multi-scale mapping relationship from fiber deviation and stress concentration induced by knots to beam cracking and load-bearing capacity decline. Third, traditional regression models struggle to reveal non-linear influence mechanisms. There are complex non-linear coupling effects among knot positions, sizes, and load-bearing capacity, but most existing studies use linear regression or empirical formulas, resulting in vague conclusions from parameter sensitivity analysis and limiting the targeted control of key defect parameters in engineering design.

Aiming at the problem that traditional research lacks multi-scale probability modeling and non-linear mechanism analysis of the influence of wood knots on the bearing capacity of wooden beams, this paper proposes a multi-scale research method integrating experimental calibration, probability modeling and intelligent analysis, and refines clear research objectives: first, quantify the influence of the random distribution of knot position and size on the bearing capacity of wooden beams through probability modeling, so as to solve the limitation that the traditional deterministic method is difficult to capture random characteristics; second, use the fully connected neural network (FCNN) algorithm to decouple the action weights of knot parameters, breaking through the bottleneck that conventional analysis is difficult to describe such complex relationships.

Through the cross-scale framework combining the wooden beam loading experiment, three groups of 1.5 million (3 × 500,000) Monte Carlo random simulations and the FCNN feature decoupling algorithm, the research systematically reveals the quantitative relationship between knot parameters and bearing capacity. Specifically, a Monte Carlo model including the random distribution of knots is constructed to break through the constraints of deterministic methods, and an FCNN algorithm is developed to quantify the influence weights of the parameters. The sensitivity analysis based on random simulation and machine learning clarifies the non-linear mechanism of the effect of knots on the bearing capacity of wooden beams. The formed “experiment–simulation–algorithm” paradigm provides an academic innovative and engineering practical solution for the reliability evaluation and defect tolerance design of wood knots, which is particularly suitable for the fields of ancient architectural heritage protection and modern wood structure engineering.

2. Materials and Methods

2.1. Loading Test and Method

2.1.1. Determination of Material Strength

Douglas-fir sawn timber was selected as the testing material [14]. The properties evaluated included density, tensile strength along the grain, compressive strength along the grain, compressive strength transverse to the grain, flexural strength, and flexural modulus of elasticity, in accordance with relevant standards [19,20,21,22,23]. The compressive specimens are shown in Figure 1a, and the flexural specimens in Figure 1b, with all dimensions in the figures expressed in millimeters (mm). The tensile specimens were prepared according to Appendix A of the Chinese standard GB 1928-2009, “General Principles of Physical and Mechanical Test Methods for Timber” [24]. Although the specimen dimensions follow Chinese codes, they exhibit minimal differences from those required by European standards [23]. Six tests were conducted for each mechanical parameter, and the average values of the obtained parameters are summarized in

Table 1. Mechanical properties of American Douglas-fir from the Rocky Mountains.

Name	Tensile Strength Parallel to Grain (MPa)	Compressive Parallel to the Grain (MPa)	Compressive Strength Perpendicular to Grain (MPa)	Flexural Strength (MPa)	Flexural Modulus of Elasticity (MPa)
Average value	93.82	34.2	6.0	87.88	13,500.24
Standard deviation	7.47	0.39	0.21	2.43	4.02

.

2.1.2. Details of Specimens

Referencing the recommended values in the historical Chinese architectural canon “Construction Method Style” and the relevant literature [25], the cross-sectional dimensions of the timber beam specimens were set at 80 mm × 120 mm, with a span of 2300 mm. A total of seven beams were fabricated across three groups, among which one beam served as a reference specimen without knots (designated as S’), and the remaining six beams incorporated timber knot defects.

It should be noted that the European standard EN 408 recommends a span-to-height ratio of not less than 18 for timber beam tests to ensure bending failure precedes shear failure [26]. In this study, the specimen span is 2300 mm, and the section height is 120 mm, resulting in a span-to-height ratio of approximately 19.2, which meets and is compatible with the testing requirements of European codes.

The design of timber knot defects involved creating artificial grooves to mimic the weakening effects of natural knots, particularly the stress concentration induced in the tension zone and the extrusion-sliding behavior in the compression zone. These artificial knots were positioned at the mid-span of the beams. Two grooves, each 2 mm wide, were manually cut at an interval of 15 mm. The groove depths were divided into three levels, corresponding to 1/10 (12 mm), 1/5 (24 mm), and 1/3 (40 mm) of the beam height, as detailed in

Table 2. Dimensions of the timber beams.

Specimen’s Type	Elevation Views			Section Views
Defect-free specimen
Specimens containing defects

(all dimensions in the schematic diagrams of Table 2 are in mm).

The defect location is denoted as follows: a value of 0 indicates the artificial knot is in the compression zone, and a value of 1 indicates placement in the tension zone. The six timber beams with artificial defects are labeled in the format “S-#-#,” where the first hash represents the zone (0 for compression, 1 for tension) and the second hash indicates the groove depth. For example, a specimen with knots in the tension zone and a groove depth of 12 mm is labeled S-1-12, and a specimen with knots in the compression zone and a depth of 40 mm is labeled S-0-40.

2.1.3. Loading Devices and Loading Protocols

A quasi-static testing method was adopted, where monotonic loading was applied to the horizontally placed timber beam through a servo-controlled loading device. The specimen had a clear span of 2160 mm, with two loading points situated 360 mm on each side of the mid-span (as shown in Figure 2a). The testing setup included a fixed beam, the specimen, a load distribution beam, an actuator, and a reaction wall. The test was carried out under a constant displacement control mode at a loading rate of 3 mm/min. The loading protocol, depicted in Figure 2b, was designed to meet the requirements of quasi-static loading. The loading was implemented in a continuous and slow incremental manner to ensure quasi-static conditions. Load and displacement data were continuously recorded during the entire test to evaluate the specimen’s load-bearing performance.

2.2. Monte Carlo Simulation

Acknowledging the inherent randomness in the occurrence of timber knots in timber beams, this study conducted three sets of Monte Carlo simulations using MATLAB (Version number: 9.12.0.1884302 (R2022a)), each comprising 500,000 trials. We plotted the relationship between the number of simulations and the mean and standard deviation of the simulation results (as shown in Figure 3). The results show that both indicators gradually stabilize when the number of simulations reaches 100,000. Considering that further increasing the number of simulations would significantly increase the computational costs of data processing and subsequent machine learning models, we selected 500,000 simulations as the final number after comprehensive consideration.

These simulations accounted for the stochastic distribution of knot positions and sizes to quantify the delineation of sensitive zones and the corresponding reduction in load-bearing capacity along the span. Artificial timber knots were modeled within the beams to simulate their effects: the reduction of the cross-sectional area in tension zones and mutual compression coupled with sliding at the knots in compression zones. Specifically, the core logic of the code used in this paper to evaluate structural failure is as follows: First, if the cross-sectional area of the simulated knot defect exceeds the total cross-sectional area of the beam, failure is directly determined, and the percentage of load-bearing capacity reduction is set to 100% (this is a condition added as a safety precaution, though such cases did not occur in actual simulations). For beams that do not fail directly, the cross-sectional area at the defect location is updated based on the defect’s location (tension or compression zone) and size, and different stress concentration factors are applied to modify the material strength. The ultimate load-bearing capacities of both the defect section and the mid-span section are then calculated. If the calculated stress in either section exceeds the material strength obtained from the material property tests mentioned earlier, the timber beam is considered to have failed. The morphology of these artificial knots is depicted in Figure 4.

The specimen configuration involved a simply supported timber beam subjected to uniform load, with end A constrained to restrict translations in the Y and Z directions, and end B constrained to restrict translation in the Y direction, as illustrated in Figure 4. This setup aimed to realistically replicate the boundary conditions and the stochastic nature of knot distribution in practical timber applications.

2.2.1. Bearing Capacity Reduction Function

Consider such a function (Equation (1)):

$$(1-{\displaystyle \frac{Bearing capacit{y}_{beam with knot defect}}{Bearing capacit{y}_{healthy beam}}})\times 100\%=Damage=F(x_1,x_2,x_3,x_4)=\left\{\begin{array}{c}{f}_c(x_2,x_3,x_4),if x_1=0\\ f_t(x_2,x_3,x_4),if x_1=1\end{array}\right.$$

(1)

• Damage: the percentage reduction in the load-carrying capacity of the beam; this value is 0% for defect-free beams.
• x₁: The positional parameter indicating the location of the timber knot relative to the beam’s cross-section. When the knot is located in the compression (upper) zone, x₁ = 0; when located in the tension (lower) zone, x₁ = 1.
• x₂: The normalized longitudinal position of the timber knot along the beam’s length. When the knot is at the end A of the beam, x₂ = 0; when at end B, x₂ = 1
• x₃: the diameter of the timber knot.
• x₄: the depth of the timber knot.

This study will evaluate how these variables influence the Damage function through 3 × 500,000 Monte Carlo simulations, analyzing the impact of x₁ to x₄ on the function’s outcomes.

2.2.2. Modeling of Stochastic Timber Knots

Timber knots are randomly generated in the tension or compression zones of a beam, following a Bernoulli distribution, with their positions assumed to be uniformly distributed along the beam’s length. However, in reality, the diameter and depth of timber knots are more appropriately modeled by a normal distribution. Referencing Pan Yuchen [27], who investigated natural timber knots in 16 beams from Zhou Fujiu’s mansion in Yangzhou City, this study employs the mean values from his findings. Specifically, the mean diameter (x₃) is set at 15 mm, and the mean depth (x₄) at 25 mm. Based on the variance (σ) of these parameters, the simulation is divided into three groups (①, ②, and ③), with detailed specifications outlined in

Table 3. Parameter settings for x₁–x₄ in the three simulation groups.

No	①	②	③
Parameter Value	$\left\{\begin{array}{c}{x}_1~B(0,0.5)\\ x_2~U(0,1)\\ x_3~N(0.015,{0.005}^2)\\ x_4~N(0.025,{0.005}^2)\end{array}\right.$	$\left\{\begin{array}{c}{x}_1~B(0,0.5)\\ x_2~U(0,1)\\ x_3~N(0.015,{0.010}^2)\\ x_4~N(0.025,{0.010}^2)\end{array}\right.$	$\left\{\begin{array}{c}{x}_1~B(0,0.5)\\ x_2~U(0,1)\\ x_3~N(0.015,{0.015}^2)\\ x_4~N(0.025,{0.015}^2)\end{array}\right.$

.

Theoretically, under a normal distribution, the diameters (x₃) and depths (x₄) of timber knots could mathematically extend to positive and negative infinity. However, this contradicts real-world engineering scenarios. To address this, additional constraints are applied to these parameters: based on Pan Yuchen’s measurements of natural timber knots in 16 timber beams from Zhou Fujiu’s mansion in Yangzhou, the maximum and minimum values are set as three times and one-third of their respective normal distribution means, as defined in Equation (2). The probability distributions for parameters (x₁) to (x₂) are graphically illustrated in Figure 5, while the distributions for (x₃) and (x₄) are systematically summarized in

Table 4. Probability density functions of variables x₃ and x₄ for groups ① through ③.

Group	①	②	③
σ	0.005	0.010	0.015
x3
x4

.

$$\left\{\begin{array}{c}{x}_3\\ x_4\end{array}\right.\in [{\displaystyle \frac{1}{3}},3]\cdot \left\{\begin{array}{c}{\mu }_{x_3}\\ {\mu }_{x_4}\end{array}\right.$$

(2)

Based on Equations (1) and (2), as well as Table 4, a MATLAB program was developed to define the geometry of the beam and generate random timber knots on it. Material parameters are adopted in accordance with Table 1. The program evaluates the tension, compression, and bending performance in the critical sections. It then calculates the reduction in the maximum load capacity of the beam caused by each generated timber knot, relative to a defect-free timber beam, while ensuring that the applied load does not surpass the overall capacity.

2.3. MATLAB-Based Machine Learning

This study utilized MATLAB to develop a machine learning algorithm that encompasses key aspects such as data preprocessing, model construction, training optimization, evaluation, validation, and result analysis. The primary goal was to build an accurate regression model capable of predicting how the position, diameter, and depth of timber knots influence damage, as well as assessing the sensitivities of these factors.

2.3.1. Machine Learning Process

(1)

Data pre-processing

First, the dataset was divided into two groups based on the location of the timber knots: compression and tension zones. Features extracted included the knot position (x₂), diameter (x₃), and depth (x₄), while the load-bearing capacity reduction rate (Damage) served as the target variable. Specifically, the validation method employed in this approach is a two-step HoldOut partitioning with fixed ratios. Initially, the ‘HoldOut’ method from ‘cvpartition’ is used to split the original dataset into a training–validation set (70%) and a test set (30%). Subsequently, the training–validation set is further divided into a training set (used for model parameter learning) and a validation set (used for monitoring model performance and tuning hyperparameters during training) at a 50:50 ratio using the same ‘HoldOut’ method. This process establishes a complete “training–validation–testing” workflow with the dataset partitioned into 35% training data, 35% validation data, and 30% test data. This traditional HoldOut validation method is straightforward, computationally efficient, and suitable for large datasets. The potential bias caused by seed-dependent partitioning is mitigated by setting ‘Shuffle’, ‘every-epoch’, which ensures data shuffling at each training epoch.

(2)

Model construction

A fully connected neural network was employed for regression prediction. The network architecture comprises the following:

• Input Layer: receives three features: knot position (x₂), diameter (x₃), and depth (x₄).
• Hidden Layers: two fully connected layers with 128 and 64 neurons, respectively, both utilizing the ReLU activation function to enhance the model’s nonlinear learning capabilities.
• Output Layer: a single neuron outputs the estimated Damage (load capacity reduction), paired with a regression layer that computes the prediction error.

This structure enables the network to learn the complex relationship between timber knot parameters and loss of bearing capacity through successive feature transformations.

(3)

Model training

The code employs a comprehensive framework for model development and evaluation, integrating robust validation strategies, hyperparameter tuning, and multi-faceted performance assessment. A stratified HoldOut validation approach with three-level dataset division (50% training, 15% validation, 15% test sets) ensures representative sampling across classes. During training, the Adam optimizer is employed with a maximum of 100 epochs and mini-batch size of 512, balancing computational efficiency and memory utilization. To prevent overfitting, the validation set (XVal, YVal) is evaluated every 30 epochs to monitor loss trends and adjust hyperparameters dynamically. The learning rate starts at 0.001 and decays by 0.1 every 50 epochs, enabling precise parameter updates in later training stages. The data undergo Z-score standardization to stabilize training, while implicit regularization is achieved through learning rate decay and early stopping indicators derived from validation/training loss divergence. Performance assessment combines quantitative metrics (RMSE, MAE, R²) computed on the test set with visual analytics: scatter plots of predictions vs. ground truth, residual distribution histograms, feature importance rankings, partial dependence plots, and 3D input space residual mappings. This multimodal evaluation rigorously tests generalization and identifies error patterns. The MATLAB ‘trainNetwork’ function orchestrates training by integrating dataset management, architectural specifications, and optimization parameters, culminating in a system that balances convergence speed with generalization capacity through adaptive hyperparameter scheduling.

The overall machine learning workflow and process are illustrated in Figure 6 below.

2.3.2. Model Evaluation and Visualization Analysis

The model’s predictive performance was assessed using root mean square error (RMSE), mean absolute error (MAE), and the coefficient of determination (R²). Scatter plots of predicted versus actual values, along with residual plots, were generated to verify the absence of systematic bias and to ensure that residual fluctuations remained within a controllable range. To further analyze the spatial distribution of errors, an error map was plotted to identify regions with high prediction inaccuracies. Partial dependence plots were constructed to illustrate the relationship between the features (x₂ to x₄) and the predicted damage, revealing the characteristic influence of each parameter. Additionally, feature importance ranking was performed to identify the most influential factors affecting model predictions.

3. Results and Discussion

3.1. Loading Experiment Results

The static load test indicated that the S’ specimen reached a maximum load of 25 kN at the point of damage. During monotonic loading, when the servo actuator was displaced to 40 mm, minor cracks appeared in the timber beams accompanied by sizzling sounds. As the displacement increased to 55 mm, the timber beams exhibited extensive cracking, and the load-bearing capacity significantly declined, indicating entry into the damage stage. When the timber knots were located in the compression zone (e.g., specimen S-0-24), cracks primarily developed horizontally (Figure 7a). Conversely, when the knot was situated in the tension zone (e.g., specimen S-1-24), cracks initially appeared at the defect site and propagated preferentially (Figure 7b) until the beam ultimately lost its load-bearing capacity (Figure 7c,d).

Figure 8 illustrates the displacement–load curves, while

Table 5. Characteristic values of loads on the curves of timber beams in a defect-free specimen and defective specimens.

Specimen	Yield Load (kN)	Breaking Load (kN)	Ultimate Load (kN)	Bending Moment at Ultimate Load (kN·m)	Percentage Decrease in Bearing Capacity (%)
S’	14.20	22.30	24.70	17.78	-
S-0-12	13.40	15.30	17.20	12.38	30.37
S-1-12	7.03	11.13	12.66	9.12	48.71
S-0-24	11.60	13.30	16.90	12.17	31.55
S-1-24	6.27	8.57	10.75	7.74	56.47
S-0-40	9.10	10.50	12.60	9.07	48.98
S-1-40	4.73	7.68	9.72	7.00	60.64

presents the corresponding load characteristics of these curves. As shown in Figure 8a, the specimen S’ exhibits an initial stage where the load steadily increases with displacement until reaching a peak value, after which the specimen experiences failure, with an ultimate load of 24.7 kN. In contrast, the load–displacement curves of specimens with timber knot defects (shown in Figure 8b,c) demonstrate lower ultimate load capacities. Notably, for specimens with knots of the same size, the specimen S-1-# exhibits a lower ultimate load capacity compared to S-0-#. Generally, the deeper the knot’s position within the timber, the sooner the member approaches its ultimate load capacity.

Specifically, the ultimate loads of specimens S-0-12, S-0-24, and S-0-40 were 17.2 kN, 16.9 kN, and 12.6 kN, respectively, representing reductions of 30.37%, 31.55%, and 48.98% relative to the defect-free timber specimen. For specimens S-1-12, S-1-24, and S-1-40, the ultimate loads were 9.9 kN, 8.4 kN, and 7.6 kN, respectively, which are 48.71%, 56.47%, and 60.64% lower compared to the defect-free specimen.

When combined with Figure 7b–d, it can be inferred that the differences are primarily due to the weakening of the effective load-bearing section caused by the knots, which directly reduces the ultimate load capacity. Knots alter the primary fiber connections of the timber, leading to decreased initial stiffness. This results in concentrated deformation and impaired load-sharing capacity within the structure, causing the load capacity to peak earlier. Furthermore, the accumulation of cracks in various regions accelerates the expansion of existing defects, weakening the structure under lower loads. Consequently, the load–displacement curves display lower peak loads and steeper post-peak declines, reflecting the compromised structural integrity caused by the presence and position of the knots.

3.2. Monte Carlo Simulation Results

Based on the functional model outlined in Section 2.2.1, three sets of 500,000 Monte Carlo simulations were performed, and the results are presented in three groups.

3.2.1. Cloud Diagram of Bearing Capacity Reduction

Table 6. Percentage reduction in bearing capacity of the timber beams.

Group	①	②	③
σ	0.005	0.010	0.015
x1 = 0
x1 = 1

displays three-dimensional scatter plots illustrating the percentage decrease in the bearing capacity of the beams within the tension and compression zones. Each point in the plots represents the outcome of a single simulation. Using the defect-free specimen S’ as a baseline for comparison, the reduction in load-carrying capacity varies from minimal to significant, with the color of each point gradually transitioning from blue to red to indicate increasing damage severity. From the data, it is evident that there is a strong correlation between the beam’s load capacity and the position of the timber knots (x₂). Specifically, the closer the knot is to the mid-span (i.e., x₂ approaching 0.5), the greater the reduction in the beam’s load capacity.

In practical structures, the core mechanical function of timber beams is to resist bending deformation, but their load-bearing characteristics vary significantly due to differences in their locations and loading conditions. Additionally, ancient buildings were constructed without systematic structural calculations, and the structures have undergone natural aging over time. As a result, even a slight reduction in the bearing capacity of wooden beams in certain positions may adversely affect the overall structural performance. Given the complexity outlined above, it is difficult to mechanically divide damage grades using uniform thresholds. Therefore, based on practical engineering experience, we have classified the reduction in bearing capacity as follows: 0–5% are classified as Class I defects, 5–25% as Class II defects, and reductions exceeding 25% as Class III defects. These classifications are color-coded in

Table 7. Distribution of the defect classes in timber beams.

Group	①	②	③
σ	0.005	0.010	0.015
x1 = 0
x1 = 1

: green for Class I, yellow for Class II, and red for Class III.

Analysis of Table 7 reveals that Class III defects, indicative of significant damage, predominantly occur in the span-center region and are evident for specimens with timber knots in both the compression and tension zones. As the positions of x₃ and x₄ increase, the range of Class III defects also expands. Specifically, when timber knots are located in the compression zone, Class III defects mainly cluster within 40% to 60% of the beam length, whereas in the tension zone, this range extends from 20% to 80%. This highlights the higher sensitivity of knots in the tension zone to the bearing capacity decline of the timber beams. Regarding Class I defects, regardless of whether the knots are in the tension or compression zones, these are typically localized within the 20% regions at both ends of the beam. For Class II defects, when timber knots are situated in the compression zone, they primarily appear within the intervals 0.2 ≤ x₂ ≤ 0.4 and 0.6 ≤ x₂ ≤ 0.8. Notably, these defects decrease rapidly as x₃ and x₄ increase. When knots are located in the tension zone, Class II defects are confined to narrow regions near x₂ ≈ 0.2 and x₂ ≈ 0.8, with their occurrence diminishing quickly as x₃ and x₄ grow larger.

The artificial timber knots used in the defective group of timber beams, as designed in Section 2.1.2, are all positioned at the mid-span (50% along the length of the beams). The diameter of each timber knot is 15 mm plus twice the width of the fiber section (2 × 2 mm), totaling 19 mm. The depths of the knots are 12 mm, 24 mm, and 40 mm, respectively.

To identify corresponding simulations with similar parameters within the Monte Carlo simulation data matrix, a search was conducted based on specified criteria: the variable x₂ should be within ±0.002 of 0.5 times the beam length, and neither x₃ nor x₄ should deviate more than ±2 mm from their original design values.

Table 8. Simulation results versus measured data.

Specimen	x1	No	x2	x3	x4	Damage (%)	Average (%)	Measured Results (%)	Absolute Error (%)
S-#-12	0	①	0.5015	17.569	10.966	31.56	31.58	30.37	+1.21
		②	0.4998	17.393	10.650	31.39
		③	0.4981	18.812	10.623	31.78
	1	①	0.4989	18.111	11.900	53.89	53.86	48.71	+5.15
		②	0.5015	18.394	11.890	53.95
		③	0.4986	18.654	10.897	53.73
S-#-24	0	①	0.5013	17.638	24.713	35.96	35.57	31.55	+4.02
		②	0.4999	17.240	24.212	35.62
		③	0.5014	17.158	22.464	35.13
	1	①	0.4985	17.066	23.516	56.10	56.49	56.47	+0.02
		②	0.5018	19.608	22.364	56.81
		③	0.5003	17.669	24.960	56.57
S-#-40	0	①	0.4989	20.806	40.970	41.23	40.45	48.98	−8.53
		②	0.5007	20.357	39.925	40.74
		③	0.4984	18.379	41.026	39.38
	1	①	0.4999	19.373	40.906	59.35	59.33	60.64	−1.31
		②	0.5002	19.190	39.741	59.15
		③	0.5005	19.589	41.213	59.48

presents the mean and absolute error of the damage predictions obtained from these simulations. The absolute errors between the simulated and measured values are all within 8.53%. The results demonstrate that the simulation outcomes are satisfactory and closely align with the experimental data.

3.2.2. Probability Distribution Histograms and Kernel Density Curves

From the probability distribution histograms and kernel density curves of all simulation results for the tensile and compressive zones of the timber beams (

Table 9. Probability distribution histograms and cumulative distribution function (CDF) plots for Groups ①–③.

Group	①	②	③
σ	0.005	0.010	0.015
Cumulative Probability
Probability Density

), several observations can be made. In simulations ① to ③, the kernel density curves for the compressive zone reach their peaks at Damage percentages of 32.125%, 30.703%, and 30.658%, respectively. Depending on the variance σ of parameters x₃ and x₄ across groups ① to ③, the Damage values are primarily distributed within the intervals of [0, 40] %, [0, 50] %, and [0, 60] %, respectively. Conversely, the kernel density curves for the tensile zone peak at Damage values of approximately 53.893%, 53.110%, and 53.137%. With differing variances σ of parameters x₃ and x₄, the Damage values in the tensile zone mainly distribute within [0, 60]%, [0, 70]%, and [0, 80]%, respectively.

These results indicate that variations in the variances of parameters x₃ and x₄ do not significantly influence the peak Damage positions in either zone. Compared to the compressive zone, the Damage distribution in the tensile zone exhibits a broader spread but remains more concentrated around its peak value. This phenomenon may be attributed to the fact that the tensile properties of wood are more sensitive to defects: knots in the tensile zone act as stress concentrators, leading to accelerated structural failure and a marked reduction in load-bearing capacity. Furthermore, the cumulative distribution function (CDF) plots reinforce this understanding: the curve for the compressive zone shows a sharp increase in probability at lower Damage levels, whereas the tensile zone curve rises more rapidly at higher Damage levels. This suggests that the probability of low Damage is higher in the compressive zone, while the probability of more severe Damage is greater in the tensile zone.

3.3. MATLAB Based Machine Learning Results

The trained model was applied to the test set to generate predictions, which were then transformed back to the original data scale using the inverse normalization formula. Predicted values were compared with the actual true values to assess the model’s performance. The root mean squared error (RMSE) was calculated to quantify the average magnitude of errors between the predicted and true values, where lower RMSE values indicate higher prediction accuracy. Additionally, the coefficient of determination (R²) was computed to evaluate the goodness-of-fit of the model to the data; values of R² closer to 1 signify stronger explanatory power.

Table 10. Evaluation of machine learning outcomes.

	RMSE	MAE	R2
Group 0 (x1 = 0)	0.20368	0.085381	0.99983
Group 1 (x1 = 1)	0.28641	0.13161	0.99984

summarizes these evaluation metrics for the two datasets, Group 0 (knot in compression zone, x₁ = 0) and Group 1 (knot in tension zone, x₁ = 1), providing a clear comparison of model performance across the different data groups.

The machine learning model demonstrates excellent performance on both datasets, with R² coefficients approaching 1 (Group 0 = 0.99983, Group 1 = 0.99984), indicating an extremely high degree of fit to the data. In terms of error metrics, Group 0’s RMSE (0.20368) and MAE (0.085381) are lower than those of Group 1 (RMSE = 0.28641, MAE = 0.13161), suggesting that the model achieves higher prediction accuracy for the subgroup with x₁ = 0.

Figure 9 illustrates the relationship between the machine learning predicted values (Predicted Values) and the Monte Carlo simulation results (True Values), with the reference line y = x included for comparison. The scatter points for both datasets are closely clustered around this line, indicating that the predicted values are very consistent with the true values. This distribution confirms that the model exhibits robust overall prediction performance.

Observing the residual plot in Figure 10, the residuals for both data groups oscillate around zero without exhibiting regular patterns such as systematic curves or periodicity. This suggests that the constructed machine learning model performs satisfactorily. Although there is slightly more fluctuation in the residuals within the region of lower predicted values, these deviations remain within an acceptable range (<6%). One possible explanation is that, while neural networks can effectively capture nonlinear relationships, more intricate feature interactions may be present in regions with low prediction values, contributing to the observed residual variability. As the predicted values increase, the residuals tend to more closely cluster around zero, indicating improved model accuracy. Overall, this pattern demonstrates that the model possesses high reliability and effectiveness in prediction.

Figure 11 showcases the model’s prediction error analysis through spatial distribution maps (subfigures (a,b)) and residual histograms with normal fitting (subfigures (c,d)). Subfigures (a,b) depict the magnitude and distribution patterns of prediction errors across varying wood knot parameter combinations. In most regions, errors remain within an extremely narrow range, demonstrating the model’s strong performance. While specific combinations (notably those with larger x₃ and x₄ values) exhibit marginally higher errors (still below 6%), the overall results confirm the model’s satisfactory predictive accuracy across diverse parameter settings.

Subfigures (c,d) present residual histograms for the compression and tension zones, respectively. The histograms are confined to the [−1, 1] x-axis range to highlight the concentration of the residuals. In the compression zone (c), residuals show a near-zero mean (Mean = −0.00) and small standard deviation (Std = 0.18), with a tight alignment to the normal distribution curve, indicating random, concentrated errors and exceptional prediction stability. The tension zone (d) exhibits a similar near-zero mean (Mean = 0.03) but slightly larger standard deviation (Std = 0.29), still maintaining a symmetric, normality-conforming distribution with errors within acceptable practical limits. Both zones confirm unbiased, normally distributed residuals, validating the model’s statistical reliability. The compression zone excels in precision, while the tension zone performs robustly with potential for further optimization. Collectively, these results underscore the model’s high accuracy and adaptability across different wood knot configurations.

3.4. Analysis of Machine Learning Results

Figure 12 illustrates the independent influence of individual features on the bearing capacity, providing insights into the relationship between each feature and the predicted values. As shown in the compression zone, the effect of x₂ on the prediction is relatively minor within approximately 0.2 times the beam length from both ends of the beam. Similarly, in the tension zone, x₂ has a limited influence within about 0.1 times the beam length from both ends. These findings are consistent with the results summarized in Table 6 and Table 7. This is also consistent with the research results of Saad and Lengyel [1].

Regarding x₃ and x₄, an increase in either feature corresponds to an increase in the predicted bearing capacity. Notably, x₃ exhibits a convex relationship with the prediction, while x₄ shows a concave trend. This indicates that as the diameter of the timber knot (x₃) increases, its contribution to the beam’s bearing capacity also increases, demonstrating an increasing marginal effect. Conversely, as the knot depth (x₄) increases, its contribution diminishes, indicating a decreasing marginal effect on bearing capacity.

This study comprehensively employs the permutation importance method and random forest algorithm to assess feature significance. For permutation importance analysis, the root mean square error (RMSE) of the baseline test set was first calculated. Each feature column was then shuffled individually, and predictions were regenerated to recalculate the RMSE. Feature importance was quantified by the increase in RMSE before and after shuffling; the larger the increase, the more critical the feature is to the model. The differences in feature importance are visually presented as a bar chart in Figure 13a.

For random forest feature analysis, features x₂, x₃, x₄ and the dependent variable Damage were first extracted from the Group 0 and Group 1 datasets. A regression model consisting of 100 decision trees was then constructed, with out-of-bag (OOB) data estimation and parallel computing enabled to optimize training efficiency. The contribution of each feature to the model’s prediction error was evaluated using the OOB permutation method and ranked in descending order of importance, as shown in Figure 13b.

As shown in Figure 13, both analytical methods indicate that for the Group 0 (x₁ = 0) and Group 1 (x₁ = 1) datasets, the importance ranking of the three features is consistently x₂ > x₃ > x₄. This suggests that the position of a timber knot has the most significant impact on the beam’s load-bearing capacity. Additionally, the higher importance of x₃ compared to x₄ implies that the width of the timber knot exerts a greater effect on the beam’s bearing capacity than its depth. This result is consistent with the conclusion verified by Saad and Lengyel through finite element models that “the location and size parameters of knots are critical factors governing the flexural capacity of timber beams” [1]. In the Group 0 (x₁ = 0) dataset, although x₂ remains the most influential feature, the effects of x₃ and x₄ are still non-negligible. By contrast, in the Group 1 (x₁ = 1) dataset, x₂’s dominance is more pronounced, while the impacts of x₃ and x₄ on the beam’s load-bearing capacity are even smaller.

4. Conclusions

This study systematically investigated the influence of timber knots on the bearing capacity of wooden beams using an integrated approach, combining monotonic loading tests, Monte Carlo simulations, and deep learning models. The main findings are summarized as follows:

(1)

Experimental verification of the timber knot weakening effect: Test results conclusively confirmed the significant weakening effect of timber knots on beam bearing capacity, particularly in the tension zone. Compared to defect-free beams, when knots were located in the compression zone, the ultimate bearing capacity decreased by 30.37% to 48.98%, representing a reduction of nearly one-third to half. When knots were situated in the tension zone, the reduction was even more pronounced, ranging from 48.71% to 60.64%, meaning a loss of nearly half to over sixty percent of capacity. Cracks preferentially initiated at defect sites caused by the knots, accelerating stiffness degradation.

(2)

Insights into random defect patterns from Monte Carlo simulations: A total of 3 × 500,000 Monte Carlo simulations were conducted to analyze the effects of defect randomness. The results reveal that timber knots in the tension zone exert a far greater impact on the beam’s bearing capacity than those in the compression zone. Specifically, when a knot is located at the mid-span of the tension zone, the maximum reduction in bearing capacity can reach a striking 75.67% (compared to a maximum reduction of 66.21% in the compression zone). Damage levels were classified into Class I, II, and III. For knots in the compression zone, Class III damage generally occurs within the 40–60% region of the normalized beam length. In contrast, for knots in the tension zone, Class III damage is distributed across a significantly wider range of 20–80%. Additionally, knots in the tension zone are more sensitive to capacity changes. Class I damage is typically distributed within 20% from the beam ends. For compression zone knots, Class II damage exhibits a bimodal distribution within the normalized length ranges of 0.2–0.4 and 0.6–0.8. Conversely, for tension zone knots, Class II damage is concentrated near the 0.2 and 0.8 marks, and these regions rapidly shrink as the knot parameters (diameter x₃ and depth x₄) increase.

(3)

Parameter sensitivity analysis via machine learning models: The fully connected neural network model identified the position of the timber knot (x₂) as the dominant factor influencing the beam’s bearing capacity, with significantly higher importance than the knot’s diameter (x₃) or depth (x₄). The model further demonstrated that increasing the knot diameter (x₃) weakens the bearing capacity in a convex pattern, indicating an increasing marginal weakening effect (i.e., each unit increase in diameter causes progressively greater capacity loss). Conversely, the influence of knot depth (x₄) gradually plateaus, reflecting a decreasing marginal effect. These insights provide a comprehensive understanding of how different parameters affect the structural performance of timber beams with knots.

5. Future Works

The present study and model exhibit the following limitations:

(1)

In Monte Carlo simulations, knot diameters and depths are assumed to follow normal distributions, with parameters derived from Pan Yuchen’s thesis on timber knots in 16 wooden beams from Yangzhou’s Zhou Fujiu Mansion. This assumption may not hold across different tree species or engineering contexts.

(2)

While artificial timber knots simulate mechanical behavior, they inadequately replicate three-dimensional morphologies and fiber deviation effects and neglect factors like knot typology, wood grain orientation, and environmental aging.

(3)

The fully connected neural network’s performance heavily depends on the normality assumption, limiting its generalization capability for non-normally distributed knot parameters, complex 3D geometries, and multi-parameter interactions in diverse scenarios.

(4)

The model lacks explicit spatial modeling of knot position–size interactions, while its two-layer fully connected architecture risks overfitting on limited datasets.

(5)

Key simplifications include omitting 3D fiber deviations and annual ring distortions, assuming simply supported beams under uniform loads, and ignoring complex boundary conditions or stress–field superposition from multiple knots.

Based on these limitations, future research will prioritize:

(1)

Data collection and model optimization: Centering on natural wood knot systems and modeling fidelity, short-term efforts will establish a geometric and material property database via batch CT scanning of natural wood specimens. Medium-term goals integrate statistical models of natural knots into the framework to evaluate prediction accuracy impacts. Long-term objectives compare artificial vs. natural knot model discrepancies to calibrate the framework and extend it to multi-factor analyses. Concurrently, machine learning attention mechanisms will capture joint distributions and nonlinear dependencies among knot parameters. MATLAB code documentation will be enhanced to enable open-source model sharing.

(2)

Model structure and algorithm improvement: Focusing on multi-scale physics integration, a cross-scale damage evolution model spanning “micro (fiber deviations, annual rings) to macro (structural mechanics)” will be developed. Integration of CT-scanned texture data will refine model accuracy. To improve knot simulation realism, solid modeling will simulate compression-zone knots, while layered modeling analyzes internal knots, strengthening the characterization of wood damage mechanisms.

(3)

Experimental design and validation: Prioritizing knot simulation technology, 3D printing techniques will replicate three-dimensional knot morphologies and fiber inclination characteristics, with mechanical equivalence verified via microscopic analysis. Experimental designs will distinguish live versus dead knot typologies and incorporate multi-factor variables (e.g., grain orientation, moisture content, microcracks) to ensure simulations reflect real-world complexity.

(4)

Engineering applications and scenario expansion: Addressing complex engineering demands, future work will investigate stress distributions under non-uniform loads and fixed/elastic supports, quantifying multi-knot stress-field superposition effects and cluster distribution impacts on load-bearing capacity. Case studies of Yangzhou’s heritage structures (e.g., Zhou Fujiu’s Residence, Wenchang Pavilion) will validate practical utility in heritage assessment. Benchmarking against international codes (Eurocode 5, NDS) will facilitate theory-to-practice translation. Additionally, long-term performance of knotted beams under creep–fatigue loads and environmental aging (humidity, temperature) will be studied to advance full-cycle analysis frameworks for engineering applications.