Tree Felling by Chainsaw: A Laboratory Assessment of the Hinge Flexibility in Relation to Its Width and to the Back-Cut Height

1. Introduction

Despite advances in mechanization, motor-manual tree felling with a chainsaw cannot be fully replaced by machines in many forestry and arboriculture contexts. Chainsaw use remains widespread in many areas of the world, such as Asia, Africa and Southeast Europe, and in private forests globally, due to the low initial investment, the versatility of the tool, and the topographical features of many sites, which often hinder the use of heavy machinery [1,2,3,4,5,6]. Beyond forestry, chainsaws are widely used in pruning, gardening, and agriculture, exposing operators to noise, exhaust fumes, hand-arm vibrations (HAV), wood dust, cutting wounds, and heavy physical workload [7,8,9,10]. Prolonged exposure is highly associated with the onset of occupational diseases in the medium and long term [11,12].

In motor-manual tree felling, the primary causes of serious accidents are due to technical reasons. About 57% of all logging accidents derive from incorrect felling direction [13], particularly in the case of hang-up trees. Gustafsson et al. [14] established that the unpredictability of falling direction is a key factor in severe and fatal incidents. Additional risks arise during the use of felling wedges to initiate tree fall, when the operator is positioned behind the trunk and under the crown, exposed to being struck by the trunk or by falling crown material [15].

In recent years, in response to these risks, several felling techniques have been developed and codified by many associations. In Europe, training frameworks are run by the Awarding Body Association International (ABA), the European Forestry and Environmental Skills Council (EFESC, European Chainsaw Standards ECS2 and ECS3), and the National Proficiency Tests Council (NPTC, City & Guilds qualifications at Levels 2 and 3) [16,17,18].

The classical chainsaw felling procedure cuts a directional notch towards the desired fall direction and a felling cut from the opposite side; together they define a residual wood section, the felling hinge, that must never be cut (Figure 1) [19]. The hinge is the tree's sole mechanical anchor during the falling phase, holding the tree and guiding it along the planned direction.

The felling cut is typically performed slightly above the base of the directional notch, leaving a step whose vertical extent is the back-cut height. This configuration is described in numerous national and international safety guidelines, although the recommended dimensions vary. The British Columbia Forest Safety Council [20] recommends a height of at least 50 mm above the undercut. The New Zealand Department of Labour [21] recommends one-tenth of the tree's diameter, with a minimum of 50 mm. Japan's Ministry of Health, Labour and Welfare [22] specifies two-thirds of the height of the notch above the undercut. The Idaho Administrative Procedures Act [23] requires the back-cut to be above the undercut level, while Husqvarna [24] allows it to be level with or slightly above the undercut.

The functions of the step thus released are: it prevents the trunk from sliding backwards off the stump during the fall, endangering the operator; it raises hinge flexibility, letting wood fibers resist longer during the fall, safely guiding direction; it reduces log damage, since fibers preferentially tear from the stump rather than from the stem.

Previous research has mainly addressed the evaluation of hinge mechanics. Among these: Guimier [25], Cutter and Baker [26], and Lindroos et al. [27] defined the hinge as a holding-wood element which provides structural resistance and guides the tree during its fall, Wojcik [28] investigated the effect of proper execution of kerfs (under-cut and felling-cut) on the direction of tree fall assessing that the highest effect on the direction of tree fall has the properly made hinge (width and height) and Lyons et al. [29] studied hinge behavior at the breaking point under three notch configurations (Humboldt, conventional, and open face). They also identified the back-cut step as an anti-kickback measure when a conventional notch is used.

Gullberg and Gullberg [30] showed that hinges exceeding 30 mm in width provide no additional guidance benefit and that excessively thick hinges impede bending, prolonging felling time [28]. Nagao and Yamada [31] quantified the effect of asymmetric hinge geometry on felling direction in Japanese cypress and showed that uneven hinges deflect the tree from its planned direction. The influence of back-cut height on hinge flexibility and elastic stiffness has not been well studied.

Given the absence of experimental evidence on the effect of back-cut height on hinge mechanical properties, this study was designed to assess variations in hinge flexibility, structural stiffness, and absorbed energy at yield in wooden beams of square cross-section under controlled laboratory conditions.

2. Materials and Methods

2.1. Sample Characterization

The classical felling technique was replicated under laboratory conditions using standardized beams with different back-cut height (S) and hinge width (C) configurations. A square cross-section was chosen instead of roundwood to standardize the geometry, maintain sample stability during loading, and improve control of the chainsaw cut, thereby enabling precise dimensions for each hinge component. To account for species-specific differences in wood density and anatomical structure, the assessment was conducted on two forest tree species, representing a hardwood and a softwood, namely European beech (Fagus sylvatica L.) and silver fir (Abies alba Mill.). For each species, 36 samples were cut from beams of 150 × 150 mm² cross-section and 600 mm length (total specimens n = 72). All green sawn timber specimens were maintained in a saturated state through water spraying or direct immersion to ensure the moisture content exceeded the fiber saturation point (>30%), replicating the state of the wood at felling.

2.2. Hinge Preparation and Geometric Parameters

The felling hinge was prepared on each sample by executing directional and felling cuts with a Stihl MS 261 chainsaw that operated inside a custom-engineered guiding framework to hold the cut consistent across all samples. Two geometric factors (in Figure 2) were varied:

Hinge width (C): Two nominal levels were targeted at preparation: a standard hinge of approximately 20 mm and a half-standard hinge of approximately 10 mm, the standard value corresponding to the customary field proportion for a hypothetical 200 mm round stem, here reduced to a square cross-section for testing. Because the cuts were executed with a chainsaw, the width of the residual hinge could not be set exactly, and it was therefore re-measured on every sample after preparation; the measured values, not the nominal ones, were used throughout the analysis. Measured widths spanned 8.5–22.0 mm, with individual samples departing from their nominal target by up to 3.0 mm. The two populations nevertheless remained fully distinct, no sample falling between 13.0 and 17.5 mm: 8.5–13.0 mm (mean 10.4 ± 1.1 mm, n = 35) for half-standard hinges and 17.5–22.0 mm (mean 19.9 ± 0.9 mm, n = 37) for standard ones. The factor was consequently coded at two levels, with the threshold placed at the upper end of that empty interval: C < 17.5 mm ("half-standard") and C ≥ 17.5 mm ("standard").

Back-cut height (S): divided into three operationally meaningful categories: almost null (S ≈ 0), regular (S ≈ C), and high step (S ≈ 2C). These categories were defined by splitting the continuous range of tested back-cut heights (0–32.4 mm) into three equal intervals, with class boundaries at 10.8 and 21.6 mm. Per-class sample counts are given in the results section.

2.3. Mechanical Testing

Applied force was measured with an electronic dynamometer (load cell) integrated into a Metro-Com universal testing machine. The device consists of a steel frame with a hydraulic piston whose downward movement applies a vertical compressive load to the sample, which rests on a secondary iron beam fixed to the frame. The rig is shown in Figure 3. A servo valve regulates oil flow and pressure to ensure the reproducibility of the loading speed from one test to the next. The load was applied to the sample so that the downward movement of the pressing head was controlled by the mold-holding frame. The sample was anchored with metal plates and threaded rods on the longitudinal axis opposite the loading point.

A spherical steel interface element with a corresponding support guide was positioned between the piston head and the sample surface, so that the load remained axial as the hinge progressively closed (Figure 3). The effective lever arm of the resisting moment is the horizontal distance A between the line of action of the load and the hinge pin, and it varied from sample to sample (mean ≈ 94 mm). The resulting experimental bending moment (T_exp) was therefore:

T_exp = F · A

(1)

Where:

F is the applied load (in N),

A is the distance between the point of the load application and the base line of the hinge (under-cut 2) in m.

Deformation was monitored using Linear Variable Displacement Transducers (LVDT). The measured Critical Bending Angles (CBA) represent a purely elastic-plastic transition within the wood fibers and not a geometric loss of equilibrium. The average experimental resisting moment (≈ 303 N·m) was well above the gravitational torque, termed the moment of the block's own weight about its basal tipping edge, of the test block itself (≈ 7.45 N·m), confirming that the device measures the intrinsic yield resistance of the wood.

2.4. Moisture Content and Density Determination

After testing, two slices approximately 20 mm thick were cut from each wooden element, perpendicular to the grain. These specimens were used to determine the moisture content at the time of testing and the wood density. The moisture content was determined in accordance with ISO 13061-1:2014 [32].

Wood density was measured on green slice specimens immediately after testing and after conditioning at 20 °C and 65% relative humidity until equilibrium moisture content was attained, as confirmed by periodic weighing. All density measurements were performed in accordance with ISO 13061-2:2014 [33].

2.5. Data Processing and Statistical Analysis

2.5.1. Signal Smoothing and Yield Point Extraction

The raw force–displacement time series were exported for each sample as individual files and pre-processed in Python (v3.13.9) using pandas, NumPy, and SciPy. The force signals were smoothed using the Savitzky–Golay filter [34], window length 11 data points, and a polynomial order of 3. The filter fits successive subsets of adjacent data points with a low-degree polynomial using the method of linear least squares and preserves the peak amplitude and curvature that simple moving-average methods smooth out [34]. The decay phase was removed by truncating the time series at the threshold of the maximum angle (0.6° for fir; 0.5° for beech).

The elastic region of each force-angle curve was identified as the interval between 20% and 40% of the maximum recorded load. The 20–40% window applies the EN 408 [35] elastic-modulus convention (fit over 0.1–0.4 F_max, spanning at least 0.2–0.3 F_max), with the lower bound raised from 10% to 20% to exclude the initial seating region and the 40% cap holding the fit below the proportional limit. A linear regression was fitted to this region to establish the elastic modulus line. The slope of this line is the bending stiffness of the hinge (N·°⁻¹): a structural property of the hinge as a mechanical element, not a material modulus of elasticity, the two being not interchangeable in this configuration (Section 3.3; Supplementary Table S7). Where fewer than three data points (the minimum for a least-squares line with df = n − 2 ≥ 1) fell within this initial window, the bounds were iteratively expanded (decreasing the lower threshold by 2 percentage points and increasing the upper threshold by 10 percentage points per iteration, up to three attempts) to ensure a determinate elastic slope estimate. In practice, the initial window always contained sufficient data (31–193 points, median 100) and the expansion was never invoked; a curve still yielding fewer than two points would have been excluded. This adaptive treatment of the fitting interval follows established practice for the automated determination of the linear region of a test record [36], standardized in ASTM E3076 [37,38]; the step sizes and the three-iteration cap are implementation choices specific to this dataset.

The Critical Bending Angle and Yield Point were determined using the offset method, consistent with approaches applied to wood-based mechanical systems (Eurocode 5 [39]; EN 408 [35]). An angular offset of 0.02° was applied to the fitted elastic line; the Yield Point was defined as the first intersection of the smoothed force-angle curve with this offset line beyond the elastic region (Figure 4). The force at that point is denoted F_yield. This method identifies the Proportional Limit, the transition from recoverable elastic deformation to permanent plastic deformation in the hinge wood fibers.

In addition to the Yield Point, the smoothed load–deformation curve was integrated with the composite Simpson rule (scipy.integrate.simpson [40]): with respect to the bending angle up to yield, which gives a force–angle integral (N·°) rather than a work; with respect to the lateral displacement up to yield, which gives the energy absorbed by the hinge (N·mm = mJ); and up to fixed lateral displacements of 1, 2, … mm (N·mm = mJ) as energy milestones describing how early in the displacement the hinge absorbs its energy. A hinge is termed aggressive when most of that energy is absorbed within the first millimeters of lateral displacement, that is, force rises steeply and early, and progressive when the same energy is spread over a larger displacement; the displacement at yield is used as the operational proxy for this distinction (Section 4.2). A wedge-referenced index was then derived from each energy as given in Eq. (2), where A is the load arm and (S + T/2) the effective hinge height, measured from the hinge to the mid-plane of the kerf where a felling wedge would bear (S the back-cut height, T the saw kerf; Figure 1). The factor k = A/(S + T/2) is the mechanical advantage of the hinge lever: moment equilibrium about the hinge requires a wedge force k times the applied load, F_wedge = k·F, while the kinematics of the same rotation reduce the wedge travel by the same factor, δ_load/δ_wedge = k. The work done at the two stations is therefore identical, as the principle of virtual work requires, and E_wedge is consequently not a work at the wedge but the absorbed energy weighted by that mechanical advantage: an index of how demanding a given hinge is to release by wedging, rising both with the energy stored in the hinge and with the leverage penalty incurred as the back-cut approaches the notch. Because (S + T/2) sits in the denominator, the decline of E_wedge with S is in part a property of the definition rather than an independent empirical effect, and the S-independent absorbed energy E (Section 3.5) is reported alongside it as the corresponding material quantity.

$$E_{\mathrm{wedge}}={\displaystyle \frac{A}{S+\frac{T}{2}}}·E$$

(2)

2.5.2. Statistical Analysis

Statistical analyses were conducted in Python (v3.13.9) with SciPy, statsmodels, and Matplotlib [40,41,42]. Normality was checked with the Shapiro–Wilk test [43] and homoscedasticity with Levene's test on the median center (the Brown–Forsythe variant) [44,45]. The responses were non-normal, and the variances differed between species, so rank-based non-parametric tests were the primary inference, with parametric models fitted to log₁₀-transformed responses as a corroboration.

Outliers were removed per species with Tukey's 1.5 × IQR rule [46] applied to the Critical Bending Angle and Yield Point distributions; the per-species rule keeps the two contrasting wood densities and strengths from leaking outliers into each other. Three records (all beech) fell outside the fences and were removed, and the single sample with C = 17.5 mm was assigned to the C ≥ 17.5 class, giving a final sample of n = 69 (36 fir, 33 beech). The headline results are unchanged under no outlier removal and under a 3 × SD per-species rule (Supplementary Table S9).

Spearman rank correlations [47] related the hinge dimensions (S, C) to the absorbed energy, within each species and C-group. Group differences were tested with the Mann–Whitney U test [48] and, across the three S classes, the Kruskal–Wallis test with Dunn’s post-hoc under Holm correction [49,50,51]. For parametric corroboration, log10-transformed responses were modelled by ordinary least squares, two-way ANOVA (Type II) and ANCOVA; the relative contribution of each geometric driver was quantified by a variance decomposition of the closed-form identity in Eq. (3). Family-wise error across the S-class post-hoc comparisons was controlled by the Holm procedure; species and C-group contrasts are reported with rank-biserial effect sizes and Spearman 95% confidence intervals (Supplementary Table S3–S4). Given the modest per-cell sizes (5–6 observations in the three-way layout, with one empty beech cell), the three-factor breakdown is treated as descriptive.

$$\log E_{\mathrm{wedge}}=\log A-\log \left(S+{\displaystyle \frac{T}{2}}\right)+\log E$$

(3)

Two composite geometric indices were derived: Z_SCW, a volume-like indicator incorporating all three dimensions S, C and W (as reported in Figure 1), and Y_CW, a cross-sectional area indicator (Eqs. 4 and 5). Statistical significance was set at α = 0.05 throughout. Family-wise error across the cell-level Spearman matrix reported in the Results is not formally controlled; per-cell correlations are treated as descriptive within each Species × C-group cell.

Z_SCW = (S + 0.01) · C · W

(4)

Y_CW = C · W

(5)

2.5.3. Field Scaling: Hinge Capacity Versus Stem Overturning Moment

The comparison of hinge capacity with the stem's own overturning moment follows the classical treatment of tree-falling mechanics, in which the stem is reduced to a rigid body rotating about the basal edge of the cut and the moments acting on it are balanced [25]. The same balance underlies the analysis of manual felling aids, where the torque a wedge, lever or jack must supply is set against the gravitational torque of the stem [27]. Here the hinge itself is placed on the resisting side of that balance, so that a laboratory yield moment can be read as a field self-release criterion.

A standing stem was idealized as a homogeneous circular cylinder (diameter D = 0.2 m; heights 10 and 20 m). Its gravitational overturning (tipping) torque about the basal edge was computed from the stem weight F_g and the lever arm D/2 (Eqs. 6 and 7).

$$F_{\mathrm{g}}=\rho ·g·\pi {\left({\displaystyle \frac{D}{2}}\right)}^2·H$$

(6)

$$T_{\mathrm{g}}=F_{\mathrm{g}}·{\displaystyle \frac{D}{2}}$$

(7)

The uniform cylinder was adopted as a conservative reference form rather than as a realistic stem profile. Because the tipping torque is taken about the basal edge, the lever D/2 is fixed by the basal diameter and is independent of stem form; taper therefore enters only through the stem mass. For a given basal diameter and height, the cylinder is the heaviest form and gives the largest T_g, hence the smallest self-release ratio R and the most self-releasing classification: the restraint shares reported here are lower bounds. Common tapered forms reduce the mass, and with it T_g, by a predictable factor: about 58% of the cylinder for a frustum narrowing from 0.20 to 0.10 m, and 50% for a paraboloid of revolution with the same basal diameter, the conventional mid-stem form. Taper, crown mass and wind loading are therefore treated as a parametric sensitivity on T_g (Section 4.4 and Supplementary Table S8) rather than as separate geometric models.

Two density bases were used: the measured conditioned density (12% MC: 448 kg·m⁻³ fir, 715 kg·m⁻³ beech) and the estimated green density of the fresh-felled stem (≈ 524 and ≈ 972 kg·m⁻³, derived from the measured green moisture contents of 40% and 69%; Supplementary Table S5). The hinge resisting moment was evaluated with the per-sample lever as M_yield (Eq. 8), that is, T_exp (Eq. 1) evaluated at the Yield Point, and each hinge was classified by the self-release ratio R (Eq. 9), where R > 1 denotes a hinge stronger than gravity (requiring mechanical assistance) and R < 1 a self-releasing hinge.

M_yield = F_yield · A

(8)

$$R={\displaystyle \frac{M_{\mathrm{yield}}}{T_{\mathrm{g}}}}$$

(9)

Beyond the binary classification, the share of hinges expected to act as a restraint was also estimated as a continuous function of hinge width. Within each species, the logarithm of the hinge yield moment was regressed on hinge width by ordinary least squares, log₁₀M_yield = a + b·C, giving a fitted mean μ(C) and a residual standard deviation σ. Treating the scatter of log₁₀M_yield about that line as normal, the estimated proportion of hinges whose moment exceeds the gravitational tipping torque at width C is P(C) = 1 − Φ([log₁₀T_g − μ(C)]/σ), where Φ is the standard normal distribution function. Confidence bands were obtained by substituting the 95% confidence limits of μ(C) into the same expression, so they express uncertainty in the fitted mean and not the residual scatter between individual hinges. The curves are evaluated for a D = 0.2 m, H = 20 m uniform-cylinder stem at the species-specific tipping torques (green: 323 N·m fir, 599 N·m beech; conditioned: 276 and 441 N·m), over the range C = 8–22 mm. The resulting estimates are reported with the field-scaling results (Section 4.4).

3. Results

All statistics below use a single analytical dataset: Tukey 1.5×IQR outlier removal applied per species to the Critical Bending Angle and Yield Point distributions (3 records removed, all beech), followed by assignment of the single C = 17.5 mm record to the C ≥ 17.5 class, giving n = 69 (36 fir, 33 beech). One further beech record with a non-physical negative integral was excluded from the force/energy correlations (n = 68).

3.1. Physical Properties of Test Samples

The average moisture content of the specimens was approximately 40% for fir (range: 34–53%) and 69% for beech (range: 59–82%). The density values of the specimens, measured after equilibrium moisture content was reached under normal climatic conditions (20 °C and 65 % RH), were 448 ± 40 kg m⁻³ for fir and 715 ± 18 kg m⁻³ for beech. Density of specimens in green condition was 524 ± 55 kg m⁻³ for fir and 972 ± 28 kg m⁻³ for beech.

3.2. Statistical Approach and Distributional Checks

Within the four Species × C-group cells, E_wedge was non-normal in all four cells (Shapiro–Wilk, p < 0.05; Supplementary Table S1) and the between-species variances differed (Levene/Brown–Forsythe, p = 0.015; Supplementary Table S2). A base-10 logarithm restored normality in three of four cells and homogenized the between-species variance. Rank-based tests were used as the primary means of inference, and log-transformed parametric models provided corroboration, following Section 2.5.2; rank-biserial effect sizes and Spearman 95% confidence intervals are given in Supplementary Table S3–S4 (Figure 5).

3.3. Hinge Flexibility, Structural Resistance and Elastic Stiffness

The Critical Bending Angle is the angular range to the proportional limit, and it is the measure of flexibility used here. It resulted higher in beech than in fir (median 0.27° vs 0.22°; Mann–Whitney p < 0.001; Table 1). Within each species, it decreased with back-cut height S (Spearman ρ = −0.68 for fir, −0.43 for beech), and it was best explained by the composite index Z_SCW (Eq. 4; ρ = −0.77 fir, −0.48 beech). Hinge width C reduced its flexibility in fir (ρ = −0.50) but not in beech (Table 2). Taller step hinges reach the elastic limit at smaller deflection angles. The lateral displacement at yield behaves in the same way as the angle: it decreases with S in both species, significantly in fir (Spearman ρ = −0.68, p < 0.001) though not in beech (ρ = −0.17, p = 0.35), and its median falls across the three back-cut classes from 0.51 mm (almost null) through 0.43 mm (regular) to 0.34 mm (high); the class difference is significant pooled (Kruskal–Wallis p < 0.001) and within fir (p < 0.001) but not within beech (p = 0.38). The Yield Point was governed mainly by hinge width C (ρ = +0.76 fir, +0.49 beech), with only a weak contribution from S that was significant in fir alone. Beech sustained higher yield loads than fir (median 3944 vs 2626 N; p = 0.003). Hinges with higher yield loads reached the elastic limit at smaller angles (ρ between CBA and Yield = −0.44, p = 0.008 for fir; −0.32, p = 0.07 for beech).

3.4. Geometry Drivers of the Wedge-Referenced Energy Index (E_wedge)

Spearman correlations within each Species × C-group cell (Table 3) show a strong and consistent negative dependence on the back-cut height S, significant in seven of the eight cells (ρ from −0.39 to −0.94). The dependence on hinge width C is, by contrast, weak and inconsistent: it reaches nominal significance in only two cells, and with opposite signs — positive in beech at C < 17.5 mm for the 1 mm milestone (ρ = +0.78, p = 0.001) and negative in fir at C < 17.5 mm at yield (ρ = −0.49, p = 0.047). The fir coefficient does not survive control for S (partial ρ = −0.39, p = 0.12; within that cell S and C are themselves correlated, ρ = +0.33), nor correction for multiple testing across the eight cells (Benjamini–Hochberg, where only the beech coefficient is retained), whereas the beech coefficient survives both (partial ρ = +0.64, p = 0.010). This instability is expected, since stratifying by C-group restricts the within-cell range of C to 2–4.5 mm against the 8.5–22.0 mm spanned by the full sample. The C effect is therefore assessed on the pooled data in the log-linear model below, where it is positive and significant. The subscript "wedge" is used because the index refers the absorbed energy to the point at which a felling wedge is driven into the back-cut, the wedge serving to rotate the stem past its point of balance — that is, to displace the center of gravity beyond the tipping edge, after which the stem's own weight completes the fall.

A joint log-linear model confirmed all three factors as independent drivers (log₁₀E_wedge ~ S + C + species; n = 68, R² = 0.75): S (β = −0.0303 dec/mm, p = 2.2 × 10⁻¹⁷), C (β = +0.0238 dec/mm, p = 6.9 × 10⁻⁴) and species (beech +0.375 dec, p = 1.7 × 10⁻⁷). Decomposing the exact identity in Eq. (3) attributes 64% of the variance of log E_wedge to the S-amplification term, 32% to the absorbed energy (which carries the C and species signal), and 5% to the species difference in the load arm A. Because S sits in the denominator of the amplification 1/(S + T/2), this dominance is in large part a property of the index's definition rather than an independent empirical effect; the empirical signal is carried by the absorbed energy, through which hinge width C and species act (Figure 6).

3.5. Energy Absorption (Hinge Toughness)

Absorbed energy to yield (Simpson integrated linear displacement curve up to Yield Point, mJ) measures hinge toughness. Unlike E_wedge, it is independent of S; it increases with hinge width C and is markedly higher in beech than fir (Mann–Whitney p < 0.001, rank-biserial 0.54; Table 4, Figure 7). Normalizing by the hinge cross-sectional area gives the per-area specific work to yield G (Eq. 10), with W the beam width (150 mm) and C the hinge width: 0.22 kJ·m⁻² for fir and 0.39 kJ·m⁻² for beech. Once the energy is expressed per unit area, both geometric dependences, on S and on C, become non-significant, while the roughly 1.8-fold species difference remains (Mann–Whitney p < 0.001, rank-biserial 0.58; Supplementary Table S6). Hinge width therefore raises absorbed energy chiefly by enlarging the cross-sectional area; the higher beech toughness is a genuine material property. The values reported here are the specific work to yield, not the classical clear-wood work of fracture (5–25 kJ·m⁻², measured to full failure); the present integrals stop at the proportional limit.

$$G={\displaystyle \frac{E}{W·C}}$$

(10)

3.6. Field Scaling: Self-Release vs Mechanical Restraint

The gravitational tipping torque of the reference stem (§2.5.3) is given in Table 5 for the measured conditioned density; the green-density case is given in Section 4.4 and Supplementary Table S5. Median hinge yield moments and the resulting share of hinges classified as restraint (R > 1) follow in Table 6.

The hinge yield moment M_yield is compared with the gravitational tipping thresholds of 10 m and 20 m stems in Figure 8, and the resulting share of hinges acting as a restraint is given by species and C-group in Figure 9. For a 20 m stem at conditioned density most short-hinge fir self-releases, but thick-hinge fir (C ≥ 17.5 mm) reaches about 53% restraint, and beech high-C hinges frequently exceed the tipping torque (82% for C ≥ 17.5 mm), requiring a wedge or jack to initiate and sustain the fall. The C-group effect dominates the field outcome; the S effect is weak because the lever A is independent of S, and the yield force depends on it only weakly, leaving M_yield nearly flat considering S variations (Figure 7 and Figure 8).

4. Discussion

4.1. Mechanical Considerations on Hinge Dimensions

The empirical anchor of the present dataset is the per-area specific work to yield G (Eq. 10), introduced in Section 3.5. G is independent of the back-cut height S (Spearman ρ ≈ 0.1, not significant) and is about twice as large in beech (0.39 kJ·m⁻²) as in fir (0.22 kJ·m⁻²; Mann–Whitney p < 0.001, rank-biserial 0.58; Section 3.5, Supplementary Table S6). Hinge width C raises absorbed energy chiefly by enlarging the cross-sectional area; the higher beech value is a genuine material property. This per-area specific work to yield is the appropriate, geometry-normalized measure of hinge toughness and the principal mechanical finding of the present dataset.

The wedge-referenced index E_wedge (Eq. 2) was constructed to express the absorbed energy E through the lever geometry of the hinge. It is a derived display quantity, not an independent observation. Because S sits in the denominator of the amplification factor 1/(S + T/2), E_wedge necessarily declines with S as a property of its own definition. A closed-form variance decomposition of the identity in Eq. (3) attributes about 64% of the variance of log E_wedge to the amplification term and about 32% to the absorbed energy. That split is a property of how the index is defined, in particular of how the saw kerf enters the effective hinge height, and is not an empirical partition of cause. What the data establish is the direction and strength of the relationship: within every Species × C-group cell the Spearman correlation between S and E_wedge is strongly negative (ρ from −0.39 to −0.94; Section 3.4). Lowering the back-cut, i.e. reducing S, therefore amplifies the wedge-referenced index for a given absorbed energy through geometry alone; it does not increase the energy the hinge absorbs.

These findings are consistent with Gullberg and Gullberg [30], who showed that excessively large hinges provide no proportional guidance benefit and impede bending, and with Wojcik [28], who observed that disproportionate hinge dimensions compromise the guide function. The present study provides an experimental quantification of these effects in terms of absorbed energy per unit cross-sectional area, with the geometry-amplified wedge index reported alongside as a derived display.

4.2. Implications for Back-Cut Height

A back-cut placed close to the notch base (S → 0) leaves the wedge a short lever arm (S + T/2): each millimeter of wedge lift then produces a larger rotation of the stem, but the wedge must develop a proportionally larger force to overcome the same hinge moment, the two effects cancelling in the work supplied (Section 2.5.1). The hinge itself responds in the opposite sense. Raising the back-cut makes it yield sooner: the Critical Bending Angle decreases with S (Spearman ρ = −0.68 for fir, −0.43 for beech; Section 3.3), as does the lateral displacement at yield (ρ = −0.38, p = 0.001), while the absorbed energy is unchanged (Section 3.5). The same energy is therefore concentrated into a shorter displacement, giving the steep, early force rise of an aggressive hinge, whereas a near-flush cut spreads it over a longer displacement and yields progressively. The median displacement at yield falls from 0.51 mm in the almost-null class to 0.43 mm in the regular class and 0.34 mm in the high class (Kruskal–Wallis p < 0.001), making it a usable quantitative proxy for the aggressive-versus-progressive distinction — the trend is consistent in both species but significant only in fir (Section 3.3). A formal operational threshold awaits future in-field validation.

These mechanics support the prescription, common to most national guidelines, that the back-cut be placed modestly above the notch base. The British Columbia Forest Safety Council [20] and New Zealand [21] recommendations of 50 mm or one-tenth of stem diameter represent a practical optimum that preserves slide-back protection without inflating the instantaneous wedge force F_wedge, while the Japanese prescription of two-thirds the notch height [22] is more conservative. The hinge yield moment M_yield, computed with the per-sample lever as Yield Point × A, showed no significant overall dependence on S (Section 3.6): the lever A is itself independent of S, and the yield force depends on it only weakly, so back-cut height has at most a minor effect on the net holding moment (a weak positive trend in fir). Back-cut height, therefore, governs the force and energy character of the hinge rather than its net holding moment. On the present evidence, a moderate back-cut, within the regular class and broadly consistent with the one-tenth-of-diameter guideline (about 50 mm for a 0.5 m stem), is the best compromise: it preserves the anti-slide step and a controlled, progressive force build-up without the aggressive early force rise of a high back-cut, and it does not compromise holding capacity, which is set by hinge width C.

4.3. Strength-Flexibility Trade-Off Among Species

A trade-off between load capacity and deformability was evident between the species. Beech hinges absorbed about twice the energy of fir and sustained higher resisting moments, consistent with the greater density of beech, but they tended towards more abrupt post-yield failure. Fir yielded at lower energies yet showed greater post-yield ductility, with gradual plastic folding that can maintain directional guidance throughout the fall arc. The tougher, stiffer beech hinge offers more resistance but a narrower margin before early, unpredictable failure; the more compliant fir hinge gives up peak capacity for sustained guidance. The distinction bears directly on felling-direction control in the two species.

4.4. Experimental Context and Scaling to Field Conditions

The apparatus was designed to isolate the hinge Yield Point from gravitational overturning. The vertical load was applied at a sample-specific lever arm equal to A (mean ≈ 94 mm), and the test block's own tipping angle (≈ 37.9°) exceeded that of a 20 m tree of the same base dimensions (≈ 0.43°) by more than 80-fold, so the sample was far from geometric instability. The measured resisting moments (tens to a few hundred N·m; Table 6) exceeded the block self-weight torque (≈ 7.45 N·m), so the measurement reflects material yield rather than structural tipping, and the Critical Bending Angles (0.08°–0.50°) represent the elastic-to-plastic transition of the wood fibers.

Scaling to representative field stems idealized as uniform circular cylinders (D = 0.2 m) clarifies the operational consequence. The operational thresholds reported in this paragraph and the next are derived from square-section bench samples and are subject to on-stump field confirmation. The samples were tested green (moisture content ≈ 40% fir, ≈ 69% beech; Section 2.4), so the self-consistent field comparison uses green densities (≈ 524 fir, ≈ 972 beech kg·m⁻³): the gravitational tipping torque T_g (Eq. 7) is 161 N·m (10 m) and 323 N·m (20 m) for fir and 300 and 599 N·m for beech. Through the self-release ratio R (Eq. 9), with the hinge resisting moment M_yield (Eq. 8; median 214 N·m fir, 422 N·m beech; Table 6), most hinges still self-release on a 20 m stem (6% fir, 21% beech act as a restraint; 11% of fir and 35% of beech hinges with C ≥ 17.5 mm). Expressed as moments, this translates directly to a field scenario: the moment a wedge or a jack must overcome is M_yield, and the deficit to supply when R < 1 is (T_g − M_yield); the yield moment is a conservative threshold (the hinge carries load past yield), so R > 1 is a lower-bound indication that assistance is prudent. A continuous restraint formula (Section 3.5, Supplementary Table S6) makes the dependence on hinge width C explicit and is summarized by the decision aid in Table 7 and plotted, with its 95% confidence band, in Figure 10. Mechanical restraint is therefore modest for fir and slender hinges but appreciable for wider beech hinges even when green, and it increases for drier, denser stems. Across cases, the design priority remains post-yield ductility for directional guidance, with a wedge or jack carried for wide-hinge beech and for drier stems. The classification remains sensitive to density and diameter (Supplementary Table S5).

These estimates assume a uniform cylinder, the heaviest stem for a given base and therefore the most self-releasing case. A realistic, tapered stem (a frustum narrowing from 0.20 to 0.10 m, about 58% of the cylinder mass) lowers the tipping torque and raises the restraint share: at green density, to 51% for fir and 61% for beech overall, and to 89% and 94% for long (C ≥ 17.5 mm) hinges. A crown mass (about 20% of stem weight near the top) only modestly offsets this (fir 49%, beech 52% overall). A modest crown wind load (for example about 200 N at 15 m, near 3000 N·m) generates an overturning moment an order of magnitude larger than either the hinge capacity or the self-weight torque and dominates the balance when present (Supplementary Table S8). The static classification is therefore conservative for a cylinder and sensitive to stem form and loading. Across all cases, hinge width C remains the dominant geometric control of whether mechanical assistance is required.

The hinge leaves its elastic range almost immediately. The Critical Bending Angle, the angular range over which deformation is still fully recoverable, has a median of 0.22° in fir and 0.27° in beech (Table 1), against the ≈ 0.43° of rotation at which a 20 m stem on the 150 × 150 mm bench section reaches its own tipping point. The typical hinge therefore yields after roughly half to two-thirds of the rotation needed for the center of gravity to cross the basal edge, and only the most flexible hinges tested (Critical Bending Angles up to 0.50°) remain elastic that far. By the time gravity takes over the fall, the hinge is already deforming plastically, and everything that follows (more than 45° of rotation to the ground, roughly two orders of magnitude beyond the elastic range) is governed by plastic behavior. Directional control is thus a post-yield property: it depends on how the hinge folds, absorbs energy and retains fiber continuity after yielding, not on its elastic stiffness or its yield load.

4.5. Limitations and Future Research

The study used square-section beams under controlled laboratory conditions, which permit precise geometric control but do not fully replicate field conditions: real stems present circular cross-sections, variable moisture, and additional crown and wind loading [52]. The square-to-circular transition raises the tipping angle (by ≈ 33% for a 0.2 m diameter versus a 150 × 150 mm section of equal height) and alters the effective gravitational torque, so the scaling relationships require recalibration with field geometry. The field thresholds reported here use the measured species densities as the self-consistent baseline; the wedge index E_wedge is an energy-based, geometry-amplified construct and is best interpreted in relative terms rather than in absolute terms. Future work should reproduce field conditions and extend the dataset across species and moisture states. The taper, crown-mass and wind effects noted here are quantified parametrically in Supplementary Table S8; a fuller dynamic treatment (eccentric crown, gusts, stem sway) remains for future work.

5. Conclusions

This study quantified how the hinge geometry (width and height) affects its mechanical response during chainsaw tree felling. The analysis used a signal-processing pipeline (Savitzky–Golay smoothing and the 0.02° offset method) and a non-parametric statistical framework with log-transformed parametric corroboration, applied to n = 69 validated samples. The principal conclusions are:

- Hinge geometry sets the wedge-referenced index E_wedge (Eq. 2; log-linear model R² = 0.75). Its decline with back-cut height S (about 64% of the variance) is largely a property of the index's definition, since S sits in the denominator of the amplification 1/(S + T/2); the empirical content (the dependence on hinge width C and species) enters through the absorbed energy (about 32%).

- Hinge toughness (absorbed energy to yield) is independent of back-cut height S, increases with hinge width C (p ≤ 0.01), and is about twice as high in beech as in fir (p < 0.001). After per-area normalization (Eq. 10), both geometric dependences become non-significant while the species contrast persists (0.39 vs 0.22 kJ·m⁻²).

- Computed with the per-sample lever A (the horizontal load arm to the hinge pivot), the hinge yield moment M_yield shows no significant overall dependence on S: the lever is independent of S and the yield force depends on it only weakly, so back-cut height tunes the force and energy character of the hinge rather than its net holding moment.

- In field scaling to a 0.2 m-diameter stem, thin-hinge fir is largely self-releasing, but thick-hinge fir reaches about 53% restraint for a 20 m stem at conditioned density (28% across all fir). Thick beech hinges frequently exceed the gravitational tipping torque at conditioned densities (up to 82% for C ≥ 17.5 mm; 45% across all beech). At the green densities measured here, thinnest hinges still self-release, but thick beech hinges already reach about 35% restraint (and thick-hinged fir about 11%). Mechanical assistance is therefore likely for thick hinges, especially in beech, and increases for drier, denser, or strongly tapered stems.

- Species differences are significant: beech hinges are tougher and stiffer but fail more abruptly, while fir hinges yield earlier yet retain directional guidance through gradual post-yield folding.

These findings provide a quantitative basis for refining hinge-sizing guidance in chainsaw operator training and for species-specific safety protocols. Field validation with round-section timber and measured wood densities is required to consolidate the present virtual scaling relationships.