Research on a Simplified Estimation Method for Wheel Rolling Resistance on Unpaved Runways

Abstract

Aiming at the practical difficulties (e.g., high cost of full-scale tests) in testing the rolling resistance of aircraft wheels on unpaved runways, this study establishes a theoretical calculation formula for wheel rolling resistance based on the Bekker model, following an analysis of the development and application of wheel–soil interaction models. Global sensitivity analysis using the Sobol’ method was performed on the theoretical formula to derive a simplified calculation model. Aircraft load simulation tests under 80 kN, 100 kN, and 120 kN loading conditions were conducted using a specialized loading vehicle to determine parameters for the simplified prediction model. The resistance values obtained from this model were then applied to calculate aircraft takeoff roll distance. The accuracy of resistance estimation was verified by comparing the calculated results with takeoff distances reported in relevant literature. This research provides a novel approach for estimating wheel rolling resistance of transport aircraft on unpaved runways and offers valuable reference for determining the required length of unpaved runways.

1. Introduction

During aircraft operations on unpaved runways, the rolling resistance encountered by the wheels significantly impacts runway length determination, a critical issue that has long attracted scholarly attention. Theoretical calculations of wheel rolling resistance rely on studies of wheel–soil interactions. Scholars have proposed various theoretical models and experimental approaches to address this challenge, primarily categorized into empirical, semi-empirical, and numerical simulation methods.

A classic example of the empirical method is the Cone Index approach developed by the U.S. Army Corps of Engineers during World War II, which provides soil strength data for wheel resistance calculations [1,2]. However, its applicability is highly limited to specific vehicles and terrains. The semi-empirical method, integrating theoretical analysis with experimental validation, has gained broader acceptance in modeling wheel–soil interactions [3,4]. A key model in this category is the pressure–sinkage model. Bernstein first proposed the rigid wheel sinkage theory [5], but its applicability was restricted to passive wheels and performed poorly for driven wheels. Bekker advanced this field by developing the widely recognized Bekker model [6] through extensive experimentation and theoretical analysis, which laid the foundation for subsequent pressure–sinkage models. Numerical simulations have also evolved into a mature framework with advancements in computational technology. For instance, Mashadi et al. established a 3D finite element model for pneumatic wheels to simulate rolling resistance, deriving a universal formula for predicting wheel resistance under varying speeds and wheel pressures [7].

In terms of experimental research, the U.S. military conducted full-scale aircraft tests to investigate wheel resistance. In the 1990s, to address the inability of existing standards to predict braking and turning behaviors of large transport aircraft on arid or semi-arid unpaved surfaces, the U.S. Army performed takeoff and landing trials with C-17 aircraft on semi-prepared runways, developing a model to predict ground taxiing behavior under such conditions [8]. In 2009, runway condition readings (RCR), runway friction coefficients (RFF and μ), and deceleration resistance were estimated by correlating C-17 stopping performance on concrete runways with soil-specific landing conditions and friction measurements, enabling landing rollout assessments [9]. While full-scale tests offer high accuracy, their exorbitant costs and logistical complexity hinder practical implementation. Pytka et al. designed a portable measurement system for evaluating rolling and braking friction coefficients on grassy airfields [10,11,12], yet their findings lacked generalizability due to limited test scales. It is worth noting that operational frameworks for paved/contaminated runways distinguish between braking traction (e.g., Runway Condition Report, RCR) and comprehensive resistance including unbraked wheel drag (e.g., Runway Condition Code, RSC under ICAO’s Global Reporting Format (GRF)) [13,14]. While this study focuses on fundamental wheel–soil interactions on dry, unpaved runways, future extensions to contaminated conditions (e.g., wet soil, snow) would require integrating RSC-based contaminant parameters to account for additional drag forces during takeoff acceleration [14,15].

This study adopts a theoretical approach. By analyzing wheel–soil interaction models, a resistance calculation equation is derived. A simplified estimation method is then established through global sensitivity analysis of resistance-influencing parameters. Field tests in loess regions were conducted to calibrate the simplified parameters, yielding an empirical formula for wheel resistance on loess-based unpaved runways. The formula’s accuracy is validated, providing a practical tool for resistance estimation in loess regions and a methodological reference for other terrains.

2. Theory and Methods

2.1. Rolling Resistance Calculation Model Based on Wheel–Soil Interaction Mechanism

2.1.1. Wheel–Soil Interaction Model

The operational dynamics of aircraft on unpaved runways are fundamentally governed by soil deformation under wheel loading, a distinctive feature that differentiates earthen surfaces from rigid pavements such as concrete. This phenomenon introduces complex rolling resistance characteristics that necessitate a rigorous mechanical model of wheel–soil interaction. Among existing modeling approaches, semi-empirical methods dominate due to their validated practicality.

The foundational work by Bernstein [5] established critical assumptions for soil–wheel interaction, proposing the seminal pressure–sinkage relationship expressed as follows:

p = kz^μ

(1)

where p denotes vertical pressure per unit contact area (kPa), z represents sinkage depth (m), k signifies the soil deformation resistance coefficient (kN/m²), and μ is a dimensionless soil property exponent. Both k and μ are derived through empirical curve fitting for specific soil types.

Building upon this framework, Bekker [6] advanced a more comprehensive pressure–sinkage formulation applicable to both cohesive and granular soils:

$$p=({\displaystyle \frac{k_c}{b}}+k_{\phi })z^n$$

(2)

Here, b indicates wheel contact width (m), n is the sinkage exponent, while k_c (kN/mⁿ⁺¹) and k_φ (kN/mⁿ⁺²) represent Bekker’s cohesive and frictional soil deformation moduli, respectively. These parameters (n, k_c, k_φ) collectively characterize the nonlinear pressure–sinkage behavior.

Subsequent refinements include Reece’s model [16], which incorporates soil unit weight (γ, kN/m³) and cohesion (c, kPa):

$$p=(c{k}_c^{\prime }+\gamma b{k}_{\phi }^{\prime }){\left({\displaystyle \frac{z}{b}}\right)}^n$$

(3)

where k_c′ and k_φ′ are dimensionless soil constants. Pope [17] further extended the theoretical framework by introducing velocity-dependent sinkage effects:

$$p=p_0{\left({\displaystyle \frac{u}{u_0}}\right)}^m$$

(4)

Here, p₀ denotes reference pressure (kPa), u and u₀ represent instantaneous and reference sinkage velocities (m/s), respectively, with m as the velocity sensitivity exponent.

Comparative analyses of these models [18,19,20,21,22] reveal that Equations (2) and (3) demonstrate superior correlation with experimental data for both sandy and clayey soils. Notably, Bekker’s formulation exhibits advantages for large-diameter wheels (D > 0.5 m) under limited sinkage conditions, achieving prediction accuracies within 12–15% of measured values [20,21,22]. The model’s parametric efficiency (n, k_c, k_φ) compared to Reece’s additional variables (c, γ) and Pope’s velocity-dependent parameters (u₀, m) enhances its practicality for runway applications.

While Bekker’s original formulation accounts for bulldozing resistance in the rolling direction [6], this component becomes negligible for large-diameter wheels operating on well-compacted unpaved runways [23]. Therefore, bulldozing resistance is excluded from the current analytical framework to focus on dominant resistance mechanisms.

2.1.2. Computational Model for Wheel Resistance

On unpaved soil surfaces, pneumatic wheels exhibit mechanical behavior analogous to rigid wheels under vertical loading [24]. Based on Bekker’s foundational assumptions:

1.

Soil reaction forces at the wheel–soil interface are strictly radial in direction;

2.

The magnitude of these radial forces equals the normal pressure measured in plate-sinkage tests at equivalent depths.

These postulates enable the derivation of force equilibrium equations from the simplified interaction model illustrated in Figure 1 [25], where the horizontal force component corresponds to the total rolling resistance:

$$F=b{\int }_0^{{\theta }_0}\sigma R\sin \theta \mathrm{d}\theta$$

(5)

$$W=b{\int }_0^{{\theta }_0}\sigma R\cos \theta \mathrm{d}\theta$$

(6)

where W denotes vertical load (N), F represents rolling resistance (N), σ signifies the radial pressure (kN/m²), b indicates wheel contact width (m), and R is the wheel radius (m).

Key parameters:

• D: Wheel outer diameter;
• F: Rolling resistance;
• z₀: Maximum sinkage depth;
• z: Local sinkage depth;
• θ: Angle between radial pressure and vertical axis;
• θ₀: Soil entry angle.

Based on the core assumption of Bekker’s model, the radial pressure (σ) acting on the tire at depth z equals the normal pressure (p) measured in plate-sinkage tests at the same depth. This equivalence leads to the following:

$$\sigma R\sin \theta =p\mathrm{d}z$$

(7)

$$\sigma R\cos \theta =p\mathrm{d}x$$

(8)

By incorporating the pressure–sinkage relationship (Equation (2)) into the vertical equilibrium equation (Equation (5)) through algebraic manipulation, the rolling resistance calculation model is derived as:

$$F=b{\int }_0^{z_0}({\displaystyle \frac{k_c}{b}}+k_{\phi })z^n\mathrm{d}z=b({\displaystyle \frac{k_c}{b}}+{ k}_{\phi }){\displaystyle \frac{z_0^{n+1}}{n+1}}{z}^n$$

(9)

Bekker correlated soil sinkage depth with vertical load through Equation (1), yielding the following expression for calculating z₀ in Equation (9) [20]:

$$z_0={\left[{\displaystyle \frac{3W}{b(3-n)(\frac{k_c}{b}+k_{\phi })\sqrt{D}}}\right]}^{{\displaystyle \frac{2}{2n+1}}}$$

(10)

where z₀ indicates maximum sinkage depth (m), W denotes vertical load (N), and D represents wheel outer diameter (m).

By combining the rolling resistance model (Equation (9)) with the maximum sinkage depth expression (Equation (10)), the consolidated equation for wheel resistance on unpaved surfaces is derived as follows:

$$F={\displaystyle \frac{1}{{(3-n)}^{\frac{2n+2}{2n+1}}{(n+1)}^{\frac{1}{2n+1}}{(\frac{k_c}{b}+k_{\phi })}^{\frac{1}{2n+1}}}}{\left({\displaystyle \frac{3W}{\sqrt{D}}}\right)}^{{\displaystyle \frac{2n+2}{2n+1}}}$$

(11)

2.2. Sobol’-Based Sensitivity Analysis of Rolling Resistance

2.2.1. Parameter Sensitivity Analysis Methodology

Equation (11) identifies two distinct classes of parameters influencing wheel resistance:

• Wheel characteristics: Load W, diameter D, and contact width b;
• Soil properties: Sinkage exponent n, cohesive modulus k_c, and frictional modulus k_φ.

The multivariate nature of these parameters introduces computational complexity, amplifying both numerical errors and computational workloads. To systematically quantify parameter influences and streamline the resistance calculation framework, this study employs Sobol’ global sensitivity analysis [26]. This approach enables the identification of dominant parameters and their interaction networks, thereby optimizing computational efficiency without sacrificing model fidelity.

The Sobol’ method, a variance decomposition-based global sensitivity analysis technique, is particularly suited for complex models exhibiting nonlinearity, high parameter dimensionality, and interaction effects. Its fundamental principle involves decomposing the variance of model outputs into contributions from individual input parameters and their combinations, thereby enabling quantitative evaluation through first-order sensitivity indices (representing isolated parameter effects) and total sensitivity indices (capturing cumulative effects, including interactions) [27,28,29].

Applied to the theoretical modeling of tire rolling resistance on unpaved surfaces, first-order indices quantify how individual parameter variations independently influence resistance predictions. Total indices serve dual purposes: (1) measuring a parameter’s overall contribution to output uncertainty and (2) revealing its interdependencies with other variables. This dual capability proves critical for resistance modeling, where parameters like vertical load and soil deformation indices exhibit strong nonlinear couplings.

Its core mechanism decomposes the total output variance $\mathbb{V}$(Y) of the model Y = f(X) (where X = {X₁, …, X_k}) into orthogonal contributions from individual parameters and their combinations:

$$Y=f\left(\mathit{X}\right)=f_0+\sum _{i=1}^k{f}_i\left(X_i\right)+\sum _{1\le i<j\le k}{f}_{\mathit{ij}}\left(X_i,X_j\right) +\dots +f_{1,...,k}(X_1, \dots , X_k)$$

(12)

The variance decomposition follows

$$\mathbb{V}\left(Y\right)=\sum _{i=1}^k{\mathbb{V}}_i+ \sum _{1\le i<j\le k}{\mathbb{V}}_{\mathit{ij}}+\dots +{ \mathbb{V}}_{1,\dots ,k}$$

(13)

Two key indices quantify parameter influence:

• First-order sensitivity index:

$$S_i={\displaystyle \frac{{\mathbb{V}}_i}{\mathbb{V}\left(Y\right)}}$$

(14)

quantifies the independent effect of parameter X_i on resistance variance.

• Total sensitivity index:

$$S_i={\displaystyle \frac{{\mathbb{V}}_i+{\sum }_{j\text{≠}i}{\mathbb{V}}_{\mathrm{ij}}+\dots }{\mathbb{V}\left(Y\right)}}$$

(15)

captures the cumulative influence of X_i, including all higher-order interactions.

The numerical workflow comprises four stages:

3.

Quasi-random sampling:

• Generate N×(2k) parameter combinations via Sobol’ sequences within defined bounds;
• Partition into base matrices A and B (N × k each).

4.

Hybrid matrix construction:

For each parameter X_i:

$${\mathit{C}}_i=\left[{\mathit{B}}_{(:,1:i-1)},{\mathit{A}}_{(:,i)},{\mathit{B}}_{(:,i+1:k)}\right]$$

(16)

5.

Model evaluation:

Compute output vectors:

• f(A): Baseline resistance values;
• f(B): Perturbed parameter effects;
• f(C_i): Isolated X_i influences.

6.

Index calculation:

• First-order sensitivity:

$$S_i={\displaystyle \frac{\frac{1}{N}{\sum }_{m=1}^{N}f{\left(\mathit{A}\right)}_{m}f{\left({\mathit{C}}_i\right)}_m-f_0^2}{\mathbb{V}\left(Y\right)}}$$

(17)

• Total sensitivity:

$$S_{\mathit{Ti}}=1-{\displaystyle \frac{\frac{1}{N}{\sum }_{m=1}^{N}f{\left(\mathit{B}\right)}_{m}f{\left({\mathit{C}}_i\right)}_m-f_0^2}{\mathbb{V}\left(Y\right)}}$$

(18)

where

$$f_0={\displaystyle \frac{1}{N}}{\sum }_{m=1}^{N}f{\left(\mathit{A}\right)}_m, \mathbb{V}\left(Y\right)={\displaystyle \frac{1}{N}}{\sum }_{m=1}^N{f}^2{\left(\mathit{A}\right)}_m-f_0^2$$

2.2.2. Sobol’ Sequence and Numerical Simulation Analysis

A global sensitivity analysis of tire rolling resistance was conducted based on the established framework. Six parameters influencing rolling resistance were assumed to follow uniform distributions within their respective ranges, as detailed in

Table 1. Parameter distributions for rolling resistance analysis.

Parameter	Physical Meaning	Lower Bound	Upper Bound
n	Soil deformation index	0.2	1.2
b/m	Wheel width	0.235	0.375
kc/(kN/mn+1)	Cohesive deformation modulus	0	80
kφ/(kN/mn+2)	Frictional deformation modulus	0	6000
W/kN	Vertical load	0	2000
D/m	Wheel diameter	0.5	1.5

[25].

A Sobol’ sequence was employed for parameter sampling within these ranges. To balance computational efficiency and sensitivity stability, 4000 samples were generated. Figure 2 presents the calculated first-order and total sensitivity indices for each parameter.

As shown in Figure 2, the horizontal axis represents six parameters influencing rolling resistance. The analysis reveals that the soil deformation index n exhibits the highest combined first-order and total sensitivity indices, indicating its dominant influence on tire resistance. Following n, parameters W, k_φ, and D demonstrate moderate sensitivity, collectively contributing significantly to resistance variations. In contrast, the sensitivity indices for b and k_c remain below 0.05, suggesting negligible impacts on resistance. Notably, the total sensitivity indices for all parameters exceed their corresponding first-order values, highlighting the presence of coupling effects among variables. Particularly for the four influential parameters (n, k_φ, W, and D), the substantial gaps between total and first-order sensitivity indices reflect strong interactive effects with other parameters.

For operational scenarios at established unpaved runways, where soil properties (n, k_c, k_φ) are typically fixed, further sensitivity analysis of aircraft-specific parameters (b, W, D) becomes essential. To address this, three representative soil types—clay soil, lean soil, and sandy loam—were selected based on their mechanical properties (

Table 2. Soil deformation parameters.

Soil Type	n	kc/(kN/mn+1)	kφ/(kN/mn+2)
Clay soil	0.5	13.19	692.15
Lean soil	0.2	16.43	1724.69
Sandy loam	0.9	52.53	1127.97

).

Under fixed soil conditions, the total and first-order sensitivity indices for b, W, and D were calculated and illustrated in Figure 3.

The sensitivity analysis reveals distinct patterns among aircraft-related parameters across different soil types. As demonstrated in Figure 3, the vertical load W consistently exhibits the highest sensitivity, followed distantly by tire diameter D, while tire width b shows negligible influence (sensitivity ≈ 0). Notably, the sensitivity magnitudes of these parameters vary slightly between soil types, with fluctuations on the order of 10⁻². A critical observation lies in the reduced disparity between total and first-order sensitivity indices for b, W, and D, which diminishes compared to the six-parameter analysis in Figure 2. This attenuation suggests that the interactive effects among the aircraft-specific parameters (b, W, D) are relatively limited, while a strong coupling remains between the aircraft parameters and soil properties (n, k_c, k_φ).

For aircraft operations on established unpaved runways, where soil characteristics remain constant, vertical load W emerges as the predominant factor governing rolling resistance. This dominance underscores the necessity of prioritizing W-resistance relationships in tire performance studies. To streamline resistance estimation, Equation (11) is simplified as follows:

$$F={ \mathit{Am}}^N$$

(19)

$$A={\displaystyle \frac{{\left(\frac{3g}{\sqrt{D}}\right)}^{\frac{2n+2}{2n+1}}}{{(3-n)}^{\frac{2n+2}{2n+1}}{(n+1)b}^{\frac{1}{2n+1}}{(\frac{k_c}{b}+{ k}_{\phi })}^{\frac{1}{2n+1}}}}$$

(20)

$$N=1+{\displaystyle \frac{1}{2n+1}}$$

(21)

where A denotes the composite coefficient encompassing soil–tire interaction characteristics, N represents the power exponent determined by the soil’s sinkage index n, and m is the aircraft mass (kg).

When both aircraft configuration and soil conditions are specified, A becomes a constant, while n serves as a soil-specific empirical parameter. This sinkage index n functions as a terrain-dependent constant across most analytical models and can be predetermined through standardized plate–sinkage tests [20,21,22]. The selection of aircraft mass m instead of vertical load W in this simplification strategically emphasizes takeoff mass as the primary operational variable. Although vertical load W fluctuates during taxiing due to aerodynamic lift variations, this study’s methodology determines low-speed resistance under static load conditions (W ≈ m·g) for input into empirical takeoff models that inherently account for lift effects during acceleration [30]. The gravitational component derived from m remains constant, making it a more stable parameter for resistance modeling. The simplified model operates under two foundational assumptions to balance accuracy and practicality. First, it equates vertical load to gravitational force, presuming minimal surface irregularities that could induce dynamic excitation [31]. This idealized loading condition holds best on well-compacted surfaces with low roughness. Second, gravitational acceleration is standardized as g = 10 m/s² to streamline calculations.

Central to the model’s predictive capability are two parameters: the soil-dependent sinkage index n and the composite coefficient A. Determining n requires controlled penetration tests to characterize soil deformation properties, while A synthesizes five interdependent variables (b, k_c, k_φ, D, n) into a single constant. Traditional methods for quantifying A accumulate uncertainties from multiple sources, including measurement errors in soil cohesion (k_c) and friction (k_φ) moduli, empirical approximations during curve fitting, and stress distribution assumptions tied to tire diameter. These compounded errors highlight the advantage of treating A as a unified parameter rather than deriving it through sequential measurements.

The model’s strength lies in its balanced compromise between simplicity and technical rigor, serving as an accessible tool for initial aircraft–terrain interaction studies when detailed soil data acquisition proves challenging. After transforming the formula, the rolling resistance of the wheel is represented in a simple and universal manner, which facilitates higher accuracy for this model.

2.3. Parameter Determination Based on Field Tests

The determination of parameters requires the relationship between tire rolling resistance and the mass of the test object. Considering factors such as the repeatability of the tests, cost, and operational complexity, Zhang Jun [32] proposed using an aircraft load-simulating vehicle to simulate wheel rolling resistance, which has been validated for its accuracy. Therefore, this study employs an aircraft load-simulating vehicle for field tests.

The test vehicle is equipped with five wheels, one of which is the main load-bearing wheel with standard tire pressure. During the test, tractors are placed at both ends of the load-simulating vehicle to ensure stable forward movement. Strain gauges are installed on the main load-bearing wheel to measure the rolling resistance. The testing process is shown in Figure 4. Due to limitations in the test site and conditions, the experiment is designed with three load levels—80 kN, 100 kN, and 120 kN—corresponding to the load on the main load-bearing wheel. The unpaved runway consists of well-compacted clay soil with a compaction degree of over 90% and good surface regularity, with zero longitudinal slope.

The test section is 20 m long, and the tractor advances the aircraft load-simulating vehicle at a speed of 10 km/h. The sampling frequency for measuring rolling resistance is set at 10 Hz, yielding approximately 80 valid data points per test. Each load condition is tested four times. The 10 Hz sampling frequency was selected primarily because it is the maximum sampling rate supported by the specific sensors employed in the measurement system. Crucially, this frequency has been demonstrated to be sufficient for capturing the continuity and essential characteristics of rolling resistance data on soil runways [33]. After processing the resistance data obtained from the aircraft load-simulating vehicle tests to remove outliers, the variation of resistance over time is plotted, as shown in Figure 5.

The key data from each group are summarized in

Table 3. Rolling resistance values under different vertical loads.

Load	Maximum (kN)	Minimum (kN)	Mean (kN)	Standard Deviation (kN)
80 kN	7.20	0.01	3.22	2.25
	4.76	0.22	2.93	1.12
	6.64	0.07	3.67	1.90
	5.85	0.05	3.25	1.44
100 kN	7.41	0.83	4.59	1.86
	9.04	0.04	4.50	2.31
	9.82	0.11	4.23	2.63
	9.47	0.20	4.36	2.42
120 kN	8.70	2.13	5.04	1.59
	9.79	2.99	6.14	1.58
	12.23	0.06	6.02	3.20
	10.27	0.01	5.73	2.43

.

The data acquired at a 10 Hz sampling frequency exhibit excellent continuity in graphical representations. On soil surfaces with 90% compaction density, maximum rolling resistance values reach 7.2 kN, 9.82 kN, and 12.23 kN under 80 kN, 100 kN, and 120 kN loader weights, respectively. While minimum resistance values show no consistent pattern, the measured resistance fluctuates significantly around central values across all loading conditions, with oscillation amplitudes expanding as loads increase. These fluctuations span 0–8 kN, predominantly concentrating between 2–8 kN. Average resistance values demonstrate a progressive increase from 3.27 kN (80 kN load) to 4.42 kN (100 kN) and 5.73 kN (120 kN), accompanied by reduced standard deviations at higher loads, indicating enhanced stability under greater vertical loading.

The preceding analysis inferred that the wheel rolling resistance is correlated with the vertical load. Additionally, the wheel rolling resistance averages under various vertical loads were obtained from the aircraft load simulation tests using a loading vehicle. The following section will utilize these data to conduct curve-fitting to explore the underlying relationship. The allometric function y = ax^b, consistent with the theoretical framework of the simplified resistance model, was selected for its inherent compatibility with observed soil mechanics behavior. This functional form enables simultaneous determination of the scaling coefficient A and the deformation exponent n through regression analysis of field test data. Soils in loess regions were categorized into clay soils, while existing studies propose n = 0.5 (N = 1.5) as a representative sinkage index for clay soils [20,21,22]. Practical limitations arise due to the empirical nature of n, which assumes idealized homogeneous soil conditions rarely achievable in field environments. For the sake of simplifying the input parameters, the unit of m is unified to tons, and the unit of F is unified to newtons.

To address these uncertainties and rigorously evaluate the exponent’s influence, a dual calibration methodology was employed:

• Data-driven parameter optimization: The exponent N was inversely derived by fitting resistance measurements to refine n, ensuring alignment with experimental observations;
• Parametric sensitivity exploration: The exponent N was systematically varied across 1.3–1.7 (0.1 increments) to quantify its impact on predictive accuracy.

This dual-strategy calibration ensures adaptability to real-world soil variability while preserving the mechanistic foundation of the original formulation. Then six candidate equations with distinct exponents were evaluated (

Table 4. Predicted rolling resistance for different fitting equations under various loads (kN).

Fitting Equation	R2	80 kN	100 kN	120 kN	200 kN	500 kN	1000 kN	1500 kN
Measured Value	—	3.27	4.42	5.73	—	—	—	—
y = 88.1x1.7	0.99632	3.02	4.42	6.02	14.35	68.11	221.30	440.89
y = 111.2x1.6	0.99579	3.10	4.43	5.93	13.42	58.14	176.24	337.17
y = 140.2x1.5	0.97742	3.17	4.43	5.83	12.54	49.57	140.20	257.56
y = 176.7x1.4	0.99972	3.25	4.44	5.73	11.71	42.25	111.49	196.68
y = 181.2x1.39	0.99977	3.26	4.45	5.73	11.66	41.66	109.18	191.83
y = 223.1x1.3	0.99513	3.33	4.45	5.64	10.96	36.07	88.82	150.46

and Figure 6), assessing their ability to reconcile theoretical predictions with measured resistance values.

As demonstrated in the analytical results, the optimized exponent N derived from resistance data fitting converges to 1.39. Within the 80–120 kN load range, predictive outputs across candidate equations exhibit marginal discrepancies, with peak accuracy occurring at N = 1.39 and progressively diminishing as N deviates bidirectionally. However, predictive divergence amplifies significantly under elevated loads: at 1500 kN, the maximum predicted value almost triples the minimum, with the N = 1.5 model overestimating resistance by 34% compared to the N = 1.39-based formulation.

Consequently, according to the test, for soil in loess regions, the following refined equation is proposed to estimate wheel rolling resistance:

$$F={181.2m}^{1.39}$$

(22)

2.4. Validation of Simplified Resistance Estimation Equations

Given the significant influence of rolling resistance on takeoff roll distance, the accuracy of the resistance equations was validated using theoretical takeoff performance calculations for a transport aircraft on unpaved runways. The aircraft’s operational parameters include a maximum takeoff mass of 152 t, four engines each providing 120 kN of thrust, a wing area (S_y) of 300 m², air density (ρ) of 1.225 kg/m³, and gravitational acceleration (g) of 10 m/s². Under idealized conditions, reference [30] calculates the theoretical takeoff roll distance of the aircraft on clay soil with 90% compaction density as 1085 m. Substituting resistance values derived from six simplified equations into this framework yielded the predicted roll distances shown in Figure 7.

As illustrated in Figure 7, the calculated runway length exhibits an increasing trend with higher values of the exponent N, tested across the range of 1.3–1.7. The optimal exponent N = 1.39, calibrated from field measurements using aircraft-load-simulated vehicle data, produced a roll distance of 936.52 m. While this result theoretically aligns closest with actual conditions at the clay test site, it deviates by 148.48 m (13.7%) from the reference value. In contrast, conventional models adopting N = 1.5, as suggested in references [20,21,22], reduced the discrepancy to 7.3%. Further analysis of expanded exponent values revealed varying error magnitudes:

• N = 1.3: 16.9% deviation;
• N = 1.4: 13.3% deviation;
• N = 1.6: 4.0% deviation;
• N = 1.7: 44.3% deviation (rendering the prediction nonviable).

These results highlight the critical sensitivity of takeoff performance predictions to exponent selection. The N = 1.5 model, despite its widespread adoption, introduces systematic overestimation errors, whereas the data-calibrated N = 1.39 provides a more contextually accurate baseline for cohesive soils. However, the 13.7% error underscores the influence of unmodeled factors such as soil moisture variability and dynamic load redistribution during high-speed taxiing. This analysis emphasizes the necessity of site-specific exponent calibration to balance theoretical assumptions with field-measured nonlinearities, particularly for heavy aircraft operations on unpaved runways.

2.5. Uncertainty Analysis of Rolling Resistance Estimation

To evaluate the prediction accuracy of the simplified resistance model $F={ \mathit{Am}}^N$, an error propagation analysis was conducted based on experimental data. The model parameters are defined as:

• F: Wheel rolling resistance (N);
• A = 181.2: Dimensionless composite coefficient (determined by experimental fitting);
• N = 1.39: Dimensionless soil deformation index (determined by experimental fitting);
• m = 152: Aircraft mass (t).

The maximum absolute error of F was computed using the total differential method:

$$\Delta F=\left|{\displaystyle \frac{\partial F}{\partial A}}\right|\Delta A+\left|{\displaystyle \frac{\partial F}{\partial N}}\right|\Delta N+\left|{\displaystyle \frac{\partial F}{\partial m}}\right|\Delta m$$

(23)

where the partial derivatives are:

$${\displaystyle \frac{\partial F}{\partial A}}={ m}^N$$

(24)

$${\displaystyle \frac{\partial F}{\partial N}}={A\cdot \mathrm{m}}^{\mathrm{N}}\cdot \ln m$$

(25)

$${\displaystyle \frac{\partial F}{\partial m}}={N\cdot A\cdot m}^{N-1}$$

(26)

According to relevant studies on determining the maximum takeoff mass of aircraft, the relative error of m is set at 1% (Δm = 1.52) [34]. Then, the impacts of ΔA and ΔN on the error are analyzed. Due to the limited sample data used to fit Figure 6a, it is impossible to conduct a quantitative analysis of ΔA and ΔN solely based on the test data in Table 3. Drawing on relevant experience [33], the relative error of A is assumed to be 5% (ΔA = 9.06), and the relative error of N is assumed to be 1% (ΔN = 0.0139). Finally, the calculated ΔF is 22.1 kN, indicating a relative error of 11.4%. Additionally, considering the errors in the empirical formula used to calculate the aircraft takeoff roll distance, it can be preliminarily concluded that a takeoff roll distance error of 13.3% is acceptable for the preliminary evaluation of unpaved runway construction on soil runways.

3. Conclusions

Based on an analytical evaluation of pressure–sinkage models in terramechanics, the Bekker formulation was selected to derive a theoretical equation for wheel rolling resistance. A global sensitivity analysis of the model parameters led to the development of a simplified resistance equation. Field experiments under vertical loads of 80 kN, 100 kN, and 120 kN provided empirical resistance values, enabling calibration of the simplified equation. Six candidate equations with adjusted power exponents were generated to evaluate predictive robustness. Finally, the accuracy of the proposed formulation was validated by incorporating resistance estimates into takeoff roll distance calculations for a transport aircraft. Key findings include:

• The Bekker-derived resistance equation demonstrates superior practicality due to its parametric simplicity and strong correlation with measured resistance values. Its ability to balance theoretical rigor with operational measurability makes it the optimal choice for resistance prediction in unpaved surface operations.
• Among six primary parameters influencing rolling resistance, vertical load W and soil sinkage index n emerge as dominant factors. When soil conditions are fixed (e.g., at a specific airfield), W becomes the sole critical operational variable. This insight justifies the simplified power-law formulation F = Am^N, which preserves essential nonlinear characteristics while simplifying the parameters compared to the original model.
• Application of the calibrated equation (N = 1.39) to takeoff performance analysis yielded a roll distance prediction of 937 m, deviating by 148 m (13.7%) from theoretical benchmarks. The 13.7% deviation falls within acceptable limits for preliminary engineering assessments, thereby confirming the model’s utility in real-world scenarios such as rapidly constructing unpaved runways after emergency events like earthquakes to promptly deliver disaster relief supplies to affected areas.

While demonstrating sufficient accuracy for cohesive soils, the model requires validation across granular soil types. Subsequent research should integrate dynamic sinkage effects and moisture-dependent parameter variations to enhance universal applicability.