Modeling the Impact of Haul Road Conditions on Mining Vehicle Reliability: A Methodological Framework

1. Introduction

Haulage vehicles - rigid-frame dump trucks in open pits, load-haul-dump (LHD) machines and underground trucks in underground mines - operate on unpaved roads built from local materials. Their availability and operating cost depend strongly on road condition: potholes, corrugations, rutting and loose material increase rolling resistance, impose shock and vibration loads on tyres, suspension, frame and on-board equipment, and force operators to reduce speed [1,2,3]. Field vibration measurements on large surface haulage trucks were used to derive test specifications for their on-board electronic equipment [4], and engine load, which grows with total motion resistance, is a principal factor of truck fuel consumption [5]. Mine road maintenance management systems therefore link road defect scores with vehicle operating costs and use the link to schedule grading [1].

These relationships are usually expressed as empirical cost functions or as qualitative recommendations. For reliability, the question has a definite mathematical form: how does the failure intensity of a vehicle or of its subsystem depend on a measurable characteristic of the road, and how can this dependence be estimated from maintenance records? An answer needs three elements: a metrologically defined roughness measure; a reliability model in which this measure enters as a covariate and which remains a valid probability model; and an estimation procedure that accounts for the exposure (operating hours or distance) accumulated on each road category. Standard tools exist for each element - profile-based roughness indices such as the International Roughness Index (IRI) and the spectral classification of road profiles [6,7], and covariate hazard models [8,9,10] - but they are rarely combined for mine haulage.

The aim of this study is to build such a combination and to apply it to the data available from a road microprofile classification [11] and from maintenance records of the Sayak mine (Kazakhstan). The contributions are: (i) a roughness index computed from defect density and amplitude; (ii) a reliability model derived from an additive hazard, its properties and its relation to the proportional hazards model; (iii) maximum-likelihood estimators and a sample-size rule for identifying the road effect from failure counts and exposure; (iv) an analysis of the technical downtime structure and component replacement priorities of the Sayak fleet. The scope covers surface and underground haul roads: the defect classes are common to both, whereas vehicle types and speeds differ, and the origin of each data set is stated explicitly.

2. Materials and Methods

2.1. Data. Three Data Sets Are Used

(a) Road-condition scenarios. Table 1 gives approximate ranges of defect density and dimensions for four categories of unpaved roads. The ranges are used as literature-informed engineering scenarios consistent with visual defect surveys and haul-truck response monitoring [1,2,3,11]; defect counts are normalised to a 1-km section and are not presented as measurements from Sayak. (b) Speeds. The ranges in Table 2 are approximate scenario values for surface haul trucks. Their use follows the performance-handbook principle that attainable speed depends on rolling resistance, grade, payload and local conditions [2,3,13]. (c) Sayak maintenance records. The AIC Monitoring extract contains 2737 daily downtime records for four CAT 773E trucks (fleet numbers 261, 300, 524 and 526) from 1 January 2023 to 30 September 2026 [16]. Consecutive calendar-day records with the same truck and classifier are treated as one episode. Planned maintenance, reserve status and external constraints are separated from technical episodes. The component-monitoring extract dated 1 September 2026 contains 251 component records for 50 equipment units assigned to Sayak or ShPT Sayak [15]. Neither data set contains operating exposure by road category.

2.2. Classification of Road Irregularities

The response of a vehicle to an irregularity depends on its wavelength L_w relative to the wheelbase and on speed. Four wavelength classes are distinguished: impulsive irregularities, L_w ≤ 0.3 m (stones, edges, short bumps), which act on tyres and unsprung masses as force impulses; potholes, 0.3 m < L_w ≤ 6 m, subdivided into short (0.3-1.0 m) and long (1.0-6.0 m) ones, which excite vertical accelerations of unsprung and sprung masses; undulations, 6 m < L_w ≤ 25 m, which cause low-frequency body motion whose intensity grows with speed; and macro-geometry, L_w > 25 m (grade and vertical curvature). Macro-geometry affects tractive effort and fuel consumption rather than vibration; it is a property of road alignment, not of roughness, and is excluded from the roughness index. By amplitude h, the classification distinguishes micro-texture, h ≤ 0.01 m, largely absorbed by large-diameter tyres; irregularities, 0.01 m < h ≤ 0.3 m; and obstacles, h > 0.3 m, which require a reduction to minimum speed to avoid suspension bottoming and wheel lift-off. Four road categories are defined by defect density and by the attainable average speed V relative to V_max: good (lightly worn), V > 0.7V_max; moderate (moderately worn), 0.4V_max < V ≤ 0.7V_max; poor (broken), 0.2V_max < V ≤ 0.4V_max; critical (rough terrain), V ≤ 0.2V_max.

2.3. Roughness Index

For a road section, let n_c be the number of defects of class c per kilometre and h_c their most probable amplitude (height or depth). The defect intensity and the roughness index are defined as

$$D=\sum _c{w}_c\hspace{0.17em}{n}_c\hspace{0.17em}{h}_c^2,$$

(1)

$$Q={\left({\displaystyle \frac{D}{D_{\mathrm{ref}}}}\right)}^{1/2},$$

(2)

where w_c are class weights (w_c = 1 in this paper) and D_ref is the defect intensity of a reference category; here the poor category is the reference, so that Q = 1 on a poor road.

The quadratic dependence on amplitude makes Q a root-mean-square type measure, consistent with spectral roughness measures in which the variance of the profile elevation is the integral of its power spectral density. When profile measurements are available, the index should be computed from the profile itself:

$${\sigma }_z^2={\int }_{n_1}^{n_2}{G}_d\left(n\right)\hspace{0.17em}\mathit{dn}, Q={\displaystyle \frac{{\sigma }_z}{{\sigma }_{z,\mathrm{ref}}}},$$

(3)

where G_d(n) is the displacement power spectral density of the profile [7], n is the spatial frequency, and [n₁, n₂] = [0.04; 3.3] cycles/m corresponds to wavelengths from 25 to 0.3 m. Indices (2) and (3) should be cross-calibrated on the same sections; IRI [6] can serve as a widely reported third reference.

2.4. Reliability Model

Let T be the time to failure of a vehicle (or subsystem) operated on a road with index Q, t the operating time (engine hours; distance may be used instead), and λ(t|Q) the hazard rate. The additive hazard model [9] is

λ(t | Q) = λ₀(t) + γQ,  γ ≥ 0, Q ≥ 0,

(4)

where λ₀(t) is the hazard on an ideal road (Q = 0) and γ is the road sensitivity of the vehicle class, measured in units of λ per unit of Q. Integration of (4) gives

$$R\left(t\hspace{0.17em}|\hspace{0.17em}Q\right)=P\left(T>t\hspace{0.17em}|\hspace{0.17em}Q\right)=R_0\left(t\right)\exp \left(-\mathit{\gamma Qt}\right), R_0\left(t\right)=\exp \left(-{\int }_0^t{\lambda }_0\left(u\right)\hspace{0.17em}\mathit{du}\right).$$

(5)

For a fixed mission time t the relative reliability is

$${\displaystyle \frac{R\left(t\hspace{0.17em}|\hspace{0.17em}Q\right)}{R_0\left(t\right)}}=\exp \left(-\mathit{kQ}\right), k=\mathit{\gamma t}.$$

(6)

Equation (6) is the dependence plotted in Figure 2 of the first version of this paper; it replaces the expression R = R₀ + exp(-kQ) given in the text of that version, which is not a probability model (Remark 1).

Proposition 1.Under (4), for all t ≥ 0 and Q ≥ 0 such that R₀(t) > 0: (a) 0 < R(t|Q) ≤ R₀(t) ≤ 1, with equality R(t|Q) = R₀(t) at Q = 0; (b) R(t|Q) is non-increasing in t and in Q; (c) for Q₂ > Q₁ the ratio R(t|Q₂)/R(t|Q₁) = exp(-γ(Q₂ - Q₁)t) depends on the indices only through their difference and decreases exponentially with t.

Proof. Statements (a) and (b) follow from (5), because γQt ≥ 0 and R₀ is a survival function; (c) follows by division of the two expressions (5). ∎

Remark 1. For R = R₀ + exp(–kQ) with 0 < R₀ < 1 and k > 0, R > 1 if and only if Q < –ln(1 - R₀)/k. For R₀ = 0.9 and k = 1 this holds for Q < 2.30, i.e. over the whole practical range of Q (Table 1).

The parameter k in (6) is therefore not a property of the suspension alone: it is the product of the road sensitivity γ and the mission time. The exponential decay of reliability under a linear growth of roughness is a consequence of the linear hazard assumption (4), not an empirical finding, and has to be tested on data (Section 2.5). When the baseline hazard is constant, λ₀(t) ≡ λ₀, the mean time to failure is

$$m\left(Q\right)={\displaystyle \frac{1}{{\lambda }_0+\mathit{\gamma Q}}}, {\displaystyle \frac{m\left(Q\right)}{m\left(0\right)}}={\displaystyle \frac{1}{1+\left(\gamma /{\lambda }_0\right)\hspace{0.17em}Q}},$$

(7)

so the dimensionless ratio γ/λ₀ is the relative increase of failure intensity per unit of Q. An alternative is the proportional hazards model [8], λ(t|Q) = λ₀(t)exp(αQ), for which

R(t | Q) = R₀(t)^exp(αQ).

(8)

For small αQ, exp(αQ) – 1 ≈ αQ, and (8) reduces to (5)–(6) with k ≈ αΛ₀(t), where Λ₀(t) = –ln R₀(t). The two models diverge at large Q and can be discriminated by information criteria once failure data by road category are available.

2.5. Parameter Estimation and Design of Observations

Let operating records be grouped by road category j = 1, …, J with index Q_j, exposure E_j (operating hours accumulated on roads of category j) and number of failures N_j. Under (4) with constant λ₀ the counts are independent Poisson variables,

N_j∼Poisson(E_j (λ₀ + γQ_j)),  j = 1,…,J,

(9)

with log-likelihood ℓ(λ₀, γ) = Σ_j[N_j ln(λ₀ + γQ_j) – E_j(λ₀ + γQ_j)] + const, which is maximised subject to λ₀ > 0, γ ≥ 0 [10]. For J = 2 and N₂/E₂ ≥ N₁/E₁ the maximum-likelihood estimators are explicit:

$$\widehat{\gamma }={\displaystyle \frac{N_2/E_2-N_1/E_1}{Q_2-Q_1}}, {\widehat{\lambda }}_0={\displaystyle \frac{N_1}{E_1}}-\widehat{\gamma }\hspace{0.17em}{Q}_1.$$

(10)

If N₂/E₂ < N₁/E₁, the constraint is active: γ̂ = 0 and λ̂₀ = ΣN_j/ΣE_j. The same scheme with a logarithmic link gives the estimator of α in (8), and subsystem parameters are obtained by applying (9) to the failures of each subsystem. Exposure is indispensable: failure counts without E_j do not identify γ.

To plan observations, consider the rate ratio θ = λ(Q₂)/λ(Q₁) between two categories and the exposure ratio ρ = E₂/E₁. By the delta method, Var(ln θ̂) ≈ 1/N₁ + 1/N₂, and a two-sided test at significance level a with power 1 – b detects θ if

$$N_1\ge {\displaystyle \frac{{\left(z_{1-a/2}+z_{1-b}\right)}^2\left(1+1/\left(\mathit{\theta \rho }\right)\right)}{{\left(\ln \theta \right)}^2}}, N_2\approx \mathit{\theta \rho }{N}_1,$$

(11)

where z_p is the p-quantile of the standard normal distribution.

2.6. Spare-Part Demand

Let the times between successive failures of a component that is replaced at failure be independent, with mean M, standard deviation σ and coefficient of variation ν = σ/M. The number of failures N(t) during operating time t is a renewal process; by the renewal central limit theorem [12], N(t) is asymptotically normal with mean t/M and variance ν²t/M. The stock that covers demand over a horizon t (lead time plus review period) with probability p is

$$n_p\left(t\right)=\lceil {\displaystyle \frac{t}{M}}+u_p\hspace{0.17em}\nu \sqrt{{\displaystyle \frac{t}{M}}}\rceil ,$$

(12)

where u_p is the normal quantile and ⌈·⌉ denotes rounding up. The normal law in (12) describes the number of failures over the horizon, not the time to failure; time to failure is described by the Weibull or lognormal laws [10].

2.7. Operational Indicators

Travel time per unit distance is 1/V. If φ is the share of travel time in the haul cycle on a good road and V_G is the speed on that road, the productivity on a road with speed V(Q) relative to a good road is

$${\displaystyle \frac{P\left(Q\right)}{P_{\mathrm{G}}}}={\displaystyle \frac{1}{1+\phi \left(V_{\mathrm{G}}/V\left(Q\right)-1\right)}}.$$

(13)

On a haul segment of length L with grade i (positive uphill) and rolling resistance coefficient f, the tractive work for gross vehicle mass m is W = mg(f + i) L. A change of f from f₁ to f₂ therefore changes the tractive energy, and at constant drivetrain efficiency the fuel use, by the factor

$${\displaystyle \frac{W_2}{W_1}}={\displaystyle \frac{f_2+i}{f_1+i}}.$$

(14)

3. Results

3.1. Roughness Index of the Road Categories

Table 1 gives the defect data with the row groups assigned to defect classes. The assignment follows from the parameter ranges: heights of 0.03–0.10 m at densities up to more than 200 per km correspond to impulsive irregularities, lengths of 0.5–5.0 m to potholes, and a small number of deep irregularities to long irregularities. This operational assignment is based on the stated dimensional ranges and is used only for the scenario calculation; it should be verified by measured longitudinal profiles in future field work. At mid-range values, (1)-(2) give D = 0.21, 2.30 and 9.19 m2/km and Q = 0.15, 0.50 and 1.00 for good, moderate and poor roads; for rough terrain, with open-ended values taken at their lower bounds, D = 25.1 m2/km and Q = 1.65 (D = 15.3 m2/km and Q = 1.29 if all its values are taken at the lower bounds). To assess the effect of the ranges, densities and amplitudes were sampled uniformly within them (400 000 draws, Dref fixed at 9.19 m2/km). The 90 % intervals of Q - 0.08-0.21, 0.35-0.68 and 0.84-1.18 - do not overlap, so the index separates the categories. The values assigned to the categories in the first version of the paper (0.2, 0.5 and 1.0) agree with (2) within the uncertainty of Table 1. On good, moderate and poor roads, potholes contribute 68-86 % of D; long irregularities become significant on poor roads (26 %) and dominate on rough terrain (about 70 %).

3.2. Operational Indicators

Table 2 combines the speed data with the quantities that follow from them. At mid-range speeds of 40, 30 and 17.5 km/h, travel time per kilometre is 1.50, 2.00 and 3.43 min, i.e. 33 % and 129 % longer on moderate and poor roads than on good ones. For a travel-time share φ = 0.5-0.7 (an illustrative range), Eq. (13) gives a productivity of 0.81-0.86 on moderate and 0.53-0.61 on poor roads relative to good roads. Reference rolling resistance coefficients for radial tyres are 2.5 % on a firm, smooth, maintained road flexing slightly under load, 4.0 % on a rutted dirt road with 25 mm tyre penetration and 5.0 % with 50 mm penetration [13]. By (14), an increase from 2.5 % to 4.0 % raises the tractive energy by 60 % on a level haul and by 14 % on an 8 % ramp; an increase to 5.0 % raises it by 100 % and 24 %, respectively. These reference values describe penetration and flexing of the road surface; their correspondence to Q has to be established by measurement (coast-down tests or on-board engine and payload data).

3.3. Technical Downtime Structure of the Sayak Fleet

The records contain 2737 daily downtime entries for four CAT 773E haul trucks. Consolidation of consecutive entries with the same truck and classifier gives 1395 episodes, of which 295 are technical (Figure 1). Tyres account for 89 technical episodes (30.2 %), followed by electrical equipment (45; 15.3 %), body and auxiliary equipment (43; 14.6 %), engine (41; 13.9 %), hydraulics (40; 13.6 %), transmission and final drives (18; 6.1 %), brakes (16; 5.4 %) and frame (3; 1.0 %). Technical downtime totals 8712.5 h. Transmission and final-drive events account for 6318.8 h (72.5 %), and tyre events for 855.6 h (9.8 %). A further 81 episodes classified as absence of spare parts account for 10739.0 h and are treated as supply-related delays rather than failures [16].

Road-induced vibration and shock can contribute to hydraulic and electrical failures through fatigue and abrasion of hoses and loosening of fittings – the reason why routing, clamping and abrasion protection are emphasised in hose assembly practice (SAE J1273) – through pressure peaks in cylinders under wheel impacts, and through chafing of wiring harnesses, wear of connector contacts and failures of sensors [4]. In the Sayak log, tyre damage is recorded explicitly under puncture or sidewall cut, tube, excessive wear and other tyre causes. Tyres form the largest group by technical episode count, while transmission and final-drive events dominate downtime hours. These patterns identify priority subsystems but do not prove that road roughness caused the events, because the records contain no road-section identifier, roughness measurement or exposure by road category. The ambiguous term “stabbing” is not used in the Sayak classifiers and has therefore been removed from the interpretation [16].

3.4. Time between failures and spare-part demand. For the illustrative spare-parts calculation retained in Table 3, the time between failures has mean M = 167.4 and standard deviation σ = 181.4. Their ratio gives ν = σ/M = 1.08; the value ν = 1.79 reported in the first version is inconsistent with M and σ (it would require σ = 300 at M = 167.4) and is not used. Three conclusions follow. First, a normal time-to-failure model is inadequate: with these moments it assigns probability Φ(–M/σ) = 0.18 to negative times. Second, ν slightly above unity is consistent with a Weibull law with shape parameter 0.92 and scale 161 (in the units of M) or with a lognormal law with logarithmic standard deviation 0.88. Third, ν > 1 is a signature of heterogeneity.

Proposition 2.Let T be exponentially distributed given a random failure rate Λ > 0 with a non-degenerate distribution and E[Λ^–2] < ∞. Then ν(T) > 1.

Proof. Since E[T|Λ] = 1/Λ and E[T²|Λ] = 2/Λ², ν² = 2E[Λ^–2]/(E[Λ^–1])² - 1, and E[Λ^–2] > (E[Λ^–1])² by Jensen’s inequality for a non-degenerate Λ. ∎

Pooling machines or operating conditions with different failure intensities - including different road conditions – thus raises ν above unity, and a Weibull shape parameter below unity should not be read as infant mortality. This is compatible with a road effect but does not identify it, because machine type and duty cycle also vary across the fleet.

Table 3 gives the stock (12) for horizons τ = t/M = 6, 12 and 30. A simulation of a Weibull renewal process with the above moments (400 000 replications) gives the same stock as (12) in all six cases, which validates the normal approximation in this range. With the inconsistent value ν = 1.79, the stock would be overstated by 2-7 units (13-27 %).

3.5. Reliability as a Function of Road Condition

Figure 2 shows the relative reliability (6) and the relative mean time between failures (7) with the intervals of Q from Table 1. For γt = 1, the probability of failure-free operation over a mission on a poor road is exp(-(1.00 - 0.15)) = 0.43 of that on a good road (Proposition 1c); for γ/λ0 = 1, the mean time between failures on a poor road is (1 + 0.15)/(1 + 1.00) = 0.58 of that on a good road. These values illustrate the model; the parameters γ and λ0 of the Sayak fleet cannot be estimated from the available records (Section 3.6).

Figure 2. Relative reliability (a) and relative mean time between failures (b) versus roughness index Q under the additive hazard model, Eqs. (6)-(7); shaded bands - 90 % intervals of Q for road categories (Table 1).

Figure 2. Relative reliability (a) and relative mean time between failures (b) versus roughness index Q under the additive hazard model, Eqs. (6)-(7); shaded bands - 90 % intervals of Q for road categories (Table 1).

3.6. Data required to identify the road effect. Figure 3 shows rule (11) for a = 0.05 and power 0.8. A 1.5-fold difference in failure intensity between two road categories is detected with about 200 failures in total at ρ = 1 (80 of them on the better road), a 1.3-fold difference with about 460, and a 1.2-fold difference with about 950. The 295 reconstructed Sayak technical episodes are of a useful order of magnitude, but they cannot be used directly in (9) because operating exposure by road category was not recorded. The following protocol makes this possible: (i) divide the haul network into sections and measure Q on each by (2) or (3), repeating the measurement after grading; (ii) record operating hours and distance by vehicle and section from dispatch or GPS data; (iii) record each failure with time, vehicle, subsystem and failure mode; (iv) estimate (9) for each subsystem, compare models (4) and (8) by the Akaike information criterion, and include payload, speed and grade as additional covariates.

4. Discussion

The results separate what the available data establish from what they do not. The roughness index (2) turns the categorical description of roads into a number that orders the categories consistently and can be measured on any section; model (4)–(6) gives the road effect a definite form with a single interpretable parameter γ; rules (10)–(11) state what has to be recorded to estimate it. Road maintenance management systems for mines [1,3] already use defect scores to relate road condition to vehicle operating costs; the present formulation adds a probability-consistent reliability component that can be estimated from the same maintenance data, separately for each subsystem.

The main limitation is the linkage between the data sets. The values of Table 1 are literature-informed scenarios rather than measurements on Sayak roads. The downtime records identify trucks, dates and classifiers but not engine-hour exposure or road sections, while the component monitor describes current installed-component status rather than a completed replacement history. Consequently, the paper does not report an estimate of γ, and statements about the size of the road effect remain conditional on its value.

Once γ and the speed and fuel relations are estimated, the economic effect of road maintenance follows from the net present value of the avoided costs:

$$\mathrm{NPV}={\displaystyle \sum _{y=1}^H}{\displaystyle \frac{\Delta C_{\mathrm{fuel},y}+\Delta C_{\mathrm{rep},y}+\Delta C_{\mathrm{down},y}+\Delta C_{\mathrm{tyre},y}}{{\left(1+r\right)}^y}}-I,$$

(15)

where the terms in the numerator are the annual savings in fuel, repairs, downtime and tyres obtained when the road index is reduced from Q₁ to Q₂ (computed from (7), (13), (14) and the measured tyre life), I is the cost of grading and compaction, r is the discount rate and H is the horizon; the payback period is the smallest H for which NPV ≥ 0.

5. Conclusions

1. A roughness index computed from the density and the squared amplitude of surface defects orders the road categories of the microprofile classification as 0.15, 0.50 and 1.00 (good, moderate, poor), with non-overlapping 90 % intervals, and can be cross-calibrated with profile-based measures.

2. An additive hazard model gives a reliability function R(t|Q) = R₀(t)exp(–γQt) that is a valid probability for all Q ≥ 0; the exponential dependence of relative reliability on Q follows from the model assumptions, and its rate equals the road sensitivity multiplied by mission time.

3. The road sensitivity is estimated by Poisson regression of failure counts on exposure by road category; about 200 failures suffice to detect a 1.5-fold difference in failure intensity between two categories.

4. For four CAT 773E trucks at Sayak, tyres account for 30.2 % of 295 reconstructed technical episodes, while transmission and final-drive events account for 72.5 % of 8712.5 technical downtime hours. The component monitor identifies 100 overdue and 15 critical replacements among 251 component records.

5. The available records do not contain exposure by road category, so the road effect on reliability is not quantified here; the proposed protocol of section-level roughness measurement and exposure recording makes its estimation possible. Future work will apply it to haul trucks and LHDs on measured road sections.