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Tractive effort

Tractive effort, also known as tractive force, is the tangential force exerted by the driving wheels of a locomotive or powered vehicle on the rail or road surface to propel the vehicle and its load forward.^[1] This force is essential for overcoming resistance from friction, gravity on inclines, and acceleration, and it is typically measured in pounds-force (lbf) or kilonewtons (kN), with 1 kN equaling approximately 224.8 lbf.^[2] In railway engineering, tractive effort is limited by the coefficient of adhesion between the wheels and the rail, preventing wheel slip, and is calculated as the product of the adhesive weight and the adhesion coefficient.^[1] The total tractive effort required to move a train includes components to accelerate the train mass, overcome rotational inertia of moving parts, counter gravitational forces on gradients, and resist aerodynamic and rolling drag, often expressed by the formula F = M \alpha + \frac{(J_w + J_m n^2) \alpha}{R} + \frac{M g G}{1000} + (A + B V + C V^2), where M is mass, \alpha is acceleration, J terms represent moments of inertia, R is wheel radius, G is gradient, V is speed, and A, B, C are resistance coefficients.^[1] For steam locomotives, starting tractive effort is derived from cylinder pressure and piston area, following TE = \frac{0.85 \times MEP \times A \times N}{D}, where MEP is mean effective pressure, A is piston area, N is number of cylinders, and D is driving wheel diameter in feet.^[3] In electric and diesel locomotives, it stems from motor torque transmitted through the drivetrain, given by TE = \frac{T \times GR}{\eta \times r}, with T as torque, GR as gear ratio, \eta as efficiency, and r as wheel radius.^[4] Tractive effort varies with speed, typically peaking at low speeds for starting and decreasing as velocity increases due to the inverse relationship with power output, where power P = TE \times V / 375 in horsepower units (with V in mph).^[5] Drawbar pull, a related metric, represents the net force available at the coupler after accounting for locomotive self-resistance, often 85-95% of total tractive effort.^[6] In road vehicles, the concept extends to automotive and off-road applications, where it determines acceleration, gradeability, and towing capacity, influenced by tire-road friction and drivetrain configuration.^[4] Engineers use tractive effort curves to optimize locomotive design for specific routes, balancing adhesion limits, fuel efficiency, and load-hauling capability.^[1]

Fundamentals

Definition

Tractive effort refers to the tangential force exerted by the driven wheels of a powered vehicle on the rail or ground surface, enabling propulsion in either the forward or backward direction. This force represents the effective push or pull generated at the point of contact between the wheels and the track or roadway, directly influencing a vehicle's ability to overcome resistance and initiate motion. In engineering contexts, it is a critical measure of a vehicle's propulsive capacity, particularly for heavy-duty applications where starting from rest or maintaining speed under load is essential.^[1]^[7]^[6] Tractive effort is inherently limited by the friction or adhesion available at the contact interface, beyond which wheel slip occurs, preventing further force application without additional measures like sanding.^[4] The concept of tractive effort originated in 19th-century railway engineering, emerging as a key parameter in the design and performance evaluation of early steam locomotives, where maximizing this force was vital for hauling heavy freight over varied terrains. This term's development reflected the era's focus on empirical testing and mechanical innovation to expand rail networks across continents.^[8]^[9] Although rooted in rail applications, tractive effort applies universally to powered vehicles, as seen in road transport where it determines an automobile's or truck's capacity to accelerate on inclines or tow loads by generating force through tire-road interaction. For instance, in both rail and road scenarios, sufficient tractive effort ensures reliable motion without excessive slippage, underscoring its role as a fundamental engineering principle across transportation modes.^[10]^[11]

Physical Principles

Tractive effort originates from the torque generated by an engine or motor, which is transmitted through the drivetrain—comprising components such as the transmission, driveshaft, differential, and axles—to the driven wheels.^[12] This process converts the rotational energy into a tangential force at the wheel's circumference, enabling the vehicle to overcome resistance and achieve propulsion.^[12] In both road and rail vehicles, the drivetrain ensures efficient torque delivery, adapting engine output to wheel rotation via gear ratios for optimal force application.^[13] At the wheel-ground or wheel-rail interface, static friction plays a pivotal role in transforming this rotational torque into linear forward force. The torque causes the wheel to tend to rotate backward relative to the contact point, but friction opposes this slip, generating a forward propulsive force on the vehicle.^[14] This frictional interaction adheres to Newton's third law of motion, where the wheel exerts a backward force on the ground or rail, and the ground or rail responds with an equal and opposite forward force on the wheel, thereby propelling the vehicle ahead.^[14] Without sufficient friction, the wheel would spin without translating torque into effective motion.^[12] The physical principles differ between rail and road vehicles primarily due to the nature of the contact interfaces and path constraints. In rail vehicles, tractive effort relies on steel wheel-rail adhesion, where the flanged wheels follow a guided path, focusing friction longitudinally for propulsion while lateral forces maintain stability on the tracks. Road vehicles, in contrast, operate on free-rolling paths with rubber tires deforming against varied surfaces, allowing for steering but introducing higher rolling resistance and more complex frictional dynamics influenced by tire viscoelasticity. These distinctions arise from the rigid, high-pressure steel-on-steel contact in rails versus the flexible, textured rubber-on-road interaction, affecting how torque is converted to sustained propulsion.

Calculations

Basic Formulas

The tractive effort (TE) generated by a vehicle's propulsion system is fundamentally derived from the engine or motor torque transmitted through the drivetrain to the wheels. The core formula for tractive effort at the wheels is given by

TE = \frac{T_e \times GR \times \eta}{r_w}

where T_e is the engine torque (in N·m or lb·ft), GR is the total gear ratio (dimensionless, product of transmission and final drive ratios), \eta is the drivetrain efficiency (accounting for losses in gears, bearings, and differentials), and r_w is the effective wheel radius (in m or ft).^[15] This expression arises from the basic relationship that tangential force at the wheel circumference equals wheel torque divided by radius, with wheel torque obtained by scaling engine torque via the gear train and efficiency. To derive this from torque basics, consider the engine producing torque T_e at its output shaft. This torque is multiplied by the overall gear reduction GR to amplify it at the axle, yielding axle torque T_a = T_e \times GR. Incorporating drivetrain losses, the effective torque becomes T_a = T_e \times GR \times \eta. The tractive effort then follows from Newton's second law applied at the contact patch: the linear force TE produces the torque at the wheel such that T_w = TE \times r_w, where T_w = T_a for a direct drive to the wheel (or further scaled if multiple stages). Thus, rearranging gives the formula above.^[16] Efficiency \eta is critical, depending on the system design and load.^[15] Tractive effort is typically expressed in newtons (N) for SI units or pounds-force (lbf) in imperial systems, consistent with rail and road vehicle applications where forces range from thousands to tens of thousands of units.^[12] For vehicles with multiple driven axles, the total tractive effort is the summation of contributions from each driven wheel or axle, assuming symmetric loading and gearing. If there are n identical driven axles, total TE_{total} = n \times TE_{single}, where TE_{single} uses the per-axle torque distribution (engine torque divided equally or per design). This adjustment ensures the formula scales appropriately for multi-axle locomotives or trucks without altering the underlying per-wheel mechanics.^[16] Actual TE is ultimately limited by adhesion between wheels and rail or road surface.

Power and Speed Relationships

The relationship between tractive effort (TE), power (P), and speed (v) in rail vehicles is fundamentally expressed as P = \text{TE} \times v, where power is the product of the force exerted by the wheels on the rail and the vehicle's velocity. In engineering practice for railways, units are often converted to horsepower (hp), pounds-force (lbf) for TE, and miles per hour (mph) for speed, yielding the formula P \, (\text{hp}) = \frac{\text{TE} \, (\text{lbf}) \times v \, (\text{mph})}{375}. This conversion factor arises from the definition of one horsepower as 550 foot-pounds per second, adjusted for the conversion of mph to feet per second (multiplying by 1.4667 ft/s per mph), resulting in the approximate divisor of 375.^[17] Under conditions of constant power output, which is typical for many modern rail propulsion systems once beyond low-speed torque-limited regimes, tractive effort varies inversely with speed: \text{TE} = \frac{P}{v}. This produces a hyperbolic relationship where TE is maximized at low speeds for strong acceleration and starting, but diminishes as speed increases to maintain the fixed power level. For instance, in high-speed electric trains, the constant power model applies above a critical speed, leading to progressively lower TE to balance energy input against motion resistance.^[18] Engine and motor characteristics significantly influence available TE through their torque and power curves. Peak torque, often occurring at lower rotational speeds, enables high initial TE for overcoming inertia and grades, while the power curve—typically peaking at intermediate speeds—governs sustained performance and the transition to speed-limited operation. In distributed traction systems, such as those in high-speed rail, motor torque-speed profiles directly shape the overall TE envelope, with efficiency losses further modulating output.^[19] For practical rail operations in the 0-100 mph range, consider a diesel-electric locomotive delivering constant 3000 hp at the rail: at 10 mph, TE ≈ 112,500 lbf, sufficient for rapid acceleration of heavy trains; at 50 mph, TE drops to ≈ 22,500 lbf; and at 100 mph, TE ≈ 11,250 lbf, prioritizing velocity over pulling force. These values highlight how unit conversions and engine limits constrain performance across typical freight and passenger speed profiles.^[17]

Graphical Analysis

Tractive Effort Curves

Tractive effort curves are graphical representations used in vehicle performance analysis, plotting tractive effort (TE) on the y-axis against vehicle speed on the x-axis. These plots depict the maximum available TE, which represents the peak pulling or pushing force the vehicle can exert at any given speed, often limited by factors such as engine output and mechanical constraints. Additionally, they include the sustained TE envelope, indicating the continuous operable force over time without overheating or excessive wear.^[20] The construction of these curves involves several key steps to integrate various performance elements. First, the base curve is derived from the engine's power output translated into force at the rail, forming the initial TE-speed relationship. For locomotives with multiple gear ratios, separate tractive effort curves are generated for each ratio, showing higher initial effort but lower top speeds for higher ratios. Finally, the adhesion limit—a horizontal line capping TE based on wheel-rail friction, typically around 25% of the vehicle's weight on drivers—is superimposed to define the practical maximum envelope. This layered approach ensures the curve reflects real-world operational boundaries.^[8]^[20] Typical tractive effort curves exhibit distinct shapes that highlight performance characteristics across speed ranges. At low speeds, the curve is typically at its maximum, limited by the adhesion to prevent wheel slip, and then decreases gradually as speed increases, approaching an asymptotic approach and flattening due to the inverse relationship with power limits, where TE diminishes as velocity rises while maintaining constant power output. The shape is influenced by power-speed relationships, as detailed in subsequent sections.^[21]^[20] The development of tractive effort curves originated in early 20th-century locomotive design, emerging from systematic performance testing to predict and optimize vehicle capabilities. Pioneering research at institutions like Purdue University from 1891 to 1906 utilized dynamometers and indicator cards to generate these plots, enabling engineers to evaluate efficiency and draw-bar pull under varying conditions for the first time in a standardized graphical format.^[21]

Interpretation of Curves

Tractive effort curves provide a graphical means to predict a railway vehicle's acceleration and overall performance by comparing the available tractive effort at various speeds against the opposing train resistance. To determine the net accelerating force, the total resistance—comprising rolling resistance, grade resistance, and curve resistance—is subtracted from the tractive effort value at a specific speed on the curve. This net force, when divided by the train's mass, yields the acceleration according to fundamental principles of dynamics. For instance, at lower speeds where tractive effort is high, the net force is typically positive, enabling rapid acceleration, while at higher speeds, it diminishes as tractive effort falls inversely with velocity under constant power conditions.^[9]^[22] A key feature of these curves is the identification of the balance speed, defined as the velocity where tractive effort precisely equals the total resistance, resulting in zero net force and thus a constant speed with no acceleration or deceleration. This point varies with route conditions: on level tangent track, it represents the maximum sustainable speed under continuous tractive effort; on upgrades or curved sections, it shifts lower due to increased resistance components. Engineers use this intersection to forecast top speeds and ensure safe train handling, such as maintaining momentum through grades. Adhesion limits may clip the curve at low speeds, restricting achievable tractive effort to prevent wheel slip.^[9]^[22] In locomotives equipped with multiple gear ratios, distinct tractive effort curves are generated for each ratio, illustrating the trade-off between pulling power and maximum speed. Higher gear ratios produce curves with greater initial tractive effort but lower top speeds, while lower ratios extend the speed range at reduced effort levels. Shift points, often marked by minimum continuous speeds where thermal or adhesion constraints are met, guide operators on when to change gears to optimize performance and avoid overload. These multi-curve analyses inform gear selection during locomotive design to match operational demands.^[23] Practically, tractive effort curves underpin locomotive specifications by enabling tonnage ratings, which quantify the maximum trailing load for given routes. By integrating curve data with route-specific resistance profiles—including grades up to several percent and curve radii—engineers calculate haulage capacities that ensure required speeds and handling characteristics, such as accelerating a loaded train from rest or sustaining speed over undulating terrain. This application is vital for efficient scheduling and safe operations across diverse railway networks.^[23]^[22]

Rail Vehicle Applications

Steam Locomotives

In steam locomotives, tractive effort is generated by the pressure of expanding steam acting on the pistons within the cylinders, which are connected to the driving wheels via connecting rods and crossheads, converting linear motion into rotational force at the wheel rims. The magnitude of this effort depends on the cylinder bore and stroke dimensions, which determine the piston's swept volume and thus the work done per cycle, but it is fundamentally limited by the boiler's capacity to generate and supply saturated or superheated steam at sufficient pressure and volume to maintain cylinder filling without significant cutoff losses. As the locomotive accelerates, the steam supply rate becomes the primary constraint, causing tractive effort to diminish rapidly beyond low speeds due to incomplete expansion and exhaust backpressure.^[24] A common approximation for starting tractive effort, valid at near-zero speeds where cutoff is minimal and mean effective pressure approaches 85% of boiler pressure, is given by TE = (0.85 × Cylinder area × Boiler pressure) / Driver diameter, with tractive effort in pounds-force, cylinder area in square inches (accounting for the number of cylinders), boiler pressure in pounds per square inch, and driver diameter in inches; this formula, often used in boiler horsepower estimates, simplifies the piston force divided by the wheel leverage while incorporating efficiency factors for valve events and friction.^[3] Historical designs evolved significantly in this regard: early locomotives produced starting tractive efforts sufficient for hauling 20-30 ton trains at 10-15 mph on level grades. By the 1940s, pinnacle articulated designs like the Union Pacific Big Boy (4-8-8-4) reached 135,000 lbf starting tractive effort through 23 3/4-inch diameter cylinders (with 32-inch stroke), 300 psi boiler pressure, and 68-inch drivers, allowing over 3,600-ton trains across the Wasatch Mountains.^[25] Compared to diesel locomotives, steam designs delivered exceptionally high starting tractive effort but exhibited lower sustained values at operational speeds due to power output drop-off as boiler steaming capacity plateaued, typically peaking at 40-60 mph before declining from exhaust restrictions. Wheel arrangement influenced this profile: freight-oriented 2-8-2 Mikado classes, with 56-63 inch drivers and 26-28 inch cylinders at 200-220 psi, often achieved 50,000-65,000 lbf starting tractive effort for heavy drag service, while passenger-focused 4-8-4 Northerns, featuring 73-80 inch drivers and similar cylinder sizes at 250-280 psi, produced 60,000-70,000 lbf but prioritized speed over low-speed pull, with both types showing characteristic effort curves that steepen downward after 20 mph.^[26]^[27]^[28]

Diesel and Electric Locomotives

In diesel and electric locomotives, tractive effort is primarily generated through electric traction motors mounted directly on the axles, which convert electrical energy from an onboard generator (in diesel-electrics) or overhead catenary (in pure electrics) into mechanical force at the wheels.^[23] This configuration enables high starting tractive efforts, typically up to 30% of the locomotive's adhesive weight on drivers, by leveraging the high torque characteristics of series-wound or AC induction motors without the mechanical limitations of direct drive systems.^[23] The resulting tractive effort curves exhibit relatively flat power output across a wide speed range, allowing sustained performance that contrasts with the rapid decline in power seen in other propulsion types. For diesel-electric locomotives, tractive effort is calculated based on the prime mover's output, transmission efficiency, and train speed, with the fundamental relationship given by

TE = \frac{\text{HP} \times 746 \times \eta}{v}

where TE is tractive effort in newtons, HP is diesel engine horsepower, 746 converts horsepower to watts, \eta is the overall transmission efficiency (typically 0.8–0.9), and v is speed in meters per second. Turbocharging enhances this by increasing air density in the engine cylinders, boosting power output by 20–50% without enlarging the engine size, thereby elevating available horsepower and thus starting and continuous tractive efforts.^[29] In pure electric locomotives, regenerative braking further optimizes tractive effort utilization by reversing the traction motors to act as generators during deceleration, converting kinetic energy back into electrical energy for storage or return to the supply system, recovering up to 20–30% of braking energy depending on infrastructure.^[30] For example, the GE Evolution Series ES44ACi diesel-electric locomotive achieves a starting tractive effort of approximately 200,000 lbf (890 kN) through its AC traction system, enabling it to haul heavy freight trains efficiently from standstill.^[31] Emerging battery-electric locomotives, as of 2025, apply similar electric traction principles with onboard energy storage, achieving comparable starting tractive efforts while reducing emissions.^[32] Compared to steam locomotives, diesel and electric designs provide superior sustained tractive effort at higher speeds due to constant horsepower output from the prime mover or external supply, maintaining effective pulling power up to 60–80 mph where steam efficiency drops sharply.^[26] Multi-unit operation, where multiple locomotives are electrically synchronized via control systems, effectively doubles or multiplies tractive effort proportionally to the number of units, allowing a single crew to manage distributed power for trains exceeding individual locomotive capacity.^[33]

Limiting Factors

Adhesion and Traction

Adhesion in rail vehicles refers to the frictional force between the steel wheels and steel rails that enables the transmission of tractive effort without slipping. The maximum tractive effort achievable is limited by this adhesion, given by the formula

TE_{\max} = \mu \times W

where \mu is the adhesion coefficient and W is the vertical weight supported by the driven axles. For dry rails under typical conditions, \mu ranges from 0.2 to 0.3, though values up to 0.5 have been observed in optimal clean and dry scenarios.^[34] Several factors influence the adhesion coefficient and help prevent wheel slip, which occurs when the demanded tractive force exceeds the available friction. Sandboxing, or the application of dry sand to the railhead ahead of the driving wheels, increases friction by introducing abrasive particles that enhance the wheel-rail interface grip. This technique, historically manual in steam locomotives, significantly improves adhesion in low-traction scenarios.^[35] Sanding also mitigates creep, the small relative motion at the contact point that can reduce effective traction if unmanaged; by altering the contact surface, it restores higher creep force levels closer to the dry rail curve.^[36] In the steam era, wheel slip was a common issue that directly led to loss of tractive effort, as uncontrolled slipping reduced the effective force transmission and risked mechanical damage to cylinders and rods. Notable incidents, such as the 1994 "Blue Peter" slip on preserved locomotive No. 60532 Blue Peter, demonstrated how sudden adhesion loss could cause prolonged wheelspin, boiler priming, and severe engine damage during acceleration. Modern locomotives employ advanced slip control systems, such as wheel slip detection modules that monitor rotational speeds and automatically adjust power output to maintain adhesion near the optimal slip ratio of 5-10%, preventing TE loss and improving efficiency.^[37] Compared to road vehicles, rail adhesion is inherently lower due to the steel-on-steel contact, with \mu typically 0.1-0.5, versus 0.7-1.0 for rubber tires on dry asphalt, reflecting the smoother, harder surfaces that limit frictional interlocking in rails.^[34]^[38] This difference necessitates specialized traction management in rail applications, where adhesion often clips the tractive effort curve at lower force levels than torque capabilities would allow.

Resistance and Environmental Effects

Train resistance represents the collective opposing forces that a rail vehicle's tractive effort must overcome to achieve motion, including rolling resistance from wheel-rail interaction, mechanical resistance from bearings and components, aerodynamic drag, and additional factors like gradient and curvature.^[39] These components are typically aggregated in empirical models such as the Davis equation, expressed as R = A + BV + CV^2, where R is the resistance force in pounds per ton, V is the speed in miles per hour, A captures speed-independent rolling and journal friction (often 1.3–3 lb/ton depending on vehicle type), B accounts for speed-linear terms like skin friction and minor mechanical losses (around 0.03–0.06 lb/ton per mph), and C models quadratic aerodynamic drag (typically 0.0005–0.0013 lb/ton per mph²).^[40] This formulation allows engineers to predict the net tractive effort required, as effective tractive effort equals total generated effort minus resistance, influencing acceleration, top speed, and energy efficiency.^[41] Rolling resistance, the dominant low-speed component, arises primarily from deformation in the wheel-rail contact patch and bearing friction, contributing 50–70% of total resistance at speeds below 50 km/h for freight trains.^[39] Aerodynamic resistance becomes predominant at higher speeds, scaling with the square of velocity and vehicle frontal area, and can account for over 80% of resistance in high-speed passenger trains exceeding 200 km/h.^[42] Curve and gradient resistances add fixed penalties, with curvature increasing effective resistance by 0.5–2 lb/ton per degree of curvature due to lateral wheel-rail forces, while a 1% grade equates to 20 lb/ton of gravitational opposition.^[43] Environmental conditions modulate these resistance components, often increasing overall opposition to motion and thus elevating the tractive effort demand. Wind, for instance, alters aerodynamic drag based on direction and speed; a headwind of 10 m/s can increase resistance by 10–20% for typical freight trains by effectively raising relative airspeed, while tailwinds provide minor relief.^[44] Temperature influences rolling resistance through changes in lubricant viscosity in axle bearings; colder conditions (below 0°C) can raise journal friction by 15–30% due to thicker oil films, as observed in early 20th-century tests on freight trains, though modern synthetic lubricants mitigate this to under 10%. Conversely, extreme heat (above 40°C) may slightly reduce rolling resistance via lower viscosity but exacerbates rail thermal expansion, indirectly increasing curve resistance through track geometry shifts.^[45] Precipitation and humidity further impact resistance, particularly through track surface effects. Rain can elevate rolling resistance by 5–15% on contaminated rails by introducing water films that increase wheel slip and deformation losses, though this is secondary to adhesion limits. Snow and ice accumulation add substantial mass and friction, potentially doubling low-speed resistance in severe winter conditions by forming insulating layers that hinder heat dissipation from bearings.^[46] Altitude affects aerodynamic terms via reduced air density, decreasing drag by about 1% per 100 m elevation gain above sea level, which marginally lowers high-speed resistance but is offset by potential power derating in internal combustion engines.^[46] These effects underscore the need for adaptive tractive effort control systems in varying environments to maintain performance.

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[PDF] Train Resistance and Railroad Emissions and Efficiency - RailTEC
Oct 24, 2008 · Grade is the resistance due to gravity acting on the train mass. The resistance is 20 lbs of force per ton of train weight for a 1 percent ...
[44]
The impact of wind on energy-efficient train control - ScienceDirect
In this paper, we focus on wind variability and propose a train resistance equation that accounts for the impact of wind speed and direction explicitly on the ...
[45]
Identifying climate-related failures in railway infrastructure using ...
Weather-related factors like temperature variations, precipitation, and humidity can have severe impacts on the operation and maintenance of land transport ...
[46]
[PDF] Train Movement and Environmental Resistance - SciTePress
With an increase in the absolute relief marks, atmospheric pressure decreases, the traction force decreases, the consequence of this may be a decrease in weight ...