Tire Load Capacity: The TRA Load Formula and a Proposed Revisions

This chapter traces the evolution of the TRA (Tire and Rim Association) load-carrying capacity formula for passenger tires, compares it with an alternative approach based on constant relative deflection, and proposes a new formula better suited to modern low-profile, large-rim tires.

Keywords: tire load capacity; TRA formula; tire deflection; tire stiffness; load index; rolling circumference

1. Introduction

Since its founding in 1903, the Tire and Rim Association (TRA) has set tire interchangeability standards covering load, dimensions, rim contours, and valves. The load formula is empirical, derived from member-company data and field experience. For a given pressure, the load limit is the maximum rated capacity; conversely, the listed pressure is the minimum recommended for that load. The formula’s goals are: (a) enable interchangeable tires across manufacturers; (b) provide progressive load ratings within a product line; (c) balance OEM requirements with real-world service.

2. History of the TRA Load Formula

Before 1928, load ratings were negotiated by manufacturers. Loads were independent of rim diameter, proportional to section width, and directly proportional to inflation pressure: L = L₀(P/P₀). In 1928, rim diameter was introduced. The first formal formula was developed by mathematician C.G. Hoover in the mid-1930s, incorporating pressure, section width, and rim diameter — designed to maintain uniform deflection across tire sizes.

The early “low-pressure passenger tire” formula was:

Revised in 1936 to:

Since then, the input parameters (section width S, rim diameter D_R, pressure P) have remained the framework; only coefficients and exponents have been adjusted for new tire types and metrication.

3. Derivation of the Base Formula

Schuring (1985) reconstructed the logic: the load-pressure relationship was made sub-linear with exponent n = 0.585, and load was assumed proportional to internal air volume V. Early tires had near-circular cross-sections where V ∝ S²(D_R + S), leading to:

Because torus volume is not strictly proportional to S², the exponent was reduced from 2 to 1.39 based on field experience:

4. The Constant K (Service Factor)

The coefficient K adjusts for tire type and service condition: truck/bus tires use K=1.00; passenger tires originally used K=1.10. The 0.425 factor was introduced to bring K near 1.00. When P-metric tires were introduced in the 1970s, metric conversion constants were merged into a revised K.

5. Pressure Exponent

The exponent 0.585 limits how quickly load grows with pressure — as pressure rises, tire stresses increase, so a sub-linear exponent caps deflection and stress. When P-metric tires were introduced, the exponent was reduced to 0.50. P-metric design load pressure rose from 26 psi to 28 psi; max load pressure from 32 psi to 35 psi.

6. Section Width (Equivalent Section Diameter)

Early tires had circular cross-sections (aspect ratio = 1.00). As tires became wider and lower-profile, the cross-section became elliptical. An equivalent section diameter S_d was introduced to preserve the circular-section calculation logic — the perimeter of an elliptical cross-section is converted to an equivalent circle.

7. Why a Formula Review Is Needed

Tire sizes have proliferated dramatically: from 91 sizes in 1975 to 145–345 by 2003; aspect ratios now range from 30 to 80; rim diameters from 12 to 22 inches. The original formula assumed circular cross-sections. Meanwhile, TPMS regulations allow tires to run at reduced pressure (increased deflection) before warning. The current formula unfairly penalizes large-rim, low-profile tires.

8. A New Research Approach: Load–Deflection and Stiffness

Industry consensus: at normal operating pressure, inflation pressure and tire dimensions (tread width, outer diameter) determine stiffness; tire structure itself contributes only 10–15%. Rhyne extended Koutny’s torus model to compute vertical stiffness directly. The tangent stiffness K_Z is a function of pressure P, contact patch width W, and outer diameter OD:

Validated against 50 tires across 34 sizes with good correlation. The contact patch width W relates to nominal section width S_N and aspect ratio AR via empirical coefficient a:

Table: Coefficient a by aspect ratio

The stiffness model was validated against 2,027 test data points across 144 sizes and 1,000+ tires (rim 13–24″, section width 175–305 mm, aspect ratio 35–80), achieving R² = 0.84 between predicted and measured stiffness.

9. TRA Load Formula vs. Deflection

Tire engineers discuss load and pressure, but what really determines tire life is deflection. Greater deflection means more strain, more heat buildup, higher operating temperature, and greater rolling loss. The static deflection under any load-pressure combination is simply:

where d = deflection (mm), L = load (kg), K_Z = tangent stiffness (kg/mm). For cross-size comparison, use relative deflection as a percentage of section height above the rim flange (SH).

10. Deflection Analysis

Applying the current TRA load formula across sizes reveals a systematic bias:

• 75-series tires at 180 and 240 kPa: as section width increases, relative deflection slightly decreases; as rim diameter increases, deflection rises significantly.
• 45-series low-profile tires: identical trend.
• Fixed section width, varying rim code: same aspect ratio geometry, only total diameter changes. Larger rim → higher relative deflection.

The current TRA formula penalizes large-rim tires: bigger rims require the tire to operate at higher deflection, accelerating fatigue and heat buildup.

11. Proposed Load Formula Based on Constant Relative Deflection

To eliminate the bias against low-profile, large-rim tires, a new formula is proposed based on constant relative deflection. Since rim flange height is a constant 17.5 mm on modern passenger rims, deflection can be expressed in terms of nominal section width and aspect ratio. Combining with the stiffness model yields a general load formula valid across all passenger tire sizes:

For maximum load calculations, use 240 kPa (the current P-metric standard-load reference pressure). Select the appropriate relative deflection percentage for the aspect ratio and compute load. Comparing the new formula with the existing TRA formula for 70-series tires confirms that the old formula underrates large-rim tires relative to the constant-deflection approach.

12. Future Work

The constant-deflection method needs further research to determine appropriate relative deflection targets for different tire categories. Whether 75-series and 30-series tires should share the same deflection threshold requires test data. The stiffness model also needs refinement for current and future tire designs. ISO has begun discussions on this approach.

Summary

The tire load formula has served for decades but needs revision for modern low-profile, large-rim tires. The proposed method based on constant relative deflection provides a physically grounded framework for evaluating current load ratings and developing future standards — covering everything from section width and aspect ratio to rim diameter and inflation pressure.

Rubber Calendering: Common Problems and Solutions

This Q&A guide covers the essential questions about rubber calendering — process principles, machine types, deflection compensation, calendering effect, pre-calendering preparation, sheet pressing, plying, profiling, fabric frictioning, and common quality defects with their root causes.

Keywords: rubber calendering; calender; calendering effect; frictioning; topping; fabric coating; rubber processing

1. What Is Calendering and How Does It Work?

Calendering is the process of forming mixed rubber compound into sheets of specified thickness and width, or coating a thin rubber layer onto textile fabric. It covers rubber sheeting, profiling, topping, frictioning, plying, cold refining, and straining operations.

Principle: Rubber is fed between two rotating work rolls, drawn into the roll gap, and squeezed into a sheet of controlled thickness and width. During calendering, the compound undergoes both viscous flow and elastic deformation — so all calendering phenomena are closely tied to rubber flow behavior and viscoelasticity.

2. Calender Types and Model Designation

Calenders are classified by:

• Number of rolls: two-roll, three-roll, four-roll.
• Purpose: sheeting calenders (uniform roll speed, 3–4 rolls); frictioning calenders (roll speed ratios, usually 3-roll); universal calenders (adjustable speed ratios via auxiliary gears); profiling calenders (one engraved roll for patterned sheets); steel cord calenders (four-roll).
• Roll arrangement: I-type, ▵-type, T-type, L-type, Z-type, S-type.

Specifications: Roll outer diameter × face length, e.g., Φ610×1730. Model example: XY-4T-1730 — XY = rubber calender; 4T = four-roll; 1730 = face length in mm.

3. Why Deflection Compensation?

Rubber passing through the roll gap exerts large separating force on the rolls. Combined with roll weight, this causes elastic bending called deflection, resulting in sheets that are thicker in the center and thinner at the edges. Compensation methods include:

• Crown (camber) method — grinding a slight convex curve onto the roll.
• Axis crossing — skewing one roll relative to the other.
• Anti-bending (prestress) method — applying external bending moment opposite to the load.

4. What Is the Calendering Effect?

The calendering effect is the directional performance difference between longitudinal and transverse directions in calendered sheets: along the calendering direction (MD), tear strength is higher, elongation is lower, and shrinkage is greater; across the direction (TD), tear strength is lower, elongation is higher, and shrinkage is lower. It is caused by orientation of rubber molecules and needle/plate-shaped filler particles along the flow direction.

Elimination: minimize needle/plate fillers (calcium carbonate, clay, talc); raise calendering temperature to increase molecular mobility. Note: some products benefit from it (rubber cables need higher MD strength); others must eliminate it (balls require uniform strength). Test specimens should always be cut in the calendering direction.

5. Pre-Calendering Preparation

(1) Heat Refining (Warming)

Compound must be heat-refined on an open mill before calendering to improve filler dispersion and plasticity. Two stages:

• Coarse refining: tight nip (2–5 mm), roll temp 40–45°C, 7–8 passes through tight nip — softens and homogenizes.
• Fine refining: wider nip (7–10 mm), roll temp 60–70°C, 6–7 passes — raises temperature for uniform high plasticity.

Mill roll speed ratio is 1:1.7 to 1:1.28 for fast warming; maintain a proper rubber bank on the rolls.

(2) Feed Supply

Continuous feeding matching output; small, frequent additions to avoid trapping air.

(3) Pre-Start Inspection

Check roll surface (clean, smooth, uniform temperature, no damage), bearings, lubrication, temperature gauges, thickness gauges, wind-up device, and motor operation.

(4) Temperature Control

Synthetic rubber calender roll temps are lower than NR; upper roll runs hotter than middle/lower rolls. Temperature depends on formulation, plasticity, process, and line speed.

(5) Fabric Drying

Fabric must be dried before coating to 1–2% moisture content. Too much moisture weakens adhesion; too dry damages fabric and reduces strength. Do not let dried fabric sit — it reabsorbs moisture.

6. Sheeting Methods

Sheeting presses warm compound into uniform sheets at equal roll speeds. Two modes:

• With rubber bank: maintain a small bank between middle and lower rolls — smooth surface, fewer bubbles, better density, but stronger calendering effect. Preferred for SBR. Do not over-fill (traps air).
• Without rubber bank: opposite; preferred for natural rubber.

7. Sheeting Process Controls

1. Roll temperature: set by compound type. High rubber content / high elasticity requires higher temp. Maintain proper temperature differential between rolls for good sheet transfer. NR sticks to hotter rolls; butyl rubber sticks to colder rolls.
2. Plasticity: too high → sticks to rolls; too low → rough surface, high shrinkage. Moderate plasticity gives smooth sheets.
3. Roll speed: faster = higher output, but must match plasticity. High plasticity allows faster speed.

8. Sheet Quality: Defects and Causes

Requirements: smooth surface, uniform thickness, no wrinkles, no pre-vulcanized particles, dense interior with no bubbles.

9. Plying (Topping)

Plying combines two thin calendered sheets into one thicker sheet — used for thick, high-quality sheets or sandwich constructions.

• Two-roll plying: simple, up to 5 mm thick, but low precision.
• Three-roll plying: pre-formed sheet/combined with freshly calendered sheet.
• Four-roll plying: two sheets simultaneously calendered and combined inline — high efficiency, good precision, but stronger calendering effect.

Requirement: both layers must have equal plasticity — otherwise delamination or bubbles. For dissimilar formulations/thicknesses, prefer simultaneous plying — combine two freshly calendered hot sheets immediately from two lines for best seal and no bubbles.

10. Profiling

Profiling forms rubber into specific cross-sections or surface patterns — e.g., shoe soles, tire components. Requirements: accurate dimensions, clear pattern, dense compound.

• High rubber content causes pattern collapse — add fillers, softeners, reclaim rubber, or factice.
• Strictly control plasticity, refining degree, and rework ratio.
• Cool the sheet immediately after leaving the calender to set the pattern before it deforms.

11. Fabric Coating: Purpose

Coating applies a thin rubber layer onto textile fabric (cord, canvas) to create carcass plies for rubber products. It bonds yarns and plies into a unified structure that shares load, while improving elasticity, water resistance, and overall product performance.

12. Fabric Coating Methods

1. Topping: two equal-speed rolls press a pre-formed rubber sheet onto the fabric surface.
2. Pressure topping: same as topping but with a rubber bank at the nip — bank pressure forces rubber into fabric gaps for better adhesion.
3. Frictioning: uses roll speed ratio to generate shear and squeeze rubber into fabric interstices for maximum bond.

Frictioning subtypes:

• Middle-roll wrap (light frictioning): fabric passes between middle and lower rolls; partial rubber works into fabric, remainder stays on middle roll. High adhesion, low rubber pickup, poor flex fatigue.
• No middle-roll wrap (heavy frictioning): all rubber works into fabric, middle roll carries no rubber. Thicker coating, better flex resistance, but lower adhesion and variable pickup.

13. Key Factors in Fabric Coating

• Roll temperature: higher temp improves flow, smoothness, and penetration → better bond; but too high risks scorch.
• Plasticity: higher plasticity = better flow and penetration = stronger bond, cleaner surface, less shrinkage.
• Roll speed: faster speed means shorter compression time, more elastic recovery, weaker bond. Match speed to plasticity. Typical: thin fabric frictioning 5–25 m/min; heavy canvas 30–40 m/min.

14. Fabric Coating Defects and Causes

Abstract

This study investigates why two identical extruder lines producing tire tread components show different quality consistency. We compared production batch sizes, die opening ratios, extrusion discharge and take-off temperatures, and line quality control methods. Results show that increasing batch sizes, optimizing operating methods, and setting proper plasticization temperatures can effectively improve tread yield.

Keywords: tire; tread extrusion; quality control; batch size; extrusion line; die opening ratio; plasticization temperature

Introduction

In tire manufacturing, mixed compound is plasticized in a screw extruder and discharged through a precision die, then stretched, cooled, and cut on a line to produce tread components. The tread is a critical tire component, and its quality control is a key focus of the semi-finished goods process — directly affecting material reuse rates.

Many factors cause tread quality variation: extrusion temperature fluctuations cause dimensional instability; die opening ratios and operating methods also contribute. The company runs two tread lines using hot-feed extruders from the same manufacturer and model (designated Line 1 and Line 2), but they show noticeably different quality consistency. This paper compares them to identify root causes and improve yield.

1. Measuring Tread Quality Variation

Using off-the-road (OTR) tire tread compound as an example, Table 1 shows the quality distribution and yield for both lines.

Table 1: Tread weight distribution and yield comparison

Note: Total pieces — Line 1: 3,098; Line 2: 4,278.

At ±100 g tolerance, Line 1 yield was only 49.81% vs. Line 2 at 77.35%. At ±200 g: 81.25% vs. 96.19%. At ±300 g: 95.26% vs. 99.98%. Both lines produce normally distributed weight, but Line 2’s distribution is much tighter; Line 1 is more spread out.

2. Root Cause Analysis

2.1 Production Batch Size

Table 2: Recent batch sizes per line (pieces)

Line 1 averaged only 245 pieces per batch, with 4 of 10 batches under 100 pieces. Frequent small batches mean more start-up and end pieces that must be reworked, increasing weight variation. Line 2 averaged 458 pieces (1.87× Line 1), with every batch over 100 pieces — fewer changeovers and less rework material.

2.2 Die Opening Ratio

Final die shoulder-width opening ratios were 89.4% (Line 1) vs. 88.4% (Line 2). Cross-sectional area was 6,987.2 mm² (Line 1) vs. 6,952.4 mm² (Line 2); piece weight was 24.39 kg vs. 24.29 kg — essentially identical. Pre-die and base compound dimensions also matched closely. Die opening ratio is not the cause of the quality difference.

2.3 Extrusion Discharge and Take-Off Temperature

Compound supply temperature was 104.5°C for both lines (within spec). But the feed throat discharge temperature was 84.8°C (Line 1) vs. 88.8°C (Line 2) — a 4°C gap. Line 1 is older equipment with poorer temperature control; lower compound temperature means less uniform plasticization, contributing to variation.

Table 3: Temperature comparison (°C)

Line 2’s cooling water is 16.1°C colder on average, and take-off temperature is 6.1°C lower. Better cooling means more stable shrinkage and consistent piece weight.

The 300 mm cut-length shrinkage test showed: Line 1 tread shrank to 281 mm (5.7% shrinkage); Line 2 tread shrank to 245 mm (18.3% shrinkage) — 12.6% more. Higher shrinkage on Line 2 allows the component to stabilize fully on the line, resulting in less weight variation.

2.4 Line Parameters

Screw speeds and line speeds were nearly identical on both lines. The key difference was shrinkage control: Line 1 had a 300 mm shrinkage of only 5.7% vs. Line 2 at 18.3%. Cut length variation was 14 mm (Line 1) vs. just 5 mm (Line 2).

Line 1 operators can adjust weigh-bridge speed, post-weigh speed, and four-zone shrinkage speed; Line 2 has an automatic floating roller before the weigh bridge that self-adjusts, removing manual control. The primary causes of Line 1’s wider distribution are smaller batch sizes and lower shrinkage rate; secondary causes are lower extrusion supply temperature and higher take-off temperature.

3. Corrective Actions

1. Increase batch sizes on Line 1 to minimize start-up/end-piece rework and weight variation.
2. Raise extruder temperature setpoints by 5–10°C at all control zones on Line 1 for more uniform compound plasticization.
3. Add cooling towers to lower cooling water temperature, reducing take-off temperature and stabilizing finished piece weight.

4. Conclusion

Comparing two identical tread extrusion lines revealed that batch size, compound plasticization temperature, and cooling efficiency are the key drivers of weight consistency. Increasing production batch sizes, raising extrusion temperatures 5–10°C, and lowering take-off temperature through better cooling dramatically improved tread yield on Line 1.