Rolling Thrust Bearings: Selection and Calculation: Worked Example

Worked Example: Tilting Pad Thrust Bearing

Problem: Design a tilting pad thrust bearing for 70,000 pounds thrust at 3,600 rpm. Shaft diameter is 6.5 inches; maximum OD available is 15 inches. Oil inlet temperature is 110°F, supply pressure is 20 psi. Maximum temperature rise of 50°F is acceptable, resulting in a viscosity of 18 centipoises. Use $c = 3.5$ Btu/gal/°F.

Step 1 — Inside diameter: $D_1 = 7$ inches (to clear shaft).

Step 2 — Outside diameter: Given maximum $D_2 = 15$ inches.

Step 3 — Radial pad width:

$$a = \frac{15 - 7}{2} = 4 \text{ inches}$$

Step 4 — Pitch-line circumference:

$$B = \pi \cdot \frac{7 + 15}{2} = 34.6 \text{ inches}$$

Step 5 — Number of pads:

$$i = \frac{34.6 \times 0.8}{4} = 6.9$$

Select $i = 6$.

Step 6 — Length of pad:

$$b = \frac{34.6 \times 0.8}{6} = 4.61 \text{ inches}$$

Make $b = 4.75$ inches.

Step 7 — Pitch-line velocity:

$$U = \frac{34.6 \times 3{,}600}{12} = 10{,}400 \text{ ft/min}$$

Step 8 — Bearing unit load:

$$p = \frac{70{,}000}{6 \times 4 \times 4.75} = 614 \text{ psi}$$

614 psi exceeds the "normal" 200 psi and even exceeds the "maximum" 500 psi from the load table. This is a heavily loaded bearing, operating at the outer edge of its capability. In practice, this is achievable with tilting pads because of their ability to optimize film geometry dynamically — but it demands careful thermal management.

Step 9 — Operating number:

$$O = \frac{1.45 \times 10^{-7} \times 18 \times 10{,}400}{5 \times 614 \times 4.75} = 1.86 \times 10^{-6}$$

Step 10 — Minimum film thickness:

From the empirical curve at $O = 1.86 \times 10^{-6}$ and $b/a = 4.75/4 \approx 1.19$:

$$\alpha = 0.30 \times 10^{-3}$$

$$h_{min} = 0.00030 \times 4.75 = 0.0014 \text{ inches}$$

✅ This exceeds 0.001 inch (acceptable for this size range).

Step 11 — Coefficient of friction:

From the empirical curve at $\alpha = 0.30 \times 10^{-3}$: $f = 0.0036$

Step 12 — Friction power loss:

$$P_f = \frac{0.0036 \times 70{,}000 \times 10{,}400}{33{,}000} = 79.4 \text{ hp}$$

79.4 hp — notable that this is lower than the tapered land design's 91 hp for the same load and speed. The tilting pad's optimized film geometry reduces friction.

Step 13 — Actual oil flow:

$$Q = 0.0591 \times 0.30 \times 10^{-3} \times 6 \times 4 \times 4.75 \times 10{,}400 = 21.02 \text{ gpm}$$

Step 14 — Temperature rise:

$$\Delta t = \frac{0.0217 \times 0.0036 \times 614}{0.30 \times 10^{-3} \times 3.5} = 45.7°F$$

The Final Verdict

$$\Delta t = 45.7°F < 50°F \quad \checkmark$$

The design is satisfactory.

Tilting Pad Design Summary

Head-to-Head: Tapered Land vs. Tilting Pad for 70,000 lb at 3,600 rpm

Both examples above were designed for the exact same application. Here's what the practitioner discovered when he compared them side by side:

the practitioner's Conclusion

The tapered land bearing had a thicker film and lower unit load — on paper, it looked like the safer choice. But it was also 17 inches in diameter versus the tilting pad's 15 inches, and it had zero tolerance for the foundation settlement that ultimately caused the failure.

The tilting pad bearing:

• Fit within the available 15-inch envelope
• Ran 12 hp cooler (79.4 vs. 91 hp)
• Operated 4.3°F below the thermal limit (45.7°F vs. 50°F)
• And most critically, could absorb the misalignment that destroyed the tapered land bearing

The tilting pad costs more upfront. It saves everything downstream.

Guide Bearings — Positioning Without Rotation

What They Are

Guide bearings are a fundamentally different application of the sliding bearing principle. Instead of supporting rotating loads, they guide linear motion — like the ways of a machine tool, the slides of a press, or the tables of milling machines.

    Typical Guide Bearing Configurations

    ┌──────────────────┐    ┌──────────────────┐
    │      Table       │    │      Table       │
    │  ┌────────────┐  │    │  ╲            ╱  │
    │  │            │  │    │    ╲        ╱    │
    │  └────────────┘  │    │      ╲    ╱      │
    │      Bed         │    │       Bed        │
    └──────────────────┘    └──────────────────┘
     Flat Way                V-Way (Dovetail)

    ┌──────────────────┐    ┌──────────────────┐
    │     Slide        │    │      Table       │
    │  ╱            ╲  │    │  ┌──┐      ┌──┐  │
    │╱                ╲│    │  │  │      │  │  │
    │      Bed         │    │  └──┘      └──┘  │
    └──────────────────┘    │       Bed        │
     Inverted V-Way          Combination Way

Operating Characteristics

Guide bearings normally operate in the boundary lubrication region. This means there is no full hydrodynamic film — the surfaces are in partial contact, relying on the lubricant's chemical properties rather than its pressure-generating capability.

Common lubrication methods for guide bearings:

• Dry — no lubricant at all (used with self-lubricating materials)
• Dry film — molybdenum disulfide (MoS₂) or tetrafluoroethylene (TFE/Teflon)
• Grease — for intermittent or slow-speed motion
• Oil — for continuous or moderate-speed applications
• Gaseous — air bearings for ultra-precision applications

Hydrostatic Guide Bearings

For applications demanding the highest precision, hydrostatic lubrication transforms the guide bearing from a boundary-lubricated slider into a virtually frictionless, zero-wear device.

How it works: External pumps supply air or oil under pressure to pockets machined into the bearing surface. This pressurized fluid creates a complete separation between the sliding surfaces — eliminating metal-to-metal contact entirely.

Benefits of hydrostatic guide bearings:

• Improved performance — friction drops dramatically
• Reduced wear — complete surface separation means no contact wear
• Increased stability — the pressurized film acts as a damper
• Sub-micron positioning capability — essential for precision machine tools

The trade-off: Hydrostatic systems require pumps, pressure regulators, filtration, and plumbing. They add cost and complexity, but for applications where precision justifies the investment, they are unmatched.

The Selection Decision Matrix — Which Bearing Type Do You Need?

Here's the decision framework the practitioner now uses for every new application:

The Decision Flow

Ask yourself these questions in order:

1. Is the load below 75 psi and primarily for positioning? → Flat Plate. Stop here.

2. Is the bearing small (< 3-inch OD) with moderate loads? → Step bearing. Lowest cost for the capability.

3. Is alignment well-controlled and guaranteed for the life of the machine? → Tapered Land. Best balance of cost and performance for large bearings with good alignment.

4. Is misalignment possible, likely, or unknown? → Tilting Pad. Pay the premium. Sleep well.

Critical Design Rules That Apply to All Types

Rule 1: Temperature Rise Must Not Exceed 50°F

This isn't a suggestion. Exceeding 50°F of temperature rise through the bearing degrades the oil, reduces viscosity below safe operating levels, and initiates a thermal runaway cycle: higher temperature → thinner film → more friction → even higher temperature.

$$\Delta t_{max} = 50°F$$

Rule 2: Film Thickness Targets

Rule 3: Oil Flow Adequacy

Always compare your required oil flow ($Q$) to your actual film flow ($Q_F$):

• If $Q_F \geq Q$ → The bearing is thermally self-sufficient. No chamfers needed.
• If $Q_F < Q$ → You need chamfers, increased taper, or external oil supply to bridge the gap.

Rule 4: Pad Proportions Matter

For step and flat plate bearings, the optimum geometry is a square pad: $a = b$. This maximizes the hydrodynamic film generation per unit of bearing area.

Rule 5: Always Use Even Numbers of Pads

This ensures balanced loading around the circumference. An odd number of pads creates an asymmetric force distribution that can excite vibration.

Your Takeaway: The Formulas You'll Use Most Often

Quick-Reference Formula Sheet

Outside Diameter (all types):

$$D_2 = \left(\frac{4W}{\pi K_g p} + D_1^2\right)^{1/2}$$

Pitch-Line Velocity:

$$U = \frac{B \cdot N}{12}$$

Required Oil Flow:

$$Q = \frac{42.4 \cdot P_f}{c \cdot \Delta t}$$

Step Bearing Film Thickness:

$$h = \frac{2.09 \times 10^{-9} \cdot i \cdot a^3 \cdot U \cdot Z}{W}$$

Step Bearing Power Loss:

$$P_f = \frac{7.35 \times 10^{-13} \cdot i \cdot a^2 \cdot U^2 \cdot Z}{h}$$

Tilting Pad Operating Number:

$$O = \frac{1.45 \times 10^{-7} \cdot Z_2 \cdot U}{5pb}$$

Tilting Pad Power Loss:

$$P_f = \frac{f \cdot W \cdot U}{33{,}000}$$

Tilting Pad Temperature Rise:

$$\Delta t = \frac{0.0217 \cdot f \cdot p}{\alpha \cdot c}$$

Tapered Land Film Thickness Factor:

$$K = \frac{5.75 \times 10^6 \cdot p}{U \cdot Y_L \cdot Z}$$

Tapered Land Film Flow:

$$Q_F = \frac{8.9 \times 10^{-4} \cdot i \cdot \delta_2 \cdot D_2^3 \cdot N \cdot Y_G \cdot Y_S^2}{D_2 - D_1}$$

Your Next Step

Pick one rotating machine in your plant or your current design project. Pull the thrust bearing specification.

Now ask yourself three questions:

1. What type of thrust bearing is installed?
2. What are the actual alignment conditions — not the ideal ones, the real ones?
3. If the alignment degraded by 0.003 inches per inch, would the bearing survive?

If the answer to question 3 makes you uncomfortable, you've just found your next engineering upgrade.

The best time to change a thrust bearing type is during the design phase. The second-best time is before the next failure.

This guide is based on established bearing engineering principles from authoritative references including Wilcock and Booser's "Bearing Design and Applications" (McGraw-Hill). All design procedures, formulas, and load ratings reflect accepted engineering practice applicable across industries and time periods.

The Complete Guide to Ball and Roller Thrust Bearing Selection, Rating Life, and Application Engineering

What Makes a Thrust Bearing a "Rolling Type"?

Before diving into the specific types, you need to understand what separates rolling-type thrust bearings from their hydrodynamic (plain) counterparts.

Rolling contact bearings substitute a rolling element — ball or roller — for a hydrodynamic or hydrostatic fluid film to carry an impressed load without wear and with greatly reduced friction. Because of their dramatically reduced starting friction compared to conventional journal bearings, they've earned the common designation of "anti-friction" bearings.

The key advantages of rolling-type thrust bearings over plain thrust bearings:

• Starting friction is low — no need to build a hydrodynamic film before load capacity develops
• Less axial space required — compact design for tight packaging constraints
• Both radial and axial loads can be carried by certain thrust bearing types
• Lubrication is simple — many configurations operate with sealed grease
• Replacement is relatively easy — standard interchangeable dimensions per AFBMA standards
• Heavy overloads can be carried momentarily — the rolling elements distribute stress across hardened raceways
• Design assistance is available from bearing supplier engineers for complex applications

Currently, multiple manufacturers produce a complete range of ball and roller thrust bearings in fully interchangeable series with standard dimensions, tolerances, and fits as specified in Anti-Friction Bearing Manufacturers Association (AFBMA) Standards.

Key Insight: Balls and rollers in these bearings are held to diametral tolerances of 0.0001 inch or less within a single bearing. This precision is essential to performance, durability, limiting runout, providing proper clearances, and ensuring smoothness of operation.

Failure trigger and engineering context

the practitioner's mistake was common. He'd inherited a machine designed by someone who viewed thrust bearings as interchangeable commodities. But thrust bearings are designed to handle thrust loads alone or in combination with radial loads — and the specific type determines which combination is permissible.

Here's what the practitioner should have known: five choices must be made when selecting any ball or roller bearing:

1. The bearing series
2. The type of bearing
3. The size of bearing
4. The method of lubrication
5. The type of mounting

These considerations are modified by anticipated operating conditions, expected life, cost, and overhaul philosophy.

When the practitioner's predecessor selected a flat-plate hydrodynamic thrust bearing for a position that experienced combined axial and radial loading with periodic shock loads, every one of those five choices was wrong.

Let's make sure you never repeat that mistake.

Types of Ball Thrust Bearings

Ball thrust bearings are the foundation of the rolling-type thrust bearing family. They use hardened steel balls running between grooved or flat washer raceways to transmit axial loads.

One-Direction Ball Thrust (Type TA)

Configuration: A shaft ring and a flat or spherical housing ring with a single row of balls between them.

Load Capability:

• Pure thrust loads in one direction only ✅
• Radial loads ❌ — Cannot carry any radial load whatsoever

AFBMA Symbol: TA (single direction, grooved raceways, flat seats)

When to Use: Applications where the axial load acts consistently in one direction and no radial component exists. Think vertical shaft positioning, preloading mechanisms, and simple axial locating devices.

Critical Limitation: If your application has even minor radial loads, a one-direction ball thrust bearing will fail prematurely. The balls will be forced to slide laterally across the raceway rather than roll, generating heat, wear, and eventual seizure.

the practitioner's Lesson: The cartoning machine shaft experienced periodic radial loads from belt tension changes. A one-direction ball thrust bearing would have failed just as catastrophically as the plain bearing — for different reasons, but with the same result.

Two-Direction Ball Thrust (Type TDA)

Configuration: A shaft ring with a ball groove machined on both sides, two separate sets of balls, and two housing rings — arranged so thrust loads in either direction can be supported.

Load Capability:

• Thrust loads in both directions ✅
• Radial loads ❌ — Still cannot carry any radial load

AFBMA Symbol: TDA (double direction, washers with grooved raceways, flat seats)

When to Use: Applications where the shaft must be axially located in both directions but no radial force component exists. Common in vertical machines with reversing thrust, screw jacks, and certain indexing mechanisms.

Design Note: The two-direction design doubles the axial envelope compared to a single-direction bearing. If space is constrained, consider whether two single-direction bearings mounted back-to-back might offer a more compact solution — though this introduces alignment complexity.

Additional Ball Thrust Variants

The Ball Thrust Quick-Decision Matrix

Types of Roller Thrust Bearings

When loads exceed what ball thrust bearings can handle — or when the application demands higher stiffness and longer life under heavy loads — roller thrust bearings take over. Roller designs offer significantly higher load capacity for a given envelope size because the line contact between rollers and raceways distributes stress over a much larger area than the point contact of balls.

Cylindrical Roller Thrust (Types TP, TPC, TR)

Configuration: Straight cylindrical rollers arranged between flat washer raceways. Several arrangements of housing and shaft washers are available.

Load Capability:

• Very high thrust loads ✅
• Radial loads ❌ — Pure thrust only
• High stiffness ✅ — Minimal axial deflection under load

AFBMA Symbols:

When to Use: Heavy-duty applications requiring maximum thrust capacity in minimum space — rolling mills, heavy presses, large gearboxes, and ship propeller thrust bearings.

Critical Design Note: Cylindrical roller thrust bearings are sensitive to misalignment. Because the rollers are straight and contact the flat raceway along their full length, even small angular errors between shaft and housing create edge loading that dramatically reduces life. The TR variant addresses this with an aligning washer, but at the cost of added complexity and axial space.

Spherical Roller Thrust

Configuration: Similar in design to the radial spherical roller bearing, but with a much larger contact angle. The rollers are barrel-shaped with one end smaller than the other.

Load Capability:

• Very high thrust load carrying capacity ✅
• Can also carry radial loads ✅ — This is the critical differentiator
• Self-aligning ✅ — Tolerates significant misalignment

AFBMA Symbol: TS (single direction, aligning flat seats, spherical rollers)

When to Use: This is the bearing the practitioner should have specified. Spherical roller thrust bearings are designed for applications with:

• Combined axial and radial loading
• Shaft deflection or misalignment between shaft and housing
• Heavy loads with potential shock components
• Situations where a self-aligning capability is essential

Why This Type Stands Alone: The barrel-shaped rollers with their asymmetric profile create a natural self-aligning action against the spherical raceway. This means the bearing continues to perform at rated capacity even when the shaft deflects under load — a condition that would destroy a cylindrical roller thrust bearing.

Engineering Reality Check: If your application has both thrust and radial loads AND you cannot guarantee perfect alignment, the spherical roller thrust bearing is almost certainly your answer. It trades some maximum thrust capacity for versatility that prevents the kind of failure the practitioner experienced.

Tapered Roller Thrust (Type TT)

Configuration: Tapered rollers arranged between conical raceways. Multiple arrangements of housing and shaft washers are used.

Load Capability:

• High thrust capacity ✅
• Can carry combined radial and thrust loads ✅
• Separable design ✅ — Simplifies mounting and maintenance

AFBMA Symbol: TT (thrust bearings)

When to Use: Applications requiring the separability of a tapered design with combined load capability. The tapered roller geometry naturally handles both axial and radial force components because the contact angle generates both thrust and radial reactions.

Tapered Roller Bearing Configurations (Radial-Thrust Types):

Speed Limitation: The roller-end thrust-flange contact in tapered roller bearings creates friction that limits both the speed they can endure and the thrust load they can carry. Reference the manufacturer's catalog before making selections for high-speed applications.

The Complete Roller Thrust Bearing Comparison

What Is Rating Life?

The Rating Life $L_{10}$ of a group of apparently identical bearings is the life in millions of revolutions that 90 percent of the group will complete or exceed. For a single bearing, $L_{10}$ also refers to the life associated with 90 percent reliability.

This is a statistical concept. It does not mean the bearing will fail at $L_{10}$ revolutions — it means there's a 10% probability of fatigue failure before reaching that life.

Thrust Ball Bearing Rating Life

The Rating Life $L_{10}$ in millions of revolutions for a thrust ball bearing is:

$$L_{10} = \left(\frac{C_a}{P_a}\right)^3$$

Where:

• $C_a$ = basic load rating (newtons or pounds)
• $P_a$ = equivalent thrust load (newtons or pounds)

For single row thrust ball bearings with balls ≤ 25.4 mm (1 inch) diameter:

$$C_a = f_c \cdot Z^{2/3} \cdot D^{1.8} \quad (\text{when } \alpha = 90°)$$

$$C_a = f_c \cdot (\cos\alpha)^{0.7} \cdot Z^{2/3} \cdot D^{1.8} \cdot \tan\alpha \quad (\text{when } \alpha \neq 90°)$$

For balls > 25.4 mm (1 inch) diameter:

$$C_a = 3.647 \cdot f_c \cdot (\cos\alpha)^{0.7} \cdot Z^{2/3} \cdot D^{1.4} \quad (\text{metric})$$

Where:

• $f_c$ = factor depending on geometry, accuracy, and material
• $Z$ = number of balls per row
• $D$ = ball diameter (mm or inches)
• $\alpha$ = nominal contact angle (degrees)

Equivalent Thrust Load for Ball Thrust Bearings

For thrust ball bearings with $\alpha \neq 90°$ under combined constant thrust and constant radial loads:

$$P_a = X \cdot F_r + Y \cdot F_a$$

Where:

• $F_r$ = applied radial load
• $F_a$ = applied axial load
• $X$ = radial load factor
• $Y$ = axial load factor

For $\alpha = 90°$: $F_r = 0$ and $Y = 1$ (pure thrust only).

X and Y Values for Thrust Ball Bearings:

For single direction bearings when $F_a/F_r > e$: Use $X$ and $Y$ values from the table.

For single direction bearings when $F_a/F_r \leq e$: Use $X = 1.18$ (45°), $X = 1.90$ (60°), $X = 3.89$ (75°) and $Y = 0.59$, $0.54$, $0.52$ respectively.

Thrust Roller Bearing Rating Life

The Rating Life $L_{10}$ in millions of revolutions for a thrust roller bearing is:

$$L_{10} = \left(\frac{C_a}{P_a}\right)^{10/3}$$

Critical Difference: Note the exponent is $10/3$ for roller bearings versus $3$ for ball bearings. This reflects the different stress distribution characteristics of line contact (rollers) versus point contact (balls).

For single row thrust roller bearings:

$$C_a = f_c \cdot l_{eff}^{7/9} \cdot Z^{3/4} \cdot D^{29/27} \quad (\text{when } \alpha = 90°)$$

$$C_a = f_c \cdot (l_{eff} \cdot \cos\alpha)^{7/9} \cdot Z^{3/4} \cdot D^{29/27} \cdot \tan\alpha \quad (\text{when } \alpha \neq 90°)$$

Where:

• $f_c$ = geometry/accuracy/material factor (from AFBMA tables)
• $l_{eff}$ = effective roller contact length (mm or inches)
• $Z$ = number of rollers in a single row
• $D$ = roller diameter (mm or inches)
• $\alpha$ = nominal contact angle (degrees)

Equivalent Thrust Load for Roller Thrust Bearings

For thrust roller bearings with $\alpha \neq 90°$:

$$P_a = X \cdot F_r + Y \cdot F_a$$

X and Y Values for Thrust Roller Bearings:

Where $e = 1.5\tan\alpha$

For $\alpha = 90°$: $F_r = 0$ and $Y = 1$ (pure axial load only).

Multi-Row Thrust Roller Bearings

For thrust roller bearings with two or more rows of rollers carrying loads in the same direction:

$$C_a = \left(Z_1 l_{eff1} + Z_2 l_{eff2} + \dots + Z_n l_{effn}\right) \cdot \left[\left(\frac{Z_1 l_{eff1}}{C_{a1}}\right)^{9/2} + \left(\frac{Z_2 l_{eff2}}{C_{a2}}\right)^{9/2} + \dots + \left(\frac{Z_n l_{effn}}{C_{an}}\right)^{9/2}\right]^{-2/9}$$

Where:

• $Z_1, Z_2, \dots Z_n$ = number of rollers in respective rows
• $C_{a1}, C_{a2}, \dots C_{an}$ = basic load rating per row
• $l_{eff1}, l_{eff2}, \dots l_{effn}$ = effective roller contact length per row

Multi-Row Thrust Ball Bearings

For thrust ball bearings with multiple rows:

$$C_a = \left(Z_1 + Z_2 + \dots + Z_n\right) \cdot \left[\left(\frac{Z_1}{C_{a1}}\right)^{10/3} + \left(\frac{Z_2}{C_{a2}}\right)^{10/3} + \dots + \left(\frac{Z_n}{C_{an}}\right)^{10/3}\right]^{-0.3}$$

The $f_c$ Factor: Where Theory Meets Manufacturing Reality

The $f_c$ factor is the bridge between theoretical bearing geometry and real-world manufacturing quality. It depends on:

• The geometry of the bearing components
• The accuracy to which the parts are made
• The material properties

Selected $f_c$ Values for Thrust Roller Bearings (α = 90°):

Metric values yield $C_a$ in newtons when $l_{eff}$ and $D$ are in mm. Inch values yield $C_a$ in pounds when $l_{eff}$ and $D$ are in inches.

For angled contact thrust roller bearings ($\alpha \neq 90°$), the $f_c$ values depend on the ratio $D\cos\alpha / d_m$ and the contact angle range. Representative values:

Important: When rollers are longer than $2.5D$, a reduction in the $f_c$ value must be anticipated due to excessive slip in the roller-raceway contact. Consult the bearing manufacturer for adjusted ratings.

Static Load Rating: The Non-Rotating Criterion

Why Static Load Rating Matters

A static load is a load acting on a non-rotating bearing. Permanent deformations appear in balls or rollers and raceways under static loads of moderate magnitude and increase gradually with increasing load.

For ball and roller bearings manufactured from hardened alloy steel, deformations occurring under maximum contact stress of 4,000 megapascals (580,000 psi) at the center of contact do not greatly impair smoothness or friction.

Static Load Rating for Thrust Roller Bearings

$$C_{oa} = 220 \left(1 - \frac{D\cos\alpha}{d_m}\right) \cdot Z \cdot l_{eff} \cdot D \cdot \sin\alpha \quad (\text{metric — newtons})$$

$$C_{oa} = 32{,}150 \left(1 - \frac{D\cos\alpha}{d_m}\right) \cdot Z \cdot l_{eff} \cdot D \cdot \sin\alpha \quad (\text{inch — pounds})$$

Static Equivalent Load for Thrust Ball Bearings

For thrust ball bearings with $\alpha \neq 90°$ under combined radial and thrust loads:

$$P_{oa} = F_a + 2.3 \cdot F_r \cdot \tan\alpha$$

This formula is valid for all load directions in double direction bearings. For single direction bearings, it is valid where $F_r/F_a \leq 0.44\cot\alpha$.

For thrust ball bearings with $\alpha = 90°$ (pure axial): $P_{oa} = F_a$