How do I estimate ball bearing life in hours from the dynamic load rating and the applied load?

C is not a working load. It is the load at which 90 percent of a large group of identical bearings would survive one million revolutions before the first signs of rolling-contact fatigue. The basic rating life follows from it:

L10 = (C / P)^p in millions of revolutions, with p = 3 for ball bearings and 10/3 for roller bearings, where P is the equivalent dynamic load.

For your case, (13 / 1.5)³ = 651 million revolutions. In hours, L10h = 10^6 × L10 / (60 × n) = 651 × 10^6 / (60 × 1450) = about 7500 hours, or about 935 working days at eight hours a day.

The cube makes life extremely sensitive to load. Doubling the load to 3 kN divides the life by eight, to about 935 hours. A 10 percent underestimate of the load costs 25 percent of the life (1.1³ = 1.33).

Turn the formula round to select a bearing. Machines in one-shift industrial use are commonly designed for 20,000 to 30,000 hours. For 20,000 hours at 1450 rpm you need 20,000 × 60 × 1450 / 10^6 = 1740 million revolutions, so C / P = 1740^(1/3) = 12.0 and C = 12.0 × 1.5 = 18 kN. That points to the next bearing series or the next bore size up.

Make sure P is right before trusting the result. For a purely radial load P equals the radial force. With an axial component, P = X × F_r + Y × F_a, with the factors X and Y taken from the catalogue according to the ratio of axial load to C0. For a belt drive, the radial load on the bearing is the vector sum of both belt strand tensions multiplied by the lever ratio of the overhang, which is usually well above the force you would work out from the transmitted torque.

The static rating C0 is a separate check for shock loads and for bearings that are loaded while stationary or turning slowly: keep the largest load well below C0 to avoid permanently denting the raceways.

Keep in mind what L10 means statistically. It is the life that 90 percent of bearings reach, so one in ten fails earlier, while the median life is several times longer. Where a higher reliability is required, the rating life standard applies a reduction factor to L10.

In practice most bearings never reach their fatigue life. They fail from contamination, lost or degraded lubricant, misalignment, damage during fitting (pressing on the wrong ring sends the fitting force through the balls) or electrical current passing through them. The modified rating life in the bearing standard includes factors for lubrication and cleanliness, and for a greased-for-life bearing the grease life can be the shorter of the two.

If the load varies over the duty cycle, do not use the peak or the simple average. Because of the cube law the correct equivalent is the cubic mean: P_m = (sum of P_i³ × time fraction_i)^(1/3). For 3 kN during 20 percent of the running time and 1 kN during the rest, P_m = (27 × 0.2 + 1 × 0.8)^(1/3) = 6.2^(1/3) = 1.84 kN, much closer to the peak than the 1.4 kN arithmetic average.

At the other extreme, a bearing needs a certain minimum load so the balls roll instead of skidding. A heavily oversized bearing running fast under almost no load can wear out from sliding damage; catalogues give the minimum load.