# Bollard Force

## Rope friction - load and effort force in a rope

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Bollards is common on quays and are used when mooring ships and boats.

The effort force in a rope can be calculated

S = F e^{-μα}(1)

where

S = effort force in the rope (N)

F = load (N)

e = 2.718..

μ = friction coefficient (approximately 0.3 - 0.5 is common for a rope around a steel or cast iron bollard)

α = angle where the rope is in contact with the bollard (radians)

### Angle - turns, degrees and radians

- 1/4 turn :
*90 degrees => α = 1/2 π* - 1/2 turn :
*180 degrees => α = π* - 1 turn :
*360 degrees => α = 2π* - 2 turns :
*720 degrees => α = 4π*

### Example - A rope with one turn around the bollard

With a friction coefficient of *0.5* the effort force in the rope can be calculated as

S = F e^{-0.5 2π}

=0.043 F(N)

As we can see - one turn around the bollard reduces the required effort force
to less than *5%* of the load.

### Example - Shoring a ship

The retardation (negative
acceleration) of a ship arriving at quay with velocity *0.05 m/s* and
stopped within *2 seconds*, can be calculated as

a = dv/dt (2)

= (0.05 m/s) / (2 s)

=0.025(m/s^{2})

With a mass of *20000 kg* the required
force F in the rope from the ship can be calculated as

F = m a (3)

= (20000 kg) (0.025 m/s^{2})

=500N

The required effort force with a half turn around a bollard with a friction coefficient of 0.4 can be calculated as

S = F e^{-0.5 2π}

= (500 N) e^{-0.4 π}

=142(N)^{ }

The Effort Force - Load Force ratio for various rope angles are indicate in the chart below:

- friction coefficient
*0.5*

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