Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 320

and summarized

Q H ¼ z S 1 z 1
2

z DS : ð 3 : 115 Þ

When the rope leaves the tackle block, using this and ( 3.113 ), the rope tensile
force is

S z ¼ Q H

z z 1

DS : ð 3 : 116 Þ

2

This equation was also developed by Matthias ( 1972 ).
The rope tensile force will be reduced further when the wire rope runs over
n stationary sheaves. The rope tensile force is then

S A ¼ S z n DS : ð 3 : 117 Þ

and using ( 3.116 )

S A ¼ Q H

z z 1

DS n DS : ð 3 : 118 Þ

2

From this equation, the minimum weight force of the hook block is

Q H ¼ z ð S A þ z 1
2

DS þ n DS Þ: ð 3 : 119 Þ

The mean loss of the rope tensile force is calculated for the mean rope force
S (as the average of the smallest S A + DS and the biggest S 1 ). The mean rope force
is

S ¼ S A þ DS þ S 1
2

: ð 3 : 120 Þ

or with ( 3.113 ) and ( 3.117 )

S ¼ S A þ z 1 þ n þ 1
2

DS ¼ S A þ z þ n
2

DS : ð 3 : 121 Þ

With ( 3.105 ), ( 3.119 ) and ( 3.121 ) and by eliminating S and DS, the minimum
mass force of the empty hook block is

c 0tot d 2 þ c 1tot S A
ð d Þ þ 1 : 33 c 1tot z þ n

z 1
2

Q H ¼ z S A þ z þ n : ð 3 : 122 Þ

2

In this equation, d is once again the nominal rope diameter, D the sheave
diameter both in mm, n the number of sheaves outside the tackle block and z the
number of bearing rope falls.