Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 197

From these the bending stress of the tape with the thickness d is

2 q E d

d

2 y 00 ¼ E d

sinh xx

r b ¼ E

sinh xx 0 :

ð 3 : 22 Þ

D

With x = x 0 in ( 3.21 ), the lever-arm y 0 is

D x 2 ¼ 2 E J

2

y 0 ¼ : ð 3 : 23 Þ

D S

The boundary angle # 0 on the contact point can also be determined with ( 3.21 )

tan # 0 ¼ y 0 ð x 0 Þ ¼ 2

D x 1 tanh xx 0 :

For x x 0 C 2.5 is tanh xx 0 & 1 with a failure smaller than 1 %. This con-
dition is fulfilled for all practical applications. With this the boundary angle is

2
D x ¼ arctan

2

# 0 ¼ arctan q

S
E J

: ð 3 : 24 Þ

D

The contact force Q is given with ( 3.14 ) and the boundary angle from ( 3.24 ).
This angle is very small; so it is nearly tan # 0 & sin # 0 and the contact force is

D

p

Q ¼ 2 S E J :

The contact angle # c for the contact bow between tape and sheave is

# c ¼ # D 2 # 0

with # D for the tape deflection angle.

Line Pressure Between Tape and Sheave

Presupposing that the tape between both of the contact points is bending limp in
the contact bow, the line pressure q (length related contact force) between the tape
and the sheave can be derived using the known tensile force F. In Fig. 3.15 , the
force F works on both cross-sections of the tape element. On the inner side,
the force dQ exists. Figure 3.16 shows the diagram of these forces. Therefore the
equation for these forces is