Page 25 - Handbook of Properties of Textile and Technical Fibres

6 Handbook of Properties of Textile and Technical Fibres

Figure 1.2 Cross-section of a
circular fiber.
δA = r•δα•δr
δr
rδα
r
δα S=r•Sin α
α

R

2 1 cos 2 a
Now sin a ¼
2
2p 1 cos 2 a R 3
R R
So that I A ¼ da r dr
0 2 0
R
a sin 2 a r
2p 4
I A ¼
2 4 4
0 0
As Sin2p ¼ 0
2p R pR
4 4
I A ¼ ¼
2 4 4
Or, if D is the fiber diameter

pD 4
I A ¼ (1.4)
64

The stiffness of the cylinder or fiber is therefore related to the fourth power of the
diameter.
To quantify the flexibility of a fiber further we can calculate the total bending of a
circular horizontal beam held at one end and loaded by a force F, as shown schemat-
ically in Fig. 1.3.
The bending moment F$l, produced by the applied force at the free end, will induce
by reaction a turning moment at the fixed end and in the opposite sense.