Page 27 - Handbook of Properties of Textile and Technical Fibres

8 Handbook of Properties of Textile and Technical Fibres

We can now write

2
d y MðxÞ
00
¼ y ¼ (1.6)
dx 2 EI A
The bending moment at a point x along the beam is given by the balance of the
moment generated by the force F at the end of the beam, of value Fl, and the opposing
moment due to the reaction at the fixed end, which has a value of Fx. So that

MðxÞ¼ Fx þ Fl

Then we can write from Eq. (1.6)

2
d y
2 $EI A ¼ MðxÞ¼ F$x þ F$l
dx
Integrating
dy Fx 2 dy
EI A ¼ þ Fl$x plus a constant but as x ¼ 0, ¼ 0 so the constant is zero.
dx 2 dx
Integrating again
l
3 2
dy Fx Flx
EI A ¼ þ

dx 6 2 0
plus a constant but as at x ¼ 0; y ¼ 0 the constant is zero.

3 3
Fl Fl 3 1 3
yðlÞEI A ¼ þ ¼ Fl þ
6 2 6 6
Fl 3
yðlÞEI A ¼
3
The minus sign reflects the downward deflection, which is at a distance of

Fl 3
jyj ¼
3EI A
From Eq. (1.4) the total deflection is

64Fl 3
jyj ¼ (1.7)
3EpD 4
We see then that the flexibility of a circular beam and hence a fiber is a function of
the reciprocal of the diameter to the fourth power.
Halving the diameter of a fiber increases its flexibility 16 times. This shows why a
very stiff material in the form of a fine fiber can still be extremely flexible.