Page 26 - Handbook of Properties of Textile and Technical Fibres

Introduction to the science of fibers 7

F
Center of gravity

l
x

F

Figure 1.3 A horizontal beam held at one end and subjected to a downwards force at the other
end will have a tendency to bend.

The bending moment at any point along the beam at a distance x from the fixed end
is given from Eq. (1.3) as:

E
MðxÞ¼ I A
r
Now let us look at Fig. 1.4.
The equation of the curve that describes the bending of the beam is given by

1 y 00
¼ (1.5)
r 3 2
0 2
1 þðy Þ
dy 1
For small deflections /0 so that we can write, from Eq. (1.5), that ¼ y 00
dx r

-y
1 = y″
ρ (1+(y′) ) 3/2
2

ρ

Figure 1.4 The bending of a beam fixed at one end.