From these equations the deflection produced by any given stress on the
chains or by a change of temperature can be calculated.
[Illustration: FIG. 71.]
36. _Deflection of Girders._-- Let fig. 71 represent a beam bent by
external loads. Let the origin O be taken at the lowest point of the bent
beam. Then the deviation y = DE of the neutral axis of the bent beam at any
point D from the axis OX is given by the relation
d²y M
--- = -- ,
dx² EI
where M is the bending moment and I the amount of inertia of the beam at D,
and E is the coefficient of elasticity. It is usually accurate enough in
deflection calculations to take for I the moment of inertia at the centre
of the beam and to consider it constant for the length of the beam. Then
dy 1
-- = ---[Integral]Mdx
dx EI
1
y = ---[Integral][Integral]Mdx².
EI
The integration can be performed when M is expressed in terms of x. Thus
for a beam supported at the ends and loaded with w per inch length M =
w(a²-x²), where a is the half span. Then the deflection at the centre is
the value of y for x = a, and is
5 wa^4
[delta] = --- ----.
24 EI