red from the vertex, then

3 . . . tan i = 2y/x.

Let the length of half the parabolic chain be called s, then

4 . . . s = x+2y squared/3x.

The following is the approximate expression for the relation between a
change [Delta]s in the length of the half chain and the corresponding
change [Delta]y in the dip:--

s+[Delta]s = x+(2/3x) {y squared+2y[Delta]y+([Delta]y) squared} =
x+2y squared/3x+4y[Delta]y/3x+2[Delta]y squared/3x,

or, neglecting the last term,

5 . . . [Delta]s = 4y[Delta]y/3x,

and

6 . . . [Delta]y = 3x[Delta]s/4y.

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

d squaredy M
--- = -- ,
dx squared 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]Mdx squared.
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(a squared-x squared), 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

The radius of curvature of the beam at D is given by the relation

R = EI/M.

[Illustration: FIG. 72.]

37. _Graphic Method of finding Deflection._--Divide the span L into any
convenient number n of equal parts of length l, so that nl = L; compute the
radii of curvature R_1, R_2, R_3 for the several sections. Let measurements
along the beam be represented according to any convenient scale, so that
calling L_1 and l_1 the lengths to be drawn on paper, we have L = aL_1; now
let r_1, r_2, r_3 be a series of radii such that r_1 = R_1/ab, r_2 =
R_2/ab, &c., where b is any convenient constant chosen of such magnitude as
will allow arcs with the radii, r_1, r_2, &c