and
6 . . . ∆y = 3x∆s/4y.
From these equations the deflection produced by any given stress on the chains or by a change of temperature can be calculated.
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 dx² | = | M 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 dx | = | 1 EI | ∫Mdx |
| y = | 1 EI | ∫∫Mdx². |
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