1550078481-Ordinary_Differential_Equations__Roberts_

304 Ordinary Differential Equations

left end of the beam of length L and let y( x) denote the vertical downward
deflection of the beam.

Load w(x)

Loaded Position

0

Figure 6.11 Deflection of a Beam

A fundamental equation of beam theory is

(37)

y"

[1 + (y')2]3/2

-M

EI

L x

where Mis the bending moment, Eis Young's modulus and I is the moment
of area of the cross section about the neutral axis. (I is also sometimes called
the moment of inertia.) Since a beam's cross section may vary, the moment
of area, I, may not be constant. Two additional fundamental equations from
beam theory are

(38)

dM

dx = V and

dV

- = -w(x)

dx

where Vis the shearing force and w(x) is the load on the beam. The fourth-

order differential equation which results from differentiating equation (37)
twice and substituting equations (38), depends upon the assumptions made
regarding y' and I.

Case 1. If y' is not assumed to be small relative to 1 (that is, if the bending
of the beam is not small), two differentiations of (37) yield the following
nonlinear fourth-order differential equation for the deflection of the beam

(39)

y<4J p-3/2gy' y"y(3) p-5/2 3 (y")3p-5/2+ 15 (y')2(y")3 p-1 ;2 = !f_ (-M)

dx^2 EI

where F = 1 + (y')^2. The final form of the differential equation to be solved
depends upon whether the moment of area of the cross section, I , is assumed
to be constant or variable.