1550078481-Ordinary_Differential_Equations__Roberts_

308

Load D iagram

A.

L

p

B.

d

c.

D. ~

E. ~ 211

Ordinary Differential Equations

Load, w(x)

r

x=JL

P , x=L

r

x=Jd

P , x=d

wo, a constant

x

wo(l - - )

L

wax

L

Bending Moment , M(x)

-P(L - x)

{

-P(d-x), O:::;x:::;d

0 , d < x:::; L

x2

• wo[L(L - x) + - ]
  2

L^2 L x x^2 x^3

-wo(- - - + - - - )

6 2 2 6L

L^2 Lx x^3

-wo(- - - +-)

3 2 6L

Figure 6.13 Bending Moments for Various Loads on a Cantilevered Beam

of Length L

An ideal column is a long, slender elastic rod which is held vertically in

place by a support at the b ase and often by an addit ional support at the top.

Suppose a weight is placed on the top of the column, thus inducing a force P

on bot h ends of the column. If P is sufficiently small , then the column will

support the weight and the column will deflect only slightly from the vertical.

However, if Pis sufficiently large, then t he column cannot support the weight

and will buckle-that is, the column will suddenly bow out from t he vertical

with large amplitude.

Consider t he column oflength L shown in Figure 6.14. Assume the column

is constrained to move in the plane. Let t he origin be located at t he base of the

column with the positive x-axis directed vertically upward and the positive

y-axis directed to the right. So y( x) represents the lateral displacement of the

column. In the eighteenth century, Euler showed t hat a column carrying a

load P satisfies the fourth-order linear differential equation

(48)

p (2)
y(4) +- Y - =O
E I
where Eis Young's modulus and I is the moment of the area of the column's
cross section. (Here I is assumed to be a positive constant.) Notice that