Engineering Optimization: Theory and Practice, Fourth Edition

6.1 Introduction 303

Figure 6.2 Finite-element model of a cantilever beam.

of the beam(F ), which can be expressed as [6.1]

F=

1

2

∫ 1

0

EI

(

d^2 w

dx^2

) 2

dx−P 0 u 3 −M 0 u 4 (E 6 )

whereEis Young’s modulus andIis the area moment of inertia of the beam. Formulate
the optimization problem in terms of the variablesx 1 =u 3 andx 2 =u 4 l or the casef
P 0 l^3 /EI = 1 andM 0 l^2 /EI = 2.

SOLUTION Since the boundary conditions are given byu 1 =u 2 = , 0 w(x)can be
expressed as

w(x)=(− 2 α^3 + 3 α^2 )u 3 + (α^3 −α^2 )lu 4 (E 7 )

so that

d^2 w

dx^2

=

6 u 3

l^2

(− 2 α+ 1 )+

2 u 4

l

( 3 α− 1 ) (E 8 )