Engineering Optimization: Theory and Practice, Fourth Edition

2.4 Multivariable Optimization with Equality Constraints 79

lie on the constraint curve,g(x 1 , x 2 ) = 0. Thus any set of variations (dx 1 , dx 2 ) that
doesnot satisfy Eq. (2.22) leads to points such asD, which do not satisfy constraint
Eq. (2.18).
Assuming that∂g/∂x 2 = , Eq. (2.22) can be rewritten as 0

dx 2 = −

∂g/∂x 1

∂g/∂x 2

(x∗ 1 , x 2 ∗)dx 1 (2.23)

This relation indicates that once the variation inx 1 (dx 1 ) s chosen arbitrarily, thei
variation inx 2 (dx 2 ) s decided automatically in order to havei dx 1 and dx 2 as a set of
admissible variations. By substituting Eq. (2.23) in Eq. (2.19), we obtain

df=

(

∂f

∂x 1

−

∂g/∂x 1

∂g/∂x 2

∂f

∂x 2

)∣∣

∣

∣

(x∗ 1 , x 2 ∗)

dx 1 = 0 (2.24)

The expression on the left-hand side is called theconstrained variationoff. Note that
Eq. (2.24) has to be satisfied for all values ofdx 1. Since dx 1 can be chosen arbitrarily,
Eq. (2.24) leads to
(
∂f
∂x 1

∂g

∂x 2

−

∂f

∂x 2

∂g

∂x 1

)∣

∣

∣

∣

(x 1 ∗, x∗ 2 )

= 0 (2.25)

Equation (2.25) represents a necessary condition in order to have(x 1 ∗, x 2 ∗) s an extremea
point (minimum or maximum).

Example 2.7 A beam of uniform rectangular cross section is to be cut from a log
having a circular cross section of diameter 2a. The beam has to be used as a cantilever
beam (the length is fixed) to carry a concentrated load at the free end. Find the dimen-
sions of the beam that correspond to the maximum tensile (bending) stress carrying
capacity.

SOLUTION From elementary strength of materials, we know that the tensile stress
induced in a rectangular beam (σ) at any fiber located a distanceyfrom the neutral
axis is given by
σ
y

=

M

I

whereMis the bending moment acting andIis the moment of inertia of the cross
section about thexaxis. If the width and depth of the rectangular beam shown in
Fig. 2.7 are 2xand 2y, respectively, the maximum tensile stress induced is given by

σmax=

M

I

y=

My

1

12 (^2 x)(^2 y)

3 =

3

4

M

xy^2

Thus for any specified bending moment, the beam is said to have m aximum tensile
stress carrying capacity if the maximum induced stress (σmax) s a minimum. Hencei
we need to minimizek/xy^2 or maximizeKxy^2 , wherek= 3 M/4 andK= 1 /k, subject
to the constraint
x^2 +y^2 =a^2