Engineering Optimization: Theory and Practice, Fourth Edition

30 Introduction to Optimization

Figure 1.13 Cross section of a reinforced concrete beam.

and the constraint on the bending moment can be expressed as [1.120]

P[MR− M≥ 0 ]=P

[

Asfs

(

d− 0. 59

Asfs

fcb

)

−M≥ 0

]

≥ 0. 95 (E 2 )

whereP[· · ·] indicates the probability of occurrence of the event [· · ·].

To ensure that the beam remains underreinforced,†the area of steel is bounded by

the balanced steel areaA(b)s as

As≤A(b)s (E 3 )

where

A(b)s = ( 0. 542 )

fc

fs

bd

600

600 +fs

Since the design variables cannot be negative, we have

d≥ 0

b≥ 0

As≥ 0 (E 4 )

Since the quantitiesM, fc, andfs are nondeterministic, the problem is a stochastic

programming problem.

1.5.7 Classification Based on the Separability of the Functions

Optimization problems can be classified as separable and nonseparable programming

problems based on the separability of the objective and constraint functions.

†If steel area is larger thanA(b)s , the beam becomes overreinforced and failure occurs all of a sudden due

to lack of concrete strength. If the beam is underreinforced, failure occurs due to lack of steel strength and

hence it will be gradual.