Engineering Optimization: Theory and Practice, Fourth Edition

Problems 55

Figure 1.24 Beam-column.

Formulate the problem of minimizing the cost of the pipeline.

1.19 A beam-column of rectangular cross section is required to carry an axial load of 25 lb
and a transverse load of 10 lb, as shown in Fig. 1.24. It is to be designed to avoid the
possibility of yielding and buckling and for minimum weight. Formulate the optimization
problem by assuming that the beam-column can bend only in the vertical (xy) plane.
Assume the material to be steel with a specific weight of 0.3 lb/in^3 , Young’s modulus of
30 × 106 psi, and a yield stress of 30,000 psi. The width of the beam is required to be at
least 0.5 in. and not greater than twice the depth. Also, find the solution of the problem
graphically.Hint:The compressive stress in the beam-column due toPyisPy/bdand
that due toPxis
Pxld
2 Izz

=

6 Pxl

bd^2

The axial buckling load is given by

(Py)cri=

π^2 EIzz

4 l^2

=

π^2 Ebd^3

48 l^2

1.20 A two-bar truss is to be designed to carry a load of 2Was shown in Fig. 1.25. Both
bars have a tubular section with mean diameterdand wall thicknesst. The material
of the bars has Young’s modulusEand yield stressσy. The design problem involves
the determination of the values ofdandtso that the weight of the truss is a minimum
and neither yielding nor buckling occurs in any of the bars. Formulate the problem as a
nonlinear programming problem.

Figure 1.25 Two-bar truss.