9.10 Additional Applications 577
Figure 9.17 Continuous beam on rigid supports.
be applicable. Accordingly, the complete bending moment distribution can be deter-
mined once the reactant support momentsm 1 , m 2 ,... , mnare known. Once the support
moments are known (chosen), the plastic limit moment necessary for each span can be
determined and the span can be designed. The bending moment at the center of theith
span is given by−Pili/4 and the largest bending moment in theith span,Mi, can be
computed as
Mi= axm
{
|mi− 1 |,|mi|,
∣
∣
∣
∣
mi− 1 +mi
2
−
Pili
4
∣
∣
∣
∣
}
, i= 1 , 2 ,... , n (9.40)
If the beam is uniform in each span, the limit moment for theith span should be greater
than or equal toMi. The cross section of the beam should be selected so that it has
the required limit moment. Thus the cost of the beam depends on the limit moment it
needs to carry. The optimization problem becomes
FindX= {m 1 , m 2 ,... , mn}Twhich minimizes
∑n
i= 1
Ri(X)
while satisfying the constraintsmi≥Mi, i= 1 , 2 ,... , n, whereRidenotes the cost of
the beam in theith span. This problem has a serial structure and hence can be solved
using dynamic programming.
9.10.2 Optimal Layout (Geometry) of a Truss
Consider the planar, multibay, pin-jointed cantilever truss shown in Fig. 9.18 [9.11,
9.12, 9.22]. The configuration of the truss is defined by thexandycoordinates of
the nodes. By assuming the lengths of the bays to be known (assumed to be unity in
Fig. 9.18) and the truss to be symmetric about thexaxis, the coordinatesy 1 , y 2 ,... , yn
define the layout (geometry) of the truss. The truss is subjected to a load (assumed to
be unity in Fig. 9.18) at the left end. The truss is statically determinate and hence the
forces in the bars belonging to bayidepend only onyi− 1 andyiand not on other
coordinatesy 1 , y 2 ,... , yi− 2 , yi+ 1 ,... , yn. Once the length of the bar and the force
developed in it are known, its cross-sectional area can be determined. This, in turn,
dictates the weight/cost of the bar. The problem of optimal layout of the truss can be
formulated and solved as a dynamic programming problem.
