Minimum-weight design of beams subjected to fixed and moving loads

Structural Optimization

“…This optimality condition, due to Save and Prager (1963) for a linear cost function and to Save (1972) for a convex cost function, is readily extended to the case where certain positions of the loading are not relevant by setting v(x, ~) == 0 for such positions ~. The condition can also be shown to be necessary for concave as well as convex cost functions.…”

“…(3.179) is fulfilled, and we conclude that we have obtained the optimal design for simultaneous fixed and moving loads. This superposition method, due to Save and Prager (1963) and extended to shells by Save and Shield (1966), holds when the mechanism for fixed loads is "compatible" with the family of mechanisms for the movable load in the sense given above. Various other problems of beams subjected to movable, concentrated, or distributed loads have been treated along the same line by Save and Prager (1963), Lamblin and Save (1971), and Lamblin (1972).…”

Minimum-weight beams with shear strain energy

“…Typical works on the strongest beam include studies by Lee et al (2009a), in which the shear effect was not included and only simple beams were considered in the numerical examples; Lee et al (2009b) also investigated the strongest arches whose maximum extreme stresses in bending are minimized. In addition, some researchers specifically investigated minimum-weight beams, which are directly related to this paper: Save and Prager (1963) studied minimumweight beams subjected to multiple fixed loads and a moving load; Gjelsvik (1971) investigated the minimum-weight design of a continuous beam, for which both the elastic and plastic design methods had been shown; Elwany and Barr (1979) researched torsional vibrations of the minimum-weight cantilever beam, where the design constraints fixed the lowest three torsional natural frequencies at specified values; Srithongchai et al (2003) studied theoretical methods of the minimum-weight simplysupported beam, in which power series solutions and matrix operator numerical methods were used to determine the beam layouts; and Meidell (2009) investigated the minimum-weight sandwich beam with honeycomb core of arbitrary density. From reviewing the open literature related to this paper, it is evident that the topics of the strongest and minimum-weight beams are still among the most attractive subjects in the field of applied mechanics.…”

Problems and Methods of Optimal Structural Design

“…71,77,and 150. A sequence of research articles on the theory of optimal design is devoted to the problems of multicriteria optimization, in particular to optimization of structures subjected to moving loads. The problem of optimization for a dynamic load was considered within the framework of plastic design by M. Save and W. Prager,(226) while the problem considering a load of short duration was considered by R. T. Shield. (230) In Ref. 222, W. Prager and R. T. Shield solve the simplest problem of multicriteria optimization for an elastic, trilayer beam that is alternately subjected to bending and tensile loads.…”

“…An analogous case arises when we consider the problem of a moving point load. (226) The design solution does not change if the load applied to the beam is regarded as a stationary point load, whose point of application is not known.…”