It is frequently true that several trial designs using the same basic structural configuration must be considered before arriving at an acceptable design either in terms of cost or weight. A systematic approach is developed for finding optimum (lightest or cheapest) designs for a wide class of elastic structures. Design parameters are treated as variables and optimization is accomplished by transforming the analysis and design cycle into the solution of a series of linear programming problems. Two examples are used as illustrations including a truss and a rigid frame each subjected to two distinct load conditions. Significant savings in weight may result from a small investment of computational effort. The amount of computation in performing a single cycle of linear programming optimization will not be excessive compared to the computations required for analyzing another trial design.