The text discusses the limits of material economy in frame structures, focusing on the minimum volume of material required to sustain given forces. It begins by describing an experiment on frame structures, where the curves show overtones and the fundamental frequency. The results indicate that the fundamental frequency is dominant, while overtones diminish more quickly. The text then delves into the mathematical formulation of the problem, using Maxwell's equations to derive the minimum volume of material required for a frame structure under given forces. The analysis shows that the minimum volume is achieved when the strains in all bars of the frame are increased by equal fractions of their lengths. The text then discusses the conditions under which this minimum volume is achieved, and provides examples of minimum frames for various force systems. It concludes with a detailed analysis of the mathematical conditions for the minimum volume of material in frame structures, and provides examples of such frames for different force systems. The text also discusses the conditions under which the minimum volume is achieved, and provides examples of such frames for different force systems.The text discusses the limits of material economy in frame structures, focusing on the minimum volume of material required to sustain given forces. It begins by describing an experiment on frame structures, where the curves show overtones and the fundamental frequency. The results indicate that the fundamental frequency is dominant, while overtones diminish more quickly. The text then delves into the mathematical formulation of the problem, using Maxwell's equations to derive the minimum volume of material required for a frame structure under given forces. The analysis shows that the minimum volume is achieved when the strains in all bars of the frame are increased by equal fractions of their lengths. The text then discusses the conditions under which this minimum volume is achieved, and provides examples of minimum frames for various force systems. It concludes with a detailed analysis of the mathematical conditions for the minimum volume of material in frame structures, and provides examples of such frames for different force systems. The text also discusses the conditions under which the minimum volume is achieved, and provides examples of such frames for different force systems.