Question: A beam, loaded in bending, must support a specified bending moment (M^{*}) without failing and be as light as possible. Section shape is a variable,

A beam, loaded in bending, must support a specified bending moment $M^{*}$ without failing and be as light as possible. Section shape is a variable, and 'failure' here means the first onset of plasticity. Derive the material index. The table summarizes the requirements.

Function Constraints Objective Free variables Light weight beam Specified failure moment M

Function Constraints Objective Free variables Light weight beam Specified failure moment M Length L specified Minimize mass m Choice of material Section shape and scale

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