Theory of monotonic transformations applied to optimal design problems

The paper presents a theory of monotonic transformations applied to optimal design problems. The method applies to design of structures which require the minimization of the value of an objective functional f over a set of nonnegative functions which obey a constraint of specified form with T as an integral operator. The case when f and T are monotonic is considered, and the derived theorem applied to examples of cantilevered beams of variable height and variable width. In conclusion, it is shown that a cantilevered beam of variable width subject to the previously derived constraint has a minimal when the width is mass given by an expression in terms of the downward acting load and the maximum effective stress.