An inequality for an integral quadratic form Original Research Article

This paper is motivated by the consideration of the stability of a taut string fixed at x = 0 and x = 1 under the weight of a symmetric distributed load. It is intuitively clear that the potential energy of the string will be decreased if portions of the load distribution are moved from the left half of the string to the other half at symmetrical positions. We shall verify this observation by deriving an integral inequality for the energy functional of the string. The mode of proof shows that similar conclusions also hold for symmetrically loaded one-dimensional mechanical systems such as the supported beam.