The conditions under which force vectors and corresponding displacement vectors become co-linear are investigated under the assumption of a linear elastic structure and for an arbitrary number of loading points. It is shown that there exist an infinite number of directions along which the load and displacement vectors in each loading point coincide. Moreover, the problem of co-linearity is analogous to the problem of finding the extreme values of the work performed on an elastic structure under the constraint that each force has a given magnitude. The result for a finite number of loading points is extended to a continuous load distribution on the boundary of an elastic structure, i.e., it is possible to find an infinite number of load distributions such that the displacement in a point on the boundary is co-linear with the boundary stress vector in that same point.
A Note on the Co-linearity of Forces and Displacements in an Elastic Structure
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Gradin, P. A., Isaksson, P., and Gulliksson, M. (August 25, 2011). "A Note on the Co-linearity of Forces and Displacements in an Elastic Structure." ASME. J. Appl. Mech. November 2011; 78(6): 061011. https://doi.org/10.1115/1.4003912
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