# Energy derivative for a construction consisting of a Timoshenko plate and a thin beam

### Abstract

The paper considers a new problem on the bending of a structure consisting of a Timoshenko plate and a thin beam. It is assumed that in the initial state the plate and the beam are in contact along the line. At the same time, the plate and the beam are bonded to each other on a part of the line, but there is no bonding on a remaining part. As a result, the deflections of the bodies may not be equal on one of the line parts. In order to prevent mutual penetration between the bodies, the non-penetration condition of the inequality type is used. Variational formulations of the problem are considered and the properties of a variational solution are studied. The main result of the work is the proof of the correctness of the energy functional derivative with respect to smooth perturbations of the original geometric configuration. An explicit formula for this derivative is also found in the work.

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*Mathematical notes of NEFU*, 28(2), pp. 68-87. doi: https://doi.org/10.25587/SVFU.2021.28.91.005.

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