The mechanical problem that occurs when a thin elastic layer (e.g., a part of a composite panel) is inserted between two tapered elastic parts and bolted together is studied. The problem considered is of relevance for situations when it is of value to get a rough estimate of the contact pressure in the elastic layer and, by extension, the contact friction. The intermediate elastic layer (i.e., panel) is assumed to behave like a Winkler foundation, and engineering beam theory is assumed to describe the behavior of the tapered outer parts. The coupling between the transverse stresses in the panel and the longitudinal stress due to an applied external tensile load in the plane of the panel is ignored. By using the influence functions for the tapered parts, one obtains an integral equation involving the compression of the panel as the only unknown function. The integral equation is solved numerically by standard procedures. An experiment was also performed in a way that differs slightly from what is covered by the more general theoretical model. This is due to practical reasons connected to the experiment. It was found that the theoretical model gave results in fair agreement with the experimental results.