Indentation of a Clamped Beam

Let us consider now a double-sided plane indentation for a clamped beam (Fig. 17.4). To simplify the problem, we will study a slender beam, neglecting shear and rotatory inertial terms. Let d(t) and P(t) express the time dependence of both the displacement of the indentor and the applied load. Furthermore la is the length of contact, and I is the half-length of the beam. The origin of coordinates will be taken at the center of the beam. Owing to the symmetry of the problem, only the solution for x 0 will be considered. 

From the conditions of the problem, and taking the deflection as positive 

On the other hand, the boundary conditions at the edge of the contact region (x = a). 

It is clear that the system formed by Eqs. (17.22), (17.24), and (17.26) is nonlinear. Two cases will be considered (1) when the applied load is known and (2) when the displacement is known. 

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