The paper deals with two coupled beams modeled by the Kirchhoff beam equation. Two unknown functions of two real variables (time and space) representing the transverse displacement of the beams satisfy a system of two semilinear equations of second order in time and fourth order in space. The equations are accompanied by boundary conditions, clamped on one side and nonlocal in time on the other side hence modeling a memory effect, and initial conditions. The authors prove the existence of a unique strong solution of the system. First, the boundary conditions are transformed by inverting a Volterra type operator. Then, a sequence of Galerkin approximations is constructed by solving a system of ODEs in time -- first locally, then showing a global existence. To pass to the limit, the Lions-Aubin lemma is applied. In the next part, the authors investigate how fast the energy decays to zero as time approaches infinity. They show that if the relaxation functions used to model the memory effect in the boundary condition tend to zero exponentially (or polynomially), the energy also tends to zero exponentially (or polynomially). The paper is well structured and shows all the technical computations.