The authors describe the long-time dynamics of a Timoshenko system consisting of two identical beams joined by a thin adhesive layer. After some transformations, the authors obtain the coupled system of three evolution equations \(\rho w_{tt}+G\varphi _{x}+g_{1}(w_{t})+f_{1}(w,\xi ,s)=h_{1}\), \(I_{\rho }\xi _{tt}-G\varphi -D\xi _{xx}+g_{2}(\xi _{t})+f_{2}(w,\xi ,s)=h_{2}\), \(I_{\rho }s_{tt}+G\varphi -Ds_{xx}+g_{3}(s_{t})+f_{3}(w,\xi ,s)=h_{3}\), posed in \((0,1)\times \mathbb{R }^{3}\). Here \(w\) is the transverse displacement, \(s\) accounts for the interfacial slip and \(\xi \) is associated to \(s\) and to the rotation angle \( \psi \) produced by the beam deflection through \(\xi =3s-\psi \). The boundary conditions \(w(0,t)=\xi (0,t)=s(0,t)=0\), \(\xi _{x}(1,t)=s_{x}(1,t)=0\) and \( 3s(1,t)-\xi _{x}(1,t)-w_{x}(1,t)=0\) are imposed, together with the initial conditions \((w,\xi ,s)\mid _{t=0}=(w_{0},\xi _{0},s_{0})\) and \((w_{t},\xi _{t},s_{t})\mid _{t=0}=(w_{1},\xi _{1},s_{1})\). The forcing terms \(f_{i}\) are supposed to be locally Lipschitz and such that there exists a \(C^{2}\) function \(F\) and \(\nabla F=(f_{1},f_{2},3f_{3})\), \(F\) satisfying further coercivity properties. The damping terms \(g_{i}\) are \(C^{1}\) functions satisfying \(g_{i}(0)=0\) and further growth and coercivity properties. The authors first write the problem in a semigroup framework in the space \(\mathcal{H} =(H_{b}^{1}(0,1))^{3}\times (L^{2}(0,1))^{3}\) with \(H_{b}^{1}(0,1)=\{u\in H^{1}(0,1)\mid u(0)=0\}\). A function \((w,\xi ,s)\) is a weak solution if it satisfies the initial conditions, it is such that \((w,\xi ,s,w_{t},\xi _{t},s_{t})\in C^{0}([0,\infty );\mathcal{H})\) and it satisfies three variational formulations deduced from the three above equations. The authors also define the notion of strong solution and they build the energy \( \mathcal{E}\) associated to this problem. In the case of a strong solution, they derive properties of the global energy \(\mathcal{E}\). They then prove the existence of a unique weak solution to this problem, and that this weak solution is a strong one if the initial condition belongs to the domain of the associated operator. The main result of the paper proves the existence of a global attractor which is equal to some unstable manifold. This global attractor is bounded in \((H^{2}(0,1)\cap H_{b}^{1}(0,1))^{3}\times (H_{b}^{1}(0,1))^{3}\) and its fractal dimension is finite. For the proof, the authors use a Lyapunov function associated to the problem and a quasi-stability property of this dynamical system.